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\begin{definition}[Definition:Well-Ordering/Definition 1] Let $\struct {S, \preceq}$ be an ordered set. The ordering $\preceq$ is a '''well-ordering''' on $S$ {{iff}} ''every'' non-empty subset of $S$ has a smallest element under $\preceq$: :$\forall T \subseteq S, T \ne \O: \exists a \in T: \forall x \in T: a \preceq x$ \end{definition}	ProofWiki
Jardon, D., Sanchis, M. (2018). POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS. Iranian Journal of Fuzzy Systems, 15(2), 1-21. doi: 10.22111/ijfs.2018.3753 D. R. Jardon; M. Sanchis. "POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS". Iranian Journal of Fuzzy Systems, 15, 2, 2018, 1-21. doi: 10.22111/ijfs.2018.3753 Jardon, D., Sanchis, M. (2018). 'POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS', Iranian Journal of Fuzzy Systems, 15(2), pp. 1-21. doi: 10.22111/ijfs.2018.3753 Jardon, D., Sanchis, M. POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS. Iranian Journal of Fuzzy Systems, 2018; 15(2): 1-21. doi: 10.22111/ijfs.2018.3753 POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS Article 2, Volume 15, Issue 2, March and April 2018, Page 1-21 PDF (448 K) Document Type: Research Paper DOI: 10.22111/ijfs.2018.3753 D. R. Jardon1; M. Sanchis* 2 1Academia de Matematicas, Universidad Autonoma de la Ciudad de Mexico, Calz. Ermita Iztapalapa s/n, Col. Lomas de Zaragoza 09620, Ciudad de Mexico , Mexico 2Institut de Matematiques i Aplicacions de Castello (IMAC), Universitat Jaume I, Campus Riu Sec, 12071-Castello, Spain We study the space of all continuous fuzzy-valued functions from a space $X$ into the space of fuzzy numbers $(\mathbb{E}\sp{1},d\sb{\infty})$ endowed with the pointwise convergence topology. Our results generalize the classical ones for continuous real-valued functions. The field of applications of this approach seems to be large, since the classical case allows many known devices to be fitted to general topology, functional analysis, coding theory, Boolean rings, etc. Fuzzy-number; Fuzzy analysis; Function space; Pointwise convergence; Dual map; Evaluation map; Fr'echet space; Grothendieck's theorem; Cardinal function [1] R. F. Arens, A topology for spaces of transformations, Ann. of Math., 47(2) (1946), 480{495. [2] A. V. Arkhangel'ski, Topological Function Spaces, Translated from the Russian by R. A. M. Hoksbergen, Mathematics and its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. [3] T. Berger and L. D. Davisson, Advances in Source Coding, International Centre for Mechanical Sciences (CISM) Courses and Lectures, No. 166, Springer-Verlag, Vienna-New York, [4] G. Bosi, J. C. Candeal, E. Indurain, E. Oloriz and M. Zudaire, Numerical representations of interval orders, Order 18(2) (2001), 171{190. [5] D. S. Bridges and G. B. Mehta, Representations of Preferences Orderings, Lecture Notes in Economics and Mathematical Systems, 422, Springer-Verlag, Berlin, 1995. [6] M. M. Choban and M. I. Ursul, Applications of the Stone Duality in the Theory of Precompact Boolean Rings, Advances in ring theory, 85{111, Trends Math., Birkhauser/Springer Basel AG, Basel, 2010. [7] L. Dengfeng, Properties of b-vex fuzzy mappings and applications to fuzzy optimization, Fuzzy Sets and Systems, 94 (1998), 253{260. [8] S. Dey, T. Mukhopadhyay, H. H. Khodaparast and S. Adhikari, Fuzzy uncertainty propagation in composites using Gram-Schmidt polynomial chaos expansion, Appl. Math. Model., 40(7{ 8) (2016), 4412{4428. [9] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications, World Scientic Publishing Co., Inc., River Edge, NJ, 1994. [10] D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci., 9(6) (1978), 613{626. [11] R. Engelking, General Topology, Translated from the Polish by the author. Second edition, Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. [12] J. J. Font, A. Miralles and M. Sanchis, On the fuzzy number space with the level convergence topology, J. Funct. Spaces Appl., 2012, Art. ID 326417, 11 pp. [13] J. J. Font and M. Sanchis, Sequentially compact subsets and monotone functions: an appli- cation to fuzzy theory, Topology Appl., 192 (2015), 113{122. [14] A. Garca-Maynez and S. Romaguera, Perfect pre-images of conally complete metric spaces, Comment. Math. Univ. Carolin., 40(2) (1999), 335{342. [15] R. Jr. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1) (1986), 31{43. [16] J. Nagata, On lattices of functions on topological spaces and of functions on uniform spaces, Osaka Math. J., 1 (1949), 166{181. [17] M. S. Osborne, Locally Convex Spaces, Graduate Texts in Mathematics, 269, Springer, Cham, [18] G. Picci and D. S. Gilliam, Dynamical systems, control, coding, computer vision. New trends, interfaces, and interplay, Papers from the Mathematical Theory of Networks and Systems Symposium (MTNS-98) held in Padova, July 6{10, 1998. Edited by Giorgio Picci and David S. Gilliam. Progress in Systems and Control Theory, 25. Birkhauser Verlag, Basel, 1999. [19] V. I. Ponomarev and V. V. Tkachuk, The countable character of X in X versus the countable character of the diagonal in X X, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (in Russian), 104(5) (1987), 16{19. [20] X. Ren and Ch. Wu, The fuzzy Riemann-Stieltjes integral, Internat. J. Theoret. Phys., 52(6) (2013), 2134{2151. [21] S. Romaguera, On conally complete metric spaces, Questions Answers Gen. Topology, 16 (1998), 165{169. [22] L. Stefanini and B. Bede, Generalized fuzzy dierentiability with LU-parametric representa- tion, Fuzzy Sets and Systems, 257 (2014), 184{203. [23] V. V. Tkachuk, A Cp-theory Problem Book. Topological and Function Spaces, Problem Books in Mathematics, Springer, New York, 2011. [24] V. V. Tkachuk, A Cp{theory Problem Book. Special Features of Function Spaces, Problem Books in Mathematics, Springer, Cham, 2014. [25] C. Wu, Function spaces and application to fuzzy analysis, Function Spaces VIII, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 79 (2008), 235{246. [26] J. F. F. Yao and J. S. Yao, Fuzzy decision making for medical diagnosis based on fuzzy number and compositional rule of inference, Fuzzy Sets and Systems, 120 (2001), 351{366. [27] G. Zhang, Y. H.Wu, M. Remias and J. Lu, Formulation of fuzzy linear programming problems as four-objective constrained optimization problems, Appl. Math. Comput., 139(2{3) (2003), Article View: 652	CommonCrawl
I am reading this paper below about optimal bid-ask spread in a market making strategy. It finds an approximation for optimal solution, but I cannot understand how it's practice to set the parameters for a sample stock (eg. AAPL). Assuming, I have this stock below, how I can find all the parameters for the optimal bid-ask spread? -> How to set $A$ and $k$, for my example stock? Which is the parameter for the tick size? Constant frequency $\Lambda$ of market buy/sell orders estimated by dividing the total volume traded over a day by the average size of market orders on that day. where $\alpha$ will be overtaken from the literature (presented in the original study), or else needs to be calculated from your specific dataset. also either overtaken from the literature or estimated on the historical values of your specific stock. where $A = \Lambda / \alpha$ and $k = \alpha K$, $K$ being a scaling parameter for the temporary impact of a market order. Not the answer you're looking for? Browse other questions tagged high-frequency market-microstructure market-making utility-theory or ask your own question. Longer term average probabilities of fills at fx ECNs?	CommonCrawl
\begin{document} \title[Variables in lambda-terms with bounded De Bruijn Indices and De Bruijn Levels]{Distribution of variables in lambda-terms with restrictions on De Bruijn indices and De Bruijn levels} \author{Bernhard Gittenberger \and Isabella Larcher} \thanks{This research has been supported by the Austrian Science Fund (FWF) grant SFB F50-03. \\ A preliminary version of this work was presented at AofA'2018. The present paper is the full version of \cite{GiLa18}.} \address{Department of Discrete Mathematics and Geometry, Technische Universit\"at Wien, Wiedner Hauptstra\ss e 8-10/104, A-1040 Wien, Austria.} \email{[email protected]} \address{Department of Discrete Mathematics and Geometry, Technische Universit\"at Wien, Wiedner Hauptstra\ss e 8-10/104, A-1040 Wien, Austria.} \email{[email protected]} \begin{abstract} We investigate the number of variables in two special subclasses of lambda-terms that are restricted by a bound of the number of abstractions between a variable and its binding lambda, the so-called De-Bruijn index, or by a bound of the nesting levels of abstractions, \textit{i.e.}, the number of De Bruijn levels, respectively. These restrictions are on the one hand very natural from a practical point of view, and on the other hand they simplify the counting problem compared to that of unrestricted lambda-terms in such a way that the common methods of analytic combinatorics are applicable. We will show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to $Cn$ and $\tilde{C}n$, respectively, where the constants $C$ and $\tilde{C}$ depend on the bound that has been imposed. So far we just derived closed formulas for the constants in case of the class of lambda-terms with bounded De Bruijn index. However, for the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of De Bruijn levels, we investigate the number of variables, as well as abstractions and applications, in the different De Bruijn levels and thereby exhibit a so-called ``unary profile'' that attains a very interesting shape. \end{abstract} \maketitle \section{Introduction} Lambda-calculus is a set of rules to manipulate lambda-terms and it is an important tool in theoretical computer science. To our knowledge, the first appearance of enumeration problems in the sense of enumerative combinatorics which are linked to lambda-calculus is found in \cite{Ha96}, where certain models of lambda-calculus are analyzed which have representations as formal power series. More recently, we observe rising interest in the quantitative properties of large random lambda-terms. The first work in this direction seems to be \cite{MTZ00}. Later David \emph{et al.} \cite{DGKRT13} investigated the proportion of normalising terms, which was also the topic of \cite{BGZ17} in a different context. Other papers dealing with certain structural properties of lambda-terms are for instance \cite{BoTa18,Gr16,SAKT17}. Since studying quantitative aspects of lambda-terms using combinatorial methods relies heavily on their enumeration, many papers are devoted to their enumeration, which itself very much depends on the particular class of terms and the definition of the term size. The enumeration may be done by contructing bijections to certain classes of maps, see e.g. \cite{MR3101704,Ze16,ZeGi15} or the use of the methodology from analytic combinatorics \cite{MR2483235}, see e.g. \cite{BGLZ17,MR2815481,bodini2015number,MR3158269,BGG18,GrLe15,MR3018087}. Another approach to gain structural insight is by random generation. Solving the enumeration problems is the basis for an efficient algorithm for this purpose, namely Boltzmann sampling \cite{DFLS04,FFP07}. The method is extendible to a multivariate setting allowing for a fine tuning according to specified structural properties of the sampled objects, as was demonstrated in \cite{BBD18,BoPo10}. The generation of lambda-terms was treated in \cite{BGT17,MR3101704,GrLe15,Pa11,Ta17,Wa05}. In \cite{bodini2015number} the authors discovered a very interesting phenomenon concerning the generating function of lambda-terms with a bounded number of De Bruijn levels, namely that the asymptotic behaviour of the coefficients of the generating function changes with the imposed bound. More precisely, the type of the dominant singularity changes from $\frac{1}{2}$ to $\frac{1}{4}$ whenever the bound belongs to a special doubly-exponentially growing sequence. This alteration of the type of the dominant singularity has a direct impact on the polynomial factor of the coefficient's asymptotics, namely it shifts from $n^{-3/2}$ to $n^{-5/2}$ for $n$ tending to infinity. This paper studies the structure of random large lambda-terms belonging to this class and thereby delivers an explanation of the above mentioned phenomenon, since it arises from the location of the variables within the lambda-term. The lambda calculus was invented by Church and Kleene in the 1930ies as a tool for the investigation of decision problems. Today it still plays an important role in computability theory and for automatic proof systems. Furthermore, it represents the basis for some programming languages, such as LISP. For a thorough introduction to lambda calculus we refer to \cite{MR3235567}. This paper does not require any preliminary knowledge of lambda calculus in order to follow the proofs. Instead we will study the basic objects of lambda calculus, namely lambda-terms, by considering them as combinatorial objects, or more precisely as a special class of directed acyclic graphs (DAGs). \begin{df}[{lambda-terms, \cite[Definition 3]{MR3063045}}] \label{def:lambda-terms} Let $\mathcal{V}$ be a countable set of variables. The set $\Lambda$ of lambda-terms is defined by the following grammar: \begin{enumerate} \item every variable in $\mathcal{V}$ is a lambda-term, \item if $T$ and $S$ are lambda-terms then $TS$ is a lambda-term, (application) \item if $T$ is a lambda-term and $x$ is a variable then $\lambda x.T$ is a lambda-term. (abstraction) \end{enumerate} \end{df} The name application arises, since lambda-terms of the form $TS$ can be regarded as functions $T(S)$, where the function $T$ is applied to $S$, which in turn can be a function itself. An abstraction can be considered as a quantifier that binds the respective variable in the sub-lambda-term within its scope. Both application and repeated abstraction are not commutative, \textit{i.e.}, in general the lambda-terms $TS$ and $ST$, as well as $\lambda x. \lambda y. M$ and $\lambda y. \lambda x. M$, are different (with the exceptions of $T=S$ and none of the variables $x$ or $y$ occurring in $M$, respectively). Each $\lambda$ binds exactly one variable (which may occur several times in the terms), and since we will just focus on a special subclass of closed lambda-terms, each variable is bound. We will consider lambda-terms modulo $\alpha$-equivalence, which means that we identify two lambda-terms if they only differ by the names of their bound variables. For example $\lambda x. ( \lambda y. (xy)) \equiv \lambda y. ( \lambda z. (yz))$. 1972 De Bruijn (\cite{de1972lambda}) introduced a representation for lambda-terms that completely avoids the use of variables by substituting them by natural numbers that indicate the number of abstractions between the variable and its binding lambda (the binding lambda is counted as well), \textit{i.e.}, $\lambda x. ( \lambda y. (xy)) = \lambda(\lambda21)$. \begin{df}[De Bruijn index, De Bruijn level] The natural numbers that represent the variables in the De Bruijn representation of a lambda-term are called De Bruijn indices. The number of nested lambdas starting from the outermost one specifies the De Bruijn level in which a variable (or De Bruijn index, respectively) is located. \end{df} For example in the lambda-term $\lambda x. x( \lambda y. (xy)) = \lambda1(\lambda21)$ the first occurrence of the variable $x$ (\textit{i.e.}, the leftmost 1 in the De Bruijn representation) is in the first De Bruijn level, while the other variables are in the second De Bruijn level. There is also a combinatorial interpretation of lambda-terms that considers them as DAGs and thereby naturally identifies two $\alpha$-equivalent terms to be equal. Combinatorially, lambda-terms can be seen as rooted unary-binary trees containing additional directed edges. Note that in general the resulting structures are not trees in the sense of graph theory, but due to their close relation to trees (see Definition \ref{def:lambda-DAG}) some authors call them lambda-trees or enriched trees. We will call them lambda-DAGs in order to emphasise that these structures are in fact DAGs, if we consider the undirected edges of the underlying tree to be directed away from its root. \begin{df}[{lambda-DAG, \cite[Definition 5]{MR3063045}}] \label{def:lambda-DAG} With every lambda-term $T$, the corresponding lambda-DAG $G(T)$ can be constructed in the following way: \begin{enumerate} \item If $x$ is a variable then $G(x)$ is a single node labeled with $x$. Note that $x$ is unbound. \item $G(PQ)$ is a lambda-DAG with a binary node as root, having the two lambda-DAGs $G(P)$ (to the left) and $G(Q)$ (to the right) as subgraphs. \item The DAG G($\lambda x.P$) is obtained from $G(P)$ in four steps: \begin{enumerate} \item Add a unary node as new root. \item Connect the new root by an undirected edge with the root of G(P). \item Connect all leaves of $G(P)$ labelled with $x$ by directed edges with the new root, where the root is start vertex of these edges. \item Remove all labels $x$ from $G(P)$. Note that now $x$ is bound. \end{enumerate} \end{enumerate} Obviously, applications correspond to binary nodes and abstractions correspond to unary nodes of the underlying Motzkin-tree that is obtained by removing all directed edges. Of course, in the lambda-DAG some of the vertices that were former unary nodes might have gained out-going edges, so they are no unary nodes in the lambda-DAG anymore. However, when we speak of unary nodes in the following, we mean the unary nodes of the underlying unary-binary tree that forms the skeleton of the lambda-DAG. \end{df} \begin{figure} \caption{The lambda-DAGs corresponding to the respective terms written below.} \label{fig:lambdaDAGs} \end{figure} \begin{figure} \caption{The lambda-DAG representing the term $\lambda x.((\lambda y. xy)(\lambda z. (z(\lambda t. tx))z))$, where the leaves are labelled with (left) the corresponding De Bruijn indices, and (right) the De Bruijn level in which they are located.} \label{fig:unarylengthheight} \label{fig:lambdatrees2} \end{figure} Since the skeleton of a lambda-DAG is a tree, we sometimes call the variables leaves (\textit{i.e.}, the nodes with out-degree zero), and the path connecting the root with a leaf (consisting of undirected edges) is called a branch. There are different approaches as to how one can define the size of a lambda-term (\cite{DGKRT13,bodini2015number, MR3018087}), but within this paper the size will be defined as the number of nodes in the corresponding lambda-DAG. As mentioned at the beginning, recently, rising interest in the number and structural properties of lambda-terms can be observed, due to the direct relationship between these random structures acting as computer programs and mathematical proofs (\cite{curry1958combinatory}). At first sight lambda-terms appear to be very simple structures, in the sense that their construction can easily be described, but so far no one has yet accomplished to derive their asymptotic number. However, the asymptotic equivalent of the logarithm of this number can be determined up to the second-order term (see \cite{MR3158269}). The difficulty of counting unrestricted lambda-terms arises due to the fact that their number increases superexponentially with increasing size. Thus, if we translate the counting problem into generating functions, then the resulting generating function has a radius of convergence equal to zero, which makes the common methods of analytic combinatorics inapplicable. This fast growth of the number of lambda-terms can be explained by the numerous possible bindings of leaves by lambdas, \textit{i.e.} by unary nodes. Consequently, lately some simpler subclasses of lambda-terms, which reduce these multiple binding possibilities, have been studied, e.g. lambda-terms with prescribed number of unary nodes (\cite{bodini2015number}), or lambda-terms in which every lambda binds a prescribed (\cite{MR3158269,MR3101704,MR3063045}) or a bounded (\cite{MR3248351,MR3101704,MR3063045}) number of leaves. In this paper we will investigate structural properties of lambda-terms that have been introduced in \cite{MR2815481} and \cite{bodini2015number}, namely at first lambda-terms with a bounded number of abstractions between each leaf and its binding lambda, which corresponds to a bounded De Bruijn index. The second class of lambda-terms that we will investigate within this paper is the class of lambda-terms with a bounded number of nesting levels of abstractions, \textit{i.e.}, lambda-terms with a bounded number of De Bruijn levels. From a practical point of view these restrictions appear to be very natural, since the number of abstractions in lambda-terms which are used for computer programming is in general assumed to be very low compared to their size (\cite{yang2011finding}). Particular interest lies in the number and distribution of the variables within these special subclasses of lambda-terms. We will show within this paper that the total number of leaves (\textit{i.e.}, variables) in lambda-DAGs with bounded De Bruijn indices as well as in lambda-terms with bounded number of De Bruijn levels is asymptotically normally distributed. For the latter class of lambda-terms we will also investigate the number of leaves in the different De Bruijn levels, which shows a very interesting behaviour. We will see that in the lower De Bruijn levels, \textit{i.e.} near the root of the lambda-DAG, there are very few leaves, while almost all of the leaves are located in the upper De Bruijn levels and these two domains will turn out to be asymptotically strictly separated. The same behaviour can be shown for unary and binary nodes, which allows us to set up a very interesting ``unary profile'' of this class of lambda-terms. For lambda-terms that are locally restricted by a bound for the De Bruijn indices the number of De Bruijn levels is not bounded and will tend to infinity for increasing size. The expected number of De Bruijn levels is unknown, which implies that the correct scaling cannot be determined. Thus, we have not been able to establish results concerning the leaves (or other types of nodes) on the different De Bruijn levels for this class of lambda-terms so far. Nevertheless, further studies on this subject seem to be very interesting. The plan of the paper is as follows: We will present the main results that have been derived in this paper, including all the definitions that are necessary for their understanding, in Section \ref{ch:mainresults}, while the subsequent sections are concerned with their proofs. In Section \ref{ch:length} we will show that the total number of variables in lambda-terms with bounded De Bruijn index is asymptotically normally distributed with mean and variance asymptotically $Cn$ and $\tilde{C}n$, respectively, where the constants $C$ and $\tilde{C}$ depend on the bound that has been imposed. Section \ref{ch:totalheight} shows the same result for lambda-terms where the number of De Bruijn levels is bounded. Finally, in the last section, Section \ref{ch:levelheight}, we show how the variables are distributed in lambda-terms with bounded number of De Bruijn levels. We will see that there are very few leaves on the lower De Bruijn levels, \textit{i.e.}, close to the root, while on the upper De Bruijn levels farther away from the root, there are many leaves. Furthermore, these two domains are strictly separated and we know exactly which is the first level containing a large number of leaves, since this level can be determined by the imposed bound of the number of De Bruijn levels. This interesting behaviour also holds for the number of binary and unary nodes. By investigating all these numbers among the different De Bruijn levels we are able to set up a so-called unary profile that shows that these special lambda-terms have a very specific shape. A random closed lambda-term with a bounded number of De Bruijn levels starts with a string of unary nodes, where the length of this string depends on the imposed bound. Then it gets slowly filled with nodes until it reaches the aforementioned separating level, where it suddenly starts to contain a lot of nodes. \section{Main results} \label{ch:mainresults} In this section we will introduce the basic definitions and summarize the main results that will be presented in this paper. First, we will investigate the total number of variables in lambda-terms with bounded De Bruijn index, \textit{i.e.}, with a bounded number of abstractions between each leaf and its binding lambda. Our first main result concerns the asymptotic distribution of the number of variables within this class of closed lambda-terms. \begin{theo} \label{theo:mainresultlength} Let $X_n$ be the total number of variables in a random closed lambda-term of size $n$ where the De Bruijn index of each variable is at most $k$. Then $X_n$ is asymptotically normally distributed with \[ \mathbb{E}{X_n} \sim \frac{k}{\sqrt{k}+2k}n, \ \ \ \ \text{and} \ \ \ \ \mathbb{V}X_n \sim \frac{k^2}{2 \sqrt{k}(\sqrt{k}+2k)^2}n, \ \ \ \ \text{as} \ n \rightarrow \infty.\] \end{theo} \begin{rem} Note that $\mathbb{E}{X_n} \longrightarrow \frac{n}{2}$ and $\mathbb{V}X_n \longrightarrow 0$ for $k \rightarrow \infty$. Since these values are known for the number of leaves in binary trees, this gives a hint that almost all leaves of a large random unrestricted lambda-term are located within an almost purely binary structure. \end{rem} Next we turn to lambda-terms with a bounded number of De Bruijn levels, \textit{i.e.} with a bounded number of unary nodes (or abstractions, respectively) in the separate branches of the corresponding lambda-DAG. \begin{theo} \label{theo:totalnumberleavesheight} Let $\rho_k(u)$ be the root of smallest modulus of the function $z \mapsto R_{j+1,k}(z,u)$, where \[R_{j+1,k}(z,u)= 1-4(k-j)z^2u-2z+2z\sqrt{1-4(k-j+1)z^2u-2z+ \sqrt{... +2z\sqrt{1-4kz^2u}}},\] and let us define $B(u)=\frac{\rho_k(1)}{\rho_k(u)}$. If $B''(1)+B'(1)-B'(1)^2 \neq 0$, then the total number of leaves in closed lambda-DAGs with at most $k$ De Bruijn levels is asymptotically normally distributed with asymptotic mean $\mu n$ and asymptotic variance $\sigma^2 n$, where $\mu = B'(1)$ and $\sigma^2=B''(1)+B'(1)-B'(1)^2$. \end{theo} \begin{rem} The requirement $B''(1)+B'(1)-B'(1)^2 \neq 0$ obviously results from the fact that otherwise the variance would be $o(n)$. However, this inequality seems to be very difficult to verify, since $B(u)=\frac{\rho_k(1)}{\rho_k(u)}$ and we do not know anything about the function $\rho_k(u)$, except for some crude bounds and its analyticity. But numerical data supports the conjecture that $B''(1)+B'(1)-B'(1)^2 \neq 0$ always holds (\textit{cf}. Table \ref{tab:initialvalues}). \end{rem} Lambda-terms with bounded number of De Bruijn levels have been studied in \cite{bodini2015number}, where a very unusual behaviour has been discovered. The asymptotic behaviour of the number of lambda-terms belonging to this subclass differs depending on whether the imposed bound is an element of a certain sequence $(N_i)_{i \geq 0}$, which will be given in Definition \ref{def:sequences}, or not. Though the behaviour of the counting sequence differs for these two cases, the result in Theorem \ref{theo:totalnumberleavesheight} concerning lambda-terms with bounded number of De Bruijn levels is the same after all. However, the method of proof is different in the two cases. For our subsequent results the distinction of cases will have an impact on the asymptotic behaviour of the counting sequence of the investigated structures. Thus, we will have to distinguish between these two cases. \begin{df}[{auxiliary sequences $(u_i)_{i \geq 0}$ and $(N_i)_{i \geq 0}$, \cite[Def.6]{bodini2015number}}] \label{def:sequences} Let $(u_i)_{i \geq 0}$ be the integer sequence defined by \[ u_0=0, \ \ \ u_{i+1}=u_i^2+i+1 \ \ \ \text{for} \ i \geq 0, \] and $(N_i)_{i \geq 0}$ by \[ N_i = u_i^2-u_i+i, \ \ \ \text{for all} \ i \geq 0. \] \end{df} In the last section we investigate the distribution of the different types of nodes in lambda-DAGs with bounded number of De Bruijn levels among the separate levels throughout the DAG. \begin{figure} \caption{Underlying Motzkin tree of e.g. the lambda term $\lambda x. ((\lambda y.yx)(\lambda z.(z(\lambda t. tx))z))$, where the different De Bruijn levels are encircled.} \label{fig:unarylevels} \end{figure} \begin{rem} Note that the De Bruijn level in which a node is located just counts the number of unary nodes in the branch connecting the root and the respective node. \end{rem} The following theorem includes the results that we will present in Section \ref{sec:leaveslevels}, where we show that the number of leaves near the root of the lambda-DAG, \textit{i.e.}, in the lower De Bruijn levels, is very low, while there are many leaves in the upper levels. Furthermore these two domains are strictly separated and the ``separating level'', \textit{i.e.}, the first level with many leaves, depends on the bound of the number of De Bruijn levels. We will show a very interesting behaviour, namely that with growing bound the number of leaves within the De Bruijn level that is directly below the critical separating level increases, until the bound reaches a certain number, which makes this adjacent leaf-filled level become the new separating level. Thus, we can observe a ``double jump'' in the asymptotic behaviour of the number of leaves within the separate levels (\textit{cf}. Figure \ref{fig:levelsdoublejump}). \begin{theo} \label{theo:levelsmeandist} Let $_{k-l}\tilde{H}_k(z,u)$ denote the bivariate generating function of the class of closed lambda-terms with at most $k$ De Bruijn levels, where $z$ marks the size and $u$ marks the number of leaves in the $(k-l)$-th De Bruijn level. Additionally, we denote its dominant singularity by $\tilde{\rho}_k(u)$, and $\tilde{B}(u)=\frac{\tilde{\rho}_k(u)}{\tilde{\rho}_k(1)}$. Then the following assertions hold: \noindent \begin{itemize}[leftmargin=] \item If $k \in (N_j,N_{j+1})$, then the average number of leaves in the first $k-j$ De Bruijn levels is $O(1)$, as the size $n \rightarrow \infty$, while it is $\Theta(n)$ for the last $j+1$ levels. In particular, if $\tilde{B}''(1)+\tilde{B}'(1)-\tilde{B}'(1)^2 \neq 0$, the number of leaves in each of the last $j+1$ De Bruijn levels is asymptotically normally distributed with mean and variance proportional to the size $n$ of the lambda-term. \item If $k = N_j$, then the average number of leaves in the first $k-j$ De Bruijn levels is $O(1)$, as $n \rightarrow \infty$, while the average number of leaves in the $(k-j)$-th level is $\Theta(\sqrt{n})$. The last $j$ De Bruijn levels have asymptotically $\Theta(n)$ leaves. In particular, if $\tilde{B}''(1)+\tilde{B}'(1)-\tilde{B}'(1)^2 \neq 0$, the number of leaves in each of the last $j$ De Bruijn levels is asymptotically normally distributed with mean and variance proportional to the size $n$ of the lambda-term.. \end{itemize} \end{theo} \begin{figure} \caption{Summary of the mean values of the number of leaves in the different De Bruijn levels in lambda-terms with at most $k$ De Bruijn levels for the case $N_j < k < N_{j+1}$ (left), and the case $k=N_j$ (right).} \label{fig:levelsdoublejump} \end{figure} Sections \ref{sec:unarylevels} and \ref{sec:binary} are concerned with the investigation of the number of unary nodes, and binary nodes respectively, among the De Bruijn levels. Using the same techniques as in Section \ref{sec:leaveslevels} we can show that their number behaves in fact very similar to the number of leaves. \begin{theo} \label{theo:levelsmeanbinary} If $k \in (N_j,N_{j+1})$, then both the average number of unary nodes and the avergae number of binary nodes in the first $k-j$ De Bruijn levels are $O(1)$, as $n \rightarrow \infty$, while they are $\Theta(n)$ in each of the last $j+1$ levels. If $k = N_j$, then both the average number of unary nodesand the average number of binary nodes in the first $k-j$ De Bruijn levels is $O(1)$, as $n \rightarrow \infty$, while the average number of nodes of the respective type in the $(k-j)$-th De Bruijn level is $\Theta(\sqrt{n})$. The last $j$ De Bruijn levels contain each asymptotically $\Theta(n)$ unary nodes, as well as $\Theta(n)$ binary nodes. \end{theo} \section{Total number of leaves in lambda-terms with bounded De Bruijn indices} \label{ch:length} In this section we investigate the asymptotic number of all leaves in closed lambda-terms with bounded De Bruijn indices. In order to get some quantitative results concerning this restricted class of lambda-terms we will use the well-known symbolic method (see \cite{MR2483235}) and therefore we introduce further combinatorial classes as it has been done in \cite{bodini2015number}: $\mathcal{Z}$ denotes the class of atoms, $\mathcal{A}$ the class of application nodes ({\textit{i.e.}, binary nodes), $\mathcal{U}$ the class of abstraction nodes (\textit{i.e.}, unary nodes), and $\hat{\mathcal{P}}^{(i,k)}$ the class of unary-binary trees such that every leaf $e$ can be labelled in $\min \{ h_u(e)+i,k \}$ ways. The classes $\hat{\mathcal{P}}^{(i,k)}$ can be specified by \[ \hat{\mathcal{P}}^{(k,k)} = k \mathcal{Z} + ( \mathcal{A} \times \hat{\mathcal{P}}^{(k,k)} \times \hat{\mathcal{P}}^{(k,k)} ) + ( \mathcal{U} \times \hat{\mathcal{P}}^{(k,k)}), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \] and \[ \hat{\mathcal{P}}^{(i,k)} = i \mathcal{Z} + ( \mathcal{A} \times \hat{\mathcal{P}}^{(i,k)} \times \hat{\mathcal{P}}^{(i,k)} ) + ( \mathcal{U} \times \hat{\mathcal{P}}^{(i+1,k)}) \ \ \ \ \ \ \ \text{for} \ \ i < k. \] Translating into generating functions with $z$ marking the size and $u$ marking the number of leaves, we get \[ \hat{P}^{(k,k)}(z,u) = kzu + z \hat{P}^{(k,k)^2}(z,u) + z\hat{P}^{(k,k)}(z,u), \] and \[ \hat{P}^{(i,k)}(z,u) = izu + z \hat{P}^{(i,k)^2}(z,u) + z\hat{P}^{(i+1,k)}(z,u), \] which yields \[ \hat{P}^{(k,k)}(z,u) = \frac{1-z- \sqrt{(1-z)^2-4z^2ku}}{2z}, \] and \[ \hat{P}^{(i,k)}(z,u) = \frac{1-\sqrt{1-4z(izu+z\hat{P}^{(i+1,k)}(z,u))}}{2z} = \frac{1-\sqrt{1-4iz^2u-4z^2\hat{P}^{(i+1,k)}(z,u)}}{2z}, \] for $i<k$. This can be written in the form \[ \hat{P}^{(i,k)}(z,u) = \frac{1- \mathbf{1}_{[i=k]}z-\sqrt{\hat{R}_{k-i+1,k}(z,u)}}{2z}, \] with \begin{align} \label{equ:radicalhatk1} \hat{R}_{1,k}(z,u) = (1-z)^2-4kuz^2, \end{align} \[ \hat{R}_{2,k}(z,u) = 1-4(k-1)z^2u-2z+2z^2+2z \sqrt{\hat{R}_{1,k}(z,u)}, \] and \begin{align} \label{equ:radicalhatrecursion} \hat{R}_{i,k}(z,u) = 1-4(k-i+1)z^2u-2z+2z \sqrt{\hat{R}_{i-1,k}(z,u)}, \ \ \ \ \ \text{for} \ \ 3 \leq i \leq k+1. \end{align} Since the class $\hat{\mathcal{P}}^{(0,k)}$ is isomorphic to the class $\mathcal{G}_{k}$ of closed lambda-terms where all De Bruijn indices are not larger than $k$, we get for the corresponding bivariate generating function \[ G_k(z,u)= \hat{P}^{(0,k)}(z,u)= \frac{1- \sqrt{\hat{R}_{k+1,k}(z,u)}}{2z}. \] From \cite{bodini2015number} we know that the dominant singularity of $G_{k}(z,1)$ comes from the innermost radicand only and consequently is of type $\frac{1}{2}$. Due to continuity arguments this implies that in a sufficiently small neighbourhood of $u=1$ the dominant singularity $\hat{\rho}_k(u)$ of $G_{k}(z,u)$ comes also only from the innermost radicand, \textit{i.e.}, $\hat{R}_{1,k}(z,u)$, and is of type $\frac{1}{2}$. By calculating the smallest positive root of $\hat{R}_{1,k}(z,u)$ we get $\hat{\rho}_k(u)= \frac{1}{1+2\sqrt{ku}}$. Now we will determine the expansions of the radicands in a neighbourhood of the dominant singularity $\hat{\rho}_k(u)$. \begin{prop} \label{prop:rjkaroundsing} Let $\hat{\rho}_k(u)$ be the root of the innermost radicand $\hat{R}_{1,k}(z,u)$, \textit{\textit{i.e.}} $\hat{\rho}_k(u) = \frac{1}{1+2\sqrt{ku}}$, where $u$ is in a sufficiently small neighbourhood of 1, \textit{i.e.} $\|u-1\|<\delta$ for $\delta>0$ sufficiently small. Then the equations \begin{align} \label{equ:r1karoundsing} \hat{R}_{1,k}(\hat{\rho}_k(u)-\epsilon,u) = \big (2 -2\hat{\rho}_k(u) +8ku\hat{\rho}_k(u) \big) \epsilon + \mathcal{O}(\|\epsilon\|^2), \end{align} and \begin{align} \label{equ:rjkaroundsing} \hat{R}_{j,k}(\hat{\rho}_k(u)-\epsilon,u) = {c}_j(u) \hat{\rho}_k(u)^2 + \frac{\sqrt{8\hat{\rho}_k(u) (1-\hat{\rho}_k(u)^2)}}{\prod_{l=2}^{j}\sqrt{c_l(u)}} \cdot \sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|), \end{align} for $2 \leq j \leq k+1$, with ${c}_1(u)=1$ and ${c}_j(u)=4(j-1)u-1+2\sqrt{c_{j-1}(u)}$ for $2 \leq j \leq k+1$, hold for $\epsilon \longrightarrow 0$ so that $\epsilon \in \mathbb{C} \setminus \mathbb{R}^{-}$, uniformly in $u$. \end{prop} \begin{proof} Using the Taylor expansion of $\hat{R}_{1,k}(z,u)$ around $\hat{\rho}_k(u)$ we obtain \[ \hat{R}_{1,k}(z,u) = \hat{R}_{1,k}(\hat{\rho}_k(u),u)+(z-\hat{\rho}_k(u)) \frac{\partial}{\partial z}\hat{R}_{1,k}(\hat{\rho}_k(u),u)+ \mathcal{O}((z-\hat{\rho}_k(u))^2). \] Per definition the first summand $\hat{R}_{1,k}(\hat{\rho}_k(u),u)$ is equal to zero. Setting $z=\hat{\rho}_k(u)-\epsilon$ and using (\ref{equ:radicalhatk1}) we obtain the first claim of Proposition \ref{prop:rjkaroundsing}. The next step is to compute an expansion of $\hat{R}_{j,k}(z,u)$ around $\hat{\rho}_k(u)$ for $2\leq j \leq k+1$. Using the recursive relation (\ref{equ:radicalhatk1}) for $\hat{R}_{2,k}(z,u)$ and the formula $\hat{\rho}_k(u)=\frac{1}{1+2\sqrt{ku}}$ yields \[ \hat{R}_{2,k}(\hat{\rho}_k(u)-\epsilon,u) = (1+4u) \hat{\rho}_k(u)^2 + 2\hat{\rho}_k(u)\sqrt{2 -2\hat{\rho}_k(u) +8ku\hat{\rho}_k(u)} \sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|). \] We set $c_2(u):=1+4u$ and $d_2(u):=2\hat{\rho}_k(u)\sqrt{2 -2\hat{\rho}_k(u) +8ku\hat{\rho}_k(u)}$ and assume that for $2 \leq j \leq k+1$ the equation $\hat{R}_{j,k}(\hat{\rho}_k(u)-\epsilon,u)=c_j(u) \hat{\rho}_k^2(u)+d_j(u) \sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|)$ holds. Now we proceed by induction. Observe that \[ \hat{R}_{j+1,k}(\hat{\rho}_k(u)-\epsilon,u) = 1- 4(k-j)\hat{\rho}_k^2(u)u-2\hat{\rho}_k(u)+ 2\hat{\rho}_k(u)\sqrt{c_j(u) \hat{\rho}_k^2+d_j(u) \sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|)}. \] Expanding, using again $\hat{\rho}_k(u)=\frac{1}{1+2\sqrt{ku}}$, and simplifying yields \[ \hat{R}_{j+1,k}(\hat{\rho}_k(u)-\epsilon,u) = 4ju\hat{\rho}_k(u)^2-\hat{\rho}_k(u)^2 +2\hat{\rho}_k(u)^2\sqrt{c_j(u)}+ \frac{d_j(u)}{\sqrt{c_j(u)}}\sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|). \] Setting $c_{j+1}(u):=4ju-1+2\sqrt{c_j(u)}$ and $d_{j+1}(u):=\frac{d_j(u)}{\sqrt{c_j(u)}}$ for $2 \leq j \leq k$, we obtain $\hat{R}_{j+1,k}(\hat{\rho}_k(u)-\epsilon,u)=c_{j+1} \hat{\rho}_k^2(u) +d_{j+1} \sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|)$. Expanding $d_{j+1}(u)$, using its recursive relation and $d_2(u)=2\hat{\rho}_k(u)\sqrt{2 -2\hat{\rho}_k(u) +8ku\hat{\rho}_k(u)}$, we get for $2 \leq j \leq k$ \[ d_{j+1}(u)= \frac{2\hat{\rho}_k(u)\sqrt{2 -2\hat{\rho}_k(u) +8ku\hat{\rho}_k(u)}}{\prod_{l=2}^j \sqrt{c_l(u)}}. \] Finally, we show that the $c_l(u)$'s are greater than zero in a neighbourhood of $u=1$. By induction it can easily be seen that they are always positive for $u=1$, since \begin{align} c_1(1)=1, \end{align} and assuming $c_{i-1}(1)<c_{i}(1)$ we get \begin{align} c_{i+1}(1)=41-1+2\sqrt{c_i(1)} > 4(i-1) + 4 - 1 + 2\sqrt{c_{i-1}(1)} = c_i(1) + 4. \end{align} Using continuity arguments we can see that the functions $c_l(u)$ have to be positive in a sufficiently small neighbourhood of $u=1$ as well, which completes the proof of (\ref{equ:rjkaroundsing}). \end{proof} \begin{theo} \label{thm:asympgleqzu} Let for any fixed $k$, $G_{k}(z,u)$ denote the bivariate generating function of the class of closed lambda-terms where all De Bruijn indices are at most $k$. Then the equation \[ [z^n]G_{k}(z,u) = \sqrt{\frac{\sqrt{ku}+2ku}{4\pi\prod_{l=2}^{k+1}c_l(u)}} (1+2\sqrt{ku})^n n^{-\frac{3}{2}} \left( 1+ O \left( \frac{1}{\sqrt{n}} \right) \right), \ \ \ \ \text{for} \ \ n \rightarrow \infty, \] with $c_1(u)=1$ and $c_j(u)=4(j-1)u-1+2\sqrt{c_{j-1}(u)}$, for $2 \leq j \leq k+1$, holds uniformly in $u$ for $\|u-1\|<\delta$, with $\delta>0$ sufficiently small. \end{theo} \begin{proof} Using $G_{k}(z,u)= \frac{1- \sqrt{\hat{R}_{k+1,k}(z,u)}}{2z}$ and (\ref{equ:rjkaroundsing}), we get for $\epsilon \in \mathbb{C} \setminus \mathbb{R}^{-}$ with $\|\epsilon\| \longrightarrow 0$ \[ G_{k}(\hat{\rho}_k(u)-\epsilon,u)= \frac{1-\sqrt{c_{k+1}(u)}\hat{\rho}_k(u)}{2\hat{\rho}_k(u)} - \frac{d_{k+1}(u)}{4\hat{\rho}_k(u)^2\sqrt{c_{k+1}(u)}} \sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|). \] Hence, \[ [z^n]G_{k}(z,u) = - \frac{d_{k+1}(u)\sqrt{\hat{\rho}_k(u)}}{4\hat{\rho}_k^2(u) \sqrt{c_{k+1}(u)}} [z^n] \sqrt{1- \frac{z}{\hat{\rho}_k(u)}} + [z^n] \mathcal{O} \left( \hat{\rho}_k(u) - z \right). \] Since $\hat{\rho}_k(u)=\frac{1}{1+2\sqrt{ku}}$ is of type $\frac{1}{2}$ and by plugging in the formula for $d_{k+1}(u)=\frac{2\hat{\rho}_k(u)\sqrt{2 -2\hat{\rho}_k(u) +8ku\hat{\rho}_k(u)}}{\prod_{l=2}^k \sqrt{c_l(u)}}$, we obtain the desired result by applying singularity analysis. \end{proof} From \cite[Theorem 1]{bodini2015number} we know the following result: \begin{align} \label{resultgleqkasymp} [z^n]G_{k}(z,1) = \sqrt{\frac{\sqrt{k}+2k}{4\pi\prod_{l=2}^{k+1}c_l(1)}} (1+2\sqrt{k})^n n^{-\frac{3}{2}} \left( 1+ O \left( \frac{1}{\sqrt{n}} \right) \right), \ \ \ \text{as} \ n \rightarrow \infty, \end{align} with $c_l(u)$ defined as in Proposition \ref{prop:rjkaroundsing}. Now we want to apply the well-known Quasi-Power Theorem. \begin{theo}[Quasi-Power Theorem, \cite{hwang1998convergence}] Let $X_n$ be a sequence of random variables with the property that \[ \mathbb{E}u^{X_n} = A(u) B(u)^{\lambda_n} \left( 1+ \mathcal{O} \left(\frac{1}{\phi_n} \right) \right) \] holds uniformly in a complex neighbourhood of $u=1$, where $\lambda_n \rightarrow \infty$ and $\phi_n \rightarrow \infty$, and $A(u)$ and $B(u)$ are analytic functions in a neighbourhood of $u=1$ with $A(1)=B(1)=1$. Set $\mu = B'(1)$ and $\sigma^2=B''(1)+B'(1)-B'(1)^2$. If $\sigma^2 \neq 0$, then \[ \frac{X_n - \mathbb{E}X_n}{\sqrt{\mathbb{V}X_n}} \rightarrow \mathcal N(0,1), \] with $\mathbb{E}X_n = \mu \lambda_n + A'(1) + \mathcal{O}(1/\phi_n))$ and $\mathbb{V}X_n = \sigma^2 \lambda_n +A''(1)+A'(1)-A'(1)^2+\mathcal{O}(1/\phi_n))$. \end{theo} Using Theorem \ref{thm:asympgleqzu} and (\ref{resultgleqkasymp}), we get for $n \longrightarrow \infty$ \[ \mathbb{E}u^{X_n} = \frac{[z^n]G_{k}(z,u)}{[z^n]G_{k}(z,1)} = \left( \frac{1+2\sqrt{ku}}{1+2\sqrt{k}} \right)^n \sqrt{ \frac{\sqrt{ku}+2ku}{2k+\sqrt{k}} \prod_{j=2}^{k+1} \frac{{c}_j(1)}{c_j(u)}} \left( 1+ O \left( \frac{1}{n} \right) \right), \] where $c_1(u)=1$ and $c_j(u)=4ju-4u-1+2\sqrt{c_{j-1}(u)}$. Thus, all assumptions for the Quasi-Power Theorem are fulfilled, and we get that the number of leaves in closed lambda-terms with De Bruijn indices at most $k$ is asymptotically normally distributed with \[ \mathbb{E}{X_n} \sim \frac{k}{\sqrt{k}+2k}n, \ \ \ \text{and} \ \ \ \mathbb{V}X_n \sim \frac{k^2}{2 \sqrt{k}(\sqrt{k}+2k)^2}n, \ \ \text{as} \ n \rightarrow \infty, \] and therefore Theorem \ref{theo:mainresultlength} is shown. \section{Total number of leaves in lambda-terms with bounded number of De Bruijn levels} \label{ch:totalheight} This section is devoted to the enumeration of leaves in closed lambda-terms with a bounded number of De Bruijn levels. As in \cite{bodini2015number} let us denote by $\mathcal{P}^{(i,k)}$ the class of unary-binary trees that contain at most $k-i$ De Bruijn levels and each leaf $e$ can be coloured with one out of $i+l(e)$ colors, where $l(e)$ denotes the De Bruijn level in which the respective leaf is located. These classes can be specified by \[ \mathcal{P}^{(k,k)} = k \mathcal{Z} + ( \mathcal{A} \times \mathcal{P}^{(k,k)} \times \mathcal{P}^{(k,k)} ), \] and \[ \mathcal{P}^{(i,k)} = i \mathcal{Z} + ( \mathcal{A} \times \mathcal{P}^{(i,k)} \times \mathcal{P}^{(i,k)} ) + ( \mathcal{U} \times \mathcal{P}^{(i+1,k)} ) \ \ \ \ \ \text{for} \ \ i < k. \] By translating into generating functions we get \[ P^{(k,k)}(z,u)= kzu + zP^{(k,k)^2}(z,u), \] and \[ P^{(i,k)}(z,u) = izu + zP^{(i,k)^2}(z,u) + zP^{(i+1,k)}(z,u) \ \ \ \ \ \text{for} \ \ i < k. \] Solving yields \[ P^{(k,k)}(z,u)=\frac{1- \sqrt{1-4kz^2u}}{2z}, \] and \[ P^{(i,k)}(z,u) = \frac{1- \sqrt{1-4iz^2u - 4z^2P^{(i+1,k)}}}{2z} \ \ \ \ \text{for} \ \ i < k. \] This can be written as \[ P^{(i,k)}(z,u) = \frac{1- \sqrt{R_{k-i+1,k}(z,u)}}{2z}, \] where \begin{align} \label{radicals1k} R_{1,k}(z,u)=1-4kz^2u, \end{align} and \begin{align} \label{radicalsrecursion} R_{i,k}(z,u)=1-4(k-i+1)z^2u-2z+2z \sqrt{R_{i-1,k}(z,u)}, \ \ \ \ \text{for} \ \ 2 \leq i \leq k+1. \end{align} For the bivariate generating function of closed lambda-terms with at most $k$ De Bruijn levels we get \[ H_{k }(z,u) = P^{(0,k)}(z,u) = \frac{1- \sqrt{R_{k+1,k}(z,u)}}{2z}. \] Thus, the generating function consists again of $k+1$ nested radicals, but as stated in Section \ref{ch:mainresults}, the counting sequence of this class of lambda-terms shows a very unusual behaviour. The type of the dominant singularity of the generating function changes when the imposed bound equals $N_j$. Thus, the subexponential term in the asymptotics of the counting sequence changes. The following result has been shown in \cite{bodini2015number}: \begin{theo}[{\cite[Theorem 3]{bodini2015number}}] \label{theo:asymphleqkueg1} Let $(u_i)_{i \geq 0}$ and $(N_i)_{i \geq 0}$ be the integer sequences defined in Definition~\ref{def:sequences} and let $H_{k}(z,1)$ be the generating function of the class of closed lambda-terms with at most $k$ De Bruijn levels. Then the following asymptotic relations hold \begin{itemize} \item[(i)] If there exists $j \geq 0$ such that $N_j < k < N_{j+1}$, then there exists a constant $h_k$ such that \[ [z^n]H_k(z,1) \sim h_k n^{-3/2} \rho_k(1)^{-n}, \ \text{as} \ n \rightarrow \infty. \] \item[(ii)] If there exists $j$ such that $k=N_j$, then \[ [z^n]H_k(z,1) \sim h_k n^{-5/4} \rho_k(1)^{-n} = h_k n^{-5/4} (2u_j)^{n}, \ \text{as} \ n \rightarrow \infty. \] \end{itemize} \end{theo} Thus, in order to investigate structural properties of this class of lambda-terms we perform a distinction of cases whether the bound $k$ is an element of the sequence $(N_i)_{i \geq 0}$ or not. \subsection{The case $N_j < k <N_{j+1}$} \label{subsec:total_height_njnj+1} From \cite{bodini2015number} we know that in this case the dominant singularity of the generating function $H_{k}(z,1)$ comes from the $(j+1)$-th radicand $R_{j+1,k}$ and is of type $\frac{1}{2}$. As in the previous section we can again use continuity arguments to guarantee that sufficiently close to $u=1$ the dominant singularity $\rho_{k}(u)$ of $H_{k}(z,u)$ comes from the $(j+1)$-th radicand $R_{j+1,k}(z,u)$ and is of type $\frac{1}{2}$. Now we will determine the expansions of the radicands in a neighbourhood of the dominant singularity. \begin{prop} \label{prop:exprjkaroundsing} Let $\rho_{k}(u)$ be the dominant singularity of $H_{k}(z,u)$, where $u$ is in a sufficiently small neighbourhood of 1, \textit{i.e.} $\|u-1\|<\delta$ for $\delta>0$ sufficiently small. Then the expansions \begin{itemize} \item[(i)] $\forall i < j+1 \ \text{(inner radicands)}: R_{i,k}(\rho_{k}(u)-\epsilon,u) = R_{i,k}(\rho_{k}(u),u)+ \mathcal{O}(\|\epsilon\|)$ \item[(ii)] $R_{j+1,k}(\rho_{k}(u)-\epsilon,u) = \gamma_{j+1}(u) \epsilon +\mathcal{O}(\|\epsilon\|^2)$, with $\gamma_{j+1}(u)=-\frac{\partial}{\partial z} R_{j+1,k}(\rho_{k}(u),u)$ \item[(iii)] $\forall i > j+1 \ \text{(outer radicands)}: R_{i,k}(\rho_{k}(u)-\epsilon,u) =a_i(u)+b_i(u)\sqrt{\epsilon}+\mathcal{O}(\|\epsilon\|),$ with $a_{i+1}(u)=1-4(k-i)\rho_{k}(u)^2u-2\rho_{k}(u)+2\rho_{k}(u)\sqrt{a_i(u)}$, and $b_{i+1}(u)=\frac{b_i(u) \rho_{k}(u)}{\sqrt{a_i(u)}}$ for $j+2 \leq i \leq k$, with $a_{j+2}(u)=1-4(k-j-1)\rho_{k}(u)^2u-2\rho_{k}(u)$ and $b_{j+2}(u)=2\rho_{k}(u)\sqrt{\gamma_{j+1}(u)}$, \end{itemize} hold for $\epsilon \longrightarrow 0$ so that $\epsilon \in \mathbb{C} \setminus \mathbb{R}^{-}$, uniformly in $u$. \end{prop} \begin{proof} \begin{itemize} \item[(i)] The first equation (for $i<j+1$) follows immediately by Taylor expansion around $\rho_{k}(u)$ and setting $z=\rho_{k}(u)-\epsilon$. \item[(ii)] The equation for $i=j+1$ follows analogously to the first case, knowing that $R_{j+1,k}(z,u)$ cancels for $z=\rho_{k}(u)$. \item[(iii)] The next step is to expand $R_{i,k}(z,u)$ around $\rho_{k}(u)$ for $i >j+1$. From the second claim of Proposition \ref{prop:exprjkaroundsing} and from the recurrence relation (\ref{radicalsrecursion}) for $R_{i,k}(z,u)$ it results \[ R_{j+2,k}(\rho_{k}(u)-\epsilon,u) = 1 - 4(k-j-1)\rho_{k}(u)^2u -2\rho_{k}(u) + 2\rho_{k}(u) \sqrt{ \gamma_{j+1}(u)}\sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|). \] We set $a_{j+2}(u):=1 - 4(k-j-1)\rho_{k}^2(u)u-2\rho_{k}(u)$ and $b_{j+2}(u):=2\rho_{k}(u) \sqrt{\gamma_{j+1}(u)}$. Now we proceed by induction. Assume $R_{i,k}(\rho_{k}(u)-\epsilon,u)= a_i(u)+b_i(u) \sqrt{\epsilon}+\mathcal{O}(\|\epsilon\|)$. We have just checked that it holds for $i=j+2$. Now we perform the induction step $i \mapsto i+1$. Using the recursion (\ref{radicalsrecursion}) for $R_{i,k}$ and plugging in the expansion $a_i(u)+b_i(u) \sqrt{\epsilon}+\mathcal{O}(\|\epsilon\|)$ for $R_{i,k}(\rho_{k}(u)-\epsilon,u)$ yields \[ R_{i+1,k}(\rho_{k}(u)-\epsilon,u)=1-4(k-i)\rho_{k}(u)^2u - 2\rho_{k}(u) +2\rho_{k}(u) \sqrt{a_i(u)} + \frac{b_i(u) \rho_{k}(u)}{\sqrt{a_i(u)}} \sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|). \] Setting $\forall i \geq j+2\ \ a_{i+1}(u):=1-4(k-i)\rho_{k}^2(u)u - 2\rho_{k}(u) +2\rho_{k}(u) \sqrt{a_i(u)}$ and $b_{i+1}(u):=\frac{b_i(u) \rho_{k}(u)}{\sqrt{a_i(u)}},$ we obtain \[ R_{i+1,k}(\rho_{k}(u)-\epsilon,u)= a_{i+1}(u)+b_{i+1}(u)\sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|). \] Expanding $b_i(u)$, using its recursive relation and $b_{j+2}(u)=2\rho_{k}(u) \sqrt{\gamma_{j+1}(u)}$ we get for $i > j+1$ \[ b_i(u) = \frac{2\rho_{k}^{i-j}(u)\sqrt{\gamma_{j+1}(u)}}{\prod_{l=j+1}^{i-1} \sqrt{a_l(u)}}. \] \end{itemize} \end{proof} We know that for sufficiently large $i$ the sequence $u_i$ is given by $u_i = \lfloor \chi^{2^i} \rfloor$, with $\chi \approx 1.36660956 \ldots$ (see \cite[Lemma 18]{bodini2015number}). Therefore we have $N_j \sim u_j^2 \sim \chi^{2^j 2}$ and $N_j<k<N_{j+1}=O(N_j^2)$, which gives $j \asymp \log \log k$. This implies that $j+1<k+1$, \textit{i.e.}, that the dominant singularity $\rho_{k}(u)$ cannot come from the outermost radical. \begin{rem} Obviously the same is true for the case $k=N_j$. Thus, the dominant singularity never comes from the outermost radical. \end{rem} Using Proposition \ref{prop:exprjkaroundsing} and $H_{k}(z,u)=\frac{1}{2z}(1-\sqrt{R_{k+1,k}(z,u)})$ we get \[ H_{k}(\rho_{k}(u)-\epsilon,u)= \frac{1-\sqrt{a_{k+1}(u)}}{2\rho_{k}(u)}-\frac{b_{k+1,k}(u)}{4\rho_{k}(u)\sqrt{a_{k+1}(u)}} \sqrt{\epsilon} + \mathcal{O}(\|\epsilon\|), \] which yields \begin{align} \label{equ_znhkzutotal} [z^n]H_{k}(z,u) = h_k(u) \rho_{k}(u)^{-n} \frac{n^{-\frac{3}{2}}}{\Gamma(-\frac{1}{2})} \left( 1+ O \left( \frac{1}{\sqrt{n}} \right) \right), \end{align} with \[ h_k(u) = -\frac{b_{k+1}(u)\sqrt{\rho_k(u)}}{4\rho_{k}(u)\sqrt{a_{k+1}(u)}}. \] Taking a look at the recursive definitions of $a_{i}(u)$ and $b_i(u)$ (see Proposition \ref{prop:exprjkaroundsing}), it can easily be seen that these functions are not equal to zero in a neighbourhood of $u=1$. We know that $a_{j+2}(1)$ is positive, since \begin{align} a_{j+2}(1)=1- 4(k-j-1)\rho_k(1)^2-2\rho_k(1)= 1- 4(k-j)\rho_k(1)^2-2\rho_k(1) + 4\rho_k^2, \end{align} and $1- 4(k-j)\rho_k(1)^2-2\rho_k(1)>0$ (see \cite{bodini2015number}). By induction we can show that the sequence $a_i:=a_i(1)$ is monotonically increasing. Let us assume that $a_{i-1}<a_i$, then we get \begin{align} a_{i+1}&>1-4(k-i)\rho_k(1)^2-2\rho_k(1)+2\rho_k(1)\sqrt{a_i} \\ &> 1- 4(k-i+1)\rho_k(1)^2-2\rho_k(1)+2\rho_k(1)\sqrt{a_{i-1}} + 4\rho_k(1)^2 > a_i + 4\rho_k(1)^2. \end{align} It is obvious that if $b_{j+2}:=b_{j+2}(1)$ is non-zero, than all the $b_i$'s, which are defined by \begin{align} b_i = \frac{\rho_k(1)b_{i-1}}{a_{i-1}}, \end{align} are non-zero. In order to prove that $b_{j+2}=2\rho_k(1) \sqrt{-\frac{\partial}{\partial z} R_{j+1,k}(\rho_k(1),1)}$ is non-zero, we also proceed by induction. Since \begin{align} R_{1,k}(z,1)=1-4kz^2, \end{align} we can see that $\frac{\partial}{\partial z} R_{1,k}(\rho_k(1),1) <0$, and assuming $\frac{\partial}{\partial z} R_{i,k}(\rho_k(1),1) < 0$ and using \begin{align} \frac{\partial}{\partial z} R_{i+1,k}(z,1)= -8(k-i)z-2+2\sqrt{R_{i,k}(z,1)}+\frac{z}{\sqrt{R_{i,k}(z,1)}} \frac{\partial}{\partial z} R_{i}(z,1), \end{align} we proved that all $b_i$'s are non-zero. Thus, we get that $h_k(u) \neq 0$. Using (\ref{equ_znhkzutotal}) and Theorem \ref{theo:asymphleqkueg1} we get for $n \longrightarrow \infty$ \begin{align} \label{equ:znhleqkzuznhleqkz1} \frac{[z^n]H_{k}(z,u)}{[z^n]H_{k}(z,1)} = \frac{h_k(u)}{h_k \Gamma(-1/2)} \left( \frac{\rho_{k}(1)}{\rho_{k}(u)} \right)^n \left( 1+ O \left( \frac{1}{n} \right) \right). \end{align} Assuming that $\sigma^2=B''(1)+B'(1)-B'(1)^2\neq 0$ with $B(u)= \frac{\rho_{k}(1)}{\rho_{k}(u)}$ we can apply the Quasi-Power Theorem. As stated in Section \ref{ch:mainresults} the proof of this assumption appears to be quite difficult, since there is only very little known about the function $\rho_{k}(u)$. However, it seems very likely that this condition will be fulfilled for arbitrary $k \in (N_j, N_{j+1})$, so that the Quasi-Power Theorem can be applied and we get that the number of leaves in lambda-terms with bounded number of De Bruijn levels is asymptotically normally distributed with asymptotic mean $\mu n$ and variance $\sigma^2 n$, respectively, where $\mu = B'(1)$ and $\sigma^2=B''(1)+B'(1)-B'(1)^2$, with $B(u)= \frac{\rho_{k}(1)}{\rho_{k}(u)}$. \begin{table}[h!] \center \small \begin{tabular}{c\|c\|c\|c} bound $k$ & $j+1$ & $B''(1)+B'(1)-B'(1)^2$ & $B'(1)$ \\ \hline \bf{1} & \bf{2} & \bf{0} & \bf{0} \\ 2 & 2 & 0.0385234386 & 0.4381229337 \\ 3 & 2 & 0.0210625856 & 0.4414407371 \\ 4 & 2 & 0.0167136805 & 0.4463973717 \\ 5 & 2 & 0.0148700270 & 0.4504258849 \\ 6 & 2 & 0.0138224393 & 0.4536185043 \\ 7 & 2 & 0.0131157948 & 0.4561987871 \\ \bf{8} & \bf{3} & \bf{0.0125868052} & \bf{0.4583333333} \\ 9 & 3 & 0.0582322465 & 0.4566104777 \\ 10 & 3 & 0.0470481360 & 0.4560418340 \\ 11 & 3 & 0.0396601986 & 0.4560810348 \\ 12 & 3 & 0.0345090124 & 0.4564489368 \\ \vdots & \vdots & \vdots & \vdots \\ 133 & 3 & 0.0077469541 & 0.4821900098 \\ 134 & 3 & 0.0077234960 & 0.4822482745 \\ \bf{135} & \bf{4} & \bf{0.0077002803} & \bf{0.4823059361} \\ 136 & 4 & 0.0132855719 & 0.4823515285 \\ 137 & 4 & 0.0131816901 & 0.4823968564 \\ 138 & 4 & 0.0130800422 & 0.4824419195 \\ 139 & 4 & 0.0129805564 & 0.4824867175 \end{tabular} \caption{Table summarizing the coefficients occurring in the variance and the mean for some initial values of $k$.} \label{tab:initialvalues} \end{table} \subsection{The case $k = N_j$} \label{subsection:k=njtotal} We know from \cite{bodini2015number} that in the case $k = N_j$ both radicands $R_{j,k}(z,1)$ and $R_{j+1,k}(z,1)$ vanish simultaneously and the dominant singularity is therefore of type $\frac{1}{4}$. This is not true for the radicands $R_{j,k}(z,u)$ and $R_{j+1,k}(z,u)$ when $u$ is in a neighbourhood of 1. Thus, we have a discontinuity at $\rho_k(1)$, which is why we do not get any uniform expansions of the radicands in a neighbourhood of $\rho_k(1)$. In order to overcome this problem we will set $u=1+\epsilon$ and investigate how the radicands behave in a neighbourhood of the dominant singularity $\rho_k(u)=\rho_k(1+\epsilon)$. Subsequently we will use the abbreviation $\rho_k:=\rho_k(1)$. \begin{lemma} \label{lem:domsingvonjradicand} For $u=1+\epsilon$ with $\epsilon \longrightarrow 0$ so that $\epsilon \in \mathbb{C} \setminus \mathbb{R}^{-}$, the dominant singularity $\rho_k(u)=\rho_k(1+\epsilon)$ of the bivariate generating function $H_k(z,1+\epsilon)$ comes from the $j$-th radicand $R_{j,k}(z,u)$. \end{lemma} \begin{proof} Setting $u=1+\epsilon$,expanding $\rho_k(u)$ around $1$ and plugging into the recursive definition of the radicands yields \begin{align} R_{j,k} $ \rho_k(1+\epsilon),1+\epsilon $ = &1-4(k-j+1)(\rho_k^2+2\rho_k\rho_k'\epsilon+ (\rho_k'^2+2\rho_k\rho_k'')\epsilon^2 +\rho_k^2\epsilon) \\ &- (2\rho_k+2\rho_k'\epsilon +2\rho_k''\epsilon^2) $1-\sqrt{R_{j-1,k}\( \rho_k(1+\epsilon),1+\epsilon $} \) + \mathcal{O} ( \|\epsilon\| ). \end{align} Using $1-4(k-j)\rho_k^2-2\rho_k=0$ and $\sqrt{R_{j-1,k}$ \rho_k(1+\epsilon),1+\epsilon $} = \sqrt{R_{j-1,k}$ \rho_k,1 $ + \mathcal{O}(\|\epsilon\|)} = 2\rho_k + \mathcal{O}(\|\epsilon\|)$, which are both shown in \cite{bodini2015number}, we get \begin{align} R_{j,k} $ \rho_k(1+\epsilon),1+\epsilon $ = &-4(k-j+1)(2\rho_k\rho_k'\epsilon+ (\rho_k'^2+2\rho_k\rho_k'')\epsilon^2) \\ &- (2\rho_k'\epsilon +2\rho_k''\epsilon^2) $1-2\rho_k + \mathcal{O}(\|\epsilon\|) $ + \mathcal{O} ( \|\epsilon\| ). \end{align} Thus, $R_{j,k} $ \rho_k(1+\epsilon),1+\epsilon $= \Theta(\|\epsilon\|)$. Using this result and again the recursive definition of the radicands results in \begin{align} R_{j+1,k} $ \rho_k(u),1+\epsilon $= 2\sqrt{R_{j,k}(\rho_k(u),1+\epsilon)} + \mathcal{O} $ \|\epsilon\| $ = \Theta(\sqrt{\|\epsilon\|}). \end{align} Thus, we see that $\|R_{j+1,k} (\rho_k(u),u)\| \gg \|R_{j,k} (\rho_k(u),u)\|$ in a neighbourhood of $u=1$, which implies that the dominant singularity has to come from the $j$-th radicand, \textit{i.e.} $R_{j,k}(\rho_k(u),u)=0$ for $u$ being sufficiently close to $1$. \end{proof} Now that we know that in this case ($k=N_j$) the dominant singularity of $H_k(z,u)$ in a neighbourhood of $u=1$ comes from the $j$-th radicand, we investigate the expansions of the radicands thoroughly for $u=1+\frac{s}{\sqrt{n}}$ in a neighbourhood with radius $\frac{t}{n}$, where $s$ and $t$ are both bounded complex numbers (\textit{cf}. Figure \ref{fig:proofidea}). \begin{lemma} \label{lem:exprjrj+1rho1} Let $z=\rho_k(u)=\rho_k(1+\frac{s}{\sqrt{n}})$ be the dominant singularity of the bivariate generating function $H_k(z,1+\frac{s}{\sqrt{n}})$ with bounded $s \in \mathbb{C}$. Then, as $n \longrightarrow \infty$, \begin{itemize} \item[(i)] $R_{j,k} \Big(\rho_k(u)$1+\frac{t}{n}$,1+\frac{s}{\sqrt{n}}\Big) = \frac{1}{n} p_j(t) + \mathcal{O} $ \frac{1}{n^{3/2}} $$, \newline with $p_j(t):=-8t(k-j+1)\rho_k^2 - 2\rho_kt+4\rho_k^2t+2t\rho_kf(\frac{t}{n})$ where $f(\frac{t}{n})$ is an analytic function around 0;\\ \item[(ii)] $R_{j+1,k}\Big(\rho_k(u)$1+\frac{t}{n}$,1+\frac{s}{\sqrt{n}}\Big) = \frac{1}{\sqrt{n}} p_{j+1}(s,t) + \mathcal{O} $ \frac{1}{n} $,$ \newline where $p_{j+1}(s,t)=2\rho_k \sqrt{p_j(t)}-4(k-j)(2\rho_k\rho_k's+\rho_k^2s)-2\rho_k's$;\\ \item[(iii)] $R_{i,k}\Big(\rho_k(u)$1+\frac{t}{n}$,1+\frac{s}{\sqrt{n}}\Big) = \hat{C}_{i} + \frac{1}{\sqrt[4]{n}} p_{i}(s,t) + O $ \frac{1}{\sqrt{n}} $ \ \ \ \ \text{for} \ i \geq j+2$, \newline where $\hat{C}_{i}$ are constants and $p_{i}(s,t)$ analytic functions $s$ and $t$. \end{itemize} \end{lemma} \begin{proof} We start with setting $u=1+\frac{s}{\sqrt{n}}$ and $z=\rho_k(u)(1+\frac{t}{n})$ with bounded $s,t \in \mathbb{C}$ (\textit{cf.} Figure \ref{fig:proofidea}), which results in \begin{align} &R_{j+1,k}$\rho_k(u)\(1+\frac{t}{n}$,1+\frac{s}{\sqrt{n}}\)= \\ &1-4(k-j)\rho_k(u)^2$1+\frac{t}{n}$^2$1+\frac{s}{\sqrt{n}}$-2\rho_k(u)$1+\frac{t}{n}$$1- \sqrt{R_{j,k}}$,\\ \text{and}\\ &R_{j,k}$\rho_k(u)\(1+\frac{t}{n}$,1+\frac{s}{\sqrt{n}}\)=\\ &1-4(k-j+1)\rho_k(u)^2$1+\frac{t}{n}$^2$1+\frac{s}{\sqrt{n}}$-2\rho_k(u)$1+\frac{t}{n}$$1- \sqrt{R_{j-1,k}} $, \end{align} where the radicand in the square root in the last bracket of both equations is of course also evaluated at $(z,u)=$\rho_k(1+\frac{s}{\sqrt{n}})(1+\frac{t}{n}),1+\frac{s}{\sqrt{n}}$$, but we will omit this notation from now on to ensure a simpler reading, \textit{\textit{i.e.}}, subsequently we will write $R_{i,k}$ instead of $R_{i,k}$\rho_k(1+\frac{s}{\sqrt{n}})\(1+\frac{t}{n}$,1+\frac{s}{\sqrt{n}}\)$. Expanding $\rho_k(1+\frac{s}{\sqrt{n}})$ around 1 and using the recursive definition for the radicands yields \begin{align} \label{equ:rjkasympnearueq1} \begin{aligned} R_{j,k} &= 1-4(k-j+1) $ \rho_k^2 + 2\rho_k \rho_k' \frac{s}{\sqrt{n}} + (\rho_k'^2+2\rho_k\rho_k'') \frac{s^2}{n} + \rho_k^2 \frac{s}{\sqrt{n}} + 2\rho_k \rho_k' \frac{s^2}{n} + \rho_k^2 \frac{2t}{n} $ \\ &-2$ \rho_k + \rho_k' \frac{s}{\sqrt{n}} + \rho_k'' \frac{s^2}{2n} + \rho_k \frac{t}{n} $ $ 1- \sqrt{R_{j-1,k}}$ + \mathcal{O} $ \frac{1}{n^{3/2}} $. \end{aligned} \end{align} From Lemma \ref{lem:domsingvonjradicand} we know that for $u$ in a sufficiently small vicinity of 1 the dominant singularity of $H_k(z,u)$ comes from the $j$-th radicand, \textit{i.e.} $R_{j,k}$ \rho_k(u),u $=0$. Expanding $R_{j,k}$ \rho_k(1+\frac{s}{\sqrt{n}}),1+\frac{s}{\sqrt{n}} $$ this yields \begin{align} &1-4(k-j+1)$ \rho_k^2 + 2\rho_k \rho_k' \frac{s}{\sqrt{n}} + (\rho_k'^2+2\rho_k\rho_k'') \frac{s^2}{n} + \rho_k^2 \frac{s}{\sqrt{n}} + 2\rho_k \rho_k' \frac{s^2}{n} $ \\ &-2$ \rho_k + \rho_k' \frac{s}{\sqrt{n}} + \rho_k'' \frac{s^2}{2n} $ $ 1- \sqrt{R_{j-1,k}\( \rho_k \( 1+\frac{s}{\sqrt{n}} $,1+\frac{s}{\sqrt{n}} \)}\) + \mathcal{O} $ \frac{1}{n^{3/2}} $=0. \end{align} Thus, Equation (\ref{equ:rjkasympnearueq1}) simplifies to \begin{align} \label{equ:rjkasympfertig} R_{j,k} = -4(k-j+1) \rho_k^2 \frac{2t}{n} - 2\rho_k \frac{t}{n} + 4\rho_k^2 \frac{t}{n} + 2\rho_k\frac{t}{n}f$\frac{t}{n}$ + \mathcal{O} $ \frac{1}{n^{3/2}} $, \end{align} where $\frac{t}{n}f$\frac{t}{n}$=\sqrt{R_{j-1,k}}-\sqrt{R_{j-1,k}$ \rho_k(1+\frac{s}{\sqrt{n}}),1+\frac{s}{\sqrt{n}}$}$, where $f$\frac{t}{n}$$ is analytic around 0. Therefore, the proof of $(i)$ is finished. Proceeding equivalently for $R_{j+1,k}$ results in \begin{align} R_{j+1,k} &= \frac{1}{\sqrt{n}} \Big( -4(k-j)(2\rho_k\rho_k's+\rho_k^2s)-2\rho_k's \Big) + 2\rho_k\sqrt{R_{j,k}} + O $ \frac{1}{n} $. \end{align} Inserting Equation (\ref{equ:rjkasympfertig}) for $R_{j,k}$ we proved the second statement of the lemma. Going one step further leads to \[ R_{j+2,k} = \hat{C}_{j+2} + \frac{1}{\sqrt[4]{n}} p_{j+2}(s,t) + O $ \frac{1}{\sqrt{n}} $, \] with $\hat{C}_{j+2}:=4\rho_k^2$ and $p_{j+2}(s,t):= 2\rho_k\sqrt{p_{j+1}(s,t)}$, where $p_{j+1}(s,t)$ is defined as in Lemma \ref{lem:exprjrj+1rho1}. Now we proceed by induction. Therefore we assume that $R_{i,k} = \hat{C}_{i} + \frac{1}{\sqrt[4]{n}} p_{i}(s,t) + O $ \frac{1}{\sqrt{n}} $$ with $i \geq j+2$. Thus, we get \begin{align} \begin{aligned} R_{i+1,k} &= 1-4(k-i) $ \rho_k^2 + 2\rho_k \rho_k' \frac{s}{\sqrt{n}} + (\rho_k'^2+2\rho_k\rho_k'') \frac{s^2}{n} + \rho_k^2 \frac{s}{\sqrt{n}} + 2\rho_k \rho_k' \frac{s^2}{n} + \rho_k^2 \frac{2t}{n} $ \\ &-2$ \rho_k + \rho_k' \frac{s}{\sqrt{n}} + \rho_k'' \frac{s^2}{2n} + \rho_k \frac{t}{n} $ $ 1- \sqrt{R_{i,k}}$ + \mathcal{O} $ \frac{1}{n^{3/2}} $. \end{aligned} \end{align} Inserting the induction hypothesis and simplifying yields \[ R_{i+1,k} = 4(i-j)\rho_k^2+2\rho_k \sqrt{\hat{C}_i} + \frac{1}{\sqrt[4]{n}} \frac{\rho_kp_i(s,t)}{\sqrt{\hat{C}_i}} + \mathcal{O} $ \frac{1}{\sqrt{n}} $. \] Setting $\hat{C}_{i+1}:= 4(i-j)\rho_k^2+2\rho_k \sqrt{\hat{C}_i}$ and $p_{i+1}(s,t):=\frac{\rho_k}{\sqrt{\hat{C}_i}} p_i(s,t)$ completes the proof. \end{proof} \begin{figure} \caption{Sketch of the idea of the proof.} \label{fig:proofidea} \end{figure} \begin{prop} \label{prop:znhkzutotal} Let $H_k(z,u)$ be the bivariate generating function of the class of closed lambda-terms with at most $k$ De Bruijn levels. Then the $n$-th coefficient of $H_k(z,1+\frac{s}{\sqrt{n}})$ with bounded $s \in \mathbb{C}$ is given by \begin{align} [z^n] H_{k}(z,1+\frac{s}{\sqrt{n}}) = C_k(s) \rho_k^{-n} n^{-\frac{5}{4}} $ 1+ O \( n^{-\frac{3}{4}} $ \), \ \ \ \text{as} \ n \longrightarrow \infty, \end{align} with a constant $C_k(s) \neq 0$. \end{prop} \begin{proof} Let us remember that $H_{k}(z,1+\frac{s}{\sqrt{n}}) = \frac{1- \sqrt{R_{k+1,k}(z,1+\frac{s}{\sqrt{n}})}}{2z}$. Thus, with the well-known Cauchy coefficient formula we get \begin{align} [z^n]H_{k} $ z, 1+\frac{s}{\sqrt{n}} $ &= \frac{1}{2i\pi} \int_{\gamma} \frac{H_{k} $ z, 1+\frac{s}{\sqrt{n}} $}{z^{n+1}} dz \\ &= \frac{1}{2i\pi} \int_{\gamma} \frac{1-\sqrt{R_{k+1,k} $ z, 1+\frac{s}{\sqrt{n}}$} }{2z^{n+2}} dz, \end{align} where $\gamma$ encircles the dominant singularity $\rho_k(u)$ as depicted in Figure \ref{fig:integrationcontours}. We denote the small Hankel-like part of the integration contour $\gamma$ that contributes the main part of the asymptotics by $\gamma_H$ (\textit{cf}. Figure \ref{fig:integrationcontours}). The curve $\gamma_H$ encircles $\rho_k(u)$ at a distance $\frac{1}{n}$ and its straight parts (that lead into the direction $\rho_k(u) \cdot \infty$) have the length $\frac{\log^2(n)}{n}$. On $\gamma \setminus \gamma_H$ we have $\|z\|=\|\rho_k(u)\| \Big\|1+ \frac{\log^2(n)}{n} + \frac{i}{n} \Big\|$. Thus, using the transformation $z=\rho(u)$ 1+ \frac{t}{n} $$, which changes $\gamma_H$ to $\tilde{\gamma}_H$, and Lemma \ref{lem:exprjrj+1rho1} and estimating the contribution of $\gamma \setminus \gamma_H$ implies that there exisits a $K>0$ such that \begin{align} [z^n]H_{k} $ z, 1+\frac{s}{\sqrt{n}} $ &= \frac{1}{2i\pi} \int_{\tilde{\gamma}_H} \frac{1-\sqrt{\hat{C}_{k+1}+\frac{1}{\sqrt[4]{n}}p_{k+1}(s,t)+\mathcal{O} $ \frac{1}{\sqrt{n}} $}}{2\rho^{n+1}e^tn} dt + \mathcal{O} $ e^{-K\log^2(n)} $\\ \label{equ:znhkmitintegral} &= \frac{1}{2i\pi} \int_{\tilde{\gamma}_H} \frac{1-\sqrt{\hat{C}_{k+1}} - \frac{1}{2\sqrt[4]{n}\sqrt{\hat{C}_{k+1}}}p_{k+1}(s,t)+\mathcal{O} $ \frac{1}{\sqrt{n}} $}{2\rho_k^{n+1}e^tn} dt+\mathcal{O} $ e^{-K\log^2(n)} $. \end{align} \begin{figure} \caption{The integral contours $\gamma$ and $\mathcal{H}$.} \label{fig:integrationcontours} \end{figure} Now, let us observe how the function $p_{k+1}(s,t)$ looks like by using the recursive definition $p_{i+1}(s,t)= \frac{\rho_k}{\sqrt{\hat{C}_i}} p_i(s,t)$ and $p_{j+2}(s,t)=2\rho_k \sqrt{2\rho_k\sqrt{p_j(t)}+q(s)}$, with a polynomial $q(s)=s$-4(k-j)(2\rho_k\rho_k'+\rho_k^2)-2\rho_k'$$ that is linear in $s$. Thus, $p_{k+1}(s,t)=D \cdot p_{j+2}(s,t)$ with a constant $D$. Inserting this into (\ref{equ:znhkmitintegral}) and splitting the integral yields \begin{align} &[z^n]H_{k} $ z, 1+\frac{s}{\sqrt{n}} $ = \\ &\frac{\rho_k^{-n}}{4i\pi\rho_kn} $ \int_{\tilde{\gamma}_H} \(1-\sqrt{\hat{C}_{k+1}}$e^{-t}dt - \int_{\tilde{\gamma}_H} \frac{De^{-t}}{2\sqrt[4]{n}\sqrt{\hat{C}_{k+1}}} \sqrt{2\rho_k\sqrt{p_j(t)}+q(s)}dt + \int_{\tilde{\gamma}_H} \mathcal{O} $ \frac{1}{\sqrt{n}} $ e^{-t}dt \). \end{align} The first integral is zero and the third integral contributes $\mathcal{O} $ \frac{1}{\sqrt{n}} $$. Thus, the main part of the asymptotics results from the second integral: There are some constants $A(s)$ and $B(s)$ such that \begin{align} - \int_{\tilde{\gamma}_H} \frac{De^{-t}}{\sqrt[4]{n}\sqrt{\hat{C}_{k+1}}}\sqrt{2\rho_k\sqrt{p_j(t)}+q(s)}dt &= - \int_{\tilde{\gamma}_H} \frac{De^{-t}}{\sqrt[4]{n}\sqrt{\hat{C}_{k+1}}}\sqrt{A(s)t+B(s)+\mathcal{O}$\frac{\log^4(n)}{n}$}dt\\ &= - \int_{\tilde{\gamma}_H} \frac{De^{-t}}{\sqrt[4]{n}\sqrt{\hat{C}_{k+1}}}\sqrt{A(s)t+B(s)}dt + \mathcal{O} $ \frac{\log^6(n)}{n} $ \\ &= - \int_{\mathcal{H}} \frac{De^{-t}}{\sqrt[4]{n}\sqrt{\hat{C}_{k+1}}}\sqrt{A(s)t+B(s)}dt + \mathcal{O} $ e^{-\tilde{K}\log^2(n)} $ \\ &\sim \tilde{C}(s)\frac{1}{\sqrt[4]{n}}. \end{align} Here $\tilde{K}$ denotes a suitable positive constant, and $\mathcal{H}$ denotes the classical Hankel curve, \emph{i.e.}, the noose-shaped curve that winds around 0 and starts and ends at $+\infty$ (\textit{cf}. Figure \ref{fig:integrationcontours}). Finally, using this result we get \begin{align} [z^n]H_{k} $ z, 1+\frac{s}{\sqrt{n}} $ = C(s) \rho_k $ 1+\frac{s}{\sqrt{n}} $^{-n} n^{-5/4} $ 1+ \mathcal{O} \( \frac{1}{\sqrt[4]{n}} $ \), \ \ \ \text{for} \ n \longrightarrow \infty, \end{align} with a constant $C(s)$ that depends on $s$. \end{proof} Now we show that the characteristic function of our standardized sequence of random variables tends to the characteristic function of the normal distribution. \begin{lemma} Let $X_n$ be the total number of variables in a random lambda-term with at most $k$ De Bruijn levels. Set $\sigma^2 :=2 \cdot $ \frac{\rho_k'(1)}{\rho_k(1)}-\frac{\rho_k''(1)}{\rho_k(1)}+\frac{\rho_k'(1)^2}{\rho_k(1)^2} $$. If $\sigma^2 \neq 0$, then \[ Z_n = \frac{X_n - \mathbb{E}X_n}{\sqrt{n}} \longrightarrow \mathcal{N} $ 0, \sigma^2 $. \] \end{lemma} \begin{proof} For the standardised sequence of random variables $Z_n$ we have with $\mu :=\frac{\mathbb{E}X_n}{n}$ \[ Z_n = \frac{X_n - \mathbb{E}X_n}{\sqrt{n}} = \frac{X_n}{\sqrt{n}}-\mu \sqrt{n}. \] Its characteristic function reads as \begin{align} \phi_{Z_n}(s) &= \mathbb{E}(e^{isZ_n}) = e^{-is\mu \sqrt{n}} \phi_{X_n}$ \frac{s}{\sqrt{n}} $ = e^{-is\mu \sqrt{n}} \mathbb{E}(e^{\frac{isX_n}{\sqrt{n}}}) = \\ &= e^{-is\mu \sqrt{n}} \frac{[z^n] H_k(z,e^{\frac{is}{\sqrt{n}}})}{[z^n]H_k(z,1)}. \end{align} From Proposition \ref{prop:znhkzutotal} we know \begin{align} \frac{[z^n] H_k(z,1+\frac{s}{\sqrt{n}})}{[z^n]H_k(z,1)} \sim C(s) $ \frac{\rho_k(1+\frac{s}{\sqrt{n}})}{\rho_k(1)} $ ^{-n}, \end{align} where the constant $C(s) \sim 1$ for $n \longrightarrow \infty$. Thus, \begin{align} \phi_{Z_n}(s) &= e^{-is\mu \sqrt{n}} \frac{[z^n] H_k(z,e^{\frac{is}{\sqrt{n}}})}{[z^n]H_k(z,1)} \sim e^{-is\mu \sqrt{n}} $ \frac{\rho_k \(1+ \frac{si}{\sqrt{n}} - \frac{s^2}{n} + \mathcal{O}\( \frac{\|s^3\|}{n^{3/2}} $ \)}{\rho_k(1)} \) ^{-n} =\\ &= e^{-is\mu \sqrt{n}} \exp $ -n \cdot \( \log\(1+ \frac{\rho_k'is}{\rho_k\sqrt{n}} - \frac{s^2}{n} \frac{\rho_k'}{\rho_k}- - \frac{s^2}{n} \frac{\rho_k''}{\rho_k} $ + \mathcal{O}$ \frac{\|s^3\|}{n^{3/2}} $ \) \) \\ &\sim e^{-is\mu \sqrt{n}} e^{-is\sqrt{n} \frac{\rho_k'}{\rho_k}} e^{s^2 $ -\frac{\rho_k'}{\rho_k}+\frac{\rho_k''}{\rho_k}-\frac{\rho_k'^2}{\rho_k^2} $}. \end{align} Since we know that the expected value of the standardised random variable is zero, we get $\mu=-\frac{\rho_k'(1)}{\rho_k(1)}+o $ \frac{1}{\sqrt{n}}$$, and thus \begin{align} \phi_{Z_n}(s) \sim e^{- \frac{s^2 \sigma^2}{2}}, \end{align} with $\sigma^2=2 \cdot $ \frac{\rho_k'(1)}{\rho_k(1)}-\frac{\rho_k''(1)}{\rho_k(1)}+\frac{\rho_k'(1)^2}{\rho_k(1)^2} $$, which completes the proof. \end{proof} Thus, we get that the total number of leaves in lambda-terms with a bounded number of De Bruijn levels is asymptotically normally distributed. \section{Unary profile of lambda-terms with bounded number of De Bruijn levels} \label{ch:levelheight} \subsection{Leaves} \label{sec:leaveslevels} The aim of this section is the investigation of the distribution of the number of leaves in the different De Bruijn levels in closed lambda-terms with bounded number of De Bruijn levels. In order to do so, let us consider that each De Bruijn level in such a lambda-term corresponds to one or more binary trees that contain different types of leaves, where the number of types corresponds to the respective level (\textit{cf.} Figure \ref{fig:unarylevels}), \textit{i.e.}, in the $i$-th De Bruijn level there may be $i$ different types of leaves. Let $\mathcal{C}$ be the class of binary trees. Using the notation from the previous sections we can specify this class by \[ \mathcal{C}= \mathcal{Z} + ( \mathcal{A} \times \mathcal{C} \times \mathcal{C}). \] Translating into bivariate generating functions $C(z,u)$ with $z$ marking the size (\textit{i.e.}, the total number of nodes) and $u$ marking the number of leaves, yields $ C(z,u)=\frac{1- \sqrt{1-4uz^2}}{2z}$. Let $_{k-l}\tilde{H}_{k}(z,u)$ be the generating function of closed lambda-terms with at most $k$ De Bruijn levels, where $z$ marks the size and $u$ marks the number of leaves on the $(k-l)$-th unary level ($0 \leq l \leq k$). Then we have \[ _{k-l}\tilde{H}_{k}(z,u) = C(z,C(z,1+ \ldots +C(z,(k-l) \cdot u+ \ldots + C(z, (k-1) +C(z,k))) \ldots) \ldots)), \] which can be written as \[ _{k-l}\tilde{H}_{k}(z,u) = \frac{1-\sqrt{\tilde{R}_{k+1,k}(z,u)}}{2z}, \] with \[ \tilde{R}_{1,k}(z,u)=1-4z^2k, \] \[\tilde{R}_{i,k}(z,u)=1-4z^2(k-i+1)-2z+2z\sqrt{\tilde{R}_{i-1,k}(z,u)}, \ \ \ \text{for} \ 2 \leq i \leq k+1, i \neq l+1, \] and \[\tilde{R}_{l+1,k}(z,u)=1-4z^2u(k-l)-2z+2z\sqrt{\tilde{R}_{l-1,k}(z,u)}. \] \begin{figure} \caption{A schematic sketch of a lambda-term with at most $k$ De Bruijn levels that exemplifies the notation that is used within this section: If we investigate the number of leaves in the $(k-l)$-th De Bruijn level, for $0 \leq l \leq k$, a factor $u$ is inserted in the recursive definition of the $(l+1)$-th radicand.} \label{fig:explamationunarylevelsnotation} \end{figure} \begin{rem} Note that the radicands $\tilde{R}_{i,k}$ that are introduced above are very similar to the radicands $R_{i,k}$ that were used in the previous section. The only difference is that now we have a $u$ only in the $(l+1)$-th radicand, while in the previous case $u$ was occurring in all radicands. Thus, from now on we will have further distinctions of cases now depending on the relative position (w.r.t. $l$) of the radicand(s) where the dominant sigularity comes from. \end{rem} This chapter consists of two sections. In the first part we will derive the mean values for the number of leaves in the different De Bruijn levels and the second part deals with the distributions of the number of leaves in these levels. \subsubsection{Mean values} Now we want to determine the mean for the number of leaves in the different De Bruijn levels, \textit{\textit{i.e.}} \[ \mathbb{E}X_n= \frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}\tilde{H}_{k}(z,1)}, \] where $X_n$ denotes the number of leaves in the $(k-l)$-th De Bruijn level of a random closed lambda-term of size $n$ with at most $k$ De Bruijn levels. In order to do so, we make the following considerations: \begin{itemize} \item $\frac{\partial}{\partial u} \tilde{R}_{i,k}(z,u)=0 \ \ \ \ \forall i < l+1$\\ \item$ \frac{\partial}{\partial u} \tilde{R}_{l+1,k}(z,u)= -4z^2(k-l)$ \\ \item $\frac{\partial}{\partial u} \tilde{R}_{i,k}(z,u)= z \cdot \frac{ \frac{ \partial}{ \partial u} \tilde{R}_{i-1,k}(z,u)}{\sqrt{\tilde{R}_{i-1,k}(z,u)}}\ \ \ \ \ \forall i > l+1$ \end{itemize} Therefore we get \begin{align} \label{equ:ablkminuslhleqk} \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \bigg\|_{u=1} = z^{k-l+1} (k-l) \prod_{i=l+1}^{k+1} \frac{1}{\sqrt{\tilde{R}_{i,k}(z,1)}}. \end{align} Again we perform a distinction of cases starting with $k$ not being an element of the sequence $(N_j)_{j \in \mathbb{N}}$. \paragraph{\textbf{The case: $N_j < k < N_{j+1}$}} Let $\tilde{\rho}_{k}(u)$ be the dominant singularity of $_{k-l}\tilde{H}_{k}(z,u)$, which we know comes from the $(j+1)$-th radicand $\tilde{R}_{j+1,k}(z,u)$. Obviously, $\tilde{\rho}_{k}(1)=\rho_{k}(1)$. Therefore we will again use the abbreviation $\rho_k:=\tilde{\rho}_k(1)$. From Proposition \ref{prop:exprjkaroundsing} we get the following expansions of the radicands for $u=1$ and $\epsilon \longrightarrow 0$ so that $\epsilon \in \mathbb{C} \setminus \mathbb{R}^{-}$: \begin{itemize} \item $\forall i < j+1 \ \text{(inner radicands)}: \tilde{R}_{i,k}(\rho_k-\epsilon,1) = \tilde{R}_{i,k}(\rho_k,1)+ \mathcal{O}(\|\epsilon\|)$,\\ \item $\tilde{R}_{j+1,k}(\rho_k-\epsilon,1) = \tilde{\gamma}_{j+1} \epsilon +\mathcal{O}(\|\epsilon\|^2)$, with $\tilde{\gamma}_{j+1}=-\frac{\partial}{\partial z} \tilde{R}_{j+1,k}(\rho_k,1)$,\\ \item $\forall i > j+1 \ \text{(outer radicands)}: \tilde{R}_{i,k}(\rho_k-\epsilon,1) =\tilde{a}_i+\tilde{b}_i\sqrt{\epsilon}+\mathcal{O}(\|\epsilon\|),$ \\ with $\tilde{a}_{i+1}=1-4(k-i)\rho_k^2-2\rho_k+2\rho_{k}\sqrt{\tilde{a}_i}$, and $\tilde{b}_{i+1}=\frac{\tilde{b}_i \rho_{k}}{\sqrt{\tilde{a}_i}}$ for $j+2 \leq i \leq k$, where $\tilde{a}_{j+2}=1-4(k-j-1)\rho_{k}^2-2\rho_k$ and $\tilde{b}_{j+2}=2\rho_k\sqrt{\tilde{\gamma}_{j+1}}$. \end{itemize} Thus, we have \begin{itemize} \item $\forall i < j+1 \ \text{(inner radicands)}: \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k-\epsilon,1)} }= \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k,1)}}+ \mathcal{O}(\|\epsilon\|)$,\\ \item $\frac{1}{\sqrt{\tilde{R}_{j+1,k}(\rho_1-\epsilon,1)}} = \frac{1}{\sqrt{\tilde{\gamma}_{j+1}}} \epsilon^{-\frac{1}{2}} +\mathcal{O} (\|\epsilon\|^{ \frac{1}{2}} )$,\\ \item $\forall i > j+1 \ \text{(outer radicands)}: \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k-\epsilon,1)}} =\frac{1}{ \sqrt{\tilde{a}_i}}-\frac{\tilde{b}_i} {2\sqrt{\tilde{a}_i^3}} \epsilon^{ \frac{1}{2}}+\mathcal{O}(\|\epsilon\|^{ \frac{3}{2}})$. \end{itemize} Now we have to perform a distinction of cases whether the De Bruijn level that we are focussing on is below the $(k-j)$-th level or not (\textit{i.e.}, whether $l$ is below $j$ or not). \paragraph{\underline{First case}: $l>j$} First let us remember that $l>j$ implies that the $u$ is inserted in a radicand that is located outside the $(j+1)$-th. From (\ref{equ:ablkminuslhleqk}) we get for $\epsilon \longrightarrow 0$ so that $\epsilon \in \mathbb{C} \setminus \mathbb{R}^{-}$ \[ \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(\rho_k-\epsilon,u) \right) \bigg\|_{u=1} = \] \[ \rho_k^{k-l+1} (k-l) $ \prod_{i=l+1}^{k+1} \frac{1}{\sqrt{\tilde{a}_i}} - \sum_{m=l+1}^{k+1} \( \frac{\tilde{b}_m}{2 \sqrt{\tilde{a}_m^3}} \prod_{i=l+1, i \neq m}^{k+1} \frac{1}{\sqrt{\tilde{a}_i}} $ \epsilon^{\frac{1}{2}} + \mathcal{O}(\|\epsilon\|^{\frac{3}{2}}) \) . \] By denoting the sum in the equation above with $\tilde{\delta}_l$ we can determine the coefficient of $z^n$ by \begin{align} [z^n] \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \bigg\|_{u=1} = - \rho_k^{k-l+1} (k-l) \tilde{\delta}_l \left( \frac{1}{\rho_k} \right)^n \frac{n^{-\frac{3}{2}}}{\Gamma(-\frac{1}{2})} \left(1+ \mathcal{O} $ \frac{1}{n} $ \right), \ \ \ \ \ \ \text{as} \ n \longrightarrow \infty, \end{align} and by using the asymptotics of the $n$-th coefficient of $_{k-l}\tilde{H}_{k}(z,1)=H_{k}(z,1)$ (see Theorem \ref{theo:asymphleqkueg1}) we finally get for the mean asymptotically as $n \longrightarrow \infty$ \[ \frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}\tilde{H}_{k}(z,1)} = \frac{- \rho_k^{k-l+1} (k-l) \tilde{\delta}_l}{h_k} $ 1+ \mathcal{O} \( \frac{1}{n} $ \). \] Thus, we showed that there is only a small number of leaves in the De Bruijn levels below the $(k-j)$-th level. More precisely, the asymptotic mean of the number of leaves is $O(1)$ for all these lower levels. \paragraph{\underline{Second case}: $l \leq j$} Similar to the first case we get \[ \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(\rho_k-\epsilon,u) \right) \bigg\|_{u=1} = \] \[ \rho_k^{k-l+1} (k-l) $ \( \prod_{i=l+1}^{j} \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k,1)}} $ $ \prod_{i=j+2}^{k+1} \frac{1}{\sqrt{\tilde{a}_i}} $ \frac{1}{\sqrt{\tilde{\gamma}_{j+1}}}\epsilon^{-\frac{1}{2}} + \ \text{const. term} + \mathcal{O}(\|\epsilon\|^{\frac{1}{2}}) \). \] By setting $\tilde{\phi}_{j+1,l}:= $ \prod_{i=l+1}^{j} \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k,1)}} $ $ \prod_{i=j+2}^{k+1} \frac{1}{\sqrt{\tilde{a}_i}} $ \frac{1}{\sqrt{\tilde{\gamma}_{j+1}}}$, we obtain for $n \longrightarrow \infty$ \[ [z^n] \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \bigg\|_{u=1} = \rho_k^{k-l+1} (k-l) \tilde{\phi}_{j+1,l} $ \frac{1}{\rho_k} $^n \frac{n^{-\frac{1}{2}}}{\Gamma(\frac{1}{2})} $ 1+ \mathcal{O} \( \frac{1}{n} $ \). \] Thus, we get for the mean asymptotically as $n \longrightarrow \infty$ \[ \frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}\tilde{H}_{k}(z,1)} = \frac{\rho_k^{k-l+1} (k-l) \Gamma(-\frac{1}{2}) \tilde{\phi}_{j+1,l}}{\Gamma(\frac{1}{2})h_k} \cdot n $ 1+ \mathcal{O} \( \frac{1}{n} $ \). \] Hence, we proved that the asymptotic mean for the number of leaves in the De Bruijn levels above the $(k-j)$-th is $\Theta(n)$. So, altogether we can see that almost all of the leaves are located in the upper $j+1$ De Bruijn levels. \paragraph{\textbf{The case: $k = N_{j}$}} Now we will deal with the second case, where the bound $k$ is an element of the sequence $(N_j)_{j \in \mathbb{N}}$. We start by determining the expansions of the radicands around the dominant singularity $\tilde{\rho}_{k}(u)$ of $_{k-l}\tilde{H}_{k}(z,u)$ for $u=1$ and $\epsilon \longrightarrow 0$ so that $\epsilon \in \mathbb{C} \setminus \mathbb{R}^{-}$ (cf. \cite[Proposition 9]{bodini2015number}): \begin{itemize} \item $\forall i < j \ \text{(inner radicands)}: \tilde{R}_{i,k}(\rho_k-\epsilon,1) = \tilde{R}_{i,k}(\rho_k,1)+ \mathcal{O}(\|\epsilon\|), $\\ \item $\tilde{R}_{j,k}(\rho_k-\epsilon,1) = \tilde{\gamma}_j \epsilon +\mathcal{O}(\|\epsilon\|^2) \ \ $ with $\tilde{\gamma}_j=-\frac{\partial}{\partial z}\tilde{R}_{j,k}(\rho_k,1)$,\\ \item $\tilde{R}_{j+1,k}(\rho_k-\epsilon,1) = 2 \tilde{\rho}_{k} \sqrt{\tilde{\gamma}_j} \epsilon^{\frac{1}{2}} + \mathcal{O}(\|\epsilon\|)$,\\ \item $\forall i > j+1 \ \text{(outer radicands)}: \tilde{R}_{i,k}(\rho_k-\epsilon,1) = \tilde{a}_i+\tilde{b}_i\epsilon^{\frac{1}{4}}+\mathcal{O}(\|\epsilon\|),$ \\ with $\tilde{a}_{i+1}=1-4(k-i)\rho_k^2-2\rho_k+2\rho_k\sqrt{\tilde{a}_i}$, and $\tilde{b}_{i+1}=\frac{\tilde{b}_i \rho_k}{\sqrt{\tilde{a}_i}}$ for $j+2 \leq i \leq k$, with $\tilde{a}_{j+2}=1-4(k-j)\rho_k^2-2\rho_k$ and $\tilde{b}_{j+2}=2\rho_k \sqrt{\tilde{\gamma}_{j}}$ . \end{itemize} Thus, we get \begin{itemize} \item $\forall i < j \ \text{(inner radicands)}: \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k-\epsilon,1)} }= \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k,1)}}+ \mathcal{O}(\|\epsilon\|)$,\\ \item $\frac{1}{\sqrt{\tilde{R}_{j,k}(\rho_k-\epsilon,1)}} = \frac{1}{\sqrt{\tilde{\gamma}_j}} \epsilon^{-\frac{1}{2}} +\mathcal{O} (\|\epsilon\|^{ \frac{1}{2}} )$,\\ \item $\frac{1}{\sqrt{\tilde{R}_{j+1,k}(\tilde{\rho}_{k}-\epsilon,1)}} = \frac{1}{\sqrt{2\rho_k}\sqrt[4]{\tilde{\gamma}_j}} \epsilon^{-\frac{1}{4}} +\mathcal{O} (\|\epsilon\|^{ \frac{1}{4}} )$,\\ \item $\forall i > j+1 \ \text{(outer radicands)}: \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k-\epsilon,1)}} =\frac{1}{ \sqrt{\tilde{a}_i}}-\frac{\tilde{b}_i} {2\sqrt{\tilde{a}_i^3}} \epsilon^{ \frac{1}{4}}+\mathcal{O}(\|\epsilon\|)$. \end{itemize} We proceed analogously to the case where $N_j < k < N_{j+1}$, with the only difference that we have to distinguish between three cases now and since for $u=1$ the $j$-th and the $(j+1)$-th radicand vanish simultaneously, we get a closed formula for the dominant singularity $\rho_k=\frac{1}{1+\sqrt{1+4(k-j)}}$. \paragraph{\underline{First case}: $l>j$} Let us again remember that $l>j$ implies that the $u$ is inserted in the $p$-th radicand with $p>j+1$. From (\ref{equ:ablkminuslhleqk}) we get for $\epsilon \in \mathbb{C} \setminus \mathbb{R}^{-}$ with $\|\epsilon\| \longrightarrow 0$ \[ \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(\rho_k-\epsilon,u) \right) \bigg\|_{u=1} = \rho_k^{k-l+1} (k-l) \prod_{i=l+1}^{k+1} $ \frac{1}{\sqrt{\tilde{a}_{i}}} - \frac{\tilde{b}_{i}}{2 \sqrt{\tilde{a}_{i}^3}} \epsilon^{\frac{1}{4}} + \mathcal{O}(\|\epsilon\|) $ = \] \[ $ \frac{1}{1+\sqrt{1+4(k-j)}} $^{k-l+1} (k-l) $ \prod_{i=l+1}^{k+1} \frac{1}{\sqrt{\tilde{a}_{i}}} - \sum_{m=l+1}^{k+1} \( \frac{\tilde{b}_{m}}{2 \sqrt{\tilde{a}_{m}^3}} \prod_{i=l+1, i \neq m}^{k+1} \frac{1}{\sqrt{\tilde{a}_{i}}} $ \epsilon^{\frac{1}{4}} + \mathcal{O}(\|\epsilon\|^{\frac{1}{2}}) \) . \] By setting $\tilde{\delta}_j := \sum_{m=l+1}^{k+1} $ \frac{\tilde{b}_{m}}{2 \sqrt{\tilde{a}_{m}^3}} \prod_{\substack{i=l+1 \\ i \neq m}}^{k+1} \frac{1}{\sqrt{\tilde{a}_{i}}} $$, extracting the $n$-th coefficient and using the asymptotics of $[z^n]_{k-l}\tilde{H}_{k}(z,1)=[z^n]H_k(z,1)=\frac{-\rho_k^{k-j-1}b_{j+2}n^{-5/4}}{4\rho_k\Gamma(-1/4)\prod_{i=j+2}^{k+1}\sqrt{a_i}}\rho_k^{-n} $ 1+ \mathcal{O} \( \frac{1}{n} $ \)$ we have for $n \longrightarrow \infty$ \[ \frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}\tilde{H}_{k}(z,1)} = \frac{-4\rho_k^{j-l+3} (k-l) \tilde{\delta}_j \prod_{m=j+2}^{k+1} \sqrt{a_m} }{b_{j+2}} $ 1 + \mathcal{O} \( n^{-\frac{1}{4}} $ \).\] Thus, as in the previous case ($k \in (N_j, N_{j+1})$) the asymptotic mean for the number of leaves in the De Bruijn levels below the $(k-j)$-th level is $O(1)$. Furthermore the constant $D_{k,l}:=\frac{-4\rho_k^{j-l+3}(1) (k-l) \tilde{\delta}_j \prod_{m=j+2}^{k+1} \sqrt{a_m} }{b_{j+2}}$ can be simplified to \begin{align} \label{equ:Dlk} D_{k,l}=\frac{k-l}{2\lambda_l} $ 1+ \frac{ \sqrt{\lambda_{l-j}}}{2\lambda_{l-j+1}} + \frac{ \sqrt{\lambda_{l-j}}}{4\lambda_{l-j+2}\sqrt{\lambda_{l-j+1}}} + \frac{ \sqrt{\lambda_{l-j}}}{8\lambda_{l-j+3}\sqrt{\lambda_{l-j+2}}\sqrt{\lambda_{l-j+1}}} + \ldots $, \end{align} with the sequence $\lambda_i$ defined by $\lambda_0=0$ and $\lambda_{i+1}=i+1+\sqrt{\lambda_i}$ for $i \geq 0$. \paragraph{\underline{Second case}: $l=j$} Thus, the $u$ is inserted in the $(j+1)$-th radicand. In this case we get \begin{align} \label{equ:meanNjleaveslevelj} \frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}\tilde{H}_{k}(z,1)} = \frac{-4\rho_k^{3} (k-j) \Gamma(-1/4) \psi_{j} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\Gamma(\frac{1}{4}) b_{j+2}} \cdot \sqrt{n} \ $ 1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \end{align} with \begin{align} \label{equ:psij} \psi_{j}= \frac{1}{\sqrt{2\rho_k} \sqrt[4]{\tilde{\gamma}_j}} \prod_{i=j+2}^{k+1} \frac{1}{\sqrt{a_{i}}}. \end{align} The constant $\hat{D}_{k,l}:=\frac{-4\rho_k^{3} (k-j) \Gamma(-1/4) \psi_{j} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\Gamma(\frac{1}{4}) b_{j+2}}$ simplifies to \begin{align} \hat{D}_{k,l}=\frac{- \Gamma(-1/4) (k-j) \sqrt{\rho_k}}{\Gamma(1/4) \sqrt{\tilde{\gamma}_j}}. \end{align} In order to get some information on the magnitude of this factor we would have to investigate $\tilde{\gamma}_j=-\frac{\partial}{\partial z}\tilde{R}_{j,k}(\rho_k,1)$, which seems to get rather involved. However, taking a look at Equation (\ref{equ:meanNjleaveslevelj}) we can see that there are already considerably more unary nodes in the $(k-j)$-th De Bruijn level, namely $\Theta(\sqrt{n})$. \paragraph{\underline{Third case}: $l \leq j$} The third case gives for $n \longrightarrow \infty$ \[ \frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}\tilde{H}_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}\tilde{H}_{k}(z,1)} = \frac{-4\rho_k^{j-l+3}(k-l) \Gamma(-1/4) \chi_{j} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\Gamma(3/4) b_{j+2}} n \ $1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \] with \[ \chi_{j}= \frac{1}{\sqrt{\tilde{\gamma}_j}} \psi_j $ \prod_{i=l+1}^{j-1} \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k,1)}} $,\] where $\psi_j$ is defined as in (\ref{equ:psij}). Thus, we proved that asymptotically there is an average of $\Theta(n)$ leaves in the upper $j$ De Bruijn levels. The constant $\tilde{D}_{k,l}:=\frac{-4\rho_k^{j-l+3}(k-l) \Gamma(-1/4) \chi_{j} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\Gamma(3/4) b_{j+2}}$ can be rewritten as \begin{align} \tilde{D}_{k,l}=\frac{-\Gamma(-1/4) (k-l) \rho_k^{j-l}}{\Gamma(3/4)\tilde{\gamma}_j} \prod_{i=l+1}^{j-1} \frac{1}{\sqrt{\tilde{R}_{i,k}(\rho_k,1)}}. \end{align} The following proposition sums up all the results that we obtained within this section. \begin{prop} \label{prop:levelsmean} Let $X_n$ denote the number of leaves in the $(k-l)$-th De Bruijn level in a random lambda-term of size $n$ with at most $k$ De Bruijn levels. If $k \in (N_j,N_{j+1})$, then we get for the asymptotic mean when $n \longrightarrow \infty$ \begin{itemize} \item in the case $l > j$: \[ \mathbb{E}X_n=\frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}H_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}H_{k}(z,1)} = C_{k,l} $ 1+ \mathcal{O} \( \frac{1}{n} $ \), \] \item and in the case $l \leq j$: \[ \mathbb{E}X_n=\frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}H_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}H_{k}(z,1)} = \tilde{C}_{k,l} \cdot n $ 1+ \mathcal{O} \( \frac{1}{n} $ \), \] \end{itemize} with constants $C_{k,l}$ and $\tilde{C}_{k,l}$ depending on $l$ and $k$. If $k =N_j$, then the asymptotic mean for $n \longrightarrow \infty$ reads as \begin{itemize} \item in the case $l > j$: \[ \mathbb{E}X_n=\frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}H_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}H_{k}(z,1)} = D_{k,l} $ 1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \] \item in the case $l = j$: \[ \mathbb{E}X_n=\frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}H_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}H_{k}(z,1)} = \hat{D}_{k,l} \cdot \sqrt{n} $ 1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \] \item and in the case $l < j$: \[ \mathbb{E}X_n = \frac{[z^n] \left( \frac{\partial}{\partial u} \ _{k-l}H_{k}(z,u) \right) \|_{u=1}}{[z^n]_{k-l}H_{k}(z,1)} = \tilde{D}_{k,l} \cdot n $ 1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \] \end{itemize} with constants $D_{k,l}$, $\hat{D}_{k,l}$ and $\tilde{D}_{k,l}$ depending on $l$ and $k$. \end{prop} All the constants occurring in Proposition \ref{prop:levelsmean} have been calculated explicitly and can be obtained for every fixed $k$. In particular, we investigated $D_{k,l}$ in order to show that for large $k$ the number of leaves in the De Bruijn levels that are closer to the root is smaller (\textit{cf}. Figure \ref{fig:summarykap5}). \begin{prop} Let us consider a random closed lambda-term of size $n$ with at most $k$ De Bruijn levels and let us consider the case $k=N_j$. Then the average number of leaves in De Bruijn level $L$, with $0 \leq L \leq k-j-1$, is asymptotically equal to a constant $C_L$, which behaves like \begin{align} C_L \sim \frac{L}{2(k-j-1-L)} \ \ \ \ \text{as} \ k \longrightarrow \infty. \end{align} \end{prop} \begin{proof} By setting $L:=k-l$ the constant $C_L$ corresponds exactly to the constant the $D_{k,l}$ of (\ref{equ:Dlk}). Thus, the proposition follows directly by investigating this constant $D_{k,l}$. The asymptotics for the sequence $\lambda_i$ ($\lambda_i \sim i, \text{as} \ i \longrightarrow \infty)$ can be obtained by bootstrapping. \end{proof} \begin{rem} Using some estimates for the $a_i's$ we can prove that the same behaviour is true for the constants $C_{k,l}$. Thus, in both cases, whether $k$ is an element of $(N_i)_{i >0}$ or not, a random closed lambda-term with at most $k$ De Bruijn levels has almost no leaves in its lowest levels if $k$ is large. \end{rem} \subsubsection{Distributions} Now that we derived the mean values for the number of leaves in the different De Bruijn levels, we are interested in their distribution. Therefore we distinguish again between the cases of $k$ being an element of the sequence $(N_i)_{i \geq 0}$ or not. \paragraph{\textbf{The case: $N_j < k < N_{j+1}$}} We know that the generating function $_{k-l}\tilde{H}_k(z,u)$ consists of $k+1$ nested radicals, where a $u$ is inserted in the $(l+1)$-th radicand counted from the innermost one. Additionally we know that for $N_j < k < N_{j+1}$ the dominant singularity $\tilde{\rho}_k(u)$ comes from the $(j+1)$-th radicand. Therefore, for $l>j$ the function $\tilde{\rho}_k(u)$ is independent of $u$, which is the reason why we do not get a quasi-power in that case. Thus, for the first $k-j$ levels of the lambda-DAG (\textit{i.e.} the case $l>j$), where there are just a few leaves, we can not say something about the distribution of the leaves so far. It might be a degenerated distribution. However, in case that $l \leq j$ (\textit{i.e.}, for the upper levels where there are a lot of leaves) we will use the Quasi-Power theorem to show that the number of leaves in the $(k-j)$-th until the $k$-th level is asymptotically normally distributed. Analogously as we did in Section \ref{subsec:total_height_njnj+1} we can show that \begin{align} \label{equ:kminusjhleqkzufracz1} \frac{[z^n] \ _{k-l}\tilde{H}_{k}(z,u)}{[z^n] \ _{k-l}\tilde{H}_{k}(z,1)} = \frac{\tilde{h}_k(u)}{h_k} $ \frac{\rho_{k}}{\tilde{\rho}_{k}(u)} $^n $ 1+ \mathcal{O} \( \frac{1}{n} $ \). \end{align} We can easily see that Equation (\ref{equ:kminusjhleqkzufracz1}) has the desired shape for the Quasi-Power Theorem. Hence, assuming that $\tilde{B}''(1)+\tilde{B}'(1)-\tilde{B}'(1)^2 \neq 0$, where $\tilde{B}(u)=\frac{\rho_{k}}{\tilde{\rho}_{k}(u)}$, the Quasi-Power Theorem can be applied, which proves that the number of leaves in a De Bruijn level that is above the $(k-j-1)$-th level is asymptotically normally distributed. \paragraph{\textbf{The case: $k = N_{j}$}} As is the previous case we do not know the distribution of the number of leaves in the lowest $k-j$ De Bruijn levels (\textit{i.e.}, the levels $0$ to $k-j-1$), due to the fact that for these levels the function $\tilde{\rho}_k(u)$ does not depend on $u$. It might also be a degenerated distribution. In Section \ref{subsection:k=njtotal} we showed that the dominant singularity comes from the $j$-th radicand when $u$ is in a neighbourhood of 1. Thus, for the case that $l=j$, where we insert a $u$ in the $j+1$-th radicand, the dominant singularity $\rho_k(u)$ does still do not depend on $u$. Therefore we also do not know the distribution of the leaves in the $(k-j)$-th De Bruijn level. It seems very unlikely that the number of leaves in this level will be asymptotically normally distributed, but further studies on this subject might be very interesting. Now we are going to show that the number of leaves in the upper $j$ De Bruijn levels (\textit{i.e.}, from the $(k-j+1)$-th to the $k$-th level) is asymptotically normally distributed. In order to do so we proceed analogously as in Section \ref{subsection:k=njtotal} for the total number of leaves. Therefore for $l<j$ we set again $z=\tilde{\rho}_k(u)(1+\frac{t}{n})$ and $u=1+\frac{s}{n}$ and obtain expansions that behave just as the ones in Lemma \ref{lem:exprjrj+1rho1}. The only differences that occur concern the constants and therefore do not alter our results for the normal distribution. Thus, Theorem \ref{theo:levelsmeandist} is proved. Figure \ref{fig:summarykap5} summarizes the results that we obtained in Section \ref{sec:leaveslevels} and illustrates a combinatorial interpretation of the occurring phenomena. \begin{figure} \caption{(1) In the $(k-j)$-th Be Bruijn level ($l=j$) are considerably more leaves than in the lower levels, but still less leaves then in the levels above. (2) With growing $k$ the $(k-j)$-th Be Bruijn level gets filled with leaves, while the number of leaves in the next level below (\textit{i.e.}, the $(k-j-1)$-th) slowly increases. (3) As soon as $k$ reaches the next element of the sequence $(N_j)_{j \geq 0}$, namely $k=N_{j+1}$ the $(k-j-1)$-th De Bruijn level immediately contains considerably more leaves than the levels below.} \label{fig:summarykap5} \end{figure} \subsection{Unary nodes} \label{sec:unarylevels} Now we want to investigate the number of unary nodes among the different De Bruijn levels. Let $C(z,u)$ again denote the generating function of the class of binary trees where $z$ marks the total number of nodes and $u$ marks the number of leaves, \textit{i.e.}, $C(z,u)=\frac{1-\sqrt{1-4z^2u}}{2z}$. The bivariate generating function $_{k-l}\bar{H}_k(z,w)$ of the class of closed lambda-terms with at most $k$ De Bruijn levels, where $z$ marks the size and $w$ the number of unary nodes in the $(k-l)$-th De Bruijn level, can then be expressed in terms of $C(z,u)$ by \begin{align} C(z,C(z,1+\ldots + C(z,(k-l)+w\cdot C(z,\ldots (k-1)+C(z,k))\ldots)\ldots)). \end{align} This can be rewritten to \begin{align} _{k-l}\bar{H}_k(z,w) = \frac{1-\sqrt{\bar{R}_{k+1,k}(z,w)}}{2z}, \end{align} with \begin{align} \bar{R}_{1,k}(z,w)&=1-4z^2k,\\ \bar{R}_{i,k}(z,w)&=1-4z^2(k-i+1)-2z+2z\sqrt{\bar{R}_{i-1,k}(z,w)} \ \ \ \text{for} \ 2 \leq i \leq k+1, \ \ i \neq l+1, \\ \bar{R}_{l+1,k}(z,w)&=1-4z^2(k-l)-2zw+2zw\sqrt{\bar{R}_{l,k}(z,w)}. \end{align} Thus, for the derivatives we get \begin{align} \frac{\partial \bar{R}_{i,k}(z,w)}{\partial w} &= 0 \ \ \ \text{for} \ i < l+1, \\ \frac{\partial \bar{R}_{l+1,k}(z,w)}{\partial w} &= -2z+2z\sqrt{\bar{R}_{l,k}(z,w)},\\ \frac{\partial \bar{R}_{i,k}(z,w)}{\partial w} &= \frac{z}{\sqrt{\bar{R}_{i-1,k}}} \frac{\partial \bar{R}_{i-1,k}(z,w)}{\partial w} \ \ \ \text{for} \ i > l+1, \end{align} which implies \begin{align} \frac{\partial _{k-l}\bar{H}_k(z,w)}{\partial w} \Big\|_{w=1} = \frac{z^{k-l}}{2} \prod_{i=l+1}^{k+1} \frac{1}{\sqrt{\bar{R}_{i,k}}} $1-\sqrt{\bar{R}_{l,k}}$. \end{align} As in the previous section we distinguish between different cases. \pagebreak \subsubsection{The case: $k=N_j$} \paragraph{\underline{First case}: $l > j+1$} Inserting the expansion of the radicands $\bar{R}_{i,k}$ and simplifying yields for $n \longrightarrow \infty$ \begin{align} [z^n]_{k-l}\bar{H}_{k}(z,w) \Big\|_{w=1} = \rho_k^{k-l}{2} \alpha_l \frac{n^{-5/4}}{\Gamma(-1/4)} \rho_k^{-n} $ 1+ \mathcal{O} \(n^{-\frac{1}{4}} $ \), \end{align} with \begin{align} \label{equ_alphal} \alpha_l= -\frac{b_l}{2\sqrt{a_l}} \prod_{i=l+1}^{k+1} \frac{1}{\sqrt{a_i}} - \sum_{m=l+1}^{k+1} \frac{b_m}{2\sqrt{a_m^3}} \prod_{\substack{i=l+1 \\ i \neq m}}^{k+1} \frac{1}{\sqrt{a_i}} $ 1-\sqrt{a_l} $, \end{align} where $a_i:=\tilde{a}_i=a_i(1)$ and $b_i:=\tilde{b}_i=b_i(1)$ are defined in the previous sections and result from the expansions of the radicands. Thus, in this case the expected value of the number of unary nodes in the $(k-l)$-th De Bruijn level reads as \begin{align} \frac{[z^n] \frac{\partial}{\partial w} _{k-l}\bar{H}_{k}(z,w)}{[z^n] _{k-l}\bar{H}_{k}(z,1)} = \frac{-2 \alpha_l \rho_k^{j-l+2} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{b_{j+2}} $ 1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \ \ \ \ \text{as} \ n \longrightarrow \infty. \end{align} Furthermore, the constant $\frac{-2 \alpha_l \rho_k^{j-l+2} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{b_{j+2}}$ can be simplified to \begin{align} 1+ $ \frac{1}{4\rho_k \lambda_{l-j}} - \frac{\sqrt{\lambda_{l-j-1}}}{2\lambda_{l-j}} $ $ 1+ \frac{\lambda_{l-j}}{2\lambda_{l-j+1}\sqrt{\lambda_{l-j}}} + \frac{\lambda_{l-j}}{2^2\lambda_{l-j+2} \sqrt{\lambda_{l-j+1}}\sqrt{\lambda_{l-j}}}+\ldots $, \end{align} with the sequence $\lambda_i$ defined by $\lambda_0=0$ and $\lambda_{i+1}=i+1+\sqrt{\lambda_i}$ for $i \geq 0$. Since the second summand is almost zero for $l$ being close to $k$ and large $k$, this implies that the number of unary nodes in these levels (close to the root) is close to one for large $k$. \paragraph{\underline{Second case}: $l = j+1$} For $n \longrightarrow \infty$ we get \begin{align} [z^n]_{k-l}\bar{H}_{k}(z,w) \Big\|_{w=1} = \rho_k^{k-l}{2} \zeta_l \frac{n^{-5/4}}{\Gamma(-1/4)} \rho_k^{-n} $ 1+ \mathcal{O} \(n^{-\frac{1}{4}} $ \), \end{align} with \begin{align} \zeta_l= -\sqrt{2\rho_k} (\rho_k \tilde{\gamma})^{1/4} \prod_{i=j+2}^{k+1} \frac{1}{\sqrt{a_i}} - \sum_{m=j+2}^{k+1} \frac{b_m}{2\sqrt{a_m^3}} \prod_{\substack{i=j+2\\ i \neq m}}^{k+1} \frac{1}{\sqrt{a_i}}. \end{align} Thus, \begin{align} \frac{[z^n] \frac{\partial}{\partial w} _{k-l}\bar{H}_{k}(z,w)}{[z^n] _{k-l}\bar{H}_{k}(z,1)} = \frac{-2 \zeta_l \rho_k^{j-l+2} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{b_{j+2}} $ 1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \ \ \ \ \text{as} \ n \longrightarrow \infty. \end{align} In this case the constant $\frac{-2 \zeta_l \rho_k^{j-l+2} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{b_{j+2}}$ simplifies to \begin{align} 1 + \frac{1}{4\rho_k} $ 1+ \frac{1}{2\lambda_2} + \frac{1}{2^2\lambda_3 \sqrt{\lambda_2}} + \ldots $. \end{align} So, the expected number of unary nodes in the $(k-j-1)$-th De Bruijn level behaves exactly like in the lower levels. Starting from the next level a change in the behaviour can be determined, as we will see in the following. \paragraph{\underline{Third case}: $l = j$} For $n \longrightarrow \infty$ we have \begin{align} [z^n]_{k-l}\bar{H}_{k}(z,w) \Big\|_{w=1} = \rho_k^{k-l}{2} \beta_l \frac{n^{-3/4}}{\Gamma(1/4)} \rho_k^{-n} $ 1+ \mathcal{O} \(n^{-\frac{1}{2}} $ \), \end{align} with \begin{align} \beta_l= \frac{1}{\sqrt{2\rho_k} \sqrt[4]{\rho_k \tilde{\gamma}_j}} \prod_{i=j+2}^{k+1} \frac{1}{\sqrt{a_i}}. \end{align} Thus, \begin{align} \frac{[z^n] \frac{\partial}{\partial w} _{k-l}\bar{H}_{k}(z,w)}{[z^n] _{k-l}\bar{H}_{k}(z,1)} = \frac{-2 \beta_l \rho_k^{2} \Gamma(-1/4) \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\Gamma(1/4) b_{j+2}} \cdot \sqrt{n} $ 1+ \mathcal{O} \( n^{-\frac{1}{2}} $ \), \ \ \ \ \text{as} \ n \longrightarrow \infty. \end{align} The constant $\frac{-2 \beta_l \rho_k^{2} \Gamma(-1/4) \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\Gamma(1/4) b_{j+2}}$ can be written as \begin{align} \frac{- \Gamma(-1/4)}{2\Gamma(1/4) \sqrt{\rho_k \tilde{\gamma}_j}}. \end{align} The expected number of unary nodes in this ``separating level'' is therefore asymptotically $\Theta(\sqrt{n})$ (as was the number of leaves). \paragraph{\underline{Fourth case}: $l < j$} For $ n \longrightarrow \infty$ we get \begin{align} [z^n]_{k-l}\bar{H}_{k}(z,w) \Big\|_{w=1} = \rho_k^{k-l}{2} \epsilon_l \frac{n^{-1/4}}{\Gamma(3/4)} \rho_k^{-n} $ 1+ \mathcal{O} \(n^{-\frac{1}{4}} $ \), \end{align} with \begin{align} \epsilon_l= \frac{1}{\sqrt{2\rho_k} \sqrt[4]{\rho_k \tilde{\gamma}_j} \sqrt{\rho_k \tilde{\gamma}_j}} \prod_{i=j+2}^{k+1} \frac{1}{\sqrt{a_i}} \prod_{m=l+1}^{j-1} \frac{1}{\sqrt{\tilde{R}_{m,k}}} $ 1- \sqrt{\bar{R}_{l,k}} $. \end{align} Thus, \begin{align} \frac{[z^n] \frac{\partial}{\partial w} _{k-l}\bar{H}_{k}(z,w)}{[z^n] _{k-l}\bar{H}_{k}(z,1)} = \frac{-2 \epsilon_l \rho_k^{j-l+2} \Gamma(-1/4) \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\Gamma(3/4) b_{j+2}} \cdot n $ 1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \ \ \ \ \text{as} \ n \longrightarrow \infty. \end{align} The constant $\frac{-2 \epsilon_l \rho_k^{j-l+2} \Gamma(-1/4) \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\Gamma(3/4) b_{j+2}}$ can be simplified to \begin{align} \frac{-\rho_k^{j-l+1} \Gamma(-1/4)}{2\Gamma(3/4) \tilde{\gamma}_j} \prod_{m=l+1}^{j-1} \frac{1}{\sqrt{\tilde{R}_{m,k}}} $ 1- \sqrt{\bar{R}_{l,k}} $. \end{align} Hence, analogously to the number of leaves, we proved that the number of unary nodes on the upper $j+1$ De Bruijn levels is $\Theta(n)$. \subsubsection{The case: $N_j<k<N_{j+1}$} This case works analogously to the previous one. Thus, we just give the results for the expected values. \paragraph{\underline{First case}: $l > j+1$} In this case, the expected value is entirely equal to the mean for the case $k=N_j$ and $l > j+1$. So, with $\alpha_l$ defined as in (\ref{equ_alphal}), we have for $ n \longrightarrow \infty$ \begin{align} \frac{[z^n] \frac{\partial}{\partial w} _{k-l}\bar{H}_{k}(z,w)}{[z^n] _{k-l}\bar{H}_{k}(z,1)} = \frac{-2 \alpha_l \rho_k^{j-l+2} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{b_{j+2}} \cdot n $ 1+ \mathcal{O} \( n^{-\frac{1}{2}} $ \). \end{align} \paragraph{\underline{Second case}: $l = j+1$} In the second case, the constant differs a little bit, but the result stays qualitatively unaltered. We get \begin{align} \frac{[z^n] \frac{\partial}{\partial w} _{k-l}\bar{H}_{k}(z,w)}{[z^n] _{k-l}\bar{H}_{k}(z,1)} = \frac{-2 \mu_l \rho_k^{j-l+2} \prod_{m=j+2}^{k+1} \sqrt{a_m}}{b_{j+2}} \cdot n $ 1+ \mathcal{O} \( n^{-\frac{1}{2}} $ \), \ \ \ \ \text{as} \ n \longrightarrow \infty, \end{align} with \begin{align} \mu_l = - \sum_{m=j+2}^{k+1} \frac{b_m}{2\sqrt{a_m^3}} \prod_{\substack{ i=j+2 \\ i \neq m}}^{k+1} \frac{1}{\sqrt{a_i}} + \sqrt{\rho_k \tilde{\gamma}_{j+1}} \prod_{i=j+2}^{k+1} \frac{1}{\sqrt{a_i}}. \end{align} \paragraph{\underline{Third case}: $l < j+1$} For $n \longrightarrow \infty$ we have \begin{align} \frac{[z^n] \frac{\partial}{\partial w} _{k-l}\bar{H}_{k}(z,w)}{[z^n] _{k-l}\bar{H}_{k}(z,1)} = \frac{-2 \theta_l \Gamma(-1/2) \rho_k^{j-l+2} \prod_{m=j+2}^{k+1} \sqrt{a_m} \prod_{s=l+1}^{j} \sqrt{\bar{R}_s}}{b_{j+2} \Gamma(1/2)} \cdot n $ 1+ \mathcal{O} \( \frac{1}{n} $ \), \end{align} with \begin{align} \theta_l = \prod_{i=j+2}^{k+1} \frac{1}{\sqrt{a_i}} \frac{1}{\sqrt{\rho_k \tilde{\gamma}_{j+1}}} $1- \sqrt{\bar{R}_{j,k}}$. \end{align} Thus, the expected number of unary nodes in the last $j+1$ De Bruijn levels is asymptotically $\Theta(n)$. \subsection{Binary nodes} \label{sec:binary} In this section we want to calculate the mean values of the number of binary nodes in the different De Bruijn levels. We denote by $C(z,v,u)$ the generating function of the class of binary trees where $z$ marks the total number of nodes, $v$ marks the number of binary nodes, and $u$ marks the number of leaves. Thus, we have \begin{align} \label{equ:gfbinarythree} C(z,v,u)= \frac{1-\sqrt{1-4z^2uv}}{2zv}. \end{align} Using this generating function, we can write the bivariate generating function $_{k-l}H_k(z,v)$ of the class of closed lambda-terms with $z$ marking the size, and $v$ marking the the number of binary nodes on the $(k-l)$-th De Bruijn level as \begin{align} \label{equ:gfrekbinary} C(z,1,C(z,1,1+C(z,1,2+\ldots +C(z,v,(k-l)+\ldots + C(z,1,k))\ldots)\ldots))). \end{align} Plugging in Equation (\ref{equ:gfbinarythree}) into (\ref{equ:gfrekbinary}) gives \begin{align} _{k-l}\breve{H}_k(z,v) = \frac{1-\sqrt{\breve{R}_{k+1,k}(z,v)}}{2z}, \end{align} with \begin{align} \breve{R}_{1,k}(z,v)&=1-4z^2k,\\ \breve{R}_{i,k}(z,v)&=1-4z^2(k-i+1)-2z+2z\sqrt{R_{i-1,k}(z,v)}, \ \ \ \text{for} \ 2 \leq i \leq k+1, \ \ i \neq l+1, \ i \neq l+2,\\ \breve{R}_{l+1,k}(z,v)&=1-4z^2(k-l)v-2zv+2zv\sqrt{\breve{R}_{l,k}(z,v)},\\ \breve{R}_{l+2,k}(z,v)&=1-4z^2(k-l-1)-\frac{2z}{v}+\frac{2z}{v}\sqrt{\breve{R}_{l+1,k}(z,v)}. \end{align} Thus, for the derivatives we get \begin{align} \frac{\partial \breve{R}_{i,k}(z,v)}{\partial v} &= 0 \ \ \ \text{for} \ i < l+1, \\ \frac{\partial \breve{R}_{l+1,k}(z,v)}{\partial v} &= -4z^2(k-l)-2z+2z\sqrt{\breve{R}_{l,k}(z,v)},\\ \frac{\partial \breve{R}_{l+2,k}(z,v)}{\partial v} &= \frac{2z}{v^2} - \frac{2z}{v^2}\sqrt{\breve{R}_{l+1,k}}+ \frac{z}{v} \frac{1}{\sqrt{\breve{R}_{l+1,k}}} \frac{\partial \breve{R}_{l+1,k}(z,v)}{\partial v}, \\ \frac{\partial \breve{R}_{i,k}(z,v)}{\partial v} &= z \frac{1}{\sqrt{\breve{R}_{i-1,k}}} \frac{\partial \breve{R}_{i-1,k}(z,v)}{\partial v} \ \ \ \text{for} \ i > l+2. \end{align} Finally, we have \begin{align} \frac{\partial _{k-l}\breve{H}_k(z,v)}{\partial v} \Big\|_{v=1}=& \prod_{i=l+2}^{k+1} \frac{1}{\sqrt{\breve{R}_{i,k}}} $ - \frac{z^{k-l-1}}{2} + \frac{\sqrt{\breve{R}_{l+1,k}}z^{k-l-1}}{2}+\frac{z^{k-l}}{2\sqrt{\breve{R}_{l+1,k}}}\right. \nonumber \\ &\qquad \left.-\frac{z^{k-l}\sqrt{\breve{R}_{l,k}}}{2\sqrt{\breve{R}_{l+1,k}}}+\frac{z^{k-l+1}(k-l)}{\sqrt{\breve{R}_{l+1,k}}} $. \label{equ:dHdvbinary} \end{align} Analogously to the previous sections we have to distinguish between different cases. For the case $k=N_j$ and $l > j+1$ we get for $n \longrightarrow \infty$ \begin{align} [z^n]_{k-l}\breve{H}_{k}(z,v)= \xi_l \frac{n^{-5/4}}{\Gamma(-1/4)} \rho_k^{-n} $ 1+ \mathcal{O} \(n^{-\frac{1}{4}} $ \), \end{align} with \begin{align} \xi_l= &- \sum_{m=l+2}^{k+1} \frac{b_m}{2\sqrt{a_m^3}} \prod_{\substack{i=l+2 \\ i \neq m}} \frac{1}{\sqrt{a_i}} $ \frac{\rho_k^{k-l-1}(\sqrt{a_{l+1}}-1)}{2} + \frac{\rho_k^{k-l+1}(k-l)}{\sqrt{a_{l+1}}} + \frac{\rho_k^{k-l}(1-\sqrt{a_l})}{2\sqrt{a_{l+1}}} $ \\ &+ \prod_{i=l+2}^{k+1} \frac{1}{\sqrt{a_i}} $ \frac{\rho_k^{k-l-1}b_{l+1}}{2\sqrt{a_{l+1}}} - \frac{\rho_k^{k-l+1}(k-l)b_{l+1}}{2\sqrt{a_{l+1}^3}} + \frac{b_{l+1}}{2\sqrt{a_{l+1}^3}}(\sqrt{a_l}-1) - \frac{\rho_k^{k-l}b_l}{4\sqrt{a_l}\sqrt{a_{l+1}}} $. \end{align} Thus, \begin{align} \frac{[z^n] \frac{\partial}{\partial v} _{k-l}\breve{H}_{k}(z,v)}{[z^n] _{k-l}\breve{H}_{k}(z,1)} = \frac{-4 \xi_l \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\rho_k^{k-j} b_{j+2}} $ 1+ \mathcal{O} \( n^{-\frac{1}{4}} $ \), \ \ \ \ \text{as} \ n \longrightarrow \infty. \end{align*} We performed a thorough investigation of the constant $ \frac{-4 \xi_l \prod_{m=j+2}^{k+1} \sqrt{a_m}}{\rho_k^{k-j} b_{j+2}}$ and showed that it is almost zero, in case $l$ is close to $k+1$ and $k$ is large,, \textit{i.e.}, if we consider a very low De Bruijn level, that is close to the root. Due to Equation (\ref{equ:dHdvbinary}) calculations get rather involved. Since the methods that are used are the same as in the previous section, we will omit further calculations. However, the results resemble the ones that we got in Section \ref{sec:leaveslevels} for the number of leaves. The only difference appears in the constants, but qualitatively also these constants behave equally. \end{document}	arXiv
Reference system origin and scale realization within the future GNSS constellation "Kepler" Susanne Glaser ORCID: orcid.org/0000-0001-6048-74371, Grzegorz Michalak2, Benjamin Männel1, Rolf König2, Karl Hans Neumayer2 & Harald Schuh1,3 Journal of Geodesy volume 94, Article number: 117 (2020) Cite this article Currently, Global Navigation Satellite Systems (GNSS) do not contribute to the realization of origin and scale of combined global terrestrial reference frame (TRF) solutions due to present system design limitations. The future Galileo-like medium Earth orbit (MEO) constellation, called "Kepler", proposed by the German Aerospace Center DLR, is characterized by a low Earth orbit (LEO) segment and the innovative key features of optical inter-satellite links (ISL) delivering highly precise range measurements and of optical frequency references enabling a perfect time synchronization within the complete constellation. In this study, the potential improvements of the Kepler constellation on the TRF origin and scale are assessed by simulations. The fully developed Kepler system allows significant improvements of the geocenter estimates (realized TRF origin in long-term). In particular, we find improvements by factors of 43 for the Z and of 8 for the X and Y component w. r. t. a contemporary MEO-only constellation. Furthermore, the Kepler constellation increases the reliability due to a complete de-correlation of the geocenter coordinates and the orbit parameters related to the solar radiation pressure modeling (SRP). However, biases in SRP modeling cause biased geocenter estimates and the ISL of Kepler can only partly compensate this effect. The realized scale enabling all Kepler features improves by 34% w. r. t. MEO-only. The dependency of the estimated satellite antenna phase center offsets (PCOs) upon the underlying TRF impedes a scale realization by GNSS. In order to realize the network scale with 1 mm accuracy, the PCOs have to be known within 2 cm for the MEO and 4 mm for the LEO satellites. Independently, the scale can be realized by estimating the MEO PCOs and by simultaneously fixing the LEO PCOs. This requires very accurate LEO PCOs; the simulations suggest them to be smaller than 1 mm in order to keep scale changes below 1 mm. Origin and scale together with orientation are the datum defining parameters of global terrestrial reference systems (TRS). The origin is defined in the long-term mean center of mass for the whole Earth's system (solid Earth and fluid envelope). The SI (Système International d'unités) meter is the unit of length and the scale is consistent with the Geocentric Coordinate Time (TCG). The orientation coincides with the orientation defined by the Bureau International de l'Heure (BIH) at the reference epoch 1984.0 (IERS Conventions 2010, Petit and Luzum 2010). The realization of the TRS is the terrestrial reference frame (TRF), for which the datum has to be properly realized by the space geodetic techniques. Global TRFs are usually determined by a combination of four space geodetic techniques: Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), Global Navigation Satellite Systems (GNSS), Satellite Laser Ranging (SLR), and Very Long Baseline Interferometry (VLBI). Since each technique has particular strengths and weaknesses resulting in a different sensitivity and suitability for the TRF-defining parameters (Sillard and Boucher 2001), only two techniques contribute currently to the datum realization, that are SLR and VLBI. The origin of the International Terrestrial Reference Frame (ITRF) is realized by SLR alone and the scale by SLR and VLBI. The unambiguous direct SLR measurements have a high sensitivity to the geocenter which is in long-term the realized origin of the ITRF. Furthermore, the non-gravitational forces have much less impact on the SLR satellites with their cannonball shape than on GNSS satellites, e.g., Meindl et al. (2013). However, it should be mentioned that special attention has to be given to range biases, satellite center-of-mass errors and signature effects affecting the SLR measurements as well as to the rather inhomogeneous global station distribution. In addition to the unique strength of VLBI to determine all Earth orientation parameters, the scale can be considered as unambiguous. The celestial reference frame of VLBI provides a stable space segment consisting of only two angles (right ascension and declination of extra-galactic radio sources) with no distance measurements to the space targets or between them. It is therefore independent of the gravitational parameter GM (G—gravitational constant, M—mass of the Earth), which has direct impact on the scale of any satellite technique. In recent ITRF solutions, including the latest ITRF2014 (Altamimi et al. 2016), GNSS and DORIS do not contribute to the realization of the origin and the scale. Nevertheless, DORIS and especially GNSS are essential for the densification of the global networks and for the combination of all four techniques. The absolutely essential requirements of 1 mm accuracy and 1 mm/decade long-term stability specified by the Global Geodetic Observing System (GGOS, Gross et al. 2009) are not fulfilled yet. The current estimated origin accuracy of ITRF2014 is 3 mm w. r. t. ITRF2008 (Altamimi et al. 2011) and the scale shows a discrepancy between SLR and VLBI of 1.37 ppb ($\sim $ 9 mm on the Earth's surface) (Altamimi et al. 2016). In view of the next realization, the planned ITRF2020, there have been continuous and promising efforts to reduce the scale discrepancy between SLR and VLBI, due to, e.g., refined range bias handling in the SLR estimation process (Appleby et al. 2016; Luceri et al. 2019) and the modeling of the gravitational deformation of the VLBI telescopes (Sarti et al. 2011; Nothnagel et al. 2019). Nevertheless, it would be desirable to have another independent technique besides SLR and VLBI for comparison and validation of TRFs. GNSS is indeed a good choice due to its excellent global station network and many satellites providing a large number of continuous observations. GNSS is, like SLR, a dynamic technique since the satellites orbit around the center of mass of the Earth's system enabling direct access to the geocenter. The information inherent in the observations, e.g., their sensitivity to geocenter motion, should be fully exploited and redundant information, potentially leading to over-constrained solutions should be avoided, in order to get a proper minimum constraint network solution (Sillard and Boucher 2001; Kotsakis 2012; Glaser et al. 2015). However, there are some limitations currently preventing GNSS from contributing to the origin and scale realization of global combined TRFs. Concerning the geocenter estimation by GNSS, Männel and Rothacher (2017) summarized five major limitations: the higher orbital height compared to, e.g., LAGEOS (SLR), the estimation of epoch-wise clock offsets, the existence of carrier-phase ambiguities, the estimation of tropospheric zenith delays and the uncertainties in the solar radiation pressure modeling. Previous studies have shown that GNSS is able under certain conditions to realize origin and scale of a TRF, especially in long-term (Heflin et al. 2002; Rülke et al. 2008; Dach et al. 2014; Glaser et al. 2015). The sensitivity of GPS/GNSS to the geocenter estimation can be improved by the inclusion of LEO (low Earth orbit) satellites as demonstrated by, e.g., König et al. (2005) using CHAMP, GRACE and SAC-C, Haines et al. (2015) using GRACE and TOPEX/Poseidon, Kuang et al. (2015) simulating different LEO orbital configurations, Männel and Rothacher (2017) combining four LEO satellites (GRACE-A, GRACE-B, OSTM/Jason-2 and GOCE), Kuang et al. (2019) using in addition the accelerometer data of GRACE and Couhert et al. (2020) using Jason-2. The common estimation of station and satellite clock offsets for every epoch and tropospheric parameters is according to Rebischung et al. (2014) a major limitation, and a GNSS without the need to estimate clock offsets would most probably facilitate an accurate geocenter estimation. Furthermore, the estimated GNSS geocenter coordinates show artifacts on the harmonics of the draconitic year (e.g., Ray et al. 2008) due to remaining systematics in the modeled GNSS orbits (Rebischung et al. 2016), especially in the solar radiation pressure modeling (Meindl et al. 2013, 2015; Rodriguez-Solano et al. 2014; Glaser et al. 2015). A possible contribution of DORIS to the TRF origin was investigated by, e.g., Couhert et al. (2018). Concerning the scale, GPS/GNSS does not contribute to the ITRF scale since present satellite antenna phase center offsets (PCOs) were derived with a fixed ITRF and are, therefore, intrinsically dependent on the SLR and VLBI scale (Schmid et al. 2007). Previous studies, e.g., Zhu et al. (2003) and Cardellach et al. (2007), provided a relation to express the dependency of the scale and the PCO of the GPS satellites. Moreover, it is possible to realize an independent scale by estimating antenna pattern of the GNSS satellites, for instance by transferring this information via LEO satellites such as GRACE and TOPEX/Poseidon (Haines et al. 2015; Männel 2016) or via calibrated antenna pattern for Galileo from the GSA (European GNSS Agency) (Rebischung 2019; Villiger et al. 2020). The realization of the network scale by GNSS is currently under investigation within the reprocessing effort of the International GNSS Service (IGS, Johnston et al. 2017) (Antenna Working Group) as contribution to the ITRF2020. It yet has to be decided how GNSS will contribute to the scale realization of ITRF2020. In this study, the potential of a future GNSS constellation on the origin and scale of global TRFs is investigated by simulations. Simulation studies are the only possibility to assess the impact of future developments of the GNSS technique on the TRF. The future constellation called "Kepler"Footnote 1 (Günther 2018) is introduced in Sect. 2.1 and the simulation setup including the different scenarios in Sect. 2.2. In Sect. 3.1 the improvements in origin, in Sect. 3.2 the improvements in scale, and in Sect. 3.3 in other parameters such as Earth rotation parameters are discussed by the future Kepler constellation w. r. t. contemporary GNSS, e.g., Galileo. Future GNSS constellation "Kepler" The future GNSS constellation Kepler is characterized by the innovative features of two-way optical inter-satellite links (ISL) and optical frequency references. This effort is related to the Helmholtz funded project called ADVANTAGEFootnote 2 (Advanced Technologies for Navigation and Geodesy, Giorgi et al. 2019b) which is a joint effort of the German Aerospace Center DLR and the German Research Center for Geosciences GFZ. The overall project goal is to establish an architecture for a future space infrastructure for navigation and geodesy. The future GNSS constellation "Kepler" proposed by the German Aerospace Center (DLR) consisting of 24 medium Earth orbit (MEO) satellites and 6 low Earth orbit (LEO) satellites. The two key technologies are two-way optical inter-satellites links and optical frequency references The Kepler constellation consists of 24 medium Earth orbit (MEO, semi-major axis: 29600 km) satellites in three orbital planes (inclination of 56$^\circ $), similar to Galileo in its final operational capability, and six LEO satellites (semi-major axis: 7600 km) in two perpendicular near-polar planes (inclination of 89.7$^\circ $). A sketch of the constellation is presented in Fig. 1. The MEO satellites transmit legacy L-band signals. The two-way optical ISL enable very precise and unambiguous inter-satellite range measurements for time synchronization within the complete constellation and support precise orbit determination. Optical frequency references are cavity-stabilized lasers, which are already used on the GRACE-FO (Gravity Recovery and Climate Experiment Follow-On) mission (Thompson et al. 2011; Dahl et al. 2017; Abich et al. 2019), and ultra-stable optical clocks carried on the LEO satellites. The frequency references are used to generate a very stable composite system time common for all satellites. The offset between the system time and the ground time scale (e.g., UTC) will be known or can be estimated with very high precision. It enables a significant reduction or even an entire elimination (fixing to known values) of the satellite clock parameters in the estimation process. The ISL connect all MEO satellites in one orbital plane and the LEO and the MEO satellites in different orbital planes according to an ISL scheduler. The ISL scheduler is based on different constraints defined by DLR. For further information about the Kepler constellation the interested reader is referred to Giorgi et al. (2019a, 2019b). Simulations were performed by utilizing the software EPOS-OC (Zhu et al. 2004) which was extended to facilitate the simulation and processing of the ISL for complex satellite constellations such as Kepler. The simulated observations are based on white noise different for the various observation types. The standard deviations of the white noise process (normally distributed random uncertainties with zero mean), see Table 1, used in the simulations are based on typical post fit RMS values from processing of real ground and LEO data. The phase measurements on the LEO satellites are usually more accurate than on the ground. White noise is usually deployed in such kind of simulations and the level of white noise in GNSS simulations is similar to, e.g., Dach et al. (2015) with 500 mm for code and 2 mm in phase and Kuang et al. (2015) with 500 mm for code and 65 mm for phase. Table 1 Standard deviation of white noise of simulated observations The precise orbit determination was performed for 1-day arcs. All the MEO and LEO satellite orbits were processed in a common adjustment applying the dynamic approach. In the simulations the code and phase hardware delays are assumed to be known (calibrated). 124 globally distributed ground stations constitute the network. The analysis period covers 10 days. This is sufficiently long in order to generate a reliable TRF (station positions only), but short enough to keep processing times at an acceptable level. The focus of this study is to investigate the impact of various simulation scenarios on different parameters. In order to get conclusions about the time variations, we will extend in further studies our simulation time span to at least three years to reliably estimate station velocities. Table 2 Simulation scenarios based on an extension of the constellation starting from MEO-only (Galileo-like) to a full Kepler constellation employing LEO satellites, inter-satellite links (ISL) and fixed (known) clocks (initial setup) In the recovery step, the simulated observations were evaluated in a precise orbit and parameter estimation process with EPOS-OC. Based on that, daily normal equation systems (NEQs) were setup for six simulation scenarios as given in Table 2. The simulation scenarios were defined to assess the individual contribution of the ISL, the LEO satellites, and the known (fixed) clocks on the TRF origin and scale. Starting from the MEO-only constellation, which is like the Galileo constellation, the MEO ISL are added, then the LEOs, then the ISL between the MEOs and the LEOs, and finally the satellite clocks are fixed to simulate the effect of known clocks with an accuracy better than the carrier phase noise, and do not have to be estimated. The last scenario features the full Kepler constellation. It should be mentioned that the fixing of satellite clocks facilitates an initial assessment of the complete features of the Kepler constellation. Prior to the combination of the daily NEQs, parameters such as orbital elements, tropospheric parameters, clocks and ambiguities (float solution) were pre-eliminated. The ambiguities of all the MEO and LEO satellites were solved as float solutions. The effect of fixed ambiguities was not analyzed within this study. For instance, Brockmann (1997) found an improvement by a factor of 3 in the Z component of the geocenter in case of an ambiguity-fixed w.r.t. an ambiguity-float solution. The daily NEQs were stacked, and finally station positions, geocenter coordinates and Earth rotation parameters (ERPs) were estimated. No-net rotation and no-net translation conditions were imposed to 39 datum stations following the simulation approach by Glaser et al. (2019). The scale was assessed by Helmert transformations w. r. t. the true simulated values. Since uncertainties in the solar radiation pressure (SRP) modeling impede an accurate origin realization by MEO-only GNSS (Meindl et al. 2013; Arnold et al. 2015), we investigated the impact of mismodeling of SRP. The empirical CODE (Center for Orbit Determination in Europe) orbit model (ECOM) consisting of nine parameters was used, Beutler et al. (1994) and Springer et al. (1999): $$\begin{aligned} D(u)= & {} D_0 + D_C \cdot \cos {(u)} + D_S \cdot \sin {(u)}\nonumber \\ Y(u)= & {} Y_0 + Y_C \cdot \cos {(u)} + Y_S \cdot \sin {(u)}\nonumber \\ B(u)= & {} B_0 + B_C \cdot \cos {(u)} + B_S \cdot \sin {(u)} \end{aligned}$$ in a Sun-fixed system with D in the direction towards the Sun, Y perpendicular to this direction along the solar panels, and B completes the right-handed system and with the constant biases $D_0,Y_0,B_0$, the harmonics $D_{[C,S]}, Y_{[C,S]}, B_{[C,S]}$, and the argument of latitude u of the satellites. The five parameter ECOM, where $D_0,Y_0,B_0,B_C,B_S$ are estimated, is commonly used. In the simulations, all ECOM parameters are based on values derived from a real GPS data analysis over 14 days (April 30–May 13, 2017) of one GPS satellite (PRN03) provided by the IGS, see Table 3. In the recovery, the nine ECOM parameters were either fixed to true simulated values (perfect SRP modeling) or fixed to values biased by 1$\sigma $ for all the satellites (SRP mismodeling). Table 3 Simulated ECOM parameters based on real GPS data analysis It should be mentioned, that the nine parameter ECOM has been further developed. For instance, Arnold et al. (2015) showed that the new extended ECOM ("ECOM2") results in more reliable geocenter estimates in Z direction. It is expected to use analytical models such as box-wing models for the Kepler constellation, as already done for the Galileo satellites. Rodriguez-Solano et al. (2014) showed that draconitic errors can significantly be reduced in the estimated parameters such as geocenter coordinates by using an adjustable box-wing model. As shown by, e.g., Bury et al. (2019), so-called hybrid models, where a priori box-wing models are introduced and three constant biases only are estimated, result in smaller formal errors of the geocenter Z component and stabilize the solution. In our simulations, the setup with ECOM is sufficient for the purpose of assessing the impact of biased SRP ECOM parameters on the geocenter estimates. The simulated ECOM values were fixed in the recovery, whereby the five ECOM parameters $D_0, Y_0, B_0, B_C, B_S$ of the "reduced ECOM" (Springer et al. 1999) were biased by 1$\varvec{\sigma }$ (see Table 3). The standard deviations, presented in Table 3, represent the scatter of the estimated values over the 14 days time period and are assumed to be realistic for this snapshot. The assumption of estimating ECOM5 parameters, instead of the full ECOM9, seems justified since it will be not necessary to estimate the full set of empirical orbit parameters if box-wing models are available. Currently in orbit modeling it is preferred to use physical instead of empirical orbit models as far as it is possible. The metadata have been released for Galileo with surface properties allowing for the generation of satellite models, such as the box-wing model. The SRP modeling of the LEO satellites was based on cannonball models and assumed to be perfect in this study. Other main orbit perturbations within the LEO orbit determination such as atmospheric drag, albedo, and time-variable gravity field were modeled and recovered in the same way; therefore no mismodeling was simulated. More information about the orbit determination and different LEO modeling errors of the Kepler constellation can be found in Michalak et al. (2020). The estimated geocenter coordinates were first evaluated by their standard deviations (i.e., formal errors) over the entire simulation period for the six different simulation scenarios (Table 2) in case of perfect models (no SRP mismodeling). As shown in Fig. 2, the extension of the constellation significantly improves the precision of the estimated geocenter coordinates, especially in Z direction. Due to the addition of the ISL between the MEO satellites within one orbital plane the precision improves by a factor of 13 in Z and 2.5 in X and Y w. r. t. the MEO-only case. This improvement is more than the expected improvement due to the larger number of observations, see Table 5 column "exp.". The addition of just the LEO satellites to the MEO constellation significantly improves the geocenter estimation from GNSS as well which is in concert with previous studies, e.g., Haines et al. (2015); Kuang et al. (2015); Männel and Rothacher (2017). We find very similar improvements as in the MEO-MEO ISL scenario by a factor of 2.5 in X and Y and even by 14 in Z for the MEO plus LEO constellation w. r. t. MEO-only. The orbital and observation geometry is improved by LEO satellites with higher inclinations, resulting in significant improvements in the formal errors of the Z component of the geocenter as already demonstrated by Kuang et al. (2015, Fig. 8). The six LEO satellites of the Kepler system are in two perpendicular planes with an inclination of 89.9$^\circ $, which improves the sensitivity of Z in the geocenter, in addition to the 24 MEO satellites with an inclination of 56$^\circ $ (Galileo-like). Finally, the full Kepler constellation with all ISL (between MEO as well as MEO and LEO satellites) and perfectly synchronized clocks shows improvements in the geocenter estimates by a factor of 43 in Z and by 8 for X and Y w. r. t. the MEO-only constellation. Standard deviations of estimated geocenter coordinates (GCC) in X, Y, Z direction in case of different simulation scenarios, see Table 2 Correlations for geocenter estimates in X, Y, Z and parameters of ECOM in case of different scenarios (Table 2) for one day and one satellite (other days and satellites show the same pattern) A correlation analysis between the estimated geocenter coordinates in X, Y, Z direction and the nine ECOM parameters was performed, the correlations for all six scenarios are presented in Fig. 3. The geocenter shows large correlations with the SRP parameters in the usual GNSS constellation, the MEO-only case. The largest correlation can be found between the Z coordinate of the geocenter and the constant bias $D_0$ in the direction towards the Sun of the SRP model, which is in accordance with Meindl et al. (2013). Based on consideration on orbital perturbations they identify the major limitation in the estimability of the geocenter by GNSS in the high correlation between the $D_0$ parameter and geocenter in Z direction. In our simulations, the scenario with the ISL between the MEO satellites already reduces the correlation between the geocenter estimates and the ECOM parameters significantly up to a complete de-correlation. Therefore, the Kepler constellation will allow a complete de-correlation of the SRP and the geocenter parameters resulting in a more reliable estimation of the origin of the reference frame. Estimated geocenter coordinates in X, Y, Z over 10 days in case of biased solar radiation pressure modeling (ECOM5 parameters are biased by 1$\sigma $ for all satellites) and different simulation scenarios (Table 2). Mean values and standard deviations in brackets The systematic effect of biased SRP parameters on the geocenter coordinates was assessed and the resulting geocenter coordinates are illustrated in Fig. 4. All the simulated ECOM values were fixed in the recovery, whereby the five ECOM parameters $D_0, Y_0, B_0, B_C, B_S$ of the "reduced ECOM" (Springer et al. 1999) were biased by $1\sigma $ (see Table 3). The biased ECOM5 model by $1\sigma $ results in biased geocenter coordinates of up to $-8.1$ mm in Z direction in the MEOs + ISL$_{\text {MEO}}$ scenario. The biased geocenter coordinates presented here are only caused by an offset introduced in the SRP modeling and not by the insensitivity of the GNSS technique to the origin. The biased ECOM can be partly compensated by the addition of the LEO satellites with perfectly simulated SRP modeling. The full Kepler constellation still shows biased geocenter coordinates of about 3 mm in X and 5 mm in Z direction. It seems that in this case the ISL do not improve the estimated geocenter coordinates. The very precise ISL observations stabilize the orbit by partly suppressing the other observations, also in the case of biased orbits. Highest priority should be always to improve the orbit modeling, and then the Kepler features such as the ISL result in an improved parameter estimation like the geocenter. It should be noted that the values of ECOM used in the simulations are average values over 14 days of real observations. They represent a certain state and are assumed to be time-invariant in this short period of time investigated in this study. In reality the ECOM parameters are time dependent since the elevation of the Sun changes w. r. t. the orbital plane in the quasi-inertial frame. With the SRP mismodeling of the MEO satellites of a certain state in our simulations (snapshot) it can be seen that the perfect LEO orbits can partly absorb the SRP bias of the MEO satellites but the ISL cannot. It can be expected that a longer time span, e.g., of at least one year, will not change the conclusions presented in this study. The quality of the scale realization of the Kepler constellation was assessed by the so-called "reference system effect" described in Sillard and Boucher (2001). They define three factors of the quality of the frame realization: the observation technique, the tracking network, and the analysis process. The last two factors were constant in our simulations, only the technique setup changed within the six simulation scenarios. All improvements in the scale can be, therefore, attributed to the amendments of the Kepler system w. r. t. the standard GNSS constellation. The resulting standard deviations of the realized scale in case of the different scenarios with perfect models are presented in Fig. 5 and the respective improvements w. r. t. the MEO-only constellation in Table 4. Here, in all scenarios the PCOs were fixed to their true values, no bias was introduced (perfect modeling). By extending the constellation, we find an improvement in the standard deviations of the realized scale. The fully-developed Kepler constellation shows the largest improvement by 34%. The Kepler constellation with the LEO satellites, the ISL between the MEO and LEO satellites and the perfect time synchronization is the most beneficial for the network scale realization. The scenarios with the ISL between the MEO satellites already show an improvement of 19%. Standard deviations of realized scale in case of different scenarios, see Table 2 Table 4 Improvement of realized scale w. r. t. MEO-only for different simulation scenarios (Table 2) Network scale change [mm] on the Earth's surface for different simulation scenarios (Table 2) and z-PCO bias of 2 cm for all MEO satellites The impact of biased satellite PCOs on the scale was evaluated assuming initially perfectly calibrated ground antennas. Since the scale of the implied frame is sensitive to the Z direction of the satellite PCOs (hereinafter z-PCO), we focus on that particular component. The bias was introduced in the adjustment in addition to the a priori PCOs used in the simulations. First, a bias of 2 cm in the z-PCO of all the MEO satellites was introduced and all PCOs (MEO and LEO) were fixed, that means, satellite PCOs were not estimated as unknown parameters. The resulting network scale change on the Earth's surface in case of the six simulation scenarios is depicted in Fig. 6. In the MEO-only scenario a scale change of about 0.9 mm (0.14 ppb) was found, which is in accordance with the relation provided by Zhu et al. (2003) (scale change [ppb] $=$ 7.8 $\cdot $ $\varDelta z$-PCO [m]). Amongst the simulation scenarios, the smallest scale difference was found for the MEO + LEO solution. The addition of the LEO satellites improves the overall orbital geometry of the constellation and seems to stabilize the solution. The MEO-only case shows the largest scale change due to the biased PCO. Compared to the MEO + LEO scenario, the solutions with the very precise ISL show a larger scale change due to biased PCO. The ISL connect the MEO satellites within one orbital plane and the MEO and LEO satellites of different planes. The ground network, which is needed to derive the network scale, is connected only via GNSS which refers directly to the z-PCO. In the MEO + LEO case the LEO orbit seems to absorb most of the MEO z-PCO bias. However, in case of the ISL more of the PCO bias propagates to the scale. The very precise ISL measurements create a stiff constellation by fixing the distance between MEO and LEO satellites and the biases get less absorbed by the orbital parameters. Network scale change [mm] on the Earth's surface for different simulation scenarios (Table 2) and z-PCO bias of 1 cm for all LEO satellites In a next setup, a smaller bias of 1 cm in the z-PCOs of the LEO satellites was introduced and all PCOs were fixed. The z-PCOs of the MEO constellation were introduced without any bias in this case. The LEO z-PCO bias of 1 cm is presented in view of a final network scale change of less than 1 mm towards the GGOS goals. The resulting network scale change is presented in Fig. 7. The MEO + LEO solution without the Kepler ISL shows the largest scale change of 2.3 mm. The network scale is more sensitive to the quality of the LEO PCOs compared to the quality of the MEO PCOs due to the lower LEO orbits. Comparing with the other scenarios with the ISL, the scale changes get significantly reduced below 0.1 mm for the full Kepler constellation. Network scale change [mm] on the Earth's surface for a MEO + LEO constellation in case of different LEO PCO bias ($-[10,5,4,3,1]$ mm). Requirement of 1 mm by GGOS is marked in red Focusing on MEO + LEO constellations, which are already in orbit unlike the proposed Kepler constellation, one can ask how accurately do the LEO PCOs need to be known for a change in the network scale below 1 mm in view of GGOS? To answer this question, we introduced different biases in the LEO z-PCOs and fixed all PCOs (no estimation of PCOs). The resulting scale changes in case of the MEO + LEO scenario and the different LEO z-PCO biases of $-[10,5,4,3,1]$ mm are depicted in Fig. 8. Therefore, to achieve a network scale accurate to the 1-mm level, the LEO $\varDelta z$-PCOs need to be known with an uncertainty of 4 mm or better. Based on various solutions, we derived an approximated relation ("rule of thumb") for the MEO plus LEO constellation: scale change [mm] $=$ 0.22 $\cdot \varDelta z$-PCO [mm], similar to the one for the MEO-only constellation provided by Zhu et al. (2003) and confirmed by our simulations. Network scale change [mm] on the Earth's surface for a MEO + LEO constellation in case of different LEO PCO bias ($-[10,5,1,0.8,0.7,0.5,0]$ mm) and an estimation of MEO PCO. Requirement of 1 mm by GGOS is marked in red Table 5 Precision of estimated parameters: increase in [%] in standard deviations s of coordinates $\mathbf {X}$ (average over all stations) and of pole coordinates ($x_p, y_p$) and UT1-UTC w. r. t. MEO-only In all results presented so far, the PCO were not estimated. Therefore, in principle, no independent network scale can be realized by GNSS due to the current intrinsic dependency to the fixed ITRF scale based on VLBI and SLR. In order to realize an independent network scale by GNSS, the MEO satellite PCOs have to be freely estimated within the adjustment together with station coordinates or to rely on their calibration on ground before launch. The estimation can be done, for instance in a MEO plus LEO constellation by fixing the LEO satellite PCOs and transferring this information to the MEO satellites. However, in this case the estimated MEO satellite PCOs and the consequent network scale completely rely on the fixed LEO satellite PCOs. To assess the impact of probable biases in the LEO satellite PCOs within an estimation of MEO PCO and other parameters, small LEO PCO biases of $-[10,5,1,0.8,0.7,0.5,0]$ mm were introduced in the MEO + LEO constellation and the resulting scale change is shown in Fig. 9. It is important to note, that in this case, the LEO $\varDelta z$-PCO needs to be known with an accuracy of 0.7 mm or better, in order to realize the network scale accurate to 1 mm. It is in agreement with an experiment with real data, where Huang et al. (2020) found a 2.7 cm shift in the terrestrial scale after adding an artificial bias of $+3$ cm on the LEO PCO when processing GNSS observations tracked by Swarm. The precision of other estimated parameters such as station positions and Earth rotation parameters are presented in Table 5 for the sake of completeness. The station positions improve by up to 8% on average for the full Kepler constellation w. r. t. the MEO-only constellation. The relative improvement in the formal errors of the different scenarios w. r. t. MEO-only is very similar in all Cartesian and local coordinate components. Therefore, only an average value of the improvements is provided. That improvement is below the expected (up to 13%) that stems from the additional observations (LEO satellites, ISL), see Table 5 column "exp.". In further studies the impact of the same number of observations and estimated parameters in every simulation scenario can be investigated, e.g., by replacing one type of observation by another and adapting the stochastic model accordingly. Standard deviations of estimated Earth rotation parameters $x_p, y_p$ and UT1-UTC in case of different scenarios, see Table 2 The improvements in the ERP go well beyond the expected improvements; 63% and 64% for x-pole and y-pole and 85% for UT1-UTC for the full Kepler constellation w. r. t. the MEO-only case. It should be mentioned that the ERP were setup as continuous piece-wise linear parameters once per day at midnight. UT1-UTC was estimated by GNSS due to fixing of the first value of the satellite arc. Comparing the different simulation scenarios, the largest leap in precision for all ERP by a single Kepler feature can be attributed to the addition of the ISL between the MEO satellites, see also Fig. 10. In case of the determination of the ERP the very precise ISL seem beneficial due to more stable orbits and a more stiff constellation. It is an important result since GNSS is the primary technique to determine polar motion. The standard deviation $s_0$ of the unit weight provides a quality criteria of the adjustment. In case of all scenarios, $s_0$ is very similar and close to 1 (as introduced a priori) demonstrating an equal quality of the solutions, even by the extension of the constellation with new additional observations (optical ISL, LEO satellites). It ensures to directly compare our six simulation scenarios from a statistical point of view. In addition, $s_0$ should get smaller in case of every simulation scenario towards the full Kepler constellation due to the larger degree of freedom (DOF $=$ "number of observation minus number of unknowns"). The MEO + LEO scenario has the smallest $s_0$ of all scenarios. However, the scenario MEO + ISL$_{\text {MEO}}$ shows a larger $s_0$ than the MEO-only case indicating that the functional and stochastic model might not yet be perfectly suited. Summary and conclusions Due to system-specific characteristics, e.g., the necessary estimation of epoch-wise satellite clocks, and remaining orbital model uncertainties, e.g., of the solar radiation pressure, GNSS do currently not contribute to the origin and scale realization of ITRF2014 but it might become possible in the future. A future GNSS constellation "Kepler" is proposed by the German Aerospace Center DLR featuring in addition to a Galileo-like MEO-only constellation, six LEO satellites in two polar planes as well as precise optical ISL and optical frequency references (Giorgi et al. 2019b). Since the accuracy requirements of GGOS on global TRFs have not been achieved yet, it is worth investigating the impact of the proposed Kepler constellation on the origin and scale realization. An ensemble of simulation scenarios with an extension of the standard Galileo constellation by the innovative Kepler features, LEO satellites in the constellation, precise ISL between the satellites and perfectly known clocks, was set up to assess the individual contribution of these system-specific characteristics. In case of the geocenter, all simulation scenarios employing the individual Kepler features result in improvements in the estimated geocenter coordinates, especially in the Z component of the geocenter, w. r. t. a MEO-only constellation. It is of special importance since the Z direction is usually considered to be the weakest geocenter component. Primarily, the individual contribution of the ISL between the MEO satellites and the addition of the LEO satellites succeed in this improvement by factors of 13 and 14 in Z and 2.5 in X and Y, respectively. The complete Kepler constellation will clearly improve the precision of the estimated geocenter by factors of 43 in Z and of 8 for X and Y w. r. t. a Galileo-like MEO-only constellation. These improvements go well beyond the expected improvement due to the larger number of observations of the full Kepler constellation w. r. t. MEO-only. Furthermore, the Kepler constellation facilitates a complete de-correlation of the geocenter coordinates and the SRP model parameters improving the reliability of the estimated geocenter. The individual contribution of the ISL between the MEO satellites in addition to the MEO-only constellation is already very promising for that. As already shown by, e.g., Meindl et al. (2013), a MEO-only constellation shows a high correlation between the geocenter and the ECOM parameters of the SRP modeling. Therefore, uncertainties in the SRP modeling lead to limitations in the estimation of the geocenter by current GNSS. The impact of SRP mismodeling was evaluated by introducing biases of 1$\sigma $ to the reduced ECOM resulting in biased geocenter coordinates of about 4 mm in the MEO-only case and of up to 8.1 mm in the scenario with the MEO plus the MEO ISL. The Kepler features as well as the complete constellation can only partly compensate for this mismodeling. Therefore, special emphasis in the utilization of an innovative future constellation like Kepler has to be placed upon the reduction of remaining uncertainties in the precise orbit modeling, especially of the SRP. The scale realization can be improved by the Kepler constellation by up to 34% w. r. t. the state-of-the-art, that is MEO-only, suggested by the standard deviations of the realized scale employing the approach of Sillard and Boucher (2001). The impact of probable uncertainties in the satellite PCOs on the network scale was investigated by introducing biased z-PCOs on the MEO and LEO satellites. We find that the z-PCOs have to be better than 2 cm in case of the MEO and better than 4 mm in case of the LEO satellites to realize a network scale to be better than 1 mm in view of GGOS. In these solutions all satellite PCO were fixed to their a priori values which are intrinsically linked to the applied ITRF. Therefore, such a network scale depends so far on the VLBI and SLR scale. An independent scale realization by GNSS is possible for instance in a MEO plus LEO constellation by estimating the MEO z-PCOs and simultaneously fixing the LEO PCOs, e.g., done by Haines et al. (2015) and Männel (2016). For this purpose, the LEO z-PCOs need to be known very precisely and our simulations suggest that they have to be better than 1 mm for a scale change below 1 mm. It might be a challenging task for pre-launch calibrations and requires special emphasis when relying on the LEO z-PCO for the scale realization by GNSS. Special attention should be also given to the risk of multipath effects of the LEO antenna. Additionally, significant improvements in the estimated ERP (63% in $x_p$, 64% in $y_p$, 85% in UT1-UTC) were identified in case of the Kepler constellation in comparison to a standard Galileo constellation. In general, the unique features of the Kepler constellation yield improvements in the estimated parameters, including the geocenter coordinates, and the network scale. As shown by the different simulation scenarios, the inclusion of the very precise ISL between the MEO satellites constitutes an important and valuable component of the Kepler constellation with regard to the TRF determination. The inclusion of LEO satellites for the TRF determination is already nowadays very beneficial, especially for the geocenter estimation due to the better orbital coverage. For instance, the Sentinel-3A/B satellites on two different orbital planes have similar inclinations but a lower altitude compared to Kepler. The Sentinel-3 satellites could be also used for co-location in space since they are equipped with SLR retroreflectors and GPS and DORIS receivers. For example, Strugarek et al. (2019) suggest to use SLR and GPS measurements to Sentinel-3A/3B for geocenter estimation. As shown in this study, the ISL as the innovative feature of Kepler between the MEOs, without LEOs, already result in a significant improvement in the geocenter estimation. Furthermore, the final Kepler constellation is one united system and will be processed accordingly which is different to a MEO + LEO constellation such as GNSS + GRACE. The precise orbit determination of current and future GNSS constellations has to be in any case continuously improved by further reducing remaining systematics. Then, the potential of the innovative features employed by the Kepler constellation can be fully exploited. Future studies comprise the refined satellite clock modeling towards most realistic simulations of the fully developed Kepler constellation. It might be also worth to study the effect of different satellite arc lengths within Kepler since, e.g., longer arcs can improve the solution of geocenter and ERP (Lutz et al. 2016) and decrease the correlation of orbital and tropospheric parameters with geocenter coordinates (Haines et al. 2015). The other space geodetic techniques will also be included to assess the benefit in multi-technique solutions like the ITRF towards the completion of the demanding GGOS requirements. 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J Geodesy 94(93). https://doi.org/10.1007/s00190-020-01417-0 Zhu S, Reigber C, König R (2004) Integrated adjustment of CHAMP, GRACE, and GPS data. J Geodesy 78(1–2):103–108. https://doi.org/10.1007/s00190-004-0379-0 Zhu SY, Massmann FH, Yu Y, Reigber C (2003) Satellite antenna phase center offsets and scale errors in GPS solutions. J Geodesy 76(11):668–672. https://doi.org/10.1007/s00190-002-0294-1 This work has been supported by the Helmholtz-Gemeinschaft Deutscher Forschungszentren e.V. under grant number ZT-0007 (ADVANTAGE, Advanced Technologies for Navigation and Geodesy). ADVANTAGE is a joint project of the German Aerospace Center (DLR) and the German Research Centre for Geosciences (GFZ) aiming at defining future systems for navigation, geodesy and metrology. The IGS (Johnston et al. 2017) is acknowledged for providing data used to derive the ECOM values for the simulations. The authors are very grateful for the detailed and valuable comments on the manuscript of three anonymous reviewers. Open Access funding enabled and organized by Projekt DEAL. GFZ German Research Centre for Geosciences, Potsdam, Germany Susanne Glaser, Benjamin Männel & Harald Schuh GFZ German Research Centre for Geosciences, Wessling, Germany Grzegorz Michalak, Rolf König & Karl Hans Neumayer Institute of Geodesy and Geoinformation Science, Chair of Satellite Geodesy, Technische Universität Berlin, Berlin, Germany Harald Schuh Grzegorz Michalak Benjamin Männel Rolf König Karl Hans Neumayer S.G. defined the study. S.G., G.M. and H.N. generated and analyzed the data. All authors contributed to the analysis, interpretation, and discussion of the results. S.G. wrote the manuscript with major contributions by R.K., B.M. and G.M and inputs from all authors. All authors read and approved the final manuscript. Correspondence to Susanne Glaser. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Glaser, S., Michalak, G., Männel, B. et al. Reference system origin and scale realization within the future GNSS constellation "Kepler". J Geod 94, 117 (2020). https://doi.org/10.1007/s00190-020-01441-0 Terrestrial reference frame Phase center offset	CommonCrawl
\begin{document} \title{Counting Signed Vexillary Permutations} \date{\today} \subjclass[2010]{Primary 05A05} \begin{abstract} We show that the number of signed permutations avoiding 1234 equals the number of signed permutations avoiding 2143 (also called vexillary signed permutations), resolving a conjecture by Anderson and Fulton. The main tool that we use is the generating tree developed by West. Many further directions are mentioned in the end. \end{abstract} \maketitle \section{Introduction}\label{sec:intro} Permutation pattern avoidance has been a popular line of research for many years. Denote the symmetric group on $n$ elements by $S_n$. We say that a permutation $w\in S_n$ \textit{avoids} a pattern $\pi\in S_k$ if there do not exist indices $1\leq a_1<\cdots<a_k\leq n$ such that $w(a_i)<w(a_j)$ if and only if $\pi(i)<\pi(j)$. Let $S_n(\pi)$ denote the set of permutations $w\in S_n$ that avoid $\pi$. Two permutations $\pi,\pi'$ are called \textit{Wilf equivalent} if $\|S_n(\pi)\|=\|S_n(\pi')\|$ for all $n$. The study of the growth rate of $\|S_n(\pi)\|$ and the study of nontrivial Wilf equivalence classes have been fruitful. One of the most famous examples of Wilf equivalence classes is all permutation patterns of length 3. Specifically, for any $\pi\in S_3$, $\|S_n(\pi)\|=C_n$, the $n^{th}$ Catalan number. Likewise, the set of permutations avoiding 1234 and the set of permutations avoiding 2143 have been traditionally well-studied and enjoy nice combinatorial properties. Their permutation matrices are shown in Figure~\ref{fig:matrix1234and2143}. \begin{figure} \caption{Permutations 1234 and 2143.} \label{fig:matrix1234and2143} \end{figure} A permutation $w$ avoids 1234 if and only if its shape under RSK has at most 3 columns, by Greene's theorem (see for example \cite{stanley1999ec2}). With some further tools in the theory of symmetric functions, 1234-avoiding permutations can be enumerated as follows: $$\|S_n(1234)\|=\frac{1}{(n+1)^2(n+2)}\sum_{j=0}^n{2j\choose j}{n+1\choose j+1}{n+2\choose j+1}.$$ This enumeration appeared in many previous works including \cite{gessel1990symmetric}, \cite{gessel1998lattice} and \cite{bousquet2002four} and is now an exercise in chapter 7 of \cite{stanley1999ec2}. A permutation that avoids 2143 is called \textit{vexillary}. Vexillary permutations can also be characterized as the permutations whose Rothe diagram, up to a permutation of its rows and columns, is the diagram for a partition \cite{manivel2001symmetric}. Moreover, their associated Schubert polynomials are flag Schur functions \cite{manivel2001symmetric}. West \cite{west1995generating} showed that $\|S_n(1234)\|=\|S_n(2143)\|$ for any $n$, so 1234 and 2143 are Wilf equivalent. The notion of Wilf equivalence can be naturally generalized to signed permutations. The signed permutation group $B_n\simeq(\mathbb{Z}/2\mathbb{Z})\wr S_n\simeq(\mathbb{Z}/2\mathbb{Z})^n\rtimes S_n$, also known as the Weyl group of type $B_n$ or $C_n$, consists of permutations $w$ on $\{-n,\ldots,-1,1,\ldots,n\}$ such that $w(i)=-w(-i)$ for all $i\in\{-n,\ldots,-1,1,\ldots,n\}$. We say that $w\in B_n$ \textit{avoids} $\pi\in S_k$ if the natural embedding of $w$ into $S_{2n}$ avoids $\pi$ in the sense of permutation pattern avoidance. For example, $w\in B_2$ given by $w(-2)=1$, $w(-1)=-2$, $w(1)=2$, $w(2)=-1$ contains $231$ and does not contain $123$, as the natural embedding $B_2\to S_{4}$ sends $w$ to $3142\in S_4$, and $3142$ contains $231$ and does not contain $123$. As a warning, this definition of pattern containment is \textbf{not} equivalent to a Weyl group element $w$ of type $B$ avoiding a type $A$ pattern, in the sense of root system pattern avoidance defined by Billey and Postnikov \cite{billey2005smoothness} to study the smoothness of Schubert varieties. In particular, let us define \begin{align} B_n(1234)=\{&w\in B_n\ \|\ \text{there do not exist}\ -n\leq a<b<c<d\leq n\ \\ &\text{such that}\ w(a)<w(b)<w(c)<w(d)\},\\ B_n(2143)=\{&w\in B_n\ \|\ \text{there do not exist}\ -n\leq a<b<c<d\leq n\ \\ &\text{such that}\ w(b)<w(a)<w(d)<w(c)\}, \end{align} i.e., the set of signed permutations avoiding 1234 and the set of signed permutations avoiding 2143 respectively, which are the main objects of interest in this paper. Analogously, $B_n(1234)$ and $B_n(2143)$ have very nice properties. In particular, the enumeration result $$\|B_n(1234)\|=\sum_{j=0}^n{n\choose j}^2C_j$$ where $C_j={2j\choose j}/(j+1)$ is the $j^{th}$ Catalan number, is given by Egge \cite{egge2010enumerating}, using techniques involving RSK and jeu-de-taquin. Geometric and combinatorial properties of signed permutations avoiding 2143, which are also called \textit{vexillary signed permutations}, are studied by Anderson and Fulton \cite{anderson2018vexillary}. They conjectured that $\|B_n(1234)\|=\|B_n(2143)\|$. The main result of this paper is to answer this conjecture positively. In fact, there are more similarities between the structures of signed permutations avoiding 1234 and signed permutations avoiding 2143. For $0\leq j\leq n$, and $\pi\in\{1234,2143\}$, define $$B_n^j(\pi):=\{w\in B_n(\pi)\ \|\ w(i)>0\ \text{for exactly }j\ \text{indices }i\in \{1,\ldots,n\}\}.$$ \begin{theorem}\label{thm:main} For $j\leq n$, $\|B_n^j(1234)\|=\|B_n^j(2143)\|$. \end{theorem} By summing over $j\in [n]$, we obtain the aforementioned conjecture by Anderson and Fulton \cite{anderson2018vexillary} as a corollary. \begin{corollary}\label{cor:main} For $n\in\mathbb{Z}_{\geq1}$, \[\|B_n(2143)\|=\|B_n(1234)\|=\sum_{j=0}^n{n\choose j}^2C_j.\] \end{corollary} As pointed out by Christian Gaetz, Theorem~\ref{thm:main} also implies a direct analogue of Corollary \ref{cor:main} for type $D$. Recall that the Weyl group of type $D_n$ can be realized as an order 2 subgroup of the Weyl group of type $B_n$. Specifically, $$D_n:=\{w\in B_n\ \|\ w(i)<0\text{ for an even number of indices }i\in\{1,\ldots,n\}\}.$$ We then say that $w\in D_n$ avoids a pattern $\pi$ if $w\in D_n\subset B_n$ avoids $\pi$. And as an analogous notation, let $D_n(\pi)$ denote the set of $w\in D_n$ that avoids $\pi$. Similarly, elements in $D_n(2143)$ are called \textit{vexillary} in type $D$ \cite{billey1998vexillary}. By summing the equality in Theorem \ref{thm:main} over $j\in [n]$ with $n-j$ even, we obtain the analogous enumeration result in type $D$. \begin{corollary} For $n\in\mathbb{Z}_{\geq1}$, $\|D_n(1234)\|=\|D_n(2143)\|$. \end{corollary} The main tool that we use in the proof of Theorem~\ref{thm:main} is the idea of generating trees developed by West \cite{west1995generating} to show that $\|S_n(1234)\|=\|S_n(1243)\|=\|S_n(2143)\|$. A generating tree is a rooted labeled tree for which the label at a vertex determines its descendants (their number and their labels). The generating trees considered by West have vertices that correspond to permutations avoiding a fixed pattern $\pi$. The descendants of a vertex corresponding to the permutation $w\in S_n$ correspond to permutations formed by inserting a new largest element $n+1$ to some location in $w$ (so that $\pi$ is still avoided). The usefulness of such generating trees stems in part from the fact that it is often possible to present an isomorphic tree with vertices labeled by only a few permutation statistics, with a simple enough succession rule to be fit for further analysis. In the case of $S_n(1234)$ versus $S_n(2143)$, West was able to find a simple description of both trees and observed that the two are naturally isomorphic, thus proving $\|S_n(1234)\|=\|S_n(2143)\|$ bijectively. There are two main difficulties in proving the simple-looking theorem (Theorem~\ref{thm:main}). First, as pointed out by Anderson and Fulton \cite{anderson2018vexillary}, the bijection between $S_n(1234)$ and $S_n(2143)$ provided by West \cite{west1995generating} does not preserve whether the permutation equals its reverse complement or not, suggesting a more careful choice of statistics for the generating trees, described in Section~\ref{sec:suc}. Second, the generating trees for $B_n^j(1234)$ and $B_n^j(2143)$ turn out to be far from isomorphic unlike the case of $S_n(1234)$ versus $S_n(2143)$ so we finish the proof by using certain generating functions in Section~\ref{sec:proof}. We end in Section~\ref{sec:open} with discussion on open problems. \section{Generating trees for 1234 and 2143 avoiding permutations}\label{sec:suc} We will start working towards an explicit generating tree for $B^j_n(1234)$ and $B^j_n(2143)$. Throughout the section, a signed permutation $w\in B_n$ should be visualized by a point graph, where the $x$-axis corresponds to the indices ${-}n,{-}n{-}1,\ldots,{-}1$, $1,\ldots,n$, and the $y$-axis corresponds to the images $w({-}n),w({-}n{-}1),\ldots,w(n)$. As a one-line notation, we will denote $w$ by $[w({-}n),w({-}n{-}1),\ldots,w({-}1)]$ in a nonstandard way, for reasons soon to be clear. A visualization of $w\in B_n$ is shown in Figure~\ref{fig:pointgraph}. \begin{figure} \caption{Visualization for $[-3,4,2,1].$} \label{fig:pointgraph} \end{figure} We first prove some simple lemmas regarding structures of signed permutations avoiding 2143 or 1234. \begin{lemma}\label{lem:struc} The following statements are true: \begin{enumerate} \item Suppose $w\in B_n^j(2143)$, where the $j$ positive indices with positive images are $1\leq i_1<i_2<\ldots<i_j\leq n$. Then $w(i_1)<w(i_2)<\ldots <w(i_j)$. \item Suppose $w\in B_n^j(1234)$, where the $j$ positive indices with positive images are $1\leq i_1<i_2<\ldots<i_j\leq n$. Then $w(i_1)>w(i_2)>\ldots>w(i_j)$. \end{enumerate} \end{lemma} \begin{proof} The proofs for the two cases are completely analogous so let's consider $w\in B_n^j(2143)$. If (1) were not true, then there exists a pair of positive indices $i<j$ with $1\leq w(j)<w(i)\leq n$. Consider the pattern forming at the indices $-j, -i, i, j$. We have $w(-i)<w(-j)<0<w(j)<w(i)$, or in other words, the pattern is $2143$. This is a contradiction with $w\in B_n^j(2143)$. \end{proof} \begin{lemma}\label{lem:ignore} Let $\pi\in \{2143, 1234\}$. If $w\in B_n$ contains $\pi$, then there exist indices ${-}n\leq i_1<i_2<i_3<i_4\leq n$ forming the pattern $\pi$ so that there exists no $i_k>0$ with $w(i_k)<0$. \end{lemma} \begin{proof} Let $w\in B_n$ and $\pi\in\{2143,1234\}$. Say $w$ contains $\pi$ at indices $i_1<i_2<i_3<i_4$. If there exists no $i_k>0$ with $w(i_k)<0$, then we are done. And if there exists no $i_k<0$ with $w(i_k)>0$, then we are also done by considering indices $-i_4<-i_3<-i_2<-i_1$ as $\pi$ equals its reverse complement. So we can without loss of generality assume that there exists $i_k>0$ with $w(i_k)<0$ and there exists $i_{\ell}<0$ with $w(i_{\ell})>0$. Here, $\ell<k$. If $\pi=1234$, this scenario is impossible since $i_{\ell}<0<i_k$ but $w(i_{\ell})>0>w(i_k)$. If $\pi=2143$, then either we have $\ell=1$, $k=2$, in which case $0<i_2<i_3<i_4$, $0<w(i_1)<w(i_4)<w(i_3)$ and the indices $-i_4<-i_3<i_3<i_4$ form 2143 or we have $\ell=3$, $k=4$, in which case $i_1<i_2<i_3<0$, $w(i_2)<w(i_1)<w(i_4)<0$ and the indices $i_1<i_2<-i_2<-i_1$ form 2143 with the desired property. \end{proof} Lemma~\ref{lem:struc} is saying that signed permutations that avoid 2143 (or 1234) have increasing (decreasing) sequence in the top right and bottom left quadrant, and moreover, Lemma~\ref{lem:ignore} allows us to ignore the contribution from the bottom right quadrant so that we can focus on the top left quadrant. Figure~\ref{fig:struct} depicts this idea. \begin{figure} \caption{Structure of signed permutations avoiding 2143 (left) or 1234 (right)} \label{fig:struct} \end{figure} \begin{example} Figure~\ref{fig:example-perm} shows an explicit signed permutation that avoids $2143$, with the dots in the bottom left and top right quadrant highlighted, and the dots in the bottom right quadrant in gray. Lemma~\ref{lem:struc} tells us that the highlighted dots form an increasing sequence. Lemma~\ref{lem:ignore} tells us that avoidance of $2143$ can be checked without considering the gray dots. \begin{figure} \caption{The $2143$-avoiding signed permutation $[-5,6,-4,7,-3,-1,2]$} \label{fig:example-perm} \end{figure} \end{example} With Lemma~\ref{lem:struc} and Lemma~\ref{lem:ignore}, we will generate $w\in B_n^j(\pi)$ by inserting elements into the top left quadrant one by one in the increasing order of their images. This idea of generating permutations is developed by West \cite{west1995generating}. For $w\in B_n^j(\pi)$, call the spaces between negative indices \textit{sites}, including the space left of $-n$ and the space right of $-1$ so that there are $n+1$ sites. Similarly, call the vertical spaces \textit{gaps}. To define insertions more formally, we introduce the auxiliary function \[ \beta_\ell(x)= \begin{cases} x & \|x\|<\ell, \\ x-1 & x<-\ell,\\ x+1 & x>\ell,\\ \end{cases} \] which can be thought as the function pushing images to their new locations when the gap between $\ell-1$ and $\ell$ gets a new image. \begin{definition}\label{def:insert} For $w\in B_n$, let $w_{\ell}^{-i}\in B_{n+1}$ be the signed permutation obtained by \textit{inserting a new element to site} $-i$ \textit{and gap} $\ell$, defined by \[ w_{\ell}^{-i}(-k)= \begin{cases} \beta_{\ell}\big(w(-k-1)\big) & i+1\leq k\leq n+1,\\ \ell & k=i,\\ \beta_{\ell}\big(w(-k)\big) & 1\leq k\leq i-1. \end{cases} \] \end{definition} Definition~\ref{def:insert} is just a formal way to express inserting an element to a specific position (and its antipode) in a signed permutation. Now we are ready to introduce the main object of interest in this section. \begin{definition} For $\pi\in\{2143,1234\}$ and $j\geq0$, let $BT^j(\pi)$, the \textit{signed permutation pattern avoidance tree}, to be a rooted tree labeled by signed permutations, defined as follows: \begin{itemize} \item its root is $[-j,\ldots,-1]$ for $\pi=2143$ and $[-1,\ldots,-j]$ for $\pi=1234$, \item the successors of $w\in B_n^j(\pi)$ are all $w_{\ell}^{-i}$'s with $1\leq i\leq n+1$ and $m<\ell\leq n+1$ that still avoid $\pi$, where $m=\max_{1\leq k\leq n}\{w(-k)\}\cup\{0\}$. \end{itemize} \end{definition} Let us briefly discuss some essential properties of this tree. Note that any permutation avoiding $\pi$ can be constructed by starting with the signed permutation in its top right and bottom left quadrants (which has the form of the corresponding root by Lemma \ref{lem:struc}) and inserting its other elements in increasing order to the top left quadrant. Hence, any permutation avoiding $\pi$ appears in $BT^j(\pi)$. Moreover, any permutation avoiding $\pi$ appears in $BT^j(\pi)$ exactly once, since there is only one way to insert its elements in increasing order. Hence, the vertices of $BT^j(\pi)$ which are $n-j$ steps away from the root correspond precisely to the signed permutations of length $n$ avoiding $\pi$ with exactly $j$ positive indices with positive images. In other words, the vertices $n-j$ steps away from the root correspond to $B^j_n(\pi)$. The main results of this section are the following. \begin{proposition}\label{prop:suc2143} The generating tree given by the following: \begin{itemize} \item the label of the root is $(j{+}1, j{+}1, j{+}1)$; \item the succession function $suc$ that takes a label as its input and outputs the set of successors is defined recursively as follows: $$suc(x,y,z)= \begin{cases} \emptyset &z=0,\\ \{(2,y{+}1,z),(3,y{+}1,z),\ldots,(x{+}1,y{+}1,z)\\ (x,x{+}1,z),(x,x{+}2,z),\ldots,(x,y,z)\}\bigcup suc(x,x,z{-}1) &z\geq1. \end{cases}$$ \end{itemize} is isomorphic as a rooted tree to $BT^j(2143)$. \end{proposition} \begin{proposition}\label{prop:suc1234} The generating tree given by the following: \begin{itemize} \item the label of the root is $(j{+}1, j{+}1, j{+}1)$; \item the succession function $suc$ that takes a label as its input and outputs the set of successors is defined recursively as follows: $$suc(x,y,z)= \begin{cases} \{(2,y{+}1,z),(3,y{+}1,z),\ldots,(x{+}1,y{+}1,z)\\ (x,x{+}1,z),(x,x{+}2,z),\ldots,(x,y,z)\} &z=1,\\ \{(2,y{+}1,z),(3,y{+}1,z),\ldots,(x{+}1,y{+}1,z)\}\\ \bigcup suc(x,y,z{-}1) &z\geq2. \end{cases}$$ \end{itemize} is isomorphic as a rooted tree to $BT^j(1234)$. \end{proposition} Before proving Proposition~\ref{prop:suc2143} and Proposition~\ref{prop:suc1234}, we first discuss some important statistics on the signed permutations of interest, that are related to the variables $x,y,z$ in the above propositions. Examples will come shortly. For $w\in B^j_n(\pi)$, a \textit{site before the first ascent (descent)} is defined to be a site such that elements to the left of this site are decreasing (increasing). In particular, if $w(-n)>\cdots>w(-1)$ (or increasing), then there are $n+1$ sites before the first ascent (descent). The number of sites before the first ascent (descent) is usually denoted via the variable $x$. For $w\in B^j_n(\pi)$, an \textit{active site} with respect to a fixed gap $\ell$, is a site such that inserting into this site and gap $\ell$ results in a signed permutation that avoids $\pi$. The number of active sites is usually denoted $y$. We will make further specifications for $\pi=2143$ and $\pi=1234$. For $w\in B_n^j(\pi)$, there are $j$ positive indices $i$ with positive images. In other words, there are $j$ elements in the top right quadrant, and they divide the top left quadrant into horizontal ``layers". Formally, the \textit{layer number} is $1+\#\{i>0:w(i)>\max_{k<0}w(k)\}$, describing the current layer that we are inserting elements into. The layer number is denoted $z$, and it ranges from $j+1$ to 1. Recall that we are constructing signed permutations in $BT^j(\pi)$ by inserting elements into the top left quadrant in increasing order of the images. Therefore, we will be saying inserting into some layer instead of inserting into some gap. The following Lemma~\ref{lem:activesites} is useful and can be observed directly so we omit the proof. \begin{lemma}\label{lem:activesites} Let $w\in B_n^j(\pi)$ and fix a layer $z$ that we are inserting elements into. If we insert the new maximal image $\ell$ in the left quadrants to some active site $-i$, then the new active sites of $w^{-i}_{\ell}$ are a subset of the old ones: here we think of the site where we inserted $\ell$ to have split into two (so the number of active sites may potentially increase by at most 1). Furthermore, if a previously active site becomes inactive, then inserting $\ell+1$ there would create a pattern $\pi$ involving both $\ell$ and $\ell+1$. \end{lemma} Now we are ready to separate the cases $\pi=2143$ and $\pi=1234$. First consider $\pi=2143$. Keep track of the following statistics on $w\in B_n^j(2143)$: \begin{itemize} \item $x$: the number of sites before the first descent, \item $y$: the number of active sites in the current layer $z$, \item $z$: the layer number, which is the lowest layer to which the maximal image of the negative indices can be inserted. \end{itemize} Figure~\ref{fig:2143} shows the statistics $x=3$, $y=5$ and $z=2$ for $w=[-6,4,-3,5,2,1]$ where we see that the layer numbers are decreasing from bottom to top and the active sites in each layer are labeled by $\times$. \begin{figure} \caption{The statistics $x=3$, $y=5$, $z=2$ for $w=[-6,4,-3,5,2,1]\in B_6^2(2143).$} \label{fig:2143} \end{figure} \begin{proof}[Proof of Proposition~\ref{prop:suc2143}] It suffices to show that if the statistics of $w$ are $x,y,z$, then the multiset of statistics of the successors of $w$ in $BT^j(2143)$ is precisely what is given in the proposition. As we are inserting new images in increasing order, there are no successors with layer number $z'>z$. Let us first determine the number of active sites $y_{z'}$ in each layer numbered $z'\leq z$. We know that the number of active sites in the layer $z$ is $y$. Let us show that the number of active sites in all layers $z'<z$ is $y_{z'}=x$. To see this, note that all sites before the first descent are active, as inserting there, the inserted element would have to be a $2$ or a $1$ in a $2143$, but this is impossible as all larger elements are in the top right quadrant, hence form an increasing sequence (so there is no $43$). Additionally, note that inserting to a site after the first descent would form a $2143$ involving the first descent as $21$, the inserted element as $4$, and the element immediately below it in the top right quadrant as $3$ (this exists since $z'<z$, so the layer was previously empty). Therefore, the active sites for layers $z'<z$ are precisely the $x$ sites before the first descent. Let us now determine the successors with layer number $z'$, given that the number of active sites in the layer $z'$ is initially $y_{z'}$. It is again easy to see that all sites before the first descent are active, and it follows from Lemma \ref{lem:activesites} that all active sites remain active after such an insertion. The new first descent appears immediately after the insertion. Hence, the successors from this case are $(2,y_{z'}+1, z'), (3, y_{z'}+1, z'), \ldots, (x+1,y_{z'}+1, z')$. Let us now consider the case of inserting to an active site after the first descent. In that case, the insertion leaves the position of the first descent unchanged. As for the active sites, all sites before the first descent are still active, all sites to the right of the first descent but to the left of the insertion are inactive (since inserting there would create an obvious $2143$), and all previously active sites to the right of the insertion remain active, since by Lemma \ref{lem:activesites}, if we also inserted to one such site and created a $2143$, it would need to involve both of the last two insertions, and these could only be $2$ and $3$, but no $4$ can be found between them. Hence, the successors from this case are $(x,x+1,z'), (x, x+2, z'), \ldots, (x, y_{z'}, z')$. We have now explicitly found the set of successors, since we have found $y_{z'}$ for each $z'\leq z$ and given the successors with each layer number $z'\leq z$ in terms of $y_{z'}$. This multiset of successors is what is given in the proposition. \end{proof} Next consider $\pi=1234$. Keep track of the following statistics on $w\in B_n^j(2134)$: \begin{itemize} \item $x$: the number of sites before the first ascent, \item $y$: the number of active sites in the top layer, \item $z$: the layer number. \end{itemize} Figure~\ref{fig:1234} shows the statistics $x=3$, $y=7$, $z=3$ for $w=[2,-3,4,-5,1,-6]$. \begin{figure} \caption{The statistics $x=3$, $y=7$, $z=3$ for $w=[2,-3,4,-5,1,-6]\in B_6^3(1234).$} \label{fig:1234} \end{figure} \begin{proof}[Proof of Proposition~\ref{prop:suc1234}] Again, it suffices to show that if the statistics of $w$ are $x,y,z$, then the multiset of statistics of the successors of $w$ in $BT^j(1234)$ is precisely what is given in the proposition. As before, there are no successors with layer number $z'>z$. The active sites for layers numbered $z'$ such that $z\geq z'>1$ are precisely the sites before the first ascent, as all sites before the first ascent are active (since the inserted element would have to be a $1$ or $2$ in a $1234$, but then there is no $34$ since all elements larger than the inserted element are in the top right quadrant and form a decreasing sequence), and all sites after the first ascent are inactive because of forming a $1234$ with the maximal element of the top right quadrant serving as a $4$. Also note that for $z>1$, all sites in the top layer are active, as if some site was inactive, there would be a $123$ in the top left quadrant, but this would also mean there exists a $1234$ with the maximal element in the top right quadrant as a $4$, which is impossible. This means that the successors with $z'$ such that $z\geq z'>1$ are precisely $(2,y+1, z'), (3,y+1, z'), \ldots, (x+1, y+1, z')$. As for successors with $z'=1$, in the top layer all sites before the first ascent are active, and inserting to such a site leaves all active sites active (the proof is analogous to what was seen for Proposition \ref{prop:suc2143}). This case gives the successors $(2, y+1, 1), (3,y+1,1), \ldots, (x+1, y+1, 1)$. If we instead insert to an active site after the first ascent, then the new active sites are precisely the ones to the left of the insertion. The proof is again analogous to what was seen for Proposition \ref{prop:suc2143}. This final case gives the successors $(x, x+1, z), (x, x+2, z), \ldots, (x,y,z)$. Putting everything together, this multiset of successors is what is given in the proposition. \end{proof} \section{Finishing the proof}\label{sec:proof} Proposition~\ref{prop:suc2143} and Proposition~\ref{prop:suc1234} allow us to translate the questions of enumerating $B_n^j(2143)$ and $B_n^j(1234)$ to questions of enumerating lattice paths in the integer lattice $\mathbb{Z}^3$ with specified rules. Respectively, let $\mathcal{P}^{2143}$ be the set of all lattice paths specified by the succession rule in Proposition~\ref{prop:suc2143} and let $\mathcal{P}^{1234}$ be the set of all lattice paths specified by the succession rule in Proposition~\ref{prop:suc1234}. We allow arbitrary starting point $(x,y,z)$ for those paths with $2\leq x\leq y$ and $1\leq z$ besides those that start at $(j+1,j+1,j+1)$. We view such a lattice path as a sequence of points connected by edges. For a path $P\in\mathcal{P}^{2143}$ and an edge $e$ of $P$ that goes from $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$, we say that $e$ is \textit{recorded} \begin{itemize} \item if $z_1=z_2$ and $y_2=y_1+1$; \item if $z_1>z_2$ (and $y_2=x_1+1$). \end{itemize} Notice that if $z_1>z_2$, then we are forced to use the succession rule of $(x_1,x_1,z_2)$ to go to $(x_2,y_2,z_2)$ and thus $y_2=x_1+1$. Analogously, for $P\in\mathcal{P}^{1234}$ and an edge $e$ of $P$ that goes from $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$, we say that $e$ is \textit{recorded} if $y_2=y_1+1$. In particular, if $z_2\geq2$, the edge is always recorded. We see from the succession rule in Section~\ref{sec:suc} that if an edge is not recorded, then the $x$-coordinates are the same for the two points connected by the edge. \begin{definition} For a path $P\in\mathcal{P}^{\pi}$, $\pi\in\{1234,2143\}$, define its \textit{signature} $\mathrm{sig}(P)$ to be the tuple consists of the $x$-coordinate of the starting point, appended with the $x$-coordinates of the ending points of recorded edges in order. \end{definition} \begin{example} Consider the following paths $P\in\mathcal{P}^{2143}$ and $P'\in\mathcal{P}^{1234}$ which are \begin{align} P=&(4,4,3)\rightarrow(3,5,3)\dashrightarrow(3,5,3)\rightarrow(4,4,2)\rightarrow(2,5,2)\dashrightarrow(2,4,2)\\ &\dashrightarrow(2,4,2)\rightarrow(2,2,1)\rightarrow(2,3,1)\dashrightarrow(2,3,1)\\ P'=&(4,4,3)\rightarrow(3,5,3)\rightarrow(4,6,3)\rightarrow(2,7,2)\rightarrow(2,8,1)\dashrightarrow(2,7,1)\\ &\dashrightarrow(2,7,1)\dashrightarrow(2,5,1)\dashrightarrow(2,4,1)\dashrightarrow(2,4,1)\rightarrow(2,5,1)\dashrightarrow(2,4,1) \end{align} where the recorded edges are written as solid arrows and the edges not recorded are written as dashed arrows. Both paths have signature $(4,3,4,2,2,2)$. \end{example} The main goal of this section is to show that for a fixed starting point $v$, a fixed signature $\gamma$ and $n\geq1$, the number of paths in $\mathcal{P}^{\pi}$ that start with $v$, have signature $\gamma$ and have length $n$ is the same for $\pi\in\{1234,2143\}$. To do this, let us define the corresponding generating functions. For $\pi\in\{1234,2143\}$, $k\geq0$, $q\geq1$, $\gamma=(\gamma_1,\ldots,\gamma_m)\in\mathbb{Z}^m$, define $\mathcal{P}^{\pi}_{k,q,\gamma}$ to be the set of paths in $\mathcal{P}^{\pi}$ that start at $(\gamma_1,\gamma_1+k,q)$ and have signature $\gamma$. For any path $P$, let its length $\ell(P)$ be the number of points that it contains. Notice that if $P$ has signature $\gamma=(\gamma_1,\ldots,\gamma_m)$, then clearly $\ell(P)\geq m$. \begin{definition} For $\pi\in\{1234,2143\}$, $k\geq0$, $q\geq1$, $\gamma\in\mathbb{Z}^m$ with $m\geq1$, define $$F^{\pi}(k,q,\gamma):=\sum_{P\in\mathcal{P}^{\pi}_{k,q,\gamma}}t^{\ell(P)-m}.$$ \end{definition} We are going to recursively compute $F^{\pi}(k,q,\gamma)$ and then compare $F^{1234}(k,q,\gamma)$ with $F^{2143}(k,q,\gamma)$. As for some notations, if $\gamma\in\mathbb{Z}^m$, write $\|\gamma\|=m$. For convention, we say $F^{\pi}(k,q,\gamma)=0$ if $q\leq0$ or $\|\gamma\|=0$. Let $\gamma'=(\gamma_2,\gamma_3,\ldots,\gamma_m)$, which is $\emptyset$ if $m=1$ and let $\gamma''=(\gamma_3,\ldots,\gamma_m)$, which is $\emptyset$ if $m\leq2$. And we will restrict our attention to only those $\gamma$'s that can be signatures of some valid paths in $\mathcal{P}^{\pi}$. Namely, we require $2\leq\gamma_{i+1}\leq\gamma_i+1$. Finally, for simplicity, let $s=1+t+t^2+\cdots=1/(1-t)$. \begin{lemma}\label{lem:2143gen} For $k\geq0$, $q\geq1$, $\gamma\in\mathbb{Z}^m$ with $m\geq1$, we have \begin{equation} F^{2143}(k,q,\gamma)=\begin{cases} s^k &\|\gamma\|=1,\\ F^{2143}(0,q{-}1,\gamma)+F^{2143}(\gamma_1{+}1{-}\gamma_2,q,\gamma') &\|\gamma\|\geq2,k=0,\\ sF^{2143}(k{-}1,q,\gamma)+sF^{2143}(\gamma_1{+}1{-}\gamma_2{+}k,q,\gamma')\\ -sF^{2143}(\gamma_1{-}\gamma_2{+}k,q,\gamma') &\|\gamma\|\geq2,k\geq1. \end{cases} \end{equation} \end{lemma} \begin{proof} We refer the readers to the succession rule in Proposition~\ref{prop:suc2143}. If $\|\gamma\|=1$, then the signature has length 1 and we are summing over paths that start at $(\gamma_1,\gamma_1+k,q)$ with no recorded edges. As soon as we decrease $q$, which is the $z$-coordinate, we need to use the succession rule for $(\gamma_1,\gamma_1,q-1)$ and then every edge is recorded so we cannot have any edges afterwards. Therefore, the only additional points on this path come from any number of $(\gamma_1,\gamma_1{+}k,q)$ followed by any number of $(\gamma_1,\gamma_1{+}k{-}1,q)$ and so on, finally ending with any number of $(\gamma_1,\gamma_1{+}1,q)$. The resulting generating function is then $(1+t+t^2+\cdots)^k=s^k$. If $k=0$, then our paths start at $(\gamma_1,\gamma_1,q)$. The next edge must be recorded. There are exactly two options: either decrease the $z$-coordinate $q$, or go directly to the next signature value $\gamma_2$ at the same $z$-coordinate. For the first option, we obtain a generating function $F^{2143}(0,q-1,\gamma)$. For the second option, we go from $(\gamma_1,\gamma_1,q)$ to $(\gamma_2,\gamma_1{+}1,q)$ and trim the signature so the corresponding generating function is $F^{2143}(\gamma_1{+}1{-}\gamma_2,q,\gamma')$. The main case is $\|\gamma\|\geq2$ and $k\geq1$. Our goal is to decrease $k$. As $k\geq1$, when we start at $(\gamma_1,\gamma_1{+}k,q)$, we are allowed to have an arbitrary number of $(\gamma_1,\gamma_1{+}k,q)$ first via unrecorded edges, which provide a factor of $s$, before we choose the next edge. Let's now compare $F^{2143}(k,q,\gamma)$ with $sF^{2143}(k{-}1,q,\gamma)$. The paths enumerated by each of them largely coincide, including those that decrease $q$ right away. The only exception is that paths that go directly from some number of $(\gamma_1,\gamma_1{+}k,q)$ to the next recorded edge ending at $(\gamma_2,\gamma_1{+}k{+}1,q)$ are counted by $F^{2143}(k,q,\gamma)$ but not by $sF^{2143}(k{-}1,q,\gamma)$; and similarly the paths that go directly to $(\gamma_2,\gamma_1{+}k,q)$ from $(\gamma_1,\gamma_1{+}k{-}1,q)$ are counted only by $sF^{2143}(k{-}1,q,\gamma)$. As a result, \begin{align} &F^{2143}(k,q,\gamma)-sF^{2143}(k{-}1,q,\gamma)\\ =&sF^{2143}(\gamma_1{+}1{-}\gamma_2{+}k,q,\gamma')-sF^{2143}(\gamma_1{-}\gamma_2{+}k,q,\gamma') \end{align} which is equivalent to the statement that we need. \end{proof} Notice that the recursive formula provided in Lemma~\ref{lem:2143gen} can determine $F^{2143}$ uniquely. \begin{lemma}\label{lem:1234gen} For $k\geq0$, $q\geq1$, $\gamma\in\mathbb{Z}^m$ with $m\geq1$, we have \begin{equation} F^{1234}(k,q,\gamma)=\begin{cases} s^k &\|\gamma\|=1,\\ F^{2143}(k,q,\gamma) &q=1,\\ F^{1234}(k,q{-}1,\gamma)+F^{1234}(\gamma_1{+}1{-}\gamma_2{+}k,q,\gamma') &\|\gamma\|\geq2,q\geq2. \end{cases} \end{equation} \end{lemma} \begin{proof} We refer the readers to the succession rule in Proposition~\ref{prop:suc1234} If $\|\gamma\|=1$, then we are considering paths that start at $(\gamma_1,\gamma_1+k,q)$ with no recorded edges. Since every edge is recorded when $q\geq2$, our only option is to decrease $q$ all the way down to 1 and then use the succession rule of $(\gamma_1,\gamma_1+k,1)$. Now we can have an arbitrary number of $(\gamma_1,\gamma_1{+}k,1)$ followed by an arbitrary number of $(\gamma_1,\gamma_1{+}k{-}1,1)$ and so on up to an arbitrary number of $(\gamma_1,\gamma_1{+}1,1)$. The generating function is thus $(1+t+t^2+\cdots)^k=s^k$. If $q=1$, the succession rules for $\mathcal{P}^{1234}$ and $\mathcal{P}^{2143}$ are the same so we have $\mathcal{P}^{1234}_{k,1,\gamma}=\mathcal{P}^{2143}_{k,1,\gamma}$. Therefore, $F^{1234}(k,1,\gamma)=F^{2143}(k,1,\gamma)$. When $q\geq2$ and $\|\gamma\|\geq2$, for a path in $\mathcal{P}^{1234}_{k,q,\gamma}$, it starts at $(\gamma_1,\gamma_1{+}k,q)$. Since $q\geq2$, all edges that keep the same $z$-coordinate $q$ are recorded. So we have exactly two options: decrease $q$ by 1, which results in the generating function $F^{1234}(k,q{-}1,\gamma)$, and go to $(\gamma_2,\gamma_1+1,q)$ indicated by the signature $\gamma$, which results in the generating function $F^{1234}(\gamma_1{+}1{-}\gamma_2{+}k,q,\gamma')$. Take the sum and we get the desired equation. \end{proof} With sufficient tools to determine the generating functions $F^{2143}$ and $F^{1234}$, we are ready to obtain their equality. \begin{lemma}\label{lem:2143gen=1234gen} For $k\geq0$, $q\geq1$, $\gamma\in\mathbb{Z}^m$ with $m\geq1$, $$F^{1234}(k,q,\gamma)=F^{2143}(k,q,\gamma).$$ \end{lemma} \begin{proof} We proceed by induction on $\|\gamma\|$, $q$ and $k$ in this order. From Lemma~\ref{lem:2143gen} and Lemma~\ref{lem:1234gen}, our statement is true when $\|\gamma\|=1$ and is also true when $\|\gamma\|\geq2$ and $q=1$. When $\|\gamma\|\geq2$, $q\geq2$ and $k=0$, from Lemma~\ref{lem:2143gen}, $$F^{2143}(0,q,\gamma)=F^{2143}(0,q-1,\gamma)+F^{2143}(\gamma_1+1-\gamma_2,q,\gamma')$$ and from Lemma~\ref{lem:1234gen}, $$F^{1234}(0,q,\gamma)=F^{1234}(0,q-1,\gamma)+F^{1234}(\gamma_1+1-\gamma_2,q,\gamma')$$ so by induction hypothesis, $F^{2143}(0,q,\gamma)=F^{1234}(0,q,\gamma)$. Now assume that $\|\gamma\|\geq2$, $q\geq2$ and $k\geq1$. With induction hypothesis and for the ease of notation, for the arguments that we already know the equality of $F^{1234}$ and $F^{2143}$, we will just write $F$ instead. By Lemma~\ref{lem:2143gen} and Lemma~\ref{lem:1234gen}, and by induction hypothesis, we have that \begin{align} F^{2143}(k,q,\gamma)=&sF(k{-}1,q,\gamma)+sF(\gamma_1{+}1{-}\gamma_2{+}k,q,\gamma')-sF(\gamma_1{-}\gamma_2{+}k,q,\gamma')\\ =&sF(k{-}1,q{-}1,\gamma)+sF(\gamma_1{-}\gamma_2{+}k,q,\gamma')\\ &+sF(\gamma_1{+}1{-}\gamma_2{+}k,q{-}1,\gamma')+sF(\gamma_1{+}2{-}\gamma_3{+}k,q,\gamma'')\\ &-sF(\gamma_1{-}\gamma_2{+}k,q{-}1,\gamma')-sF(\gamma_1{+}1{-}\gamma_3{+}k,q,\gamma'')\\ =&F(k,q{-}1,\gamma)+F(\gamma_1{+}1{-}\gamma_2{+}k,q,\gamma')\\ =&F^{1234}(k,q,\gamma) \end{align} as desired. We also see that the above argument goes through when $\|\gamma'\|=1$, in which case $\gamma''=\emptyset$. Therefore, the induction step is established so we obtain the desired lemma. \end{proof} With the main technical lemma (Lemma~\ref{lem:2143gen=1234gen}), Theorem~\ref{thm:main} becomes immediate. \begin{proof}[Proof of Theorem~\ref{thm:main}] For $\pi\in\{1234,2143\}$ and $j\leq n$, $$B_n^j(\pi)=\sum_{\gamma_1=j+1}[t^{n-j-\|\gamma\|+1}]F^{\pi}(0,j+1,\gamma).$$ Since $F^{1234}(0,j+1,\gamma)=F^{2143}(0,j+1,\gamma)$, $B_n^j(1234)=B_n^j(2143)$. \end{proof} \section{Open questions}\label{sec:open} There are still many interesting questions to be asked. Firstly, the proof provided in Section~\ref{sec:proof} is semi-bijective. With recursive formulas provided in Lemma~\ref{lem:2143gen} and Lemma~\ref{lem:1234gen}, we are able to obtain the equality of $F^{1234}(k,q,\gamma)=F^{2143}(k,q,\gamma)$. However, is there an explicit bijection between paths in $\mathcal{P}^{1234}_{k,q,\gamma}$ and $\mathcal{P}^{2143}_{k,q,\gamma}$ that is length-preserving? Secondly, for a fixed $j\geq0$, it is desirable to obtain an explicit formula for the generating function $$\sum_{n=j}^{\infty}B_n^j(\pi)t^{n-j}$$ for either $\pi\in\{1234,2143\}$. The case $j=0$ is the generating function for 1234 (or 2143) avoiding permutations $\sum_{n}S_n(1234)t^n$, which is studied in \cite{bousquet2002four} and already has a complicated form. Thirdly, can our techniques be further generalized? We make the following conjecture. \begin{conjecture}\label{conj:12345} For $j\leq n$, $\|B_n^j(12345)\|=\|B_n^j(21354)\|$. \end{conjecture} We have checked Conjecture~\ref{conj:12345} for $n\leq 7$. Notice that when $j=0$, the statement holds \cite{west1990permutations} and when $j=n$, it is not hard to see that both sides equal the Catalan number $C_j$. More generally, it is known that the identity element $1,2,\ldots,k$ and $\pi=2,1,3,\ldots$, $k{-}2,k,k{-}1$ are Wilf equivalent in the sense of permutations \cite{west1990permutations}. So are they Wilf equivalent in signed permutations? Finally, we note that some results in this paper fit nicely into the framework of $\mathrm{st}$-Wilf equivalence introduced by Sagan and Savage \cite{sagansavage}. In the classical case of the symmetric group $S_n$ and a permutation statistic $\mathrm{st}\colon S_n\to \mathcal{S}$ (where $\mathcal{S}$ is some set), the patterns $\pi_1,\pi_2$ are said to be $\mathrm{st}$-Wilf equivalent if for all $n\geq 1$ and $\sigma\in \mathcal{S}$, the number of $w\in S_n(\pi_1)$ with $\mathrm{st}(w)=\sigma$ is equal to the analogous number for $\pi_2$. The analogous definition for $B_n$ is clear; furthermore, our Theorem \ref{thm:main} and Lemma \ref{lem:2143gen=1234gen} can be restated as instances of $\mathrm{st}$-Wilf equivalence. But there are many other important permutation statistics on $B_n$, and proving corresponding $\mathrm{st}$-Wilf equivalences for permutation patterns in signed permutations is an interesting direction for further investigation. \section*{Acknowledgments} This research was carried out as part of the 2019 Summer Program in Undergraduate Research (SPUR) of the MIT Mathematics Department. The authors thank Prof.\,Alex Postnikov for suggesting the project and Christian Gaetz, Prof.\,Ankur Moitra and Prof.\,David Jerison for helpful conversations. \end{document}	arXiv
EECT Home Asymptotic for the perturbed heavy ball system with vanishing damping term June 2017, 6(2): 155-175. doi: 10.3934/eect.2017009 Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction Alex H. Ardila Department of Mathematics, IME-USP, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil Received July 2016 Revised February 2017 Published April 2017 Figure(1) In this paper we study the one-dimensional logarithmic Schrödin-\break ger equation perturbed by an attractive $δ^{\prime}$ -interaction $i{\partial _t}u + \partial _x^2u + {\rm{ }}{\gamma ^\prime }(x)u + u{\mkern 1mu} {\rm{Log\|}}u\|2 = 0,(x,t) \in \mathbb{R} \times \mathbb{R} ,$ where $γ>0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive $δ^{\prime}$-interaction case, the set of the ground state is completely determined. 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Physics And Astronomy (7) Slavic Review (2) American Slavic and East European Review (1) Combinatorics, Probability and Computing (1) Mathematika (1) Transactions of the International Astronomical Union (1) Association for Slavic, East European, and Eurasian Studies (3) Women and Politics Section-APSA (1) Radiolysis of solid-state nitrogen heterocycles provides clues to their abundance in the early solar system Phillip G. Hammer, Ruiqin Yi, Isao Yoda, H. James Cleaves, Michael P. Callahan Journal: International Journal of Astrobiology / Volume 18 / Issue 4 / August 2019 We studied the radiolysis of a wide variety of N-heterocycles, including many of biological importance, and find that the majority are remarkably stable in the solid-state when subjected to large doses of ionizing gamma radiation from a 60Co source. Degradation of N-heterocycles as a function of dose rate and total dose was measured using high-performance liquid chromatography with UV detection. Many N-heterocycles show little degradation when γ-irradiated up to a total dose of ~1 MGy, which approximates hundreds of millions of years' worth of radiation emitted in meteorite parent bodies due to slow radionuclide decay. Extrapolation of these results suggests that these N-heterocyclic compounds would be stable in dry parent bodies over solar system timescales. We suggest that the abundance of these N-heterocycles as measured presently in carbonaceous meteorites is largely reflective of their abundance at the time aqueous alteration stopped in their parent bodies and the absence of certain compounds in present-day samples is either due to the formation mechanisms or degradation which occurred during periods of aqueous alteration or thermal metamorphism. Has Breeding Improved Soybean Competitiveness with Weeds? Devin J. Hammer, David E. Stoltenberg, Jed B. Colquhoun, Shawn P. Conley Journal: Weed Science / Volume 66 / Issue 1 / January 2018 Soybean yield gain over the last century has been attributed to both genetic and agronomic improvements. Recent research has characterized how breeding efforts to improve yield gain have also secondarily impacted agronomic practices such as seeding rate, planting date, and fungicide use. To our knowledge, no research has characterized the relationship between weed–soybean interference and genetic yield gain. Therefore, the objectives of this research were to determine whether newer cultivars would consistently yield higher than older cultivars under increasingly competitive environments, and whether soybean breeding efforts over time have indirectly increased soybean competitiveness. Field research was conducted in 2014, 2015, and 2016 in which 40 maturity group (MG) II soybean cultivars released between 1928 and 2013 were grown season-long with three different densities of volunteer corn (0, 2.8, and 11.2 plants m−2). Soybean seed yield of newer cultivars was higher than older cultivars at each volunteer corn density (P<0.0001). Soybean seed yield was also higher in the weed-free treatment than at low or high volunteer corn seeding rates. However, soybean cultivar release year did not affect late-season volunteer corn shoot dry biomass at either seeding rate of 2.8 or 11.2 seeds m−2. The results indicate that while soybean breeding efforts have increased yield potential over time, they have not increased soybean competitiveness with volunteer corn. These results highlight the importance of other cultural practices such as planting date and crop row spacing for weed suppression in modern soybean production systems. Diagnostic utility of brain activity flow patterns analysis in attention deficit hyperactivity disorder J. Biederman, P. Hammerness, B. Sadeh, Z. Peremen, A. Amit, H. Or-ly, Y. Stern, A. Reches, A. Geva, S. V. Faraone A previous small study suggested that Brain Network Activation (BNA), a novel ERP-based brain network analysis, may have diagnostic utility in attention deficit hyperactivity disorder (ADHD). In this study we examined the diagnostic capability of a new advanced version of the BNA methodology on a larger population of adults with and without ADHD. Subjects were unmedicated right-handed 18- to 55-year-old adults of both sexes with and without a DSM-IV diagnosis of ADHD. We collected EEG while the subjects were performing a response inhibition task (Go/NoGo) and then applied a spatio-temporal Brain Network Activation (BNA) analysis of the EEG data. This analysis produced a display of qualitative measures of brain states (BNA scores) providing information on cortical connectivity. This complex set of scores was then fed into a machine learning algorithm. The BNA analysis of the EEG data recorded during the Go/NoGo task demonstrated a high discriminative capacity between ADHD patients and controls (AUC = 0.92, specificity = 0.95, sensitivity = 0.86 for the Go condition; AUC = 0.84, specificity = 0.91, sensitivity = 0.76 for the NoGo condition). BNA methodology can help differentiate between ADHD and healthy controls based on functional brain connectivity. The data support the utility of the tool to augment clinical examinations by objective evaluation of electrophysiological changes associated with ADHD. Results also support a network-based approach to the study of ADHD. By Lenard A. Adler, Pinky Agarwal, Rehan Ahmed, Jagga Rao Alluri, Fawaz Al-Mufti, Samuel Alperin, Michael Amoashiy, Michael Andary, David J. Anschel, Padmaja Aradhya, Vandana Aspen, Esther Baldinger, Jee Bang, George D. Baquis, John J. Barry, Jason J. S. Barton, Julius Bazan, Amanda R. Bedford, Marlene Behrmann, Lourdes Bello-Espinosa, Ajay Berdia, Alan R. Berger, Mark Beyer, Don C. Bienfang, Kevin M. Biglan, Thomas M. Boes, Paul W. Brazis, Jonathan L. Brisman, Jeffrey A. Brown, Scott E. Brown, Ryan R. Byrne, Rina Caprarella, Casey A. Chamberlain, Wan-Tsu W. Chang, Grace M. Charles, Jasvinder Chawla, David Clark, Todd J. Cohen, Joe Colombo, Howard Crystal, Vladimir Dadashev, Sarita B. Dave, Jean Robert Desrouleaux, Richard L. Doty, Robert Duarte, Jeffrey S. Durmer, Christyn M. Edmundson, Eric R. Eggenberger, Steven Ender, Noam Epstein, Alberto J. Espay, Alan B. Ettinger, Niloofar (Nelly) Faghani, Amtul Farheen, Edward Firouztale, Rod Foroozan, Anne L. Foundas, David Elliot Friedman, Deborah I. Friedman, Steven J. Frucht, Oded Gerber, Tal Gilboa, Martin Gizzi, Teneille G. Gofton, Louis J. Goodrich, Malcolm H. Gottesman, Varda Gross-Tsur, Deepak Grover, David A. Gudis, John J. Halperin, Maxim D. Hammer, Andrew R. Harrison, L. Anne Hayman, Galen V. Henderson, Steven Herskovitz, Caitlin Hoffman, Laryssa A. Huryn, Andres M. Kanner, Gary P. Kaplan, Bashar Katirji, Kenneth R. Kaufman, Annie Killoran, Nina Kirz, Gad E. Klein, Danielle G. Koby, Christopher P. Kogut, W. Curt LaFrance, Patrick J.M. Lavin, Susan W. Law, James L. Levenson, Richard B. Lipton, Glenn Lopate, Daniel J. Luciano, Reema Maindiratta, Robert M. Mallery, Georgios Manousakis, Alan Mazurek, Luis J. Mejico, Dragana Micic, Ali Mokhtarzadeh, Walter J. Molofsky, Heather E. Moss, Mark L. Moster, Manpreet Multani, Siddhartha Nadkarni, George C. Newman, Rolla Nuoman, Paul A. Nyquist, Gaia Donata Oggioni, Odi Oguh, Denis Ostrovskiy, Kristina Y. Pao, Juwen Park, Anastas F. Pass, Victoria S. Pelak, Jeffrey Peterson, John Pile-Spellman, Misha L. Pless, Gregory M. Pontone, Aparna M. Prabhu, Michael T. Pulley, Philip Ragone, Prajwal Rajappa, Venkat Ramani, Sindhu Ramchandren, Ritesh A. Ramdhani, Ramses Ribot, Heidi D. Riney, Diana Rojas-Soto, Michael Ronthal, Daniel M. Rosenbaum, David B. Rosenfield, Durga Roy, Michael J. Ruckenstein, Max C. Rudansky, Eva Sahay, Friedhelm Sandbrink, Jade S. Schiffman, Angela Scicutella, Maroun T. Semaan, Robert C. Sergott, Aashit K. Shah, David M. Shaw, Amit M. Shelat, Claire A. Sheldon, Anant M. Shenoy, Yelizaveta Sher, Jessica A. Shields, Tanya Simuni, Rajpaul Singh, Eric E. Smouha, David Solomon, Mehri Songhorian, Steven A. Sparr, Egilius L. H. Spierings, Eve G. Spratt, Beth Stein, S.H. Subramony, Rosa Ana Tang, Cara Tannenbaum, Hakan Tekeli, Amanda J. Thompson, Michael J. Thorpy, Matthew J. Thurtell, Pedro J. Torrico, Ira M. Turner, Scott Uretsky, Ruth H. Walker, Deborah M. Weisbrot, Michael A. Williams, Jacques Winter, Randall J. Wright, Jay Elliot Yasen, Shicong Ye , G. Bryan Young, Huiying Yu, Ryan J. Zehnder Edited by Alan B. Ettinger, Albert Einstein College of Medicine, New York, Deborah M. Weisbrot, State University of New York, Stony Brook Book: Neurologic Differential Diagnosis Print publication: 17 April 2014, pp xi-xx By Waiel Almoustadi, Brian J. Anderson, David B. Auyong, Michael Avidan, Michael J. Avram, Roland J. Bainton, Jeffrey R. Balser, Juliana Barr, W. Scott Beattie, Manfred Blobner, T. Andrew Bowdle, Walter A. Boyle, Eugene B. Campbell, Laura F. Cavallone, Mario Cibelli, C. Michael Crowder, Ola Dale, M. Frances Davies, Mark Dershwitz, George Despotis, Clifford S. Deutschman, Brian S. Donahue, Marcel E. Durieux, Thomas J. Ebert, Talmage D. Egan, Helge Eilers, E. Wesley Ely, Charles W. Emala, Alex S. Evers, Heidrun Fink, Pierre Foëx, Stuart A. Forman, Helen F. Galley, Josephine M. Garcia-Ferrer, Robert W. Gereau, Tony Gin, David Glick, B. Joseph Guglielmo, Dhanesh K. Gupta, Howard B. Gutstein, Robert G. Hahn, Greg B. Hammer, Brian P. Head, Helen Higham, Laureen Hill, Kirk Hogan, Charles W. Hogue, Christopher G. Hughes, Eric Jacobsohn, Roger A. Johns, Dean R. Jones, Max Kelz, Evan D. Kharasch, Ellen W. King, W. Andrew Kofke, Tom C. Krejcie, Richard M. Langford, H. T. Lee, Isobel Lever, Jerrold H. Levy, J. Lance Lichtor, Larry Lindenbaum, Hung Pin Liu, Geoff Lockwood, Alex Macario, Conan MacDougall, M. B. MacIver, Aman Mahajan, Nándor Marczin, J. A. Jeevendra Martyn, George A. Mashour, Mervyn Maze, Thomas McDowell, Stuart McGrane, Berend Mets, Patrick Meybohm, Charles F. Minto, Jonathan Moss, Mohamed Naguib, Istvan Nagy, Nick Oliver, Paul S. Pagel, Pratik P. Pandharipande, Piyush Patel, Andrew J. Patterson, Robert A. Pearce, Ronald G. Pearl, Misha Perouansky, Kristof Racz, Chinniampalayam Rajamohan, Nilesh Randive, Imre Redai, Stephen Robinson, Richard W. Rosenquist, Carl E. Rosow, Uwe Rudolph, Francis V. Salinas, Robert D. Sanders, Sunita Sastry, Michael Schäfer, Jens Scholz, Thomas W. Schnider, Mark A. Schumacher, John W. Sear, Frédérique S. Servin, Jeffrey H. Silverstein, Tom De Smet, Martin Smith, Joe Henry Steinbach, Markus Steinfath, David F. Stowe, Gary R. Strichartz, Michel M. R. F. Struys, Isao Tsuneyoshi, Robert A. Veselis, Arthur Wallace, Robert P. Walt, David C. Warltier, Nigel R. Webster, Jeanine Wiener-Kronish, Troy Wildes, Paul Wischmeyer, Ling-Gang Wu, Stephen Yang Edited by Alex S. Evers, Washington University School of Medicine, St Louis, Mervyn Maze, University of California, San Francisco, Evan D. Kharasch, Washington University School of Medicine, St Louis Book: Anesthetic Pharmacology Print publication: 10 March 2011, pp viii-xiv By G. David Adamson, Majed Al Hudhud, Baris Ata, Pedro N. Barri, Christopher L. R. Barratt, Elisabet Clua, C. Dechanet, H. Déchaud, Didier Dewailly, Marion Dewailly, David K. Gardner, Linda Hammer Burns, B. Hédon, Wayland Hsiao, Vanessa J. Kay, Gab Kovacs, Robert I. McLachlan, Vicki Nisenblat, Robert J. Norman, W. Ombelet, Edouard Poncelet, Shauna Reinblatt, Anthony J. Rutherford, Peter N. Schlegel, Wendy B. Shelly, F. Shenfield, Joe Leigh Simpson, Anna Smirnova, Seang Lin Tan, George A. Thouas, Geoffrey Trew, P. C. Wong, Cheng Toh Yeong Edited by Gab Kovacs Book: The Subfertility Handbook Published online: 06 December 2010 Print publication: 11 November 2010, pp ix-xii Building-up a database of spectro-photometric standards from the UV to the NIR J. Vernet, F. Kerber, V. Mainieri, T. Rauch, F. Saitta, S. D'Odorico, R. Bohlin, V. Ivanov, C. Lidman, E. Mason, A. Smette, J Walsh, R. Fosbury, P. Goldoni, P. Groot, F. Hammer, L. Kaper, M. Horrobin, P. Kjaergaard-Rasmussen, F. Royer Journal: Proceedings of the International Astronomical Union / Volume 5 / Issue H15 / November 2009 Print publication: November 2009 We present results of a project aimed at establishing a set of 12 spectro-photometric standards over a wide wavelength range from 320 to 2500 nm. Currently no such set of standard stars covering the near-IR is available. Our strategy is to extend the useful range of existing well-established optical flux standards (Oke 1990, Hamuy et al. 1992, 1994) into the near-IR by means of integral field spectroscopy with SINFONI at the VLT combined with state-of-the-art white dwarf stellar atmospheric models (TMAP, Holberg et al. 2008). As a solid reference, we use two primary HST standard white dwarfs GD71 and GD153 and one HST secondary standard BD+17 4708. The data were collected through an ESO "Observatory Programme" over ~40 nights between February 2007 and September 2008. Towards further understanding of the co-morbidity between attention deficit hyperactivity disorder and bipolar disorder: a MRI study of brain volumes J. Biederman, N. Makris, E. M. Valera, M. C. Monuteaux, J. M. Goldstein, S. Buka, D. L. Boriel, S. Bandyopadhyay, D. N. Kennedy, V. S. Caviness, G. Bush, M. Aleardi, P. Hammerness, S. V. Faraone, L. J. Seidman Published online by Cambridge University Press: 15 October 2007, pp. 1045-1056 Although attention deficit hyperactivity disorder (ADHD) and bipolar disorder (BPD) co-occur frequently and represent a particularly morbid clinical form of both disorders, neuroimaging research addressing this co-morbidity is scarce. Our aim was to evaluate the morphometric magnetic resonance imaging (MRI) underpinnings of the co-morbidity of ADHD with BPD, testing the hypothesis that subjects with this co-morbidity would have neuroanatomical correlates of both disorders. Morphometric MRI findings were compared between 31 adults with ADHD and BPD and with those of 18 with BPD, 26 with ADHD, and 23 healthy controls. The volumes (cm3) of our regions of interest (ROIs) were estimated as a function of ADHD status, BPD status, age, sex, and omnibus brain volume using linear regression models. When BPD was associated with a significantly smaller orbital prefrontal cortex and larger right thalamus, this pattern was found in co-morbid subjects with ADHD plus BPD. Likewise, when ADHD was associated with significantly less neocortical gray matter, less overall frontal lobe and superior prefrontal cortex volumes, a smaller right anterior cingulate cortex and less cerebellar gray matter, so did co-morbid ADHD plus BPD subjects. Our results support the hypothesis that ADHD and BPD independently contribute to volumetric alterations of selective and distinct brain structures. In the co-morbid state of ADHD plus BPD, the profile of brain volumetric abnormalities consists of structures that are altered in both disorders individually. Attention to co-morbidity is necessary to help clarify the heterogeneous neuroanatomy of both BPD and ADHD. The first galaxies: instrument requirements and concept study for OWL J.-G. Cuby, J.-P. Kneib, F. Hammer, E. Prieto, M. Marteaud, P. Vola, P. Jagourel, P.-E. Blanc, T. Fusco Journal: Proceedings of the International Astronomical Union / Volume 1 / Issue S232 / November 2005 A highlight science case for the European ELT is: First light - The First Galaxies and the Ionization State of the Early Universe. It aims at understanding the formation and evolution of the first sources of light at the end of the Dark Ages and of the re-ionization of the Universe. The corresponding instrument requirements are: a few tens of integral field units with spatial sampling $\sim$20mas and individual fields of ${\sim}1''$ over a wide field of view of $5' \times 5'$ or larger. Multi-Object Adaptive Optics is required to locally provide significant image quality enhancement. Spectroscopic observations are required in the near IR domain with a spectral resolution of a few 1000. MOMFIS is a preliminary instrument concept designed for OWL around this science case. The instrument concept and preliminary design are presented. Development efforts are estimated, as well as development risks and required R&D activities. Adaptive optics concept for multi-object 3D spectroscopy on ELTs B. Neichel, T. Fusco, M. Puech, J-M. Conan, M. Le Louarn, E. Gendron, F. Hammer, G. Rousset, P. Jagourel, P. Bouchet In this paper, we present a first comparison of different Adaptive Optics (AO) concepts to reach a given scientific specification for 3D spectroscopy on Extremely Large Telescope (ELT). We consider that a range of 30%–50% of Ensquarred Energy (EE) in H band (1.65$\mu$m) and in an aperture size from 25 to 100mas is representative of the scientific requirements. From these preliminary choices, different kinds of AO concepts are investigated: Ground Layer Adaptive Optics (GLAO), Multi-Object AO (MOAO) and Laser Guide Stars AO (LGS). Using Fourier based simulations we study the performance of these AO systems depending on the telescope diameter. Cosmological star formation history J. Bouvier, J.-P. Zahn, F. Hammer Journal: European Astronomical Society Publications Series / Volume 3 / 2002 I review the recent instrumental progresses which allow to directly observe galaxy properties at the earliest epochs of the Universe. This leads to the derivation of fundamental quantities, such as the epoch of massive galaxy formation or the universal history of heavy element formation. Star formation history shows a gradual decline since the last 8-9 Gyr. The bulk of present-day stellar mass and metal content was formed at redshifts 0.5 to ~3, which is consistent with hierarchical scenarii of galaxy formation. The numerous uncertainties related to these determinations are emphasized, including those linked to stellar physics. Commission 10: Solar Activity: (Activite Solaire) G. Ai, A. Benz, K. P. Dere, O. Engvold, N. Gopalswamy, R. Hammer, A. Hood, B. V. Jackson, I. Kim, P. C. Marten, G. Poletto, J. P. Rozelot, A. J. Sanchez, K. Shibata, L. van Driel-Geztelyi Journal: Transactions of the International Astronomical Union / Volume 24 / Issue 1 / 2000 The Sun's activity has been evolving in the ascending phase of Solar Cycle 23 since 1996. Similarly, the research on solar activity is also in the ascending phase of a new active period. Numerous new results have been obtained from a large amount of space and ground observations covering a wide spectral range. In particular, observations with YOHKOH, SOHO, and TRACE have revealed a multitude of phenomena and processes in the solar atmosphere which provide us a new picture of the Sun. The Effect of Ion-Bombardment on the Formation of Voids During Deposition of a-Ge:H F. Origo, P. Hammer, D. Comedi, I. Chambouleyron Journal: MRS Online Proceedings Library Archive / Volume 507 / 1998 Published online by Cambridge University Press: 10 February 2011, 477 The role of substrate ion bombardment on the structural and H bonding properties of hydrogenated amorphous germanium (a-Ge:H) films was studied by infrared (ir) spectroscopy. A Kaufman type ion source was used to produce an Ar1 beam directed towards a Ge target for a- Ge:H ion beam sputtering deposition in a H2-containing vacuum chamber. A low energy (100 eV) H2 ++Ar+ beam obtained from an additional ion source was allowed to impinge directly on the substrate during film growth at various beam currents. It was found that substrate bombardment by 100 eV ions favors the formation of voids, as deduced from the increasing contribution of the surface-like Ge-H stretching mode to the ir spectrum with increasing ion current. The void density was reduced below the ir detection limit by totally removing the ion beam on the substrate while keeping all other parameters fixed. For this condition, we observe no or very small surface-like contributions to the ir spectra, irrespective of substrate temperature (25-260°C) or growth rate used. A narrowing of the infrared Ge-H stretching mode peak is observed with increasing deposition temperature, indicating a concomitant tendency towards a more ordered structure. Edited by Paul E. Micevych, University of California, Los Angeles, Ronald P. Hammer, Jr, Tufts University, Massachusetts Book: Neurobiological Effects of Sex Steroid Hormones Published online: 15 October 2009 Print publication: 26 May 1995, pp ix-x 6 - Sex steroid regulation of hypothalamic opioid function By Ronald P. Hammer, Sun Cheung Print publication: 26 May 1995, pp 143-159 The discovery that the hypothalamus is responsible for controlling both reproductive hormones and behavior suggested various mechanisms by which hormonal and behavioral cycles are inexorably linked, and even co-regulated. While our knowledge about this process has grown dramatically, our understanding of the essential control circuits that operate during normal reproduction, or fail in abnormal functioning, is still limited. Various neurotransmitter candidates have been proposed as essential elements of the systems that regulate reproduction. However, few are involved in so many aspects of reproduction as are the opioid peptides, which play a critical or supporting role in (a) controlling hormonal cycling in females (Akabori and Barraclough 1986; Kalra 1985; Wiesner et al. 1984), (b) regulating reproductive behavior in males (Hughes et al. 1988; Matuszewich and Dornan 1992; Myers and Baum 1979) and females (Pfaus and Pfaff 1992; Sirinathsinghji 1986; Wiesner and Moss 1986a), and even (c) modulating mesolimbic dopamine release mediated by reinforcing sexually relevant olfactory stimuli (Mitchell and Gratton 1991). Gonadal steroid regulation of hypothalamic opioids represents an important feedback system by which to control reproduction. Hypothalamic (opioid) circuits regulate hormonal releasing hormones that control pituitary secretion. This regulation in turn affects gonadal steroid hormones, which act centrally to alter opioid function and facilitate reproductive behavior. Since such feedback is vitally important for the regulation of hormonal and behavioral events during the estrous cycle, most of this discussion will be limited to opioid action in females. Many of the experiments that we will describe utilized models of hormone manipulation to investigate natural regulation of hypothalamic opioid systems in animals, primarily rodents. Print publication: 26 May 1995, pp v-viii Print publication: 26 May 1995, pp xiii-xvi Many books are subject to the fundamental questions "Why this topic?" and "Why now?" Scientific texts are perhaps most susceptible because they often present similar topics. As a partial answer to these questions, we paraphrase P. B. Medawar in his Advise to a Young Scientist: We have tried to prepare the kind of book that we ourselves would like to read and have as a reference. In recent years, the field of reproductive neuroendocrinology has experienced a renaissance brought about by the application of cellular and molecular biological techniques. We have made significant progress in understanding the mechanisms that underlie central nervous system control of reproductive behavior. This progress has been well documented at various meetings and in individual papers. We felt it was necessary, therefore, to offer a collection of essays by some of those who have contributed to this renaissance. We hasten to add that the chapters in this volume do not necessarily reflect all of the vital issues of behavioral neuroendocrinology. Rather, they represent brief reviews by and current data from a number of productive scientists in this field. Because of a limitation of space, several important topics are not discussed or are only briefly presented in this volume. These include the spinal nucleus of the bulbocavernosus system, cell membrane steroid receptors, interactions of steroids with γ-aminobutyric acid receptors, the songbird neural circuitry, as well as the insect and amphibian models of reproduction and metamorphosis. Each of these models has proved to be extremely useful for studying the effects of sex steroid hormones on the nervous system. Part II - Sex steroid interactions with specific neurochemical circuits Print publication: 26 May 1995, pp xi-xii	CommonCrawl
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Further Evidence of Linkage at the tva and tvc Loci in the Layer Lines and a Possibility of Polyallelism at the tvc Locus Ghosh, A.K.;Pani, P.K. 601 https://doi.org/10.5713/ajas.2005.601 PDF KSCI Three lines of White Leghorn (WL) chickens (IWJ, IWG and IWC) maintained at Central Avian Research Institute, Izatnagar (UP), were used for chorioallantoic membrane (CAM) and liver tumour (LT) assay. Eleven-day-old embryos of each line were partitioned into three groups and inoculated with 0.2 ml of subgroup A, subgroup C and an equal mixture of subgroup A and C Rous sarcoma virus (RSV). Subgroup virus receptor on the cell surface membrane for subgroup A is coded for by tumour virus a (tva) locus and for subgroup C by tumour virus c (tvc) locus. The random association of the genes at the tva and tvc loci in IWJ and IWC line was assessed and the $x^2$-values for phenotypic classes were found to be significant, indicating the linkage between the tva and tvc loci. The linkage value was estimated to be 0.09 on pooled sex and pooled line basis. On the basis of four subclass tumour phenotypes a 4-allele model was proposed for tva locus having $a^{s1}$, $a^{s2}$, $a^{r1}$ and $a^{r2}$ alleles and the frequencies were calculated as 0.47, 0.13, 0.13 and 0.27 for IWJ line, 0.31, 0.33, 0.14 and 0.22 for IWG line and 0.44, 0.11, 0.21 and 0.24 for IWC line, respectively. Similarly, for tvc locus the frequencies of four alleles i.e. $c^{s1}$, $c^{s2}$, $c^{r1}$ and $c^{r2}$ were calculated as 0.42, 0.20, 0.21 and 0.17 for IWJ line, 0.42, 0.17, 0.27 and 0.14 for IWG line and 0.30, 0.21, 0.16 and 0.33 for IWC line, respectively. The $x^2$-values for all classes of observations were not significant (p>0.05), indicating a good fit to the 4-allele model for the occurrence of 4-subclass tumour phenotypes for tva and tvc loci. On the basis of the 2-allele model both tva and tvc locus carries three genotypes each. But, on the basis of the 4-allele model tva and tvc locus carries 10 genotypes each. The interaction between A-resistance and C-resistance (both CAM and LT death) was ascertained by taking the 10 genotypes of tva locus and 3 genotypes of tvc locus by pooling the lines and partitioning the observations into 3 classes. The $x^2$-values for the genotypic classes of CAM (-) LT (+) and CAM (-) LT (-) phenotypes to mixed virus (A+C) infection were found to be highly significant (p<0.01), indicating increased resistance, which indicates the joint segregation of $a^r$ and $c^r$ genes, suggesting the existence of close linkage between the tva and tvc loci. Therefore, an indirect selection approach using subgroup C viruses can be employed to generate stocks resistant to subgroup A LLV, obviating contamination with the most common agent causing LL in field condition. Modeling the Productivity of a Breeding Sheep Flock for Different Production Systems Kamalzadeh, A. 606 Individual production traits, such as reproduction and mortality rates, are partial measures, but may be used to evaluate the performance of different systems by comparing the rate of flock growth and potential offtake. The productivity of two existing sheep production systems, one extensive, one intensive, was compared with an alternative semi-intensive system. The future flock sizes, offtakes and structures were predicted based on the age structure of the flock and age-specific reproduction, mortality and growth rates. The measurements were illustrated with reference to growth of a sheep flock of different age and sex categories. The flock was in a socalled dynamic situation. During the dry period, the digestible organic matter intake of the animals in the intensive system and both extensive and semi-extensive systems was 36 and 20.1 g kg$^{-0.75}$ d$^{-1}$, respectively. During the cold period, the digestible organic matter intake of the animals in extensive, intensive and semi-extensive systems was 34, 34.5 and 41 g kg$^{-0.75}$ d$^{-1}$, respectively. During the dry period, the animals in the both extensive and semi-intensive systems lost in body weight at a rate of 19 g per day, but the rate of gain in body weight of the animals in intensive system was 57 g per day. During the cold period, the animals in extensive, intensive and semiintensive systems gained in body weight at rates of 56, 67 and 97 g per day, respectively. The higher gain of animals during the cold period in the semi-intensive system was related to a sustained higher intake of low-quality roughage and more efficient use of the available feed. Compared to the intensive system, the annual concentrate input of the semi-intensive system was about 48% lower for each livestock unit. The productivity of the semi-intensive system was higher than that of the extensive system. Genetic Contribution of Indigenous Yakutian Cattle to Two Hybrid Populations, Revealed by Microsatellite Variation Li, M.H.;Nogovitsina, E.;Ivanova, Z.;Erhardt, G.;Vilkki, J.;Popov, R.;Ammosov, I.;Kiselyova, T.;Kantanen, J. 613 Indigenous Yakutian cattle' adaptation to the hardest subarctic conditions makes them a valuable genetic resource for cattle breeding in the Siberian area. Since early last century, crossbreeding between native Yakutian cattle and imported Simmental and Kholmogory breeds has been widely adopted. In this study, variations at 22 polymorphic microsatellite loci in 5 populations of Yakutian, Kholmogory, Simmental, Yakutian-Kholmogory and Yakutian-Simmental cattle were analysed to estimate the genetic contribution of Yakutian cattle to the two hybrid populations. Three statistical approaches were used: the weighted least-squares (WLS) method which considers all allele frequencies; a recently developed implementation of a Markov chain Monte Carlo (MCMC) method called likelihood-based estimation of admixture (LEA); and a model-based Bayesian admixture analysis method (STRUCTURE). At population-level admixture analyses, the estimate based on the LEA was consistent with that obtained by the WLS method. Both methods showed that the genetic contribution of the indigenous Yakutian cattle in Yakutian-Kholmogory was small (9.6% by the LEA and 14.2% by the WLS method). In the Yakutian-Simmental population, the genetic contribution of the indigenous Yakutian cattle was considerably higher (62.8% by the LEA and 56.9% by the WLS method). Individual-level admixture analyses using STRUCTURE proved to be more informative than the multidimensional scaling analysis (MDSA) based on individual-based genetic distances. Of the 9 Yakutian-Simmental animals studied, 8 showed admixed origin, whereas of the 14 studied Yakutian-Kholmogory animals only 2 showed Yakutian ancestry (>5%). The mean posterior distributions of individual admixture coefficient (q) varied greatly among the samples in both hybrid populations. This study revealed a minor existing contribution of the Yakutian cattle in the Yakutian-Kholmogory hybrid population, but in the Yakutian-Simmental hybrid population, a major genetic contribution of the Yakutian cattle was seen. The results reflect the different crossbreeding patterns used in the development of the two hybrid populations. Additionally, molecular evidence for differences among individual admixture proportions was seen in both hybrid populations, resulting from the stochastic process in crossing over generations. Correlations of Genic Heterozygosity and Variances with Heterosis in a Pig Population Revealed by Microsatellite DNA Marker Zhang, J.H.;Xiong, Y.Z.;Deng, C.Y. 620 Correlation of microsatellite heterozygosity with performance or heterosis was reported in wild animal populations and domestic animal populations, but the correlation with heterosis in a crossbreeding F$_1$ pig population remained uncertain. To explore this, we had random selected and mated Yorkshire${\times}$Meishan (F, n = 82) and their reciprocal (G, n = 47) to F$_1$, and used the two straightbreds as control groups (Yorkshire = 34, Meishan = 55), and observed the heterosis of birth weight (BWT), average daily gain (ADG) and feed and meat ratio (FMR). Two Kinds of measurement-individual heterozygosity (IH) and individual mean d$^2$ (lg value, ID) were used as index of heterozygosity and variance from 39 microsatellite marker loci to perform univariate regression analysis against heterosis. We detected significant correlation of IH with BWT in all of F$_1$ (F+G) and in F. We observed significant correlation of ID with ADG in all of F$_1$ (F+G), and with FMR in all of F$_1$ (F+G) and in F. There was significant maternal effect on heterosis, which was indicated by significant difference of means and distribution of heterosis between F and G. This difference was consistent with distributions of IH and ID, and with difference of means in F and G. From this study, it would be suggested that the two kinds of genetic index could be used to explore the genetic basis of heterosis in crossbreeding populations but could not determine which is better. Inducible Nitric Oxide Synthase Expression and Luteal Cell DNA Fragmentation of Porcine Cyclic Corpora Lutea Tao, Yong;Fu, Zhuo;Xia, Guoliang;Lei, Lei;Chen, Xiufen;Yang, Jie 626 Nitric oxide (NO) derived from inducible nitric oxide synthase (iNOS) is involved in cell apoptosis, which contributes to luteal regression and luteolysis in some species. In large domestic animals, no direct evidence for the relationship between NO and cell apoptosis in the process of corpus luteum regression is reported. The present study was conducted to investigate the localization of iNOS on porcine corpora lutea (CL) during the oestrus cycle and its relation to cell DNA fragmentation and CL regression. According to morphology, four luteal phases throughout the estrous cycle were defined as CL1, CL2, CL3 and CL4. By isoform-specific antibody against iNOS, the immunochemial staining was determined. Luteal cell DNA fragmentation was determined by flow cytometry. The results showed that no positive staining for iNOS was in CL1 and that iNOS was produced but limited to the periphery of CL2, while in the CL3, the spreading immunochemical staining was found inside the CL. No iNOS positive staining was detected in CL4. Meanwhile, DNA fragmentation increased dramatically when CL developed from CL2 to CL3 (p<0.05). In CL4, higher proportion of luteal cells still had fragmented DNA than that of luteal cells from CL1 or CL2 (p<0.05). These results indicate that iNOS expression is closely related to luteal cell apoptosis and then to luteal regression. Effect of Removal of Follicles through Repeated Transvaginal Follicle Aspiration on Subsequent Follicular Populations in Murrah Buffalo (Bubalus bubalis Akshey, Y.S.;Palta, P.;Manik, R.S.;Vivekananad, Vivekananad;Chauhan, M.S. 632 This study was conducted to investigate the effects of removal of all ovarian follicles through repeated transvaginal follicle aspiration (TVFA) on the subsequent follicular populations in buffaloes. This information is crucial for determining the optimum time interval between successive aspirations for recovering oocytes from live buffaloes through Transvaginal Oocyte Retrieval (TVOR). The oestrus of cycling buffaloes (n=5) were synchronized by a single PGF injection schedule. All the ovarian follicles were removed once every 7 days for 6 weeks through TVFA, starting from Day 7 of the oestrous cycle (Day 0 = day of oestrus). The number and size of individual ovarian follicles was recorded at Day 3 and Day 5 (Day 0 = day of TVFA) through transrectal ultrasonography. The follicles were classified on the basis of their diameter as small (3-5 mm), medium (6-9 mm) and large ($\geq$10 mm). There was no difference in the number of small and medium follicles, and the number of total follicles between Day 3 and Day 5. However, the number of large follicles was significantly higher (p<0.05) at Day 5 than that at Day 3. There was a significant (p<0.05) decrease in the proportion of small follicles and an increase (p<0.05) in the proportion of large follicles from Day 3 to Day 5, with no change in the proportion of medium follicles. The number of total follicles at Day 3 or Day 5 did not differ during the 6 TVFA sessions. It can be concluded that an interval of 3 days is more suitable than that of 5 days between successive aspirations for recovering oocytes through TVOR in a twice weekly schedule and that repeated removal of follicles through TVFA does not adversely affect the number of total follicles 3 or 5 days after TVFA. Influence of Varying Levels of Dietary Undegraded Intake Protein Intake on Nutrient Intake, Body Weight Change and Reproductive Parameters in Postpartum Awassi Ewes Haddad, S.G.;Kridli, R.T.;Al-Wadi, D.M. 637 The objective of this study was to evaluate the effect of dietary undegradable protein (UP) level on body weight change, nutrient intake, milk production and postpartum reproductive performance of Awassi ewes. Twenty-seven multiparous Awassi ewes (initial body weight = 53.3${\pm}$1.6 kg) were randomly assigned to three dietary treatments (9 ewes/treatment) for 62 days using a completely randomized design. Experimental diets were isonitrogenous (15.5% CP), isocaloric, and were formulated to contain 17.9 (LUP), 27.1 (MUP), and 34.0% (HUP) of the dietary CP as UP. On day 10${\pm}$3 (day 0 = parturition) ewes were housed in individual pens for 5 weeks. Feed offered and refused was recorded daily. At the end of this period, animals were removed from their pens and combined into 3 separate groups (LUP, MUP and HUP). One fertile, harnessed ram was allowed with each group for 34 days. Rams were rotated every 2 days among the three groups. Each group was offered the corresponding experimental diet. Organic matter, CP, UP and metabolizable energy intakes were higher (p<0.05) for ewes fed the HUP diet compared with ewes fed the LUP and MUP diets. Ewes fed the HUP diet gained more (p<0.05) weight compared with ewes fed the MUP diet (7.3 vs. 2.1 kg), while ewes fed the LUP diet lost an average of 2.1 kg. Pregnancy rate of ewes fed the HUP diet was 100%, compared with 66 and 33% for ewes fed the MUP and LUP diets, respectively. Lambing rate was greater (p<0.05) for ewes fed HUP (8/9) diet compared with ewes fed the MUP (4/9) and LUP (3/9) diets. These results indicate that Awassi ewes receiving adequate dietary UP level consume more feed and are capable of returning to estrus shortly after parturition and are capable of producing two lamb crops per year. Effect of Sample Preparation on Prediction of Fermentation Quality of Maize Silages by Near Infrared Reflectance Spectroscopy Park, H.S.;Lee, J.K.;Fike, J.H.;Kim, D.A.;Ko, M.S.;Ha, Jong Kyu 643 Near infrared reflectance spectroscopy (NIRS) has become increasingly used as a rapid, accurate method of evaluating some chemical constituents in cereal grains and forages. If samples could be analyzed without drying and grinding, then sample preparation time and costs may be reduced. This study was conducted to develop robust NIRS equations to predict fermentation quality of corn (Zea mays) silage and to select acceptable sample preparation methods for prediction of fermentation products in corn silage by NIRS. Prior to analysis, samples (n = 112) were either oven-dried and ground (OD), frozen in liquid nitrogen and ground (LN) and intact fresh (IF). Samples were scanned from 400 to 2,500 nm with an NIRS 6,500 monochromator. The samples were divided into calibration and validation sets. The spectral data were regressed on a range of dry matter (DM), pH and short chain organic acids using modified multivariate partial least squares (MPLS) analysis that used first and second order derivatives. All chemical analyses were conducted with fresh samples. From these treatments, calibration equations were developed successfully for concentrations of all constituents except butyric acid. Prediction accuracy, represented by standard error of prediction (SEP) and $R^2_{v}$ (variance accounted for in validation set), was slightly better with the LN treatment ($R^2$ 0.75-0.90) than for OD ($R^2$ 0.43-0.81) or IF ($R^2$ 0.62-0.79) treatments. Fermentation characteristics could be successfully predicted by NIRS analysis either with dry or fresh silage. Although statistical results for the OD and IF treatments were the lower than those of LN treatment, intact fresh (IF) treatment may be acceptable when processing is costly or when possible component alterations are expected. Effect of Lactobacillus buchneri 40788 and Buffered Propionic Acid on Preservation and Nutritive Value of Alfalfa and Timothy High-moisture Hay Baah, J.;McAllister, T.A.;Bos, L.;Herk, F. Van;Charley, R.C. 649 The effects of Lactobacillus buchneri 40788 and buffered propionic acid on preservation, intake and digestibility of alfalfa (Medicago sativa) and timothy (Phleum pratense) hay were investigated. During baling, forages were treated with L. buchneri 40788 (1.2${\times}$10$^6$ CFU/g) as a liquid (LLB) or as a granular preparation (GLB), with buffered propionic acid (10 mL/kg, BPA), or left untreated (control). Triplicate 500 kg round bales of each treatment were put up at two moisture levels for each forage: 17%${\pm}$0.33% and 20%${\pm}$0.30% for timothy and 17%${\pm}$0.20% and 19%${\pm}$0.27% for alfalfa (mean${\pm}$SD). Bales were sampled for chemical and microbiological analyses after 0, 30 and 60 d of storage. Compared to controls, all preservatives reduced (p<0.05) heating of both forages at all moisture levels with the exception of alfalfa baled at 19% moisture. After 60 d of storage, GLB reduced (p<0.05) moulds in 17% timothy hay as compared to other treatments, but at 20% moisture, moulds were reduced in LLB- and BPA-treated timothy as compared to controls. In alfalfa at 17% moisture, total bacteria were lower (p<0.05) in GLB-treated bales than LLB or control bales, but yeast and total bacteria were only reduced in BPA-treated alfalfa at 19% moisture. In situ DM disappearance of timothy (both moisture levels) and alfalfa (19% moisture level) increased (p<0.05) with LLB treatment compared to control. Digestibility of both forages did not differ (p>0.05) among treatments, however, voluntary DM intake of LLB-treated timothy (1.32 kg/d) was 22.3% higher (p<0.05) than control, and 14.1% higher than BPA-treated timothy. Treating timothy and alfalfa hay with L. buchneri 40788 or buffered propionic acid may improve the nutritive value of the hay when baled at 17 to 20% moisture. Effect of Pre-partum Feeding of Crossbred Cows on Growth Performance, Metabolic Profile and Immune Status of Calves Panigrahi, B.;Pandey, H.N.;Pattanaik, A.K. 661 The effects of pre-partum feeding management in terms of birth weight, growth, metabolic profile and immunity of calves were studied using 24 crossbred (Bos taurus${\times}$Bos indicus) cows, divided into three equal groups. The dietary treatments included feeding of either 3.0 kg concentrate/head/d throughout the 60 d pre-partum (T$_1$), or 3.0 kg concentrate during 60-22 d pre-partum and thereafter at an increased allowance at 0.25 kg/d during the next 21 d till it reached 1% of live weight (T$_2$). The third group of cows was fed similar to T$_2$, except that the concentrate feeding during 60-22 d pre-partum was reduced to 2.0 kg (T$_3$). All the groups had access to ad libitum green fodder throughout. The results revealed that the mean daily dry matter (DM) intake by the cows was similar (p>0.05) among the three groups during the 60 days of the pre-partum but T2 animals tended to gain more live weight (41.25 kg) than T$_1$ (38.12 kg) and T$_3$ (36.25 kg). The body condition score of the cows did not change appreciably over the experimental period. The mean birth weight of the calves was 24.00${\pm}$1.10, 24.63${\pm}$1.17 and 23.25${\pm}$1.19 kg for the three groups, respectively, with the corresponding average daily gain of 154.2, 155.0 and 169.7 g during the subsequent 60 days; both these parameters did not vary significantly ascribable to prepartum feeding regimens of their dams. The total immunoglobulin (Ig) concentration in the colostrum was 6.31${\pm}$0.34, 5.80${\pm}$0.21 and 6.13${\pm}$0.30 g/dl for the three groups, respectively, showing no influence of dietary treatments. The mean serum Ig levels (T$_1$ 2.10${\pm}$0.09, T$_2$ 2.05${\pm}$0.09 and T$_3$ 2.10${\pm}$0.12 g/dl) of calves at 5 d of age were similar among the dietary groups as was the case with various serum biochemical constituents. It is concluded that the variations in pre-partum dietary management elicited no significant influence on the calf performance including the immune status. Evaluation on Nutritional Value of Field Crickets as a Poultry Feedstuff Wang, Dun;Zhai, Shao Wei;Zhang, Chuan Xi;Bai, Yao Yu;An, Shi Heng;Xu, Ying Nan 667 The proximate analysis, amino acid content and true amino acid digestibility and TMEn for poultry of adult Field crickets Gryllus testaceus Walker, were investigated. The insect was also used as partial replacement of protein supplements in the broiler diet on an equal CP percentage and TMEn basis. The results indicated that the adult insect contained: crude protein 58.3%; fat 10.3%, chitin 8.7% and ash 2.96% on dry matter basis, respectively. The total amounts of methionine, cystine and lysine in the Field crickets were 1.93%, 1.01% and 4.79%, respectively, and their true digestibility coefficients, determined in cecectomized roosters, were 94.1%, 85% and 96%, respectively. The TMEn of this insect meal was 2,960 kcal/kg determined in cecectomized roosters. When cornsoybean meal diets were formulated on an equal CP percentage and TMEn basis, up to 15% Field cricket could replace control diet without any adverse affects on broiler weight gain, feed intake or gain:feed ratio from 8 to 20 d posthatching. Evaluation of the Genetic Diversities and the Nutritional Values of the Tra (Pangasius hypophthalmus) and the Basa (Pangasius bocourti) Catfish Cultivated in the Mekong River Delta of Vietnam Men, L.T.;Thanh, V.C.;Hirata, Y.;Yamasaki, S. 671 A total of 50 individual catfish, the Tra (Pangasius hypophthalmus) cultivated in either floating cages (Tra-c) or in ponds (Tra-p) and the Basa (Pangasius bocourti) raised in three floating cages, were collected in two of the Mekong Delta provinces. The caudal fin of each individual fish was used for protein electrophoresis employing the SDS-PAGE method. The one fillet sides were used as a representative sample to determine the dry matter (DM), crude protein (CP), ether extract (EE) and amino acids (AAs). The catfish oil was extracted from the belly fats, and the fatty acid (FA) composition was analyzed. There were 21 bands of the Tra and the Basa. Protein bands of the two varieties were 28.6-33.3% polymorphic, while polymorphic individuals of the Tra ranged from 80.0 to 100.0%, and the Basa was 90.0% polymorphic. The phenotypic diversity (Ho) of the Tra ranged from 1.71 to 1.80, while the Basa ranged as high as 2.14%. Diversity values (H$_{EP}$) for genetic diversity markers were equal in the Tra and the Basa. The sum of the effective number of alleles (SENA) of both varieties ranged from 3.40 to 3.83 for the Basa and the Tra, respectively. The lower values of Ho and SENA, as compared with those of the fresh water prawn (Macrobrachium equidens) in the area, would suggest that the species with the low values will become extinct due to inbreeding; the gene pools of each observed population were below a suitable threshold. Many of the differences in the nutritional values of the Tra-c, the Tra-p and the Basa were measured; their nutrient values were comparable to fishmeal or fish oil. Most of the DM, CP, and EE were higher in the Tra, especially in the Tra-c. The essential AA content, especially that of lysine, was highest in the Tra-c, next highest in the Tra-p, and lowest in the Basa. Therefore, the amino acid patterns were closer to the ideal patterns in the same sequences. In contrast, the essential FAs were concentrated in the Basa fish oil. It was found that suitable selection of parents for seed production is required to avoid inbreeding. Catfish may be valuable sources of nutrition for both humans and animals, and the differences in their nutritional values by variety and/or management must be taken into account. Growth Performance, Humoral Immune Response and Carcass Characteristics of Broiler Chickens Fed Alkali Processed Karanj Cake Incorporated Diet Supplemented with Methionine Panda, K.;Sastry, V.R.B.;Mandal, A.B. 677 A study was conducted to see the effect of dietary incorporation of alkali (1.5% NaOH, w/w) processed solvent extracted karanj cake (SKC) supplemented with methionine on growth performance, humoral immune response and carcass characteristics of broiler chickens from 0 to 8 weeks of age. One hundred and twenty, day- old broiler chicks were wing banded, vaccinated against Marek' disease and distributed in a completely randomized design (CRD) into 3 groups of 40 chicks each, which was further replicated to 4 and fed on diet containing soybean meal and those of test groups were fed diets containing alkali (1.5% NaOH) treated SKC partially replacing soybean meal nitrogen of reference diet (12.5%) without or with supplementation of methionine (0.2%). Individual body weight of chicks and replicate-wise feed intakes were recorded at weekly intervals throughout the experimental period. Feed consumption from 1 to 14, 28, 42 and 56 d of age was recorded for each replicate and feed conversion efficiency (weight gain/feed intake) for the respective period was calculated. Mortality was monitored on daily basis. On 28$^{th}$ day of experimental feeding, two birds of each replicate in each dietary group (8 birds/diet) were inoculated with 0.1 ml of a 1.0% suspension of sheep red blood cells (SRBC) and the antibody titre (log 2) was measured after 5 days by the microtitre haemmagglutination procedure. After 42 days of experimental feeding, a retention study of 4 days (43-47 d) duration was conducted on all birds to determine the retention of various nutrients such as DM, N, Ca, P and GE. On 43$^{rd}$ day of experimental feeding, one representative bird from each replicate of a dietary treatment (4/dietary group) was sacrificed, after fasting for two hours with free access to water, through cervical dislocation to observe the weight of dressed carcass, primal cuts (breast, thigh, drumstick, back, neck and wing), giblet (liver, heart and gizzard), abdominal fat and digestive organs. The body weight gain of chicks fed reference diet and those fed diet incorporated with NaOH treated SKC (12.5% replacement) with or without methionine supplementation was comparable during 0 to 4 weeks of age. However, dietary incorporation of alkali processed SKC replacing 12.5% nitrogen moiety of soybean meal resulted in growth retardation, subsequently as evidenced by significantly (p<0.05) lowered body weight gain during 0 to 6 weeks of age in birds fed diet incorporated with alkali processed SKC at 6.43% without methionine as compared to those supplemented with methionine or reference diet. Dietary incorporation of alkali (1.5% NaOH) processed SKC replacing 12.5% of soybean meal nitrogen in the diet of broiler chickens had no adverse effect on feed conversion ratio during all the weeks of experimental feeding. The humoral immune response (HIR) as measured by the antibody titre in response to SRBC inoculation was comparable among all the dietary groups. No significant difference in the intake and retention of DM, N, Ca, P or GE was noted among the chicks fed reference and alkali processed SKC incorporated diets with or without methionine supplementation. None of the carcass traits varied significantly due to dietary variations, except the percent weight of liver and giblet. The percent liver weight was significantly (p<0.05) higher in the birds fed diet incorporated with alkali processed SKC as compared to that in other two groups. Thus solvent extracted karanj cake could be incorporated after alkali (1.5% NaOH, w/w) processing at an enhanced level of 6.43%, replacing 12.5% of soybean meal nitrogen, in the broiler diets up to 4 weeks of age, beyond which the observed growth depression on this diet could be alleviated by 0.2% methionine supplementation. Age-dependent Changes of Differential Gene Expression Profile in Backfat Tissue between Hybrids and Parents in Pigs Ren, ZH.Q.;Xiong, Yuanzhu;Deng, CH.Y.;Zuo, B.;Liu, Y.G.;Lei, M.G. 682 Large White, an introduced European pig breed, and Meishan, a Chinese indigenous pig breed, were hybridized directly and reciprocally and a total of 260 pigs, including purebreds, Large White and Meishan, and their hybrids, White${\times}$Meishan (LM) and Meishan${\times}$Large White (ML) pigs, were bred in our laboratory. The mRNA differential display PCR (DD-PCR) was used to detect the age-dependent changes of differential gene expression in backfat tissue between hybrids and parents. Some measures were taken to reduce the false positives in our experiment. Among the total of 2,686 bands obtained, 1,952 bands (about 72.67%) were reproducible and eight patterns (fifteen kinds) of gene expression were observed. The percentage of differentially expressed genes between hybrids and parents is 56.86% at the age of four months and 57.71% at the age of six months. This indicated that the differences of gene expression between hybrids and their parents were very obvious. U-test was used to compare the patterns of gene expression between the age of four and six months, and results showed that bands occurring in only one hybrid and bands displayed in one hybrid and one parent were significantly different at p<0.05, and bands visualized in only two hybrids were significantly different at p<0.01. These indicated that differential gene expression between hybrids and parents changed at different ages. Relationship between Intersequence Pauses, Laying Persistency and Concentration of Prolactin during the Productive Period in White Leghorn Hens Reddy, I.J.;David, C.G.;Singh, Khub 686 Prolactin is considered to influence the taking of pauses in between ovulatory sequences in White Leghorn hens. Therefore modulating concentrations of prolactin using bromocriptine - a dopamine agonist during early life (17 to 36 weeks of age) could overcome the inhibitory effects of high concentration of prolactin on ovarian activity. The effect of modulation of prolactin concentration on egg production, sequence length and inter sequence pauses were studied by analyzing the oviposition records from 19 to 72 weeks were studied and compared with untreated controls. Bromocriptine administered subcutaneously (100 $\mu$g kg$^{-1}$ body weight or orally through feed (640 $\mu$g day$^{-1}$ bird$^{-1}$) resulted in a steady and sustained decrease in prolactin levels (p<0.01) during and after the withdrawal of treatment up to one reproductive cycle (72 weeks of age). The treated birds had comparatively longer sequences (p<0.01) and fewer pauses (p<0.01). Egg production increased (p<0.01) by fourteen per cent through subcutaneous administration and eleven per cent through oral feeding, over the control birds. It is concluded that the physiological pauses that occur during ovulatory sequences can be disrupted effectively using bromocriptine. Prolactin levels are modulated which may interfere with the follicular recruitment and subsequent oviposition thereby improve egg laying potential of the bird. Dietary Manipulation of Lean Tissue Deposition in Broiler Chickens Choct, M.;Naylor, A.J.;Oddy, V.H. 692 Two experiments were conducted to examine the effect of graded levels of dietary chromium and leucine, and different fat sources on performance and body composition of broiler chickens. The results showed that chromium picolinate at 0.5 ppm significantly (p<0.05) lowered the carcass fat level. Gut weight and carcass water content were increased as a result of chromium treatment. Body weight, plucked weight, carcass weight, abdominal fat pad weight, breast yield and feed efficiency were unaffected by chromium treatment. Leucine did not interact with chromium to effect lean growth. Dietary leucine above the recommended maintenance level (1.2% of diet) markedly (p<0.001) reduced the breast muscle yield. The addition of fish oil to broiler diets reduced (p<0.05) the abdominal fat pad weights compared to birds on linseed diets. Fish oil is believed to improve lean growth through the effects of long chain polyunsaturated fatty acids in lowering the very low-density lipoprotein levels and triglyceride in the blood, in the meantime increasing glucose uptake into the muscle tissue in blood and by minimizing the negative impact of the immune system on protein breakdown. The amount of fat in the diet (2% or 4%) did not affect body composition. Effects of Graded Levels of Dietary Saccharomyces cerevisiae on Growth Performance and Meat Quality in Broiler Chickens Zhang, A.W.;Lee, B.D.;Lee, K.W.;Song, K.B.;An, G.H.;Lee, C.H. 699 An experiment was conducted to investigate the effects of various dietary levels of Saccharomyces cerevisiae (SC) on the growth performance and meat quality (i.e., tenderness and oxidative stability) of Ross broiler chickens. Two hundred and forty dayold broiler chicks were fed four experimental diets with graded levels of SC at 0.0, 0.3, 1.0 and 3.0%. Each treatment consisted of six cages with 10 chicks per cage. Feed and water were provided ad libitum throughout the experiment that lasted for 5 wk. Birds were switched from starter to finisher diets at 3 wk of age. The average BW gains of broiler chickens increased (linear p<0.05) during either 0-3 or 0-5 wk of age as dietary SC levels increased. A linear effect (p<0.05) of SC on feed intake during either 4-5 wk or 0-5 wk of ages was also monitored. The addition of SC to the control diet significantly lowered shear forces in raw breast, raw thigh, and boiled drumstick meats (linear p<0.05). Upon incubation, 2-thio-barbituric acid-reactive substances (TBARS) values increased gradually in breast and thigh meats while more dramatic increase was noted in skin samples. The TBARS values of either breast or thigh meats were not significantly affected (p>0.05) by dietary treatments up to 10 d of incubation. At 15 d of incubation, TBARS values of breast and thigh meats from all SC-treated groups were significantly lower (p<0.05) than those of the control. It appears that dietary SC could enhance growth performance of broiler chickens, and improve tenderness and oxidative stability of broiler meats. Performance and Carcass Composition of Growing-finishing Pigs Fed Wheat or Corn-based Diets Han, Yung-Keun;Soita, H.W.;Thacker, P.A. 704 The objective of this experiment was to compare corn and wheat in finishing pig diets in order to determine whether performance, carcass quality, fatty acid composition or fat colour is altered by choice of cereal grain. A total of 126 crossbred pigs were used in this experiment. At the start of the experiment, a portion of the experimental animals were assigned to receive a wheat-based diet formulated using soybean meal as the sole source of supplementary protein. The remainder of the pigs were assigned to a corn-based diet formulated to supply a similar level of lysine (0.65%) and energy (3,300 kcal/kg DE). At two week intervals, a portion of the pigs on the corn-based diet were switched to the wheat-based diet so that a gradient was produced with pigs being fed the corn and wheatbased diets for different proportions of the finishing period ranging from 100% on wheat to 100% on corn. There were no significant differences in the growth rate of pigs fed the two diets (p = 0.834). Pigs fed wheat tended to consume slightly less feed (p = 0.116) and had a significantly improved feed conversion (p = 0.048) compared with pigs fed corn. Choice of cereal did not affect dressing percentage (p = 0.691), carcass value index (p = 0.146), lean yield (p = 0.134), loin fat (p = 0.127) or loin lean (p = 0.217). Fatty acid composition of backfat was unaffected by the cereal grain fed (p>0.05). Total saturated fatty acid content was 33.31% for both corn and wheat fed pigs (p = 0.997) while the polyunsaturated fatty acid content was 12.01% for corn fed pigs and 11.21% for wheat fed pigs (p = 0.257). The polyunsaturated/saturated ratio was 0.36 for pigs fed corn and 0.34 for pigs fed wheat (p = 0.751). Hunter Lab Colour Scores indicated no difference either in the whiteness or yellowness of the fat. In conclusion, wheat can substitute for corn in growingfinishing pig rations without detrimental effects on pig performance. There were no differences in either the fatty acid composition of backfat or in backfat colour indicating that the decision to use wheat vs. corn needs to be made on economic grounds rather than being based on their effects on fat quality. Study of the Microbial and Chemical Properties of Goat Milk Kefir Produced by Inoculation with Taiwanese Kefir Grains Chen, Ming-Ju;Liu, Je-Ruei;Lin, Chin-Wen;Yeh, Yu-Tzu 711 One of the prerequisites for the successful implementation of industrial-scale goat kefir production is to understand the effects of different kefir grains and culture conditions on the microbial and chemical properties of the goat kefir. Thus, the objectives of the present study were to evaluate the characteristics of kefir grains in Taiwan on the microbial and chemical properties of goat milk kefir, as well as to understand the influence of culture conditions on production of medium chain-length triglycerides (MCT). Kefir grains were collected from households in northern Taiwan. Heat-treated goat milk was inoculated with 3-5% (V/W) kefir grains incubated at 15, 17.5, 20 or 22.5$^{\circ}C$ for 20 h, and the microflora count, ethanol content, and caproic (C6), caprylic (C8), and capric acid (C10) levels measured at 4 h intervals. Our results indicate that incubation with kefir grains results in 10$^6$-10$^7$ CFU/ml microflora count and 1.18 g/L of ethanol content at 20 h of fermentation. Incubation with 5% kefir grain at 20-22.5$^{\circ}C$ produces the highest MCT levels. Expression of Serum and Muscle Endocrine Factors at Antemortem and Postmortem Periods and Their Relationship with Pig Carcass Grade Kim, W.K.;Kim, M.H.;Ryu, Y.H.;Ryu, Y.C.;Rhee, M.S.;Seo, D.S.;Lee, C.Y.;Kim, B.C.;Ko, Y. 716 Carcass weight and backfat thickness are primary yield grading factors. Insulin-like growth factor (IGF)-I/-II, transforming growth factor $\beta$1 (TGF-$\beta$1), and epidermal growth factor (EGF) regulate the proliferation and differentiation of cells including adipocytes. Also, interleukin (IL)-2/-6, cortisol, and dehydroepiandrosterone-sulfate (DHEA-S) are known to be related to muscle growth and fat depth. However, the relationships between endocrine factors and carcass grade have not been studied. Therefore, this study aimed to measure the concentrations of endocrine factors in serum and muscle, and to investigate the relationship of endocrine factors with carcass grade. A total of 60 crossbred gilts (Duroc${\times}$Yorkshire${\times}$Landrace) were used. Blood from the jugular vein was collected at antemortem (7 days before slaughter) and postmortem periods, and M. Longissimus was collected at 45 min and 24 h after slaughter. The concentrations of IGF-I/-II, EGF, TGF-$\beta$1, IL-2/-6, cortisol and DHEA-S were analyzed by radioimmunoassay (RIA) or enzyme-linked immunosorbent assay (ELISA). In general, IGF and EGF concentrations in serum and muscle of grade A carcasses were found to be higher than those of grade C carcasses at antemortem and postmortem periods, whereas the pattern of TGF-$\beta$1 concentration was reversed. In particular, the concentrations of muscle IGF-I (24 h postmortem) and serum TGF-$\beta$1 (antemortem) were significantly different between grades A and C (p<0.05). The present results indicate that serum and muscle growth factors affect carcass weight and backfat thickness, and indirectly suggest the possibility that carcass grade could be predicted by expression of serum and/or muscle growth factors. Characteristics of Gene Structure of Bovine Ghrelin and Influence of Aging on Plasma Ghrelin Kita, K.;Harada, K.;Nagao, K.;Yokota, H. 723 Ghrelin is a novel growth-hormone-releasing acylated peptide, which has been purified and identified in rat stomach. In the present study, the full-length sequence of bovine ghrelin cDNA was cloned by RT-PCR. The bovine ghrelin cDNA sequence derived in the present study included a 348 bp open reading frame and a 137 bp 3'UTR. The putative amino acid sequence of bovine prepro-ghrelin consisted of 116 amino acids, which contained the 27-amino acid ghrelin. The sequence analysis of the bovine ghrelin gene revealed that an intron existed between Gln$^{13}$ and Arg$^{14}$ of ghrelin. This exon-intron boundary matched the GT-AG rule of the splicing mechanism. Compared with rats, which have two tandem CAG sequences in the 3'end of intron, bovine ghrelin genome has only one CAG sequence. Therefore, although rats can produce 28 amino acid-ghrelin and 27 amino acid-des-Gln$^{14}$-ghrelin by alternative splicing, ruminant species, including bovines, might be able to produce only one type of ghrelin peptide, des-Gln$^{14}$-ghrelin. The influence of aging on plasma ghrelin concentration was also examined. Plasma ghrelin concentration increased after birth to approximately 600 days of age, and then remained constant. Characteristics of Bovine Lymphoma Caused by Bovine Leukemia Virus Infection in Holstein-Friesian Dairy Cattle in Korea Yoon, S.S.;Bae, Y.C.;Lee, K.H.;Han, B.;Han, H.R. 728 The frequency and distribution of lymphoma caused by bovine leukemia virus (BLV) infection in various organs were investigated. Lymphoma samples were obtained from slaughtered cattle or from cattle submitted to the National Veterinary Research and Quarantine Service, Korea. Thirty female Holstein-Friesian dairy cattle aged over three years with the BLV-associated lymphoma were studied. None of the Korean native cattle (Hanwoo) had lymphoma in this study however. Lymphoma tissues were gray to pink in color, soft in consistency, and bulged from the cut surface. In advanced lymphoma tissues, there was great variety in the appearance of involved structures due to hemorrhage, necrosis, and/or calcification. Neoplastic tissues were observed in lymph nodes in all lymphoma cases. Intestine (96.4%), heart (88.9%), stomach (73.1%), and diaphragm (62.5%) were frequently involved with lymphoma. However, there was no lymphoma detected in liver. Large neoplastic masses, sometimes reaching the size of over 20 cm, were found in the abdominal cavities. It is suggested that metastasis of lymphomas occurs mainly via lymph based on gross observations; neoplasia may have been initiated in the serosal surface of the lung, heart, peritoneum, and numerous hollow organs in the abdominal cavity. Also many organs in the abdominal and thoracic cavity were affected by neoplastic tissues simultaneously. Characteristics observed in this study could be used as criteria to differentiate BLV-associated lymphoma from other nodular lesions in the slaughterhouse and as fundamental data to make clear the mechanism of metastasis or pathogenesis of EBL. Polymorphism of the Promoter Region of Hsp70 Gene and Its Relationship with the Expression of HSP70mRNA, HSF1mRNA, Bcl-2mrna and Bax-AMrna in Lymphocytes in Peripheral Blood of Heat Shocked Dairy Cows Cai, Yafei;Liu, Qinghua;Xing, Guangdong;Zhou, Lei;Yang, Yuanyuan;Zhang, Lijun;Li, Jing;Wang, Genlin 734 The blood samples were collected from dairy cows at the same milking stage. The single-strand conformation polymorphism (PCR-SSCP) method was used to analyze for polymorphism at the 5'flanking region of the hsp70 gene. The mRNA expression levels of HSP70, HSF1, Bcl-2 and Bax-$\alpha$ at different daily-mean-temperature were analyzed by relative quantitative RTPCR. The DNA content, cell phase and the ratio of apoptosis of lymphocytes in peripheral blood of dairy cattle at different daily-meantemperature were determined by FCM. The PCR-SSCP products of primer pair 1 showed polymorphisms and could be divided into four genotypes: aa, ab, ac, cc, with the cis-acting element (CCAAT box) included. Mutations in the hsp70 5'flanking region (468-752 bp) had different effects on mRNA expression of HSP70, HSF1, Bcl-2 and Bax-$\alpha$. The ac genotypic cows showed higher expressions of HSP70mRNA, HSF1mRNA and Bcl-2mRNA/Bax-$\alpha$mRNA and lower ratio of apoptosis. These mutation sites can be used as molecular genetic markers to assist selection for anti-heat stress cows. Separation and Purification of Angiotensin Converting Enzyme Inhibitory Peptides Derived from Goat's Milk Casein Hydrolysates Lee, K.J.;Kim, S.B.;Ryu, J.S.;Shin, H.S.;Lim, J.W. 741 To investigate the basic information and the possibility of ACE-inhibitory peptides for antihypertension materials, goat's caisin (CN) was hydrolyzed by various proteolytic enzymes and ACE-inhibitory peptides were separated and purified. ACE-inhibition ratios of enzymatic hydrolysates of goat's CN and various characteristics of ACE-inhibitory peptides were determined. ACE-inhibition ratios of goat's CN hydrolysates were shown the highest with 87.84% by pepsin for 48 h. By Sephadex G-25 gel chromatograms, Fraction 3 from goat's CN hydrolysates by pepsin for 48 h was confirmed the highest ACE-inhibition activity. Fraction 3 g and Fraction 3 gh from peptic hydrolysates by RP-HPLC to first and second purification were the highest in ACE-inhibition activity, respectively. The most abundant amino acid was leucine (18.83%) in Fraction 3 gh of ACE-inhibitory peptides after second purification. Amino acid sequence analysis of Fraction 3 gh of ACE-inhibitory peptides was shown that the Ala-Tyr-Phe-Tyr, Pro-Tyr-Tyr and Tyr-Leu. IC$_{50}$ calibrated in peptic hydrolysates at 48 h, Fraction 3, Fraction 3 g and Fraction 3 gh from goat's CN hydrolysates by pepsin for 48 h were 29.89, 3.07, 1.85 and 0.87 g/ml, respectively. Based on the results of this experiment, goat's CN hydrolysates by pepsin were shown to have ACE-inhibitory activity. Nutritional and Tissue Specificity of IGF-I and IGFBP-2 Gene Expression in Growing Chickens - A Review - Kita, K.;Nagao, K.;Okumura, J. 747 Nutritional regulation of gene expression associated with growth and feeding behavior in avian species can become an important technique to improve poultry production according to the supply of nutrients in the diet. Insulin-like growth factor-I (IGF-I) found in chickens has been characterized to be a 70 amino acid polypeptide and plays an important role in growth and metabolism. Although it is been well known that IGF-I is highly associated with embryonic development and post-hatching growth, changes in the distribution of IGF-I gene expression throughout early- to late-embryogenesis have not been studied so far. We revealed that the developmental pattern of IGF-I gene expression during embryogenesis differed among various tissues. No bands of IGF-I mRNA were detected in embryonic liver at 7 days of incubation, and thereafter the amount of hepatic IGF-I mRNA was increased from 14 to 20 days of incubation. In eyes, a peak in IGF-I mRNA levels occurred at mid-embryogenesis, but by contrast, IGF-I mRNA was barely detectable in the heart throughout all incubation periods. In the muscle, no significant difference in IGF-I gene expression was observed during different stages of embryogenesis. After hatching, hepatic IGF-I gene expression as well as plasma IGF-I concentration increases rapidly with age, reaches a peak before sexual maturity, and then declines. The IGF-I gene expression is very sensitive to changes in nutritional conditions. Food-restriction and fasting decreased hepatic IGF-I gene expression and refeeding restored IGF-I gene expression to the level of fed chickens. Dietary protein is also a very strong factor in changing hepatic IGF-I gene expression. Refeeding with dietary protein alone successfully restored hepatic IGF-I gene expression of fasted chickens to the level of fed controls. In most circumstances, IGF-I makes a complex with specific high-affinity IGF-binding proteins (IGFBPs). So far, four different IGFBPs have been identified in avian species and the major IGFBP in chicken plasma has been reported to be IGFBP-2. We studied the relationship between nutritional status and IGFBP-2 gene expression in various tissues of young chickens. In the liver of fed chickens, almost no IGFBP-2 mRNA was detected. However, fasting markedly increased hepatic IGFBP-2 gene expression, and the level was reduced after refeeding. In the gizzard of well-fed young chickens, IGFBP-2 gene expression was detected and fasting significantly elevated gizzard IGFBP-2 mRNA levels to about double that of fed controls. After refeeding, gizzard IGFBP-2 gene expression decreased similar to hepatic IGFBP-2 gene expression. In the brain, IGFBP-2 mRNA was observed in fed chickens and had significantly decreased by fasting. In the kidney, IGFBP-2 gene expression was observed but not influenced by fasting and refeeding. Recently, we have demonstrated in vivo that gizzard and hepatic IGFBP-2 gene expression in fasted chickens was rapidly reduced by intravenous administration of insulin, as indicated that in young chickens the reduction in gizzard and hepatic IGFBP-2 gene expression in vivo stimulated by malnutrition may be, in part, regulated by means of the increase in plasma insulin concentration via an insulin-response element. The influence of dietary protein source (isolated soybean protein vs. casein) and the supplementation of essential amino acids on gizzard IGFBP-2 gene expression was examined. In both soybean protein and casein diet groups, the deficiency of essential amino acids stimulated chickens to increase gizzard IGFBP-2 gene expression. Although amino acid supplementation of a soybean protein diet significantly decreased gizzard IGFBP-2 mRNA levels, a similar reduction was not observed in chickens fed a casein diet supplemented with amino acids. This overview of nutritional regulation of IGF-I and IGFBP-2 gene expression in young chickens would serve for the establishment of the supply of nutrients to diets to improve poultry production.	CommonCrawl
Overspill In nonstandard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any internal subset A of N, if 1. 1 is an element of A, and 2. for every element n of A, n + 1 also belongs to A, then A = N If N were an internal set, then instantiating the internal induction principle with N, it would follow N = N which is known not to be the case. The overspill principle has a number of useful consequences: • The set of standard hyperreals is not internal. • The set of bounded hyperreals is not internal. • The set of infinitesimal hyperreals is not internal. In particular: • If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive non-infinitesimal (or appreciable) hyperreal. • If an internal set contains N it contains an unlimited (infinite) element of N. Example These facts can be used to prove the equivalence of the following two conditions for an internal hyperreal-valued function ƒ defined on R. $\forall \epsilon \in \mathbb {R} ^{+},\exists \delta \in \mathbb {R} ^{+},\|h\|\leq \delta \implies \|f(x+h)-f(x)\|\leq \varepsilon $ and $\forall h\cong 0,\ \|f(x+h)-f(x)\|\cong 0$ The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive ε, $\forall {\mbox{ positive }}\delta \cong 0,\ (\|h\|\leq \delta \implies \|f(x+h)-f(x)\|<\varepsilon ).$ Applying overspill, we obtain a positive appreciable δ with the requisite properties. These equivalent conditions express the property known in nonstandard analysis as S-continuity (or microcontinuity) of ƒ at x. S-continuity is referred to as an external property. The first definition is external because it involves quantification over standard values only. The second definition is external because it involves the external relation of being infinitesimal. References • Robert Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer. Infinitesimals History • Adequality • Leibniz's notation • Integral symbol • Criticism of nonstandard analysis • The Analyst • The Method of Mechanical Theorems • Cavalieri's principle Related branches • Nonstandard analysis • Nonstandard calculus • Internal set theory • Synthetic differential geometry • Smooth infinitesimal analysis • Constructive nonstandard analysis • Infinitesimal strain theory (physics) Formalizations • Differentials • Hyperreal numbers • Dual numbers • Surreal numbers Individual concepts • Standard part function • Transfer principle • Hyperinteger • Increment theorem • Monad • Internal set • Levi-Civita field • Hyperfinite set • Law of continuity • Overspill • Microcontinuity • Transcendental law of homogeneity Mathematicians • Gottfried Wilhelm Leibniz • Abraham Robinson • Pierre de Fermat • Augustin-Louis Cauchy • Leonhard Euler Textbooks • Analyse des Infiniment Petits • Elementary Calculus • Cours d'Analyse	Wikipedia
A global mass budget for positively buoyant macroplastic debris in the ocean Laurent Lebreton1,2, Matthias Egger ORCID: orcid.org/0000-0002-1791-18421 & Boyan Slat1 Scientific Reports volume 9, Article number: 12922 (2019) Cite this article An Author Correction to this article was published on 30 January 2020 This article has been updated Predicted global figures for plastic debris accumulation in the ocean surface layer range on the order of hundreds of thousands of metric tons, representing only a few percent of estimated annual emissions into the marine environment. The current accepted explanation for this difference is that positively buoyant macroplastic objects do not persist on the ocean surface. Subject to degradation into microplastics, the major part of the mass is predicted to have settled below the surface. However, we argue that such a simple emission-degradation model cannot explain the occurrence of decades-old objects collected by oceanic expeditions. We show that debris circulation dynamics in coastal environments may be a better explanation for this difference. The results presented here suggest that there is a significant time interval, on the order of several years to decades, between terrestrial emissions and representative accumulation in offshore waters. Importantly, our results also indicate that the current generation of secondary microplastics in the global ocean is mostly a result of the degradation of objects produced in the 1990s and earlier. Finally, we propose a series of future emission scenarios until 2050, discussing the necessity to rapidly reduce emissions and actively remove waste accumulated in the environment to mitigate further microplastic contamination in the global ocean. Since mass production of synthetic polymers started in the 1950s, plastic waste has been accumulating and degrading in terrestrial and oceanic environments1. Particularly, positively buoyant plastic objects are accumulating at the surface of the oceans, transported by currents, wind and waves, reaching remote subtropical oceanic gyres2,3,4,5. In 2010, annual emissions of plastic waste from land into the ocean were estimated to range between 4.8 and 12.7 million metric tons6. Integrated from the 1950s when plastic was first introduced in our societies, and assuming emissions proportional to global plastic production, the total accumulated mass on the ocean surface layer should be as high as tens of millions metric tons. Synthetic polymers with a density lower than sea water represent over 65.5% of the current global plastic production7. Yet, a major fraction of positively buoyant plastic is missing as current estimates of >250,000 metric tons8 are far from the predicted tens of millions of metric tons that should be floating in the global ocean by now. The answer to the missing plastic question could be the combination of three possible explanations. Firstly, the input of plastic into the ocean could be overestimated. Evaluation of inputs rely on reported country-scale statistics on municipal waste generation9 for which a fraction is assumed to reach the ocean. However, the dynamics of release into the marine environment are poorly known. Other assessments considering emissions from rivers as a function of rainfall and plastic waste generation predict smaller inputs, yet still on the orders of million metric tons10,11. Secondly, the total plastic mass currently floating on the ocean surface could be underestimated. Commonly, surface concentrations of plastic pollution are monitored using neuston nets (generally <1 m sampling width) with mesh size typically in the sub-millimeter scale, requiring large sampling effort due to spatial and temporal heterogeneity12. This data is then coupled with global dispersal models to assess the total mass of plastic debris on the ocean surface layer13,14. Previous work has demonstrated that combining surface trawl data with visual sightings for the occurrence of larger debris (>200 cm) results in greater estimates8. More recently, a study using larger surface trawls (6 m sampling width) and aerial imagery (~360 m sampling width) instead of visual sightings to calculate the mass contribution of debris respectively >5 cm and >50 cm in size, has shown that accepted concentration figures could be underestimated by a factor of four to sixteen15 in the North Pacific. This may also be true in other parts of the world but even with a four to sixteen fold underestimation, this would result in a predicted standing mass of several hundreds of thousands to a few million metric tons, which is still one to two orders of magnitude lower than the tens of million metric tons of plastic that are expected to have entered the ocean since the 1950s. Thirdly, positively buoyant plastic could be removed from the sea surface. Floating plastic debris at sea undergoes fouling that may result in a loss of buoyancy in seawater. If the debris density increases sufficiently, buoyant plastic will be transported to deeper water depths. In deeper and different environments (e.g. aphotic zone), debris may experience rapid defouling and resurface for a repetition of the same cycle of events16. However, in shallower depths, debris could eventually reach the marine benthic sediment or directly strand on shorelines. Furthermore, plastic at sea is degrading into smaller particles due to photodegradation, mechanical abrasion and oxidation17. The observed size distribution of plastic debris collected at the ocean surface shows that the smaller fraction of microplastics (<0.1 cm) is underrepresented when compared to expected degradation rates from larger objects8,13. These smaller particles could potentially be disappearing faster from the sea surface from ingestion by marine life18 and incorporation into marine snow19, aggregation20,21, or sinking from biofouling22. Unfortunately, current sampling techniques are very limited for the in-situ detection of sub-millimeter sized particles and far less is known for particles smaller than microplastics (<0.05 cm). Assuming the distribution of plastic mass per size class had not reached equilibrium, one could also argue that the smaller size fraction could still be in formation but remains underrepresented as the emissions overwhelms degradation rates. Researching these questions, a recent study presented a global model for emission, degradation and settling of macroplastics (>0.5 cm) and microplastics (<0.5 cm) in the ocean. Assuming no settling for macroplastics, the results suggested that 99.8% of the plastic mass that has entered the marine environment since 1950 has degraded into micro- and nanoplastics and has subsequently settled below the surface layer23. The authors predicted that under a zero-emission scenario, almost all plastic would be removed from the ocean surface layer within three years. Yet, accumulation of plastic in the global ocean has mostly been mapped for surface waters but very little information is known on where the underwater plastic may have accumulated. Moreover, while the latter explanation could solve the question of the missing ocean plastic, it raises new questions. Dispersal models and observations suggest that debris released from terrestrial sources and accumulating in offshore oceanic gyres requires on average a minimum of several years to reach such remote oceanic areas24,25. Thus, if positively buoyant plastic were persisting less than three years on the surface layer, accumulation in offshore subtropical gyres would not occur. Furthermore, a recent study analyzing the age of plastic objects found in the North Pacific subtropical gyre by identifying production dates on collected macroplastics at sea reported a significant number of decades-old objects, dating as far back as the 1970s15. The degradation of plastic initiated by solar UV radiation is severely retarded when floating in seawater17. While the production date does not necessarily inform on the disposal date, the relative age distribution of objects found at sea, assuming a sufficiently large sample size, should be representative of the discard of plastic objects from the different consumer and market sectors, and their associated product lifetime7. Here, we propose a new global ocean surface mass balance budget model for positively buoyant macroplastics. The principal objective of this study is to identify the key processes governing the fate of marine litter based on field evidence and to orient future research. Different model parameterizations are tested to predict the mass of positively buoyant plastic in offshore surface waters. We constrain our model parameters with field data, dispersal model outputs and recent estimates from the literature. We create a simple whole-ocean emission-transport-degradation model, by including probabilities of debris stranding/settling and recirculation into coastal environments. We propose a convergent model to explain (1) the discrepancies between current accepted figures for plastic emissions and standing stock on the ocean surface layer, and (2) the occurrence of decades-old debris in subtropical oceanic gyres. Using our convergent parameterized model, we predict future mass of positively buoyant plastic in the ocean under several emission scenarios. Global ocean surface mass balance model We introduce a simple box model to quantify buoyant macroplastics (>0.5 cm) at the surface of the global ocean. We consider synthetic polymer production data from 1950 to 2015 and compute the fraction of positively buoyant polymers (65.5% of total) used by different market sectors associated with product lifetime distributions7. Starting in 1950 and for every year, we compute the plastic population age distribution of material mass that is reaching end of lifetime and is therefore discarded. We define the global mass of plastic D, produced in year y0 and discarded in year y as follow: $$D(y,{y}_{0})=\mathop{\sum }\limits_{\sigma }^{\begin{array}{c}market\,\\ sectors\end{array}}\,Production({y}_{0})\,\ast \,Market\,Share(\sigma )\,\ast \,LifeTime(y-{y}_{0},\sigma )$$ With Production(y0), global plastic production for year y0, MarketShare(σ), percentage of market share in global plastic production for sector σ and, Lifetime(y-y0, σ), probability density function of product lifetime for sector σ and plastic age y-y0 as proposed by Geyer et al.7 with market sectors including "Packaging", "Transportation", "Building and Construction", "Electrical/Electronic", "Consumer & Institutional Products", "Industrial Machinery" and "Other". More information on the market share and probability density functions for lifetime of plastic objects is provided in Supplementary Table 1. Some of the discarded plastic waste may enter the global ocean in coastal areas6. Once at sea, buoyant plastic may strand back on the shoreline or sink from fouling-induced loss of buoyancy. At deeper depths, debris may experience rapid defouling followed by resurfacing as floating debris16. In shallower depth, however, debris has a higher chance to reach the seabed. In this framework, we define the coastal waters as the area with bathymetry between high tide line and the euphotic depth (typically shallower than 200 m water depth). Our model considers that when in the coastal environment a fraction of floating plastic mass is captured by the landmass, undergoing repeated episodes of stranding and release at the shoreline or settling and resurfacing from the seabed. Some of the floating plastic remaining at the surface may be transported to offshore waters. As time passes, the mass of stranded, settled and floating plastic degrades into microplastics, thus leaving the model domain. Our model primarily focuses on buoyant macroplastics and considers mass loss from degradation into microplastics as a permanent sink. We divide the global marine environment in three surface domains (Fig. 1): the shoreline (S), the coastal surface waters (C) and the offshore surface waters (O). Our model includes 6 mass compartments: SM, CM, OM for macroplastics and Sm, Cm, Om for secondary microplastics. The model conserves mass. For any year between 1950 to present, the accumulated mass of plastic that has been introduced in the global ocean is equal to the sum of these 6 mass compartments. Predicting quantities of positively buoyant macroplastics (>0.5 cm) in the ocean environment using a global ocean emission-transport-degradation model. Every year, a fraction i of discarded plastic material is emitted into the coastal surface layer (CM). Material present in coastal waters can strand or settle around shorelines (SM) with probability s and material from SM can leak back into CM with release probability r. Material from CM can escape the continental shelf and enter the ocean surface layer OM with transport probability t. Finally, fractions dS, dC and dO of macroplastics present in the marine environment enter a permanent sink by degradation into microplastics (<0.5 cm) from the shoreline (Sm), the coastal surface layer (Cm) and the offshore surface layer (Om). The processes are repeated annually from 1950 to 2015. The model is initiated with input into coastal environments starting in 1950. Then from year y-1 to year y, we compute the net mass input of plastic produced in year y0, into the surface waters of the global coastal environment: $$\Delta (y,{y}_{0})=i\,\ast \,D(y,{y}_{0})+\,r\,\ast \,(1-{d}_{S})\,\ast \,{S}_{M}(y-1,\,{y}_{0})+(1-{d}_{C})\,\ast \,{C}_{M}(y-1,{y}_{0})$$ The first term constitutes the direct inputs from discarded plastic leaking into the environment. Model parameter i is the annual mass fraction of discarded plastic reaching the coastal ocean. The two other terms represent respectively material released from the shoreline SM and existing floating material in CM left from previous years that has not yet degraded into microplastics. Model parameters dC and dS are regarded as the annual mass fraction of respectively floating and stranded or settled macroplastics degrading into microplastics, while r is the annual mass fraction of stranded or settled macroplastics that is released back into surface waters of coastal environments. The resulting mass of plastic produced in year y0 that is present in coastal surface waters during year y is then computed as: $${C}_{M}(y,{y}_{0})=(1-s)\,\ast \,(1-t)\,\ast \,\Delta (y,{y}_{0})$$ Where s is the annual mass fraction of floating plastic that strands and settles around shorelines and t the annual mass fraction of remaining floating plastic that is transported offshore. The total mass of plastic produced in year y0 and stranded or settled around the global shoreline during year y is the sum of newly stranded or settled material and previously accumulated plastic that was not degraded into microplastics and not released back into coastal waters: $${S}_{M}(y,{y}_{0})=(1-r)\,\ast \,(1-{d}_{S})\,\ast \,{S}_{M}(y-1,{y}_{0})+s\,\ast \,\,\varDelta (y,{y}_{0})$$ Finally, the total mass of plastic produced in year y0 and floating in offshore waters during year y is the sum of previously accumulated plastic that has not degraded into microplastics and new debris leaked from the coastal waters. $${O}_{M}(y,{y}_{0})=\,\,(1-{d}_{O})\,\ast \,{O}_{M}(y-1,{y}_{0})+t\,\ast \,(1-s)\ast \,\varDelta (y,{y}_{0})$$ With dO the degradation rate for macroplastics in offshore surface waters. For each year, the three mass sink terms are populated with input from degradation into microplastics from coastal, shoreline and offshore environments. $${C}_{m}(y,{y}_{0})=(1+{d}_{C})\,\ast \,{C}_{M}(y-1,{y}_{0})$$ $${S}_{m}(y,{y}_{0})=(1+{d}_{S})\,\ast \,{S}_{M}(y-1,{y}_{0})$$ $${O}_{m}(y,{y}_{0})=(1+{d}_{O})\,\ast \,{O}_{M}(y-1,{y}_{0})$$ In this study, we assumed the degradation rates dC, dS and dO to be equal. The degradation term is called thereafter d. $$d={d}_{C}={d}_{S}={d}_{O}$$ We note that in nature, these values may be different particularly for the shoreline where the degradation rate could be greater than in surface waters. These values will also likely differ between polymer composition and dimension of objects. We acknowledge that global degradation into secondary microplastics is far more complex than described by our model. With current available data, however, we are limited to propose a whole-ocean average degradation rate for the total macroplastic mass. Specifications of degradation rates by environments and polymer types will require more experimental research. Another major assumption here is that model parameters do not show any interannual variability and that the dynamics of degradation, stranding, release and recirculation into the coastal environment is independent from the age and characteristics of plastic objects. A crucial parameter is the fraction i of new plastic waste generated on land that reaches the ocean. This parameter has a substantial influence when constraining the values of s (stranding on shoreline), r (release from shoreline) and t (offshore transport). We constrain parameter i by using estimates of global input from land into the ocean for 2010 with 4.8 to 12.7 million metric tons of input6. For a global plastic waste generation of 274 million metric tons in 20107, this translates to a fraction of annual discarded plastic reaching the ocean ranging from i = 1.7–4.6%. We therefore used this reported range to define the confidence interval for the results presented here. Age distribution of ocean plastic To study the persistency of macroplastics, the age distribution of plastic in the different compartments of our model is compared to the age distribution of plastic debris collected in a large oceanic gyre. In 2015, a multivessel expedition collected marine plastics debris floating in the Great Pacific Garbage Patch located in the North Pacific subtropical gyre15. The expedition landed 664 kg of positively buoyant macroplastics (debris larger than 0.5 cm) back to shore. Of the 83,144 collected pieces (>0.5 cm), 427 had a recognizable inscription for which 11 languages and 50 dates of production could be identified. Here, we consider the distribution of production dates found on these samples to be representative of plastic age distribution in oceanic gyres. By exploring all possible combinations of our five model parameters ranging from 0% to 100% at every 1%, we observe that only the degradation rate d significantly impacts the relative age distribution of positively buoyant macroplastics in offshore surface waters (Fig. 2). This is because degradation into microplastics is the only permanent sink considered by our model i.e. it is the only natural mechanism that removes material from our model domain. Therefore, we use the observed plastic age distribution to constrain the parameter d. Note that when fixing parameters s, r and t to 0%, we reproduce a global ocean emission-degradation model for macroplastics introduced in a previous study23. We compare the modelled decadal distribution with our samples for a degradation rate d ranging from 0% to 100%. Only one plastic object with production date in the 2010s was identified in the samples which were collected in 2015. We explain this by the minimum time taken for objects to reach the area that we estimate from dispersal model trajectories to be between 5 to 10 years to be representative of sources. Therefore, accounting for a minimum delay of 5 years, we compared observed and modelled decadal distributions of production dates from the 1950s to the 2000s. We computed the sum of squared residuals between observed and modelled age distribution by decades and varied model parameter d for minimization. Best fit was found for d = 3% of mass of positively buoyant macroplastics annually degraded into microplastics (Supplementary Fig. 1). A small degradation rate is in good agreement with field experiments, estimating a mass loss ranging from 0.65% to 1.9% of total mass, depending on polymers, for samples immersed at sea for a period of 12 months26. Comparison between observed and predicted plastic age distribution. Observations correspond to the relative age distribution of macroplastics collected from the North Pacific subtropical gyre in 201515. This distribution is derived from production date labels identified on debris (N = 50). Model predicted age distributions are given for a range of the degradation variable d (3%, 10%, 50% and 90%) and parameters s, r and t set to 0%, reproducing a simple global emission-degradation model. Whiskers extend to all possible values by computing all possible combinations of s, r, and t varying from 0% to 100%, showing that the plastic age distribution is mostly sensitive to the degradation rate parameter d. We compute the least square sum for decadal distribution of observed and modelled plastic from 1950s to 2000s; minimum value is found for d = 3% (Supplementary Fig. 1). Pearson p-test values for d = 3%, 10%, 50%, 90% are respectively p = 0.009, 0.0133, 0.0361 and 0.0310. Underrepresentation of objects produced between 2010 and 2015 (the year when the samples were collected) is explained by the minimum time required for plastic debris to travel to oceanic gyres (~5 years). Model sensitivity analysis We study model convergence by varying parameters s, r and t from 0% to 100%. The model is considered convergent when the confidence interval of model predicted mass floating on the global ocean surface (e.g. CM + OM) includes values on the order of hundreds of thousand metric tons of material (i.e. <106 metric tons). The model is generally converging for large values of stranding probability (s) and low values of offshore transport (t). Converging values for coastal release (r) are inversely proportional to the value of stranding probability (s). This was to be expected, given that these parameters are intrinsically connected as their difference measures the capture efficiency of the continental mass. Note that parameter s must be higher than r to reproduce accumulation on the world's beaches. To constrain the model parameters s and t, we investigate trajectories of Lagrangian particles from a global dispersal model reproducing 20-year of surface circulation25,15. Particles are released from significant point sources (Fig. 3) near the coast based on population27 and waste management data6. A proportional number of particles is attributed to each country based on the estimated amount of mismanaged waste the country's coastal population generates over a year. The particle release locations are derived from coastal population density and the timing of release is randomly distributed throughout the year. Particles are advected using different model forcing components, including sea surface currents, stokes drift and variable influences of wind. Lagrangian dispersal model source locations and global ocean surface model domains. Amplitude and location of model particle sources are derived from predicted inputs of plastic from land into the ocean6 and population changes from 1993 to 201227. The separation between coastal and offshore surface waters in our model is shown with areas of respectively light and dark blue color. Coastal surface waters represent the continental shelf with bottom depths shallower than the photic zone (i.e. depths <200 m). For stranding probability (s), we follow particles from their day of release until they spend two consecutive days near the shoreline. The model parameter (s) is defined as the fraction of model particles that have spent at least two consecutive days near the shoreline after one year since their initial release over the total number of particles present in the model. A particle is considered near the shoreline when it is located at a distance smaller than the hydrodynamic model cell size from a land cell (1/16°, several kms depending on latitude). With no wind influence and after one year, 96% of the 2,510,918 particles investigated have spent at least two consecutive days near the shoreline. This value increases when adding wind forcing, for a windage coefficient of 2% (of 10 m height wind speed value), 98% of particles have transited around the coast for at least two days. This increase in beaching probability for high windage debris is in good agreement with observations reporting mainly low-windage debris accumulating in oceanic gyres24. Here, we considered that if a particle spends more than two consecutive days in contact to the shoreline it is likely stranded as it would have gone through at least one full tidal cycle. We used these results to define the stranding probability in our model with s = 96–98%. For offshore transport probability (t), we locate the fraction of particles that stay on the continental shelf one year after release. The continental shelf is defined in our model by a water depth shallower than 200 m. We used gridded bathymetry data from the General Bathymetric Chart of the Ocean28 to determine ocean water depth. After one year of release, between 32% and 34% of modelled non-beaching particles have escaped the continental shelf. Values decrease with windage coefficient, except for the first month which we attribute to a rapid presorting of model particles depending on emissions location. We used these results to estimate the annual offshore transport probability with t = 32–34%. Using midpoint values for s and t (i.e. 97% and 33%, respectively), a coastal release parameter r = 1% results in convergence (Fig. 4) and explains the discrepancies between emissions estimate and observed mass on the global ocean surface layer. An overview of the model parameters, with description and selected values is given in Table 1. Sensitivity analysis and model convergence. (a) Stranding probability (s) determined from Lagrangian particle trajectories starting in coastal environments. The value of s is defined as the percentage of particles present in the model that have spent two consecutive days in close proximity (<1/16°) to the shoreline. Intervals correspond to different forcing scenarios with influence of wind ranging from 0% (thick dark line) to 2% (thin dark line) of 10 m height wind speed. Probability of stranding increases with windage coefficient. (b) Offshore transport (t) estimated from the same Lagrangian trajectories. The parameter t is defined as the percentage of model particles that are located outside the continental shelf (>200 m water depth) after one year of release. Intervals also correspond to different wind forcing scenarios ranging from 0% (thick dark line) to 2% (thin dark line). Generally, the probability of transport to offshore waters decreases with windage coefficient. (c) Sensitivity analysis and model convergence. Parameters s, t, and r vary from 0% to 100%. The model is considered convergent (blue colored area) when the confidence range for mass estimate on the global ocean surface layer (OM + CM) overlaps with values below 106 metric tons, assuming a degradation rate d = 3% and emission rates i = 1.7–4.6%. Our model is converging with estimated value s = 97%, t = 33%, and r = 1% (black diamond). Table 1 Whole-ocean model parameters description and corresponding selected values for this study. Mass and age of buoyant plastics in the ocean environment Under this convergent parameterized model, we provide an alternative explanation for the large differences between total predicted emissions of buoyant plastic since 1950 (70.0–189.3 million metric tons considered by our model) and total mass floating on the global ocean in 2015 (less than 1% of global emissions since 1950 with a predicted 0.61–1.65 million metric tons). A large part (66.8%) of all the buoyant macroplastic (>0.5 cm) released into the marine environment since the 1950s is stored by the world's shoreline with debris stranded, settled and/or buried, undergoing episodes of capturing and resurfacing. We estimated for 2015, this represented 46.7–126.4 million metric tons of macroplastic. Finally, a significant mass fraction (32.3%) may already have degraded into microplastics (<0.5 cm) with 22.3–60.4 million metric tons from the shoreline and 0.29–0.80 million metric tons from the ocean. Figure 5 shows the modeled age distribution in 2015 for plastics introduced in the ocean environment. Most buoyant plastic (79%) present in the coastal surface layer is originating from objects less than 5 years old. For the offshore surface layer, where older macroplastic objects have had more time to accumulate, plastic younger than 5 years accounts for only 26% of the buoyant plastic mass. Macroplastics older than 15 years contribute nearly half of the total mass (47%). Finally, the modeled age distribution of secondary microplastics generated from the degradation of macroplastics shows that most (74%) of the degraded plastic mass in the ocean comes from objects produced in the 1990s (27%) and earlier (47%). Modelled age distribution of plastic in the global ocean environment for the year 2015. Annual midpoint mass estimate by age of buoyant plastic distributed between microplastic and macroplastics at the shoreline, the coastal waters and offshore waters. Note the different scales of the x-axes. Future emission scenarios Using our convergent parameterized model, we assessed future scenarios from 2015 to 2050. We defined three source scenarios with (1) emissions increasing with average 2005–2015 global plastic production's annual growth rate, (2) emissions stagnating at 2020 levels and (3) no more emissions from 2020 onwards. Figure 6 shows the total projected mass per scenario and per model compartment. Our model predicts that for a business-as-usual scenario, where no effort is given to mitigating emissions, the quantities of buoyant macroplastics at the surface of the ocean and coastline could quadruple by the year 2050 (midpoint value of 4.5 million metric tons for OM + CM and 342.5 million metric tons for SM). By then, a predicted midpoint of respectively 3.0 (OM + CM) and 231.6 (SM) million metric tons of plastic will have degraded into microplastics. If emissions of plastics into the oceans are kept constant from the year 2020 onwards, the mass of buoyant macroplastics on the global ocean surface and coastlines continues to increase, although at a slower rate due to the degradation of older objects into smaller particles. The latter, however, cannot compensate for annual inputs and the resulting macroplastic mass is increasing. If sources are stopped from the year 2020 onwards, floating and stranded mass of macroplastics decrease by 2050 to respectively 59% and 57% of their 2020 levels. The mass of microplastics in the ocean and on beaches, however, more than doubles from 2020 levels as material left in the environment is slowly degrading. Future projections for accumulated mass of buoyant macroplastics (top) and degraded material (microplastics, bottom) from the ocean surface layer under three scenarios for emissions. (Red) Emissions are increasing at average 2005–2015 growth rate, (Dark blue) emissions are constant from 2020 and (Light blue) emissions are stopped from 2020. Solid lines represent mid-point estimates while shaded areas represent uncertainties (see Section 2.1). In this study, we introduce a simple global ocean surface box model for positively buoyant macroplastic that gives a plausible explanation for (1) the differences between estimated annual emissions of plastic into the marine environment and the predicted standing mass of plastic at the surface of the ocean, and for (2) the observation of significant number of decades-old objects in offshore subtropical waters. Based on field evidence and using a simple model, we offer an alternative explanation to the missing plastic question by identifying the key processes governing the fate of floating macroplastics. We argue that plastic accumulated in offshore surface waters is highly persistent. Accumulated quantities are less than initially expected because of the capacity for the global landmass to trap and filter marine litter inducing a delay -likely on the order of decades- for fragmented buoyant plastic to reach offshore accumulation zones. A rapid degradation sink term of >90% per year, as previously proposed to answer the difference between emissions and surface measurements23, cannot reproduce observations of plastic age distribution at sea. Instead, our model suggests that stranding, settling and resurfacing in coastal environments must be playing a major role in the removal of buoyant macroplastics from the surface of the ocean. Our model predicts that most of the plastic mass that has entered the marine environment since the 1950s has not disappeared from the ocean surface by degradation but is stranded or settled on its way to offshore waters, possibly slowly circulating between coastal environments with repeated episodes of beaching, fouling, defouling and resurfacing. Most of the modeled macroplastic mass floating in coastal waters is composed of relatively new objects while older objects are better represented in the open ocean. This is in good agreement with observations as the most common type of plastic litter found in the subtropical offshore waters are unidentified, thick, polyethylene or polypropylene plastic fragments15. This suggests that only certain types of plastic have the capacity to persist for a sufficient amount of time to eventually reach these accumulation zones. We hypothesize that a natural sorting for plastic debris is occurring in coastal environments, characterized with the capacity for the shoreline to capture the major part of floating material and where only a small fraction eventually escapes and accumulates in offshore waters. There is very little information on the amplitude of the different mechanisms governing the capture and the release of marine litter by the landmass. To obtain a convergent model, the stranding parameter (s) must be much greater than the release parameter (r). This informs us that the capture mechanisms should be dominating the release mechanisms, and therefore that the landmass likely is storing a major fraction of the missing plastic debris. Particularly, debris buried under sediments could be stored for unknown duration29 and eventually be transported to deeper water depths through sedimentary gravity flows30. Furthermore, our model predicts that the microplastic contamination resulting from the formation of secondary particles in the marine environment is mostly representative of the degradation of objects from the 1990s and earlier. The comparison between our predicted value for the year 2014 (0.28–0.75 million metric tons), which does not account for direct input of microplastics from terrestrial sources, with estimates of microplastic concentration at the surface of the global ocean for the same year (0.093–0.236 million metric tons14), suggests that at least two-third of these microplastics have disappeared from the ocean surface layer likely by settling, ingestion, aggregation, stranding or degradation into even smaller particles. The relative removal contribution of these different mechanisms is largely unknown and would require more field observations and laboratory experiments. Thus, this current model cannot be directly translated to microplastics. Additionally, sources from terrestrial emissions of primary and secondary microplastics would need to be accounted for. However, the framework presented here can be refined to specific polymers or market sectors as well as geographic locations to examine plastic fluxes and design efficient mitigation strategies for source reduction and cleanup. Our model was intentionally built using simple assumptions with only five varying parameters. We acknowledge that there are some uncertainties associated with using such a simple model approach as the values of these parameters likely differs between polymer or object types, or geographically and temporally. For instance, we assume the annual input rates to be directly proportional to global plastic consumption which may not be entirely true as waste generation rates may have changed since the 1950s. However, the objective of this study is to describe exchange processes by predicting orders of magnitude of mass quantities within the marine compartments considered by our model. The values of our five model parameters should be regarded as annual averages for the whole mass of positively buoyant macroplastic available in the marine environment. Here, we show how dispersal models can be used to constrain such model parameters. Assessing the age distribution of ocean plastic debris is useful when formulating mass balance budgets. We recommend the systematic collection of origin and age indicators on plastic debris found in the environment as it helps to better understand the source, transport and fate of plastic pollution at sea. Monitoring the distribution of microplastics within the water column and on the seabed may help in assessing long-term fate of microplastics at sea, thus allowing extending this current mass budget model to microplastics. Finally, we show that comparing current emission estimates of plastic into the marine environment with data recently collected offshore is misleading, as there is a time lag of likely several years to decades between the two metrics. These results are somewhat alarming as even with an extremely ambitious scenario (no further emissions in the ocean by 2020), the level of microplastics in the ocean could double by mid-century as already accumulated plastic waste slowly degrades into smaller pieces. This information is important as it shows that mitigating microplastic pollution in the global ocean requires two major components: (1) drastically reducing emissions of plastic pollution in the coming years and (2) actively engaging in removal operations of plastic waste from the marine environment to reduce further generation of secondary microplastics for the decades to come. This conclusion can likely by applied to other natural environments. Without proper handling and management of accumulated plastic waste, the legacy of the last 70 years of throw-away society will live on through the generation of ever smaller synthetic polymer fragments in soils, freshwater ecosystems and eventually the ocean. The systematic removal of plastic waste from the natural environment should be encouraged and coordinated at a global scale. An amendment to this paper has been published and can be accessed via a link at the top of the paper. Barnes, D. K. A., Galgani, F., Thompson, R. C. & Barlaz, M. Accumulation and fragmentation of plastic debris in global environments. Philosophical Transactions of the Royal Society B: Biological Sciences 364, 1985–1998 (2009). Moore, C. J., Moore, S. L., Leecaster, M. K. & Weisberg, S. B. A comparison of plastic and plankton in the North Pacific Central Gyre. Marine Pollution Bulletin. 42, 1297–1300 (2001). Law, K. L. et al. Plastic accumulation in the North Atlantic subtropical gyre. Science 329, 1185–1188 (2010). Law, K. L. et al. Distribution of surface plastic debris in the eastern Pacific Ocean from an 11-year data set. Environmental Science & Technology 48, 4732–4738 (2014). Goldstein, M. C., Titmus, A. J. & Ford, M. Scales of sptial heterogeneity of plastic marine debris in the northeast Pacific Ocean. Plos One 8, e80020 (2013). ADS Article Google Scholar Jambeck, J. R. et al. Plastic waste inputs from land into the ocean. Science 347, 768–771 (2015). Geyer, R., Jambeck, J. R. & Law, K. L. Production, use, and fate of all plastics ever made. Science Advances 3, e1700782 (2017). Eriksen, M. et al. Plastic pollution in the world's oceans: more than 5 trillion plastic pieces weighing over 250,000 tons afloat at sea. Plos One 9, e111913 (2014). Hoornweg, D. & Bhada-Tata, P. What a waste: a global review of solid waste management. World Bank, http://hdl.handle.net/10986/17388 (2012). Lebreton, L. et al. River plastic emissions to the world's oceans. Nature Communications 8, 15611 (2017). Schmidt, C., Krauth, T. & Wagner, S. Export of plastic debris by rivers into the sea. Environmental Science & Technology. 51(21), 12246–12253 (2017). Ryan, P. G., Moore, C. J., Van Franeker, J. A. & Moloney, C. L. Monitoring the abundance of plastic debris in the marine environment. Philosophical Transaction of the Royal Society B 364, 1999–2012 (2009). Cózar, A. et al. Plastic debris in the open ocean. PNAS 111, 10239–10244 (2014). Van Sebille, E. et al. A global inventory of small floating plastic debris. Environmental Research Letters 10, 124006 (2015). Lebreton, L. et al. Evidence the the great pacific garbage patch is rapidly accumulating plastic. Scientific Reports 8, 4666 (2018). Ye, S. & Andrady, A. Fouling of floating plastic debris under Biscayne Bay exposure conditions. Marine Pollution Bulletin 22, 608–613 (1991). Andrady, A. Microplastics in the marine environment. Marine Pollution Bulletin 62, 1596–1605 (2011). Laist, D. W. Impacts of marine debris: entanglement of marine life in marine debris including a comprehensive list of species with entanglement and ingestion records. In Marine Debris, Sources, Impacts and Solutions (eds Coe, J. M. & Rogers, D. B.). Springer, 99–139 (1997). Porter, A., Lyons, B. P., Galloway, T. S. & Lewis, C. Role of marine snows in microplastic fate and bioavailability. Environmental Science & Technology 52, 7111–7119 (2018). Long, M. et al. Interactions between microplastics and phytoplankton aggregates: impact on their respective fates. Marine Chemistry 175, 39–46 (2015). Michels, J., Stippkugel, A., Lenz, M., Wirtz, K. & Engel, A. Rapid Aggregation of Biofilm-covered Microplastics with marine biogenic particles. Proceedings of The Royal Society B 285, 1885 (2018). Fazey, F. M. C. & Ryan, P. G. Biofouling on buoyant marine plastics: an experimental study into the effect of size on surface longevity. Environmental Pollution 210, 354–360 (2016). Koelmans, A. A., Kooi, M., Law, K. L. & Van Sebille, E. All is not lost: deriving a top-down mass budget of plastic at sea. Environmental Research Letters 12, 114028 (2017). Maximenko, N., Hafner, J., Kamachi, M. & MacFadyen, A. Numerical simulations of debris drift from the Great Japan Tsunami of 2011 and their verification with observational reports. Marine Pollution Bulletin 132, 5–25 (2018). Lebreton, L., Greer, S. D. & Borrero, J. C. Numerical modelling of floating debris in the world's oceans. Marine Pollution Bulletin 64, 653–661 (2012). Artham, T. et al. Biofouling and stability of synthetic polymers in sea water. International Biodeterioration & Biodegradation 63, 884–890 (2009). Yetman, G., Gaffin, S. R. & Xing, X. Global 15 × 15 minute grids of the downscaled population based on the SRES B2 scenario, 1990 and 2025, NASA Socioeconomic Data and Applications Center (SEDAC), https://doi.org/10.7927/H4HQ3WTH (2004). GEBCO. The General Bathymetric Chart of The Ocean. GEBCO_2014 grid available at, www.gebco.net/data_and_products/gridded_bathymetry_data/ (2014). Lavers, J. L., Dicks, L., Dicks, M. R. & Finger, A. Significant plastic accumulation on the Cocos (Keeling) Islands, Australia. Sci. Rep. 9, 7102 (2019). Pierdomenico, M., Casalbore, D. & Chiocci, F. L. Massive benthic litter funnelled to deep sea by flash-flood generated hyperpycnal flows. Sci. Rep. 9, 5330 (2019). The Ocean Cleanup Foundation, Rotterdam, The Netherlands Laurent Lebreton, Matthias Egger & Boyan Slat The Modelling House Limited, Raglan, New Zealand Laurent Lebreton Matthias Egger L.L., M.E. and B.S. designed the study. L.L. developed the model, wrote the manuscript, and prepared figures and tables. All authors reviewed the manuscript. Correspondence to Laurent Lebreton. Lebreton, L., Egger, M. & Slat, B. A global mass budget for positively buoyant macroplastic debris in the ocean. Sci Rep 9, 12922 (2019). https://doi.org/10.1038/s41598-019-49413-5 Accepted: 24 August 2019 Achievements in the production of bioplastics from microalgae Young-Kwon Park Jechan Lee Phytochemistry Reviews (2022) Incorporating terrain specific beaching within a lagrangian transport plastics model for Lake Erie Juliette Daily Victor Onink Matthew J. Hoffman Microplastics and Nanoplastics (2021) Addressing the importance of microplastic particles as vectors for long-range transport of chemical contaminants: perspective in relation to prioritizing research and regulatory actions Todd Gouin Seagrasses provide a novel ecosystem service by trapping marine plastics Anna Sanchez-Vidal Miquel Canals Marta Veny Scientific Reports (2021) Behavioural Mechanisms of Microplastic Pollutants in Marine Ecosystem: Challenges and Remediation Measurements Megha Bansal Deenan Santhiya Jai Gopal Sharma Water, Air, & Soil Pollution (2021) About Scientific Reports Guide to referees Guest Edited Collections Scientific Reports Top 100 2019 Scientific Reports Top 10 2018 Editorial Board Highlights 10th Anniversary Editorial Board Interviews Scientific Reports (Sci Rep) ISSN 2045-2322 (online)	CommonCrawl
How to prevent superluminal traveling idiots from wrecking half of the universe? Linked but not a duplicate of this question. An example of the problem During the last Star-Wars movie, I was dumbfounded by the sheer stupidity of admiral Holdo's move, during this particularly visual scene: While most people I know don't seem to realize how incredibly reckless this was, I'd like to offer one of my favorite quotes ever to summarize the situation: Gunnery Chief: This, recruits, is a 20-kilo ferrous slug. Feel the weight. Every five seconds, the main gun of an Everest-class dreadnought accelerates one to 1.3 percent of light speed. It impacts with the force of a 38-kiloton bomb. That is three times the yield of the city buster dropped on Hiroshima back on Earth. That means Sir Isaac Newton is the deadliest son-of-a-bitch in space. Now! Serviceman Burnside! What is Newton's First Law? First Recruit: Sir! An object in motion stays in motion, sir! Gunnery Chief: No credit for partial answers, maggot! First Recruit: Sir! Unless acted on by an outside force, sir! Gunnery Chief: Damn straight! I dare to assume you ignorant jackasses know that space is empty. Once you fire this husk of metal, it keeps going till it hits something. That can be a ship, or the planet behind that ship. It might go off into deep space and hit somebody else in ten thousand years. If you pull the trigger on this, you're ruining someone's day, somewhere and sometime. That is why you check your damn targets! That is why you wait for the computer to give you a damn firing solution! That is why, Serviceman Chung, we do not "eyeball it!" This is a weapon of mass destruction. You are not a cowboy shooting from the hip! Second Recruit: Sir, yes sir! (Emphasis mine - quote from an anonymous human military on the Citadel in Mass Effect 2) So, for those who've been sleeping in the back of the classroom during their physics class, Holdo accelerating a ship at a speed faster than light and ramming it into another ship is the equivalent of a cosmic shotgun. (Hence the pretty little light streaks scattering in a cone from the impact point and instantly destroying Star Destroyers). A single bolt of 1g from this ship now has a bare minimum kinetic energy of $E = 0.5 \times m \times C^2 = 0.5 \times 0.001 \times 9 \times 10^{16} = 4.5 \times 10^{13} J$. 1 (A kiloton is $4,184 \times 10^{12} J$, so the absolute minimum energy on impact, if physics didn't break at that point, as explained in comment is around 10 kilotons. Hiroshima's Little Boy was estimated between 12 and 15 kilotons). To be honest, I'm not even sure this law holds at speeds higher than light, and I don't have the theoretical knowledge to even make an educated guess. Given that some several-years travel at the speed of light could be done in mere hours in the SW universe, I'd posit that those numbers are way higher, and each bolt (not even talking about chunks of the ship) are now delayed orbital strikes aiming god knows where. At this point, you might as well charge the Rebellion for crime against the universe. No wonder why the Yuuzhan-Vongs paid us a visit. The other question explains perfectly the problem. As soon as your universe includes FTL travel, an idiot somewhere is gonna make this mistake (on purpose or not) and a lot of people are gonna pay for it. Following the example above, I struggle to create any universe with FTL travel, because each war would mean I'd have to wipe half of the celestial map. A few propositions to protect a planet or a system from this kind of incident includes: giving the person using FTL the means to avoid said incident, trusting them to understand the risks, pre-emptive strike, or (my favorite) clouds of space dust. Now, the two first answers are made irrelevant by idiocy. Holdo knew the risks and had the computers telling her not to do it. By hubris, despair or idiocy, someone in a space battle will end up pushing the red button. (You don't even have to sacrifice a pilot. Guided FTL rockets are the end-game) Pre-emptive strike seems a bit radical. While they're targeted and shouldn't cause collateral damage, you can't just destroy every planet where FTL "might" happen. The cloud of dust is useful to protect a single system or planet, but there is no way to effectively shield the universe, unless you want to fill all empty space with space dust. Is there any way to devise a universe with FTL travel without realistically condemning half of said universe to utter destruction by FTL strikes? (Not asking how to shield a sole planet / a sole civilisation from a dumb accident) Note that any reactionary counter measure suggests you know that a danger is coming your way. The problem of FTL is that the danger travels faster than the information. You'll "see" the explosion way after the upper deck of the destroyer tore through your planet. And the one behind. And the one behind the one behind. (But maybe quantum entanglement can help. I've read somewhere about research being done on the topic to communicate faster than light, but I don't understand the principle behind it). To clarify what I'm asking for, I'm looking for references of universes with non-destructive faster than light space-travel, effective countermeasures covering the universe, or anything that allows you to write a story including both FTL and idiots without dramatic consequences. 1 Fixed thanks to elPolloLoco's answer. Don't do maths absent-mindedly during lunch break without double checking the data. weapons space-travel weapon-mass-destruction faster-than-light security NyakouaiNyakouai $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – L.Dutch ♦ $\begingroup$ "To be honest, I'm not even sure this law holds at speeds higher than light"—it does not: any massful particle requires infinite energy to get at the speed of light, and there isn't more than infinite energy available for accelerating it further. $\endgroup$ – Jan Hudec $\begingroup$ There are a number of problems with this question, though the leading two answers offer reasonable outs for a Worldbuilder. However, the biggest problem is a substantial misapprehension of scale, that any worldbuilder should correct before proceeding. You see Space is big. Really, really, really big. So big that the numbers that you site are nigh-trivial in comparison. Consider a supernova, they are many orders of magnitude larger, more energetic and more dangerous than any gun made by humans. But how much of your world, is gone because of supernovas? Add 0.01% onto that. $\endgroup$ – RBarryYoung $\begingroup$ To elaborate on what @JanHudec said, special relativity has it at $E=m \frac{1}{\sqrt{1-v^2/c^2}} c^2 - mc^2$, so the energy goes up asymptotically as you approach the speed of light. $\endgroup$ – AJMansfield $\begingroup$ You have much bigger problem anyway. Whatever energy your ships have must have been provided by the engines. If you have engines capable of providing that sort of energy, there are many more creative ways of releasing it in a very destructive manner. $\endgroup$ But in most stories, FTL doesn't actually follow Newton's 1st law; everybody knows that once your hyperdrive fails, you're dead in the water. Otherwise, the Enterprise could hit warp 9.9 and easily coast across the galaxy. And that actually makes sense in the warp-style of FTL: FTL travel is only possible within the field generated by a functioning FTL drive. In fact, within the reference frame of that field, matter still isn't moving faster than light! Once the drive is destroyed, the field collapses and that matter resumes its non-FTL velocity. But regardless, there are many other FTL concepts that operate completely differently. See Hyperspace, Jump drives, Jumpgates, and more in Wikipedia's scifi section. BoomChuckBoomChuck $\begingroup$ Yes! With this type of setup, what you have is a ship approaching an object at whatever it's "impulse speed" is, but wrapped in a bubble/field/whatever that changes the space around it. When any other object (another ship, planet, etc) enters that bubble/field and collides with the ship, the speed of the collision is the same as it would have been if neither of them were in the bubble at all, nothing more than "ramming speed". Could still cause massive local destruction of the two colliding objects, but nothing like a relativistic shotgun $\endgroup$ – Harthag $\begingroup$ A lot of those other FTL concepts operate similarly. You travel FTL in a different layer of subspace (or w/e), not in real space. Or through a temporary, artificial wormhole (or equivalent). All real space travel is sublight (and usually quite far below light speed). So the weapon potential is limited to dropping a package behind enemy lines, not ramming. $\endgroup$ – jaxad0127 $\begingroup$ As a matter of fact, warp bubbles solve nothing, only change threat profile. What happens when someone flies through a planet or star? I can tell you because star trek styled warp drive is not prohibited by general relativity and some researchers looked into that: grossly simplifying, matter scooped up by bubble is converted into radiation and released at destination. We are talking supernova type event. $\endgroup$ – M i ech $\begingroup$ "matter scooped up by bubble is converted into radiation and released at destination" Converting 100% of the mass of a single atom into light would release such a massive amount of energy... Every speck of dust you fly through would turn into an ultra death laser. I'd like to read that research, if you have a link. $\endgroup$ – BoomChuck $\begingroup$ Explained succinctly here in Futurama. $\endgroup$ – eyeballfrog 1. Space is mostly empty If you pull the trigger on this, you're ruining someone's day, somewhere and sometime Well, not necessarily. There's so much empty space between anything interesting in the universe, that it's very likely that those "shots from a cosmic shotgun" will just keep traveling through the cosmic void, never even approaching anything worth considering. Additionally (as @DanDryant pointed out), objects traveling faster than the speed of light can escape any gravity well, and their trajectories wouldn't even change much when passing through one. This means that superluminal objects are less of a risk compared to "slow" moving objects - the first need a direct hit on their first and single pass through a system, while the latter can get caught in the gravity well - going around and around, increasing their odds of hitting the massive object in its center or some other objects caught in that well. If you are specifically worried about space battles, where missing the tiny enemy fighter ships can mean bombarding the planet you are defending (or its sun!), just have these battles typically occurring far away from planets. From a worldbuilder's perspective, you can achieve this by two different strategies, used independently or combined: 2. Defensive technologies or circumstances Intelligent races will just have to invent and deploy defensive technologies to survive the "superluminal idiots" of the universe. Just as the Chinese Empire built The Great Wall of China to surround their entire territory. Any FTL capable civilization will need some sort of defense: Planetary shields that block, disintegrate or teleport any unauthorized/unexpected approaching objects. Scattering artificial "cosmic dust" / interceptor drones / etc. around important systems / sites / stations - anything that can absorb or dampen FTL projectiles. Planetary or system-wide Warp Inhibitors, slowing down anything (or anything unauthorized) traveling towards it. Strategic positions (inside nebulae, next to black holes etc. etc.) making any FTL dangerous - these can help even non star faring races to survive. 3. Limitations or variants on real-space FTL As the worldbuilder, you can tweak physics or the nature of FTL so it is more difficult to weaponize, at least against settled planets: No FTL next to gravity wells - this was used widely in Larry Niven's Known Space universe. FTL only works on relatively "flat" space - forcing space battles to commence at civilized speeds or far away from solar systems. FTL is warp based rather than simply traveling very fast in real-space (that's the gist of Boomchuck's answer). This provides ships the ability to travel astronomical distances in reasonable time without gaining an insane amount of kinetic energy. It also means that debris of an FTL collision won't be traveling at superluminal speeds (though they can still be dangerously fast...). A typical trope of this approach demands an intact "warp bubble" around the FTL ship - which is dispersed if it interacts with another bubble (and or gravity wells etc.). FTL is teleportation based. Your interstellar civilizations use stargates or hyper-jumps to skip from A to B without traversing the distance between them. FTL is done in Hyper space rather than real space - a ship slips to another dimension, where it cannot interact with anything in this dimension, and where the physics are different to allow fast traveling (or distances are different there - same thing really). Ships may each have their own exclusive variant dimension, or they may interact with other hyperspace travelers (possibly only using a compatible technology / frequency etc.). Finally, keep in mind the principle sometimes called the Kzinti Lesson: "A reaction drive's efficiency as a weapon is in direct proportion to its efficiency as a drive." If you have a functional interstellar drive, it will be very difficult to completely prevent its weaponization. Atomic Rockets have an entry about propulsion systems in their exotic weapons page - it's really worth reading. G0BLiNG0BLiN $\begingroup$ It also helps that if you're accelerating things to significant fractions of the speed of light, you're pretty much guaranteed to exceed escape velocity for the system, regardless of the direction of your velocity vector. The only thing you need to avoid is a direct hit, which is incredibly unlikely thanks to the Space is Really Big situation. And, even if you're fighting around a planet, any velocity you add is presumably going to be in addition to your existing orbital velocity, so you're unlikely to hit the planet unless you deliberately try to. $\endgroup$ – Dan Bryant $\begingroup$ How empty is the space? Well the density gives it away: ~10^-29 g/cm3. Density of lowest density element is hydrogen with ~10^-4 g/cm3. So basically your chances of hitting something when fired in a random direction in open space is 10^-25. Of course if you are firing towards a cruiser orbiting a planet, that is a different story. $\endgroup$ – Cem Kalyoncu $\begingroup$ The Kzinty Lession reminds me of Maxim #24 from Schlock Mercenary: "Any sufficiently advanced technology is indistinguishable from a big gun." $\endgroup$ – Mason Wheeler $\begingroup$ Luckily for us, ancient colonies on Earth deployed a planetary shield that's still functional today. We now call it "the atmosphere". $\endgroup$ – Vaelus $\begingroup$ +1 for mentioning Atomic Rockets. Anyone asking a question about space should be force to read Atomic Rockets first. $\endgroup$ – Tom First of all, this is wrong: "A single bolt of 1g from this ship now has a bare minimum kinetic energy: $$E = m \times C^2 = 0.001 \times 9 \times 10^6 = 9 \times 10^3 J$$ Kinetic energy is calculated as $0.5mv^2$ $v$ being $c = (3 \times 10^8) v^2 = 9 \times 10^{16}$ So you get $$E = (10^{-3}) \times 9 \times (10^{16}) \times 0.5 = 4.5 \times (10^{13}) J$$ But it doesn't really matter does it? Now here is the thing: This formula is not suited for speed close to $c$. That is because close to the speed of light, the mass of the object changes.[Edit: the mass of the object does not change, the formula is still useless at speeds close to c, reasons to be found in comments] Maybe you once heard that it takes an infinite amount of energy to bring an object with mass to the speed of light? Using this formula, it wouldn't. Now let's get to your problem. Obviously having all kind of people flying WMDs is kind of problematic and wouldn't do the galaxy any good. FTL by just flying really fast is already a violation of physics, so just leave it out. You need your characters to be able to move between starsystems or even galaxies in little time? Give them some kind of warp- engine or wormholes or some other magic travelling aid that moves them from A to B without accelerating them and so without putting insane amounts of energy into them. Edit: See the Ender series for example. I think FTL is introduced in "speaker of the dead" or "Ender in exile". elPolloLocoelPolloLoco $\begingroup$ I looked for the value quickly and the final result seemed me weak. Thanks for pointing out the mistakes $\endgroup$ – Nyakouai $\begingroup$ "Now here is the thing: This formula is not suited for speed close to c. That is because close to the speed of light, the mass of the object changes. Maybe you once heard that it takes an infinite amount of energy to bring an object with mass to the speed of light? Using this formula, it wouldn't." I remember reading your mass increases when nearing C, but I never learned why. So assuming your bolt of 1g now has an exponentially increased mass and is travelling faster than C, as far as I understand it, the kinetic energy is way off the charts tby breaking laws of physics). $\endgroup$ $\begingroup$ Me neither. It is being discussed on physics stockexchange: physics.stackexchange.com/questions/1686/… There is no faster than c traveling. To accelerate a body with mass to exactly c, the amount of energy needed is already infinite $\endgroup$ – elPolloLoco $\begingroup$ In the Ender series, FTL is introduced in "Children of the Mind". It's an ability that Jane "learns", IIRC $\endgroup$ $\begingroup$ FWIW the formula for kinetic energy, which applies even at speeds close to that of light, is $mc^2(1/\sqrt{1 - v^2/c^2} - 1)$. In this formula, $m$ (the mass) is an intrinsic property of the object and does not change depending on its speed. $\endgroup$ – David Z Is there any way to devise a universe with FTL travel without realistically condemning half of said universe to utter destruction by FTL strikes? My personal favorite answer to this is E.E. "Doc" Smith's FTL approach from the Lensman series: Inertialess drives. Basically, you turn the inertialess drive on and immediately assume the velocity at which all forces acting on your ship cancel each other out. If you have a reasonably-powerful engine pushing in one direction, this will mean moving at an extremely high multiple of the speed of light, because that's what it will take for the drag from interstellar gases to match the power of your engine. But, if you run into a planet, you stop instantly with no impact effect, because the resistive force of the planet is enough to cancel out your engine's thrust. This also makes most projectile or energy weapons useless when used from a lone attacker, as the impact of the attacker's guns or the light pressure of their lasers will simply push the defender away with no damage inflicted. The only effective means of space combat is for large fleets to form up as cylinders, cones, or spheres around the enemy and crush them with simultaneous weapons fire from all directions. When the inertialess drive is turned off, you will immediately resume your original vector. Although this eliminates the problems of superluminal impacts and accidental planetary devastation, it still allows for deliberate planet-killer attacks, such as finding another planet moving on the opposite vector, taking it inertialess, and inserting it into the target world's orbit. The kinetic energy of a 20kg slug at 0.13c is peanuts compared to a head-on collision between two planets at typical orbital velocities. And, if that's not destructive enough, in the later books of the series, they start finding antimatter planets to do this with because, sometimes, there's no such thing as too much overkill. Dave SherohmanDave Sherohman $\begingroup$ Ahem. ITKM There is no "overkill." There is only "open fire" and "reload." $\endgroup$ – Martin Bonner supports Monica $\begingroup$ @MartinBonner More applicable is Rule 24: Any sufficiently advanced technology is indistinguishable from a big gun. $\endgroup$ – Ray $\begingroup$ and not just antimatter planets (that they create.) The ultimate overkill is they find planets from an alternate universe where the light speed barrier is reversed (ie. all "normal" matter is superluminal.) They drag a couple into normal space, and drop them inert (ie not under inertialess drive) aimed at a planet (and its sun) so they get by a planetary-mass superluminal projectile each. (hilariously, it;s noted that even this wasn't enough, as the big bad guys would have been able to duplicate the weapon within weeks, and then that would have been all she wrote.) $\endgroup$ – Wenlocke $\begingroup$ This was going to be my answer. The fact that an inertialess object just stops when any outside force is exacted upon it is beautiful. Though, as you point out there are so many other dangerous shenanigans you can get up to with Inertialess drive. $\endgroup$ – ShadoCat $\begingroup$ There's No Kill Like Overkill $\endgroup$ – Bob Jarvis - Reinstate Monica Kinetic Energy at High Velocities If you design a universe that permits faster than light travel, you will need to invent laws for how kinetic energy works when traveling faster than light. For centuries, we used the Newtonian formula for kinetic energy: $KE = \frac{m v^2}{2}$. And that worked well. It isn't wrong; it's just an approximation that only holds for velocities well below the speed of light. But for high velocities, we need to use Einstein's version instead: $KE = \frac{m c^2}{\sqrt{1 - v^2/c^2}} - m c^2$. (m is the rest mass in that formula, not the relative mass.) We could also use this for low velocities if we wanted to; it's approximately equal to $\frac{m v^2}{2} + \frac{3m}{8}\frac{v^4}{c^2} + \frac{6m}{16}\frac{v^6}{c^4} + ...$, so all terms except $\frac{m v^2}{2}$ will be very close to zero. But for that same reason, we don't need to bother with any term except the first when $v << c$, so it'd be overkill. We instead use the simplest approximation that works well at the sort of speeds we're looking at. Now let's look at what those equations mean in practice. Suppose we have a 1 kg mass and see how much energy these predict at various speeds. Low speed: 10 m/s If it's moving at 10 m/s, then using the Newtonian approximation, we get a kinetic energy of $50 J$. If we use the Einsteinian version, we get $50.00000000028 J$. Both versions are exceedingly accurate, and so we might as well use the simpler one. Relativistic speed: 0.99c The Newtonian approximation gives us $3.64 \times 10^{16} J$. The Einsteinian one gives us $2.58 \times 10^{17} J$. The Newtonian version is eight times too small; at these velocities, we can no longer use that approximation. FTL speed: 2c The Newtonian approximation gives $1.8 \times 10^{17} J$, but we don't care because we've long since passed the velocities where it's remotely accurate. The Einsteinian approximation gives us $-1.038\;i \times 10^{17} J$. Since our kinetic energy is now both negative and imaginary, it's safe to say that this approximation is no longer accurate (or even meaningful) at faster than light velocities. (It's not even defined at exactly light speed; there's a division by 0.) In our universe, the Einsteinian equation may not be an approximation; it may be completely accurate at all velocities for all we know. But if FTL is possible in your universe, than it must be an approximation that is only valid at low speeds. I can't tell you what equation you should use for calculating kinetic energy at FTL speeds; I can just say that the one you're using will not give accurate results in our universe, and the best approximation known to modern science for our universe doesn't even permit FTL, so that won't be right, either. Decide how you want kinetic energy to work at FTL speeds, and then be consistent. Energy Scale You don't actually have to make up a new equation if you don't want to, though. You can safely assume that Newtonian physics hold at any velocity in your universe without creating the sort of problems that you're envisioning. $10^{13} J$ may sound like a lot, but that's by puny Earthling standards. We've tested nukes that are 20,000 times larger than that, and we're all still here. Let's consider something with a bit more energy than a mere hydrogen bomb. Suppose that Newtonian physics holds at high speed, and that we've got a ship the size of the Titanic moving at light speed. It would have a kinetic energy of $2.337 \times 10^{24} J$. That's a decent amount of energy, but it's 100 times less than the Sun outputs every second, and 100 million times less than what we'd need to blow up the Earth. (Which in turn means that the scene you mention was not the most energetic event we've seen in Star Wars.) To put things into serious perspective, a supernova can output $10^{44} J$. That's $10^{20}$ times more than our ship's kinetic energy. Which is to say, our ship colliding with something outputs at least 100,000,000,000,000,000,000 times less energy than something that happens in the universe on a fairly regular basis. $\begingroup$ You need to correct Einstein's version of the KE equation above. Where it should read "mv^2" it reads "mc^2". It's a simple typo, we all make them. The good thing is they're easily fixed. Otherwise it's a great answer. Plus one for putting numbers into FTL physics. $\endgroup$ – a4android $\begingroup$ @a4android $mc^2$ is actually correct. $m$ is the rest mass, so it should more properly be displayed $m_0$, but I left it as $m$ so as to more easily compare it to the Newtonian version. $m_0/\sqrt{1-v^2/c^2}$ is the relative mass. $m_0/\sqrt{1-v^2/c^2} \times c^2$ is the total mass energy, and $m_0 c^2$ is the rest mass energy, which we subtract from the total to leave just the kinetic energy. Thus, $m_0 c^2/\sqrt{1-v^2/c^2} - m_0 c^2$. ($mv^2/2$ emerges in the Taylor expansion of that formula.) (source. chapter 15) $\endgroup$ $\begingroup$ Your clarification is much appreciated. May I suggest editing your answer, but specifying that the mass is the rest mass. That would improve it and not confuse dumbos like yours truly. $\endgroup$ Space is big and very very empty... but it isn't totally empty. There is a fine mist of dust and gas and atoms between the stars. It's not much. The cold depths of interstellar space have a purer vacuum than almost anything you'll find in a lab on earth. But if you go fast enough it really matters. In the solar system with the solar wind the density of atoms is 2x10^7 per cubic meter, mostly hydrogen or helium. Outside the solar system it varies quit a bit by temperature and charge but if you take a steel ball and throw it into the cold dark night... Lets see what happens if you hit something, easy (scaled down to keep the photographer alive) But what if you miss... The 20-kilo ferrous slug leaves the ships railgun at 1.3% of the speed of light. The slug is pure iron. The sphere has a radius of 8.464 centimeters. It's going at 1.3% of the speed of light so lets wait about 1.3 light years away and see what it looks like when it arrives. Hmm. It's a little bit late... not very late but a little bit... In order to reach us the sphere has had to pass through 2.768×10^20 cubic centimeters of "empty" space A Cold Neutral Medium (CNM) in interstellar space has about 20-50 atoms per cubic cm mostly hydrogen or helium. We'll assume 25 hydrogen atoms per cm^3 to make this easier. So while it's traveled it's hit (2.768×10^20 *25) atoms of hydrogen... about 11.58 mg (milligrams) of hydrogen. Assuming it doesn't hit a grain of sand long the way and turn itself into a cloud of atoms. That's not a lot... but by the time the slug has traveled 2,245,230 light years it's struck about 20kg of hydrogen atoms. Notice that that's it's own weight in hydrogen. That's a long way, almost the distance to Andromeda. So by then it's going about half it's original speed and some of the iron has likely ablated away from those little impacts. The universe doesn't fill up with high speed debris because every piece of debris experiences a tiny and subtle drag from hitting the fine mist of atoms between the stars. MurphyMurphy $\begingroup$ You managed to make an absolutely coherent answer with a really good explanation as why the problem isn't one while missing the significant bit about the projectile being FTL and not the slug of the quote. (Though I'm curious to know if the answer differs at speed higher than C since laws of physics get messy/do not exist) $\endgroup$ $\begingroup$ @Nyakouai projectiles can't be ftl. It's like saying a rock is going "slower than stopped". Other issues include the problem that anything that allows something to go FTL is automatically a time machine as well. Even in most scifi, you need a magi-teck FTL engine to keep going FTL. The debris from any impacts or explosions will always be limited to slower than light because they would require infinite energy to send them flying out from the explosion faster than that. Any object thrown out then experiences the same issue as the slug. $\endgroup$ – Murphy $\begingroup$ @Nyakouai If you edit the laws of physics to be Newtonian rather than Einsteinian with no speed limit at the speed of light, no relativity and no time travel... then the same logic applies. it still hits dust and a fine mist of atoms and gradually slows while it crosses the universe. $\endgroup$ tl;wr: Based on Star Wars canon: Gravity sucks you out of hyperspace. At least in the written Star Wars universe it is canon, that while you may travel faster than light (no details known on how that works), ships can not fly through gravity wells. Meaning, before a ship travelling FTL hits a sun or a planet, it gets sucked out of hyperspace and has to travel STL. The same should go for any matter/particles/shipwreck parts which are propelled to FTL by any possible freak "accident". Ships are usually not large enough to generate a gravitic field of their own, so basically after disabling all security measures etc. Admiral Holdo would have been able to execute the pictured manoeuvre, resulting in an at least close-to-lightspeed blast of shrapnels, shredding half of the fleet. And while that shrapnels may race through space for a very long time and distance, at least planets are rather safe from them, due to their solar systems gravity wells. Any debris would slow down to anything STL and at least partially burn in the atmosphere. Sure, there could still be some damage, but the total number of freak accidents and thus shrapnels is to be estimated rather small. The lone ship in space could get unlucky though, to be honest. Thought on the side: It would be interesting to know, whether the Death Stars were massive enough to create their own gravity field. The movies are - sadly - another topic. At least in Episode VII Disney wrecked the canon when the Millenium Falcon travelled through hyperspace right above the surface of Starkiller Base, below its energy shields. A jump actually impossible since the gravity should have sucked the Falcon out of hyperspace long before it got there. I can't cite numbers here, but I estimate that, when it comes to planets you have to be at least in a geostationary orbit or higher in order to be able to jump into hyperspace - meaning, you'd be about the same distance from the planet, when you get sucked out of hyperspace. Suns are even another topic, since they are even more massive and have very large gravity wells. E.g. in either one of the Thrawn Trilogy books or one of the X-Wing-series books a New Republic spy with an Imperial background gets kind of pressed back into Imperial service, placing them on the command ship of either Thrawn or an Imperial Warlord. They manage to delete some navigational data, leading to the Star Destroyer being sucked out of hyperspace by a suns' gravity and stranding the ship effectively for several hours or even days, throwing off the imperial time plan and thus saving the day. Also Grand Admiral Thrawn himself makes very good use of the few "Interdictor" ships he controls. Those are star destroyers equipped with powerful gravity well projectors, able to pull ships from hyperspace. Sure, you have to know where the enemy will pass through, but the man was brilliant and thus several times trapped unprepared New Republic convoys or whole fleets right before the guns of his fleet, the most prominent - and also last - example being the battle of Bilbringi, during which Thrawn is murdered by one of his alien bodyguards. ErikErik $\begingroup$ "At least in the written Star Wars universe it is canon" isn't that not canon any more? $\endgroup$ – VLAZ $\begingroup$ One movie contradicts it, the others don't really speak about it. $\endgroup$ – Erik $\begingroup$ When Disney acquired the Star Wars license, they declared the Expanded Universe materials non-canon and rebranded them as "Legends". In-universe they would be the equivalent of hearsay or "my friend's third cousin's aunt once heard". So the only "true" materials right now, as far as SW universe is concerned, are the movies. $\endgroup$ $\begingroup$ Sounds like politicians declaring climate change isn't real. $\endgroup$ $\begingroup$ Nothing really like that. $\endgroup$ It's not "faster than light", it's "shorter than space" If your FtL drive functions by moving your ship outside of normal space, and then back into normal space some distance away, then countermeasures are easy: any device that prevents reentry into normal space makes that space impregnable to FtL intrusion. Schlock Mercenary In the earliest days of the comic, FtL travel was done using Wormgates - enormous portals that ships could move between instantaneously. In this case, the energy cost of wormholes increased exponentially with the size of the wormhole, meaning that any wormhole large enough to transport a ship required its own infrastructure - hence the wormgates. Then the protagonists invented a new FtL system called the teraport. It functions by creating a ridiculous number of tiny wormholes, and then pushing an object's individual molecules through the wormholes and reassembling them on the other side. (Hence the name 'tear-apart' 'teraport'). Because of the way the energy costs scaled, the teraport was extremely energy-efficient compared to a wormgate, and a device capable of teraporting a person could be held in your hand. This immediately revolutionized warfare, because it turns out that being able to drop a bomb anywhere you want on your enemy's ship (or house) makes a battle very short. Of course, they can do the same... The Teraport Wars came to an end with the invention of Teraport Area Denial, which projects a field which prevents the opening of the tiny wormholes that the teraport uses. (higher energy wormholes can punch through a TAD field, but the whole reason the teraport is usable is because of how low energy the individual wormholes are). And since the teraport process requires ripping the objects being moved into infinitesimal chunks and then reassembling them, interrupting the process can be rather messy for those involved. Arcanist LupusArcanist Lupus $\begingroup$ +1 for quoting a fundamental principle from Gallifreyan transdimensional engineering. $\endgroup$ – A. I. Breveleri One of the quirks of travelling at such speeds in "normal space" is that a stationary rocks are effectively travelling at that speed relative to you as you crash into them. That's true of any particle or object between your point of departure and your point of arrival. What's important to note is that physics doesn't care whether I punch you in the face or you headbutt my fist, the outcome is the same. In other words, in a universe with normal space ftl, it is essential to have some easily accessible form of resistance to simple objects at relativistic speeds. What this means is that the mass driver described in the question is a useless weapon in ship to ship combat as it wouldn't even reach the same level of damage as hitting a rock on your way to the battle. Since ships tend to be aimed at planets and boosted up to such speeds, it's perhaps not unreasonable to shield your planets as well. You don't need to worry about species that don't think of this, or don't have a solution to it, as they'll either never make it out past their own Oort cloud or blow up their own planet long before they much trouble the rest of us. SeparatrixSeparatrix $\begingroup$ In Isaac Asimov's novel Nemesis anything travailing at > c has the sign of g flipped. So FTL objects repel rather than attract. $\endgroup$ – JGNI $\begingroup$ @JGNI, but gravity propagates at the speed of light so that wouldn't affect objects in the ship's path $\endgroup$ – Separatrix Independently of all the in-universe specific scenarios described and calculated so far, I think in any setup where your technology allows you to accelerate matter close to c (or above) in normal space, you will have the rough equivalent to today's landmine and explosives disposal units. Your rookie Gunner shot the Everest main gun by mistake? Call in the Sweepers ! They will calculate the round's position in normal space and, while you slap the Gunner, will move to intercept and dispose of the round diligently with means appropiate to the technological background of your Universe. For Mass Effect, that would actually include calculating the Gate it is going to be closest to for practical interception and schedule it for sometime in a few decades. For Niven's Known Space, you can be much more accurate and intercept inside Oort's cloud in hours or days. Say you had a mayor battle with lots of potential debris and lost rounds. Sweepers of winning faction will calculate where the spherical front of debris and rounds is, based on used weapons known speed, the sectors of the sphere posing a risk to the elliptical plane of nearby populated stars, and setup clean up patrols spanning for years or decades. Would not be too different from the efforts to clean up the Zone Rouge in France post WWI. You do not need to clean up the whole expanding sphere, just the parts heading to known settelments, the rest can be integrated in navigational charts with big red signs "Everest rounds on the loose. Drop off to normal space at your own risk". To deal with unreported accidents, the odd Evil Scientist, and forgotten civilization's debris from centuries ago, you will have to trust in local defense measures with integrated protocols to deal with incoming garbage at c speed as pointed in answers above. Why is any of this never described in stories? Because everyone in-universe gives it for granted and nobody cares to worry or followup, same as we do not see the news opening with every landmine field or random bomb from 50 years ago neutralized... unless you have to evacuate half of Hamburg because of it. It would actually make for nice background story material. SeretbaSeretba $\begingroup$ Personally, my favorite answer. It's a fun and also new possibilities for plot hooks. May not go in depth about relativity problems, but really under-appreciated and deserve more recognition, in my opinion. $\endgroup$ Since no one answered it I'll go with the FTL answer from a special universe you might be acquainted with: our own. The Alcubierre drive is derived from the equations of Relativity, and would allow the object to travel faster than light without all the problems that would normally entail. Below you can first find a publication with the benefits quoted in this answer, but I suggest you read the article as well. Below that is a video of PBS Space-time about the subject, I dont know how reliable they are but apparently the speaker does research in Quasars so the channel should have more reliability than the average site. https://www.researchgate.net/publication/258317793_The_Status_of_the_Warp_Drive Notable are the advantages in this paper: Benefit 1: Removal of interstellar distance barrier, as no longer restricted to subluminal speed limitations. Get faster than light travel, as measured by distant observer outside of disturbed region. This will allow missions to the nearby stars and closer examination of astrophysical phenomena than is possible today. Benefit 2: It is a conventional transport scheme, in that it requires no 'tearing' of space or non-trivial topologies (i.e. wormholes) and does not require the transmission of copies of objects across space as a means of getting to the destination (i.e. teleportation).Warp drive is a simple transport from origin to destination through space. Benefit 3: No time dilation effects, as usually expected with other space propulsion schemes due to special relativity. This is because the vehicle could be moving at subluminal speeds so that clocks on board would remain synchronized with the origin and destination. Benefit 4: No relativistic mass increase of vehicle, since ship is at the centre of warp bubble is at rest with respect to locally flat space. Benefit 5: No requirement for rocket type propulsion to achieve near light speed, which usually restricts the maximum speed attainable due to special relativistic effects such as infinite thrust for infinite masses. Benefit 6: Technological and economic benefits to mankind. (1) (PDF) The Status of the Warp Drive. Available from: https://www.researchgate.net/publication/258317793_The_Status_of_the_Warp_Drive [accessed Jan 29 2019]. For some background information you can watch this: https://www.youtube.com/watch?v=94ed4v_T6YM DemiganDemigan Idk, man. Just use a normal Alcubierre drive or basically any Space-time warping FTL. Since these drives move the space around the ship instead of the ship itself, a ship with a failed drive will exit warp at the same (sublight) speed with which it entered, which may even be a state of rest. And as far as the SW portion is concerned, I would like you to know that Shit-do ramming was completely lore-inaccurate, and a result of Disney, as usual, confusing hyperdrives with warp drives. Hyperdrives send a ship into a separate dimension, where they move much larger distances in realspace with every unit they move in hyperspace. Of course, objects which generate large mass shadows (think the gravity wells of planets, stars or black holes) generate "mass shadows" in hyperspace, which pull a ship back into normal spacetime of entered and may even destroy it (this is why ships cannot jump anywhere near a planet). The Supremacy, though large, will not generate a mass shadow even close to what is required, and even if it did, the Raddus would only collide at sublight speeds. Thus, that scene broke lore very badly. Budhaditya GhoshBudhaditya Ghosh $\begingroup$ Maybe there's a brief moment of warp speed before the ship gets into the hyperspace dimension? That would explain the directional flash that comes right after or before a ship goes into or out of hyperspace. So perhaps Raddus was rapidly accelerating but never made it into hyperspace. $\endgroup$ – LukeN $\begingroup$ The distance was too great to not make it into Hyperspace. There was a scene (probably in Rogue One) where a Rebel Cruiser successfully jumps to hyperspace right in front of an ISD. $\endgroup$ – Budhaditya Ghosh From a Traveller pen-and-paper RPG point of view: FTL travel is more like "wrinkle in time" travel than "really really fast" travel. The jump drive steps you into jumpspace, you fly along at no net change in realspace velocity or heading, and a week later you pop out of jumpspace as if nothing happened, except you're now a few light-years away from where you started. Almost like a side-effect of something else happening. That said, even Traveller struggles with the fraction-of-C-rock bomb effect. rjerje $\begingroup$ You don't need FTL for that to be a problem. You just need a sufficiently powerful drive and enough time. $\endgroup$ – Keith Morrison $\begingroup$ +1 Too right. So the original question is not "really" about FTL either! $\endgroup$ – rje Sir! An object in motion stays in motion... unless acted on by an outside force, sir! You don't need space dust. There are already known outside forces in space which cause drag. The solar wind from our sun is an example. Whatever speed the ship pieces were at, they would not continue at that speed indefinitely. From a practical "world building" view this isn't widely known/understood/appreciated, so I'd go with "the lack of a functioning hyperdrive" making this irrelevant, instead of the physics.stackexchange.com view. To be blunt, some things are best ignored. Like if you allow a FTL bullet and miss a ship but hit a planet (or maybe it deflects off the ship) the inhabitants who aren't hit still have to deal with the burst of x-rays, gamma rays, and scattered particles caused when the near light speed debris hits their atmosphere. You can see the effects of a baseball traveling at 0.9c here (https://what-if.xkcd.com/1/) the physics of which appear to be legitimate and then extrapolate in your head what an item of larger mass would do. J. Chris ComptonJ. Chris Compton For this answer I'm assuming you want a ship to be able to get to a destination FTL instead of just being able to move FTL. If you just want a way to give FTL travel but none of the kinetic headaches why not just have a drive that folds space on itself and the ship transposes itself into the new location. It would drastically reduce the effectiveness of using FTL as a Military weapon. Defense against it would be a generator that creates a field that causes improper folding and sends the FTL somewhere else. Making all/Most/Some planets equipped with these generators would prevent drunk Aliens from transposing whatever they want anywhere on the planet. Wouldn't be able to use it to rob a bank. If that is still to powerful just make it so that any type of FTL needs an end anchor point to guide it and that using it without it would just scatter your energy across the universe because of [Insert Tchnobable] TolureTolure The FTL drive of the ship might have a, for now, undiscovered physical mechanism to propell the ship beyond c. However this mechanism works in detail, it might only be able to affect a confined system (like the spaceship). So even if your superluminar vessel "hits" another ship or even just an asteroid: it could transfer absurd amounts of engergy/momentum, but the striken object might not accelerate beyond c itself. So the excess of kinetic energy might just be instantly transformed into heat or radiation. At worst you may have a flaming ball of plasma shooting near the speed of light towards a random destination but this isn't really "unusual" in our universe? ZeodynZeodyn Consider that a black hole is already dragging in light, it has "FTL" effects. Perhaps your black holes are your inter-galactic defenders. The shrapnel will be subject to getting chomped up by sentient and hungry black holes roaming within and without galaxies. Nikola RadevicNikola Radevic Holdo's maneuver doesn't make sense, even in the SW universe. (I didn't let it ruin the movie for me, but it's dumb.) You can't go FTL and stay in normal space; this is basic science that every geek knows. So you can't ram another ship using your FTL engines. You could maybe purposefully exit from FTL in the same (normal) space occupied by another ship. I would think that would do a lot of damage, cause reactor breaches, decompression, etc. Just not a spectacular ramming. So, the answer to your question is that your "problem" is a non-problem. BTW, a spaceship firing a kinetic weapon with that much energy would have to SUPPLY that energy. BTW, energy from kinetic rounds doesn't necessarily all get deposited in the target, but instead punches holes through it. The energy only gets deposited if the round encounters armor thick enough to absorb all that energy. So, the proper defense against sci-fi railguns is to build ships with fairly thin walls, many compartments, multiple redundancies, and explosion failsafes. Also, make ships very maneuverable. The counter to this defense is to have the ammo spread out into many pieces after launch -- a railgun shotgun. The counter to the railgun shotgun is...armor. ;-) dmmdmm Don't worry, the odds of actually hitting anyone are quite low Your concern is accidental ultra-relativistic (or even superluminal) shrapnel, produced by a collision with a ship using a warp drive. I assume each piece of shrapnel has the destructive force of a nuclear bomb, as per your numbers in the question. A direct hit would ruin a city or spaceship, but it is far too low yield to damage an astronomical body. If the shrapnel is able to crack a planet, then the spaceship's warp drive would have a power output exceeding the Death Star. If your garden-variety FTL drives can power the Death Star a hundred times over, then you have bigger security concerns than mere shrapnel. Therefore, I shall assume nuclear bomb levels of destructive force. For this shrapnel to actually cause harm, it needs to strike infrastructure or populated territories. However, space is really big. And even more importantly, things in space are also really big. It is a reasonable bet that the pieces of shrapnel are going to hit something eventually. Even considering drag from the interstellar medium as Murphy did in his answer, any piece of debris of substantial mass has an effective range on the order of a million lightyears. But the question is what they will hit. A lot of the material universe, at least that which isn't made of dust and free gases, is made of stars and planets. So let us take our solar system as an example and assume that the distribution of objects is similar in our solar system as it will be elsewhere. The probability of hitting an object is proportional to its cross-sectional area. The objects in our solar system have cross-sectional areas, from largest down, The Sun: $1.519695684 \times 10^{12}$ km$^2$ Jupiter: $1.535468464\times 10^{10}$ km$^2$ Saturn: $1.065303332\times 10^{10}$ km$^2$ Uranus: $2.020769922\times 10^9$ km$^2$ Neptune: $1.904568191\times 10^9$ km$^2$ Earth: $1.27516118\times 10^8$ km$^2$ Venus: $1.15058579\times 10^8$ km$^2$ Mars: $3.6092848\times 10^7$ km$^2$ Ganymede: $2.1746610\times 10^7$ km$^2$ Titan: $2.0830723\times 10^7$ km$^2$ Mercury: $1.8699187\times 10^7$ km$^2$ Callisto: $1.8254256\times 10^7$ km$^2$ Io: $1.0423372\times 10^7$ km$^2$ The Moon: $9.484174\times 10^6$ km$^2$ We could keep going down the list, but the objects just keep getting smaller. Here, we have $1.519695684 \times 10^{12}$ km$^2$ of star, $2.993305607\times 10^{10}$ km$^2$ of gas giants, $2.97366732\times 10^8$ km$^2$ of rocky planets, and I have listed $8.0739135\times 10^7$ km$^2$ of moons, although there are more moons which would increase this number by a bit. The total area in this list is $1.550006846\times 10^{12}$ km$^2$. We can see that, by area, the solar system is overwhelmingly composed of the Sun, at 98%. The gas giants make up only 1.9%. Rocky bodies like Earth make up only 0.1%. The odds of hitting an isolated spaceship or space-station are so astronomically low as to not be a concern. We can reasonably assume that gas giants and stars are not populated (at least, not by anything more than some space stations), so we only have 1 piece of shrapnel in a 1000 hitting anything which might have people on it. Looking at the picture of Admiral Holdo's manoeuvre, it looks like there might be a few thousand pieces of shrapnel there. So we've gone from 'exterminate half the known universe' to 'hit a couple of planets'. That's a substantial improvement on the original prognosis. However, the situation gets better. Most inhabited planets are likely to have atmospheres. These atmospheres would absorb a lot of the energy of this shrapnel via impact-driven nuclear fusion (relevant xkcd). Any small pieces of shrapnel would not even make it to the surface. Only the big pieces would be of any threat to a planetary population. The majority of the ultra-relativistic shrapnel is likely to be made of small pieces, since larger pieces are harder to accelerate. So we could probably reduce the number of impacts by a factor of a few. Thus we go from 'hit a couple of planets' to 'maybe hit the surface of a planet'. If you ask about planets without atmospheres, it is probably reasonable to assume that only a small portion of planets without atmospheres have any sizeable habitation, and even then their population densities are likely much smaller. These factors are probably similar to the mitigating factors of an atmosphere. However, even if you hit the surface of a planet, you are still quite likely to miss people. Earth is 70% ocean and 30% land, with people living only on the land. And of that area, only 3% of it is urbanised. Which means if you were to hurl your shrapnel at a random point on the Earth's surface, you would have only a 3-in-10 chance of hitting any people at all and only a 0.9% chance of striking an urban area causing major loss of life. Some sci-fi planets are substantially more urbanised, but some are also substantially less urbanised, so it probably balances out. Over the course of this question, I have downgraded the severity of Admiral Holdo's manoeuvre from 'universal cataclysm' to 'moderately small chance of nuking a single city'. Granted, this still isn't good, so the unnamed Gunnery Chief is still right to tell his recruits not to fire blindly into space. But it is hardly the universe-breaking apocalypse you had first thought. I assumed here that many planets are inhabited (a la Star Wars). Since you have so many populated planets, the loss of a single random city on a random planet is hardly going to dent the galactic population. If you assume that only a small number of planets are inhabited, then the chances of you accidentally hitting an inhabited planet go down proportionally. Now, if hundreds of such collisions took place in every single space battle, then we would have cause for concern. But having that as your intentional strategy would be deliberately reckless and unethical. However, if we assume that only the occasional idiot or desperate hero makes such a collision, then the risks are, on a universal scale, quite low. I still wouldn't like to be in one of those highly unlucky cities destroyed by a piece of stray shrapnel, but it is a far, far better prognosis than 'condemn half the universe to utter destruction'. BBeastBBeast $\begingroup$ Long story short, this question got so much negative backlash that I'm not even bothering anymore with it. I'll just point "I assume each piece of shrapnel has the destructive force of a nuclear bomb, as per your numbers in the question." -> Factually wrong as I said "bare minimum" if you use standard equation and base light speed. Since ships travel faster than light, energy is, in standard physics, higher by an order of magnitude. In relativistic physics, it's, from the other answers, impossible/infinite. So the yield is quite relevant. Thanks for your input tho, but this question is dead. $\endgroup$ $\begingroup$ Congratulations! Welcome to Worldbuilding, BBeast, the probability of hitting anything will be infinitesimal. Commonsense, really. $\endgroup$ Energy is conserved. This is one of the laws of physics that was discovered by Newton. The rest of Newton's physics has been changed with both relativity and quantum mechanics, but this basic fact stands solid. FTL travel must mean that physics as we know it is not complete, but the safest bet is that energy will still be conserved in the new physics. Of course, you as the author can just state it as a fact, if you want to. What does this mean? It means you can't get more energy out of an FTL collision than you put in as starship fuel. This makes the universe safe. The most energy-dense fuel we know is anti-matter. If put in contact with the same amount of ordinary matter, both will convert completely in energy. The maximum can be found using Einstein's famous $E=mc^2$. You end up with energies that can be very bad news to a single planet, but not noticeable to a star. However, simply storing the same amount of anti-matter and having an accident with that would be just as bad as having star ship shrapnel rain down. Stig HemmerStig Hemmer I have read a book (i don't remember which one, but it was from the age of Asimov's first books), in which this problem was neatly solved: FTL in that universe used some kind of "physics bubble" (compare it to the warp field in star trek), which extended a few light-seconds around the producing vessel. These bubbles were unable to interact, so any time two ships would be closer than that light second to each other, the FTL would automatically fail, dropping the ships into sub-luminar speeds untill they got far enough away. Making FTL work in a way similar to this will indeed limit the dangers of such idiots, since super-luminar collisions are impossible by definition (for reference, a light-second is roughly the distance between the earth and the moon) ThisIsMeThisIsMe $\begingroup$ Welcome to Worldbuilding, ThisIsMe! If you have a moment, please take the tour and visit the help center to learn more about the site. You may also find Worldbuilding Meta and The Sandbox useful. Here is a meta post on the culture and style of Worldbuilding.SE, just to help you understand our scope and methods, and how we do things here. Have fun! $\endgroup$ – Gryphon Robotech! I am surprised that I have not seen this answer yet, but if anyone remembers the Americanized Anime show from the 1980s, Robotech, they had a unique and interesting take on FTL travel. Basically an FTL drive in the Robotech universe was a device which would create a sphere of energy around itself and essentially teleport anything inside that sphere to a location potentially many light years away. This is similar to the "space fold" concept. The ship is not achieving superluminal velocity at all, it is simply exchanging it's place with whatever is at a different location. In the very first episode of the series, we see the potential to weaponize this type of FTL drive when the SDF-1 is forced to make an emergency space fold while still inside the Earth's atmosphere, not very far above an island in the South Pacific. The result is that the ship inadvertently teleports a significant chunk of the island with it into space. Clearly, this kind of thing could damage planets, cut enemy ships in half, etc, but the total potential for destruction is fairly limited by the size of the field which can be generated by the drive. In an interesting side note, the size of the FTL field that can be generated seems to be proportional to the size of the ship creating the field, which kind of makes sense (it's never spelled out in the series or completely explained) which may explain the incredibly large size of some of the ships that exist in the Robotech universe. In any case, this basic concept: that the ship's engines (whatever allows it to move through space) are a totally separate system from the FTL "fold drive" which basically teleports the ship to a different location is a foundation which can be used to plan out a science fantasy/space opera universe. I have no idea how the creators of Robotech intended to deal with the admittedly unlikely event of two ships choosing to fold to the exact same location at the same time. I think they didn't think it through that far. I would suggest that either both folds fail, or there would be a huge explosion: whatever is more interesting for the story. JBiggsJBiggs Simply posit the following for your FTL drive: The vessel goes into an alternate space or alternate phase or whatever you want to call it, in which all FTL travel takes place; In this space or phase, distances are somehow "shorter" than regular space, so that travel can be accomplished far more quickly, even by a ship travelling at subliminal speeds. It takes a certain amount of energy to put a vessel into FTL space; the energy is linear to the mass of the vessel, and inversely proportional to the flatness of space. A vessel can only exit FTL space at a point whose escape velocity in normal space is equal to the escape velocity of the entrance point. All of this means that your thrusters only need to get you far enough out to where your FTL drive will work, and no more (so nobody will build them). There won't be a danger from ships traveling at relativistic speeds because there won't be any ships going this fast in normal space in the first place. EvilSnackEvilSnack Peace and Lack of Access Right now, we have access to technologies that could wipe out humanity, and yet we haven't. The reasons are twofold: there haven't been any major (direct) wars between nuclear powers, and nobody else has access to these technologies. To translate these across, we need: Firstly: humanity is reasonably united, and hasn't encountered any hostile alien races, so there haven't been any significant wars to wipe out massive chunks of the map. Secondly: FTL technology is absurdly difficult and expensive to manufacture, to the point of being the sole domain of of major governments/maybe one or two massive corporations. Thus, no idiots flying around at FTL speeds wiping out planets. That doesn't mean you can't have civilian interstellar spaceflight: maybe the FTL technology is in the form of stable wormholes maintained by the government, or space stations that sit at convenient junction points, extracting a fee for accelerating ships up to FTL velocities towards another such station, and decelerating any ship that gets thrown towards it at FTL speeds back down to something safe. Whatever. Thirdly: making FTL systems is very obvious, and there is an active and efficient investigation force that identifies anybody attempting to do so and prevents them with extreme prejudice. Fourthly: whatever they are, the FTL facilities that do exist are extremely secure and well-defended, preventing terrorist attacks from taking them out. Between these, we have a situation in which the only people with the capacity to use FTL systems to cause massive amounts of chaos don't have any reason to want to do so, and the people who do have reason to want to do so have easier routes to doing so. There is no actual FTL speed You don't need to worry about FTL debris if your FTL transportation method actually requires a functioning device to arrive at the destination before the light does, and it doesn't require the ship to make any sorts of relativistic speed records either. Any kind that does some sort of fourth-dimensional shortcut method (Hyperspace, wormholes, travel through the hell dimension, wrinkling of the spacetime to actually shorten the distance, and so forth) will do just nicely, and the only thing you'll have to worry about is telefragging of something on the arrival - but just push the allowed exit points outside of gravity wells, and the chances of such a collision instantly are reduced to being an astronomically rare event, since you now can accidentally intersect only with rocks and other ships, and the results of that would be significantly tamer. Darth BiomechDarth Biomech Not the answer you're looking for? Browse other questions tagged weapons space-travel weapon-mass-destruction faster-than-light security . How would advanced aliens protect themselves from idiots with FTL? The handwavium drive is broken! Is there any hope? How would FTL travel appear through window of ship? Will This Violate Causality? Does my idea of FTL make any sense? Law enforcement in FTL civilization What mandates travel in an interstellar society? What would one experience travelling through an invisible Krasnikov tube, and how much could they be used to cut travel times? What FTL models are appropriate given my constraints? Effects of being hit by an object going at FTL speeds	CommonCrawl
\begin{document} \title{\bf Forward-Backward Evolution Equations\\ and Applications\thanks{This work is supported in part by NSF Grant DMS-1406776.}} \author{Jiongmin Yong\thanks{Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA ([email protected]).} } \maketitle \noindent{\bf Abstract:} Well-posedness is studied for a special system of two-point boundary value problem for evolution equations which is called a {\it forward-backward evolution equation} (FBEE, for short). Two approaches are introduced: A decoupling method with some brief discussions, and a method of continuation with some substantial discussions. For the latter, we have introduced Lyapunov operators for FBEEs, whose existence leads to some uniform {\it a priori} estimates for the mild solutions of FBEEs, which will be sufficient for the well-posedness. For some special cases, Lyapunov operators are constructed. Also, from some given Lyapunov operators, the corresponding solvable FBEEs are identified. \noindent\bf AMS Mathematics Subject Classification. \rm 34D10, 34G20, 35K90, 35L90, 47D06, 47J35, 49K27. \noindent{\bf Keywords}. Forward-backward evolution equations, decoupling field, Lyapunov operator, method of continuation. \maketitle \section{Introduction} In this paper, we consider the following system of evolution equations: \begin{equation}\label{FBEE1}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\negthinspace \negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)+b(t,y(t),\psi(t)),\\ \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-g(t,y(t),\psi(t)),\end{array}\qquad t\in[0,T],\\ \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=h(y(T)),\end{array}\right.\end{equation} where $A:{\cal D}(A)\subseteq X\to X$ generates a $C_0$-semigroup $e^{At}$ on a real Hilbert space $X$ (identified with its dual $X^$), with $$\big(e^{At}\big)^=e^{A^{\negthinspace }t},\qquad t\ge0,$$ being the adjoint semigroup generated by $A^$ (the adjoint operator of $A$), and $b$, $g$, and $h$ being suitable maps. The above could be called a {\it two-point boundary value problem}, mimicking a similar notion for ordinary differential equations. We see that the equation for $y(\cdot)$ is an initial value problem which should be solved forwardly, and the equation for $\psi(\cdot)$ is a terminal value problem which should be solved backwardly. Therefore, inspired by the so-called {\it forward-backward stochastic differential equations} (FBSDEs, for short, see \cite{Ma-Yong 1999, Yong 1997, Yong 2010} for details), we prefer to call (\ref{FBEE1}) a {\it forward-backward evolution equation} (FBEE, for short). In common occasions, two-point boundary value problem is related to certain eigenvalue problems, for which the well-posedness of the problem might not be the goal, instead, one might be more interested in the existence of solutions, not necessarily the uniqueness. See, for examples, \cite{Bailey-Shampine-Waltman 1968,Zettl 2005,DeCoster-Habets 2006}, and see also \cite{Dower-McEneaney 2015} for some other considerations. Whereas, in this paper, we are interested in the well-posedness of (\ref{FBEE1}). On the other hand, our system has a special structure, involving one {\it forward} evolution equation and one {\it backward} evolution equation. Hence, we use the name FBEE to distinguish the current situation from other situations in the literature. A pair of functions $(y(\cdot),\psi(\cdot))$ is called a {\it strong solution} of (\ref{FBEE1}) if these functions are differentiable almost everywhere, with the property $$y(t)\in{\cal D}(A),\quad\psi(t)\in{\cal D}(A^),\qquad\hbox{\rm a.e.{ }} t\in[0,T],$$ and the equations are satisfied almost everywhere. A pair $(y(\cdot),\psi(\cdot))$ is called a {\it mild solution} (or a {\it weak solution}) to FBEE (\ref{FBEE1}) if the following system of integral equations are satisfied: \begin{equation}\label{mild}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle y(t)=e^{At}x+\int_0^te^{A(t-s)}b(s,y(s),\psi(s))ds,\\ \noalign{ }\displaystyle\psi(t)=e^{A^(T-t)}h(y(T))+\int_t^Te^{A^(s-t)}g(s,y(s),\psi(s))ds, \end{array}\right.\quad t\in[0,T].\end{equation} Note that in the case $A$ is bounded, (\ref{FBEE1}) and (\ref{mild}) are actually equivalent, and thus, a mild solution $(y(\cdot),\psi(\cdot))$ is actually a strong solution. Our study of the above system is mainly motivated by the study of optimal control theory. It is known that for a standard optimal control problem of an evolution equation with, say, a Bolza type cost functional, by applying the Pontryagin maximum/minimum principle, one will obtain an {\it optimality system} of the above form whose solution will give a candidate for the optimal trajectory and its adjoint (\cite{Li-Yong 1995}). Therefore, solvability of the above type system is important, at least for optimal control theory of evolution equations. Roughly speaking, when $T$ is small enough, or the Lipschitz constants of the involved functions are small enough, one can show that FBEE (\ref{FBEE1}) will have a unique mild solution, by means of contraction mapping theorem. On the other hand, if (\ref{FBEE1}) is the optimality system (obtained via Pontryagin maximum/minimum principle) of a corresponding optimal control problem which admits an optimal control, then this FBEE admits a mild solution, which might not be unique. Further, if the corresponding optimal control has an optimal control and the optimality system admits a unique mild solution, then this solution can be used to construct the optimal control(s). Hence, under proper conditions, FBEE (\ref{FBEE1}) could admit a (unique) mild solution, without restriction on the length of the time horizon $T$, and/or the size of the Lipschitz constants of the involved functions. This is actually the case if the FBEE is the optimality system of a linear-quadratic (LQ, for sort) optimal control problem satisfying proper conditions (\cite{Li-Yong 1995}). In this paper, we will study the (unique) solvability of FBEE (\ref{FBEE1}) under some general conditions. Two approaches will be introduced: decoupling method and method of continuation. The former is inspired by the so-called {\it invariant embedding} which can be traced back to \cite{Ambarzumyan 1943, Chandrasekhar 1950, Bellman-Wing 1975}. Such a method was used in the study of FBSDEs (see \cite{Ma-Protter-Yong 1994, Ma-Yong 1999}, for details). The latter is inspired by the method of continuity for elliptic partial differential equations (see, e.g. \cite{Gilbarg-Trudinger}), and FBSDEs (\cite{Hu-Peng 1995, Yong 1997, Peng-Wu 1999, Yong 2010}). Due to the nature of FBEE (\ref{FBEE1}), some technical difficulties exist in applying either of these methods. We will briefly present some main idea of the decoupling method and will relatively more carefully present the method of continuation. The rest of this paper is organized as follows. In Section 2, we will present some preliminary results, including a main motivation from optimal control theory. Linear FBEEs are carefully discussed in Section 3. In Section 4, a brief description on the decoupling method will be given. In Section 5, we will introduce the so-called {\it Lyapunov operator} which is adopted from \cite{Yong 2010} (for FBSDEs). The existence of Lyapunov operators lead to some uniform {\it a priori} estimates for the mild solutions of our FBEE. Well-posedness of FBEEs will be established in Section 6. In Section 7, we will construct some Lyapunov operators through which some well-posed FBEEs will be identified. In Section 8, we briefly discussed some extensions of our main results. In Section 9, several illustrative examples will be presented. Finally, some concluding remarks will be made in Section 10. \section{Preliminaries} Throughout of this paper, we let $X$ be a separable real Hilbert space, with the norm $\\|\cdot\\|$ and the inner product $\big\langle\cdot\,,\cdot\big\rangle$. We identify the dual $X^$ with $X$. The set of all bounded linear operators from $X$ to itself is denoted by ${\cal L}(X)$. The set of all self-adjoint operators on $X$ is denoted by $\mathbb{S}(X)$ and the set of all positive semi-definite operators on $X$ is denoted by $\mathbb{S}^+(X)$. For the notational simplicity, when there is no confusion, we will not distinguish between $\lambda$ and $\lambda I$ (for any $\lambda\in\mathbb{R}$). For example, we use $\lambda-A$ to denote $\lambda I-A$. Also, if $F$ is in $\mathbb{S}^+(X)$, we denote it by $F\ges0$; if $F-cI\ges0$, we simply denote it by $F\geqslant c$, and $F\leqslant c$ means $-F\geqslant-c$. Next, we denote $$C([0,T];X)=\Big\{y:[0,T]\to X\bigm\|y(\cdot)\hbox{ is continuous}\Big\},$$ and $$\\|y(\cdot)\\|_\infty=\sup_{t\in[0,T]}\\|y(t)\\|,\qquad\forall y(\cdot)\in C([0,T];X).$$ For convenience and definiteness of our presentation, we introduce the following standing assumptions: {\bf(H0)$'$} $A:{\cal D}(A)\subseteq X\to X$ generates a $C_0$-semigroup $e^{At}$ on $X$. {\bf(H0)} In addition to (H0)$'$, either \begin{equation}\label{case1}A^=A,\end{equation} with the spectrum $\sigma(A)\subseteq\mathbb{R}$ of $A$ satisfying \begin{equation}\label{si>0}\sup\sigma(A)\equiv\sup{\mathop{\rm Re}\,}\sigma(A)=-\sigma_0<0,\end{equation} or \begin{equation}\label{case2}A^=-A,\end{equation} for which it holds: $\sigma(A)\subseteq i\mathbb{R}$ and thus \begin{equation}\label{si=0}\sup{\mathop{\rm Re}\,}\sigma(A)=\sigma_0=0.\end{equation} Case (\ref{case1}) corresponds to the heat equation (or second order parabolic equations) with proper lower order terms and proper boundary conditions. Case (\ref{case2}) corresponds to the wave equation (or second order hyperbolic equations) with proper boundary conditions, without damping. Some extensions of the results presented in this paper are possible. But for the moment, we prefer not to get into the most general situations, for the simplicity of our presentation. We should keep in mind that for the case $A^=A$, one has $\sigma_0>0$ and for the case $A^=-A$, one has $\sigma_0=0$. Let us now look at our main motivation of studying our FBEEs. Consider the following controlled system: \begin{equation}\label{state2.5}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle \dot y(t)=Ay(t)+f(t,y(t),u(t)),\qquad t\in[0,T],\\ \noalign{ }\displaystyle y(0)=x,\end{array}\right.\end{equation} with cost functional of Bolza type: \begin{equation}\label{2.2}J(x;u(\cdot))=\int_0^Tf^0(s,y(s),u(s))ds+f^1(y(T)).\end{equation} In the above, $f:[0,T]\times X\times U\to X$, $f^0:[0,T]\times X\times U\to\mathbb{R}$, $f^1:X\to\mathbb{R}$ are suitable maps, with $U$ being a separable metric space. We call $x\in X$ an {\it initial state}, $u(\cdot)$ a {\it control}, and $y(\cdot)$ a {\it state trajectory}, respectively. Denote $${\cal U}=\big\{u:[0,T]\to U\bigm\|u(\cdot)\hbox{ is measurable}\big\}.$$ This is the set of all {\it admissible controls}. Under some mild conditions, for any $x\in X$ and $u(\cdot)\in{\cal U}$, {\it state equation} (\ref{state2.5}) admits a unique {\it mild solution} $y(\cdot)\equiv y(\cdot\,;x,u(\cdot))$, i.e., the solution to the following integral equation: \begin{equation}\label{2.3}y(t)=e^{At}x+\int_0^te^{A(t-s)}f(s,y(s),u(s))ds,\qquad t\in[0,T],\end{equation} and the cost functional $J(x;u(\cdot))$ is well-defined. Then one can pose the following optimal control problem. \bf Problem (C). \rm For any initial state $x\in X$, find a $\bar u(\cdot)\in{\cal U}$ such that \begin{equation}\label{2.4}J(x;\bar u(\cdot))=\inf_{u(\cdot)\in{\cal U}}J(x;u(\cdot)).\end{equation} Any $\bar u(\cdot)\in{\cal U}$ satisfying (\ref{2.4}) is called an {\it optimal control}, the corresponding $\bar y(\cdot)\equiv y(\cdot\,;x,\bar u(\cdot))$ is called an {\it optimal state trajectory} and $(\bar y(\cdot),\bar u(\cdot))$ is called an {\it optimal pair}. With the above setting, we have the following standard result. To simplify the presentation, we assume that the involved maps $f,f^0,f^1$ have all the required measurability and smoothness. The readers are referred to \cite{Li-Yong 1995} for details. \bf Proposition 2.1. (Pontryagin's Minimum Principle) \sl Let {\rm(H0)$'$} hold and let $(\bar y(\cdot),\bar u(\cdot))$ be an optimal pair of Problem {\rm(C)}. Then the following {\it minimum condition} holds: \begin{equation}\label{min2.9}\begin{array}{ll} \noalign{ }\displaystyle\big\langle\psi(t),f(t,y(t),u(t))\big\rangle+f^0(t,y(t),u(t))=\min_{u\in U}\big[\big\langle\psi(t),f(t,y(t),u)\big\rangle+f^0(t,y(t),u)\big],\qquad t\in[0,T],\end{array}\end{equation} where $\psi(\cdot)$ is the mild solution to the following {\it adjoint equation}: \begin{equation}\label{adjoint2.10}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-f_y(t,\bar y(t),\bar u(t))^\psi(t)-f^0_y(t, \bar y(t),\bar u(t)),\quad t\in[0,T],\\ \noalign{ }\displaystyle\psi(T)=f^1_y(\bar y(T)),\end{array}\right.\end{equation} i.e., the following holds: \begin{equation}\label{}\begin{array}{ll} \noalign{ }\displaystyle\psi(t)=e^{A^(T-t)}f_y^1(\bar y(T))+\int_t^T e^{A^(s-t)}\[f_y(s,\bar y(s),\bar u(s))^\psi(s)+f^0_y(s,\bar y(s),\bar u(s))\]ds,\quad t\in[0,T].\end{array}\end{equation} \rm Note that (\ref{state2.5}) and (\ref{adjoint2.10}) form a system with the minimum condition (\ref{min2.9}) bringing in the coupling. Suppose there exists a map $\varphi:[0,T]\times X\times X\to U$ such that $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle\psi,f(t,y,\varphi(t,y,\psi))\big\rangle+f^0(t,y,\varphi(t,y,\psi))=\min_{u\in U}\[\big\langle\psi,f(t,y,u)\big\rangle+f^0(t,y,u)\Big].\end{array}$$ Then we obtain the following system (dropping the bar in $\bar y(\cdot)$) \begin{equation}\label{2.7}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)+f(t,y(t),\varphi(t,y(t),\psi(t))),\qquad t\in[0,T],\\ [1mm] \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-f_y(t,y(t),\varphi(t,y(t),\psi(t)))^\psi(t) -f^0_y(t,y(t),\varphi(t,y(t),\psi(t))),\qquad t\in[0,T],\\ [1mm] \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=f^1_y(y(T)).\end{array}\right.\end{equation} This is called the {\it optimality system} of Problem (C), which is an FBEE of form (\ref{FBEE1}) with $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle b(t,y,\psi)=f(t,y,\varphi(t,y,\psi)),\\ \noalign{ }\displaystyle g(t,y,\psi)=f_y(t,y,\varphi(t,y,\psi))^\psi+f^0_y(t,y,\varphi(t,y,\psi)),\\ \noalign{ }\displaystyle h(y)=f^1_y(y).\end{array}\right.$$ If $(y(\cdot),\psi(\cdot))$ is a mild solution of FBEE (\ref{2.7}), then $y(\cdot)$ will be a candidate of optimal trajectory and $\varphi(\cdot\,,y(\cdot),\psi(\cdot))$ will be a candidate of optimal control. Let us now look an interesting special case of the above. To this end, we let $U$ also be a real Hilbert space, and $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle f(t,y,u)=F(t,y)+B(t)u,\qquad f^1(y)=G(y),\\ \noalign{ }\displaystyle f^0(t,y,u)=Q(t,y)+\big\langle S(t)y,u\big\rangle+{\,1\over\,2}\big\langle R(t)u,u\big\rangle,\end{array}\right.$$ for some suitable maps $F(\cdot\,,\cdot)$, $B(\cdot)$, $G(\cdot)$, $Q(\cdot\,,\cdot)$, $S(\cdot)$, and $R(\cdot)$. Then the state equation becomes \begin{equation}\label{2.8}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)+F(t,y(t))+B(t)u(t),\qquad t\in[0,T],\\ \noalign{ }\displaystyle y(0)=x,\end{array}\right.\end{equation} and the cost functional takes the form: \begin{equation}\label{2.9}\begin{array}{ll} \noalign{ }\displaystyle J(x;u(\cdot))=\int_0^T\[Q(y(t))+\big\langle S(t)y(t),u(t)\big\rangle+{1\over2}\big\langle R(t)u(t),u(t)\big\rangle\]dt+G(y(T)).\end{array}\end{equation} Note that the right-hand side of the state equation is affine in $u(\cdot)$ and the integrand in the cost functional is up to quadratic in $u(\cdot)$. We therefore refer to the corresponding optimal control problem as an {\it affine-quadratic optimal control problem} (AQ problem, for short). For finite-dimensional case, general AQ problem was studied in \cite{Wang-Yong 2014}. In current case, the adjoint equation reads $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-F_y(t,y(t))^\psi(t)-Q_y(t,y(t))-S(t)^u(t), \quad t\in[0,T],\\ \noalign{ }\displaystyle\psi(T)=G_y(y(T)).\end{array}\right.$$ By the minimum condition $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle\psi(t),B(t)u(t)\big\rangle+\big\langle S(t)y(t),u(t)\big\rangle+{1\over2}\big\langle R(t)u(t),u(t)\big\rangle\\ \noalign{ }\displaystyle=\min_{u\in U}\[\big\langle\psi(t),B(t)u\big\rangle+\big\langle S(t),u\big\rangle+{1\over2}\big\langle R(t)u,u\big\rangle\Big],\qquad t\in[0,T],\end{array}$$ we obtain, assuming the invertibility of $R(t)$, $$u(t)=-R(t)^{-1}\big[B(t)^\psi(t)+S(t)y(t)\big],\qquad t\in[0,T].$$ Therefore, the corresponding optimality system reads as follows: $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)+F(t,y(t))-B(t)R(t)^{-1}S(t)y(t)-B(t)R(t)^{-1}B(t)^\psi(t),\\ [1mm] \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-\big[Q_y(t,y(t))-S(t)^R(t)^{-1}S(t)y(t)\big] -\big[F_y(t,y(t))^-S(t)^R(t)^{-1}B(t)^\big]\psi(t),\\ [1mm] \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=G_y(y(T)).\end{array}\right.$$ When $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle y\mapsto F(t,y)\hbox{ is linear},\\ \noalign{ }\displaystyle y\mapsto Q(t,y),~y\mapsto G(y)\hbox{ are convex},\end{array}\right.$$ the corresponding optimal control problem is referred to as {\it linear-convex problem}, which was studied in \cite{You 1987a, You 1987b, You 1997}. See also \cite{You 2002} for some investigations on finite-dimensional two-person zero-sum differential games of linear state equation with non-quadratic payoff/cost functional where the convexity of $y\mapsto Q(t,y)$ and $y\mapsto G(y)$ were not assumed. Further, if $$F(t,y)\equiv0,\quad Q(t,y)={1\over2}\big\langle Q(t)y,y\big\rangle,\quad G(y) ={1\over2}\big\langle Gy,y\big\rangle,$$ for some $Q:[0,T]\to\mathbb{S}(X)$ and $G\in\mathbb{S}(X)$, the problem is reduced to a classical LQ problem. In this case, the optimality system becomes the following linear FBEE: \begin{equation}\label{FBEE-LQ}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=\big[A-B(t)R(t)^{-1}S(t)\big]y(t)-B(t)R(t)^{-1}B(t)^\psi(t),\\ [1mm] \noalign{ }\displaystyle\dot\psi(t)=-\big[A\negthinspace -\negthinspace B(t)R(t)^{-1}S(t)\big]^\psi(t)\negthinspace -\negthinspace \big[Q(t)\negthinspace -\negthinspace S(t)^R(t)^{-1}S(t)\big]y(t),\\ [1mm] \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=Gy(T).\end{array}\right.\end{equation} It is known that under the following conditions: \begin{equation}\label{}R(t)\geqslant\delta I,\quad Q(t)-S(t)^R(t)^{-1}S(t)\ges0,\quad t\in[0,T],\quad G\ges0,\end{equation} the map $u(\cdot)\mapsto J(x;u(\cdot))$ for the current LQ problem is uniformly convex, and the above linear FBEE (\ref{FBEE-LQ}) admits a unique mild solution $(y(\cdot),\psi(\cdot))$ (\cite{Li-Yong 1995}). Next, we note that under (H0)$'$, by Hille-Yosida's theorem, there exist $M\ges1$ and $\omega\in\mathbb{R}$ such that \begin{equation}\label{H-Y1}\\|(\lambda-A)^{-n}\\|\leqslant{M\over(\lambda-\omega)^n},\qquad\forall\lambda>\omega,~n\ges1,\end{equation} and the {\it Yosida approximation} $A_\lambda$ of $A$ is well-defined: \begin{equation}\label{Yosida}A_\lambda=\lambda A(\lambda-A)^{-1},\qquad\qquad\lambda>\omega.\end{equation} By making a shifting and absorbing a relevant term into $b(t,y,\psi)$ (see (\ref{FBEE1})), we may assume that $\omega=0$ in the above. Then by \cite{Pazy 1983}, we may assume the following: \begin{equation}\label{2.19}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\\|e^{A_\lambda t}\\|\leqslant M,\qquad\forall t\ges0,\\ \noalign{ }\displaystyle\lim_{\lambda\to\infty}\\|A_\lambda x-Ax\\|=0,\qquad\forall x\in{\cal D}(A),\\ \noalign{ }\displaystyle\lim_{\lambda\to\infty}\sup_{t\in[0,T]}\\|e^{A_\lambda t}x-e^{At}x\\|=0,\qquad\forall x\in X.\end{array}\right.\end{equation} Now, let us look at (H0). It is clear that under condition (\ref{case1}), one has \begin{equation}\label{A<-d}\big\langle Ax,x\big\rangle\leqslant-\sigma_0\\|x\\|^2,\qquad\forall x\in{\cal D}(A),\end{equation} and under (\ref{case2}), one has \begin{equation}\label{A=0}\big\langle Ax,x\big\rangle=0,\qquad\forall x\in{\cal D}(A).\end{equation} The following simple result is concerned with the Yosida approximation $A_\lambda$ of $A$, under (H0). \bf Proposition 2.2. \sl If $(\ref{case1})$ holds, then \begin{equation}\label{3.5}\big\langle A_\lambda x,x\big\rangle\leqslant-{\lambda\sigma_0\over\lambda+\sigma_0}\\|x\\|^2,\qquad\forall x\in X,\quad\lambda>0.\end{equation} If $(\ref{case2})$ holds, then \begin{equation}\label{3.6}\big\langle(A_\lambda+A_\lambda^)x,x\big\rangle\les0,\qquad\forall x\in X,\quad\lambda>0.\end{equation} \it Proof. \rm Under (\ref{case1}), $A$ admits the following spectral decomposition (\cite{Dunford-Schwartz}): \begin{equation}\label{spectral}Ax=\int_{\sigma(A)}\mu dE_\mu x,\qquad\forall x\in{\cal D}(A),\end{equation} where $\mu\mapsto E_\mu$ is the projection-valued measure associated with $A$, and $\sigma(A)\subseteq(-\infty,-\sigma_0]$ is the spectrum of $A$. Consequently, $$A_\lambda=\lambda A(\lambda-A)^{-1}=\int_{\sigma(A)}{\lambda\mu\over\lambda-\mu}dE_\mu.$$ Since the map $\mu\mapsto{\lambda\mu\over\lambda-\mu}$ is increasing on $(-\infty,-\sigma_0]$, we have $$\big\langle A_\lambda x,x\big\rangle=\int_{\sigma(A)}{\lambda\mu\over\lambda-\mu}d\\|E_\mu x\\|^2\leqslant-{\lambda\sigma_0\over\lambda+\sigma_0} \\|x\\|^2,\qquad\forall x\in X.$$ Now, let (\ref{case2}) hold. We let $\mathbb{X}=X+iX$ be the complexification of $X$, i.e., $$\mathbb{X}=\big\{x+iy\bigm\|x,y\in X\big\},$$ with the following definition of addition, scalar multiplication, and inner product: $$\begin{array}{ll} \noalign{ }\displaystyle(x+iy)+(\widetilde x+i\widetilde y)=(x+\widetilde x)+i(y+\widetilde y),\qquad\forall x,y,\widetilde x,\widetilde y\in X,\\ \noalign{ }\displaystyle(\alpha+i\beta)(x+iy)=(\alpha x-\beta y)+i(\alpha y+\beta x),\qquad\forall\alpha,\beta\in\mathbb{R},~x,y\in X,\\ \noalign{ }\displaystyle\big\langle x+iy,\widetilde x+i\widetilde y\big\rangle=\big\langle x,\widetilde x\big\rangle+\big\langle y,\widetilde y\big\rangle+i\big(\big\langle y,\widetilde x\big\rangle-\big\langle x,\widetilde y\big\rangle\big),\qquad\forall x,\widetilde x,y,\widetilde y\in X.\end{array}$$ Naturally extend $A$ to ${\bf A}:{\cal D}({\bf A})\subseteq\mathbb{X}\to\mathbb{X}$ as follows $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle{\cal D}({\bf A})={\cal D}(A)+i{\cal D}(A)\subseteq\mathbb{X},\\ \noalign{ }{\bf A}(x+iy)=Ax+iAy,\qquad\forall x+iy\in{\cal D}({\bf A}).\end{array}\right.$$ Then under (\ref{case2}), we have $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle{\bf A}(x+i y),\widetilde x+i\widetilde y\big\rangle=\big\langle Ax+iAy,\widetilde x+i\widetilde y\big\rangle=\big\langle Ax,\widetilde x\big\rangle+\big\langle Ay,\widetilde y\big\rangle+i\big(\big\langle Ay,\widetilde x\big\rangle-\big\langle Ax,\widetilde y\big\rangle\big)\\ \noalign{ }\displaystyle=-\big\langle x,A\widetilde x\big\rangle-\big\langle y,A\widetilde y\big\rangle-i\big(y,A\widetilde x\big\rangle-\big\langle x,A\widetilde y\big)=-\big\langle\widetilde x+i\widetilde y,{\bf A}(x+iy)\big\rangle,\end{array}$$ which implies that $${\bf A}^=-{\bf A}.$$ Hence, ${\bf A}$ admits the following spectral decomposition: $${\bf A} z=\int_{\sigma({\bf A})}\mu dE_\mu z,\qquad\forall z\in{\cal D}({\bf A}),$$ with $\sigma({\bf A})\subseteq i\mathbb{R}$. Consequently, for any $\lambda>0$, $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle({\bf A}_\lambda+{\bf A}_\lambda^)z,z\big\rangle =\int_{\sigma({\bf A})}\[{\lambda\mu\over\lambda-\mu}-{\lambda\mu\over\lambda+\mu}\]d\\|E_\mu z\\|^2\\ \noalign{ }\displaystyle\qquad\qquad\qquad=-\int_{\sigma({\bf A})}{2\lambda\|\mu\|^2\over\lambda^2+\|\mu\|^2}\,d\\|E_\mu z\\|^2\les0,\qquad z=x+iy\in\mathbb{X}.\end{array}$$ Note that for any $x\in X$, if $$(\lambda-{\bf A})^{-1}x=\widetilde z=\widetilde x+i\widetilde y,$$ with $\widetilde x,\widetilde y\in X$, then $$x=(\lambda-{\bf A})(\widetilde x+i\widetilde y)=(\lambda-A)\widetilde x+i(\lambda-A)\widetilde y.$$ Hence, we must have $$\widetilde x=(\lambda-A)^{-1}x,\qquad\widetilde y=0.$$ Consequently, $${\bf A}_\lambda x=A_\lambda x,\qquad\forall x\in X.$$ Likewise, $${\bf A}_\lambda^x=A^_\lambda x,\qquad\forall x\in X.$$ Hence, (\ref{3.6}) follows. \signed {$\sqr69$} To conclude this section, let us introduce some assumptions on the coefficients of FBEE (\ref{FBEE1}). {\bf(H1)} The maps $b,g:[0,T]\times X\times X\to X$ and $h:X\to X$ are continuous, and the map $(y,\psi)\mapsto(b(t,y,\psi),g(t,y,\psi),h(y))$ is locally Lipschitz. {\bf(H2)} In addition to (H1), let the map $(y,\psi)\mapsto(b(t,y,\psi),g(t,y,\psi),h(y))$ be uniformly Lipschitz and of uniformly linear growth, i.e., there exists a constant $L>0$ such that \begin{equation}\label{2.1}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\\|b(t,0,0)\\|\leqslant L,\qquad\qquad t\in[0,T],\\ [1mm] \noalign{ }\displaystyle\\|b(t,y,\psi)-b(t,\bar y,\bar\psi)\\|\leqslant L\\|y-\bar y\\|+L\\|\psi-\bar\psi\\|,\qquad\forall t\in[0,T],~y,\bar y,\psi,\bar\psi\in X,\end{array}\right.\end{equation} \begin{equation}\label{2.2}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\\|g(t,0,0)\\|\leqslant L,\qquad\qquad t\in[0,T],\\ [1mm] \noalign{ }\displaystyle\\|g(t,y,\psi)-g(t,\bar y,\bar\psi)\\|\leqslant L\\|y-\bar y\\|+L\\|\psi-\bar\psi\\|,\qquad\forall t\in[0,T],~y,\bar y,\psi,\bar\psi\in X,\end{array}\right.\end{equation} and \begin{equation}\label{}\\|h(y)-h(\bar y)\\|\leqslant L\\|y-\bar y\\|,\qquad\forall y,\bar y\in X.\end{equation} {\bf(H3)} In addition to (H1), let the map $(y,\psi)\mapsto(b(t,y,\psi),g(t,y,\psi),h(y))$ be Fr\'echet differentiable, with continuous Fr\'echet derivatives. Note that (H3) is neither stronger nor weaker than (H2), since the Fr\'echet derivatives $b_x,b_\psi,g_x,g_\psi,h_y$, if they exist, are not necessarily uniformly bounded. We let ${\cal G}_1$, ${\cal G}_2$, ${\cal G}_3$ be the set of all $(b,g,h)$ satisfying (H1), (H2), and (H3), respectively. Any $(b,g,h)\in{\cal G}_1$ uniquely {\it generates} an FBEE (1.1) (without mentioning the well-posedness). Hence, any $(b,g,h)\in{\cal G}_1$ is called the {\it generator} of an FBEE of form (\ref{FBEE1}). \section{Linear FBEEs} In this section, we consider the following linear FBEE: \begin{equation}\label{linear}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\negthinspace \negthinspace \negthinspace \begin{array}{ll}\dot y(t)=Ay(t)+B_{11}(t)y(t)+B_{12}(t)\psi(t)+b_0(t),\\ \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-B_{21}(t)y(t)-B_{22}(t)\psi(t)-g_0(t),\end{array}\quad t\in[0,T],\\ \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=Hy(T)+h_0,\end{array}\right.\end{equation} with \begin{equation}\label{Bbg}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle B_{ij}(\cdot)\in L^\infty(0,T;{\cal L}(X)),\qquad i,j=1,2,\\ \noalign{ }\displaystyle b_0(\cdot),g_0(\cdot)\in L^1(0,T;X),\quad H\in{\cal L}(X),\quad h_0\in X.\end{array}\right.\end{equation} The above is a special case of (\ref{FBEE1}). A pair $(y(\cdot),\psi(\cdot))$ is called a mild solution to (\ref{linear}) if the following holds: $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle y(t)=e^{At}x+\int_0^te^{A(t-s)}\big[B_{11}(s)y(s)+B_{12}(s)\psi(s)+b_0(s)\big]ds,\\ \noalign{ }\displaystyle\psi(t)=e^{A^(T-t)}\big[Hy(T)+h_0\big]+\int_t^Te^{A^(s-t)}\big[B_{21}(s)y(s) +B_{22}(s)\psi(s)+g_0(s)\big]ds,\end{array}\quad t\in[0,T].\right.$$ Our first result is the following. \bf Proposition 3.1. \sl Let {\rm(H0)$'$} and {\rm(\ref{Bbg})} hold. Then FBEE {\rm(\ref{linear})} admits a mild solution if the following operator: $$\begin{array}{ll} \noalign{ }\displaystyle\psi(\cdot)\mapsto\psi(\cdot)-\int_0^T$\Phi_{22}(T,\cdot)H\Phi_{11}(T,s) +\int_{s\vee\cdot}^T\Phi_{22}(r,\cdot)B_{21}(r)\Phi_{11}(r,s)dr$B_{12}(s)\psi(s)ds\end{array}$$ is invertible on $C([0,T];X)$, where $\Phi_{11}(\cdot\,,\cdot)$ and $\Phi_{22}(\cdot\,,\cdot)$ are evolution operators generated by $A+B_{11}(\cdot)$ and $A^+B_{22}(\cdot)$, respectively. \it Proof. \rm By the variation of constants formula, we have $$y(t)=\Phi_{11}(t,0)x+\int_0^t\Phi_{11}(t,s)\big[B_{12}(s)\psi(s)+b_0(s)\big]ds,$$ and $$\begin{array}{ll} \noalign{ }\displaystyle\psi(t)=\Phi_{22}(T,t)\big[Hy(T)+h_0\big]+\int_t^T\Phi_{22}(s,t)\big[B_{21}(s)y(s) +g_0(s)\big]ds\\ \noalign{ }\displaystyle\qquad=\Phi_{22}(T,t)\Big\{H\[\Phi_{11}(T,0)x+\int_0^T\Phi_{11}(T,s) $B_{12}(s)\psi(s)+b_0(s)$ds\Big]+h_0\Big\}\\ \noalign{ }\displaystyle\qquad\quad+\int_t^T\Phi_{22}(s,t)\[B_{21}(s)$\Phi_{11}(s,0)x+\int_0^s\Phi_{11}(s,r) \big[B_{12}(r)\psi(r)+b_0(r)\big]dr$+g_0(s)\]ds\\ \noalign{ }\displaystyle\qquad=\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace $\Phi_{22}(T,t)H\Phi_{11}(T,s)+\negthinspace \negthinspace \int_{s\vee t}^T\Phi_{22}(r,t)B_{21}(r)\Phi_{11}(r,s)dr$B_{12}(s)\psi(s)ds\\ \noalign{ }\displaystyle\qquad\quad+$\Phi_{22}(T,t)H\Phi_{11}(T,0)+\int_t^T\Phi_{22}(s,t)B_{21}(s)\Phi_{11}(s,0)ds$x \\ \noalign{ }\displaystyle\qquad\quad+\int_0^T$\Phi_{22}(T,t)H \Phi_{11}(T,s)+\int_{s\vee t}\Phi_{22}(r,t)B_{21}(r)\Phi_{11}(r,s)dr$b_0(s)ds\\ \noalign{ }\displaystyle\qquad\quad+\int_t^T\Phi_{22}(s,t)g_0(s)ds+\Phi_{22}(T,t)h_0,\qquad t\in[0,T].\end{array}$$ The above is a Fredholm integral equation for $\psi(\cdot)$ of the second kind. By our assumption, it has a unique solution. Then our result follows. \signed {$\sqr69$} Next, we consider a special case: $A^=-A$. For such a case, we have that \begin{equation}\label{AA}\mathbb{A}\equiv\begin{pmatrix}A&0\\ 0&-A^\end{pmatrix}=\begin{pmatrix}A&0\\ 0&A\end{pmatrix}\end{equation} generates a $C_0$-group $e^{\mathbb{A} t}\equiv\begin{pmatrix}\sc e^{At}&\sc0\\ \sc0&\sc e^{At}\end{pmatrix}$ on $X\times X$. Hence, if we denote \begin{equation}\label{BB}\mathbb{B}(t)=\begin{pmatrix}B_{11}(t)&B_{12}(t)\\ -B_{21}(t)&-B_{22}(t)\end{pmatrix},\end{equation} then $\mathbb{A}+\mathbb{B}(\cdot)$ generates an evolution operator $\widehat\Phi(\cdot\,,\cdot)$ on $X\times X$. The following result concerns the well-posedness of the corresponding linear FBEE. \bf Proposition 3.2. \sl Let $A^=-A$ and {\rm(\ref{Bbg})} hold. Then linear FBEE {\rm(\ref{linear})} admits a unique mild solution $(y(\cdot),\psi(\cdot))$ for any $h_0\in X$ if and only if \begin{equation}\label{-1}\left[(-H,I)\widehat\Phi(T,0)\begin{pmatrix}0\\ I\end{pmatrix}\right]^{-1}\in{\cal L}(X\times X).\end{equation} \it Proof. \rm Suppose (\ref{linear}) admits a unique mild solution. Then we have \begin{equation}\label{}\begin{pmatrix}y(t)\\ \psi(t)\end{pmatrix}=\widehat\Phi(t,0)\begin{pmatrix}x\\ \psi(0)\end{pmatrix}+\int_0^t\widehat\Phi(t,s)\begin{pmatrix}b_0(s)\\-g_0(s) \end{pmatrix}ds,\qquad t\in[0,T],\end{equation} with $\psi(0)$ undetermined. By the condition at $t=T$, we have $$\begin{array}{ll} \noalign{ }\displaystyle h_0=-Hy(T)+\psi(T)=(-H,I)\begin{pmatrix}y(T)\\ \psi(T)\end{pmatrix}\\ \noalign{ }\displaystyle\quad=(-H,I)\widehat\Phi(T,0)\begin{pmatrix}x\\ \psi(0)\end{pmatrix}+\int_0^T(-H,I)\widehat\Phi(t,s)\begin{pmatrix}b_0(s)\\-g_0(s) \end{pmatrix}ds\\ \noalign{ }\displaystyle\quad=(-H,I)\widehat\Phi(T,0)\begin{pmatrix}0\\ I\end{pmatrix}\psi(0)+(-H,I)\widehat\Phi(T,0)\begin{pmatrix}I\\ 0\end{pmatrix}x+\int_0^T(-H,I)\widehat\Phi(t,s)\begin{pmatrix}b_0(s)\\-g_0(s) \end{pmatrix}ds.\end{array}$$ Hence, in order for any $h_0\in X$ the above uniquely determines $\psi(0)$, we need (\ref{-1}). Conversely, if (\ref{-1}) holds, one obtains \begin{equation}\label{psi(0)}\begin{array}{ll} \noalign{ }\displaystyle\psi(0)=\left[(-H,I)\widehat\Phi(T,0)\begin{pmatrix}0\\ I\end{pmatrix}\right]^{-1}\[h_0-(-H,I)\widehat\Phi(T,0)\begin{pmatrix}I\\ 0\end{pmatrix}x-\int_0^T(-H,I)\widehat\Phi(t,s)\begin{pmatrix}b_0(s)\\-g_0(s) \end{pmatrix}ds\Big].\end{array}\end{equation} From this we obtain the mild solution $(y(\cdot),\psi(\cdot))$ of FBEE (\ref{linear}). \signed {$\sqr69$} We note that in principle, condition (\ref{-1}) is checkable, although it might be practically complicated. We also note that, in the above, the condition that $A^=-A$, or $e^{At}$ is a group, plays an essential role. It seems that if $e^{At}$ is not a group, the arguments used above will not work (since $\widehat\Phi(\cdot\,,\cdot)$ in the above might not be defined). \rm We now return to the general linear FBEE (\ref{linear}) (without assuming (H0)). Suppose $(y(\cdot),\psi(\cdot))$ is a strong solution to linear FBEE (\ref{linear}). Inspired by the well-known invariant imbedding idea (\cite{Bellman-Wing 1975,Ma-Protter-Yong 1994, Yong-Zhou 1999,Ma-Wu-Zhang-Zhang 2014}), we suppose that the following relation holds: $$\psi(t)=\mathbb{P}(t)y(t)+p(t),\qquad t\in[0,T],$$ for some Fr\'echet differentiable functions $\mathbb{P}:[0,T]\to{\cal L}(X)$ and $p:[0,T]\to X$. Then, formally, we should have $$\begin{array}{ll} \noalign{ }\displaystyle-A^[\mathbb{P}(t)y(t)+p(t)]-B_{21}(t)y(t)-B_{22}(t)[\mathbb{P}(t)y(t)+p(t)]-g_0(t) =\dot\psi(t)\\ \noalign{ }\displaystyle=\dot\mathbb{P}(t)y(t)+\mathbb{P}(t)$Ay(t)+B_{11}(t)y(t) +B_{12}(t)[\mathbb{P}(t)y(t)+p(t)]+b_0(t)$+\dot p(t)\\ \noalign{ }\displaystyle=$\dot\mathbb{P}(t)+\mathbb{P}(t)A+\mathbb{P}(t)B_{11}(t)+\mathbb{P}(t)B_{12}(t)\mathbb{P}(t)$y(t) +\mathbb{P}(t)B_{12}(t)p(t)+\mathbb{P}(t)b_0(t)+\dot p(t).\end{array}$$ Hence, $$\begin{array}{ll} \noalign{ }\ds0=$\dot\mathbb{P}(t)+\mathbb{P}(t)A+A^\mathbb{P}(t)+\mathbb{P}(t)B_{11}(t)+B_{22}(t)\mathbb{P}(t) +\mathbb{P}(t)B_{12}(t)\mathbb{P}(t)+B_{21}(t)$y(t)\\ \noalign{ }\displaystyle\qquad+\dot p(t)+A^p(t)+$\mathbb{P}(t)B_{12}(t)+B_{22}(t)$p(t)+\mathbb{P}(t)b_0(t)+g_0(t).\end{array}$$ This suggests that we choose $\mathbb{P}(\cdot)$ satisfying the following: \begin{equation}\label{Riccati}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot\mathbb{P}(t)+\mathbb{P}(t)A+A^\mathbb{P}(t)+\mathbb{P}(t)B_{11}(t)+B_{22}(t)\mathbb{P}(t) +\mathbb{P}(t)B_{12}(t)\mathbb{P}(t)+B_{21}(t)=0,\qquad t\in[0,T],\\ \noalign{ }\displaystyle\mathbb{P}(T)=H,\end{array}\right.\end{equation} and choose $p(\cdot)$ satisfying \begin{equation}\label{p}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot p(t)+A^p(t)+$\mathbb{P}(t)B_{12}(t)+B_{22}(t)$p(t)+\mathbb{P}(t)b_0(t)+g_0(t)=0,\qquad t\in[0,T],\\ \noalign{ }\displaystyle p(T)=h_0.\end{array}\right.\end{equation} Equation (\ref{Riccati}) is called a {\it differential Riccati equation}. Any $\mathbb{P}:[0,T]\to{\cal L}(X)$ is called a {mild solution} of (\ref{Riccati}) if the following integral equation is satisfied: \begin{equation}\label{Riccati2}\begin{array}{ll} \noalign{ }\displaystyle\mathbb{P}(t)=e^{A^(T-t)}He^{A(T-t)}+\int_t^Te^{A^(s-t)}\[\mathbb{P}(s)B_{11}(s) +B_{22}(s)\mathbb{P}(s)\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad+\mathbb{P}(s)B_{12}(t)\mathbb{P}(s)+B_{21}(s)\]e^{A(s-t)}ds,\quad t\in[0,T].\end{array}\end{equation} Note that if $A$ is bounded, (\ref{Riccati}) and (\ref{Riccati2}) are equivalent. Further, recalling that $\Phi_{11}(\cdot\,,\cdot)$ and $\Phi_{22}(\cdot\,,\cdot)$ are the evolution operators generated by $A+B_{11}(\cdot)$ and $A^+B_{22}(\cdot)$, respectively, one sees that (\ref{Riccati2}) is also equivalent to the following: \begin{equation}\label{Riccati3}\begin{array}{ll} \noalign{ }\displaystyle\mathbb{P}(t)=\Phi_{22}(T,t)H\Phi_{11}(T,t)+\int_t^T\Phi_{22}(s,t) \[\mathbb{P}(s)B_{12}(t)\mathbb{P}(s)+B_{21}(s)\Big]\Phi_{11}(s,t)ds,\quad t\in[0,T].\end{array}\end{equation} Having the above derivation, we now present the following result. \bf Proposition 3.3. \sl Let {\rm(H0)$'$} and {\rm(\ref{Bbg})} hold. Let Riccati equation {\rm(\ref{Riccati})} admit a unique mild solution $\mathbb{P}:[0,T]\to{\cal L}(X)$. Then linear FBEE {\rm(\ref{linear})} admits a unique mild solution $(y(\cdot),\psi(\cdot))$. \it Proof. \rm For any $\lambda>0$, consider the following: $$\begin{array}{ll} \noalign{ }\displaystyle\mathbb{P}_\lambda(t)=e^{A_\lambda^(T-t)}He^{A_\lambda(T-t)}+\int_t^Te^{A_\lambda^(s-t)} \widehat\mathbb{Q}(s)e^{A_\lambda(s-t)}ds,\quad t\in[0,T],\end{array}$$ where, with the mild solution $\mathbb{P}(\cdot)$ of the Riccati equation (\ref{Riccati}), $$\widehat\mathbb{Q}(s)=\mathbb{P}(s)B_{11}(s) +B_{22}(s)\mathbb{P}(s)+\mathbb{P}(s)B_{12}(t)\mathbb{P}(s)+B_{21}(s).$$ Clearly, $\mathbb{P}_\lambda(\cdot)$ is uniformly bounded (by noting (\ref{2.19})). Moreover, for any $x\in X$, $$\begin{array}{ll} \noalign{ }\displaystyle\\|\mathbb{P}_\lambda(t)x-\mathbb{P}(t)x\\|\le\\|e^{A^_\lambda(T-t)}He^{A_\lambda(T-t)}x-e^{A^(T-t)}H e^{A(T-t)}x\\|\\ \noalign{ }\displaystyle\qquad+\int_t^T\\|e^{A_\lambda^(s-t)}\widehat\mathbb{Q}(s)e^{A_\lambda(s-t)}x- e^{A^(s-t)}\widehat\mathbb{Q}(s)e^{A(s-t)}x\\|ds\\ \noalign{ }\displaystyle\leqslant\\|e^{A^_\lambda(T-t)}H\big[e^{A_\lambda(T-t)}x-e^{A(T-t)}x\big]\\| +\\|e^{A^_\lambda(T-t)}He^{A(T-t)}x-e^{A^(T-t)}H e^{A(T-t)}x\\|\\ \noalign{ }\displaystyle\qquad+\int_t^T$\\|e^{A_\lambda^(s-t)}\widehat\mathbb{Q}(s)\big[e^{A_\lambda(s-t)}x- e^{A(s-t)}x\big]\\|+\\|e^{A_\lambda^(s-t)}\widehat\mathbb{Q}(s)e^{A(s-t)}x- e^{A^(s-t)}\widehat\mathbb{Q}(s)e^{A(s-t)}x\\|$ds\\ \noalign{ }\displaystyle\leqslant K\\|e^{A_\lambda(T-t)}x-e^{A(T-t)}x\\|+\\|e^{A^_\lambda(T-t)}He^{A(T-t)}x-e^{A^(T-t)}H e^{A(T-t)}x\\|\\ \noalign{ }\displaystyle\qquad+\int_t^T$K\\|e^{A_\lambda(s-t)}x- e^{A(s-t)}x\\|+\\|e^{A_\lambda^(s-t)}\widehat\mathbb{Q}(s)e^{A(s-t)}x- e^{A^(s-t)}\widehat\mathbb{Q}(s)e^{A(s-t)}x\\|$ds\to0.\end{array}$$ Hereafter, $K>0$ represents a generic constant which can be different from line to line. Note that $\mathbb{P}_\lambda(\cdot)$ also solves the following Lyapunov equation: \begin{equation}\label{}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot\mathbb{P}_\lambda(t)+\mathbb{P}_\lambda(t)A_\lambda+A_\lambda^\mathbb{P}_\lambda(t)+\widehat\mathbb{Q}(t)=0,\qquad t\in[0,T],\\ \noalign{ }\displaystyle\mathbb{P}_\lambda(T)=H.\end{array}\right.\end{equation} Now, we let $p_\lambda(\cdot)$ be the solution of the following: \begin{equation}\label{p_l}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot p_\lambda(t)\negthinspace +\negthinspace A_\lambda^p_\lambda(t)\negthinspace +\negthinspace \big[\mathbb{P}_\lambda(t)B_{12}(t)\negthinspace +\negthinspace B_{22}(t)\big]p_\lambda(t)\negthinspace +\negthinspace \mathbb{P}_\lambda(t)b_0(t) \negthinspace +\negthinspace g_0(t)=0,\qquad t\in[0,T],\\ \noalign{ }\displaystyle p_\lambda(T)=h_0,\end{array}\right.\end{equation} It is clear that $$\\|p_\lambda(\cdot)\\|_\infty<\infty.$$ We estimate $$\begin{array}{ll} \noalign{ }\displaystyle\\|p_\lambda(t)-p(t)\\|\leqslant\\|e^{A_\lambda^(T-t)}h_0-e^{A^(T-t)}h_0\\|\\ \noalign{ }\displaystyle\qquad\qquad+\int_t^T\negthinspace \negthinspace \Big\{\Big\\|e^{A_\lambda^(s-t)}$\mathbb{P}_\lambda(s)B_{12}(s)+B_{22}(s)$ p_\lambda(s)-e^{A^(s-t)}$\mathbb{P}(s)B_{12}(s)+B_{22}(s)$p(s)\Big\\|\\ \noalign{ }\displaystyle\qquad\qquad+\Big\\|e^{A_\lambda^(s-t)}$\mathbb{P}_\lambda(s)b_0(s)+g_0(s)$-e^{A^(s-t)} $\mathbb{P}(s)b_0(s)+g_0(s)$\Big\\|\Big\}ds\\ \noalign{ }\displaystyle\leqslant\\|e^{A_\lambda^(T-t)}h_0-e^{A^(T-t)}h_0\\|+\int_t^T\Big\{\Big\\|e^{A_\lambda^(s-t)}$\mathbb{P}_\lambda(s)B_{12}(s) +B_{22}(s)$$p_\lambda(s)-p(s)$\Big\\|\\ \noalign{ }\displaystyle\qquad\qquad+\Big\\|$e^{A^_\lambda(s-t)}-e^{A^(s-t)}$$\mathbb{P}(s)B_{12}(s)+B_{22}(s)$p(s)\Big\\|\\ \noalign{ }\displaystyle\qquad\qquad+\Big\\|e^{A_\lambda^(s-t)}$\mathbb{P}_\lambda(s)-\mathbb{P}(s)$$B_{12}(s)+B_{22}(s)$p(s)\Big\\|\\ \noalign{ }\displaystyle\qquad\qquad+\Big\\|$e^{A_\lambda^(s-t)}-e^{A^(s-t)}$$\mathbb{P}(s)b_0(s)+g_0(s)$\Big\\| +\Big\\|e^{A_\lambda^(s-t)} $\mathbb{P}_\lambda(s)-\mathbb{P}(s)$b_0(s)\Big\\|\Big\}ds\\ \noalign{ }\displaystyle\leqslant\\|e^{A_\lambda^(T-t)}h_0-e^{A^(T-t)}h_0\\|\\ \noalign{ }\displaystyle\qquad\quad\int_t^T\Big\{K\\|p_\lambda(s)-p(s)\\|+\Big\\|\big(e^{A^_\lambda(s-t)}-e^{A^(s-t)}\big) \big(\mathbb{P}(s)B_{12}(s)+B_{22}(s)\big)p(s)\Big\\|\\ \noalign{ }\displaystyle\qquad\quad+\Big\\|\big(e^{A_\lambda^(s-t)}-e^{A^(s-t)}\big)\big(\mathbb{P}(s)b_0(s)+g_0(s)\big) \Big\\|\\ \noalign{ }\displaystyle\qquad\quad+K$\Big\\|\big(\mathbb{P}_\lambda(s)-\mathbb{P}(s)\big)\big(B_{12}(s)+B_{22}(s)\big)p(s)\Big\\| \negthinspace +\negthinspace \Big\\|\big(\mathbb{P}_\lambda(s)-\mathbb{P}(s)\big)b_0(s)\Big\\|$\Big\}ds.\end{array}$$ Then by Gronwall's inequality, we have $$\begin{array}{ll} \noalign{ }\displaystyle\\|p_\lambda(t)-p(t)\\|\leqslant K\\|e^{A_\lambda^(T-t)}h_0-e^{A^(T-t)}h_0\\|\\ \noalign{ }\displaystyle\qquad\qquad+K\int_t^T\Big\{\Big\\|$e^{A^_\lambda(s-t)}-e^{A^(s-t)}$$\mathbb{P}(s)B_{12}(s)+B_{22}(s)$p(s)\Big\\|\\ \noalign{ }\displaystyle\qquad\qquad+\Big\\|$e^{A_\lambda^(s-t)}-e^{A^(s-t)}$$\mathbb{P}(s)b_0(s)+g_0(s)$\Big\\|\\ \noalign{ }\displaystyle\qquad\qquad+\Big\\|\big(\mathbb{P}_\lambda(s)-\mathbb{P}(s)\big)\big(B_{12}(s)+B_{22}(s)\big)p(s)\Big\\| +\Big\\|\big(\mathbb{P}_\lambda(s)-\mathbb{P}(s)\big)b_0(s)\Big\\|\Big\}ds\to0.\end{array}$$ Now, let $y_\lambda(\cdot)$ be the solution to the following: $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y_\lambda(t)=A_\lambda y_\lambda(t)+B_{11}(t)y_\lambda(t)+B_{12}(t)\big[\mathbb{P}_\lambda(t)y_\lambda(t)+p_\lambda(t)\big]+b_0(t),\quad t\in[0,T],\\ \noalign{ }\displaystyle y_\lambda(0)=x.\end{array}\right.$$ By the convergence of $\mathbb{P}_\lambda(\cdot)\to\mathbb{P}(\cdot)$ and $p_\lambda(\cdot)\to p(\cdot)$, we have $$\lim_{\lambda\to\infty}\\|y_\lambda(\cdot)-y(\cdot)\\|_\infty=0.$$ Define $$\psi_\lambda(t)=\mathbb{P}_\lambda(t)y_\lambda(t)+p_\lambda(t).$$ Then one has $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y_\lambda(t)=A_\lambda y_\lambda(t)+B_{11}(t)y_\lambda(t)+B_{12}(t)\psi_\lambda(t)+b_0(t),\\ \noalign{ }\displaystyle y_\lambda(0)=x,\end{array}\right.$$ and $$\begin{array}{ll} \noalign{ }\displaystyle\dot\psi_\lambda(t)=\dot\mathbb{P}_\lambda(t)y_\lambda(t)+\mathbb{P}_\lambda(t)\dot y_\lambda(t)+\dot p_\lambda(t)\\ \noalign{ }\displaystyle=-$\mathbb{P}_\lambda(t)A_\lambda+A_\lambda^\mathbb{P}_\lambda(t)+\mathbb{P}(t)B_{11}(t) +B_{22}(t)\mathbb{P}(t)+\mathbb{P}(t)B_{12}(t)\mathbb{P}(s)+B_{21}(t)$y_\lambda(t)\\ \noalign{ }\displaystyle\qquad+\mathbb{P}_\lambda(t)\Big\{A_\lambda y_\lambda(t)+B_{11}(t)y_\lambda(t)+B_{12}(t)\[\mathbb{P}_\lambda(t)y_\lambda(t)+p_\lambda(t)\Big] +b_0(t)\Big\}\\ \noalign{ }\displaystyle\qquad-\[A_\lambda^p_\lambda(t)+$\mathbb{P}_\lambda(t)B_{12}(t)+B_{22}(t)$p_\lambda(t)+\mathbb{P}_\lambda(t)b_0(t) +g_0(t)\Big]\\ \noalign{ }\displaystyle=-A_\lambda^\big[\mathbb{P}_\lambda(t)y_\lambda(t)+p_\lambda(t)\big]-B_{21}(t)y_\lambda(t) -B_{22}(t)\big[\mathbb{P}(t)y_\lambda(t)+p_\lambda(t)\big]-g_0(t)\\ \noalign{ }\displaystyle\qquad+\big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]B_{11}(t)y_\lambda(t)+\big[\mathbb{P}_\lambda(t)B_{12}(t)\mathbb{P}_\lambda(t) -\mathbb{P}(t)B_{12}(t)\mathbb{P}(t)\big]y_\lambda(t)\\ \noalign{ }\displaystyle=-A_\lambda^\psi_\lambda(t)-B_{21}(t)y_\lambda(t) -B_{22}(t)\psi_\lambda(t)-g_0(t)+R_\lambda(t),\end{array}$$ where $$R_\lambda(t)=$\big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]B_{11}(t)+\big[\mathbb{P}_\lambda(t)B_{12}(t)\mathbb{P}_\lambda(t) -\mathbb{P}(t)B_{12}(t)\mathbb{P}(t)\big]$y_\lambda(t).$$ Since $$\begin{array}{ll} \noalign{ }\displaystyle\\|R_\lambda(t)\\|\leqslant K\\|y_\lambda(t)-y(t)\\|+\Big\\|$\big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]B_{11}(t)+\big[\mathbb{P}_\lambda(t)B_{12}(t)\mathbb{P}_\lambda(t) -\mathbb{P}(t)B_{12}(t)\mathbb{P}(t)\big]$y(t)\Big\\|\\ \noalign{ }\displaystyle\leqslant K\\|y_\lambda(t)-y(t)\\|+\\|\big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]B_{11}(t)y(t)\\|\\ \noalign{ }\displaystyle\quad+\\|\big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]B_{12}(t)\mathbb{P}(t)y(t)\\|+\\|\mathbb{P}_\lambda(t)B_{12}(t) \big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]y(t)\\|\\ \noalign{ }\displaystyle\leqslant K\\|y_\lambda(t)-y(t)\\|+\\|\big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]B_{11}(t)y(t)\\|\\ \noalign{ }\displaystyle\quad+\\|\big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]B_{12}(t)\mathbb{P}(t)y(t)\\| +K\\|\big[\mathbb{P}_\lambda(t)-\mathbb{P}(t)\big]y(t)\\|\to0,\end{array}$$ we see that $$\psi_\lambda(\cdot)\to\psi(\cdot),$$ and $(y(\cdot),\psi(\cdot))$ is a mild solution to FBEE (\ref{linear}). \signed {$\sqr69$} Now, a natural question is when the Riccati equation (\ref{Riccati}) admits a mild solution. Let us rewrite the Riccait equation as follows: \begin{equation}\label{Riccati4}\begin{array}{ll} \noalign{ }\displaystyle\mathbb{P}(t)=\Phi_{22}(T,t)H\Phi_{11}(T,t)+\int_t^T\Phi_{22}(s,t)B_{21}(s)\Phi_{11}(s,t)ds\\ \noalign{ }\displaystyle\qquad\qquad+\int_t^T\Phi_{22}(s,t) \mathbb{P}(s)B_{12}(s)\mathbb{P}(s)\Phi_{11}(s,t)ds,\qquad t\in[0,T].\end{array}\end{equation} A trivial case is $B_{12}(\cdot)=0$ for which the above equation is linear and it always has a solution, under (H0)$'$ and (\ref{Bbg}). In general, we have the following result. \bf Proposition 3.4. \sl Suppose {\rm(H0)$'$} and $(\ref{Bbg})$ hold. {\rm(i)} Equation {\rm(\ref{Riccati4})} admits at most one solution $\mathbb{P}(\cdot)\in C([0,T];{\cal L}(X))$. {\rm(ii)} Suppose in addition that \begin{equation}\label{=}B_{22}(t)=B_{11}(t)^,\quad t\in[0,T],\end{equation} and \begin{equation}\label{ge0}H\in\mathbb{S}^+(X),\qquad -B_{12}(\cdot),B_{21}(\cdot)\in L^\infty(0,T;\mathbb{S}^+(X)).\end{equation} Then Riccati equation {\rm(\ref{Riccati4})} admits a unique solution $\mathbb{P}(\cdot)\in C([0,T];\mathbb{S}^+(X))$. \rm \it Proof. \rm (i) Suppose $\mathbb{P}(\cdot),\widetilde\mathbb{P}(\cdot)\in C([0,T];{\cal L}(X))$ are two solutions to (\ref{Riccati4}). Then $\widehat\mathbb{P}(\cdot)\equiv\mathbb{P}(\cdot)-\widetilde\mathbb{P}(\cdot)$ satisfies the following: $$\begin{array}{ll} \noalign{ }\displaystyle\widehat\mathbb{P}(t)=\int_t^T\Phi_{22}(s,t)\[\mathbb{P}(s)B_{12}(s)\mathbb{P}(s)-\widetilde\mathbb{P}(s)B_{12}(s) \widetilde\mathbb{P}(s)\Big]\Phi_{11}(s,t)ds\\ \noalign{ }\displaystyle\qquad=\int_t^T\Phi_{22}(s,t)\[\widehat\mathbb{P}(s)B_{12}(s)\mathbb{P}(s)+\widetilde\mathbb{P}(s)B_{12}(s) \widehat\mathbb{P}(s)\Big]\Phi_{11}(s,t)ds.\end{array}$$ Here, we note that $\mathbb{P}(\cdot)=\widehat\mathbb{P}(\cdot)+\widetilde\mathbb{P}(\cdot)$. Hence $$\begin{array}{ll} \noalign{ }\displaystyle\mathbb{P}(s)B_{12}(s)\mathbb{P}(s)-\widetilde\mathbb{P}(s)B_{12}(s)\widetilde\mathbb{P}(s) =\[\widehat\mathbb{P}(s)+\widetilde\mathbb{P}(s)\]B_{12}(s)\mathbb{P}(s)-\widetilde\mathbb{P}(s)B_{12}(s)\[\mathbb{P}(s)-\widehat\mathbb{P}(s)\Big]\\ \noalign{ }\displaystyle=\widehat\mathbb{P}(s)B_{12}(s)\mathbb{P}(s)+\widetilde\mathbb{P}(s)B_{12}(s)\widehat\mathbb{P}(s).\end{array}$$ Then by Gronwall's inequality, we obtain that $$\widetilde\mathbb{P}(\cdot)=\mathbb{P}(\cdot).$$ (ii) Under our conditions, we have $$A^+B_{22}(\cdot)=\big[A+B_{11}(\cdot)\big]^.$$ Hence, $$\Phi_{22}(\cdot\,,\cdot)=\Phi_{11}(\cdot\,,\cdot)^,$$ and (\ref{Riccati}) can be written as \begin{equation}\label{Riccati}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot\mathbb{P}(t)+\mathbb{P}(t)\big[A+B_{11}(t)+B_{12}(t)\mathbb{P}(t)\big] +\big[A+B_{11}(t)+B_{12}(t)\mathbb{P}(t)\big]^\mathbb{P}(t)\\ \noalign{ }\displaystyle\qquad+\mathbb{P}(t)\big[-B_{12}(t)\big]\mathbb{P}(t)+B_{21}(t)=0,\qquad t\in[0,T],\\ \noalign{ }\displaystyle\mathbb{P}(T)=H,\end{array}\right.\end{equation} We now let $$\mathbb{P}_0(t)=0,\qquad t\in[0,T],$$ and let $\mathbb{P}_{n+1}(\cdot)$ be the mild solution of the following Lyapunov equation: \begin{equation}\label{Riccati}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot\mathbb{P}_{n+1}(t)+\mathbb{P}_{n+1}(t)\big[A+B_{11}(t)+B_{12}(t)\mathbb{P}_n(t)\big] +\big[A+B_{11}(t)+B_{12}(t)\mathbb{P}_n(t)\big]^\mathbb{P}_{n+1}(t)\\ \noalign{ }\displaystyle\qquad+\mathbb{P}_n(t)\big[-B_{12}(t)\big]\mathbb{P}_n(t)+B_{21}(t)=0,\qquad t\in[0,T],\\ \noalign{ }\displaystyle\mathbb{P}_{n+1}(T)=H,\end{array}\right.\end{equation} Observe the following: $$\begin{array}{ll} \noalign{ }\ds0=\dot\mathbb{P}_{n+1}(t)-\dot\mathbb{P}_n(t)+\big[\mathbb{P}_{n+1}(t)-\mathbb{P}_n(t)\big] \big[A+B_{11}(t)\big]+\big[A+B_{11}(t)\big]^\big[\mathbb{P}_{n+1}(t)-\mathbb{P}_n(t)\big]\\ \noalign{ }\displaystyle\qquad+\mathbb{P}_{n+1}(t)B_{12}(t)\mathbb{P}_n(t)+\mathbb{P}_n(t)B_{12}(t)\mathbb{P}_{n+1}(t) -\mathbb{P}_n(t)B_{12}(t)\mathbb{P}_{n-1}(t)-\mathbb{P}_{n-1}(t)B_{12}(t)\mathbb{P}_n(t)\\ \noalign{ }\displaystyle\qquad-\mathbb{P}_n(t)B_{12}(t)\mathbb{P}_n(t)+\mathbb{P}_{n-1}(t)B_{12}(t)\mathbb{P}_{n-1}(t)\\ \noalign{ }\displaystyle\quad=\dot\mathbb{P}_{n+1}(t)-\dot\mathbb{P}_n(t)+\big[\mathbb{P}_{n+1}(t)-\mathbb{P}_n(t)\big] \big[A+B_{11}(t)+B_{12}(t)\mathbb{P}_n(t)\big]\\ \noalign{ }\displaystyle\qquad+\big[A+B_{11}(t)+B_{12}(t)\mathbb{P}_n(t)\big]^\big[\mathbb{P}_{n+1}(t)-\mathbb{P}_n(t)\big] +\big[\mathbb{P}_n(t)-\mathbb{P}_{n-1}(t)\big]B_{12}(t)\big[\mathbb{P}_n(t)-\mathbb{P}_{n-1}(t)\big].\end{array}$$ This implies that $$\mathbb{P}_{n+1}(t)\leqslant\mathbb{P}_n(t),\qquad t\in[0,T],\qquad n\ges1.$$ On the other hand, from (\ref{Riccati}), one has $$\mathbb{P}_n(t)\ges0,\qquad\forall t\in[0,T],~n\ges1.$$ Hence, by \cite{Riesz-Nagy 1955, Muscat 2014} for any $t\in[t,T]$, there exists a $\mathbb{P}(t)\in\mathbb{S}^+(X)$ such that \begin{equation}\label{limit}\lim_{n\to\infty}\\|\mathbb{P}_n(t)x-\mathbb{P}(t)x\\|=0,\qquad\forall x\in X.\end{equation} Note that for any $x\in X$, $$\begin{array}{ll} \noalign{ }\displaystyle\mathbb{P}_{n+1}(t)x=\Phi_{11}(T,t)^H\Phi_{11}(T,t)x +\int_t^T\Phi_{11}(s,t)^B_{21}(s)\Phi_{11}(s,t)xds\\ \noalign{ }\displaystyle\qquad\qquad\qquad+\int_t^T\Phi_{11}(s,t)^\[\mathbb{P}_{n+1}(s)B_{12}(s)\mathbb{P}_n(s) +\mathbb{P}_n(s)B_{12}(s)\mathbb{P}_{n+1}(s)\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad-\mathbb{P}_n(s)B_{12}(s)\mathbb{P}_n(s)\Big]\Phi_{11}(s,t)xds,\qquad t\in[0,T].\end{array}$$ Thus, making use of (\ref{limit}), we obtain that $\mathbb{P}(\cdot)$ is a mild solution to (\ref{Riccati4}). \signed {$\sqr69$} Let us look at linear FBEE (\ref{FBEE-LQ}) resulting from linear-quadratic optimal control problem. We rewrite (\ref{FBEE-LQ}) below: \begin{equation}\label{FBEE-LQ}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)-B(t)R(t)^{-1}S(t)y(t)-B(t)R(t)^{-1}B(t)^\psi(t),\\ [1mm] \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-\big[Q(t)-S(t)^R(t)^{-1}S(t)\big]y(t) +S(t)^R(t)^{-1}B(t)^\psi(t),\\ [1mm] \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=Gy(T).\end{array}\right.\end{equation} Note that $$R(t)\geqslant\delta I,\qquad\forall t\in[0,T],$$ which leads to the existence of $R(t)^{-1}$. Hence, $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle B_{11}(t)=-B(t)R(t)^{-1}S(t),\qquad B_{12}(t)=-B(t)R(t)^{-1}B(t)^,\\ \noalign{ }\displaystyle B_{21}(t)=\big[Q(t)-S(t)^R(t)^{-1}S(t)\big],\qquad B_{22}(t)=-S(t)^R(t)^{-1}B(t)^.\end{array}\right.$$ Then in the case that $$G\ges0,\qquad Q(t)-S(t)^R(t)^{-1}S(t)\ges0,\qquad t\in[0,T],$$ the corresponding Riccati equation admits a unique solution and linear FBEE (\ref{FBEE-LQ}) admits a solution. \section{Decoupling Method --- A Brief Description} We note that in the previous section, the essence of the approach by means of Riccati equation is to use the ansatz $$\psi(t)=\mathbb{P}(t)y(t)+p(t),\qquad t\in[0,T].$$ Inspired by this, we now look at nonlinear cases. For FBEE (\ref{FBEE1}), suppose $$\psi(t)=\mathbb{K}(t,y(t)),\qquad t\in[0,T],$$ for some $\mathbb{K}(\cdot\,,\cdot)$. Let $\{\zeta_n\}_{n\ge1}$ be an orthonormal basis of $X$. Then $$\mathbb{K}(t,y)=\sum_{n=1}^\infty\big\langle\mathbb{K}(t,y),\zeta_n\big\rangle\zeta_n\equiv\sum_{n=1}^\infty k^n(t,y)\zeta_n,$$ with $k^n:[0,T]\times X\to\mathbb{R}$. Suppose $(t,y)\mapsto\mathbb{K}(t,y)$ is Fr\'echet differentiable. Then, so is $(t,y)\mapsto k^n(t,y)=\big\langle\mathbb{K}(t,y),\zeta_n\big\rangle$, and for any $z\in X$, $$\begin{array}{ll} \noalign{ }\displaystyle{\mathbb{K}(t,y+\delta z)-\mathbb{K}(t,y)\over\delta}=\sum_{n=1}^\infty{k^n(t,y+\delta z)-k^n(t,y)\over\delta}\zeta_n\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\to\sum_{n=1}^\infty\big\langle k^n_y(t,y),z\big\rangle\zeta_n\equiv\[\sum_{n=1}^\infty\zeta_n\otimes k_y^n(t,y)\]z.\end{array}$$ This means \begin{equation}\label{K_y}\mathbb{K}_y(t,y)=\sum_{n=1}^\infty\zeta_n\otimes k_y^n(t,y),\qquad(t,y)\in[0,t]\times X.\end{equation} Note that $\mathbb{K}_y(t,y)$ is independent of the choice of $\{\zeta_n\}_{n\ge1}$. We now present the following result. \bf Proposition 4.1. \sl Let {\rm(H0)$'$} and {\rm(H2)} hold. Let $\mathbb{K}:[0,T]\times X\to X$ be Fr\'echet differentiable satisfying the following: \begin{equation}\label{K}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\mathbb{K}_t(t,y)\negthinspace +\negthinspace \mathbb{K}_y(t,y)\big[Ay\negthinspace +\negthinspace b(t,y,\mathbb{K}(t,y))\big] \negthinspace +\negthinspace A^\mathbb{K}(t,y)\negthinspace +\negthinspace g(t,y,\mathbb{K}(t,y))=0,\quad(t,y)\in[0,T]\times{\cal D}(A),\\ \noalign{ }\displaystyle\mathbb{K}(T,y)=h(y),\qquad y\in X.\end{array}\right.\end{equation} Let $y(\cdot)$ be a classical solution to the following: \begin{equation}\label{closed}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)+b(t,y(t),\mathbb{K}(t,y(t)),\qquad t\in[0,T],\\ \noalign{ }\displaystyle y(0)=x,\end{array}\right.\end{equation} and \begin{equation}\label{psi=K}\psi(t)=\mathbb{K}(t,y(t)),\qquad t\in[0,T].\end{equation} Then $(y(\cdot),\psi(\cdot))$ is a strong solution of FBEE {\rm(\ref{FBEE1})}. \it Proof. \rm Note that as a part of requirement for $\mathbb{K}(\cdot\,,\cdot)$ being a solution to (\ref{K}), one has $$\mathbb{K}(t,y)\in{\cal D}(A^),\qquad\forall(t,y)\in[0,T]\times X.$$ By (\ref{psi=K}), we have $$\begin{array}{ll} \noalign{ }\displaystyle\dot\psi(t)=\mathbb{K}_t(t,y(t))+\mathbb{K}_y(t,y(t))\[Ay(t)+b(t,y(t),\mathbb{K}(t,y(t))\Big]\\ \noalign{ }\displaystyle\qquad=-A^\mathbb{K}(t,y(t))-g(t,y(t),\mathbb{K}(t,y(t))=-A^\psi(t)-g(t,y(t),\psi(t)),\end{array}$$ and $$\psi(T)=\mathbb{K}(T,y(T))=h(y(T)).$$ Hence, our claim follows. \signed {$\sqr69$} It is seen that thanks to the map $\mathbb{K}(\cdot\,,\cdot)$, the original FBEE (\ref{FBEE1}) is decoupled into (\ref{closed}) and (\ref{psi=K}). Because of this, we introduce the following notion: \bf Definition 4.2. \rm A map $\mathbb{K}:[0,T]\times X\to X$ is called a {\it decoupling field} of FBEE (\ref{FBEE1}) if it is a solution to (\ref{K}). Now the natural question is when one can solve equation (\ref{K}). The linear case has been treated in the previous section. To look at the nonlinear case, let us further assume that $A^=A$ and it has a sequence of eigenvalues $$0>-\sigma_0=\lambda_1\geqslant\lambda_2\geqslant\lambda_2\cdots,$$ with the corresponding eigenfunctions $\{\zeta_n\}_{n\ges1}$ which form an orthonormal basis for $X$. Then $$\mathbb{K}(t,y)=\sum_{n=1}^\infty\big\langle\mathbb{K}(t,y),\zeta_n\big\rangle\zeta_n\equiv\sum_{n=1}^\infty k^n(t,y)\zeta_n.$$ Hence, $$\begin{array}{ll} \noalign{ }\ds0=\mathbb{K}_t(t,y)\negthinspace +\negthinspace \mathbb{K}_y(t,y)\big[Ay\negthinspace +\negthinspace b(t,y,\mathbb{K}(t,y))\big] \negthinspace +\negthinspace A^\mathbb{K}(t,y)\negthinspace +\negthinspace g(t,y,\mathbb{K}(t,y))\\ \noalign{ }\displaystyle\quad=\sum_{n=1}^\infty$k^n_t(t,y)+\big\langle k^n_y(t,y),Ay+b(t,y,\mathbb{K}(t,y))\big\rangle+\lambda_nk^n(t,y)+\big\langle g(t,y,\mathbb{K}(t,y)),\zeta_n\big\rangle$\zeta_n.\end{array}$$ Therefore, we obtain a coupled system of countably many equations: \begin{equation}\label{kn}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle k^n_t(t,y)+\big\langle k^n_y(t,y),Ay+b(t,y,\mathbb{K}(t,y))\big\rangle+\lambda_nk^n(t,y)\\ \noalign{ }\displaystyle\qquad\qquad+\big\langle g(t,y,\mathbb{K}(t,y)),\zeta_n\big\rangle=0,\qquad(t,y)\in[0,T]\times X,\\ \noalign{ }\displaystyle k^n(T,y)=\big\langle h(y),\zeta_n\big\rangle,\qquad y\in X.\end{array}\right.\end{equation} Let us look at a special case. Suppose $$\mathbb{K}(t,y)=k^1(t,y)\zeta_1,\qquad(t,y)\in[0,T]\times X,~\zeta_1\in{\cal D}(A^).$$ Then \begin{equation}\label{k}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle k^1_t(t,y)+\big\langle k^1_y(t,y),Ay+b(t,y,k^1(t,y)\zeta_1)\big\rangle+\lambda_1k^1(t,y)\\ \noalign{ }\displaystyle\qquad\qquad+\big\langle g(t,y,k^1(t,y)\zeta_1),\zeta_1\big\rangle =0,\qquad(t,y)\in[0,T]\times X,\\ \noalign{ }\displaystyle k^1(T,y)=h^1(y),\qquad y\in X,\end{array}\right.\end{equation} and \begin{equation}\label{k=0}\big\langle g(t,y,k^1(t,y)\zeta_1),\zeta_n\big\rangle=0,\qquad n>1.\end{equation} To guarantee (\ref{k=0}), we assume that \begin{equation}\label{g4.7}g(t,y,\hbox{\rm span$\,$}\{\zeta_1\})\subseteq\hbox{\rm span$\,$}\{\zeta_1\}.\end{equation} We note that (\ref{k}) is a first order Hamilton-Jacobi equation in the Hilbert space $X$, involving an unbounded operator $A$. Therefore, it is possible to study the existence of viscosity solution of it. When the viscosity solution has certain regularity, one might be able to obtain a decoupling field $\mathbb{K}(t,y)=k^1(t,y)\zeta_1$ for our FBEE. Apparently, this is merely a very special case for the general FBEEs, and it already looks complicated. Hence, there is a very long way to go in this direction to establish a satisfactory theory (for nonlinear FBEEs). We hope to report some further results in this direction in our future publications. \section{Lyapunov Operators and a Priori Estimates} We now look at the solvability by another method, called {\it method of continuity}. We first look at the following linear FBEE: \begin{equation}\label{FBEE2}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)+B_{11}(t)y(t)+B_{12}(t)\psi(t)+b_0(t),\qquad t\in[0,T],\\ \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-B_{21}(t)y(t)-B_{22}(t)\psi(t)-g_0(t),\qquad t \in[0,T],\\ \noalign{ }\displaystyle\hbox{$y(0)$ and $\psi(T)$ are given},\end{array}\right.\end{equation} with $B_{ij}:[0,T]\to{\cal L}(X)$. Let \begin{equation}\label{AB}\mathbb{A}=\begin{pmatrix}\sc A&\sc0\\ \sc0&\sc-A^\end{pmatrix},\quad\mathbb{B}(t)=\begin{pmatrix}\sc B_{11}(t)&\sc B_{12}(t)\\ \sc-B_{21}(t)&\sc-B_{22}(t)\end{pmatrix}.\end{equation} Then FBEE (\ref{FBEE2}) can be written as \begin{equation}\label{}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}\sc\dot y(t)\\ \sc\dot\psi(t)\end{pmatrix}=\big[\mathbb{A}+\mathbb{B}(t)\big]\begin{pmatrix}\sc y(t)\\ \sc\psi(t)\end{pmatrix}+\begin{pmatrix}\sc b_0(t)\\ \sc-g_0(t)\end{pmatrix},\qquad t\in[0,T],\\ [4mm] \noalign{ }\displaystyle\hbox{$y(0)$ and $\psi(T)$ are given}.\end{array}\right.\end{equation} We introduce the following {\it Lyapunov differential equation} for operator-valued function $\Pi(\cdot)$: \begin{equation}\label{Pi}\dot\Pi(t)+\Pi(t)[\mathbb{A}-\mathbb{M}(t)]+[\mathbb{A}-\mathbb{M}(t)]^\Pi(t)+\mathbb{Q}(t)=0,\quad t\in[0,T],\end{equation} where $$\Pi(t)=\begin{pmatrix}\sc P(t)&\sc\Gamma(t)^\\ \sc\Gamma(t)&\sc\bar P(t)\end{pmatrix},\quad\mathbb{M}(t)=\begin{pmatrix}\sc M(t)&\sc0\\ \sc 0&\sc-\bar M(t)^\end{pmatrix},\quad \mathbb{Q}(t)=\begin{pmatrix}\sc Q_0(t)&\sc\Theta(t)^\\ \sc\Theta(t)&\sc\bar Q_0(t)\end{pmatrix},$$ with $M,\bar M,\Theta:[0,T]\to{\cal L}(X)$ and $Q_0,\bar Q_0:[0,T]\to\mathbb{S}(X)$ to be properly chosen later. We may equivalently write (\ref{Pi}) as follows: \begin{equation}\label{L1}\dot P(t)+P(t)[A-M(t)]+[A^-M(t)^]P(t)+Q_0(t)=0,\end{equation} \begin{equation}\label{L2}\dot{\bar P}(t)-\bar P(t)[A^-\bar M(t)^]-[A-\bar M(t)]\bar P(t)+\bar Q_0(t)=0,\end{equation} \begin{equation}\label{Psi}\dot\Gamma(t)+\Gamma(t)[A-M(t)]-[A-\bar M(t)]\Gamma(t)+\Theta(t)=0,\end{equation} and \begin{equation}\label{Psi}\dot\Gamma(t)^-\Gamma(t)^[A^-\bar M(t)^]+[A^-M(t)^]\Gamma(t)^+\Theta(t)^=0.\end{equation} Let us first look at (\ref{L1}) and (\ref{L2}). Operator-valued functions $P(\cdot)$ and $\bar P(\cdot)$ are mild solutions to (\ref{L1}) and (\ref{L2}), respectively, if the following hold: \begin{equation}\label{L1a}\begin{array}{ll} \noalign{ }\displaystyle P(t)\negthinspace =\negthinspace e^{A^(T-t)}P(T)e^{A(T-t)}-\negthinspace \negthinspace \int_t^T\negthinspace \negthinspace e^{A^(s-t)}[P(s)M(s)\negthinspace +\negthinspace \negthinspace M(s)^P(s)\negthinspace -\negthinspace Q_0(s)]e^{A(s-t)}ds,~t\negthinspace \in\negthinspace [0,T],\end{array}\end{equation} and \begin{equation}\label{L2a}\begin{array}{ll} \noalign{ }\displaystyle\bar P(t)\negthinspace =\negthinspace e^{At}\bar P(0)e^{A^t}-\negthinspace \negthinspace \int_0^t\negthinspace \negthinspace e^{A(t-s)}[\bar P(s)\bar M(s)^\negthinspace \negthinspace +\negthinspace \bar M(s)\bar P(s)\negthinspace +\negthinspace \bar Q_0(s)]e^{A^(t-s)}ds,~t\negthinspace \in\negthinspace [0,T].\end{array}\end{equation} We use the above definition simply because when $A$ is bounded, (\ref{L1}) is equivalent to (\ref{L1a}), and (\ref{L2}) is equivalent to (\ref{L2a}). Further, if we let $\Phi(\cdot\,,\cdot)$ and $\bar\Phi(\cdot\,,\cdot)$ be the evolution operators generated by $A-M(\cdot)$ and $A-\bar M(\cdot)$, respectively, then $P(\cdot)$ and $\bar P(\cdot)$ admit the following representation: \begin{equation}\label{L1a}P(t)=\Phi(T,t)^P(T)\Phi(T,t)\negthinspace +\negthinspace \negthinspace \int_t^T\negthinspace \negthinspace \Phi(s,t)^Q_0(s)\Phi(s,t)ds,\qquad t\in[0,T],\end{equation} and \begin{equation}\label{L2a}\bar P(t)=\bar\Phi(t,0)\bar P(0)\bar\Phi(t,0)^-\int_0^t\negthinspace \negthinspace \bar\Phi(t,s)\bar Q_0(s)\bar\Phi(t,s)^ds,\qquad t\in[0,T].\end{equation} This yields that if $$P(T),-\bar P(0),Q_0(t),\bar Q_0(t)\ges0,\qquad t\in[0,T],$$ then \begin{equation}\label{P>0}P(t)\ges0,\quad\bar P(t)\les0,\qquad t\in[0,T].\end{equation} Now, let us look at (\ref{Psi}) and (\ref{Psi}), which are equivalent. We assume that (H0) holds. Therefore, we have two cases to discuss. \it Case 1. \rm Let (\ref{case1}) hold. In this case, since $A$ is dissipative, the appearance of the term $\Gamma(t)A-A\Gamma(t)$ makes (\ref{Psi}) and (\ref{Psi}) difficult to solve in general. To overcome this, we require that for all $t\in[0,T]$, $\Gamma(t):{\cal D}(A)\to{\cal D}(A)$ and \begin{equation}\label{PsiA=APsi}\Gamma(t)Ax=A\Gamma(t)x,\qquad t\in[0,T],~x\in{\cal D}(A).\end{equation} Then both (\ref{Psi}) and (\ref{Psi}) are reduced to the following: \begin{equation}\label{G1}\dot\Gamma(t)-\Gamma(t)M(t)+\bar M(t)\Gamma(t)+\Theta(t)=0,\end{equation} which admits a unique solution as long as, say, $M(\cdot)$, $\bar M(\cdot)$ and $\Theta(\cdot)$ are bounded and $\Gamma(T)\in{\cal L}(X)$ is given. Actually, if $\Psi(\cdot\,,\cdot)$ and $\bar\Psi(\cdot\,,\cdot)$ are evolution operators generated by $-M(\cdot)$ and $\bar M(\cdot)$, respectively, then \begin{equation}\label{G}\Gamma(t)=\bar\Psi(T,t)\Gamma(T)\Psi(T,t)+\int_t^T\bar\Psi(s,t)\Theta(s)\Psi(s,t) ds,\quad t\in[0,T].\end{equation} Note that under (\ref{case1}), $A$ admits a spectral decomposition $$A=\int_{\sigma(A)}\mu dE_\mu,$$ with $\sigma(A)\subseteq(-\infty,-\sigma_0]$ being the spectral of $A$, if \begin{equation}\label{3.24}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\Gamma(T)=\int_{\sigma(A)}\gamma(\mu)dE_\mu,\quad\Theta(t)=\int_{\sigma(A)}\theta(t,\mu)dE_\mu,\\ \noalign{ }\displaystyle M(t)=\int_{\sigma(A)}m(t,\mu)dE_\mu,\quad\bar M(t)=\int_{\sigma(A)}\bar m(t,\mu)dE_\mu,\end{array}\right.\end{equation} for some suitable maps $\gamma:\sigma(A)\to\mathbb{R}$ and $\theta,m,\bar m:[0,T]\times\sigma(A)\to\mathbb{R}$, then $$\Gamma(t)=\int_{\sigma(A)}$e^{\int_t^T[\bar m(s,\mu)-m(s,\mu)]ds}\gamma(\mu) +\int_t^Te^{\int_t^\tau[\bar m(s,\mu)-m(s,\mu)]ds} \theta(\tau,\mu)d\tau$dE_\mu.$$ Hence, (\ref{PsiA=APsi}) holds in this case. In particular, if \begin{equation}\label{3.25}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\gamma(\mu)=\gamma,\quad\theta(t,\mu)=\theta(t),\\ \noalign{ }\displaystyle m(t,\mu)=m(t),\quad\bar m(t,\mu)=\bar m(t),\end{array}\right.\quad(t,\mu)\in[0,T]\times\sigma(A),\end{equation} we have $$\Gamma(t)=$e^{\int_t^T[\bar m(s)-m(s)]ds}\gamma+\int_t^Te^{\int_t^\tau[\bar m(s)-m(s)]ds} \theta(\tau)d\tau$I.$$ Also, as a special case of (\ref{3.24}), if \begin{equation}\label{3.26}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\Gamma(t)=\gamma(t)A_\lambda,\qquad\Theta(t)=\theta(t)A_\lambda,\\ \noalign{ }\displaystyle M(t)=m(t)A_\lambda,\qquad\bar M(t)=\bar m(t)A_\lambda,\end{array}\right.\qquad t\in[0,T],\end{equation} for some suitable scalar functions $\gamma(\cdot),\theta(\cdot),m(\cdot),\bar m(\cdot)$, then $$\Gamma(t)=$e^{\int_t^T[\bar m(s)-m(s)]ds}\gamma+\int_t^Te^{\int_t^\tau[\bar m(s)-m(s)]ds} \theta(\tau)d\tau$A_\lambda,$$ for which (\ref{PsiA=APsi}) will also hold. \it Case 2. \rm Let (\ref{case2}) hold. Then $e^{At}$ is a group. Consequently, $e^{-At}$ is well-defined. Similar to the case of $P(\cdot)$, a map $\Gamma(\cdot)$ is called a mild solution to equation (\ref{Psi}), if the following holds: \begin{equation}\label{3.29}\begin{array}{ll} \noalign{ }\displaystyle\Gamma(t)=e^{-A(T-t)}\Gamma(T)e^{A(T-t)}\\ \noalign{ }\displaystyle\qquad\quad+\negthinspace \negthinspace \int_t^T\negthinspace \negthinspace e^{-A(s-t)}[\bar M(s)\Gamma(s)\negthinspace -\negthinspace \Gamma(s)M(s)\negthinspace +\negthinspace \Theta(s)]e^{A(s-t)}ds,\quad t\in[0,T].\end{array}\end{equation} By recalling the evolution operators $\Phi(\cdot\,,\cdot)$ and $\bar\Phi(\cdot\,,\cdot)$ generated by $A-M(\cdot)$ and $A-\bar M(\cdot)$, (noting that in the current case, $\bar\Phi(s,t)^{-1}$ exists) we have \begin{equation}\label{3.26}\Gamma(t)=\bar\Phi(T,t)^{-1}\Gamma(T)\Phi(T,t)+\int_t^T\bar\Phi(s,t)^{-1} \Theta(s)\Phi(s,t)ds,\quad t\in[0,T].\end{equation} We point out that in the current case, (\ref{PsiA=APsi}) is not needed. However, if (\ref{3.24}) holds and we are working in a complex Hilbert space, we will still have (\ref{PsiA=APsi}) and $\Gamma(\cdot)$ can also be given by (\ref{G}). In what follows, when we say a mild solution $\Pi(\cdot)$ of (\ref{Pi}), we mean that $P(\cdot)$ and $\bar P(\cdot)$ are given by (\ref{L1a}) and (\ref{L2a}), respectively, and $\Gamma(\cdot)$ is defined by by (\ref{G}) such that (\ref{PsiA=APsi}) holds for the case $A=A^$ and $\Gamma(\cdot)$ is defined by (\ref{3.26}) for the case $A^=-A$ (since we prefer to stay with a real Hilbert space). The following is the main result of this section and it will play an important role below. \bf Proposition 5.1. \sl Let $(y(\cdot),\psi(\cdot))$ be a mild solution of linear FBEE $(\ref{FBEE2})$ and $\Pi(\cdot)$ be a mild solution of Lyapunov differential equation $(\ref{Pi})$. Then \begin{equation}\label{}\begin{array}{ll} \noalign{ }\displaystyle\big\langle\Pi(T)\begin{pmatrix}y(T)\\ \psi(T)\end{pmatrix},\begin{pmatrix}y(T)\\ \psi(T)\end{pmatrix}\big\rangle-\big\langle\Pi(0)\begin{pmatrix}y(0)\\ \psi(0)\end{pmatrix},\begin{pmatrix}y(0)\\ \psi(0)\end{pmatrix}\big\rangle\\ \noalign{ }\displaystyle=\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \[\negthinspace \big\langle\negthinspace $\Pi(t)[\mathbb{B}(t)\negthinspace +\negthinspace \mathbb{M}(t)]\negthinspace +\negthinspace [\mathbb{B}(t)\negthinspace +\negthinspace \mathbb{M}(t)]^\Pi(t) \negthinspace -\negthinspace \mathbb{Q}(t)$\negthinspace \begin{pmatrix}y(t)\\ \psi(t)\end{pmatrix},\negthinspace \begin{pmatrix}y(t)\\ \psi(t)\end{pmatrix}\big\rangle\\ \noalign{ }\displaystyle\qquad\qquad+2\big\langle\Pi(t)\begin{pmatrix}b_0(t)\\ -g_0(t)\end{pmatrix},\begin{pmatrix}y(t)\\ \psi(t)\end{pmatrix}\big\rangle\]dt.\end{array}\end{equation} \it Proof. \rm For $\lambda>0$, let $\Phi_\lambda(\cdot\,,\cdot)$ and $\bar\Phi_\lambda(\cdot\,,\cdot)$ be the evolution operators generated by $A_\lambda-M(\cdot)$ and $A_\lambda-\bar M(\cdot)$, respectively. Define \begin{equation}\label{L1b}P_\lambda(t)\negthinspace =\negthinspace \Phi_\lambda(T,t)^P(T)\Phi_\lambda(T,t)+\int_t^T\Phi_\lambda(s,t)^ Q_0(s)\Phi_\lambda(s,t)ds,\qquad t\in[0,T],\end{equation} and \begin{equation}\label{L2b}\bar P_\lambda(t)=\bar\Phi_\lambda(t,0)\bar P(0)\bar\Phi_\lambda(t,0)^-\negthinspace \negthinspace \int_0^t\negthinspace \negthinspace \bar\Phi_\lambda(t,s)\bar Q_0(s)\bar\Phi_\lambda(t,s)^ds,\qquad t\in[0,T].\end{equation} For the case $A^=A$, we have (\ref{G}) with (\ref{PsiA=APsi}) which leads to $$\Gamma(t)A_\lambda=A_\lambda\Gamma(t),\qquad t\in[0,T].$$ For the case $A^=-A$, we define \begin{equation}\label{}\Gamma_\lambda(t)=\negthinspace \bar \Phi_\lambda(T,t)^{-1}\Gamma(T)\Phi_\lambda(T,t)\negthinspace +\negthinspace \negthinspace \negthinspace \int_t^T\negthinspace \negthinspace \negthinspace \bar\Phi_\lambda(s,t)^{-1}\Theta(s) \Phi_\lambda(s,t)ds,\qquad t\in[0,T].\end{equation} A direct computation shows that $$\dot\Pi_\lambda(t)+\Pi_\lambda(t)[\mathbb{A}_\lambda-\mathbb{M}(t)]+[\mathbb{A}_\lambda-\mathbb{M}(t)]^\Pi_\lambda(t) +\mathbb{Q}(t)=0,$$ where $$\mathbb{A}_\lambda=\begin{pmatrix}\sc A_\lambda&\sc0\\\sc0&\sc-A^_\lambda\end{pmatrix},\quad\mathbb{Q}(t)=\begin{pmatrix}\sc Q_0(t)& \sc\Theta(t)^\\ \sc\Theta(t)&\sc\bar Q_0(t)\end{pmatrix},$$ and $$\Pi_\lambda(t)=\begin{pmatrix} \sc P_\lambda(t)&\sc\Gamma_\lambda(t)^\\ \sc\Gamma_\lambda(t)&\sc\bar P_\lambda(t)\end{pmatrix},$$ with $\Gamma_\lambda(\cdot)=\Gamma(\cdot)$ for the case $A^=A$. At the same time, we let $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}\sc\dot y_\lambda(t)\\ \sc\dot\psi_\lambda(t)\end{pmatrix}=\big[\mathbb{A}_\lambda+\mathbb{B}(t)\big] \begin{pmatrix}\sc y_\lambda(t)\\ \sc\psi_\lambda(t)\end{pmatrix}+\begin{pmatrix}\sc b_0(t)\\ \sc-g_0(t) \end{pmatrix},\qquad t\in[0,T],\\ [2mm] \noalign{ }\displaystyle y_\lambda(0)=y(0),\qquad\psi_\lambda(T)=\psi(T).\end{array}\right.$$ Then $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle\Pi_\lambda(T)\begin{pmatrix}y_\lambda(T)\\ \psi_\lambda(T)\end{pmatrix},\begin{pmatrix}y_\lambda(T)\\ \psi_\lambda(T)\end{pmatrix}\big\rangle-\big\langle\Pi_\lambda(0)\begin{pmatrix}y_\lambda(0)\\ \psi_\lambda(0)\end{pmatrix},\begin{pmatrix}y_\lambda(0)\\ \psi_\lambda(0)\end{pmatrix}\big\rangle\\ \noalign{ }\displaystyle=\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \[\big\langle-$\Pi_\lambda(t)[\mathbb{A}_\lambda\negthinspace -\negthinspace \mathbb{M}(t)]\negthinspace +\negthinspace [\mathbb{A}_\lambda\negthinspace -\negthinspace \mathbb{M}(t)]^\Pi_\lambda(t) \negthinspace +\negthinspace \mathbb{Q}(t)$\begin{pmatrix}y_\lambda(t)\\ \psi_\lambda(t)\end{pmatrix},\begin{pmatrix}y_\lambda(t)\\ \psi_\lambda(t)\end{pmatrix}\big\rangle\\ \noalign{ }\displaystyle\qquad+\negthinspace \big\langle\big\{\Pi_\lambda(t)[\mathbb{A}_\lambda+\mathbb{B}(t)]\negthinspace +\negthinspace [\mathbb{A}_\lambda +\mathbb{B}(t)]^\Pi_\lambda(t)\big\} \begin{pmatrix}y_\lambda(t)\\ \psi_\lambda(t)\end{pmatrix},\begin{pmatrix}y_\lambda(t)\\ \psi_\lambda(t)\end{pmatrix}\big\rangle\\ \noalign{ }\displaystyle\qquad+2\big\langle\Pi_\lambda(t)\begin{pmatrix}b_0(t)\\ -g_0(t)\end{pmatrix},\begin{pmatrix}y_\lambda(t)\\ \psi_\lambda(t)\end{pmatrix}\big\rangle\]dt\\ \noalign{ }\displaystyle=\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \[\big\langle\negthinspace $\Pi_\lambda(t)[\mathbb{B}(t)\negthinspace +\negthinspace \mathbb{M}(t)]\negthinspace +\negthinspace [\mathbb{B}(t)\negthinspace +\negthinspace \mathbb{M}(t)]^\Pi_\lambda(t)\negthinspace -\negthinspace \mathbb{Q}(t)$ \begin{pmatrix}y_\lambda(t)\\ \psi_\lambda(t)\end{pmatrix},\begin{pmatrix}y_\lambda(t)\\ \psi_\lambda(t)\end{pmatrix}\big\rangle\\ \noalign{ }\displaystyle\qquad+2\big\langle\Pi_\lambda(t)\begin{pmatrix}b_0(t)\\ -g_0(t)\end{pmatrix},\begin{pmatrix}y_\lambda(t)\\ \psi_\lambda(t)\end{pmatrix}\big\rangle\]dt.\end{array}$$ Passing to the limit, we obtain our result. \signed {$\sqr69$} Next, we let $${\cal G}_0=\Big\{(b_0,g_0,h_0)\bigm\|b_0(\cdot),g_0(\cdot)\in L^2(0,T;X),~ h_0\in X\Big\}.$$ For any $x\in X$ and $(b,g,h)\in{\cal G}_1$, $(b_0,g_0,h_0)\in{\cal G}_0$, and $\rho\in[0,1]$, consider the following FBEE: It is easy to see that when \begin{equation}\label{rho1}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\negthinspace \negthinspace \negthinspace \begin{array}{ll}\dot y^\rho(t)=Ay^\rho(t)+\rho b(t,y^\rho(t),\psi^\rho(t)) +b_0(t),\\ \noalign{ }\displaystyle\dot\psi^\rho(t)=-A^\psi^\rho(t)-\rho g(t,y^\rho(t),\psi^\rho(t)) -g_0(t),\end{array}\qquad t\in[0,T],\\ \noalign{ }\displaystyle y^\rho(0)=x,\qquad\psi^\rho(T)=\rho h(y^\rho(T))+h_0.\end{array}\right.\end{equation} It is easy to see that for $\rho=0$, (\ref{rho1}) is a trivial decoupled FBEE which admits a unique mild solution, and for $\rho=1$, (\ref{rho1}) is essentially the same as (although it looks a little more general than) FBEE (1.1). We will show that under certain conditions, there exists an absolute constant $\varepsilon>0$ such that when (\ref{rho1}) is (uniquely) solvable for some $\rho\in[0,1)$, it must be (uniquely) solvable for (\ref{rho1}) with $\rho$ replaced by $(\rho+\varepsilon)\land1$. Then by repeating the same argument, we obtain the (unique) solvability of (\ref{FBEE1}) over $[0,T]$. Such an argument is called a {\it method of continuation} (see \cite{Yong 1997}). In doing so, the key is to establish an {\it a priori} estimate for the mild solutions to (\ref{rho1}), uniform in $\rho\in[0,1]$. To this end, we need to make some preparations. For any $\lambda>0$, we introduce the following approximate system of FBEE (\ref{rho1}): \begin{equation}\label{rho2}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\negthinspace \negthinspace \negthinspace \begin{array}{ll}\dot y^\rho_\lambda(t)=A_\lambda y^\rho_\lambda(t)+\rho b(t,y^\rho_\lambda(t),\psi^\rho_\lambda(t))+b_0(t),\\ \noalign{ }\displaystyle\dot\psi^\rho_\lambda(t)=-A_\lambda^\psi^\rho_\lambda(t)-\rho g(t,y^\rho_\lambda(t),\psi^\rho_\lambda(t))-g_0(t),\end{array}\qquad t\in[0,T],\\ \noalign{ }\displaystyle y^\rho_\lambda(0)=x,\qquad\psi^\rho_\lambda(T)=\rho h(y^\rho_\lambda(T))+h_0.\end{array}\right.\end{equation} Suppose for the initial condition $x\in X$, the generator $(b,g,h)\in{\cal G}_4$ and $(b_0,g_0,h_0)\in{\cal G}_0$, FBEE (\ref{rho2}) admits a solution $(y_\lambda^\rho(\cdot),\psi_\lambda^\rho(\cdot))$. Also, let $(\bar y_\lambda^\rho(\cdot),\bar \psi_\lambda^\rho(\cdot))$ be a solution of $(\ref{rho2})$ with $(b,g,h)$, $(b_0,g_0,h_0)$, and $x$ respectively replaced by $(\bar b,\bar g,\bar h)\in{\cal G}_1$, $(\bar b_0,\bar g_0,\bar h_0)\in{\cal G}_0$, and $\bar x\in X$. Define \begin{equation}\label{4.4}\widehat y(\cdot)=\bar y_\lambda^\rho(\cdot)-y_\lambda^\rho(\cdot),\quad \widehat\psi(\cdot)=\bar\psi_\lambda^\rho(\cdot)-\psi_\lambda^\rho(\cdot).\end{equation} Denote \begin{equation}\label{4.8}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\widetilde b_y(t)\negthinspace =\negthinspace \negthinspace \int_0^1b_y(t,y_\lambda^\rho(t)+\alpha\widehat y(t), \psi_\lambda^\rho(t)+\alpha\widehat\psi(t))d\alpha,\\ \noalign{ }\displaystyle\widetilde b_\psi(t)\negthinspace =\negthinspace \negthinspace \int_0^1b_\psi(t,y_\lambda^\rho(t)+\alpha\widehat y(t),\psi_\lambda^\rho(t)+\alpha\widehat\psi(t)))d\alpha,\\ \noalign{ }\displaystyle\widetilde g_y(t)\negthinspace =\negthinspace \negthinspace \int_0^1g_y(t,y_\lambda^\rho(t)+\alpha\widehat y(t), \psi_\lambda^\rho(t)+\alpha\widehat\psi(t))d\alpha,\\ \noalign{ }\displaystyle\widetilde g_\psi(t)\negthinspace =\negthinspace \negthinspace \int_0^1g_\psi(t,y_\lambda^\rho(t)+\alpha\widehat y(t),\psi_\lambda^\rho(t)+\alpha\widehat\psi(t)))d\alpha,\\ \noalign{ }\displaystyle\widetilde h_y=\int_0^1h_y(y_\lambda^\rho(T)+\alpha\widehat y(T))d\alpha,\end{array}\right.\end{equation} and set \begin{equation}\label{5.30}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\delta b(t)=\bar b(t,\bar y_\lambda^\rho(t),\bar\psi_\lambda^\rho(t))-b(t,\bar y_\lambda^\rho(t),\bar\psi_\lambda^\rho(t)),\quad\delta b_0(t)=\bar b_0(t)-b_0(t),\\ \noalign{ }\displaystyle\delta g(t)=\bar g(t,\bar y_\lambda^\rho(t),\bar\psi_\lambda^\rho(t))-g(t,\bar y_\lambda^\rho(t),\bar\psi_\lambda^\rho(t)),\quad\delta g_0(t)=\bar g_0(t)-g_0(t),\\ \noalign{ }\displaystyle\delta h=\bar h(\bar y_\lambda^\rho(T))-h(\bar y_\lambda^\rho(T)),\quad\delta h_0=\bar h_0-h_0,\quad\widehat x=\bar x-x.\end{array}\right.\end{equation} Then $(\widehat y(\cdot),\widehat\psi(\cdot))$ satisfies \begin{equation}\label{rho h}\left\{\begin{array}{ll} \noalign{ }\displaystyle\negthinspace \negthinspace \negthinspace \begin{array}{ll}\dot{\negthinspace \negthinspace \widehat y}(t)=A_\lambda\widehat y(t)+\rho\widetilde b_y(t)\widehat y(t) +\rho\widetilde b_\psi(t)\widehat\psi(t)+\rho\delta b(t)+\delta b_0(t),\\ \noalign{ }\displaystyle\dot{\negthinspace \negthinspace \negthinspace \widehat\psi}(t)=-A_\lambda^\widehat\psi(t)\negthinspace -\negthinspace \rho\widetilde g_y(t)\widehat y(t) \negthinspace -\negthinspace \rho\widetilde g_\psi(t)\widehat\psi(t)\negthinspace -\negthinspace \rho\delta g(t)\negthinspace -\negthinspace \delta g_0(t), \end{array}\qquad t\in[0,T],\\ \noalign{ }\displaystyle\negthinspace \negthinspace \widehat y(t)=\widehat x,\qquad\widehat\psi(T)=\rho\widetilde h_y\widehat y(T)+\delta h+\delta h_0.\end{array}\right.\end{equation} For the above linear FBEE, we have the following result. \bf Proposition 5.2. \sl Let $(b,g,h)\in{\cal G}_3$ and \begin{equation}\label{l(bg)}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle L_{by}(t)\negthinspace \triangleq\sup_{(y,\psi)\in X\times X}\[\max\sigma$ {b_y(t,y,\psi)+b_y(t,y,\psi)^\over2}$\Big]^+,\\ \noalign{ }\displaystyle L_{g\psi}(t)\triangleq\sup_{(y,\psi)\in X\times X}\[\max\sigma${ g_\psi(t,y,\psi)+g_\psi(t,y,\psi)^\over2}$\Big]^+,\end{array}\right.\quad t\in[0,T],\end{equation} where $\sigma(\Lambda)$ is the spectrum of the operator $\Lambda\in{\cal L}(X)$. Let $(y^\rho_\lambda(\cdot),\psi^\rho_\lambda(\cdot))$ be a solution to FBEE $(\ref{rho2})$, and $(\bar y^\rho_\lambda(\cdot),\bar\psi^\rho_\lambda(\cdot))$ be a solution of {\rm(\ref{rho2})} corresponding to $(\bar b,\bar g, \bar h)\in{\cal G}_1$, $(b_0,g_0,h_0)\in{\cal G}_0$ and $\bar x\in X$. Then \begin{equation}\label{\|hy\|}\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat y(\cdot)\\|_\infty\leqslant\rho\int_0^Te^{\rho\int_s^TL_{by}(\tau)d\tau} \\|\widetilde b_\psi(s)\widehat\psi(s)\\|ds+K\[\\|\widehat x\\|+\int_0^T$\\|\delta b(s)\\|+\\|\delta b_0(s)\\|$ds\Big],\end{array}\end{equation} and \begin{equation}\label{\|hpsi\|}\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat\psi(\cdot)\\|_\infty\leqslant\rho\[e^{\rho\int_0^TL_{g\psi}(\tau)d\tau}\\|\widetilde h_y\widehat y(T)\\| +\int_t^T\negthinspace \negthinspace \negthinspace e^{\rho\int_0^sL_{g\psi}(\tau)d\tau}\\|\widetilde g_y(s)\widehat y(s)\\|ds\Big]\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad+K\[\\|\delta h\\|+\\|\delta h_0\\|+\int_0^T$\\|\delta g(s)\\|+\\|\delta g_0(s)\\|$ds\Big].\end{array}\end{equation} \rm The proof is straightforward and for reader's convenience, a proof is presented in the appendix. We note that in the above proposition, it is only assumed that $(b,g,h)\in {\cal G}_3$ (the set of all generators satisfying (H3)). Therefore, the Fr\'echet derivatives $b_y,b_\psi$, and so on are not necessarily bounded. However, it is still possible that $$\int_0^T$\\|\widetilde b_\psi(s)\widehat\psi(s)\\|+\\|\widetilde g_y(s)\widehat y(s)\\|$ds<\infty.$$ On the other hand, in the case $(b,g,h)\in{\cal G}_4$, we have $$\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat y(\cdot)\\|_\infty\leqslant\rho\int_0^Te^{\rho\int_s^TL_{by}(\tau)d\tau} \\|\widetilde b_\psi(s)\widehat\psi(s)\\|ds+K\[\\|\widehat x\\|+\int_0^T$\\|\delta b(s)\\|+\\|\delta b_0(s)\\|$ds\Big]\\ \noalign{ }\displaystyle\qquad\qquad\leqslant\rho$\int_0^Te^{\rho\int_s^TL_{by}(\tau)d\tau}\\|\widetilde b_\psi(s)\\|ds$ \\|\widehat\psi(\cdot)\\|_\infty+K\[\\|\widehat x\\|+\int_0^T$\\|\delta b(s)\\|+\\|\delta b_0(s)\\|$ds\Big],\end{array}$$ and $$\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat\psi(\cdot)\\|_\infty\leqslant\rho\[e^{\rho\int_0^TL_{g\psi}(\tau)d\tau}\\|\widetilde h_y\widehat y(T)\\| +\int_0^T\negthinspace \negthinspace \negthinspace e^{\rho\int_0^sL_{g\psi}(\tau)d\tau}\\|\widetilde g_y(s)\widehat y(s)\\|ds\Big]\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad+K\[\\|\delta h\\|+\\|\delta h_0\\|+\int_0^T$\\|\delta g(s)\\|+\\|\delta g_0(s)\\|$ds\Big]\\ \noalign{ }\displaystyle\leqslant\negthinspace \rho^2\negthinspace \[e^{\rho\negthinspace \int_t^T\negthinspace \negthinspace L_{g\psi}(\tau)d\tau}\\|\widetilde h_y\\|\negthinspace +\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \negthinspace e^{\rho\negthinspace \int_0^s\negthinspace \negthinspace L_{g\psi}(\tau)d\tau}\\|\widetilde g_y(s)\\|ds\Big]\[\negthinspace \int_0^T\negthinspace \negthinspace e^{\rho\negthinspace \int_s^T\negthinspace \negthinspace L_{by}(\tau)d\tau}\\|\widetilde b_\psi(s)\\|ds\Big]\\|\widehat\psi(\cdot)\\|_\infty\\ \noalign{ }\displaystyle\quad+K\[\\|\widehat x\\|\negthinspace +\negthinspace \\|\delta h\\|\negthinspace +\negthinspace \\|\delta h_0\\|\negthinspace +\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \negthinspace $\\|\delta b(s)\\|\negthinspace +\negthinspace \\|\delta b_0(s)\\|\negthinspace +\negthinspace \\|\delta g(s)\\|\negthinspace +\negthinspace \\|\delta g_0(s)\\|$ds\Big].\end{array}$$ Hence, when the following holds: \begin{equation}\label{<1}\begin{array}{ll} \noalign{ }\displaystyle\rho^2\negthinspace \[e^{\rho\negthinspace \int_0^T\negthinspace \negthinspace L_{g\psi}(\tau)d\tau}\\|\widetilde h_y\\| \negthinspace +\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \negthinspace e^{\rho\negthinspace \int_0^s\negthinspace \negthinspace L_{g\psi}(\tau)d\tau}\\|\widetilde g_y(s)\\|ds\Big]\[\negthinspace \int_0^T\negthinspace \negthinspace e^{\rho\negthinspace \int_s^T\negthinspace \negthinspace L_{by}(\tau)d\tau}\\|\widetilde b_\psi(s)\\|ds\Big]\negthinspace \negthinspace <\1n1,\end{array}\end{equation} FBEE (\ref{rho2}) admits a unique mild solution $(y_\lambda^\rho(\cdot),\psi_\lambda^\rho(\cdot))$, by means of contraction mapping theorem. It is not hard to see that condition (\ref{<1}) holds when one of the following holds: $\bullet$ The parameter $\rho=0$, this is a trivial case, for which the FBEE is linear and decoupled. $\bullet$ The time duration $T$ is small enough. $\bullet$ The coupling is weak enough in the sense that the Lipschitz constant of $b(t,y,\psi)$ with respect to $\psi$ (the bound of $b_\psi(\cdot)$), and/or the Lipschitz constants of $g(t,y,\psi)$ and $h(y)$ with respect to $y$ (the bounds of $g_y(\cdot)$ and $h_y(\cdot)$) are small enough. An extreme case is that $b(t,y,\psi)$ is independent of $\psi$, or $g(t,y,\psi)$ and $h(y)$ are independent of $y$, which corresponds to the decoupled case. From Proposition 5.2, we see that due to the coupling, in general, one can only obtain an estimate of $\widehat y(\cdot)$ in terms of $\widehat\psi(\cdot)$, and an estimate of $\widehat\psi(\cdot)$ in terms of $\widehat y(\cdot)$. In order to obtain an a priori estimate on the whole $(\widehat y(\cdot),\widehat\psi(\cdot))$, we need either have an estimate for $$\int_0^T\\|\widetilde b_\psi(s)\widehat\psi(s)\\|^2ds$$ independent of $\widehat y(\cdot)$, or have an estimate for $$\\|\widetilde h_y\widehat y(T)\\|^2+\int_0^T\\|\widetilde g_y(s)\widehat y(s)\\|^2ds$$ independent of $\widehat\psi(\cdot)$. We now search conditions under which this is possible. To this end, we introduce the following notions. \bf Definition 5.3. \rm A continuous function $\Pi(\cdot)\equiv\begin{pmatrix}\sc P(\cdot) &\sc\Gamma(\cdot)^\\ \sc\Gamma(\cdot)&\sc\bar P(\cdot)\end{pmatrix}:[0,T]\to\mathbb{S}(X\times X)$ is called a type (I) {\it Lyapunov operator} of the generator $(b,g,h)\in{\cal G}_3$ if there exist $\mathbb{Q}:[0,T]\to\mathbb{S}(X\times X)$ and $\mathbb{M}:[0,T]\to{\cal L}(X\times X)$ with $$\mathbb{Q}(t)=\begin{pmatrix}\sc Q_0(t)&\sc\Theta(t)^\\ \sc\Theta(t)&\sc\bar Q_0(t)\end{pmatrix},\quad\mathbb{M}(t)=\begin{pmatrix}\sc M(t)&\sc0\\ \sc0&\sc-\bar M(t)^\end{pmatrix},\quad t\in[0,T],$$ such that $\Pi(\cdot)$ is a mild solution to the Lyapunov differential equation \begin{equation}\label{Pi}\dot\Pi(t)+\Pi(t)[\mathbb{A}-\mathbb{M}(t)]+[A-\mathbb{M}(t)]^\Pi(t)+\mathbb{Q}(t)=0,\quad t\in[0,T].\end{equation} and for some constants $\mu,K>0$, the following are satisfied: \begin{equation}\label{I}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\;\Pi(0)+\begin{pmatrix} -K&0\cr0&0\end{pmatrix}\les0,\\ [4mm] \noalign{ }\displaystyle\begin{pmatrix}I&\rho h_y(y)^\\ 0&I\end{pmatrix}\negthinspace \Pi(T)\negthinspace \begin{pmatrix}I&0\\ \rho h_y(y)&I\end{pmatrix}\negthinspace \negthinspace +\negthinspace \negthinspace \begin{pmatrix}-\mu h_y(y)^h_y(y)&0\cr0&K\end{pmatrix}\negthinspace \negthinspace \geqslant\1n0,\qquad\forall y\negthinspace \in\negthinspace X,~\rho\negthinspace \in\negthinspace [0,1],\\ [5mm] \noalign{ }\displaystyle\negthinspace \begin{pmatrix}\mathbb{H}^\rho(t,\Pi(t),y,\psi)-\mathbb{Q}(t) +\mu\begin{pmatrix} g_y(t,y,\psi)^g_y(t,y,\psi)&0\cr0&0\end{pmatrix}&\Pi(t)\cr \Pi(t)&-K\end{pmatrix}\negthinspace \negthinspace \leqslant\1n0,\\ [5mm] \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,y,\psi)\in[0,T]\times X\times X,~\rho\in[0,1],\end{array}\right.\end{equation} where \begin{equation}\label{Hrho}\begin{array}{ll} \noalign{ }\displaystyle\mathbb{H}^\rho(t,\Pi,y,\psi)=\rho\big[\Pi\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^* \Pi\big]+\Pi\mathbb{M}(t)+\mathbb{M}(t)^\Pi,\\ [1mm] \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\forall(t,\Pi,y,\psi)\in[0,T]\times\mathbb{S}(X)\times X\times X,~\rho\in[0,1],\end{array}\end{equation} and $$\mathbb{B}(t,y,\psi)=\begin{pmatrix}\sc b_y(t,y,\psi)&\sc b_\psi(t,y,\psi)\\ \sc-g_y(t,y,\psi)&\sc-g_\psi(t,y,\psi)\end{pmatrix}, \qquad\forall(t,y,\psi)\in[0,T]\times X\times X.$$ If (\ref{I}) is replaced by the following: \begin{equation}\label{II}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\Pi(0)+\begin{pmatrix}-K&0\cr0&0\end{pmatrix}\les0,\\ [3mm] \noalign{ }\displaystyle\begin{pmatrix}I&\rho h_y(y)^\\ 0&I\end{pmatrix}\Pi(T)\begin{pmatrix}I&0\\ \rho h_y(y)&I\end{pmatrix}+\begin{pmatrix}0&0\cr0&K\end{pmatrix}\ges0,\quad\forall y\in X,~\rho\in[0,1],\\ [5mm] \noalign{ }\displaystyle\begin{pmatrix}\mathbb{H}^\rho(t,\Pi(t),y,\psi)-\mathbb{Q}(t) +\mu\begin{pmatrix}0&0\cr0&b_\psi(t,y,\psi)^* b_\psi(t,y,\psi)\end{pmatrix}&\Pi(t)\cr\Pi(t)&-K\end{pmatrix}\negthinspace \negthinspace \leqslant\1n0,\\ [5mm] \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,y,\psi)\in[0,T]\times X\times X,~\rho\in[0,1],\end{array}\right.\end{equation} then $\Pi(\cdot)$ is called a type (II) {\it Lyapunov operator} of $(b,g,h)$. If $\Pi(\cdot)$ is either a type (I) or Type (II) Lyapunov operator of $(b,g,h)$, we simply call it a Lyapunov operator of $(b,g,h)$. The existence of a Lyapunov operator gives some kind of compatibility of the coefficients in FBEE (\ref{FBEE1}), which will guarantee the well-posedness of the FBEE. We will carefully discuss properties and existence of Lyapunov operators a little later. First, we present the following result gives the (uniform) stability of mild solutions to (\ref{rho1}) when the generator $(b,g,h)\in{\cal G}_3$ admits a Lyapunov operator. \bf Proposition 5.4. \sl Let $(b,g,h)\in{\cal G}_3$ admit a Lyapunov operator $\Pi(\cdot)$ of either type (I) or (II). For any $\rho\in[0,1]$, and $x\in X$, let $(y^\rho(\cdot),\psi^\rho(\cdot))$ be a mild solution of FBEE $(\ref{rho1})$ with some $(b_0(\cdot),g_0(\cdot),h_0)\in{\cal G}_0$, and let $(\bar y^\rho(\cdot),\bar\psi^\rho(\cdot))$ be a mild solutions of FBEE $(\ref{rho1})$ corresponding to another generator $(\bar b,\bar g,\bar h)\in{\cal G}_1$, and some $(\bar b_0(\cdot),\bar g_0(\cdot),\bar h_0)\in{\cal G}_0$, $\bar x\in X$. Then \begin{equation}\label{stability1}\negthinspace \negthinspace \negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\\|\bar y^\rho(\cdot)-y^\rho(\cdot)\\|_\infty+\\|\bar\psi^\rho(\cdot)-\psi^\rho(\cdot)\\|_\infty \leqslant K\Big\{\\|\bar x-x\\|^2\negthinspace \negthinspace +\\|\bar h_0-h_0\\|^2 +\\|\bar h(\bar y^\rho(T))-h(\bar y^\rho(T))\\|^2\\ \noalign{ }\displaystyle\qquad+\int_0^T\Big(\\|\bar b(s,\bar y^\rho(s),\bar\psi^\rho(s))- b(t,\bar y^\rho(t),\bar\psi^\rho(s))\|^2+\\|\bar b_0(s)-b_0(s)\\|^2\\ \noalign{ }\displaystyle\qquad\quad+\\|\bar g(s,\bar y^\rho(s),\bar\psi^\rho(s))-g(s,\bar y^\rho(s),\bar\psi^\rho(s))\\|^2+\\|\bar g_0(s)-g_0(s)\\|^2\Big)ds\Big\},\end{array}\end{equation} uniformly in $\rho\in[0,1]$. In particular, if $(\bar b,\bar g,\bar h)=(b,g,h)$, then \begin{equation}\label{stability2}\negthinspace \negthinspace \negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\\|\bar y^\rho(\cdot)-y^\rho(\cdot)\\|_\infty^2+\\|\bar\psi^\rho(\cdot)-\psi^\rho(\cdot) \\|_\infty^2\\ \noalign{ }\displaystyle\leqslant\negthinspace K\Big\{\\|\bar x-x\\|^2\negthinspace \negthinspace +\negthinspace \\|\bar h_0-h_0\\|^2\negthinspace \negthinspace +\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \Big(\\|\bar b_0(s)\negthinspace -\negthinspace b_0(s)\\|^2\negthinspace \negthinspace +\negthinspace \\|\bar g_0(t)\negthinspace -\negthinspace g_0(s)\\|^2\Big)ds\Big\},\end{array}\end{equation} uniformly in $\rho\in[0,1]$. \it Proof. \rm Recall notations in (\ref{4.4})--(\ref{5.30}) and noting Proposition 5.1, we have (suppressing $s$ when it has no ambiguity) $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle\begin{pmatrix}\sc I&\sc\rho(\widetilde h_y)^\\ \sc0&\sc I\end{pmatrix}\Pi(T)\begin{pmatrix}\sc I&\sc0\\ \sc\rho\widetilde h_y&\sc I\end{pmatrix}\begin{pmatrix}\sc\widehat y(T)\cr\sc\rho\delta h+\delta h_0\end{pmatrix},\begin{pmatrix}\sc\widehat y(T)\cr\sc\rho\delta h+\delta h_0\end{pmatrix}\big\rangle-\big\langle\Pi(0)\begin{pmatrix}\sc\widehat x\cr\sc\widehat \psi(0)\end{pmatrix},\begin{pmatrix}\sc\widehat x\cr\sc\widehat\psi(0) \end{pmatrix}\big\rangle\\ [4mm] \noalign{ }\displaystyle=\big\langle\Pi(T)\begin{pmatrix}\sc\widehat y(T)\cr\sc\rho\widetilde h_y\widehat y(T)+\rho\delta h+\delta h_0\end{pmatrix},\begin{pmatrix}\sc\widehat y(T)\cr\sc\rho\widetilde h_y\widehat y(T)+\rho\delta h+\delta h_0\end{pmatrix}\big\rangle-\big\langle\Pi(0)\begin{pmatrix}\sc\widehat x\cr\sc\widehat \psi(0)\end{pmatrix},\begin{pmatrix}\sc\widehat x\cr\sc\widehat\psi(0) \end{pmatrix}\big\rangle\\ [4mm] \noalign{ }\displaystyle=\big\langle\Pi(T)\begin{pmatrix}\sc\widehat y(T)\cr\sc\widehat\psi(T)\end{pmatrix},\begin{pmatrix}\sc\widehat y(T)\cr\sc\widehat\psi(T)\end{pmatrix}\big\rangle-\big\langle \Pi(0)\begin{pmatrix}\sc\widehat x\cr\sc\widehat\psi(0)\end{pmatrix},\begin{pmatrix}\sc\widehat x\cr\sc\widehat\psi(0) \end{pmatrix}\big\rangle\\ [4mm] \noalign{ }\displaystyle=\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \negthinspace \Big\{\negthinspace \big\langle\negthinspace $\widetilde\mathbb{H}^\rho-\mathbb{Q}$\begin{pmatrix}\sc\widehat y\cr\sc\widehat\psi\end{pmatrix},\begin{pmatrix}\sc\widehat y\cr\sc\widehat \psi\end{pmatrix}\big\rangle+2\big\langle\Pi\begin{pmatrix}\sc\widehat y\cr\sc\widehat\psi\end{pmatrix},\begin{pmatrix}\sc\rho\delta b+\delta b_0\cr\sc\rho\delta g+\delta g_0\end{pmatrix}\big\rangle\Big\}ds\\ [4mm] \noalign{ }\displaystyle=\int_0^T\big\langle\begin{pmatrix}\sc\widetilde\mathbb{H}^\rho-\mathbb{Q}&\sc\Pi\cr\sc \Pi&\sc0\end{pmatrix}\begin{pmatrix}\sc\widehat y\cr\sc\widehat\psi\cr\sc\rho\delta b+\delta b_0\cr\sc\rho\delta g+\delta g_0\end{pmatrix},\begin{pmatrix}\sc\widehat y\cr\sc\widehat\psi\cr\sc\rho\delta b+\delta b_0\cr\sc\rho\delta g+\delta g_0\end{pmatrix}\big\rangle ds,\end{array}$$ where $$\widetilde\mathbb{H}^\rho(t,\Pi,y,\psi)=\rho\big[\Pi\widetilde\mathbb{B}+\widetilde\mathbb{B}^ \Pi\big]+\Pi\mathbb{M}(t)+\mathbb{M}(t)^\Pi,$$ and $$\widetilde\mathbb{B}=\begin{pmatrix}\widetilde b_y&\widetilde b_\psi\\ -\widetilde g_y&-\widetilde g_\psi\end{pmatrix},$$ with $\widetilde b_y$, etc. given by (\ref{4.8}). Consequently, in the case that $\Pi(\cdot)$ is a type (I) Lyapunov operator of $(b,g,h)$, we have $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle\Pi(T)\begin{pmatrix}\sc\widehat y(T)\cr\sc\rho\widetilde h_y\widehat y(T)+\rho\delta h+\delta h_0\end{pmatrix},\begin{pmatrix}\sc\widehat y(T)\cr\sc\rho\widetilde h_y\widehat y(T)+\rho\delta h+\delta h_0\end{pmatrix}\big\rangle-\big\langle \Pi(0)\begin{pmatrix}\sc\widehat x\cr\sc\widehat \psi(0)\end{pmatrix},\begin{pmatrix}\sc\widehat x\cr\sc\widehat\psi(0)\end{pmatrix}\big\rangle\\ \noalign{ }\displaystyle\geqslant\mu\\|\widetilde h_y\widehat y(T)\\|^2-K\[\\|\widehat x\\|^2+\\|\delta h\\|^2+\\|\delta h_0\\|^2\Big],\end{array}$$ and $$\begin{array}{ll} \noalign{ }\displaystyle\int_0^T\big\langle\begin{pmatrix}\sc\widetilde\mathbb{H}^\rho-\mathbb{Q}&\sc\Pi\cr\sc \Pi&\sc0\end{pmatrix}\begin{pmatrix}\sc\widehat y\cr\sc\widehat\psi\cr\sc\rho\delta b+\delta b_0\cr\sc\rho\delta g+\delta g_0\end{pmatrix},\begin{pmatrix}\sc\widehat y\cr\sc\widehat\psi\cr\sc\rho\delta b+\delta b_0\cr\sc\rho\delta g+\delta g_0\end{pmatrix}\big\rangle ds\\ \noalign{ }\displaystyle\leqslant\int_0^T$-\mu\\|\widetilde g_y(s)\widehat y(s)\\|^2+K\\|\rho\delta b(s)+\delta b_0(s)\\|^2+K\\|\rho\delta g(s)+\delta g_0(s)\\|^2$ds.\end{array}$$ Hence, $$\begin{array}{ll} \noalign{ }\displaystyle\\|\widetilde h_y\widehat y(T)\\|^2+\int_0^T\\|\widetilde g_y(s)\widehat y(s)\\|^2ds\leqslant K\[\\|\widehat x\\|^2+\\|\delta h\\|^2+\\|\delta h_0\\|^2\\ \noalign{ }\displaystyle\qquad\qquad\qquad+\int_0^T$\\|\delta b(s)\\|^2+\\|\delta b_0(s)\\|^2+\\|\delta g(s)\\|^2+\\|\delta g_0(s)\\|^2$ds\Big].\end{array}$$ Combining the above with (\ref{\|hy\|}) and (\ref{\|hpsi\|}), we obtain (\ref{stability1}). On the other hand, in the case that $\Pi(\cdot)$ is a type (II) Lyapunov operator for $(b,g,h)$, we have $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle\Pi(T)\begin{pmatrix}\widehat y(T)\cr\rho\widetilde h_y\widehat y(T)+\rho\delta h+\delta h_0\end{pmatrix},\begin{pmatrix}\widehat y(T)\cr\rho\widetilde h_y\widehat y(T)+\rho\delta h+\delta h_0\end{pmatrix}\big\rangle-\big\langle \Pi(0)\begin{pmatrix}\widehat x\cr\widehat \psi(0)\end{pmatrix},\begin{pmatrix}\widehat x\cr\widehat\psi(0)\end{pmatrix}\big\rangle\\ [4mm] \noalign{ }\displaystyle\geqslant-K\[\\|\widehat x\\|^2+\\|\delta h\\|^2+\\|\delta h_0\\|^2\Big],\end{array}$$ and $$\begin{array}{ll} \noalign{ }\displaystyle\int_0^T\big\langle\begin{pmatrix}\sc\widetilde\mathbb{H}^\rho-\mathbb{Q}&\sc\Pi\cr\sc \Pi&\sc0\end{pmatrix}\begin{pmatrix}\sc\widehat y\cr\sc\widehat\psi\cr\sc\rho\delta b+\delta b_0\cr\sc\rho\delta g+\delta g_0\end{pmatrix},\begin{pmatrix}\sc\widehat y\cr\sc\widehat\psi\cr\sc\rho\delta b+\delta b_0\cr\sc\rho\delta g+\delta g_0\end{pmatrix}\big\rangle ds\\ \noalign{ }\displaystyle\leqslant\int_0^T$-\mu\\|\widetilde b_\psi(s)\widehat\psi(s)\\|^2+K\\|\rho\delta b(s)+\delta b_0(s)\\|^2+K\\|\rho\delta g(s)+\delta g_0(s)\\|^2$ds.\end{array}$$ Hence, $$\begin{array}{ll} \noalign{ }\displaystyle\int_0^T\\|\widetilde b_\psi(s)\widehat\psi(s)\\|^2ds\leqslant K\[\\|\widehat x\\|^2+\\|\delta h\\|^2+\\|\delta h_0\\|^2\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\int_0^T$\\|\delta b(s)\\|^2+\\|\delta b_0(s)\\|^2+\\|\delta g(s)\\|^2+\\|\delta g_0(s)\\|^2$ds\Big].\end{array}$$ Then, combining the above with (\ref{\|hy\|}) and (\ref{\|hpsi\|}), we again obtain (\ref{stability1}). \signed {$\sqr69$} \section{Well-Posedness of FBEEs via Lyapunov Operators} We now state and prove the following theorem concerning the well-posedness of FBEE (\ref{FBEE1}). \bf Theorem 6.1. \sl Let $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$ admit a type {\rm(I)} or {\rm(II)} Lyapunov operator $\Pi(\cdot)$. Then FBEE $(\ref{FBEE1})$ admits a unique mild solution $(y(\cdot),\psi(\cdot))$. Moreover, the following estimate holds: \begin{equation}\label{2.38}\begin{array}{ll} \noalign{ }\displaystyle\\|y(\cdot)\\|_\infty+\\|\psi(\cdot)\\|_\infty\leqslant K\[\\|x\\|+\\|h(0)\\|+\int_0^T$\\|b(s,0,0)\\|+\\|g(s,0,0)\\|$ds\Big].\end{array}\end{equation} Further, if $(\bar y(\cdot),\bar\psi(\cdot))$ is a mild solution of FBEE $(1.1)$ corresponding to $(\bar b,\bar g,\bar h)\in{\cal G}_2\cap{\cal G}_3$, then the following stability estimate holds: \begin{equation}\label{stability3}\begin{array}{ll} \noalign{ }\displaystyle\\|\bar y(\cdot)-y(\cdot)\\|_\infty+\\|\bar \psi(\cdot)-\psi(\cdot)\\|_\infty\leqslant K\Big\{\\|\bar x-x\\|+\\|\bar h(\bar y(T))-h(\bar y(T))\\|\\ \noalign{ }\displaystyle\qquad\qquad\qquad+\int_0^T$\\|\bar b(s,\bar y(s),\bar\psi(s))-b(s,\bar y(s),\bar\psi(s))\\|+\\|\bar g(s,\bar y(s),\bar \psi(s)-g(s,\bar y(s),\bar\psi(s)\\|$ds\Big\}.\end{array}\end{equation} \it Proof. \rm Let $(b_0,g_0,h_0)\in{\cal G}_0$. Let $\rho\in[0,1)$. Suppose the following (coupled) FBEE admits a unique mild solution $(y^\rho(\cdot),\psi^\rho(\cdot))$: \begin{equation}\label{lambda3}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y^\rho(t)=Ay^\rho(t)+\rho b(t,y^\rho(t),\psi^\rho(t))+b_0(t),\\ \noalign{ }\displaystyle\dot\psi^\rho(t)=-A^\psi^\rho(t)+\rho g(t,y^\rho(t),\psi^\rho(t)) +g_0(t),\\ \noalign{ }\displaystyle y^\rho(0)=x,\qquad\psi^\rho(T)=\rho h(y^\rho(T))+h_0,\end{array}\right.\end{equation} and the following estimate holds: \begin{equation}\label{}\begin{array}{ll} \noalign{ }\displaystyle\\|y^\rho(\cdot)\\|_\infty+\\|\psi^\rho(\cdot)\\|_\infty\leqslant K\Big\{\\|x\\|+\\|h(0)\\|+\\|h_0\\|\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\int_0^T\negthinspace \negthinspace $\\|b(s,0,0)\\|+\\|b_0(s)\\|+\\|g(s,0,0)\\|+\\|g_0(s)\\|$ds\Big\}.\end{array}\end{equation} Now, let $\varepsilon>0$ such that $\rho+\varepsilon\in[0,1]$. Consider the following coupled FBEE: \begin{equation}\label{3.3}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle \dot y^{\rho+\varepsilon}(t)=Ay^{\rho+\varepsilon}(t)+(\rho+\varepsilon)b(t,y^{\rho+\varepsilon}(t),\psi^{\rho+\varepsilon}(t))+b_0(t),\\ \noalign{ }\displaystyle \dot\psi^{\rho+\varepsilon}(t)=-A^\psi^{\rho+\varepsilon}(t)+(\rho+\varepsilon)g(t,y^{\rho+\varepsilon}(s),\psi^{\rho+\varepsilon}(t)) +g_0(t),\\ \noalign{ }\displaystyle y^{\rho+\varepsilon}(t)=x,\qquad\psi^{\rho+\varepsilon}(T)=(\rho+\varepsilon) h(y^{\rho+\varepsilon}(T))+h_0,\end{array}\right.\end{equation} To obtain the (unique) solvability of the above problem, we introduce the following sequence of problems: \begin{equation}\label{}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle y^{\rho+\varepsilon,0}(\cdot)=\psi^{\rho+\varepsilon,0}(\cdot)=0,\\ \noalign{ }\displaystyle\dot y^{\rho+\varepsilon,k+1}(t)=Ay^{\rho+\varepsilon,k+1}(t)+\rho b(t,y^{\rho+\varepsilon,k+1}(t),\psi^{\rho+\varepsilon,k+1}(t))\\ \noalign{ }\displaystyle\qquad\qquad\qquad\quad+\varepsilon b(t,y^{\rho+\varepsilon,k}(t),\psi^{\rho+\varepsilon,k}(t))+b_0(t),\\ \noalign{ }\displaystyle\dot\psi^{\rho+\varepsilon,k+1}(t)=-A^\psi^{\rho+\varepsilon,k+1}(t)+\rho g(t,y^{\rho+\varepsilon,k+1}(t),\psi^{\rho+\varepsilon,k+1}(t))\\ \noalign{ }\displaystyle\qquad\qquad\qquad\quad+\varepsilon g(t,y^{\rho+\varepsilon,k}(t),\psi^{\rho+\varepsilon,k}(t))+g_0(t),\\ \noalign{ }\displaystyle y^{\rho+\varepsilon,k+1}(0)=x,\\ \noalign{ }\displaystyle\psi^{\rho+\varepsilon,k+1}(T)=\rho h(y^{\rho+\varepsilon,k+1}(T))+\varepsilon h(y^{\rho+\varepsilon,k}(T))+h_0.\end{array}\right.\end{equation} By our assumption, inductively, for each $k\ges0$, as long as $(y^{\rho+\varepsilon,k}(\cdot),\psi^{\rho+\varepsilon,k}(\cdot))\in C([t,T];X\times X)$, the above FBEE admits a unique mild solution $(y^{\rho+\varepsilon,k+1}(\cdot),$ $\psi^{\rho+\varepsilon,k+1}(\cdot))\in C([t,T];X\times X)$. Further, \begin{equation}\label{6.7}\begin{array}{ll} \noalign{ }\displaystyle\\|y^{\rho+\varepsilon,k+1}(\cdot)\\|_\infty+\\|\psi^{\rho+\varepsilon,k+1}(\cdot)\\|_\infty\\ \noalign{ }\displaystyle\leqslant\negthinspace \negthinspace K\Big\{\\|x\\|+\\|h(0)\\|+\\|\varepsilon h(y^{\rho+\varepsilon,k+1}(T))+h_0\\|\\ \noalign{ }\displaystyle\qquad\quad+\int_0^T\negthinspace \negthinspace \Big(\\|b(s,0,0)\\|+\\|\varepsilon b(s,y^{\rho+\varepsilon,k}(s),\psi^{\rho+\varepsilon,k}(s))+ b_0(s)\\|\)ds\\ \noalign{ }\displaystyle\qquad\quad+\int_0^T\negthinspace \negthinspace \Big(\\|g(s,0,0)\\|+\\|\varepsilon g(s,y^{\rho+\varepsilon,k}(s),\psi^{\rho+\varepsilon,k}(s))+g_0(t)\\|\)ds\Big\}\\ \noalign{ }\displaystyle\leqslant K\Big\{\\|x\\|+\\|h(0)\\|+\\|h_0\\|+\varepsilon \\|y^{\rho+\varepsilon,k}(T)\\|\\ \noalign{ }\displaystyle\qquad\quad+\int_0^T\negthinspace \negthinspace \negthinspace \Big[\\|b(s,0,0)\\|+\\|b_0(s)\\|+\varepsilon $\\|y^{\rho+\varepsilon,k}(s)\|+\\|\psi^{\rho+\varepsilon,k}(s)\\|$\]ds\\ \noalign{ }\displaystyle\qquad\quad+\int_0^T\negthinspace \negthinspace \negthinspace \[\\|g(s,0,0)\\|+\\|g_0(s)\\| +\negthinspace \varepsilon$\\|y^{\rho+\varepsilon,k}(s)\\|+\\|\psi^{\rho+\varepsilon,k}(s)\\|$\]ds\Big\}\\ \noalign{ }\displaystyle\leqslant K\varepsilon$\\|y^{\rho+\varepsilon,k}(\cdot)\\|_\infty+\\|\psi^{\rho+\varepsilon,k}(\cdot)\\|_\infty$\\ \noalign{ }\displaystyle\qquad+K\Big\{\\|x\\| +\\|h(0)\\|+\\|h_0\\|+\int_0^T\negthinspace \negthinspace \negthinspace $\\|b(s,0,0)\\|+\\|b_0(s)\\|+ \\|g(s,0,0)\\|+\\|g_0(s)\\|$ds\Big\}.\end{array}\end{equation} Now, since $(b,g,h)$ admits a type (I) or type (II) Lyapunov operator $P(\cdot)$, by Proposition 5.3, we obtain $$\begin{array}{ll} \noalign{ }\displaystyle\\|y^{\rho+\varepsilon,k+1}(\cdot)-y^{\rho+\varepsilon,k}(\cdot)\\|_\infty+\\|\psi^{\rho+\varepsilon,k+1}(\cdot)-\psi^{\rho+\varepsilon,k}(\cdot) \\|_\infty\\ \noalign{ }\displaystyle\leqslant\varepsilon K\Big\{\\|h(y^{\rho+\varepsilon,k}(T))-h(y^{\rho+\varepsilon,k-1}(T))\\|\\ \noalign{ }\displaystyle\qquad+\int_0^T\Big[\\| b(s,y^{\rho+\varepsilon,k}(s),\psi^{\rho+\varepsilon,k}(s))-b(s,y^{\rho+\varepsilon,k-1}(s),\psi^{\rho+\varepsilon,k-1}(s))\\|\\ \noalign{ }\displaystyle\qquad\quad+\\|g(s,y^{\rho+\varepsilon,k}(s),\psi^{\rho+\varepsilon,k}(s))-g(s,y^{\rho+\varepsilon,k-1}(s),\psi^{\rho+\varepsilon,k-1}(s)) \\|\Big]ds\Big\}\\ \noalign{ }\displaystyle\leqslant\varepsilon K_0$\\|y^{\rho+\varepsilon,k}(\cdot)-y^{\rho+\varepsilon,k-1}(\cdot)\\|_\infty+\\|\psi^{\rho+\varepsilon,k}(\cdot)-\psi^{\rho+\varepsilon,k-1}(\cdot) \\|_\infty$.\end{array}$$ Here, $K_0>0$ is an absolute constant (independent of $k\ges1$). Thus, taking $\varepsilon>0$ small enough so that $\varepsilon K_0\leqslant{1\over2}$, we obtain $$\lim_{k\to\infty}$\\|y^{\rho+\varepsilon,k}(\cdot)-y^{\rho+\varepsilon}(\cdot)\\|_\infty+\\|\psi^{\rho+\varepsilon,k}(\cdot)- \psi^{\rho+\varepsilon}(\cdot)\\|_\infty$=0,$$ with $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle y^{\rho+\varepsilon}(\cdot)=\sum_{k=1}^\infty\Big[y^{\rho+\varepsilon,k}(\cdot)-y^{\rho+\varepsilon,k-1}(\cdot)\Big],\\ \noalign{ }\displaystyle \psi^{\rho+\varepsilon}(\cdot)=\sum_{k=1}^\infty\Big[\psi^{\rho+\varepsilon,k}(\cdot) -\psi^{\rho+\varepsilon,k-1}(\cdot)\Big].\end{array}\right.$$ which is the unique mild solution of FBEE (\ref{3.3}). Further, let $k\to\infty$ in (\ref{6.7}), we obtain $$\begin{array}{ll} \noalign{ }\displaystyle\\|y^{\rho+\varepsilon}(\cdot)\\|_\infty+\\|\psi^{\rho+\varepsilon}(\cdot)\\|_\infty\leqslant K\varepsilon$\\|y^{\rho+\varepsilon}(\cdot)\\|_\infty+\\|\psi^{\rho+\varepsilon}(\cdot)\\|_\infty$\\ \noalign{ }\displaystyle\qquad\qquad+K\Big\{ \\|x\\|+\\|h(0)\\|+\\|h_0\\|+\int_t^T\negthinspace \negthinspace \negthinspace \Big(\\|b(s,0,0)\\|+\\|b_0(s)\\|+\\|g(s,0,0)\\|+\\|g_0(s)\\|\Big)^2\negthinspace ds\Big]\Big\}.\end{array}$$ Note that the constant $K$ in front of $\varepsilon$ above is universal. Then choose an $\varepsilon>0$ satisfying $K\varepsilon\le{1\over2}$ so that the first term on the right hand side can be absorbed into the left hand, leading to the following: $$\begin{array}{ll} \noalign{ }\displaystyle\\|y^{\rho+\varepsilon}(\cdot)\\|_\infty+\\|\psi^{\rho+\varepsilon}(\cdot)\\|_\infty\leqslant K\Big\{\\|x\\|+\\|h(0)\\| +\\|h_0\\|\\ \noalign{ }\displaystyle\qquad\qquad+\int_t^T\Big(\\|b(s,0,0)\\|+\\|b_0(s)\\|+\\|g(s,0,0)\\|+\\|g_0(s)\\|\)ds\Big\}.\end{array}$$ Continuing the above procedure, we obtain the solvability of the following coupled FBEE: $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle \dot y(t)=Ay(t)+b(t,y(t),\psi(t))+b_0(t),\\ \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-g(t,y(t),\psi(t))-g_0(t),\\ \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=h(y(T))+h_0,\end{array}\right.$$ with the mild solution $(y(\cdot),\psi(\cdot))$ satisfying $$\begin{array}{ll} \noalign{ }\displaystyle\\|y(\cdot)\\|_\infty+\\|\psi(\cdot)\\|_\infty\leqslant K\Big\{\\|x\\|+\\|h(0)\\|+\int_0^T\Big(\\|b(s,0,0)\\|+\\|b_0(s)\\|+ \\|g(s,0,0)\\|+\\|g_0(s)\\|\Big)ds\Big\}.\end{array}$$ Thus, in particular, by taking $(b_0,g_0,h_0)=0$, we obtain the solvability of FBEE (1.1) with estimate \begin{equation}\label{5.15}\begin{array}{ll} \noalign{ }\displaystyle\\|y(\cdot)\\|_\infty\negthinspace +\negthinspace \\|\psi(\cdot)\\|_\infty\negthinspace \leqslant\negthinspace K\Big\{\\|x\\|\negthinspace +\negthinspace \\|h(0)\\|\negthinspace +\negthinspace \negthinspace \int_0^T\negthinspace \negthinspace \Big(\\|b(s,0,0)\\|\negthinspace +\negthinspace \\|g(s,0,0)\\|\Big)ds\Big\}.\end{array}\end{equation} Now, let $(\bar y(\cdot),\bar\psi(\cdot))$ be a mild solution to (1.1) corresponding to $(\bar b,\bar g,\bar h)\in{\cal G}_2\cap{\cal G}_3$. Then, $(\bar y(\cdot)-y(\cdot),\bar\psi(\cdot)-\psi(\cdot))$ satisfies a linear FBEE with the generator admitting a type (I) or type (II) Lyapunov operator $\Pi(\cdot)$, the same as that for the generator $(b,g,h)$. Hence, applying (\ref{5.15}), we obtain the following stability estimate: \begin{equation}\label{5.16}\begin{array}{ll} \noalign{ }\displaystyle\\|\bar y(\cdot)-y(\cdot)\\|_\infty+\\|\bar\psi(\cdot)-\psi(\cdot)\\|_\infty\leqslant K\Big\{\\|\bar x-x\\|+\\|\bar h(\bar y(T))-h(\bar y(T))\\|\\ \noalign{ }\displaystyle\qquad\quad+\int_0^T$\\|\bar b(s,\bar y(s),\bar\psi(s))-b(s,\bar y(s),\bar\psi(s))\\|+\\|\bar g(s,\bar y(s),\bar\psi(s))-g(s,\bar y(s),\bar\psi(s))\\|$ds\Big\}.\end{array}\end{equation} This proves the theorem. \signed {$\sqr69$} \section{Construction of Lyapunov Operators and Solvable FBEEs} In this section, we will construct some Lyapunov operators, through which we obtain well-posedness of corresponding FBEEs. First of all, we prove the following result which is practically more convenient to use than the definition. \bf Theorem 7.1. \sl Let {\rm(H0)} hold and let $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$. Let $\Pi(\cdot)\equiv\begin{pmatrix}\sc P(\cdot)&\sc\Gamma(\cdot)^\\ \sc\Gamma(\cdot)&\sc\bar P(\cdot)\end{pmatrix}$ be a mild solution to linear Lyapunov differential equation $(\ref{Pi})$ for some \begin{equation}\label{MQ}\mathbb{M}(\cdot)\equiv\begin{pmatrix}\sc M(\cdot)&\sc0\\ \sc0&\sc-\bar M(\cdot)^\end{pmatrix},\qquad\mathbb{Q}(\cdot)\equiv\begin{pmatrix}\sc Q_0(\cdot)&\Theta(\cdot)^\\ \Theta(\cdot)&\sc\bar Q_0(\cdot)\end{pmatrix}.\end{equation} Then $\Pi(\cdot)$ is both a Lyapunov operator of types {\rm(I)} and {\rm(II)} for $(b,g,h)$ if the following hold: \begin{equation}\label{t,T}\bar P(t)\leqslant-\delta,\qquad P(T)\geqslant\delta,\end{equation} \begin{equation}\label{T}P(T)+h_y(y)^\Gamma(T)\negthinspace +\negthinspace \Gamma(T)^h_y(y)+h_y(y)^\bar P(T)h_y(y)\geqslant\delta,\qquad\forall y\in X,\end{equation} and \begin{equation}\label{t<s<T}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\Pi(t)\mathbb{M}(t)+\mathbb{M}(t)^\Pi(t)-\mathbb{Q}(t)\leqslant-\delta,\qquad t\in[0,T],\\ [1mm] \noalign{ }\displaystyle\Pi(t)\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\Pi(t)+\Pi(t)\mathbb{M}(t)+\mathbb{M}(t)^ \Pi(t)-\mathbb{Q}(t)\leqslant-\delta,\quad(t,y,\psi)\negthinspace \in\negthinspace [0,T]\negthinspace \times \negthinspace X\negthinspace \times\negthinspace X.\end{array}\right.\end{equation} for some $\delta>0$, with $$\mathbb{B}(t,y,\psi)=\begin{pmatrix}\sc b_y(t,y,\psi)&\sc b_\psi(t,y,\psi)\\ \sc-g_y(t,y,\psi)&\sc-g_\psi(t,y,\psi)\end{pmatrix},\qquad(t,y,\psi)\in[0,T] \times X\times X.$$ \rm Note that $\delta>0$ appears in (\ref{t,T})--(\ref{t<s<T}) does not have to be the same. But, we can always make them the same by shrinking $\delta$ if necessary. \it Proof. \rm First of all, in order $\Pi(\cdot)$ to be a type (I) Lyapunov operator of the generator $(b,g,h)$, one needs (\ref{I}). Hence, at $t=0$, one needs $$0\geqslant\Pi(0)+\begin{pmatrix}\sc -K&\sc0\cr\sc0&\sc0\end{pmatrix}=\begin{pmatrix}\sc P(0)-K&\sc\Gamma(0)^\cr\sc\Gamma(0)&\sc\bar P(0)\end{pmatrix},$$ for some $K>0$, which will be ensured by the following: \begin{equation}\label{}\bar P(0)\leqslant-\delta,\end{equation} for some $\delta>0$. Next, at $t=T$, one needs \begin{equation}\label{6.5}\begin{array}{ll} \noalign{ }\ds0\leqslant\begin{pmatrix}\sc I&\sc\rho h_y(y)^\\ \sc0&\sc I\end{pmatrix}\Pi(T)\begin{pmatrix}\sc I&\sc0\\ \sc\rho h_y(y)&\sc I\end{pmatrix}+\begin{pmatrix}\sc-\mu h_y(y)^* h_y(y)&\sc0\cr\sc0&\sc K\end{pmatrix}\\ [4mm] \noalign{ }\displaystyle\quad=\negthinspace \rho^2\negthinspace \begin{pmatrix}\sc0&\sc h_y(y)^\\ \sc0&\sc0\end{pmatrix}\negthinspace \Pi(T)\negthinspace \begin{pmatrix}\sc0&\sc0\\ \sc h_y(y)&\sc0\end{pmatrix}\negthinspace \negthinspace +\negthinspace \rho\[\negthinspace \begin{pmatrix}\sc0&\sc h_y(y)^\\ \sc0&\sc0\end{pmatrix}\negthinspace \Pi(T)\negthinspace +\negthinspace \Pi(T)\negthinspace \begin{pmatrix}\sc0&\sc0\\ \sc h_y(y)&\sc0\end{pmatrix}\negthinspace \Big]\\ [4mm] \noalign{ }\displaystyle\qquad\quad+\Pi(T)+\begin{pmatrix}\sc-\mu h_y(y)^* h_y(y)&\sc0\cr\sc0&\sc K\end{pmatrix},\qquad\qquad\forall y\negthinspace \in\negthinspace X,~\rho\negthinspace \in\negthinspace [0,1],\end{array}\end{equation} for some $\mu,K>0$. If we are able to show the following (which will be done below) \begin{equation}\label{bar P(T)<0}\bar P(T)\les0,\end{equation} then \begin{equation}\label{bar P(T)<0}\begin{pmatrix}0&h_y(y)^\\ 0&0\end{pmatrix}\Pi(T)\begin{pmatrix}0&0\\ h_y(y)&0\end{pmatrix}=\begin{pmatrix}h_y(y)^\bar P(T)h_y(y)&0\\ 0&0\end{pmatrix}\les0.\end{equation} Hence, (\ref{6.5}) is true if and only if it is true for $\rho=0,1$, i.e., $$0\leqslant\Pi(T)+\begin{pmatrix}-\mu h_y(y)^ h_y(y)&0\cr0&K\end{pmatrix}=\begin{pmatrix}P(T)-\mu h_y(y)^h_y(y)&\Gamma(T)^\cr\Gamma(T)&\bar P(T)+K\end{pmatrix},$$ and $$\begin{array}{ll} \noalign{ }\ds0\leqslant\begin{pmatrix}0&h_y(y)^\\ 0&0\end{pmatrix}\Pi(T)\begin{pmatrix}0&0\\ h_y(y)&0\end{pmatrix}+\[\begin{pmatrix}0&h_y(y)^\\ 0&0\end{pmatrix}\Pi(T)+\Pi(T)\begin{pmatrix}0&0\\ h_y(y)&0\end{pmatrix}\Big]\\ [4mm] \noalign{ }\displaystyle\qquad+\Pi(T)+\begin{pmatrix}-\mu h_y(y)^* h_y(y)&0\cr0& K\end{pmatrix}\\ [4mm] \noalign{ }\displaystyle\quad=\negthinspace \negthinspace \begin{pmatrix}P(T)+h_y(y)^\Gamma(T)+\Gamma(T)^ h_y(y)+\negthinspace h_y(y)^\bar P(T)h_y(y)-\mu h_y(y)^h_y(y)&\Gamma(T)^\negthinspace +h_y(y)^\bar P(T)\cr\Gamma(T)+\bar P(T)h_y(y)&\bar P(T)+K\end{pmatrix}.\end{array}$$ By choosing $\mu>0$ small enough and $K>0$ large enough, we see that the above is implied by the following: $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}P(T)&\Gamma(T)^\cr\Gamma(T)&\bar P(T)+K\end{pmatrix}\geqslant\delta,\\ [4mm] \noalign{ }\displaystyle\begin{pmatrix}P(T)+h_y(y)^\Gamma(T)+\Gamma(T)^* h_y(y)+\negthinspace h_y(y)^\bar P(T)h_y(y)&\Gamma(T)^+h_y(y)^\bar P(T)\cr\Gamma(T)+\bar P(T)h_y(y)&\bar P(T)+K\end{pmatrix}\geqslant\delta,\end{array}\right.$$ for some $\delta>0$. This will further be implied by $$\begin{array}{ll} \noalign{ }\displaystyle P(T)\geqslant\delta,\quad P(T)\negthinspace +\negthinspace h_y(y)^\Gamma(T)\negthinspace +\negthinspace \Gamma(T)^h_y(y)\negthinspace +\negthinspace h_y(y)^\bar P(T)h_y(y)\negthinspace \geqslant\negthinspace \delta,\end{array}$$ for some $\delta>0$. Hence, to summarize, at $t=0,T$, it suffices to have (\ref{t,T})--(\ref{T}). Now, we look at $t\in(0,T)$. One needs $$\begin{pmatrix}\mathbb{H}^\rho(t,y,\psi)-\mathbb{Q}(t)+\mu\begin{pmatrix} g_y(t,y,\psi)^g_y(t,y,\psi)&0\cr0&0\end{pmatrix}&\Pi(t)\cr \Pi(t)&-K\end{pmatrix}\negthinspace \negthinspace \leqslant\1n0,$$ for some $\mu,K>0$. The left-hand side is affine in $\rho$. Hence, the above is true for all $\rho\in[0,1]$ if and only if it is true for $\rho=0,1$, i.e., $$\negthinspace \negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}\Pi(t)\mathbb{M}(t)+\mathbb{M}(t)^\Pi(t)-\mathbb{Q}(t) +\mu\begin{pmatrix} g_y(t,y,\psi)^g_y(t,y,\psi)&0\cr0&0\end{pmatrix}&\Pi(t)\cr \Pi(t)&-K\end{pmatrix}\negthinspace \negthinspace \leqslant\1n0,\\ [2mm] \noalign{ }\displaystyle\begin{pmatrix}\Pi(t)\mathbb{B}(t,y,\psi)\negthinspace +\negthinspace \mathbb{B}(t,y,\psi)^\negthinspace \Pi(t) \negthinspace +\negthinspace \Pi(t)\mathbb{M}(t)\negthinspace +\negthinspace \mathbb{M}(t)^\negthinspace \Pi(t)-\mathbb{Q}(t) \negthinspace +\negthinspace \mu\negthinspace \begin{pmatrix}g_y(t,y,\psi)^\negthinspace g_y(t,y,\psi)&0\cr0&0\end{pmatrix}&\Pi(t)\cr \Pi(t)&-K\end{pmatrix}\negthinspace \negthinspace \negthinspace \leqslant\1n0,\end{array}$$ for some $\mu,K>0$. These are implied by the following: (by letting $\mu>0$ small enough) \begin{equation}\label{6.7}\begin{pmatrix}\Pi(t)\mathbb{M}(t)+\mathbb{M}(t)^\Pi(t)-\mathbb{Q}(t) &\Pi(t)\cr\Pi(t)&-K\end{pmatrix}\negthinspace \negthinspace \leqslant-\delta,\end{equation} and \begin{equation}\label{6.8}\begin{pmatrix}\Pi(t)\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\Pi(t) +\Pi(t)\mathbb{M}(t)+\mathbb{M}(t)^\Pi(t)-\mathbb{Q}(t)&\Pi(t)\cr \Pi(t)&-K\end{pmatrix}\negthinspace \negthinspace \leqslant-\delta,\end{equation} for some $\delta>0$. It is clear that (\ref{6.7})--(\ref{6.8}) hold for some large $K>0$ if (\ref{t<s<T}) holds. Further, we note that the first condition in (\ref{t<s<T}) implies $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle-P(t)M(t)-M(t)^P(t)+Q_0(t)\geqslant\delta,\\ [2mm] \noalign{ }\displaystyle\bar P(t)\bar M(t)^+\bar M(t)\bar P(t)+\bar Q_0(t)\geqslant\delta.\end{array}\right.$$ Hence, for any $t\in[t,T]$, using $P(T)\geqslant\delta$ and $\bar P(0)\leqslant-\delta$, we obtain \begin{equation}\label{L1a}\begin{array}{ll} \noalign{ }\displaystyle P(t)=e^{A^(T-t)}P(T)e^{A(T-t)}+\negthinspace \int_t^T\negthinspace \negthinspace e^{A^(s-t)}[-P(s)M(s)\negthinspace -\negthinspace \negthinspace M(s)^P(s)\negthinspace +\negthinspace Q_0(s)]e^{A(s-t)}ds\ges0,\end{array}\end{equation} and \begin{equation}\label{L2a}\begin{array}{ll} \noalign{ }\displaystyle\bar P(t)=e^{At}\bar P(0)e^{A^t}-\negthinspace \int_0^t\negthinspace \negthinspace e^{A(t-s)}[\bar P(s)\bar M(s)^\negthinspace \negthinspace +\negthinspace \bar M(s)\bar P(s)\negthinspace +\negthinspace \bar Q_0(s)]e^{A^(t-s)}ds\les0.\end{array}\end{equation} In particular, (\ref{bar P(T)<0}) holds. This proves that under (\ref{t,T})--(\ref{t<s<T}), $\Pi(\cdot)$ is a type (I) Lyapunov operator for the generator $(b,g,h)$. We can similarly prove that under (\ref{t,T})--(\ref{t<s<T}), $\Pi(\cdot)$ is also a type (II) Lyapunov operator for the generator $(b,g,h)$. \signed {$\sqr69$} \bf Corollary 7.2. \sl Let {\rm(H0)} hold and let $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$. Let $\Pi(\cdot)\equiv\begin{pmatrix}\sc P(\cdot)&\sc\Gamma(\cdot)^\\ \sc\Gamma(\cdot)&\sc\bar P(\cdot)\end{pmatrix}$ be a mild solution to $(\ref{Pi})$ for some $\mathbb{M}(\cdot)$ and $\mathbb{Q}(\cdot)$ of form {\rm(\ref{MQ})} such that {\rm(\ref{t,T})--(\ref{T})} hold. Suppose the following hold: \begin{equation}\label{t<s<T}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\Pi(t)\mathbb{M}(t)+\mathbb{M}(t)^\Pi(t)-\mathbb{Q}(t)\leqslant-\delta-\varepsilon,\qquad t\in[0,T],\\ [1mm] \noalign{ }\displaystyle\Pi(t)\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\Pi(t)\leqslant\varepsilon,\qquad\quad(t,y,\psi)\negthinspace \in\negthinspace (0,T)\negthinspace \times \negthinspace X\negthinspace \times\negthinspace X,\end{array}\right.\end{equation} for some $\delta,\varepsilon>0$. Then $\Pi(\cdot)$ is both a Lyapunov operator of types {\rm(I)} and {\rm(II)} for $(b,g,h)$. In particular, this is the case if the following holds: \begin{equation}\label{t<s<T}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\mathbb{Q}(t)\geqslant\delta+\varepsilon,\qquad t\in[0,T],\\ \noalign{ }\displaystyle\Pi(t)\mathbb{B}(t,y,\psi)\negthinspace +\negthinspace \mathbb{B}(t,y,\psi)^\Pi(t)\negthinspace \leqslant\varepsilon, \quad(t,y,\psi)\negthinspace \in\negthinspace (0,T)\negthinspace \times\negthinspace X\negthinspace \times\negthinspace X,\end{array}\right.\end{equation} for some $\delta,\varepsilon>0$. \it Proof. \rm It is clear that (\ref{t<s<T}) implies (\ref{t<s<T}). In particular, by letting $\mathbb{M}(\cdot)=0$, we see that (\ref{t<s<T}) implies (\ref{t<s<T}). \signed {$\sqr69$} We note that condition (\ref{t<s<T}) is more convenient to check than (\ref{t<s<T}). Next, we look at some concrete special cases of Theorem 7.1, which will be more practically useful. We first present the following result. \bf Lemma 7.3. \sl Let {\rm(H0)} hold and $p_1,\bar p_0,q_0,\bar q_0,\theta,\gamma\in\mathbb{R}$ and $m,\bar m>-\sigma_0$. Let \begin{equation}\label{MM}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle M(t)=mI,\quad\bar M(t)=\bar mI,\\ \noalign{ }\displaystyle Q_0(t)=q_0I,\quad\bar Q_0(t)=\bar q_0I,\quad\Theta(t)=\theta I.\end{array}\right.\end{equation} The mild solution $\Pi(\cdot)$ of {\rm(\ref{Pi})} satisfying \begin{equation}\label{7.16}P(T)=p_1I,\quad\bar P(0)=-\bar p_0I,\quad\Gamma(T)=\gamma I,\end{equation} is given by the following: In the case $A^=A$, for any $t\in[0,T]$, \begin{equation}\label{Pi(A=A)}\begin{array}{ll} \noalign{ }\displaystyle\Pi(t)\negthinspace =\negthinspace \begin{pmatrix}\sc p_1e^{2(A-m)(T-t)}&\sc\big[\gamma e^{(\bar m-m)(T-t)} +\theta{e^{(\bar m-m)(T-t)}-1\over\bar m-m}\big]I\\ \sc\big[\gamma e^{(\bar m-m)(T\negthinspace -\negthinspace s)}\negthinspace +\theta{e^{(\bar m-m)(T-t)}-1\over \bar m-m}\big]I&\sc-\bar p_0e^{2(A-\bar m)t}\end{pmatrix}\\ [4mm] \noalign{ }\displaystyle\qquad\qquad+{1\over2}\negthinspace \begin{pmatrix}\sc q_0(A-m)^{-1}[e^{2(A-m)(T-t)}-I]&\sc0\\ \sc0&\sc-\bar q_0(A-\bar m)^{-1}[e^{2(A-\bar m)t}-I]\end{pmatrix}\negthinspace ,\end{array}\end{equation} and in the case $A^=-A$, for $t\in[0,T]$, \begin{equation}\label{Pi(A=-A)}\Pi(t)\negthinspace =\negthinspace \negthinspace \begin{pmatrix}\sc \big[p_1e^{-2m(T-t)}+{q_0\over2m}(1-e^{-2m(T-t)})\big]I&\sc\big[\gamma e^{(\bar m-m)(T\negthinspace -\negthinspace s)}\negthinspace +\theta{e^{(\bar m-m)(T-t)}-1\over\bar m-m}\big]I\\ \\ \sc\big[\gamma e^{(\bar m-m)(T-t)}+\theta{e^{(\bar m-m)(T-t)}-1\over\bar m-m}\big]I&\sc-\big[\bar p_0e^{-2\bar mt}+{\bar q_0\over2\bar m}(1-e^{-2\bar mt})\big]I\end{pmatrix}\negthinspace \negthinspace .\end{equation} In the above, the following convention is adopted: \begin{equation}\label{convention}{1-e^{-\alpha\beta}\over\alpha}\equiv\beta,\qquad\hbox{if }\alpha=0.\end{equation} In particular, if \begin{equation}\label{=}\bar m=m,\end{equation} then in the case $A^=A$, for $m\in\mathbb{R}(A)$ and $t\in[0,T]$, \begin{equation}\label{Pi(A=A)}\begin{array}{ll} \noalign{ }\displaystyle\Pi(t)\negthinspace =\begin{pmatrix}\sc p_1e^{2(A-m)(T-t)}&\sc [\gamma+\theta(T-t)]I\\ \sc[\gamma+\theta(T-t)]I&\sc-\bar p_0e^{2(A-m)t}\end{pmatrix}\\ [4mm] \noalign{ }\displaystyle\qquad\qquad+{1\over2}\negthinspace \begin{pmatrix}\sc q_0(A-m)^{-1}[e^{2(A-m)(T-t)}-I]& \sc0\\ \sc0&\sc-\bar q_0(A-m)^{-1}[e^{2(A-m)t}-I]\end{pmatrix}\negthinspace ,\end{array}\end{equation} and in the case $A^=-A$, for $t\in[0,T]$, \begin{equation}\label{Pi(A=-A)*}\begin{array}{ll} \noalign{ }\displaystyle\Pi(s)\negthinspace =\begin{pmatrix}\[p_1e^{-2m(T-s)}+q_0{1-e^{2m(T-s)}\over2m}\]I&\sc[\gamma+\theta(T-t)]I\\ \sc[\gamma+\theta(T-t)]I&\sc-\[\bar p_0e^{-2m(s-t)}+\bar q_0{1-e^{-2mt}\over2m}\]I\end{pmatrix}.\end{array}\end{equation} Further, if $m=\bar m=0$, then, for $A^=A$, $t\in[0,T]$, \begin{equation}\label{Pi(A=A)}\Pi(t)=\begin{pmatrix}\sc (p_1+{q_0\over2}A^{-1})e^{2A(T-t)}+{q_0\over2}A^{-1}&\sc[\gamma +\theta(T-t)]I\\ \sc[\gamma+\theta(T-t)]I&\sc-(\bar p_0+{\bar q_0\over2}A^{-1})e^{2At}+{\bar q_0\over2}A^{-1} \end{pmatrix},\end{equation} and for $A^=-A$, $t\in[0,T]$, \begin{equation}\label{Pi(A=-A)}\Pi(t)=\begin{pmatrix}\sc[p_1+q_0(T-t)]I&\sc[\gamma +\theta(T-t)]I\\ \\ \sc[\gamma+\theta(T-t)]I&\sc-[\bar p_0+\bar q_0t]I\end{pmatrix}.\end{equation} \it Proof. \rm With the choice of (\ref{MM}), we have \begin{equation}\label{6.23P}\begin{array}{ll} \noalign{ }\displaystyle P(t)=e^{(A-m)^(T-t)}P(T)e^{(A-m)(T-t)}+\int_t^Te^{(A-m)^(s-t)}Q_0(s)e^{(A-m)(s-t)}ds\\ \noalign{ }\displaystyle\qquad=p_1e^{(A-m)^(T-t)}e^{(A-m)(T-t)}+q_0\int_t^Te^{(A-m)^(s-t)}e^{(A-m)(s-t)}ds,\quad t\in[0,T],\end{array}\end{equation} and \begin{equation}\label{6.24barP}\begin{array}{ll} \noalign{ }\displaystyle\bar P(t)=e^{(A-\bar m)t}\bar P(0)e^{(A-\bar m)^t}-\int_0^t\negthinspace \negthinspace e^{(A-\bar m)(t-s)}\bar Q_0(s)e^{(A-\bar m)^(t-s)}ds\\ \noalign{ }\displaystyle\qquad=-\bar p_0e^{(A-\bar m)t}e^{(A-\bar m)^t}-\bar q_0\int_0^te^{(A-\bar m)(t-s)}e^{(A-\bar m)^(t-s)}ds,\quad t\in[0,T].\end{array}\end{equation} Also, in the current case, $\Gamma(\cdot)$ satisfies $$\dot\Gamma(t)+(\bar m-m)\Gamma(t)+\theta I=0,$$ which leads to (with $\Gamma(T)=\gamma I$) \begin{equation}\label{6.25G}\begin{array}{ll} \noalign{ }\displaystyle\Gamma(t)=\[\gamma e^{(\bar m-m)(T-t)}+\theta\int_t^Te^{(\bar m-m)(s-t)}dr\]I\\ \noalign{ }\displaystyle\qquad=\[\gamma e^{(\bar m-m)(T-t)}+\theta{e^{(\bar m-m)(T-t)}-1\over \bar m-m}\]I=\Gamma(t)^,\quad t\in[0,T].\end{array}\end{equation} When $\bar m-m=0$, the above is understood as \begin{equation}\label{6.26G}\Gamma(t)=\big[\gamma+\theta(T-t)\big]I,\qquad t\in[0,T].\end{equation} We now look at two cases. In the case $A^=A$, (\ref{6.23P}) and (\ref{6.24barP}) become \begin{equation}\label{6.27P}P(t)=p_1e^{2(A-m)(T-t)}+{q_0\over2}(A-m)^{-1}\big[e^{2(A-m)(T-t)} -I\big],\quad t\in[0,T],\end{equation} and \begin{equation}\label{6.28barP}\bar P(t)=-\bar p_0e^{2(A-\bar m)t}-{\bar q_0\over2}(A-\bar m)^{-1}\big[e^{2(A-\bar m)t}-I\big],\quad t\in[0,T].\end{equation} In the case $A^=-A$, (\ref{6.23P}) and (\ref{6.24barP}) become \begin{equation}\label{6.29P}\begin{array}{ll} \noalign{ }\displaystyle P(t)\negthinspace =\negthinspace $p_1e^{-2m(T-t)}\negthinspace +\negthinspace q_0\int_t^Te^{-2m(s-t)}ds$I\negthinspace =\negthinspace $p_1e^{-2m(T-t)}\negthinspace +\negthinspace {q_0\over2m}(1-e^{-2m(T-t)})$I,\quad t\in[0,T],\end{array}\end{equation} with the above understood as follows when $m=0$, \begin{equation}\label{6.30P}P(t)=$p_1+q_0(T-t)$I,\qquad t\in[0,T],\end{equation} and \begin{equation}\label{6.31barP}\begin{array}{ll} \noalign{ }\displaystyle\bar P(t)=-$\bar p_0e^{-2\bar mt}+\bar q_0\int_0^te^{-2\bar m(t-s)}ds$I=-$\bar p_0e^{-2\bar mt}+{\bar q_0\over2\bar m}(1-e^{-2\bar mt})$I,\qquad t\in[0,T],\end{array}\end{equation} with the above understood as follows when $\bar m=0$, \begin{equation}\label{6.32barP}\bar P(t)=-$\bar p_0+\bar q_0t$I,\qquad t\in[0,T].\end{equation} The rest conclusions are clear. \signed {$\sqr69$} Combining Theorem 7.1 or Corollary 7.2 with Lemma 7.3, we can present many concrete cases for which the corresponding FBEEs are well-posed. For the simplicity of presentation, we only consider below the case that (\ref{=}) holds. First, we present a simple lemma. \bf Lemma 7.4. \rm Let $$f(\kappa)=\alpha e^{-\kappa}+\beta{1-e^{-\kappa}\over\kappa},\qquad\kappa>0,$$ with $\alpha,\beta>0$. Then $\kappa\mapsto f(\kappa)$ is decreasing on $(0,\infty)$ and $$0=\lim_{\kappa\to\infty}f(\kappa)=\inf_{\kappa>0}f(\kappa)<\sup_{\kappa>0}f(\kappa)=\lim_{\kappa\to0}f(\kappa)=\alpha+\beta.$$ \it Proof. \rm We note that $$f'(\kappa)=-\alpha e^{-\kappa}+\beta{e^{-\kappa}\kappa-(1-e^{-k})\over\kappa}=-${\alpha\over e^\kappa}+{e^\kappa-1-\kappa\over\kappa^2e^\kappa}$<0.$$ Then our conclusion follows immediately. \signed {$\sqr69$} \bf Theorem 7.5. \sl Let {\rm(H0)} hold and let $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$. Suppose there are constants $p_1,\bar p_0,q_0,\bar q_0,\delta,\bar\delta,\varepsilon>0$, $m>-\sigma_0$, and $\gamma,\theta\in\mathbb{R}$ such that \begin{equation}\label{6.33}\begin{array}{ll} \noalign{ }\displaystyle p_1I+\gamma[h_y(y)+h_y(y)^]-\[\bar p_0e^{-2(\sigma_0+m)(T-t)}+{\bar q_0(1-e^{-2(\sigma_0+m)(T-t)}) \over2(\sigma_0+m)}\]h_y(y)^h_y(y)\geqslant\delta,\end{array}\end{equation} \begin{equation}\label{6.34}\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}q_0{\sigma_0+me^{-2(\sigma_0+m)(T-t)}\over\sigma_0+m}-2mp_1e^{-2(\sigma_0+m)(T-t)} &\theta\\ \theta&\bar q_0{\sigma_0+me^{-2(\sigma_0+m)t}\over\sigma_0+m}-2m\bar p_0e^{-2(\sigma_0+m)t} \end{pmatrix}\geqslant\bar\delta+\varepsilon,\\ [5mm] \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall t\in[0,T].\end{array}\end{equation} Then the corresponding FBEE is well-posed if one of the following holds: {\rm(i)} In the case $A^=A$, it holds that \begin{equation}\label{PiB+BPi}\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}p_1e^{2(A-m)(T-t)}&0\\ 0&-\bar p_0e^{2(A-m)t}\end{pmatrix}\mathbb{B}(t,y,\psi) +\mathbb{B}(t,y,\psi)^\begin{pmatrix}p_1e^{2(A-m)(T-t)}&0\\ 0&-\bar p_0e^{2(A-m)t}\end{pmatrix}\\ \noalign{ }\displaystyle+[\gamma+\theta(T-t)]\[\begin{pmatrix}0&I\\ I&0\end{pmatrix}\mathbb{B}(t,y,\psi) +\mathbb{B}(t,y,\psi)^\begin{pmatrix}0&I\\ I&0\end{pmatrix}\Big]\\ [4mm] \noalign{ }\displaystyle+{1\over2}\[\begin{pmatrix}q_0(A-m)^{-1}[e^{2(A-m)(T-t)}-I]&0\\ 0&-\bar q_0(A-m)^{-1}[e^{2(A-m)t}-I]\end{pmatrix}\mathbb{B}(t,y,\psi)\\ \noalign{ }\displaystyle+\mathbb{B}(t,y,\psi)^\negthinspace \negthinspace \begin{pmatrix} q_0(A-m)^{-1}[e^{2(A-m)(T-t)}-I] &0\\ 0&-\bar q_0(A-m)^{-1}[e^{2(A-m)t}-I]\end{pmatrix}\negthinspace \Big]\negthinspace \negthinspace \leqslant\negthinspace \varepsilon.\end{array}\end{equation} {\rm(ii)} In the case $A^=-A$, it holds that \begin{equation}\label{PiB+BPi}\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}p_1e^{-2m(T-t)}&0\\ 0&-\bar p_0e^{-2mt}\end{pmatrix}\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\begin{pmatrix}p_1e^{-2m(T-t)}&0\\ 0&-\bar p_0e^{-2mt}\end{pmatrix}\\ \noalign{ }\displaystyle+[\gamma+\theta(T-t)]\[\begin{pmatrix}0&I\\ I&0\end{pmatrix}\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\begin{pmatrix}0& I\\ I&\sc0\end{pmatrix}\Big]\\ \noalign{ }\displaystyle+\begin{pmatrix}q_0{1-e^{-2m(T-t)}\over2m}I&0\\ 0&-\bar q_0{1-e^{-2mt}\over2m}I\end{pmatrix}\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\begin{pmatrix} q_0{1-e^{-2m(T-t)}\over2m}I& 0\\ 0&-\bar q_0{1-e^{-2mt}\over2m}I\end{pmatrix}\leqslant\varepsilon.\end{array}\end{equation} \it Proof. \rm (i) We consider the case $A^=A$. First of all, by taking \begin{equation}\label{6.38}P(T)=p_1I>0,\qquad\bar P(0)=-\bar p_0I<0,\end{equation} we see that (\ref{t,T}) holds. Next, to get (\ref{T}), we look at the following (recalling (\ref{6.31barP})): $$\begin{array}{ll} \noalign{ }\displaystyle P(T)+h_y(y)^\Gamma(T)\negthinspace +\negthinspace \Gamma(T)^h_y(y)+h_y(y)^\bar P(T)h_y(y)\\ \noalign{ }\displaystyle=p_1I+\gamma[h_y(y)+h_y(y)^]-h_y(y)^$\bar p_0e^{2(A-m)T}+{\bar q_0\over2}(A-m)^{-1}\big[e^{2(A-m)T}-I\big]$h_y(y).\end{array}$$ Let us estimate the quadratic term in $h_y(y)$ on the right hand side of the above. To this end, we observe the following: For any $\tau>\sigma_0$ (recall $m>-\sigma_0$), $$\begin{array}{ll} \noalign{ }\displaystyle \bar p_0e^{-2(\tau+m)T}+{\bar q_0T(1-e^{-2(\tau+m)T})\over2(\tau+m)T}\equiv\bar p_0e^{-\kappa}+{\bar q_0T(1-e^{-\kappa})\over\kappa}\equiv f(\kappa),\end{array}$$ with $\kappa=2(\tau+m)T\ges2(\sigma_0+m)T>0$. By Lemma 7.4, we have $$\begin{array}{ll} \noalign{ }\displaystyle\sup_{\kappa\ges2(\sigma_0+m)T}f(\kappa)=f(2(\sigma_0+m)T)=\bar p_0e^{-2(\sigma_0+m)T}+{\bar q_0(1-e^{-2(\sigma_0+m)T})\over2(\sigma_0+m)}.\end{array}$$ By the spectral decomposition of $A$, making use of (\ref{6.33}), one has $$\begin{array}{ll} \noalign{ }\displaystyle P(T)+h_y(y)^\Gamma(T)\negthinspace +\negthinspace \Gamma(T)^h_y(y)+h_y(y)^\bar P(T)h_y(y)\\ \noalign{ }\displaystyle=p_1I+\gamma[h_y(y)+h_y(y)^]-h_y(y)^$\bar p_0e^{2(A-m)T}+{\bar q_0\over2}(A-m)^{-1} \big[e^{2(A-m)T}-I\big]$h_y(y)\\ \noalign{ }\displaystyle\geqslant p_1I+\gamma[h_y(y)+h_y(y)^]-\[\bar p_0e^{-2(\sigma_0+m)T}+{\bar q_0(1-e^{-2(\sigma_0+m)T}) \over2(\sigma_0+m)}\]h_y(y)^h_y(y)\geqslant\delta,\end{array}$$ which gives (\ref{T}). Next, note that (in the case $\bar m=m$) \begin{equation}\label{6.40}-\Pi(t)\mathbb{M}(t)-\mathbb{M}(t)^\Pi(t)+\mathbb{Q}(t)=\begin{pmatrix}\sc-2mP(t) +q_0I&\sc\theta I\\ \sc\theta I& \sc2m\bar P(t)+\bar q_0I\end{pmatrix}.\end{equation} We now look at the following (noting (\ref{6.27P}) and (\ref{6.28barP})): $$\begin{array}{ll} \noalign{ }\displaystyle-2mP(t)+q_0I=\negthinspace -2mp_1e^{2(A-m)(T-t)}\negthinspace -\negthinspace mq_0(A\negthinspace -\negthinspace m)^{-1}\big[e^{2(A-m)(T-t)}\negthinspace -\negthinspace I\big] \negthinspace \negthinspace +\negthinspace q_0I\\ \noalign{ }\displaystyle=-2mp_1e^{2(A-m)(T-t)}+q_0\big[A-me^{2(A-m)(T-t)}\big](A-m)^{-1},\end{array}$$ and $$\begin{array}{ll} \noalign{ }\ds2m\bar P(t)+q_0I=-2m\bar p_0e^{2(A-m)t} -m\bar q_0(A-m)^{-1}\big[e^{2(A-m)t}-I\big]\negthinspace \negthinspace +\negthinspace \bar q_0I\\ \noalign{ }\displaystyle=-2m\bar p_0e^{2(A-m)t}+\bar q_0\big[A-me^{2(A-m)t}\big](A-m)^{-1}.\end{array}$$ Similar to the above, for any $\tau>\sigma_0$ and $m>-\sigma_0$, we have ($\tau+m>0$) $$\begin{array}{ll} \noalign{ }\displaystyle-2mp_1e^{-2(\tau+m)(T-t)}+{q_0(\tau+me^{-2(\tau+m)(T-t)})\over\tau+m}\\ \noalign{ }\displaystyle=-2mp_1e^{-2(\tau+m)(T-t)}+q_0+{2mq_0(T-t)(e^{-2(\tau+m)(T-t)}-1) \over2(\tau+m)(T-t)}\\ \noalign{ }\displaystyle=q_0-\[2mp_1e^{-\kappa}-{2mq_0(T-t)(1-e^{-\kappa})\over\kappa}\Big]\equiv q_0- f(\kappa),\end{array}$$ with $\kappa=2(\tau+m)(T-t)>2(\sigma_0+m)(T-t)$. By Lemma 7.4 again, we have \begin{equation}\label{6.41}\begin{array}{ll} \noalign{ }\displaystyle-2mp_1e^{-2(\tau+m)(T-t)}+{q_0(\tau+me^{-2(\tau+m)(T-t)})\over\tau+m}\\ \noalign{ }\displaystyle\qquad\geqslant q_0-\sup_{\kappa\ges2(\sigma_0+m)(T-t)}f(\kappa)=q_0-f(2(\sigma_0+m)(T-t))\\ \noalign{ }\displaystyle\qquad=q_0-2mp_1e^{-2(\sigma_0+m)(T-t)}+{2mq_0(e^{-2(\sigma_0+m)(T-t)}-1)\over 2(\sigma_0+m)}\\ \noalign{ }\displaystyle\qquad=q_0\[1-{2m(1-e^{-2(\sigma_0+m)(T-t)})\over2(\sigma_0+m)}\Big] -2mp_1e^{-2(\sigma_0+m)(T-t)}\\ \noalign{ }\displaystyle\qquad=q_0{\sigma_0+me^{-2(\sigma_0+m)(T-t)}\over\sigma_0+m}-2mp_1e^{-2(\sigma_0+m)(T-t)}\\ \noalign{ }\displaystyle\qquad\geqslant q_0-\lim_{\kappa\to0}f(\kappa)=q_0[1+2\sigma_0(T-t)]+2\sigma_0p_1.\end{array}\end{equation} Similarly, \begin{equation}\label{6.42}\begin{array}{ll} \noalign{ }\displaystyle-2m\bar p_0e^{-2(\tau+m)t}+{\bar q_0(\tau+me^{-2(\tau+m)t})\over\tau+m}\geqslant\bar q_0\[1-{2m(1-e^{-2(\sigma_0+m)t})\over2(\sigma_0+m)}\Big]-2m\bar p_0 e^{-2(\sigma_0+m)t}\\ \noalign{ }\displaystyle\qquad\geqslant\bar q_0{\sigma_0+me^{-2(\sigma_0+m)t}\over\sigma_0+m}-2m\bar p_0 e^{-2(\sigma_0+m)t}\geqslant\bar q_0[(1+2\sigma_0t]+2\sigma_0\bar p_0.\end{array}\end{equation} Consequently, using the spectral decomposition of $A$, we have $$\begin{array}{ll} \noalign{ }\displaystyle-2mP(t)+q_0I\geqslant q_0{\sigma_0+me^{-2(\sigma_0+m)(T-t)}\over\sigma_0+m} -2mp_1e^{-2(\sigma_0+m)(T-t)},\\ [2mm] \noalign{ }\ds2m\bar P(t)+\bar q_0I\geqslant\bar q_0{\sigma_0+me^{-2(\sigma_0+m)t}\over\sigma_0+m}-2m\bar p_0 e^{-2(\sigma_0+m)t}.\end{array}$$ Hence, $$\begin{array}{ll} \noalign{ }\displaystyle-\Pi(t)\mathbb{M}(t)-\mathbb{M}(t)^\Pi(t)+\mathbb{Q}(t)=\begin{pmatrix}-2mp_1e^{2(A-m)(T-t)}&\theta I\\ \theta I&-2m\bar p_0e^{2(A-m)t}\end{pmatrix}\\ [4mm] \noalign{ }\displaystyle\qquad+\begin{pmatrix}q_0[A-me^{2(A-m)(T-t)}](A-m)^{-1}&0\\ 0&\bar q_0[A-me^{2(A-m)t}](A-m)^{-1}\end{pmatrix}\\ [4mm] \noalign{ }\displaystyle\geqslant\begin{pmatrix}\big[q_0{\sigma_0+me^{-2(\sigma_0+m)(T-t)}\over\sigma_0+m}\big]I&\theta I\\ \theta I&\big[\bar q_0{\sigma_0+me^{-2(\sigma_0+m) t}\over\sigma_0+m}\big]I \end{pmatrix}\\ [4mm] \noalign{ }\displaystyle\qquad-2m\begin{pmatrix}p_1e^{-2(\sigma_0+m)(T-t)}I&0\\ 0&\bar p_0e^{-2(\sigma_0+m)(T-t)}I\end{pmatrix}\geqslant\delta,\qquad\forall t\in[0,T],\end{array}$$ for some $\delta>0$, provided (\ref{6.34}) holds. Then by (\ref{PiB+BPi}), together with the representation of $\Pi(\cdot)$ from Lemma 7.3, we have $$\Pi(t)\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\Pi(t)\leqslant\varepsilon.$$ Hence, Corollary 7.2 applies. (ii) We now consider the case $A^=-A$. Again, we still have (\ref{t,T}) by (\ref{6.33}). Next, for the current case, recalling (\ref{6.28barP}), $$\begin{array}{ll} \noalign{ }\displaystyle P(T)+h_y(y)^\Gamma(T)\negthinspace +\negthinspace \Gamma(T)^h_y(y)+h_y(y)^\bar P(T)h_y(y)\\ \noalign{ }\displaystyle=p_1I+\gamma[h_y(y)+h_y(y)^]-$\bar p_0e^{-2m(T-t)}+{\bar q_0\over2m}(1-e^{-2m(T-t)})$h_y(y)^h_y(y)\geqslant\delta.\end{array}$$ This leads to (\ref{6.33}) with $\sigma_0=0$. In the current case, we still have (\ref{6.40}). We observe the following (noting (\ref{6.29P}) and (\ref{6.31barP})): $$\begin{array}{ll} \noalign{ }\displaystyle-2mP(t)+q_0I=\negthinspace -2m$p_1e^{-2m(T-t)}+{q_0\over2m}(1-e^{-2m(T-t)})$I +\negthinspace q_0I\\ \noalign{ }\displaystyle=-2mp_1e^{-2m(T-t)}+q_0e^{-2m(T-t)}=(q_0-2mp_1)e^{-2m(T-t)},\end{array}$$ and $$\begin{array}{ll} \noalign{ }\ds2m\bar P(t)+\bar q_0I=-2m$\bar p_0e^{-2mt}+{\bar q_0\over2m}(1-e^{-2mt})$I +\bar q_0I\\ \noalign{ }\displaystyle=-2m\bar p_0e^{-2mt}+\bar q_0e^{-2mt}=(\bar q_0-2m\bar p_0)e^{-2mt},\end{array}$$ Hence, $$\begin{array}{ll} \noalign{ }\displaystyle-\Pi(t)\mathbb{M}(t)-\mathbb{M}(t)^\Pi(t)+\mathbb{Q}(t)\\ \noalign{ }\displaystyle=\begin{pmatrix}(q_0-2mp_1)e^{-2m(T-t)}I&\theta I\\ \theta I&(\bar q_0-2m\bar p_0)e^{-2mt}I\end{pmatrix}\geqslant\delta,\quad\forall t\in[0,T],\end{array}$$ for some $\delta>0$, provided $$\begin{pmatrix}(q_0-2mp_1)e^{-2m(T-t)}&\theta\\ \theta&(\bar q_0-2m\bar p_0)e^{-2mt}\end{pmatrix}>0,\quad\forall t\in[0,T],$$ which is implied by (\ref{6.34}) with $\sigma_0=0$. The rest proof is obvious. \signed {$\sqr69$} \bf Corollary 7.6. \sl Let {\rm(H0)} hold, and let $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$. Let $p_1,\bar p_0,q_0,\bar q_0,\delta,\bar\delta,\varepsilon>0$, $\gamma\in\mathbb{R}$ such that \begin{equation}\label{6.41}p_1+\gamma[h_y(y)+h_y(y)^]-(\bar p_0+\bar q_0T)h_y(y)^h_y(y)\geqslant\delta,\end{equation} and \begin{equation}\label{6.42}\begin{pmatrix}q_0[1+2\sigma_0(T-t)]+2\sigma_0p_1&\theta\\ \theta&\bar q_0[1+2\sigma_0t]+2\sigma_0\bar p_0\end{pmatrix}\geqslant\bar\delta+\varepsilon,\qquad t\in[0,T].\end{equation} Then the corresponding FBEE is well-posed if one of the following holds: {\rm(i)} For the case $A^=A$, the following holds: \begin{equation}\label{6.43}\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}p_1e^{2(A+\sigma_0)(T-t)}& 0\\ 0&-\bar p_0e^{2(A+\sigma_0)t}\end{pmatrix}\mathbb{B}(t,y,\psi) +\mathbb{B}(t,y,\psi)^\begin{pmatrix}p_1e^{2(A+\sigma_0)(T-t)}&0\\ 0&-\bar p_0e^{2(A+\sigma_0)t}\end{pmatrix}\\ \noalign{ }\displaystyle+[\gamma+\theta(T-t)]\[\begin{pmatrix}0&I\\ I&0\end{pmatrix}\mathbb{B}(t,y,\psi) +\mathbb{B}(t,y,\psi)^\begin{pmatrix}0&I\\ I&0\end{pmatrix}\Big]\\ [4mm] \noalign{ }\displaystyle+\[\begin{pmatrix}q_0(T-t)\eta\big((A+\sigma_0)(T-t)\big)&0\\ 0&-\bar q_0t\eta\big((A+\sigma_0)t\big)\end{pmatrix}\mathbb{B}(t,y,\psi)\\ \noalign{ }\displaystyle+\mathbb{B}(t,y,\psi)^\negthinspace \negthinspace \begin{pmatrix}\sc q_0 (T-t)\eta\big((A+\sigma_0)(T-t)\big)&0\\ 0&-\bar q_0t\eta\big((A+\sigma_0)t\big)\end{pmatrix}\negthinspace \Big]\negthinspace \negthinspace \leqslant\negthinspace \varepsilon,\end{array}\end{equation} where \begin{equation}\label{eta}\eta(\kappa)=\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle{e^\kappa-1\over\kappa},\qquad\kappa\ne0,\\ \noalign{ }\ds1,\qquad\qquad\kappa=0.\end{array}\right.\end{equation} {\rm(ii)} For the case $A^=-A$, the following holds: \begin{equation}\label{6.44}\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}\sc[p_1+q_0(T-t)]I&\sc[\gamma+\theta(T-t)]I\\ \sc[\gamma+\theta(T-t)]I&\sc-[\bar p_0+\bar q_0t]I\end{pmatrix}\mathbb{B}(t,y,\psi)\\ [4mm] \noalign{ }\displaystyle+\mathbb{B}(t,y,\psi)^\begin{pmatrix}\sc[p_1+q_0(T-t)]I&\sc[\gamma+\theta(T-t)]I\\ \sc[\gamma+\theta(T-t)]I&\sc-[\bar p_0+\bar q_0t]I\end{pmatrix}\leqslant\negthinspace \varepsilon.\end{array}\end{equation} \it Proof. \rm By letting $m\to-\sigma_0$ in (\ref{6.33}) and (\ref{6.34}), we have (\ref{6.41})--(\ref{6.42}). Thus, when (\ref{6.41})--(\ref{6.42}) hold, for $m$ sufficiently closes to $-\sigma_0$, (\ref{6.33})--(\ref{6.34}) hold. (i) For the case $A^=A$, we note that $${1\over2}(A-m)^{-1}[e^{2(A-m)(T-t)}-I]=\int_{\sigma(A)}{e^{2(\mu-m)(T-t)}-1 \over2(\mu-m)}dE_\mu.$$ By the definition of $\eta(\cdot)$, we have that $${e^{2(\mu-m)(T-t)}-1\over2(\mu-m)}=(T-t)\eta\big(2(\mu-m)(T-t)\big),$$ and $$\begin{array}{ll} \noalign{ }\displaystyle\lim_{m\to-\sigma_0}{1\over2}(A-m)^{-1}[e^{A-m)(T-t)}-I] =(T-t)\int_{\sigma(A)}\eta\big((\mu+\sigma_0)(T-t)\big)dE_\mu\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=(T-t)\eta\big((A+\sigma_0)(T-t)\big).\end{array}$$ Then sending $m\to-\sigma_0$ in (\ref{PiB+BPi}), we obtain (\ref{6.43}). Now, for the case $A^=-A$, by sending $m\to0$, we obtain (\ref{6.44}). \signed {$\sqr69$} Although the conditions stated in Theorem 7.5 and Corollary 7.6 still look lengthy, they are practically checkable. To illustrate this, let us look at an interesting situation covered. \bf Corollary 7.7. \sl Let {\rm(H0)} hold and $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$. Let \begin{equation}\label{hy>0}h_y(y)+h_y(y)^\ges0,\qquad\forall y\in X.\end{equation} Let \begin{equation}\label{7.48}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle B_{11}(t,y,\psi)=B_{22}(t,y,\psi)^,\\ \noalign{ }\displaystyle B_{12}(t,y,\psi)+B_{12}^(t,y,\psi)\les0,\\ \noalign{ }\displaystyle B_{21}(t,y,\psi)+B_{21}(t,y,\psi)^\geqslant\delta.\end{array}\right.\qquad(t,y,\psi)\in[0,T]\times X\times X,\end{equation} for some $\delta>0$, and \begin{equation}\label{7.49}\bar p_0(t)B_{22}+B_{22}^\bar p_0(t)\les0,\qquad\forall t\in[0,T],\end{equation} where \begin{equation}\label{}\bar p_0(t)=\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\bar p_0e^{2(A+\sigma_0)t}+\bar q_0t\eta\big((A+\sigma_0)t\big),\qquad \hbox{if }A^=A,\\ \noalign{ }\displaystyle[\bar p_0+\bar q_0t]I,\qquad\qquad\qquad\qquad\qquad\quad\hbox{if }A^=-A,\end{array}\right.\end{equation} with $p_1,q_0,\bar p_0,\bar q_0>0$ and $\eta(\cdot)$ is defined by {\rm(\ref{eta})}. Then the FBEE generated by $(b,g,h)$ is well-posed. \it Proof. \rm First of all, by (\ref{hy>0}), and the boundedness of $h_y(\cdot)$ (since $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$), we can find $p_1>0$ large so that (\ref{6.41}) holds, and $\gamma>0$ is allowed to be arbitrarily large. Also, by letting $\theta=0$, we see that (\ref{6.42}) holds, as long as $q_0,\bar q_0>0$. Next, we define \begin{equation}\label{}p_1(t)=\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle p_1e^{2(A+\sigma_0)(T-t)}+q_0(T-t)\eta\big((A+\sigma_0)(T-t)\big),\quad \hbox{if }A^=A,\\ \noalign{ }\displaystyle[p_1+q_0(T-t)]I,\qquad\qquad\qquad\qquad\qquad\qquad\quad\hbox{if } A^=-A.\end{array}\right.\end{equation} Then according to Corollary 7.6, the FBEE is well-posed if the following holds: $$\begin{array}{ll} \noalign{ }\displaystyle\varepsilon\geqslant\begin{pmatrix}\sc p_1(t)&\sc\gamma I\\ \sc\gamma I&\sc-\bar p_0(t)\end{pmatrix}\begin{pmatrix}\sc B_{11}&\sc B_{12} \\ \sc-B_{21}&\sc-B_{22} \end{pmatrix}+\begin{pmatrix}\sc B_{11}^&\sc-B_{21}^\\ \sc B_{12}^&\sc-B_{22}^\end{pmatrix} \begin{pmatrix}\sc p_1(t)&\sc\gamma I\\ \sc\gamma I&\sc-\bar p_0(t)\end{pmatrix}\\ \noalign{ }\displaystyle=\begin{pmatrix}\sc p_1(t)B_{11}-\gamma B_{21}&\sc p_1(t)B_{12}-\gamma B_{22}\\ \sc\gamma B_{11}+\bar p_0(t)B_{21}&\sc\gamma B_{12}+\bar p_0(t)B_{22}\end{pmatrix}+\begin{pmatrix}\sc B_{11}^p_1(t)-\gamma B_{21}^&\sc\gamma B_{11}^+B_{21}^\bar p_0(t)\\ \sc B_{12}^p_1(t)-\gamma B_{22}^&\sc\gamma B_{12}^+B_{22}^\bar p_0(t)\end{pmatrix}\\ \noalign{ }\displaystyle=\begin{pmatrix}\sc p_1(t)B_{11}+B_{11}^p_1(t)-\gamma[B_{21}+B_{21}^]&\sc p_1(t)B_{12}+B_{21}^\bar p_0(t)\\ \sc B_{12}^p_1(t)+\bar p_0(t)B_{21}&\sc\bar p_0(t)B_{22}+B_{22}^\bar p_0(t)+\gamma[B_{12}+B_{12}^]\end{pmatrix}.\end{array}$$ This is equivalent to the following: $$\begin{pmatrix}\sc\varepsilon+\gamma[B_{21}+B_{21}^]- [p_1(t)B_{11}+B_{11}^p_1(t)]&\sc-[p_1(t)B_{12}+B_{21}^\bar p_0(t)]\\ \sc-[B_{12}^p_1(t)+\bar p_0(t)B_{21}]&\sc\varepsilon-\gamma[B_{12}+B_{12}^]-[\bar p_0(t)B_{22}+B_{22}^\bar p_0(t)]\end{pmatrix}\ges0,$$ which is implied by $$\gamma[B_{21}+B_{21}^]- [p_1(t)B_{11}+B_{11}^p_1(t)]>0,$$ and $$\begin{array}{ll} \noalign{ }\displaystyle\varepsilon-\gamma[B_{12}+B_{12}^]-[\bar p_0(t)B_{22}+B_{22}^\bar p_0(t)]\\ \noalign{ }\displaystyle-[B_{12}^p_1(t)+\bar p_0(t)B_{21}]$\gamma[B_{21}+B_{21}^]- [p_1(t)B_{11}+B_{11}^p_1(t)]$^{-1}[p_1(t)B_{12}+B_{21}^\bar p_0(t)]\ges0.\end{array}$$ Note that $$\begin{array}{ll} \noalign{ }\displaystyle\Big\\|$\gamma[B_{21}+B_{21}^]- [p_1(t)B_{11}+B_{11}^p_1(t)]$^{-1}\Big\\|\\ \noalign{ }\displaystyle\leqslant{1\over\gamma}\\|(B_{21}+B_{21}^)^{-1}\\|\Big\\|$I-{1\over\gamma}(B_{21}+B_{21}^)^{-1} [p_1(t)B_{11}+B_{11}^p_1(t)]$^{-1}\Big\\|\\ \noalign{ }\displaystyle\leqslant{1\over\gamma}\\|(B_{21}+B_{21}^)^{-1}\\|{1\over1-{1\over\gamma}\\|(B_{21}+B_{21}^)^{-1} [p_1(t)B_{11}+B_{11}^p_1(t)]\\|}\\ \noalign{ }\displaystyle={\\|(B_{21}+B_{21}^)^{-1}\\|\over\gamma-\\|(B_{21}+B_{21}^)^{-1} [p_1(t)B_{11}+B_{11}^p_1(t)]\\|}.\end{array}$$ Therefore, it suffices to have (note (\ref{7.48})--(\ref{7.49})) $$\begin{array}{ll} \noalign{ }\displaystyle\varepsilon\geqslant{\\|B_{12}^p_1(t)+\bar p_0(t)B_{21}\\|^2\\|(B_{21}+B_{21}^)^{-1}\\|\over\gamma-\\|(B_{21}+B_{21}^)^{-1} [p_1(t)B_{11}+B_{11}^p_1(t)]\\|},\end{array}$$ which can be achieved by letting $\gamma>0$ sufficiently large. Then our conclusion follows. \signed {$\sqr69$} Let us look at some more cases. \bf Corollary 7.8. \sl Let {\rm(H0)} hold. Suppose $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$ such that \begin{equation}\label{>0}h_y(y)+h_y(y)^\ges0,\qquad\forall y\in X,\end{equation} and \begin{equation}\label{<-d}\begin{pmatrix}0&I\\ I&0\end{pmatrix}\mathbb{B}(t,y,\psi) +\mathbb{B}(t,y,\psi)^\begin{pmatrix}0&I\\ I&0\end{pmatrix}\leqslant-\delta,\quad\forall(t,y,\psi)\in[0,T]\times X\times X,\end{equation} for some $\delta>0$. Then the corresponding FBEE is well-posed. \it Proof. \rm First, by letting $p_1,\bar p_0,q_0,\bar q_0>0$, with $q_0,\bar q_0>0$ suitably large, we will have (\ref{6.42}). Then letting $p_1>0$ large, we will have (\ref{6.41}). Next, by noting $$0\leqslant\eta(\kappa)\les1,\qquad\forall\kappa\le0,$$ we see that under the condition $(b,g,h)\in{\cal G}_4$, either $A^=A$ or $A^=-A$, we always have the boundedness of all the terms involved in the left-hand sides of (\ref{6.43}) and (\ref{6.44}), respectively. Hence, under condition (\ref{<-d}), we can find $\gamma>0$ large enough so that (\ref{6.43}) and (\ref{6.44}) holds, respectively. Due to the condition (\ref{>0}), by letting $\gamma>0$ large, (\ref{6.41}) will not be affected. Then Corollary 7.6 applies to get the well-posedness of the corresponding FBEE. \signed {$\sqr69$} Note that (\ref{<-d}) is equivalent to the following: $$\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}-[g_y(t,y,\psi)+g_y(t,y,\psi)^]& b_y(t,y,\psi)^-g_\psi(t,y,\psi)\\ b_y(t,y,\psi)-g_\psi(t,y,\psi)^& b_\psi(t,y,\psi)+b_\psi(t,y,\psi)^\end{pmatrix}\leqslant-\delta,\qquad\forall(t,y,\psi)\in[0,T]\times X\times X.\end{array}$$ This is further equivalent to the uniform monotonicity of the following map $$\begin{pmatrix}\sc y\\ \sc\psi\end{pmatrix}\mapsto\begin{pmatrix}\sc g(t,y,\psi)\\\sc -b(t,y,\psi) \end{pmatrix},$$ in the sense that for some $\delta>0$, $$\begin{array}{ll} \noalign{ }\displaystyle\big\langle\begin{pmatrix}\sc g(t,y,\psi)-g(t,\bar y,\bar\psi)\\ \sc -b(t,y,\psi)+b(t,\bar y,\bar\psi)\end{pmatrix},\begin{pmatrix}\sc y-\bar y\\ \sc\psi-\bar\psi\end{pmatrix}\big\rangle\geqslant\delta\big(\\|y-\bar y\\|^2+\\|\psi-\bar\psi\\|^2),\qquad\forall t\in[0,T],~y,\bar y,\psi,\bar\psi\in X.\end{array}$$ It is possible to cook up many other cases from Theorem 7.5 and/or Corollary 7.6, for which the corresponding FBEEs are well-posed. Let us list some of them here. \bf Corollary 7.9. \sl Let {\rm(H0)} hold and $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$. Then the corresponding FBEE is well-posed if one of the following holds: {\rm(i)} For some $\delta,\varepsilon>0$, $$I+h_y(y)+h_y(y)^\geqslant\delta,\qquad\forall y\in X.$$ In the case $A^=A$, for all $(t,y,\psi)\in[0,T]\times X\times X$, $$\begin{pmatrix}e^{2(A+\sigma_0)(T-t)}& I\\ I&0\end{pmatrix}\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\begin{pmatrix} e^{2(A+\sigma_0)(T-t)}&I\\ I&0\end{pmatrix}\les0,$$ and in the case $A^=-A$, for all $(t,y,\psi)\in[t,T]\times X\times X$, $$\begin{pmatrix}I&I\\ I&0\end{pmatrix}\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\begin{pmatrix}I& I\\ I&0\end{pmatrix}\les0.$$ {\rm(ii)} For some $\delta,\varepsilon>0$, $$I-h_y(y)^h_y(y)\geqslant\delta,\qquad\forall y\in X.$$ In the case $A^=A$, for all $(t,y,\psi)\in[0,T]\times X\times X$, $$\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}e^{2(A+\sigma_0)(T-t)}&0\\ 0& e^{2(A+\sigma_0)t}\end{pmatrix}\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\begin{pmatrix}e^{2(A+\sigma_0)(T-t)}&0\\ 0&e^{2(A+\sigma_0)t}\end{pmatrix}\les0,\end{array}$$ and in the case $A^=-A$, for all $(t,y,\psi)\in[0,T]\times X\times X$, \begin{equation}\label{}\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\le0.\end{equation} {\rm(iii)} For some $\delta,\varepsilon>0$, \begin{equation}\label{}I+h_y(y)+h_y(y)^-h_y(y)^h_y(y)\geqslant\delta,\qquad\forall y\in X.\end{equation} In the case $A^=A$, for all $(t,y,\psi)\in[0,T]\times X\times X$, \begin{equation}\label{}\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}e^{2(A+\sigma_0)(T-t)}&I\\ I& e^{2(A+\sigma_0)t}\end{pmatrix}\mathbb{B}(t,y,\psi) +\mathbb{B}(t,y,\psi)^\begin{pmatrix}e^{2(A+\sigma_0)(T-t)}&I\\ I&e^{2(A+\sigma_0)t}\end{pmatrix}\les0,\end{array}\end{equation} and in the case $A^=-A$, for all $(t,y,\psi)\in[0,T]\times X\times X$, \begin{equation}\label{}\begin{pmatrix}I&I\\ I&I\end{pmatrix}\mathbb{B}(t,y,\psi)+\mathbb{B}(t,y,\psi)^\begin{pmatrix}I& I\\ I&I\end{pmatrix}\les0.\end{equation} \it Proof. \rm (i) We take $p_1=\gamma>0$ large enough and take $\bar p_0,q_0,\bar q_0>0$ small enough, $\theta=0$. Then we may apply Corollary 7.6 to get our claim. (ii) and (iii) can be proved similarly. \signed {$\sqr69$} Inspired by the above result, it is easy for us to prove many other results of similar nature. We prefer not to get into exhausting details. \section{More General Cases} In this section, we will briefly consider some more general cases. First of all, we consider the case $(b,g,h)\in{\cal G}_2$, i.e., the generator $(b,g,h)$ only satisfies (H2), and might not be Fr\'echet differentiable in $(y,\psi)$. Such a situation happens in many optimal control problems. To study such a case, let us recall some results from \cite{Pales-Zeidan 2007}. Let $f:X\to X$ be Lipschitz continuous and $\bar y\in X$. For any linear subspace $L\subseteq X$, we define {\it $L$-G\^ateaux-Jacobian $D_Lf(\bar y)\in{\cal L}(L;X)$} by the following (if the limit exists): $$D_Lf(\bar y)(x)=f'(\bar y;x)=\lim_{t\to0}{f(\bar y+tx)-f(\bar y)\over t},\qquad\forall x\in X.$$ The set of all points $\bar y\in X$ for which $D_Lf(\bar y)$ exist is denoted by $\Omega_L(f)$. Next, we let $$\partial_Lf(\bar y)=\bigcap_{\delta>0}\mathop{\overline{\rm co}}\Big\{D_Lf(y)\bigm\|y\in\Omega_L(f),\\|y-\bar y\\|\leqslant\delta\Big\}$$ and define the {\it generalized Jacobian} of $f(\cdot)$ at $\bar y$ by the following: $$\partial f(\bar y)=\Big\{\Psi\in{\cal L}(X;X)\bigm\|\Psi\big\|_L\in\partial_Lf(\bar y),~\forall L\hbox{ subspace of }X\Big\}.$$ For any $y,z\in X$, define $y\otimes z:{\cal L}(X)\to\mathbb{R}$ by $$(y\otimes z)(\Psi)=\big\langle\Psi(y),z\big\rangle,\qquad\forall\Psi\in{\cal L}(X).$$ Then $y\otimes z\in{\cal L}({\cal L}(X);\mathbb{R})$. Let $$X\otimes X=\hbox{\rm span$\,$}\{y\otimes z\bigm\|y,z\in X\}\subseteq{\cal L}(X)^.$$ The weak topology induced by $X\otimes X$ on ${\cal L}(X)$ is called the {\it weak$^$-operator-topology}, denoted by $\beta(X)$. The following can be found in \cite{Pales-Zeidan 2007}. \bf Proposition 8.1. \sl If $f:X\to X$ is Lipschitz near $\bar y$, then $\partial f(\bar y)$ is non-empty, bounded, and $\beta(X)$-compact. \rm More interestingly, we have the following {\it mean-value theorem} (see \cite{Pales-Zeidan 2007}, Theorem 4.4). \bf Proposition 8.2. \sl Let $f:X\to X$ be locally Lipschitz. Then for any $y,\bar y\in X$, $$f(y)-f(\bar y)\in\[\mathop{\overline{\rm co}}$\bigcup_{\lambda\in[0,1]}\partial f(\bar y+\lambda(y-\bar y))$\Big](y-\bar y).$$ \rm With the above preparation, we now consider the case that $(b,g,h)\in{\cal G}_2$. Naturally, we need only to define $$\mathbb{B}(s,y,\psi)=\begin{pmatrix}\sc b_y&\sc b_\psi\\ \sc-g_y&\sc-g_\psi\end{pmatrix},$$ with $$\begin{array}{ll} \noalign{ }\displaystyle b_y\in\partial_yb(t,y,\psi),\quad b_\psi\in\partial_\psi b(t,y,\psi),\quad g_y\in\partial_y g(t,y,\psi),\quad g_\psi\in\partial_\psi g(t,y,\psi).\end{array}$$ Then, all the results from previous sections for $(b,g,h)\in{\cal G}_2\cap{\cal G}_3$ can be carried over properly to the case $(b,g,h)\in{\cal G}_2$. Next, we consider $(b,g,h)\in{\cal G}_3$, i.e., the generator $(b,g,h)$ may be not globally Lipschitz with respect to $y$ and/or $\psi$, or equivalently, the Fr\'echet derivative of $(y,\psi)\mapsto(b(t,y,\psi),g(t,y,\psi),h(y))$ might be not bounded. In such cases, a priori uniform boundedness of the mild solution $(y(\cdot),\psi(\cdot))$ could play an essential role. Let us indicate one such a case. To this end, we introduce the following {\bf(H3)$'$} In addition to (H3), there is a non-decreasing function $f:[0,\infty)\to[0,\infty)$ such that \begin{equation}\label{}\begin{array}{ll} \noalign{ }\displaystyle\\|b_y(t,y,\psi)\\|+\\|b_\psi(t,y,\psi)\\|+\\|g_y(t,y,\psi)\\| +\\|g_\psi(t,y,\psi)\\|+\\|h_y(y)\\|\leqslant f(\\|y\\|+\\|\psi\\|),\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,y,\psi)\in[0,T] \times X\times X,\end{array}\end{equation} Moreover, \begin{equation}\label{}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\big\langle b(t,y,\psi),y\big\rangle\leqslant L(1+\\|y\\|^2),\\ \noalign{ }\displaystyle\big\langle g(t,y,\psi),\psi\big\rangle\leqslant L(1+\\|\psi\\|^2),\end{array}\right.\qquad\forall(t,y,\psi)\in[0,T]\times X\times X.\end{equation} Under (H3)$'$, if $(y^\rho_\lambda(\cdot),\psi^\rho_\lambda(\cdot))$ is a solution to (\ref{rho2}), then $$\begin{array}{ll} \noalign{ }\displaystyle\\|y^\rho_\lambda(t)\\|^2=\\|x\\|^2+2\int_0^t\big\langle y^\rho_\lambda(s),A_\lambda y^\rho_\lambda(s)+\rho b(s,y^\rho_\lambda(s),\psi^\rho_\lambda(s))+b_0(s) \big\rangle ds\\ \noalign{ }\displaystyle\qquad\qquad\leqslant\\|x\\|^2+2L\int_0^t$1+\\|y^\rho_\lambda(s)\\|^2+\\|y^\rho_\lambda(s)\\|\, \\|b_0(s)\\|$ds.\end{array}$$ Then by Gronwall's inequality, we have $$\\|y_\lambda^\rho(\cdot)\\|_\infty\leqslant K$1+\\|x\\|+\int_0^T\\|b_0(r)\\|dr$.$$ Similarly, $$\begin{array}{ll} \noalign{ }\displaystyle\\|\psi^\rho_\lambda(t)\\|^2=\\|\psi^\rho_\lambda(T)\\|^2 -2\int_t^T\big\langle\psi^\rho_\lambda(s), -A^_\lambda\psi^\rho_\lambda(s)-\rho g(s,y^\rho_\lambda(s),\psi^\rho_\lambda(s))-g_0(s)\big\rangle ds\\ \noalign{ }\displaystyle\qquad\qquad\leqslant\\|\psi^\rho_\lambda(T)\\|^2+2L\int_t^T$1+\\|\psi^\rho_\lambda(s)\\|^2 +\\|\psi^\rho_\lambda(s)\\|\,\\|g_0(s)\\|$ds.\end{array}$$ Hence, it follows from Gronwall's inequality that $$\begin{array}{ll} \noalign{ }\displaystyle\\|\psi^\rho_\lambda(\cdot)\\|_\infty\leqslant K$1+\\|\psi^\rho_\lambda(T)\\|+\int_0^T \\|g_0(s)\\|ds$\\ \noalign{ }\displaystyle\qquad\qquad\leqslant K$1+\\|h(0)\\|+f(\\|y^\rho_\lambda(T)\\|)\\|y^\rho_\lambda(T)+\\|h_0\\| +\int_0^T\\|g_0(s)\\|ds$\leqslant K.\end{array}$$ Consequently, the relevant proofs will go through as if (H4) is assumed. For concrete PDEs, there are some other ways to obtain uniform boundedness of the (weak) solutions to the system. We will see some of such below. \section{Several Illustrative Examples} In this section, we look at several examples. \bf Example 9.1. (Linear-Convex Optimal Control Problem) \rm Consider an optimal control problem with a linear state equation: $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)+Bu(t),\\ \noalign{ }\displaystyle y(0)=x,\end{array}\right.$$ and with the cost functional $$J(x;u(\cdot))=\int_0^T$Q(y(t))+{1\over2}\big\langle Ru(t),u(t)\big\rangle$ds+G(y(T)),$$ where $y\mapsto Q(y)$ and $y\mapsto G(y)$ are $C^2$ and convex. Then Pontryagin minimum principle leads to the optimality system: \begin{equation}\label{FBEE(LC)}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)-BR^{-1}B^\psi(t),\\ \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-Q_y(y(t)),\\ \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=G_y(y(T)).\end{array}\right.\end{equation} In this case, we have $$\begin{array}{ll} \noalign{ }\displaystyle b(t,y,\psi)=-BR^{-1}B^\psi,\quad g(t,y,\psi)=Q_y(y),\quad h(y)=G_y(y).\end{array}$$ Thus, $$\begin{array}{ll} \noalign{ }\displaystyle b_y(t,y,\psi)=0,\qquad b_\psi(t,y,\psi)=-BR^{-1}B^,\\ \noalign{ }\displaystyle g_y(t,y,\psi)=Q_{yy}(y),\qquad g_\psi(t,y,\psi)=0,\qquad h_y(y)=G_{yy}(y).\end{array}$$ Then $$\mathbb{B}(t,y,\psi)=\begin{pmatrix}\sc0&\sc-BR^{-1}B^\\ \sc-Q_{yy}(y)&\sc0\end{pmatrix},$$ Hence, under conditions $$R\geqslant\delta,\qquad M\geqslant G_{yy}(y)\ges0,\quad M\geqslant Q_{yy}(y)\geqslant\delta,\quad\forall y\in X,$$ for some $M,\delta>0$, all the conditions of Corollary 7.7 hold, and the FBEE (\ref{FBEE(LC)}) admits a unique mild solution. A further special case is the following: $$Q(y)={1\over2}\big\langle Qy,y\big\rangle,\quad G(y)={1\over2}\big\langle Gy,y\big\rangle,$$ for some $Q,G\in\mathbb{S}^+(X)$. In this case, the FBEE can be written as \begin{equation}\label{FBEE(LQ)}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)-BR^{-1}B^\psi(t),\\ \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-Qy(t),\\ \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=Gy(T).\end{array}\right.\end{equation} Hence, according to the above, when $$R\geqslant\delta,\qquad G\ges0,\qquad Q\geqslant\delta,$$ for some $\delta>0$, the FBEE (\ref{FBEE(LQ)}) admits a unique mild solution. \bf Example 9.2. (AQ Problem) \rm For the simplicity of presentation, we let $S(\cdot)=0$, and assume that all the involved functions are time-independent. Then the optimality system reads \begin{equation}\label{FBEE-AQ}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\dot y(t)=Ay(t)+F(y(t))-BR^{-1}B^\psi(t),\\ [1mm] \noalign{ }\displaystyle\dot\psi(t)=-A^\psi(t)-Q_y(y(t))-F_y(y(t))^\psi(t),\\ [1mm] \noalign{ }\displaystyle y(0)=x,\qquad\psi(T)=G_y(y(T)),\end{array}\right.\end{equation} with $A^=A$ or $A^=-A$. Thus, \begin{equation}\label{}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle b(t,y,\psi)=F(y)-BR^{-1}B^\psi,\\ [1mm] \noalign{ }\displaystyle g(t,y,\psi)=Q_y(y)+F_y(y)^\psi,\\ [1mm] \noalign{ }\displaystyle h(y)=G_y(y).\end{array}\right.\end{equation} Let $\{\xi_n\}_{n\ge1}$ be an orthonormal basis of $X$, under which we may let $$F(y)=\sum_{n=1}^\infty\big\langle F(y),\xi_n\big\rangle\xi_n\equiv\sum_{n=1}^\infty f^n(y)\xi_n.$$ Then $$F_y(y)z=\lim_{\delta\to0}{F(y+\delta z)-F(y)\over\delta}=\sum_{n=1}^\infty\big\langle f_y^n(y),z\big\rangle\xi_n\equiv \sum_{n=1}^\infty[\xi_n\otimes f_y^n(y)]z.$$ Thus, $$F_y(y)=\sum_{n=1}^\infty\xi_n\otimes f_y^n(y),$$ and $$F_y(y)^\psi=\sum_{n=1}^\infty[f^n_y(y)\otimes\xi_n]\psi=\sum_{n=1}^\infty f^n_y(y)\big\langle\xi_n,\psi\big\rangle.$$ Hence, $$[F_y(y)^\psi]_y=\sum_{n=1}^\infty f^n_{yy}(y)\big\langle\xi_n,\psi\big\rangle\in\mathbb{S}(X), \qquad\forall y\in X.$$ Consequently, $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle b_y(s,y,\psi)=F_y(y),\quad b_\psi(s,y,\psi)=-BR^{-1}B^,\\ \noalign{ }\displaystyle g_y(s,y,\psi)=Q_{yy}(y)+[F_y(y)^\psi]_y,\quad g_\psi(s,y,\psi)=F_y(y)^=b_y(s,y,\psi)^,\\ \noalign{ }\displaystyle h_y(y)=G_{yy}(y).\end{array}\right.$$ Then $$\mathbb{B}(s,y,\psi)=\begin{pmatrix}\sc F_y(y)&\sc-BR^{-1}B^\\ \sc-Q_{yy}(y)-[F_y(y)^\psi]_y&\sc-F_y(y)^ \end{pmatrix}\equiv\begin{pmatrix}\sc B_{11}&\sc B_{12}\\ \sc-B_{21}&\sc-B_{22}\end{pmatrix}$$ From this, we can calculate $$\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}0&I\\ I&0\end{pmatrix}\mathbb{B}(s,y,\psi)+\mathbb{B}(s,y,\psi)^* \begin{pmatrix}0&I\\ I&0\end{pmatrix}\\ \noalign{ }\displaystyle=\begin{pmatrix}\sc-[g_y(s,y,\psi)+g_y(s,y,\psi)^]&\sc b_y(s,y,\psi)^-g_\psi(s,y,\psi)\\ \sc b_y(s,y,\psi)-g_\psi(s,y,\psi)^&\sc b_\psi(s,y,\psi)+b_\psi(s,y,\psi)^\end{pmatrix}\\ \noalign{ }\displaystyle=-2\begin{pmatrix}\sc Q_{yy}(y)+[F_y(y)^\psi]_y&\sc0\\ \sc0&\sc BR^{-1}B^\end{pmatrix}\end{array}$$ Next, we note that if $\psi(\cdot)$ is a mild solution to the backward evolution equation in (\ref{FBEE-AQ}), we have $$\begin{array}{ll} \noalign{ }\displaystyle\\|\psi(t)\\|^2=\\|\psi(T)\\|^2+2\int_t^T\big\langle[A+F_y(y(s))]\psi(s)+Q_y(y(s)),\psi(s)\big\rangle ds\\ \noalign{ }\displaystyle\leqslant\\|G_y(y(T)))\\|^2+2\int_t^T\\|Q_y(y(s))\\|\,\\|\psi(s)\\|ds\\ \noalign{ }\displaystyle\leqslant\\|G_y(\cdot)\\|_\infty^2+2\int_t^T\\|Q_y(\cdot)\\|_\infty\\|\psi(s)\\|ds\equiv\varphi(t).\end{array}$$ Then $$\dot\varphi(t)=-2\\|Q_y(\cdot)\\|_\infty\\|\psi(t)\\|\geqslant-2\\|Q_y(\cdot)\\|_\infty\sqrt{\varphi(t)},$$ which leads to $$$\sqrt{\varphi(t)}$'\geqslant-\\|Q_y(\cdot)\\|_\infty.$$ Then $$\begin{array}{ll} \noalign{ }\displaystyle\\|\psi(t)\\|=\sqrt{\varphi(t)}=\sqrt{\varphi(T)}-\int_t^T$\sqrt{\varphi(s)}$'ds\leqslant\\|G_y(\cdot)\\|_\infty+\\|Q_y(\cdot)\\|_\infty T,\qquad\forall t\in[0,T].\end{array}$$ Hence, $$\begin{array}{ll} \noalign{ }\displaystyle\\|F_y(y)^\psi\\|=\\|F_y(y)\\|\,\\|\psi\\|\leqslant\\|F_y(y)\\|$\\|G_y(\cdot)\\|_\infty +\\|Q_y(\cdot)\\|_\infty T$,\\ \noalign{ }\displaystyle\qquad\qquad\qquad\qquad\qquad\forall\,\\|\psi\\|\leqslant$\\|G_y(\cdot)\\|_\infty +\\|Q_y(\cdot)\\|_\infty T$.\end{array}$$ Consequently, if we assume $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle G_{yy}(y)\ges0,\quad\forall y\in X,\qquad BR^{-1}B^\geqslant\delta,\\ \noalign{ }\displaystyle Q_{yy}(y)\geqslant\\|F_y(y)\\|$\\|G_y(\cdot)\\|_\infty +\\|Q_y(\cdot)\\|_\infty T$+\delta,\qquad\forall y\in X,\end{array}\right.$$ for some $\delta>0$, then (\ref{FBEE-AQ}) admits a unique mild solution, by Corollary 7.8. \bf Example 9.3. (Optimal Control of a Parabolic PDE). \rm We now consider an optimal control problem for a parabolic equation. Such a problem was studied in \cite{Stojanovic 1989}. The controlled state equation reads: \begin{equation}\label{}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle y_t=\Delta y-(\lambda+u)y+f,\qquad\hbox{in }(0,T)\times\Omega,\\ \noalign{ }\displaystyle y\big\|_{\partial\Omega}=0,\\ \noalign{ }\displaystyle y(0,x)=y_0(x),\qquad x\in\Omega,\end{array}\right.\end{equation} where $y(t,x)$ is the state and $u(t,x)$ is the control, and $\Omega\subseteq \mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$. The cost functional is the following: \begin{equation}\label{}J(u(\cdot))={1\over2}\int_0^T\int_\Omega$L\|y-y_d\|^2+Nu^2$dxdt+{1\over2}\int_\Omega M\|y(T,x)-z(x)\|^2dx.\end{equation} We assume that $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle f(t,x)\ges0,\quad y_d(t,x)\les0,\qquad(t,x)\in(0,T)\times\Omega,\\ [2mm] \noalign{ }\displaystyle y_0(x)\ges0,\quad z(x)\les0,\qquad x\in\Omega.\end{array}\right.$$ According to \cite{Stojanovic 1989}, optimal control exists and the optimality system reads: \begin{equation}\label{}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle y_t=\Delta y-\lambda y-{1\over N}\psi y^2+f,\qquad\hbox{in }(0,T)\times\Omega,\\ \noalign{ }\displaystyle\psi_t=-\Delta\psi+\lambda\psi+{1\over N}y\psi^2-L(y-y_d),\qquad\hbox{in }(0,T)\times\Omega,\\ \noalign{ }\displaystyle y\big\|_{\partial\Omega}=\psi\big\|_{\partial\Omega}=0,\\ [2mm] \noalign{ }\displaystyle y(0,x)=y_0(x),\quad\psi(T,x)=M\big(y(T,x)-z(x)\big),\qquad x\in\Omega,\end{array}\right.\end{equation} Then we have $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle b(s,y,\psi)=-\lambda y-{1\over N}\psi y^2+f,\\ [2mm] \noalign{ }\displaystyle g(s,y,\psi)=-\lambda\psi-{1\over N}y\psi^2+L(y-y_d).\end{array}\right.$$ Hence, $$\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle b_y=-\lambda-{2\over N}y\psi,\qquad b_\psi=-{1\over N}y^2,\\ \noalign{ }\displaystyle g_y=L-{1\over N}\psi^2,\qquad g_\psi=-\lambda-{2\over N}y\psi=b_y.\end{array}\right.$$ Then $$\mathbb{B}(t,y,\psi)=\begin{pmatrix}\sc-\lambda-{2\over N} y\psi&\sc-{1\over N}y^2\\ \sc-L+{1\over N}\psi^2&\sc\lambda+{2\over N}y\psi\end{pmatrix}.$$ Thus, $$\begin{array}{ll} \noalign{ }\displaystyle\begin{pmatrix}\sc0&\sc I\\ \sc I&0\end{pmatrix}\mathbb{B}(t,y,\psi) +\mathbb{B}(t,y,\psi)\begin{pmatrix}\sc0&\sc I\\ \sc I&0\end{pmatrix}=2\begin{pmatrix}\sc-L+{1\over N}\psi^2& \sc0\\ \sc0&\sc-{1\over N}y^2\end{pmatrix}\les0,\end{array}$$ provided $\psi$ is bounded (which was shown in \cite{Stojanovic 1989}) and $N$ is large enough. \section{Concluding Remarks} We have discussed the well-posedness of FBEEs which is mainly motivated by the optimality systems of optimal control problems for infinite dimensional evolution equations. We have presented some basic results from two approaches: the decoupling method and the method of continuity. It is seen that the theory is far from mature and many challenging questions are left open. Here is a partial list of these: $\bullet$ In the direction of decoupling method, it is widely open that how one can construct decoupling field, through solving a PDE in Hilbert space. $\bullet$ In the direction of method of continuity, more careful analysis is need to make the stated condition easier to use. $\bullet$ More general generators $A$ other than $A^=A$ and $A^=-A$. Also, taking into account of PDEs, the generator $(b,g,h)$ might be unbounded (involving differential operators). \section{Appendix.} \rm \it Proof of Proposition 4.2. \rm First, we have $$\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat y(s)\\|^2\negthinspace =\negthinspace \\|\widehat x\\|^2\negthinspace \negthinspace +\negthinspace \negthinspace \int_t^s\negthinspace \negthinspace $2\big\langle\widehat y(r),A_\lambda\widehat y(r)\negthinspace +\negthinspace \rho\widetilde b_y(r)\widehat y(r)\negthinspace +\negthinspace \rho\widetilde b_\psi(s)\widehat\psi(s) \negthinspace +\negthinspace \delta b(r)\negthinspace +\negthinspace \delta b_0(r)\big\rangle$dr\\ \noalign{ }\displaystyle\leqslant\negthinspace \\|\widehat x\\|^2\negthinspace \negthinspace +\negthinspace \negthinspace \negthinspace \int_t^s\negthinspace \negthinspace \[\negthinspace \big\langle$A_\lambda+A_\lambda^+\rho\big(\widetilde b_y(r)\negthinspace +\negthinspace \widetilde b_y(r)^\big)$\widehat y(r),\widehat y(r)\big\rangle\negthinspace +2\\|\widehat y(r)\\|\\|\rho\widetilde b_\psi(r)\widehat\psi(r)\negthinspace +\negthinspace \rho\delta b(r)\negthinspace +\negthinspace \delta b_0(r)\\|\]dr\\ \noalign{ }\displaystyle\leqslant\\|\widehat x\\|^2\negthinspace \negthinspace +2\negthinspace \negthinspace \int_t^s\negthinspace \negthinspace \[\rho L_{by}(r)\\|\widehat y(r)\\|^2+\\|\widehat y(r)\\|\\|\rho\widetilde b_\psi(r)\widehat\psi(r)\negthinspace +\negthinspace \rho\delta b(r)\negthinspace +\negthinspace \delta b_0(r)\\|\]dr\equiv\varphi(s)^2.\end{array}$$ Then $$\begin{array}{ll} \noalign{ }\displaystyle\varphi(s)\dot\varphi(s)=\rho L_{by}(s)\\|\widehat y(s)\\|^2+\\|\widehat y(s)\\|\\|\rho\widetilde b_\psi(s)\widehat\psi(s)\negthinspace +\negthinspace \rho\delta b(s)\negthinspace +\negthinspace \delta b_0(s)\\|\\ \noalign{ }\displaystyle\qquad\qquad\leqslant\rho L_{by}(s)\varphi(s)^2+\varphi(s)\\|\rho\widetilde b_\psi(s)\widehat\psi(s)\negthinspace +\negthinspace \rho\delta b(s)\negthinspace +\negthinspace \delta b_0(s)\\|\\ \noalign{ }\displaystyle\qquad\qquad\equiv a_2(s)\varphi(s)^2+a_1(s)\varphi(s).\end{array}$$ Consequently, $$\dot\varphi(s)\leqslant a_2(s)\varphi(s)+a_1(s).$$ Hence, by Gronwall's inequality, $$\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat y(s)\\|\leqslant\varphi(s)\le e^{\int_t^sa_2(\tau)d\tau}\\|\widehat x\\|+\int_t^se^{\int_r^sa_2(\tau)d\tau}a_1(r)dr\\ \noalign{ }\displaystyle=e^{\rho\int_t^sL_{by}(\tau)d\tau}\\|\widehat x\\|+\int_t^se^{\rho\int_r^sL_{by}(\tau)d\tau}\\|\rho\widetilde b_\psi(r)\widehat\psi(r)\negthinspace +\negthinspace \rho\delta b(r)\negthinspace +\negthinspace \delta b_0(r)\\|dr\\ \noalign{ }\displaystyle\leqslant\rho\int_t^se^{\rho\int_r^sL_{by}(\tau)d\tau}\\|\widetilde b_\psi(r)\widehat\psi(r)\\|dr+K\[\\|\widehat x\\|+\int_t^s$\\|\delta b(r)\\|+\\|\delta b_0(r)\\|$dr\Big].\end{array}$$ This proves (\ref{\|hy\|}). We now prove (\ref{\|hpsi\|}). One has $$\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat\psi(T)\\|^2-\\|\widehat\psi(s)\\|^2\\ \noalign{ }\displaystyle=\negthinspace \negthinspace \int_s^T2\big\langle\widehat\psi(r),-A_\lambda^\widehat\psi(r)-\rho\widetilde g_y(r)\widehat y(r)\negthinspace -\negthinspace \rho\widetilde g_\psi(r)\widehat\psi(r)-\negthinspace \rho\delta g(r)\negthinspace -\negthinspace \delta g_0(r)\negthinspace \big\rangle dr\\ \noalign{ }\displaystyle\geqslant-2\int_s^T\negthinspace \negthinspace \negthinspace $\big\langle \rho[\widetilde g_\psi(s)+\widetilde g_\psi(s)^]\widehat\psi(s),\widehat\psi(s)\big\rangle +\big\langle\widehat\psi(s),\rho\widetilde g_y(s)\widehat y(s)+\rho\delta g(s)+\delta g_0(s)\big\rangle$ds\\ \noalign{ }\displaystyle\ges2\int_t^T\Big\{-\rho L_{g\psi}(r)\\|\widehat\psi(r)\\|^2-\\|\widehat\psi(r)\\|\,\\|\rho\widetilde g_y(r)\widehat y(r)+\rho\delta g(r)+\delta g_0(r)\\|\Big\}ds,\end{array}$$ which leads to $$\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat\psi(s)\\|^2\leqslant\\|\widehat\psi(T)\\|^2+2\int_s^T\Big\{\rho L_{g\psi}(r)\\|\widehat\psi(r)\\|^2+\\|\widehat\psi(r)\\|\,\\|\rho\widetilde g_y(r)\widehat y(r)+\rho\delta g(r)+\delta g_0(r)\\|\Big\}dr\equiv\varphi(s)^2.\end{array}$$ Then $$\begin{array}{ll} \noalign{ }\displaystyle\varphi(s)\dot\varphi(s)=-\rho L_{g\psi}(r)\\|\widehat\psi(s)\\|^2+\\|\widehat\psi(s)\\|\,\\|\rho\widetilde g_y(s)\widehat y(s)+\rho\delta g(s)+\delta g_0(s)\\|\\ \noalign{ }\displaystyle\qquad\qquad\geqslant-\rho L_{g\psi}(r)\varphi(s)^2+\\|\rho\widetilde g_y(s)\widehat y(s)+\rho\delta g(s)+\delta g_0(s)\\|\varphi(s)\equiv-a_2(s)\varphi(s)^2-a_1(s)\varphi(s).\end{array}$$ Thus, $$\dot\varphi(s)+a_2(s)\varphi(s)\geqslant-a_1(s).$$ $$$e^{-\int_s^Ta_2(\tau)d\tau}\varphi(s)$'\geqslant-a_1(s)e^{-\int_s^Ta_2(\tau)d\tau}$$ $$\varphi(T)-e^{-\int_s^Ta_2(\tau)d\tau}\varphi(s)\geqslant-\int_s^Ta_1(r)e^{-\int_r^Ta_2(\tau)d\tau}dr.$$ Hence, $$\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat\psi(s)\\|\leqslant\varphi(s)\leqslant e^{\int_s^Ta_2(\tau)d\tau}\varphi(T)+\int_s^Ta_1(r)e^{\int_s^ra_2(\tau)d\tau}dr\\ \noalign{ }\displaystyle\leqslant e^{\rho\int_s^TL_{g\psi}(\tau)d\tau}\\|\widetilde h_y\widehat y(T)+\rho\delta h+\delta h_0\\|+\int_s^Te^{\rho\int_s^rL_{g\psi}(\tau)d\tau}\\|\rho\widetilde g_y(r)\widehat y(r)+\rho\delta g(r)+\delta g_0(r)\\|dr\\ \noalign{ }\displaystyle\leqslant\rho\[e^{\rho\int_s^TL_{g\psi}(\tau)d\tau}\\|\widetilde h_y\widehat y(T)\\|+\int_s^Te^{\rho\int_s^rL_{g\psi}(\tau)d\tau}\\|\widetilde g_y(r)\widehat y(r)\\|dr\Big]\\ \noalign{ }\displaystyle\qquad+K\[\\|\delta h\\|+\\|\delta h_0\\|+\int_s^T$\\|\delta g(r)\\|+\\|\delta g_0(r)\\|$dr\Big].\end{array}$$ This completes the proof. \signed {$\sqr69$} Note that if we let $(y^\rho_0(\cdot),\psi^\rho_0(\cdot))$ be the mild solution of the following: \begin{equation}\label{rho0}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\negthinspace \negthinspace \negthinspace \begin{array}{ll}\dot y_0^\rho(s)=Ay_0^\rho(s)+\rho b(s,0,0)+b_0(s),\\ \noalign{ }\displaystyle\dot\psi_0^\rho(s)=-A^\psi_0^\rho(s)-\rho g(s,0,0)-g_0(s),\end{array}\qquad s\in[t,T],\\ \noalign{ }\displaystyle y_0^\rho(t)=x,\qquad\psi^\rho_0(T)=\rho h(0)+h_0.\end{array}\right.\end{equation} Then $$y^\rho_0(s)=e^{A(s-t)}x+\int_t^se^{A(s-r)}[\rho b(r,0,0)+b_0(r)]dr.$$ Hence, $$\\|y^\rho_0(\cdot)\\|_\infty\leqslant\\|x\\|+\int_t^T\\|\rho b(r,0,0)+b_0(r)\\|dr.$$ Also, $$\psi^\rho_0(s)=e^{A^(T-s)}\psi^\rho_0(T)+\int_s^Te^{A^(r-s)}[\rho g(r,0,0)+g_0(r)]dr.$$ Hence, $$\\|\psi^\rho_0(\cdot)\\|_\infty\leqslant\\|\rho h(0)+h_0\\|+\int_t^T\\|\rho g(r,0,0)+g_0(r)\\|dr.$$ Now, taking $$\begin{array}{ll} \noalign{ }\displaystyle\bar b(s,y,\psi)=b(s,0,0),\quad\bar b_0(s)=b_0(s),\\ \noalign{ }\displaystyle\bar g(s,y,\psi)=g(s,0,0),\quad\bar g_0(s)=g_0(s),\\ \noalign{ }\displaystyle\bar h(y)=h(0),\qquad\bar h_0=h_0.\end{array}$$ Then \begin{equation}\label{}\left\{\negthinspace \negthinspace \begin{array}{ll} \noalign{ }\displaystyle\delta b(s)=b(s,0,0)-b(s,y_\lambda^\rho(s),\psi_\lambda^\rho(s)),\quad\delta b_0(s)=\bar b_0(s)-b_0(s)=0,\\ \noalign{ }\displaystyle\delta g(s)=g(s,0,0)-g(s,y_\lambda^\rho(s),\psi_\lambda^\rho(s)),\quad\delta g_0(s)=\bar g_0(s)-g_0(s)=0,\\ \noalign{ }\displaystyle\delta h=h(0)-h(y_\lambda^\rho(T)),\quad\delta h_0=\bar h_0-h_0=0,\quad\widehat x=x-\bar x=0.\end{array}\right.\end{equation} \begin{equation}\label{\|hy\|}\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat y(\cdot)\\|_\infty\leqslant\rho\int_t^Te^{\rho\int_r^TL_{by}(\tau)d\tau}\\|\widetilde b_\psi(r)\widehat\psi(r)\\|dr.\end{array}\end{equation} and \begin{equation}\label{\|hpsi\|*}\begin{array}{ll} \noalign{ }\displaystyle\\|\widehat\psi(\cdot)\\|_\infty\leqslant\rho\[e^{\rho\int_t^TL_{g\psi}(\tau)d\tau}\\|\widetilde h_y\widehat y(T)\\|+\int_t^T\negthinspace \negthinspace \negthinspace e^{\rho\int_t^rL_{g\psi}(\tau)d\tau}\\|\widetilde g_y(r)\widehat y(r)\\|dr\Big].\end{array}\end{equation} \end{document}	arXiv
\begin{document} \allowdisplaybreaks \def\mathbb R} \def\ff{\frac} \def\ss{\sqrt{\mathbb R} \def\ff{\frac} \def\ss{\sqrt} \def\B{\mathbf B} \def\mathbb N} \def\kk{\kappa} \def\m{{\bf m}{\mathbb N} \def\kk{\kappa} \def\m{{\bf m}} \def\varepsilon}\def\ddd{D^{\varepsilon}\def\ddd{D^} \def\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho} \def\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma{\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma} \def\nabla} \def\pp{\partial} \def\E{\mathbb E{\nabla} \def\pp{\partial} \def\E{\mathbb E} \def\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D} \def\sigma} \def\ess{\text{\rm{ess}}{\sigma} \def\ess{\text{\rm{ess}}} \def\begin} \def\beq{\begin{equation}} \def\F{\scr F{\begin} \def\beq{\begin{equation}} \def\F{\scr F} \def\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}{\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}} \def\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega{\text{\rm{e}}} \def\uparrow{\underline a} \def\OO{\Omega} \def\oo{\omega} \def\tilde} \def\Ric{\text{\rm{Ric}}{\tilde} \def\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}{\text{\rm{Ric}}} \def\text{\rm{cut}}} \def\P{\mathbb P} \def\ifn{I_n(f^{\bigotimes n}){\text{\rm{cut}}} \def\P{\mathbb P} \def\ifn{I_n(f^{\bigotimes n})} \def\scr C} \def\G{\scr G} \def\aaa{\mathbf{r}} \def\r{r{\scr C} \def\G{\scr G} \def\aaa{\mathbf{r}} \def\r{r} \def\text{\rm{gap}}} \def\prr{\pi_{{\bf m},\varrho}} \def\r{\mathbf r{\text{\rm{gap}}} \def\prr{\pi_{{\bf m},\varrho}} \def\r{\mathbf r} \def\mathbb Z} \def\vrr{\varrho} \def\ll{\lambda{\mathbb Z} \def\vrr{\varrho} \def\ll{\lambda} \def\scr L}\def\Tt{\tt} \def\TT{\tt}\def\II{\mathbb I{\scr L}\def\Tt{\tilde} \def\Ric{\text{\rm{Ric}}} \def\TT{\tilde} \def\Ric{\text{\rm{Ric}}}\def\II{\mathbb I} \def{\rm in}}\def\Sect{{\rm Sect}} \def\H{\mathbb H{{\rm in}}\def\Sect{{\rm Sect}} \def\H{\mathbb H} \def\scr M}\def\Q{\mathbb Q} \def\texto{\text{o}} \def\LL{\Lambda{\scr M}\def\Q{\mathbb Q} \def\texto{\text{o}} \def\LL{\Lambda} \def{\rm Rank}} \def\B{\scr B} \def\i{{\rm i}} \def\HR{\hat{\R}^d{{\rm Rank}} \def\B{\scr B} \def{\rm in}}\def\Sect{{\rm Sect}} \def\H{\mathbb H{{\rm i}} \def\HR{\hat{\mathbb R} \def\ff{\frac} \def\ss{\sqrt}^d} \def\rightarrow}\def\l{\ell}\def\iint{\int{\rightarrow}\def\l{\ell}\def\iint{\int} \def\scr E}\def\no{\nonumber{\scr E}\def\no{\nonumber} \def\scr A}\def\V{\mathbb V}\def\osc{{\rm osc}{\scr A}\def\V{\mathbb V}\def\osc{{\rm osc}} \def\scr B}\def\Ent{{\rm Ent}}\def\3{\triangle}\def\H{\scr H{\scr B}\def\Ent{{\rm Ent}}\def\3{\triangle}\def\H{\scr H} \def\scr U}\def\8{\infty}\def\1{\lesssim}\def\HH{\mathrm{H}{\scr U}\def\8{\infty}\def\1{\lesssim}\def\HH{\mathrm{H}} \def\scr T{\scr T} \title{{f Path Independence of Additive Functionals for SDEs under $G$-framework} \begin{abstract} The path independence of additive functionals for SDEs driven by the $G$-Brownian motion is characterized by nonlinear PDEs. The main result generalizes the existing ones for SDEs driven by the standard Brownian motion. \end{abstract} \noindent AMS subject Classification:\ 60H10, 60H15. \\ \noindent Keywords: additive functional; $G$-SDEs; $G$-Brownian motion; nonlinear PDE \vskip 2cm \section{Introduction} Stochastic differential equations (SDEs) under the linear probability space have been widely used in modeling financial markets and economic phenomena \cite{B08,BS}. However, in many practical situations, most of the financial activities take place with uncertainty \cite{BR}, for which a fundamental theory of SDEs driven by the $G$-Brownian motion ($G$-SDEs) has been developed in \cite{peng2, peng1, peng4}. Since then $G$-SDEs have received much attention, see for instance \cite{HJP} on the Feyman-Kac formula, \cite{HJ,HJY} on the stochastic control, \cite{FWZ,FOZ} on the ergodicity, \cite{RYS,WG} on the stochastic stability, and \cite{GRR} on the $G$-SPDEs. In the equilibrium financial market, there exists a risk neutral measure which admits a path independent density with respect to the real world probability \cite{HC}. To construct such risk neutral measures, the path independence of additive functionals for SDEs has been investigated extensively; see \cite{Wang} for the pioneer work. Subsequently, \cite{Wang} has been extended in \cite{Wang3,QW,Wang2} for finite dimensional SDEs, and in \cite{QW2, Wang1} for infinite dimensional SPDEs, where \cite{Wang2} allows the SDEs involved to be degenerate. Recently, \cite{RW} investigated the path independence of additive functionals for a class of distribution dependent SDEs. Nevertheless, all of these papers only focus on linear probability spaces. To fill this gap, in this paper, we intend to characterize the path independence of additive functionals for $G$-SDEs. To this end, below we recall some basic facts on the $G$-Brownian motion. For a positive integer $d$, let $(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d, \langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\cdot,\cdot\>,\|\cdot\|)$ be the $d$-dimensional Euclidean space, $\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\otimes\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ the family of all $d\times d$-matrices, $\mathbb{S}^d$ the collection of all symmetric $d\times d$-matrices, ${\bf0}_d\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ the zero vector, ${\bf0}_{d\times d}\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\otimes \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ the zero matrix, and ${\bf{I}}_{d\times d}\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\otimes \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ the identity matrix. For a matrix $A$, let $A^$ be its transpose and $\\|A\\|_{\rm HS}=(\mbox{trace}(AA^))^{1/2}$ be its Hilbert-Schmidt norm (or Frobenius norm). For a number $a\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt$, $a^+$ and $a^-$ stipulate its positive part and negative part, respectively. For $\sigma_1, \sigma_2 \in \mathbb{S}^d,$ the notation $\sigma_1\leq \sigma_2$ (res. $\sigma_1 < \sigma_2$) means that $\sigma_2- \sigma_1$ is non-negative (res. positive) definite, and we let $$ [{\sigma_1}, {\sigma_2}] :=\{\gamma\|\gamma\in \mathbb{S}^d, {\sigma_1}\leq\gamma \leq {\sigma_2}\}.$$ Let $C^{1,2}(\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d;\mathbb R} \def\ff{\frac} \def\ss{\sqrt)$ be the collection of all continuous functions $V:\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow\mathbb R} \def\ff{\frac} \def\ss{\sqrt $ which are once differentiable w.r.t. the first argument, twice differentiable w.r.t. the second argument, and all these derivatives are joint continuous. Write $\nabla} \def\pp{\partial} \def\E{\mathbb E$ and $\nabla} \def\pp{\partial} \def\E{\mathbb E^2$ by the gradient operator and Hessian operator, respectively. For any fixed $T>0$, $$\OO_T=\{\omega\|[0,T]\ni t\mapsto\omega_t\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d \mbox{ is continuous with }\omega(0)={\bf0}_d \}$$ endowed with the uniform topology. Let $B_t(\omega)=\omega_t, \omega\in\OO_T, $ be the canonical process. Set $$L_{ip}(\Omega_T):=\{\varphi(B_{t_1}, \cdots, B_{t_n}), n\in\mathbb{N}, t_1,\cdot \cdot \cdot, t_n\in [0,T],\varphi\in C_{b,lip}(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^{d}\otimes\mathbb R} \def\ff{\frac} \def\ss{\sqrt^n)\},$$ where $C_{b,lip}(\mathbb{R}^{d}\otimes\mathbb R} \def\ff{\frac} \def\ss{\sqrt^n)$ denotes the set of bounded Lipschitz functions $f:\mathbb{R}^{d}\otimes \mathbb R} \def\ff{\frac} \def\ss{\sqrt^n\rightarrow}\def\l{\ell}\def\iint{\int\mathbb R} \def\ff{\frac} \def\ss{\sqrt$. Let $G: \mathbb{S}^d \rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt $ be a monotonic, sublinear and homogeneous function; see e.g. \cite[p16]{peng4}. Throughtout the paper, we always assume that $G: \mathbb{S}^d \rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt $ is non-degenerate, i.e., there exists some $\delta>0$ such that \begin{equation} \label{Gnon} G(A)-G(B)\geq \frac{\delta}{2} \mbox{trace}[A-B],\ A\geq B, A,B \in \mathbb{S}^d. \end{equation} For any $\xi \in L_{ip}(\Omega_T)$, i.e., $$\xi(\omega)=\varphi(\omega(t_1),\cdot \cdot \cdot,\omega(t_{n})), \ 0=t_0< t_1<\cdot \cdot \cdot<t_n=T,$$ the conditional $G$-expectation is defined by $$ \bar{\mathbb{E}}_t[\xi]:=u_k(t,\omega(t);\omega(t_1),\cdot \cdot \cdot,\omega(t_{k-1}) ), \ \xi \in L_{ip}(\Omega_T), \ t \in [t_{k-1}, t_k), \ k=1, \cdot\cdot\cdot, n, $$ where $(t,x)\mapsto u_k(t,x; x_1, \cdot\cdot\cdot,x_{k-1})$, $k=1,\cdot \cdot\cdot,n$, solves the following $G$-heat equation \begin{equation}\label{Gheat} \begin{cases} \partial_tu_k+G(\partial_x^2u_k)=0, \ (t,x) \in [t_{k-1}, t_k)\times \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d, \ k=1,\cdot \cdot\cdot,n,\\ u_k(t_k,x;x_1,\cdot\cdot\cdot,x_{k-1})= u_{k+1}(t_k,x;x_1,\cdot\cdot\cdot,x_{k-1},x_k), \ k=1, \cdot\cdot\cdot, n-1,\\ u_n(t_n,x;x_1,\cdot\cdot\cdot,x_{n-1})=\varphi(x_1,\cdot\cdot\cdot,x_{n-1},x), \ k=n. \end{cases} \end{equation} Since $G$ is non-degenerate, the solution of \eqref{Gheat} satisfies $u_k\in C^{1,2}(\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d;\mathbb R} \def\ff{\frac} \def\ss{\sqrt);$ see \cite[Appendix $C$, Theorem 4.5, p127]{peng4}. The corresponding $G$-expectation of $\xi$ is defined by $\bar{\mathbb{E}}[\xi]=\bar{\mathbb{E}}_0[\xi]$. Then the canonical process $B_t(\omega):=\omega_t$ is called a $G$-Brownian motion in $(\Omega_T,L_{G}^p(\Omega_T),\bar{\mathbb{E}})$, where $L_G^{p}(\Omega_T$) is the completion of $L_{ip}(\Omega_T)$ under the norm $(\bar{\mathbb{E}}[\|\cdot\|^p])^{\frac{1}{p}}$, $p\geq1. $ By definition, we have $G(A)=\frac{1}{2}\bar{\mathbb{E}}\langle AB_1,B_1\rangle$, $A\in\mathbb{S}^d$. The function $G$ is called the generating function corresponding of the $d$-dimensional $G$-Brownian motion $(B_t)_{t\ge0}$. According to \cite{peng4}, there exists a bounded, convex, and closed subset $\Theta\subset \mathbb{S}^m$ such that \begin{equation}\label{GA} G(A)=\frac{1}{2}\sup _{Q\in \Theta}\mbox{trace}[AQ], \ A\in\mathbb{S}^d. \end{equation} In particular, for 1-dimensional $G$-Brownian motion $(B_t)_{t\ge0}$, one has $G(a)=(\overline{\sigma}^2a^+-\underline{\sigma}^2a^-)/2, a\in \mathbb{R}, $ where $\overline{\sigma}^2: = \bar{\mathbb{E}} [B^2_1]\geq -\bar{\mathbb{E}} [-B^2_1 ]= :\underline{\sigma}^2>0$. Let \begin{equation}\label{equa11} M_{G}^{p,0}([0,T])= \Big\{\eta_t:=\sum_{j=0}^{N-1} \xi_{j} I_{[t_j, t_{j+1})}(t); ~\xi_{j}\in L_{G}^p(\Omega_{t_{j}}), N\in\mathbb{N},\ 0=t_0<t_1<\cdots <t_N=T \Big\}. \end{equation} Let $M_G^p([0,T])$ and $H_G^p([0,T])$ be the completion of $M_G^{p,0}([0,T])$ under the norm $$\\|\eta\\|_{M_G^p([0,T])}:=\left(\bar{\mathbb{E}}\int_{0}^{T}\|\eta_{t}\|^p\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\right)^{\frac{1}{p}},\ \\|\eta\\|_{H_G^p([0,T])}:=\left\{\bar{\mathbb{E}}\left(\int_{0}^{T}\|\eta_{t}\|^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\right)^\frac{p}{2}\right\}^{\frac{1}{p}},$$ respectively. We need to point out that if $p=2$, then $M_G^p([0,T])=H_G^p([0,T])$. Denote by $M_G^p([0,T];\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$ all $d$-dimensional stochastic processes $\eta_t=(\eta_t^1, \cdot \cdot \cdot, \eta_t^d),$ $t\geq0$ with $\eta_t^i\in M_G^p([0,T]).$ Let $ H_G^1([0,T]; \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d) $ be all $d$-dimensional stochastic processes $\zeta_t=(\zeta_t^{1}, \cdot\cdot\cdot,\zeta_t^{d}), t\geq0$ with $\zeta^{i}\in H_G^1([0,T]). $ Furthermore, we also need the Choquet capacity associated with the $G$-expectation. Let $\mathcal{M}$ be the collection of all probability measures on $(\Omega_T, \mathcal{B}(\Omega_T))$. According to \cite{15}, there exists a weakly compact subset $\mathcal{P}\subset \mathcal{M}$ such that $$\bar{\mathbb{E}}[X]=\sup_{P\in \mathcal{P}}\mathbb{E}_P[X], \ X\in L_{ip}(\Omega_T).$$ Then the associated Choquet capacity is defined by $$c(A)=\sup_{P\in \mathcal{P}}P(A), \ A\in \mathcal{B}(\Omega_T).$$ A set $A\subset \Omega_T$ is called polar if $c(A)=0$, and we say that a property holds quasi-surely (q.s.) if it holds outside a polar set. In this paper, we consider the following $G$-SDE \begin{equation}\label{kerner} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D X_t=b(t,X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t+ \sum_{i,j=1}^dh_{ij}(t,X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i,B^j\rangle_t+\langle\sigma(t,X_t),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t\rangle, \end{equation} where $b, h_{ij}=h_{ji}:[0,T]\times \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ and $\sigma} \def\ess{\text{\rm{ess}}:[0,T] \times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\otimes \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$, $B_t$ is a $d$-dimensional $G$-Brwonian motion, and $\langle B^i,B^j\rangle_t$ stands for the mutual variation process of the $i$-th component $B^i_t$ and the $j$-th component $B^j_t$. To ensure the existence and uniqueness of the solution of \eqref{kerner} in $ M_G^2([0,T]; \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$, we assume \begin{equation}\label{Lip} \begin{split} & \|b(t, x)-b(t, y)\|+\sum_{i,j=1}^d \|h_{ij}(t, x)-h_{ij}(t, y)\|+\\|\sigma(t, x)-\sigma(t, y)\\|_{\rm HS}\leq K\|x-y\|, \end{split} \end{equation} for some constant $ K\geq0$ and all $t\in[0,T]$, $ x$, $y\in \mathbb{R}^d;$ see \cite[Theorem 1.2, p82]{peng4}. Now we recall from \cite{RW} the following notions for the path independence of additive functionals. \begin{defn}\label{def} For $ f=(f_{ij}):\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow\mathbb{S}^d$ and $g:\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$, the additive functional $(A_{s,t}^{f,g})_{0\leq s\leq t}$ is defined by \begin{equation}\label{eq2} A_{s,t}^{f,g}=\beta\int_s^t G(f)(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r+\aa\sum_{i,j=1}^d\int_s^t f_{ij}(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i,B^j\rangle_r +\int_s^t\langle g(r, X_r), \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\rangle, \end{equation} where $ \beta,\aa \in\mathbb{ R}$ are two parameters, $f=f^\ast$, and $(X_r)_{r\ge0}$ solves \eqref{kerner}. \end{defn} \begin{defn}\label{defn1} The additive functional $ A_{s,t}^{f,g}$ is said to be path independent, if there exists a function $ V:\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow\mathbb R} \def\ff{\frac} \def\ss{\sqrt $ such that for any $s\in [0,T]$ and any solution $(X_t)_{t\in[s,T]}$ to \eqref{kerner} from time $s$, it holds \begin{equation}\label{eq1} A_{s,t}^{f,g} =V(t,X_t)-V(s,X_s), \ t\in[s,T]. \end{equation} \end{defn} In terms of {\bf Definition} \ref{defn1}, the path independence of the additive functional $A_{s,t}^{f,g}$ means that $A_{s,t}^{f,g}$ depends only on $X_s$ and $X_t$ but not the path $(X_r)_{s<r< t},$ for any solution $(X_r)_{r\in[s,T]}$ to \eqref{kerner} from times $s$ and any $t\in(s, T]$. The aim of this paper is to provide sufficient and necessary characterizations for the path independence of the additive functional $A_{s,t}^{f,g}$. To see that \eqref{eq2} covers additive functionals investigated in existing references for the path independence under the linear probability space, let $\Theta$ in \eqref{GA} be a singleton: $\Theta={Q}$, and $\bar{\mathbb{E}}=\E$ be a linear expectation. Then the associated $G$-Brownian motion $B_t$ becomes the classical zero-mean normal distributed with covariance $Q$. Specially, let $\underline{\sigma}^2=\overline{\sigma}^2={\bf1}_{d\times d}$, i.e., $Q={\bf1}_{d\times d}$, $G(A) =\frac{1}{2}\mbox{trace}(A)$, $A\in \mathbb{S}^d$, we have $\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \langle B^i,B^j\rangle_r=\delta_{ij}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r$, where $\delta_{ij}$ is a indicative function, $1\leq i,j \leq d$, and \eqref{eq2} reduces to \begin{equation} A_{s,t}^{f,g}=\left(\aa+\frac{\beta}{2}\right)\sum_{i=1}^d\int_s^t f_{ii}(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r +\int_s^t\langle g(r, X_r), \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\rangle. \end{equation} Taking ${\bf{f}}(r, X_r)=\left(\aa+\frac{\beta}{2}\right)\sum_{i=1}^d f_{ii}(r, X_r)$, this goes back to the additive functional studied in \cite{RW}: \begin{equation} A_{s,t}^{f,g}:=\int_s^t {\bf{f}}(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r +\int_s^t\langle g(r, X_r), \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\rangle. \end{equation} In particular, when ${\bf{f}}=\frac{1}{2}\|g\|^2$, we have \begin{equation}\label{eq3} A_{s,t}^{\frac{1}{2}\|g\|^2,g} =\frac{1}{2}\int_s^t\|g\|^2(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r+\int_s^t\langle g(r, X_r),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\rangle , \ 0 \leq s\leq t, \end{equation} which corresponds to the Girsanov transform $\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\mathbb{Q}_{s,t}:=\exp\{{-A_{s,t}^{\frac{1}{2}\|g\|^2,g}}\}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\mathbb{P}$. To make the solution $X_t$ of \eqref{kerner} a martingale under $\mathbb{Q}_{s,t}$, we reformulate \eqref{kerner} as \begin{equation} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D X_t=\{b+ h\}(t,X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t+\langle\sigma(t,X_t),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t\rangle, \end{equation} where \begin{equation}\label{eq4} h(t,u):=\sum_{i=1}^dh_{ii}(t,u), ~~ (t,u)\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d. \end{equation} When $\sigma$ is invertible, taking $g=\sigma} \def\ess{\text{\rm{ess}}^{-1}(b+h)$ in \eqref{eq3}, we have \begin{equation} A_{s,t}^{\frac{1}{2}\|g\|^2,g}=\frac{1}{2}\int_s^t\|\sigma} \def\ess{\text{\rm{ess}}^{-1}(b+h)\|^2(r,X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r+\int_s^t\langle (\sigma} \def\ess{\text{\rm{ess}}^{-1}(b+h))(r,X_r),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\rangle , \ 0 \leq s\leq t. \end{equation} Then, by the Girsanov theorem, $(X_r)_{s \leq r\leq t}$ is a martingale under $\mathbb{Q}_{s,t}$, which fits well the requirement of risk netural measure in finance. The path independence of this particular additive functional has been investigated in \cite{Wang3, RW,Wang,Wang1,Wang2}. \begin{rem} \label{rem1} When $\alpha \neq 0$, \eqref{eq2} is equivalent to \begin{equation}\label{md} \begin{split} \alpha^{-1} A_{s,t}^{f,g}&=\alpha^{-1}\beta \int_s^t G(f)(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r+\sum_{i,j=1}^m\int_s^t f_{ij}(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i,B^j\rangle_r\\ &\quad +\int_s^t\langle \alpha^{-1}g(r, X_r),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\rangle. \end{split} \end{equation} So, in this case, the path independence of the additive functional \eqref{eq2} can be reduced to the case of $\alpha=1.$ However, the case for $\aa=0$ also includes interesting examples (see Example \ref{exa1} below), so it is reasonable to consider $A_{s,t}^{f,g}$ in \eqref{eq2} with two parameters $\alpha $ and $ \beta$. \end{rem} The remainder of the paper is organized as follows. In Section 2, following the line of \cite{Song2, Song}, we present a decomposition theorem for multidimensional $G$-semimartingales. In Section 3, we characterize the path independence of $A^{f,g}_{s,t}$ using nonlinear PDEs, so that main results in \cite{Wang3,Wang,Wang1,Wang2} are extended to the present nonlinear expectation setting. Finally, in Section 4, we provide an example to illustrate the main result for $\alpha=0$ as mentioned in Remark \ref{rem1}. \section{A Decomposition Theorem} This part is essentially due to \cite{Song2, Song}. Set $\delta_n(t):=\sum_{i=1}^{n-1}(-1)^i{\bf 1}_{(\frac{i}{n}, \frac{i+1}{n}]}(t)$, $t\in[0,T].$ For any $A, B\in \mathbb{S}^d$, let $(A,B)_{\rm HS}=\mbox{trace}(AB)$ and $\\|A\\|_{\rm HS}=\ss{(A,A)_{\rm HS}}$. Then $(\mathbb{S}^d, \langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\cdot,\cdot\>_{\rm HS}, \\|\cdot\\|_{\rm HS})$ is a Hilbert space; see e.g. \cite{RWu}. Let $\mbox{\mbox{spec}}(\cdot)$ be the spectrum of a matrix $\cdot$, and let $\<B\>_t=(\<B^i,B^j\>_t)_{ij}$. From now on, we consider \begin{equation}\label{G(A)} G(A):=\frac{1}{2}\sup _{\gamma\in [\underline{\sigma}, \bar{\sigma}]}\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\gamma^2,A\>_{\rm HS}, \ A\in\mathbb{S}^d, 0<\underline{\sigma}<\overline{\sigma} \ \rm{are \ two\ matrices\ in} \ \mathbb{S}^d. \end{equation} Consequently, $\underline{\sigma}^2<\frac{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D}{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t}\langle B\rangle_t\leq\bar{\sigma}^2,$ and \eqref{Gnon} holds for $\delta=\lambda_0(\underline{\sigma}^2)$, where $\lambda_0(\underline{\sigma}^2)=\min\{\ll\in\mbox{spec}(\underline\sigma} \def\ess{\text{\rm{ess}}^2)\}$. Let $ M_G^1([0,T]; \mathbb{S}^d) $ be all symmetric $d\times d$ matrices $\eta_t=(\eta_t^{ij})_{d\times d}$ with $\eta^{ij}\in M_G^1([0,T]), $ and $\\|\eta\\|_{M_G^1([0,T]; \mathbb{S}^d)}=\bar{\mathbb{E}}\int_{0}^{T}\\|\eta_{t}\\|_{\rm HS}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t.$ Let $c_0=\min\{\ll\in\mbox{spec}((\bar\sigma} \def\ess{\text{\rm{ess}}^2-\underline\sigma} \def\ess{\text{\rm{ess}}^2)/2)\}$, $C_0=\ff{1}{2}\\|\bar\sigma} \def\ess{\text{\rm{ess}}^2-\underline\sigma} \def\ess{\text{\rm{ess}}^2\\|_{\rm HS}$. To make the content self-contained, we cite from \cite{Song2} some well-known results and restated them as follows. \begin{lem}\label{lem01} {\rm Let $G$ be in \eqref{G(A)}. For any $\eta\in M_G^1([0,T]; \mathbb{S}^d)$, the limit $$\\|\eta\\|_{\mathbb{M}_G}:=\lim_{n\rightarrow}\def\l{\ell}\def\iint{\int \infty}\bar{\mathbb{E}} \int_0^T \delta_n(s)(\eta_s,\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B\rangle_s)_{\rm HS}$$ exists. Moreover, $\\|\cdot\\|_{\mathbb{M}_G}$ defines a norm on $M_G^1([0,T]; \mathbb{S}^d), $ and for any $0<\epsilon\leq c_0$, it holds that, $$\epsilon\\|\eta\\|_{M_{G_\epsilon}^1([0,T]; \mathbb{S}^d)}\leq \\|\eta\\|_{\mathbb{M}_G}\leq C_0\\|\eta\\|_{M_G^1([0,T]; \mathbb{S}^d)},$$ where $G_\epsilon(A):=\frac{1}{2}\sup_{\gamma\in[\underline{\sigma}_{\epsilon}, \bar{\sigma}_{\epsilon}]}\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\gamma^2,A\>_{\rm HS},$ $\underline{\sigma}_{\epsilon}^2:=\underline{\sigma}^2+\epsilon\,{\bf I}_{d\times d}$, and $\bar{\sigma}_{\epsilon}^2:=\bar{\sigma}^2-\epsilon\,{\bf I}_{d\times d}$. } \end{lem} With Lemma \ref{lem01} in hand, we have the following corollary which will play a crucial role in the analysis below. \begin{cor}\label{cor01} {\rm Let $G$ be in \eqref{G(A)}, and let $\eta\in M_G^1([0,T]; \mathbb{S}^d),$ $\zeta \in M_G^1([0,T])$. If $$\int_0^t (\eta_s,\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B\rangle_s)_{\rm HS}=\int_0^t \zeta_s\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,~~~t\in[0,T],$$ then $$\bar{\mathbb{E}} \int_0^T \\|\eta_s\\|_{\rm HS}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s=\bar{\mathbb{E}} \int_0^T \|\zeta_s\|\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s=0.$$} \end{cor} \begin{proof} According to Lemma \ref{lem01} and \cite[Theorem 3.3 (i)]{Song}, we deduce that \begin{equation} \\|\eta\\|_{\mathbb{M}_G}=\lim_{n\rightarrow}\def\l{\ell}\def\iint{\int \infty}\bar{\mathbb{E}} \int_0^T \delta_n(s)(\eta_s,\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B\rangle_s)_{\rm HS} =\lim_{n\rightarrow}\def\l{\ell}\def\iint{\int \infty}\bar{\mathbb{E}} \int_0^T \delta_n(s)\zeta_s\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s=0. \end{equation} Recall from Lemma \ref{lem01} that $\\|\eta\\|_{\mathbb{M}_G}$ is a norm, then, $c$-q.s., $\eta_t\equiv{\bf0}_{d\times d}$, a.e. $t\in [0,T]$. Therefore, $\bar{\mathbb{E}} \int_0^T \\|\eta_s\\|_{\rm HS}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s =0,$ which leads to $\bar{\mathbb{E}} \int_0^T \|\zeta_s\|\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s =0$. \end{proof} Consider the following It\^{o} process in $(\Omega_T,L_G^p(\Omega_T),\bar{\mathbb{E}})$ \begin{equation}\label{Peng} X_t=\int_0^t \xi_r\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r+\sum_{i,j=1}^d\int_0^t \eta_r^{ij} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i,B^j\rangle_r +\sum_{i=1}^d\int_0^t \zeta_r^i \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r^i,~~~t\geq0, \end{equation} where $\xi, \eta^{ij} \in M_G^1([0,T];\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$ with $\eta^{ij} =\eta^{ji}$, and $\zeta^i\in H_G^1([0,T];\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$. Now we can state the following decomposition theorem. \begin{thm}\label{thm} \label{Thm2.3} For $G$ in \eqref{G(A)} and let $X_t $ be in \eqref{Peng}. Then $X_t={\bf0}_d$ for all $t\in[0,T] $ if and only if on $\Omega_T\times [0,T]$ it holds $c\times \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t-q.s. \times a.e.$, $\xi_t={\bf0}_d, \eta_t^{ij}={\bf0}_{d}, \zeta_t^i={\bf0}_{d}$, $i,j=1, \cdot\cdot\cdot,d.$ \end{thm} \begin{proof} The proof of the sufficiency is trivial, it suffices to prove the necessity. Assume $X_t={\bf0}_{d}$ for $t\in[0,T]$. Then \eqref{Peng} is equivalent to \begin{equation}\label{reduced} \int_0^t\xi_s^k\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s+ \sum_{i,j=1}^d\int_0^t \eta_r^{kij} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i,B^j\rangle_r +\sum_{j=1}^d\int_0^t \zeta_s^{kj} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s^j={\bf0}_{d}, ~k=1,\cdots,d, \ t\in[0,T], \end{equation} where $\xi^k_\cdot$ (resp. $\eta_\cdot^{kij})$ denotes the $k$-th component of the column vector $\xi$ (resp. $\eta_\cdot^{ij})$. Taking quadratic processes w.r.t. $\int_0^\cdot \zeta_s^{ki} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s^i $ on both side of \eqref{reduced}, we deduce that \begin{equation} \begin{split} {0}&= \sum_{i,j=1}^d \left\langle \int_0^\cdot \zeta_s^{kj} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s^j,\int_0^\cdot \zeta_s^{ki} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s^i \right\rangle_t = \left\langle \sum_{i=1}^d \int_0^\cdot \zeta_s^{ki }\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_s^i \right\rangle_t\\ &= \sum_{i,j=1}^d\int_0^t \zeta_s^{kj} \zeta_s^{ki} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i, B^j\rangle_s=\int_0^t ((\zeta_s^k)^\zeta_s^k, \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B \rangle_s)_{\rm HS}\\ &=\int_0^t \langle \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B \rangle_s(\zeta_s^k )^, (\zeta_s^k)^\rangle \geq\int_0^t \langle \underline{\sigma}^2 (\zeta_s^k )^, (\zeta_s^k)^\rangle \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \geq0 \end{split} \end{equation} with $\zeta^k:=(\zeta^{k1},\cdots,\zeta^{kd})$. Since $\underline{\sigma}^2>0,$ this implies $\zeta_t={\bf0}_{d\times d}$, a.e. $t\in [0,T]$. It remains to show that $\xi_t={\bf0}_{d}$ and $\eta_t={\bf0}_{d}$. In fact, since $\zeta_t={\bf0}_{d}$, we have \begin{equation} \int_0^t-\xi_s^k\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s= \sum_{i,j=1}^m\int_0^t \eta_s^{kij} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i,B^j\rangle_s= \int_0^t(\eta_s^k, \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B\rangle_s)_{\rm HS}, \ k=1, \cdot\cdot\cdot, d,~~t\in[0,T] \end{equation} with $\eta_\cdot^k=(\eta_\cdot^{kij})_{ij}$. By Corollary \ref{cor01}, this implies \begin{equation} \bar{\mathbb{E}}\int_0^T\|\xi_s^k\|\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s= \bar{\mathbb{E}} \int_0^T\\|\eta_s^k\\|_{\rm HS} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s=0, \ k=1, \cdot\cdot\cdot, d. \end{equation} Thus, we conclude that $c$-q.s. for a.e. $t\in [0,T],$ $\eta_t^k= {\bf0}_{d\times d}$ and $\xi_t^k=0, \ k=1, \cdot\cdot\cdot, d.$ Therefore, $\xi_t=\eta_t^{ij}={\bf0}_{d}$. \end{proof} \section{Characterization of Path Independence} The main result of the paper is the following. \begin{thm}\label{theorem1} Let $G$ be in \eqref{G(A)}. Then $A_{s,t}^{f,g}$ is path independent in the sense of \eqref{eq1} for some $V\in C^{1,2}(\mathbb{R}_{+}\times\mathbb{R}^d)$ if and only if \begin{equation}\label{thm2} \begin{cases} \frac{\partial }{\partial t}V(t,x) =\beta G(f)(t,x)-\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,b\>(t,x),\\ \aa f_{ij}(t,x)=\Big(\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,h_{ij}\>+ \frac{1}{2}\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\sigma} \def\ess{\text{\rm{ess}}_i,(\nabla} \def\pp{\partial} \def\E{\mathbb E^2V)\sigma} \def\ess{\text{\rm{ess}}_j\> \Big)(t,x), \\ g(t,x)=(\sigma} \def\ess{\text{\rm{ess}}^\nabla} \def\pp{\partial} \def\E{\mathbb E V)(t,x), \ \ \ (t,x) \in [0,T]\times \mathbb{R}^d, \end{cases} \end{equation} where $i,j=1,\cdots,d$, and $\sigma_i$ stands for the $i$-th column of $\sigma$. \end{thm} \begin{proof} We first prove the necessity. For any $(s,x)\in[0,T]\times \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$, let $(X_t)_{t\geq s} $ solves \eqref{kerner} with $X_s=x$. Since $(A_{s,t}^{f,g})_{t\in[s,T]}$ is path independent in the sense of \eqref{eq1}, it follows that \begin{eqnarray}\label{2} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D V(t,X_t)&=&\beta G(f)(t,X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t+\aa\sum_{i,j=1}^d f_{ij}(t, X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i,B^j\rangle_t\\ \nonumber &&+\langle g(t, X_t),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t\rangle, \ t\in [s,T]. \end{eqnarray} On the other hand, by It\^{o}'s formula, we derive that \begin{equation}\label{1} \begin{split} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D V(t,X_t)&=\left(\frac{\partial }{\partial t}V+\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,b\> \right)(t,X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t +\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma(\sigma} \def\ess{\text{\rm{ess}}^\nabla} \def\pp{\partial} \def\E{\mathbb E V)(t,X_t),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t\> \\ &\quad+\sum_{i,j=1}^d\Big(\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,h_{ij}\>+ \frac{1}{2}\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\sigma} \def\ess{\text{\rm{ess}}_i,(\nabla} \def\pp{\partial} \def\E{\mathbb E^2V)\sigma} \def\ess{\text{\rm{ess}}_j\> \Big)(t,X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \langle B^i, B^j\rangle_t, \ t\in [s,T]. \end{split} \end{equation} Since coeffieients $b,h$ and $\sigma$ satisfy the Lipschitz condition in \eqref{Lip}, and the solution of \eqref{kerner} satisfies $X_t\in M_G^2([0,T]; \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$, it's not difficult to verify $\left(\frac{\partial }{\partial t}V+\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,b\> \right)(t,X_t)\in M_G^1([0,T]) $, $\Big(\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,h_{ij}\>+ \frac{1}{2}\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\sigma} \def\ess{\text{\rm{ess}}_i,(\nabla} \def\pp{\partial} \def\E{\mathbb E^2V)\sigma} \def\ess{\text{\rm{ess}}_j\> \Big)(t,X_t)\in M_G^1([0,T])$, and $(\sigma} \def\ess{\text{\rm{ess}}_i^\ast\nabla} \def\pp{\partial} \def\E{\mathbb E V)(t,X_t)\in H_G^1([0,T])$, thus hypotheses of Theorem \ref{Thm2.3} are satisfied. Combining \eqref{2} and \eqref{1}, and applying Theorem \ref{thm} for the process $V(t,X_t)$, we obtain $c$-q.s. for a.e. $t\in[s,T]$, \begin{equation}\label{3} \begin{cases} \Big(\frac{\partial }{\partial t}V+\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,b\> \Big)(t,X_t)= \beta G(f)(t,X_t),\\ \aa f_{ij}(t,X_t)=\Big(\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,h_{ij}\>+ \frac{1}{2}\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\sigma} \def\ess{\text{\rm{ess}}_i,(\nabla} \def\pp{\partial} \def\E{\mathbb E^2V)\sigma} \def\ess{\text{\rm{ess}}_j\> \Big)(t,X_t), \\ g(t,X_t)=(\sigma} \def\ess{\text{\rm{ess}}^\nabla} \def\pp{\partial} \def\E{\mathbb E V)(t,X_t). \end{cases} \end{equation} Since all terms in \eqref{3} are continuous in $t$, these equations hold $c$-q.s. at $t=s$, so by $X_s=x$, we have \begin{equation}\label{4} \begin{cases} \Big(\frac{\partial }{\partial t}V+\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,b\> \Big)(s,x)= \beta G(f)(s,x),\\ \aa f_{ij}(s,x)=\Big(\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,h_{ij}\>+ \frac{1}{2}\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\sigma} \def\ess{\text{\rm{ess}}_i,(\nabla} \def\pp{\partial} \def\E{\mathbb E^2V)\sigma} \def\ess{\text{\rm{ess}}_j\> \Big)(s,x), \\ g(s,x)=(\sigma} \def\ess{\text{\rm{ess}}^\nabla} \def\pp{\partial} \def\E{\mathbb E V)(s,x). \end{cases} \end{equation} Due to the arbitrariness of $s$ and $x$, we prove \eqref{thm2}. Next, for the sufficiency, taking advantage of \eqref{thm2}, we deduce from \eqref{1} that \eqref{2} holds true. By taking stochastic integration we prove \eqref{eq1}, and therefore complete the proof. \end{proof} Let us comparison this result with known ones in the linear expectation setting. \begin{rem}\label{rem} Comparing with the Girsanov transform in the linear expectation setting as mentioned in Introduction, we take for instance $\alpha=1, \beta=-1$, $h_{ij}=0$ and \begin{equation}\label{ast} f_{ii}=\frac{1}{d}\|g\|^2, \ 1\leq i\leq d. \end{equation} When $\underline{\sigma}^2=\bar{\sigma}^2={\bf1}_{d\times d}$, this goes back to the classic linear expectation, $(B_t)_{t\ge0}$ is a $d$-dimensional standard Brownian motion defined on the probability space $(\Omega,\F,\P)$, we have $\langle B^i,B^j \rangle_r=\delta_{ij}r$, and \begin{equation}\label{G} G(f)=\frac{\|g\|^2}{2d}\mbox{trace}[{\bf{1}}_{d\times d}]=\frac{{\|g\|^2}}{2}. \end{equation} So \begin{eqnarray}\label{A} A_{s,t}^{{ f},g}:&=&\beta \int_s^t G(f)(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r+\alpha\sum_{i,j=1}^d\int_s^t f_{ij}(r, X_r)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\langle B^i,B^j\rangle_r\\ \nonumber &&+\int_s^t\langle g(r, X_r),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\rangle\\ \nonumber &=& \frac{1}{2}\int_s^t \|g(r, X_r)\|^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r +\int_s^t\langle g(r, X_r),\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_r\rangle \end{eqnarray} gives the weighted probability $\exp\{{-A_{s,t}^{f,g}}\}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\mathbb{P}$ in the Girsanov theorem. By taking $\alpha=1, \beta=-1$, $h_{ij}=0$ and $f$ in \eqref{ast}, the assertion of Theorem \ref{theorem1} becomes that $A_{s,t}^{f,g}$ is path independent in the sense of \eqref{eq1} for some $V\in C^{1,2}(\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\times\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d;\mathbb R} \def\ff{\frac} \def\ss{\sqrt)$ if and only if \begin{equation}\label{8} \begin{cases} \frac{\partial }{\partial t}V(t,x) =-\frac{1}{2} G\Big((\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\sigma} \def\ess{\text{\rm{ess}}_i,(\nabla} \def\pp{\partial} \def\E{\mathbb E^2V)\sigma} \def\ess{\text{\rm{ess}}_j\> )_{1\leq i,j\leq d}\Big)(t,x)-\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E V,b\>(t,x),\\ f_{ij}(t,x)=\frac{1}{2}\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\sigma} \def\ess{\text{\rm{ess}}_i,(\nabla} \def\pp{\partial} \def\E{\mathbb E^2V)\sigma} \def\ess{\text{\rm{ess}}_j\> (t,x), \\ g(t,x)=(\sigma} \def\ess{\text{\rm{ess}}^\nabla} \def\pp{\partial} \def\E{\mathbb E V)(t,x), \ \ \ (t,x) \in [0,T]\times \mathbb{R}^d. \end{cases} \end{equation} It is easy to see that this generalizes the main results derived in \cite{Wang3, Wang,Wang1,Wang2} where $h\equiv0$ and $g$ is given by $\sigma} \def\ess{\text{\rm{ess}}^{-1}b$, under additional condition ensuring the existence of $\sigma} \def\ess{\text{\rm{ess}}^{-1}b$, i.e., $b$ takes value in $\{ \sigma} \def\ess{\text{\rm{ess}} v: v\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\}$. However, since $G$ is a linear function in the linear expectation case, Theorem \ref{theorem1} does not directly apply to existing results, but extends them to the non-degenerate $G$-setting. \end{rem} Moreover, the nonlinear PDE included in \eqref{theorem1} covers the $G$-heat equation as a special example. \begin{rem} When $h=b=0$, $\alpha=1$, and $\beta=-2$, the PDE in \eqref{thm2} for $V$ reduces to the following $G$-heat equation $$\frac{\partial }{\partial t} {V}(t,x) +G\Big(\big(\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\sigma} \def\ess{\text{\rm{ess}}_i,(\nabla} \def\pp{\partial} \def\E{\mathbb E^2V)\sigma} \def\ess{\text{\rm{ess}}_j\> (t,x)\big)_{1\leq i,j\leq m}\Big)=0,$$ which is one of main motivations for the study of $G$-Brownian motion. \end{rem} \section{An Example with $\alpha=0$} Now we provide an example to demonstrate our main result for $\alpha=0$. As indicated in Remark \ref{rem1} that when $\alpha\neq0 $ the study can be reduced to $\alpha=1.$ \begin{exa} \label{exa1} Let $d=1$, $\alpha=0,$ and $ \beta=2.$ By Theorem \ref{theorem1}, $A_{s,t}^{f,g}$ is path independent if and only if \begin{equation} \label{a} \begin{cases} f(t,x)= \frac{1}{2}G^{-1}\left(\frac{\partial V}{\partial t}+b\frac{\partial V}{\partial x}\right)(t,x),\\ \left(h\frac{\partial V}{\partial x}\right)(t,x)+\frac{1}{2}\left(\sigma} \def\ess{\text{\rm{ess}}^2\frac{\partial^2 V}{\partial x^2}\right)(t,x)=0,\\ g(t,x)=\left(\sigma} \def\ess{\text{\rm{ess}}^\frac{\partial V}{\partial x}\right)(t,x), \ \ \ (t,x) \in [0,T]\times \mathbb{R}. \end{cases} \end{equation} We may solve $V$ by using $\mathcal{L}_t$-Harmonic function: \begin{equation}\label{ex1} \mathcal{L}_tV_0(x)=0, \ V_0\in C^{1,2}(\mathbb R} \def\ff{\frac} \def\ss{\sqrt\rightarrow}\def\l{\ell}\def\iint{\int\mathbb R} \def\ff{\frac} \def\ss{\sqrt), \ t\geq0, \end{equation} where $\mathcal{L}_t=h(t,x) \frac{\partial }{\partial x}+\frac{1}{2}\sigma} \def\ess{\text{\rm{ess}}^2(t,x)\frac{\partial^2 }{\partial x^2}$. For any $\mathcal{L}_t$-Harmonic function $V_0$, $t\geq0,$ let $V(t,x)=\varphi(t)V_0(x)$ for some $\varphi\in C^{1,2}(\mathbb R} \def\ff{\frac} \def\ss{\sqrt_+\rightarrow}\def\l{\ell}\def\iint{\int\mathbb R} \def\ff{\frac} \def\ss{\sqrt)$. Then $V$ solves the above PDE in \eqref{a}. Therefore, $A_{s,t}^{f,g}$ is path independent if \begin{equation}\label{ex2} \begin{cases} f(t,x)= \frac{1}{2}G^{-1}\left(\varphi'(t)V_0(x) +b(t,x)\varphi(t) V_0'(x)\right),\\ g(t,x)=\sigma} \def\ess{\text{\rm{ess}}(t,x)\varphi(t) V_0'(x), \ \ \ (t,x) \in [0,T]\times \mathbb{R}. \end{cases} \end{equation} To present specific choices of $V_0$, let $h$ and $\sigma} \def\ess{\text{\rm{ess}}$ do not depend on $t$. Then \eqref{ex1} becomes \begin{equation} h(x)V_0'(x)+\frac{1}{2}\sigma} \def\ess{\text{\rm{ess}}^2(x)V_0''(x)=0. \end{equation} When $\sigma} \def\ess{\text{\rm{ess}}^2(x)\neq 0,$ this is equivalent to \begin{equation} V_0''(x)=-2\frac{h(x)}{\sigma} \def\ess{\text{\rm{ess}}^2(x)}V_0'(x). \end{equation} Thus, \begin{equation} V_0(x)=V_0(0)+V_0'(0)\int_0^x\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-2\int_0^u\frac{h(r)}{\sigma} \def\ess{\text{\rm{ess}}^2(r)}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D u. \end{equation} In particular, when $\sigma} \def\ess{\text{\rm{ess}}(x)=1, h(x)=x$, we have \begin{equation} V_0(x)=V_0(0)+V_0'(0)\int_0^x\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-u^2}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D u, \end{equation} which is related to the Gaussian distribution. \end{exa} \paragraph{Acknowledgement.} The authors are grateful to Professor Feng-Yu Wang for his guidance, valuable suggestion and comments on earlier versions of the paper, as well as Professor Yongsheng Song for his patient help and corrections. \begin} \def\beq{\begin{equation}} \def\F{\scr F{thebibliography}{99} \bibitem{B08} Bishwal J P N, Parameter Estimation in Stochastic Differential Equations, Springer, Berlin, 2008. \bibitem{BS} Black F, Scholes M, The pricing of options and corporate liabilities, J Polit Econ, 1973, 81: 637--654. \bibitem{BR} Bai X P, Buckdahn R, Inf-convolution of $G$-expectations, Sci China Math, 2010, 53: 1957-1970. \bibitem{15} Denis L, Hu M S, Peng S G, Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion pathes, Potential Anal, 2011, 34: 139--161. \bibitem{FWZ} Feng C R, Wu P Y, Zhao H Z, Ergodicity of Invariant Capacity, 2018, arXiv: 1806.03990. \bibitem{FOZ} Feng C R, Qu B Y, Zhao H Z, A Sufficient and Necessary Condition of PS-ergodicity of Periodic Measures and Generated Ergodic Upper Expectations, 2018, arXiv: 1806.03680. \bibitem{GRR} Gu Y F, Ren Y, Sakthive R, Square-mean pseudo almost automorphic mild solutions for stochastic evolution equations driven by $G$-Brownian motion, Stoch Anal Appl, 2016, 34: 528--545. \bibitem{HJP} Hu M S, Ji S L, Peng S G, Song Y S, Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by $G$-Brownian motion, Stochastic Process Appl, 2014, 124: 1170--1195. \bibitem{HJ} Hu M S, Ji S L, Dynamic programming principle for stochastic recursive optimal control problem driven by a $G$-Brownian motion, Stochastic Process Appl, 2017, 127: 107--134. \bibitem{HJY} Hu M S, Ji S L, Yang S Z, A stochastic recursive optimal control problem under the $G$-expectation framework, Appl Math Optim, 2014, 70: 253--278. \bibitem{peng2} Peng S G, $G$-Brownian motion and dynamic risk measures under volatility uncertainty, 2007, arXiv: 0711.2834v1. \bibitem{peng1} Peng S G, $G$-expectation, $G$-Brownian motion and related stochastic calculus of It\^{o} type, in: Stochastic Analysis and Applications, in: Abel Symp., vol. 2, Springer, Berlin, 2007, pp.541--567. \bibitem{peng4} Peng S G, Nonlinear expectations and stochastic calculus under uncertainty, 2010, arXiv: 1002.4546v1. \bibitem{P} Peng S G, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math, 2017, 47: 1223--1254, doi: 10.1360/N012016-00209. \bibitem{Song2} Peng S G, Song Y S, Zhang J F, A complete representation theorem for $G$-martingales, Stochastics, 2014, 86: 609--631. \bibitem{Wang3} Wu J L, Yang W, On stochastic differential equations and a generalised Burgers equation, pp 425--435 in Stochastic Analysis and Its Applications to Finance: Essays in Honor of Prof. Jia-An Yan (eds. 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\begin{document} \title{\bf Evolutionary {\boldmath $\Gamma$} \begin{abstract} A notion of evolutionary $\Gamma$-convergence of weak type is introduced for sequences of operators acting on time-dependent functions. This extends the classical definition of $\Gamma$-convergence of functionals due to De Giorgi. The $\Gamma$-compactness of equi-coercive and equi-bounded sequences of operators is proved. Applications include the {\it structural compactness and stability\/} of quasilinear flows for pseudo-monotone operators. \end{abstract} \noindent{\bf Keywords:} $\Gamma$-convergence \noindent{\bf AMS Classification (2000):} 35K60, 47H05, 49J40, 58E. \section{Introduction} \label{intro} \noindent In this note we deal with a notion of {\it evolutionary $\Gamma$-convergence of weak type,\/} which extends De Giorgi's basic definition; see \cite{DeFr} and e.g.\ the monographs \cite{At}, \cite{Bra1}, \cite{Bra2}, \cite{Da}. Here we formulate $\Gamma$-convergence for operators (rather than functionals) that act on time-dependent functions ranging in a Banach space $X$. This definition of evolutionary $\Gamma$-convergence is quite different from that of \cite{SaSe} and from that of \cite{DaSa}, \cite{Mi1}, \cite{Mi2}. In those works $\Gamma$-convergence is set for almost any $t\in {}]0,T[$, whereas here it is assumed just in the weak topology of $L^1(0,T)$, and thus is a substantially weaker notion. This definition fits the rather general framework of $\bar\Gamma$-convergence, defined in Chap.~16 of \cite{Da}; see also references therein. However \cite{Da} does not encompass Theorem~\ref{teo.comp} ahead, which is the main achievement of this work and is at the basis of the results of \cite{Vi17}; see also the survey \cite{ViLinc}. After results of Brezis and Ekeland \cite{BrEk}, Nayroles \cite{Na} and Fitzpatrick \cite{Fi}, a class of first-order quasilinear flows can be given a variational formulation, without assuming monotonicity of the operator. In the parallel work \cite{Vi17}, Theorem~\ref{teo.comp} is applied to prove structural compactness and structural stability of the corresponding Cauchy problem. This applies to a large number of PDEs of mathematical physics, see \cite{Vi17}. We proceed as follows. In Section~\ref{sec.evol} we define evolutionary $\Gamma$-convergence of weak type. In Theorem~\ref{teo.comp} we prove compactness for this convergence. In Theorem~\ref{teo.comp'} we apply Theorem~\ref{teo.comp} to the structural compactness and stability of nonmonotone flows. \section{Definition of evolutionary {\boldmath $\Gamma$}-convergence of weak type} \label{sec.evol} \noindent In this section we extend De Giorgi's basic notion of $\Gamma$-convergence to operators (rather than functionals) that act on time-dependent functions ranging in a Banach space $X$. \noindent{\bf Functional set-up.} Let $X$ be a real separable and reflexive Banach space, $p\in [1,+\infty[$, $T>0$, and \begin{equation}\label{eq.evol.mu} \begin{split} &\hbox{ $\mu$ be a positive and finite measure on $]0,T[$, } \\ &\hbox{ that is absolutely continuous w.r.t.\ the Lebesgue measure. } \end{split} \end{equation} Examples of interest will be the Lebesgue measure of $]0,T[$, and $\mu$ such that $d\mu(t) = (T-t) \, dt$. Let us set \begin{equation}\label{eq.evol.defgamma.1} \begin{split} &L^p_\mu(0,T) = \Big\{\mu\hbox{-measurable }v: {}]0,T[{}\to \erre: \!\!\int_0^T \! \|v(t)\|^p \, d\mu(t)< +\infty \Big\}, \\ &L^p_\mu(0,T;X) = \Big\{\mu\hbox{-measurable }w: {}]0,T[{}\to X: \!\!\int_0^T \! \\|w(t)\\|_X^p \, d\mu(t)< +\infty \Big\}, \end{split} \end{equation} and equip either space with the respective graph norm. Let us equip $L^p_\mu(0,T;X)$ with a topology $\tau$ that is intermediate between the weak and the strong topology. \footnote{ The generality of this topology is instrumental to the application to structural stability, see \cite{Vi17}. A reader interested just in evolutionary $\Gamma$-convergence might go through this section assuming that $\mu$ is the Lebesgue measure and that $\tau$ is the weak topology. } For any operator $\psi: L^p_\mu(0,T;X)\to L^1_\mu(0,T):w\mapsto \psi_w$, let us set \begin{equation}\label{eq.evol.crochet} [\psi,\xi](w) = \int_0^T \psi_w(t) \, \xi(t) \, d\mu(t) \qquad\forall w\in L^p_\mu(0,T;X), \forall \xi\in L^\infty(0,T). \end{equation} \noindent{\bf Evolutionary $\Gamma$-convergence of weak type.} Let $\{\psi_n\}$ be a bounded sequence of operators $L^p_\mu(0,T;X)\to L^1_\mu(0,T)$; by this we mean that, for any bounded subset $A$ of $L^p_\mu(0,T;X)$, the set $\{\psi_{n,w}: w\in A, n\in \enne\}$ is bounded in $L^1_\mu(0,T)$. If $\psi$ is also an operator $L^p_\mu(0,T;X)\to L^1_\mu(0,T)$, we shall say that \begin{equation}\label{eq.evol.defgamma.1+} \begin{split} &\hbox{ $\psi_n$ sequentially $\Gamma$-converges to }\psi \\ &\hbox{ in the topology $\tau$ of $L^p_\mu(0,T;X)$ and } \\ &\hbox{ in the weak topology of }L^1_\mu(0,T) \end{split} \end{equation} if and only if, denoting by $L^\infty_+(0,T)$ the cone of the nonnegative functions of $L^\infty(0,T)$, \begin{equation}\label{eq.evol.defgamma.2} [\psi_n, \xi] \mbox{ sequentially $\Gamma\tau$-converges to $[\psi,\xi]$ in } L^p_\mu(0,T;X), \; \forall \xi\in L^\infty_+(0,T). \end{equation} We shall say that a sequence {\it $\Gamma\tau$-converges\/} if it $\Gamma$-converges with respect to $\tau$, that a functional is {\it $\tau$-lower semicontinuous\/} if it is lower semicontinuous with respect to $\tau$, and so on. By the classical definition of sequential $\Gamma$-convergence, \eqref{eq.evol.defgamma.2} means that for any $\xi\in L^\infty_+(0,T)$ \begin{equation}\label{eq.evol.defgamma.3} \begin{split} &\forall w\in L^p_\mu(0,T;X), \forall \hbox{ sequence $\{w_n\}$ in }L^p_\mu(0,T;X), \\ &\hbox{if \ $w_n\to\!\!\!\!\!\!{}_{_\tau}\;\; w$ in $L^p_\mu(0,T;X)$ \ then \ } \liminf_{n\to+ \infty} \; [\psi_n,\xi](w_n) \ge [\psi,\xi](w), \end{split} \end{equation} \vskip-0.4truecm \begin{equation}\label{eq.evol.defgamma.4} \begin{split} &\forall w\in L^p_\mu(0,T;X), \exists\hbox{ sequence $\{w_n\}$ of $L^p_\mu(0,T;X)$ such that } \\ &w_n\to\!\!\!\!\!\!{}_{_\tau}\;\; w \hbox{ in $L^p_\mu(0,T;X)$ \ and \ } \lim_{n\to +\infty} \; [\psi_n,\xi](w_n) = [\psi,\xi](w). \end{split} \end{equation} By the properties of ordinary $\Gamma$-convergence, \eqref{eq.evol.defgamma.2} entails that \begin{equation}\label{eq.evol.defgamma.5} [\psi, \xi] \hbox{ is sequentially $\tau$-lower semicontinuous in }L^p_\mu(0,T;X), \; \forall \xi\in L^\infty_+(0,T). \end{equation} \begin{remarks}\rm (i) This definition of {\it evolutionary $\Gamma$-convergence\/} is not equivalent either to that of \cite{SaSe} or to that of \cite{DaSa}, \cite{Mi1}, \cite{Mi2}. In those works $\Gamma$-convergence is actually assumed for almost any $t\in {}]0,T[$, whereas here it is just weak in $L^1_\mu(0,T)$. (ii) The present definition is based on testing the sequence $\{\psi_{n,w}\}$ on functions of time, and may equivalently be reformulated in terms of set-valued functions as follows. Denoting by ${\cal L}(0,T)$ the $\sigma$-algebra of the Lebesgue-measurable subsets of $]0,T[$, \eqref{eq.evol.defgamma.2} is tantamount to \begin{equation}\label{eq.evol.defgamma.2'} \int_A \psi_{n,w}(t) \, d\mu(t) \to \int_A \psi_w(t) \, d\mu(t) \qquad\forall A\in {\cal L}(0,T). \end{equation} As the elements of ${\cal L}(0,T)$ are in one-to-one correspondence with the characteristic functions of $L^\infty(0,T)$, this equivalence can be checked by mimicking the argument based on the Lusin theorem, that we use in the proof below. (iii) By restating the definition \eqref{eq.evol.defgamma.1+}, \eqref{eq.evol.defgamma.2} as in the previous remark, it fits the rather general framework of $\bar\Gamma$-convergence, that is defined in Chap.~16 of \cite{Da}, see also references therein. The results of that monograph however do not encompass the theorem of $\Gamma$-compactness of the next section. (iv) Although we consider generic operators $\psi: L^p(0,T;X)\to L^1(0,T)$, our main concern is for superposition (i.e., Nemytski\u{\i}-type) operators of the form \begin{equation}\label{eq.evol.super} \begin{split} &\psi_w(t) = \varphi(t,w(t)) \qquad\forall w\in L^p_\mu(0,T;X),\hbox{ for a.e.\ }t\in {}]0,T[, \\ &\varphi: {}]0,T[{} \times X\to \erre^+ \hbox{ \ being a {\it normal function.\/} } \end{split} \end{equation} (By this we mean that $\varphi$ is globally measurable and $\varphi(t,\cdot)$ is lower semicontinuous for a.e.\ $t\in {}]0,T[$.) $\Box$ \end{remarks} \section{Evolutionary {\boldmath $\Gamma$}-compactness of weak type} \label{sec.comp} \noindent In this section we prove a theorem of compactness for the evolutionary $\Gamma$-convergence that we just defined. This will be based on a result of Hiai \cite{Hi}, that we now display. As a preparation, let us say that a functional $\Phi: L^p_\mu(0,T;X)\to \erre$ is invariant by translations if, denoting by $\widetilde w$ the function obtained by extending $w$ to $\erre$ with null value, \begin{equation}\label{eq.evol.invar} \begin{split} &\Phi\big(\widetilde w(\cdot +s)\big\|_{[0,T]}\big) = \Phi(w) \\[1mm] &\forall w\in L^p_\mu(0,T;X), \forall s>0 \text{ such that $w=0$ a.e.\ in }]s,T[. \end{split} \end{equation} The next result will play a role in the present analysis. \begin{lemma} [\cite{Hi}] \label{lemma.Hiai} Let $X$ be a real separable Banach space and $p\in [1,+\infty[$. Let a functional $\Phi: L^p_\mu(0,T;X)\to \erre\cup \{+\infty\}$ ($\Phi \not\equiv +\infty$) be lower semicontinuous and also {\rm additive,\/} in the sense that \begin{equation}\label{eq.evol.add} \begin{split} &\forall w_1,w_2\in L^p_\mu(0,T;X), \\ &\mu\big( \{t\in {}]0,T[{}: w_1(t)w_2(t) \not= 0\} \big) =0 \quad\Rightarrow\quad \Phi(w_1 +w_2) = \Phi(w_1) + \Phi(w_2). \end{split} \end{equation} \indent Then there exists a normal function $\varphi: {}]0,T[{} \times X\to \erre\cup\{+\infty\}$ such that \begin{eqnarray} \begin{split}\label{aaa} &\varphi(t,\cdot) \not\equiv +\infty \quad\forall t\in ]0,T[, \\ &\varphi(\cdot,0) =0 \quad\hbox{ a.e.\ in }{}]0,T[, \\ &\Phi(w) = \int_0^T \varphi(t, w(t)) \, d\mu(t) \qquad\forall w\in L^p_\mu(0,T;X). \end{split} \end{eqnarray} \indent Moreover, if $\Phi$ is convex then $\varphi(t,\cdot)$ is also convex for a.e.\ $t\in{}]0,T[$, and the function $\varphi$ is unique, up to modification on sets of the form $N\times X$ with $\mu(N)=0$. \\ \indent Finally, if the functional $\Phi$ is invariant by translations, then $\varphi$ does not depend on $t$. \end{lemma} This lemma may be compared e.g.\ with Section~2.4 of \cite{But}, which however deals with a finite-dimensional space $X$. We are now ready to state and prove the main result of this note. \begin{theorem} \label{teo.comp} Let $X$ be a real separable and reflexive Banach space, $p\in [1,+\infty[$, $T>0$, and $\{\varphi_n\}$ be a sequence of normal functions ${}]0,T[{} \times X\to \erre^+$. Assume that this sequence is equi-coercive and equi-bounded, in the sense that \begin{equation}\label{eq.evol.equibc} \begin{split} &\exists C_1,C_2,C_3 >0: \forall n,\hbox{for a.e.\ }t\in {}]0,T[, \forall w\in X, \\ &C_1 \\|w\\|_X^p\le \varphi_n(t,w) \le C_2\\|w\\|_X^p +C_3, \end{split} \end{equation} and that \begin{equation}\label{eq.evol.nul} \varphi_n(t,0) =0 \qquad\hbox{ for a.e.\ }t\in {}]0,T[,\forall n. \end{equation} \indent Let $\mu$ fulfill \eqref{eq.evol.mu}. Let $\tau$ be a topology on $L^p_\mu(0,T;X)$ that either coincides or is finer than the weak topology, and such that \begin{equation}\label{eq.evol.gamcom} \begin{split} &\hbox{ for any sequence $\{F_n\}$ of functionals } L^p_\mu(0,T;X)\to \erre^+\cup \{+\infty\}, \\ &\hbox{ if \ }\sup_{n\in\enne} \big\{\\|w\\|_{L^p_\mu(0,T;X)}: w\in L^p_\mu(0,T;X), F_n(w) \le C \big\}<+\infty, \\ &\hbox{ then \ $\{F_n\}$ has a sequentially $\Gamma\tau$-convergent subsequence. } \end{split} \end{equation} \indent Then there exists a normal function $\varphi: {}]0,T[{} \times X\to \erre^+$ such that $\varphi(\cdot,0) =0$ a.e.\ in $]0,T[$, and such that, defining the operators $\psi,\psi_n: L^2_\mu(0,T;X)\to L^1_\mu(0,T)$ for any $n$ as in \eqref{eq.evol.super}, possibly extracting a subsequence, \begin{equation}\label{eq.evol.tesi} \begin{split} &\hbox{ $\psi_n$ sequentially $\Gamma$-converges to }\psi \\ &\hbox{ in the topology $\tau$ of $L^p_\mu(0,T;X)$ and } \\ &\hbox{ in the weak topology of $L^1_\mu(0,T)$ (cf.\ \eqref{eq.evol.defgamma.1+}). } \end{split} \end{equation} \indent Moreover, if $\varphi_n$ does not depend on $t$ for any $n$, then the same holds for $\varphi$. \end{theorem} \noindent{\bf Proof.\/} For the reader's convenience, we split this argument into several steps. (i) First we show that, denoting by $C^0_+([0,T])$ the cone of the nonnegative functions of $C^0([0,T])$, \begin{equation}\label{eq.evol.conv} \begin{split} &\forall \xi\in C^0_+([0,T]), \exists g_\xi: L^p_\mu(0,T;X)\to \erre^+ \hbox{ such that } \\ &[\psi_n,\xi] \hbox{ $\Gamma\tau$-converges to }g_\xi. \end{split} \end{equation} The separable Banach space $C^0([0,T])$ has a countable dense subset $M$, e.g., the family of polynomials with rational coefficients. Let us denote by $M_+$ the cone of the nonnegative elements of $M$. For any $\xi\in M_+$, by \eqref{eq.evol.equibc} a suitable subsequence $\{[\psi_{n'},\xi]\}$ weakly $\Gamma$-converges to a function $g_\xi: L^p_\mu(0,T;X)\to \erre$, and \begin{equation}\label{eq.evol.a} \begin{split} &C_1 \int_0^T \\|w(t)\\|_X^p \, \xi(t) \, d\mu(t) \le g_\xi(w) \\ &\le C_2\int_0^T \\|w(t)\\|_X^p \, \xi(t) \, d\mu(t) + C_3 \int_0^T \xi(t) \, d\mu(t) \qquad\forall w\in L^p_\mu(0,T;X). \end{split} \end{equation} A priori the selected subsequence $\{[\psi_{n'},\xi]\}$ might depend on $\xi$. However, because of the countability of $M_+$, via a diagonalization procedure one can select a subsequence that is independent of $\xi\in M_+$. (Henceforth we shall write $\psi_n$ in place of $\psi_{n'}$, dropping the prime.) For that subsequence thus \begin{equation}\label{eq.evol.a=} \begin{split} &\forall \xi\in M_+, \exists g_\xi: L^p_\mu(0,T;X)\to \erre^+ \hbox{ such that } \\ &[\psi_n,\xi]\hbox{ \ $\Gamma\tau$-converges to \ $g_\xi$ in }L^p_\mu(0,T;X), \end{split} \end{equation} that is, for any $\xi\in M_+$, \begin{equation} \begin{split}\label{eq.evol.b=} &\forall w\in L^p_\mu(0,T;X), \forall \hbox{ sequence $\{w_n\}$ in }L^p_\mu(0,T;X), \\ &\hbox{if \ $w_n \to\!\!\!\!\!\!{}_{_\tau}\;\; w$ in $L^p_\mu(0,T;X)$ \ then \ } \liminf_{n\to+ \infty} \; [\psi_n,\xi](w_n) \ge g_\xi(w), \end{split} \end{equation} \vskip-0.4truecm \begin{equation} \begin{split}\label{eq.evol.c=} &\forall w\in L^p_\mu(0,T;X), \exists\hbox{ sequence $\{w_n\}$ of $L^p_\mu(0,T;X)$ such that } \\ &w_n \to\!\!\!\!\!\!{}_{_\tau}\;\; w \hbox{ in $L^p_\mu(0,T;X)$ \ and \ } \lim_{n\to + \infty} \; [\psi_n,\xi](w_n) = g_\xi(w). \end{split} \end{equation} (The recovery sequence $\{w_n\}$ in \eqref{eq.evol.c=} may depend on $\xi$.) As any $\xi\in C^0_+([0,T])$ is the uniform limit of some sequence $\{\xi_m\}$ in $M$, for any bounded sequence $\{w_n\}$ in $L^p_\mu(0,T;X)$ \begin{equation} \begin{split} &\sup_n \big\| [\psi_n,\xi](w_n) - [\psi_n,\xi_m](w_n) \big\| \overset{\eqref{eq.evol.crochet}}{=} \sup_n \Big\|\int_0^T \psi_{n,w_n}(t) \, [\xi(t) - \xi_m(t)] \, d\mu(t) \Big\| \\ &\overset{\eqref{eq.evol.super}}{\le} \\| \xi - \xi_m \\|_{C^0([0,T])} \sup_n \int_0^T \varphi_n(t,w) \, d\mu(t) \\ &\overset{\eqref{eq.evol.equibc}}{\le}\\| \xi - \xi_m \\|_{C^0([0,T])} \sup_n \bigg\{ C_2 \!\! \int_0^T \\|w_n(t)\\|_X^p \, d\mu(t) + C_3 \mu(]0,T[) \bigg\} \qquad\forall m. \end{split} \end{equation} By the density of $M_+$ in $C^0_+([0,T])$, \eqref{eq.evol.conv} then follows. (ii) Next we extend \eqref{eq.evol.conv} to any $\xi\in L^\infty_+(0,T)$. By the classical Lusin theorem, for any $\xi\in L^\infty_+(0,T)$ there exists a sequence $\{\xi_m\}$ in $C^0_+([0,T])$ such that, setting $A_m =\{t\in [0,T]: \xi_m(t) \not= \xi(t)\}$, \begin{equation}\label{eq.evol.Lusin} \begin{split} &\\|\xi_m\\|_{C^0([0,T])} \le \\|\xi\\|_{L^\infty(0,T)} \quad\forall m, \\ &\mu(A_m)\to 0. \end{split} \end{equation} Hence \begin{equation} \begin{split} &\sup_n \big\| [\psi_n,\xi](w_n) - [\psi_n,\xi_m](w_n) \big\| \overset{\eqref{eq.evol.crochet}}{=} \sup_n \Big\|\int_0^T \psi_{n,w_n}(t) \, [\xi(t) - \xi_m(t)] \, d\mu(t) \Big\| \\ &\overset{\eqref{eq.evol.super},\eqref{eq.evol.equibc}}{\le} \\|\xi - \xi_m\\|_{L^\infty(0,T)} \bigg\{ C_2 \sup_n \!\! \int_{A_m} \\|w_n(t)\\|_X^p \, d\mu(t) + C_3 \mu(A_m) \bigg\} \qquad\forall m. \end{split} \end{equation} As $w_n\to\!\!\!\!\!\!{}_{_\tau}\;\; w$ in $L^p_\mu(0,T;X)$ (see \eqref{eq.evol.c=}), the sequence $\{\\|w_n(\cdot)\\|_X^p\}$ is equi-integrable. By this property and by \eqref{eq.evol.Lusin}$_2$, \[ \sup_n \int_{A_m} \\|w_n(t)\\|_X^p \, d\mu(t)\to 0 \qquad\text{ as }m\to \infty. \] \eqref{eq.evol.conv} is thus extended to any $\xi\in L^\infty_+(0,T)$. (iii) Next we prove that \begin{equation}\label{eq.evol.due} \begin{split} &\exists \psi: L^p_\mu(0,T;X)\to L^1_\mu(0,T) \hbox{ such that } \\ &g_\xi(w) = [\psi_w,\xi] \qquad\forall w\in L^p_\mu(0,T;X),\forall \xi\in L^\infty_+(0,T). \end{split} \end{equation} (Here some care is needed, since $L^\infty_+(0,T)$ is not a linear space.) Let us fix any $\xi\in L^\infty_+([0,T])$, any $w\in L^p_\mu(0,T;X)$, and any sequence $\{w_n\}$ as in \eqref{eq.evol.c=}. By the boundedness of $\{w_n\}$ and by \eqref{eq.evol.equibc}, the sequence $\{\psi_{n,w_n}\}$ = $\{\varphi_n(\cdot,w_n)\}$ is bounded in $L^1_\mu(0,T)$ and is equi-integrable. There exists then a function $\gamma \in L^1_\mu(0,T)$ such that, possibly extracting a subsequence, $\psi_{n,w_n}\rightharpoonup \gamma$ in $L^1_\mu(0,T)$. \footnote{ We denote the strong and weak convergence respectively by $\to$ and $\rightharpoonup$. } Thus \begin{equation}\label{eq.evol.o} [\psi_n,\xi](w_n) =\int_0^T \psi_{n,w_n}(t) \, \xi(t) \, d\mu(t) \to \int_0^T \gamma(t) \, \xi(t) \, d\mu(t) \qquad\forall \xi\in L^\infty(0,T). \end{equation} Thus by \eqref{eq.evol.c=} \begin{equation}\label{eq.evol.q} g_\xi(w) = \int_0^T \gamma(t) \, \xi(t) \, d\mu(t) \qquad\forall \xi\in L^\infty_+(0,T). \end{equation} Therefore $\gamma$ is determined by $w\in L^p_\mu(0,T;X)$, and is independent of the specific sequence $\{w_n\}$ that fulfills \eqref{eq.evol.c=}. This defines an operator \begin{equation}\label{eq.evol.f} \psi: L^p_\mu(0,T;X)\to L^1_\mu(0,T): w \mapsto \psi_w = \gamma. \end{equation} The equality \eqref{eq.evol.q} thus reads \begin{equation}\label{eq.evol.r} g_\xi(w) =\int_0^T \psi_w(t) \, \xi(t) \, d\mu(t) \qquad\forall \xi\in L^\infty_+(0,T), \forall w\in L^p_\mu(0,T;X). \end{equation} Recalling the definition \eqref{eq.evol.crochet}, we see that this completes the proof of \eqref{eq.evol.due}. (iv) Finally, we show that there exists a normal function $\varphi: {}]0,T[{} \times X\to \erre^+$ such that the operator $\psi$ that we just defined in \eqref{eq.evol.f} is as in \eqref{eq.evol.super}. By \eqref{eq.evol.nul}, for any $n$ the functional \begin{equation} \begin{split} &\Phi_n: L^p_\mu(0,T;X)\to \erre^+: \\ &w\mapsto \int_0^T \psi_{w,n}(t) \, d\mu(t) =\int_0^T \varphi_n(t,w_n(t)) \, d\mu(t) \end{split} \end{equation} is additive in the sense of \eqref{eq.evol.add} below. This property then also holds for the limit functional \begin{equation} \Phi: L^p_\mu(0,T;X)\to \erre^+: w\mapsto \int_0^T \psi_w(t) \, d\mu(t). \end{equation} By selecting $\xi\equiv 1$ in \eqref{eq.evol.defgamma.5}, we get that $\Phi$ is lower semicontinuous. By Lemma~\ref{lemma.Hiai} then there exists a normal function $\varphi$ as we just specified. $\Box$ \section{Applications} \label{sec.appl} \noindent In this section we briefly illustrate how the notion of evolutionary $\Gamma$-convergence of weak type can be applied to prove the structural compactness and structural stability of flows of the form \begin{equation}\label{eq.flow1} D_tu + \alpha(u)\ni h \qquad\hbox{ in $V'$, a.e.\ in time }(D_t := {\partial/\partial t}); \end{equation} here $V$ is a Hilbert space, and $\alpha:V\to {\cal P}(V')$ is a {\it semi-monotone\/} operator. This is a particular case of the class of generalized pseudo-monotone operators of Browder and Hess \cite{BrHe}, and includes mappings of the form \begin{equation}\label{eq.equicoer} H^1_0(\Omega)\to {\cal P}(H^{-1}(\Omega)): v\mapsto - \nabla \cdot \vec\gamma(v,\nabla v), \end{equation} with $\vec\gamma$ continuous w.r..t. the first argument, and maximal monotone w.r..t. the second one. We refer to \cite{ViLinc} for a more expanded outline and to \cite{Vi17} for a detailed presentation. Under suitable restrictions, there exists a topology $\tau$ as above in Section~\ref{sec.evol}, such that \begin{equation}\label{eq.fitzp.nonconv} \begin{split} &\varphi:V \!\times\! V' \to \erre\cup \{+\infty\} \text{ is lower semicontinuous w.r.t.\ }\tau, \\ &\varphi(v,v^) \ge \langle v^,v\rangle \qquad\forall (v,v^)\in V \!\times\! V', \\ &\varphi(v,v^) = \langle v^,v\rangle \quad\Leftrightarrow\quad v^\in \alpha(v). \end{split} \end{equation} After defining the functional \begin{equation} \begin{split} \label{eq.BENfun} \Phi(v,v^) = \int_0^T [\varphi(v,v^ -D_tv) - \langle v^,v\rangle] \, d\mu(t) + {1\over2}\int_0^T \\|v(T)\\|_H^2 \, dt - {T\over2} \\|u(0)\\|_H^2 & \\ \forall (v,v^)\in L^2(0,T;V \!\times\! V'),& \end{split} \end{equation} one can show that \begin{equation}\label{eq.equiv} \eqref{eq.flow1} \quad\Leftrightarrow\quad \Phi(u,u^*) = \inf \Phi =0. \end{equation} This provides a (nonstandard) variational structure of the flow, and paves the way to the use of a notion of evolutionary $\Gamma$-convergence of weak type. \noindent{\bf Structural compactness and structural stability.} We define {\it structural stability\/} as robustness to perturbations of the structure of the problem, e.g.\ operators in differential equations. These notions have obvious applicative motivations, as data and operators are accessible just with some approximation. See e.g.\ \cite{ViLinc} and \cite{Vi17}. Let us briefly illustrate these notions for a problem of the form $Au\ni h$, $A$ being a multi-valued operator acting in a Banach space and $h$ a datum. Given bounded families $\{h_n\}$ and $\{A_n\}$, we formulate the stability of the problem via two properties: (i) {\it structural compactness:\/} existence of convergent sequences of data $\{h_n\}$ and of operators $\{A_n\}$ (in a sense to be specified); (ii) {\it structural stability:\/} if $A_nu_n\ni h_n$ for any $n$, $A_n\to A$, $h_n\to h$ and $u_n\to u$, then $u$ is a solution of the asymptotic problem: $Au\ni h$. Structural compactness and structural stability of minimization principles can adequately be dealt with via De Giorgi's theory of $\Gamma$-convergence. Next we outline how this can be extended to flows, after these have been variationally formulated as in \eqref{eq.BENfun} and \eqref{eq.equiv}. \begin{theorem} [\cite{Vi17}] \label{teo.comp'} Let $V$ be a real separable Hilbert space, and $\mu$ be the measure on $]0,T[$ that fulfills \eqref{eq.evol.mu}. Let $\{\varphi_n\}$ be a sequence of normal functions ${}]0,T[{} \times V \!\times\! V'\to \erre^+$ such that \begin{eqnarray} &\varphi_n(t,\cdot) \in {\cal F}(V) \qquad\hbox{for a.e.\ }t\in {}]0,T[, \forall n, \label{eq.comstab.reppn} \\ &\begin{split} &\exists C_1,C_2,C_3 >0: \forall n,\hbox{for a.e.\ }t\in {}]0,T[, \forall w\in V \!\times\! V', \\ &C_1 \\|w\\|_{V \!\times\! V'}^2\le \varphi_n(t,w) \le C_2\\|w\\|_{V \!\times\! V'}^2 +C_3, \end{split} \label{eq.comstab.equibc'} \\ &\varphi_n(t,0) =0 \qquad\hbox{ for a.e.\ }t\in {}]0,T[,\forall n, \label{eq.comstab.nul'} \end{eqnarray} and define the operators $\psi_n: L^2_\mu(0,T;V \!\times\! V')\to L^1_\mu(0,T)$ by \begin{equation}\label{eq.comstab.super'} \psi_{n,w}(t) = \varphi_n(t,w(t)) \qquad\forall w\in L^2_\mu(0,T;V \!\times\! V'),\hbox{ for a.e.\ }t\in {}]0,T[,\forall n. \end{equation} \indent Then there exists a normal function $\varphi: {}]0,T[{} \times V \!\times\! V'\to \erre^+$ such that \begin{equation}\label{eq.comstab.repp} \varphi(t,\cdot) \in {\cal F}(V) \qquad\hbox{for a.e.\ }t\in {}]0,T[, \end{equation} and such that, defining the corresponding operator $\psi: L^2_\mu(0,T;V \!\times\! V')\to L^1_\mu(0,T)$ as in \eqref{eq.comstab.super'}, possibly extracting a subsequence \begin{equation}\label{eq.comstab.tesi'} \begin{split} &\hbox{ $\psi_n$ sequentially $\Gamma$-converges to }\psi \\ &\hbox{ in the topology $\widetilde\pi$ of $L^2_\mu(0,T;V \!\times\! V')$ and } \\ &\hbox{ in the weak topology of $L^1_\mu(0,T)$ (cf.\ \eqref{eq.evol.defgamma.1+}). } \end{split} \end{equation} \indent Moreover, if $\varphi_n$ does not depend on $t$ for any $n$, then the same holds for $\varphi$. \end{theorem} This result provides the structural compactness and structural stability of flows of the form \begin{equation}\label{eq.flow2} D_tu - \nabla \cdot \vec\gamma(v,\nabla v) \ni h \qquad\hbox{ in $H^{-1}(\Omega)$, a.e.\ in time,} \end{equation} with $\vec\gamma$ as above, see \cite{Vi17}. This can also be extended to doubly-nonlinear flows, see \cite{ViLinc}. \centerline{\bf Acknowledgment} The author is a member of GNAMPA of INdAM. This research was partially supported by a MIUR-PRIN 2015 grant for the project ``Calcolo delle Variazioni" (Protocollo 2015PA5MP7-004). This author is indebted to Giuseppe Buttazzo, who brought Hiai's paper \cite{Hi} to his attention. \baselineskip=10.truept \baselineskip=12truept Author's address: Augusto Visintin \par Universit\`a degli Studi di Trento \par Dipartimento di Matematica \par via Sommarive 14, \ 38050 Povo (Trento) - Italia \par Tel +39-0461-281635 (office), +39-0461-281508 (secretary) \par Fax +39-0461-281624 \par Email: [email protected] \end{document}	arXiv
Why not use the Lagrangian, instead of the Hamiltonian, in nonrelativistic QM? Undergraduate classical mechanics introduces both Lagrangians and Hamiltonians, while undergrad quantum mechanics seems to only use the Hamiltonian. But particle physics, and more generally quantum field theory seem to only use the Lagrangian, e.g. you hear about the Klein-Gordan Lagrangian, Dirac Lagrangian, Standard Model Lagrangian and so on. Why is there a mismatch here? Why does it seem like only Hamiltonians are used in undergraduate quantum mechanics, but only Lagrangains are used in quantum field theory? quantum-mechanics quantum-field-theory path-integral lagrangian-formalism hamiltonian-formalism knzhou RevoRevo $\begingroup$ Both methods are equivalent and are used, to tell the truth. Momenta and coordinates had been used before QM in the old (Bohr) quantization, remember quantization of the phase space $\int dpdq$. $\endgroup$ – Vladimir Kalitvianski Mar 4 '12 at 17:57 $\begingroup$ Perhaps it's worth noting that the Lagrangian/path integral approach is very poorly suited to the study of bound state problems. Just try the hydrogen atom with the Lagrangian approach, even Feynman couldn't do it! $\endgroup$ – KF Gauss Jun 1 '18 at 16:57 In order to use Lagrangians in QM, one has to use the path integral formalism. This is usually not covered in a undergrad QM course and therefore only Hamiltonians are used. In current research, Lagrangians are used a lot in non-relativistic QM. In relativistic QM, one uses both Hamiltonians and Lagrangians. The reason Lagrangians are more popular is that it sets time and spacial coordinates on the same footing, which makes it possible to write down relativistic theories in a covariant way. Using Hamiltonians, relativistic invariance is not explicit and it can complicate many things. So both formalism are used in both relativistic and non-relativistic quantum physics. This is the very short answer. HeidarHeidar $\begingroup$ There is also an covariant Hamiltonian formalism of field theory, in which the phase space is infinite dimensional or one uses the language of multisymplectic. Either way the mathematics is too sophisticated to be covered in (under)graduate courses. $\endgroup$ – Tobias Diez Mar 5 '12 at 23:10 $\begingroup$ Why must path integrals be used in order to use the Lagrangian formalism for QM? $\endgroup$ – Stan Shunpike Feb 6 '15 at 4:32 $\begingroup$ @StanShunpike If you apply the Lagrangian formalism to quantum mechanics you end up with the path integral formalism. That's how Feynman discovered it. If you use the Hamiltonian formalism, you end up with the usual canonical formulation. If you use the Newtonian formalism, well, you end up with Bohmian mechanics. C.f. physicstravelguide.com/frameworks $\endgroup$ – Tim Dec 13 '17 at 10:00 As Weinberg points in his QFT book, in the Hamiltonian formalism it is easier to check the unitarity of the theory because unitarity is directly related to evolution, while in the Lagrangian formalism the symmetries that mix space with time are more explicit. Therefore the Hamiltonian formalism is usually more convenient in non-relativistic and galilean quantum theories. Diego MazónDiego Mazón $\begingroup$ What do you mean by more explicit? $\endgroup$ – Stan Shunpike Feb 6 '15 at 4:31 $\begingroup$ @StanShunpike In order for a theory to be Poincare invariant, the Lagrangian needs to be a Poincare scalar, what it is easy to see. The equivalent condition in the Hamiltonian formalism is that there is a Poincare algebra with the Hamiltonian as the zero component of the 4-momentum. This condition needs to be checked, as it is not elemental to see. $\endgroup$ – Diego Mazón Feb 8 '15 at 2:00 I would say because of the way you efficiently solve problems as well as pedagogy. Both are used in both cases though. The Hamiltonian operator approach emphasises the spectrum aspects of quantum mechanics, which the student is introduced to at this point $-$ but here is a Lagrangian $$\mathcal{L}\left(\psi, \mathbf{\nabla}\psi, \dot{\psi}\right) = \mathrm i\hbar\, \frac{1}{2} (\psi^{}\dot{\psi}-\dot{\psi^{}}\psi) - \frac{\hbar^2}{2m} \mathbf{\nabla}\psi^{} \mathbf{\nabla}\psi - V( \mathbf{r},t)\,\psi^{}\psi$$ for the Schrödinger equation $$\frac{\partial \mathcal{L}}{\partial \psi^{}} - \frac{\partial}{\partial t} \frac{\partial \mathcal{L}}{\partial\frac{\partial \psi^{}}{\partial t}} - \sum_{j=1}^3 \frac{\partial}{\partial x_j} \frac{\partial \mathcal{L}}{\partial\frac{\partial \psi^{*}}{\partial x_j}} = 0.$$ The Lagrangian (density) is especially relevant for the path integral formulation, and in some way closer to bring out symmetries of a field theory. Noether theorem and so on. $-$ but I remember Peskin & Schröders book on quantum field theory starts out with the Hamiltonian approach and introduces path integral methods only 300 pages in. Nikolaj-KNikolaj-K I think the Hamiltonian approach is emphasized in undergraduate due more to habit and the influence of Dirac, rather than due to any profound mathematical reason. The Hamiltonian is also easier to teach because it is compatible with classical intuitions of time. Historically, Dirac argued strongly for the primacy of the Hamiltonian, literally until shortly before his death. My own interpretation of an oblique reprimand of the Lagrangian that Dirac made in his Lectures on Quantum Mechanics (1966) (a great read!) is that Dirac was unhappy with the fame that Feynman was acquiring, although Dirac was always so reserved in expressing discontent with other physicists that it's very hard to say for sure. Dirac's downplaying the value of Lagrangian approach is of course highly ironic, since it was Dirac who first showed that the classical Lagrangian can be applied to QM, in an early paper [1]. It was that same paper that many years later inspired and unleashed Feynman's remarkable QED work. [1] P. A. M. Dirac, The Lagrangian in Quantum Mechanics, Phys. Zs. Sowjetunion 3 (1933) No. 1; reprinted in: J. Schwinger (Ed.), Selected Papers on Quantum Electrodynamics, 1958, No. 26 Emilio Pisanty Terry BollingerTerry Bollinger $\begingroup$ Dirac didn't give a hoot about fame. He famously wanted to reject the Nobel prize, but was told that would just make him more famous. He was annoyed for the same reason that he gave up the path integral--- he couldn't figure out what to do in the case that the Hamiltonian wasn't quadratic in the momenta. This was ignored by Feynman as well, and it is only resolved by a more general view of path integration than that available in Feynman's work. The quadratic momentum case is enough for field theory, unfortunately, so people don't notice that the formalism as usually presented is incomplete. $\endgroup$ – Ron Maimon Mar 24 '12 at 7:54 $\begingroup$ @RonMaimon interesting, thanks! I was not aware of that specific concern by Dirac. You wouldn't happen to have a quick reference on that, would you?... And overall, the more original work I've read by Dirac, the more my jaw drops. He was an amazing and (I think) under-appreciated thinker, even given his substantial fame. $\endgroup$ – Terry Bollinger Mar 24 '12 at 8:12 $\begingroup$ Bolinger: Dirac is a great physicist, he is the founder of high energy physics, but I think people already recognize that. The story I read regarded the infamous Pocono conference (or shelter island, I forget which is which) where Feynman presented path-integrals and diagrams. Dirac commented that this formalism is not apparently unitary. In his lectures on field theory from the 1960s, he makes the case that quadratic momenta are the only thing the path integral handles. I might be getting the cites wrong, I read it a long, long time ago. $\endgroup$ – Ron Maimon Mar 24 '12 at 8:33 $\begingroup$ @RonMaimon There is a path integral formulation of the Hamiltonian formalism. It is equivalent to the usual formulation for theories quadratic in the momenta. $\endgroup$ – orbifold May 16 '12 at 20:55 $\begingroup$ @orbifold: And many people said (stupidly) that the p-q path integral is not well defined because p and q don't commute (for example, Sidney Coleman used to say this, it's totally wrong). Further, Dirac would have considered the p-q path integral to be equivalent to the canonical formalism (which it is), since it picks out a p-q decomposition and a time-decomposition. It's only Feynman's form (after doing the p integral) that is covariant under relativity. $\endgroup$ – Ron Maimon May 17 '12 at 0:16 In few words Unitarity of evolution operator U(t) is easy to see with Hamiltonian formalism. Lorentz invariance of S-matrix (scattering matrix) is easy to see with Lagrangian formalism. Manuel G. C.Manuel G. C. 5111 silver badge11 bronze badge It is not true "that QFT and particle physics rely instead on Lagrangian" The generator of time translations in quantum theory is the Hamiltonian, not the Lagrangian; therefore, we need a Hamiltonian to study evolution of the quantum system. As mentioned in the Volume 1 of Weinberg's textbook on QFT, chapter 7: It is the Hamiltonian formalism that is needed to calculate the S-matrix (whether by operator or path-integral methods) but it is not always easy to choose Hamiltonians that yield a Lorentz-invariant S-matrix. The point of the Lagrangian formalism is that it makes it easy to satisfy Lorentz invariance and other symmetries: a classical theory with a Lorentz-invariant Lagrangian density will when canonically quantized lead to a Lorentz-invariant quantum theory. That is, we shall see here that such a theory allows the construction of suitable quantum mechanical operators that satisfy the commutation relations of the Poincaré algebra, and therefore leads to a Lorentz-invariant S-matrix. Therefore the usual recipe consists on postulating some Lagrangian, checking it satisfies certain basic properties, then deriving a Hamiltonian from that Lagrangian, and finally using this Hamiltonian to compute the elements of the S-matrix. juanrgajuanrga In my opinion, the existing choice between the canonical (Hamiltonian) and path-integral (Lagrangian) formalisms is a far-reaching consequence of particle-wave dualism in QM. The first emphasizes the spectral aspects, the second can be viewed as a deep generalization of Fermat's principle for rays propagation in optics. Since most of experiments in particle physics represent some sort of scattering, the wave aspects are usually more important, hence, the Lagrangian formalism is much more adequate for the practical use. SergeiSergei Not the answer you're looking for? Browse other questions tagged quantum-mechanics quantum-field-theory path-integral lagrangian-formalism hamiltonian-formalism or ask your own question. Why does Quantum Field Theory use Lagrangians rather than Hamiltonians? Why not formulate Quantum Mechanics using Lagrangians? Conceptual difference between Hamiltonian and Lagrangian formulation Why we are not using Lagrangian instead of Hamiltonian in non-relativistic quantum mechanics? What exactly are Hamiltonian Mechanics (and Lagrangian mechanics) What is the difference between configuration space and phase space? Eigenvalues of the Lagrangian? Is there a mathematical reason for the Lagrangian to be Lorentz invariant? Lagrangian mechanics vs Hamiltonian mechanics What is "momentum density" and why it important to QFT? Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations Is the Lagrangian density in field theory real? Why is the Hamiltonian the Legendre transform of the Lagrangian? Where does Field Theory come from?	CommonCrawl
Word Processing in Groups Word Processing in Groups is a monograph in mathematics on the theory of automatic groups; these are a type of abstract algebra whose operations are defined by the behavior of finite automata. The book's authors are David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Mike Paterson, and William Thurston. Widely circulated in preprint form, it formed the foundation of the study of automatic groups even before its 1992 publication by Jones and Bartlett Publishers (ISBN 0-86720-244-0).[1][2][3] Topics The book is divided into two parts, one on the basic theory of these structures and another on recent research, connections to geometry and topology, and other related topics.[1] The first part has eight chapters. They cover automata theory and regular languages, and the closure properties of regular languages under logical combinations; the definition of automatic groups and biautomatic groups; examples from topology and "combable" structure in the Cayley graphs of automatic groups; abelian groups and the automaticity of Euclidean groups; the theory of determining whether a group is automatic, and its practical implementation by Epstein, Holt, and Sarah Rees; extensions to asynchronous automata; and nilpotent groups.[1][2][4] The second part has four chapters, on braid groups, isoperimetric inequalities, geometric finiteness, and the fundamental groups of three-dimensional manifolds.[1][4] Audience and reception Although not primarily a textbook, the first part of the book could be used as the basis for a graduate course.[1][4] More generally, reviewer Gilbert Baumslag recommends it "very strongly to everyone who is interested in either group theory or topology, as well as to computer scientists." Baumslag was an expert in a related but older area of study, groups defined by finite presentations, in which research was eventually stymied by the phenomenon that many basic problems are undecidable. Despite tracing the origins of automatic groups to early 20th-century mathematician Max Dehn, he writes that the book studies "a strikingly new class of groups" that "conjures up the fascinating possibility that some of the exploration of these automatic groups can be carried out by means of high-speed computers" and that the book is "very likely to have a great impact".[2] Reviewer Daniel E. Cohen adds that two features of the book are unusual, and welcome: First, that the mathematical results that it presents all have names, not just numbers, and second, that the cost of the book is low.[3] Years later, in 2009, mathematician Mark V. Lawson wrote that despite its "odd title" the book made automata theory, once the domain of computer scientists, respectable among mathematicians, and that it became part of "a quiet revolution in the diplomatic relations between mathematics and computer science".[5] References 1. Apanasov, B. N., "Review of Word Processing in Groups", zbMATH, Zbl 0764.20017 2. Baumslag, Gilbert (1994), "Review of Word Processing in Groups", Bulletin of the American Mathematical Society, New Series, 31 (1): 86–91, doi:10.1090/S0273-0979-1994-00481-1, MR 1568123 3. Cohen, D. E. (November 1993), "Review of Word Processing in Groups", Bulletin of the London Mathematical Society, 25 (6): 614–616, doi:10.1112/blms/25.6.614 4. Thomas, Richard M. (1993), "Review of Word Processing in Groups", Mathematical Reviews, MR 1161694 5. Lawson, Mark V. (December 2009), "Review of A Second Course in Formal Languages and Automata Theory by Jeffrey Shallit", SIAM Review, 51 (4): 797–799, JSTOR 25662348	Wikipedia
Chebyshevskii Sbornik Chebyshevskii Sb.: Most published authors of the journal 1. N. N. Dobrovol'skii 41 2. N. M. Dobrovol'skii 35 3. A. E. Gvozdev 28 4. V. N. Chubarikov 26 5. A. V. Shutov 22 6. D. V. Gorbachev 19 7. A. P. Laurinčikas 17 8. V. G. Chirskii 16 9. V. N. Bezverkhnii 15 10. V. G. Durnev 15 11. I. Yu. Rebrova 15 12. O. A. Matveeva 14 13. V. N. Kuznetsov 14 14. O. V. Zetkina 13 15. O. V. Kuzovleva 13 16. P. L. Ivankov 12 17. D. V. Malii 12 18. O. A. Pikhtilkova 12 19. L. A. Tolokonnikov 12 20. M. N. Dobrovol'skii 11 21. A. R. Esayan 11 22. V. I. Ivanov 11 23. A. N. Sergeev 11 24. N. A. Shchuchkin 11 25. S. A. Gritsenko 10 26. N. V. Budarina 9 27. A. I. Nekritsukhin 9 28. D. Siauciunas 9 29. S. V. Vostokov 8 30. I. V. Dobrynina 8 31. A. A. Zhukova 8 32. G. M. Zhuravlev 8 33. A. Ya. Belov 7 34. O. A. Gorkusha 7 35. E. Deza 7 36. L. P. Dobrovol'skaya 7 37. A. I. Zetkina 7 38. I. A. Martyanov 7 39. A. N. Privalov 7 40. A. V. Rodionov 7 41. V. L. Usol'tsev 7 42. Yu. N. Shteinikov 7 10 most published authors of the journal Most cited authors of the journal 1. N. N. Dobrovol'skii 125 2. N. M. Dobrovol'skii 105 3. L. P. Dobrovol'skaya 53 5. M. N. Dobrovol'skii 37 8. I. N. Balaba 24 9. I. Yu. Rebrova 23 10. D. K. Sobolev 23 11. V. N. Soboleva 23 12. D. N. Azarov 22 15. V. N. Bezverkhnii 18 16. E. I. Yushina 18 17. A. E. Gvozdev 17 18. G. M. Zhuravlev 17 20. V. Yu. Matveev 16 24. A. A. Abrosimova 14 25. D. V. Gorbachev 13 27. M. A. Korolev 13 28. E. S. Makarov 13 29. V. L. Usol'tsev 13 30. A. D. Bruno 12 31. I. V. Dobrynina 12 32. O. V. Inchenko 12 33. N. N. Mot'kina 12 34. V. I. Subbotin 12 36. A. D. Breki 11 37. N. M. Korobov 11 38. A. S. Simonov 10 39. O. E. Bocharova 9 40. S. S. Demidov 9 41. Do Duc Tam 9 42. A. P. Laurinčikas 9 43. E. A. Morozova 9 45. E. V. Trikolich 9 46. G. V. Fedorov 9 10 most cited authors of the journal Most cited articles of the journal L. P. Dobrovol'skaya, M. N. Dobrovol'skii, N. M. Dobrovol'skii, N. N. Dobrovol'skii Chebyshevskii Sb., 2012, 13:4, 4–107 16 2. On the residual finiteness of $p$-groups D. N. Azarov 3. The zeta-function is the monoid of natural numbers with unique factorization N. N. Dobrovol'skii Chebyshevskii Sb., 2017, 18:4, 188–208 11 4. On the error of approximate integration over modified grids L. P. Dobrovol'skaya, N. M. Dobrovol'skiĭ, A. S. Simonov Chebyshevskii Sb., 2008, 9:1, 185–223 10 5. Hypothesis about "barier series" for the zeta-functions of monoids with the exponential sequence of primes N. N. Dobrovolsky, M. N. Dobrovolsky, N. M. Dobrovolsky, I. N. Balaba, I. Yu. Rebrova Chebyshevskii Sb., 2018, 19:1, 106–123 9 6. Number-theoretic method in approximate analysis S. S. Demidov, E. A. Morozova, V. N. Chubarikov, I. Yu. Rebrova, I. N. Balaba, N. N. Dobrovol'skii, N. M. Dobrovol'skii, L. P. Dobrovol'skaya, A. V. Rodionov, O. A. Pikhtil'kova Chebyshevskii Sb., 2017, 18:4, 6–85 9 7. On hyperbolic Hurwitz zeta function N. M. Dobrovolsky, N. N. Dobrovolsky, V. N. Soboleva, D. K. Sobolev, L. P. Dobrovol'skaya, O. E. Bocharova Chebyshevskii Sb., 2016, 17:3, 72–105 9 8. The arithmetic and geometry of one-dimensional quasilattices A. V. Shutov 9. On monoids of natural numbers with unique factorization into prime elements N. N. Dobrovolsky 10. On number of zeros of the Riemann zeta function that lie in «almost all» very short intervals of neighborhood of the critical line Do Duc Tam 11. Universal generalization of the continued fraction algorithm A. D. Bruno Chebyshevskii Sb., 2015, 16:2, 35–65 8 12. The zeta function of monoids with a given abscissa of absolute convergence 13. Application of plasticity theory of dilating media to sealing processes of powders of metallic systems E. S. Makarov, A. E. Gvozdev, G. M. Zhuravlev, A. N. Sergeev, I. V. Minaev, A. D. Breki, D. V. Malii 14. Almost nilpotent varieties in different classes of linear algebras O. V. Shulezhko 15. About the modern problems of the theory of hyperbolic zeta-functions of lattices N. M. Dobrovol'skii 16. Arithmetic properties of polyadic integers V. G. Chirskii N. M. Dobrovol'skii, N. N. Dobrovol'skii, E. I. Yushina 18. Bounded remainder sets on a two-dimensional torus A. A. Abrosimova 19. On a family of two-dimensional bounded remainder sets 20. The two-dimensional Hecke–Kesten problem 21. On joint universality of Dirichlet $L$-functions A. Laurinčikas 22. Algorithms for computing optimal coefficients L. P. Bocharova Chebyshevskii Sb., 2007, 8:1, 4–109 7 23. Discrepancy of two-dimensional Smolyak grids Chebyshevskii Sb., 2007, 8:1, 110–152 7 24. Methods of estimating of incomplete Kloosterman sums M. A. Korolev 25. Ob odnom klasse sil'no simmetrichnyh mnogogrannikov V. I. Subbotin 26. On non-linear Kloosterman sum 27. The rate of convergence of the average value of the full rational arithmetic sums V. N. Chubarikov 28. The universal formal group that defines the elliptic function of level 3 V. M. Buchstaber, E. Yu. Bunkova 29. On the virtual residuality root-class residuality of generalized free products and hnn-extension of groups D. V. Goltsov 30. On certain properties of polyadic expansions V. G. Chirskii, V. Y. Matveev 31. On representations of positive integers 32. On the virtual residuality a finite $p$-groups of descending HNN-extension 33. On the linear independence of some functions P. L. Ivankov 34. Continued fractions for quadratic irrationalities from the field $\mathbb{Q}(\sqrt{5})$ E. V. Trikolich, E. I. Yushina 35. On finite special continued fractions O. A. Gorkusha Chebyshevskii Sb., 2008, 9:1, 80–107 6 36. On irrationality measures of some values of the Gauss function E. S. Sal'nikova 37. On some properties of special polynomials N. M. Korobov 38. On generalized Jacobians and rational continued fractions in the hyperelliptic fields V. S. Zhgoon 39. On varieties with identities of one generated free metabelian algebra A. B. Verevkin, S. P. Mishchenko 40. On the boundary behavior of a class of Dirichlet series V. N. Kuznetsov, O. A. Matveeva 41. Hyperbolic zeta function of lattice over quadratic field N. M. Dobrovol'skii, N. N. Dobrovol'skii, V. N. Soboleva, D. K. Sobolev, E. I. Yushina 42. Elementary of the complete rational arithmetical sums 43. $\mathrm{BR}$-sets 44. Structure of discriminant set of real polynomial A. B. Batkhin 45. О некоторых обобщениях сильно симметричных многогранников 46. On matrix decomposition of one reduced cubic irrational N. M. Dobrovol'skii, D. K. Sobolev, V. N. Soboleva 47. Representations of positive integers in DBNS V. G. Chirskii, R. F. Shakirov 49. On the computation of some singular series S. A. Gritsenko, N. N. Mot'kina 50. About convex polyhedrons with equiangular and parquet faces A. V. Timofeenko 10 most cited articles of the journal Most requested articles of the journal \| this week \| this month \| all time \| 1. Materials and technologies for production products by additive manufacturing A. N. Kubanova, A. N. Sergeev, N. M. Dobrovolskii, A. E. Gvozdev, P. N. Medvedev, D. V. Maliy 2. Number theory and applications in cryptography S. V. Vostokov, R. P. Vostokova, S. V. Bezzateev 3. Essentially Baer modules T. H. N. Nhan 4. Estimates the Bergman kernel for classical domains É. Cartan's J. Sh. Abdullayev 5. Topical problems concerning Beatty sequences A. V. Begunts, D. V. Goryashin 6. G. M. Fikhtengol'ts and the teaching of the mathematical analysis in the Russia during the first half of the XXth century S. S. Demidov, S. S. Petrova 7. The lattice of definability. Origins and directions of research A. L. Semenov, S. F. Soprunov 8. Combinatorics on words, facrordynamics and normal forms I. A. Reshetnikov 9. About three-dimensional nets of Smolyak II N. N. Dobrovol'skii, D. V. Gorbachev, V. I. Ivanov 10. From the algebraic methods of Diophantus–Fermats–Euler to the arithmetic of algebraic curves: about the history of diophantine equations after Euler T. A. Lavrinenko, A. A. Belyaev Total publications: 1088 Scientific articles: 960 Citations: 850 Cited articles: 329 2020 SJR 0.273 2018 CiteScore 0.320 Most published authors Most cited authors Most requested articles	CommonCrawl
pisoir Last seen Jul 11 at 15:33 TeX - LaTeX 657 657 33 silver badges1212 bronze badges MathOverflow 166 166 66 bronze badges Server Fault 101 101 11 bronze badge 18 Plus sign in the middle of a ring 12 Prime numbers of the form $1+p+p^2+\dots+p^n$ 6 Ring as a submolecule 5 amino acid -- peptides chemfig conic-sections 9 What is a fewnomial? Feb 1 '15 7 Analytical solution to $a^x+b^x=x$ Dec 13 '13 7 "Polynomials" with non-integer exponents May 13 '14 6 Does $1^{\frac{-i\ln 2}{2\pi}}$ equal 2? May 12 '15 5 Number of zeros in polynomial-exponential sums Feb 24 '14 4 Number of inflection points of $\log f(x)$ vs. $f(x)$ Jan 13 '14 3 Shift of the position of extrema Jan 20 '16 3 The derivative of the determinant of a Kronecker product Apr 21 '14 3 Trying to understand the function $y = x^x$ Jan 30 '16 3 Evaluating the Cauchy product of $\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{n+1}$ and $\sum_{n=0}^{\infty}\frac{1}{3^n} $ Dec 23 '18	CommonCrawl
On the geometry of the p-Laplacian operator DCDS-S Home On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities August 2017, 10(4): 815-835. doi: 10.3934/dcdss.2017041 Effective acoustic properties of a meta-material consisting of small Helmholtz resonators Agnes Lamacz and Ben Schweizer , Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87,44227 Dortmund, Germany * Corresponding author: Ben Schweizer Received March 2016 Revised October 2016 Published April 2017 We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions $u^\varepsilon : \Omega_\varepsilon \to \mathbb{R}$ to a Helmholtz equation in the limit $\varepsilon \to 0$ with the help of two-scale convergence. The domain $\Omega_\varepsilon $ is obtained by removing from an open set $\Omega\subset \mathbb{R}^n$ in a periodic fashion a large number (order $\varepsilon ^{-n}$) of small resonators (order $\varepsilon $). The special properties of the meta-material are obtained through sub-scale structures in the perforations. Keywords: Helmholtz equation, homogenization, resonance, perforated domain, frequency dependent effective properties. Mathematics Subject Classification: 78M40, 35P25, 35J05. Citation: Agnes Lamacz, Ben Schweizer. Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 815-835. doi: 10.3934/dcdss.2017041 G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. 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Sketch of the geometry around the end-point of the channel Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361 Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks & Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461 Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473 Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains. Networks & Heterogeneous Media, 2008, 3 (1) : 97-124. doi: 10.3934/nhm.2008.3.97 Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-31. doi: 10.3934/dcdsb.2019063 Jiann-Sheng Jiang, Kung-Hwang Kuo, Chi-Kun Lin. Homogenization of second order equation with spatial dependent coefficient. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 303-313. doi: 10.3934/dcds.2005.12.303 Gregory A. Chechkin, Tatiana P. Chechkina, Ciro D'Apice, Umberto De Maio. Homogenization in domains randomly perforated along the boundary. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 713-730. doi: 10.3934/dcdsb.2009.12.713 Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks & Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189 S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604 Ciro D'Apice, Umberto De Maio, Peter I. Kogut. Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 283-314. doi: 10.3934/dcdsb.2009.11.283 John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333 Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Specified homogenization of a discrete traffic model leading to an effective junction condition. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2173-2206. doi: 10.3934/cpaa.2018104 Sang-Yeun Shim, Marcos Capistran, Yu Chen. Rapid perturbational calculations for the Helmholtz equation in two dimensions. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 627-636. doi: 10.3934/dcds.2007.18.627 Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden. Minimum free energy in the frequency domain for a heat conductor with memory. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 793-816. doi: 10.3934/dcdsb.2010.14.793 Sheree L. Arpin, J. M. Cushing. Modeling frequency-dependent selection with an application to cichlid fish. Mathematical Biosciences & Engineering, 2008, 5 (4) : 889-903. doi: 10.3934/mbe.2008.5.889 Gleb G. Doronin, Nikolai A. Larkin. Kawahara equation in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 783-799. doi: 10.3934/dcdsb.2008.10.783 T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675 Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050 Agnes Lamacz Ben Schweizer	CommonCrawl
The difference in effective light penetration may explain the superiority in photosynthetic efficiency of attached cultivation over the conventional open pond for microalgae Junfeng Wang1, Jinli Liu2 & Tianzhong Liu1 The 'attached cultivation' technique for microalgae production, combining the immobilized biofilm technology with proper light dilution strategies, has shown improved biomass production and photosynthetic efficiency over conventional open-pond suspended cultures. However, how light is transferred and distributed inside the biofilm has not been clearly defined yet. In this research, the growth, photosynthetic oxygen evolution, and specific growth rate for microalgal cells in both open-pond and attached cultivation were studied to determine the effective light penetration at different phases of the cultivation. As a result, the light conditions inside the culture broth as well as the biofilm were revealed for the first time. Results showed that outdoor, in a conventional 20-cm deep open pond, all of the algal cells were fully illuminated in the first 3 days of cultivation. As the biomass concentration increased from day 4 to day 10, the light could only effectively penetrate 45.5% of the open-pond depth, and then effective light penetration gradually decreased to 31.1% at day 31, when the biomass density reached a maximum value of 0.45 g L−1 or 90 g m−2. In the attached cultivation system, under nitrogen-replete condition, almost 100% of the immobilized algal cells inside the biofilm were effectively illuminated from day 0 through day 10 when the biomass density increased from 8.8 g m−2 to 107.6 g m−2. Higher light penetration efficiency might be the reason why, using attached cultivation, observed values for photosynthetic efficiency were higher than those recorded in conventional open-pond suspended cultures. Microalgae are a group of photosynthetic microorganisms represented by more than 40,000 species, many of which can produce high-value bioactive compounds [1]. In recent years, due to the price increase of fossil fuel as well as the concerns on environment deterioration, the global R&D investment on microalgae biofuel vastly increased. The reasons for this are that microalgae have high photosynthetic efficiency and high oil content and are believed to be the most promising feedstock for environment-friendly renewable liquid biofuel thanks to the fact that they do not compete with food crops for arable land and water supply [2-7]. However, until now, the success of microalgae cultures only happened in some small-scale tests and no success in biofuel production or CO2 mitigation has yet been achieved on a commercial scale anywhere in the world with microalgae cultivation [8,9]. The primary reason for this is that even when state-of-the-art techniques and designs are adopted on a large scale for open ponds or closed photobioreactors (PBRs), microalgal cell growth rate values currently cannot reach those obtained in highly controlled lab environments. The highest recorded biomass productivities for open ponds and PBRs are around 40 g dry mass m−2 day−1 and long-term averaged productivity is 10~20 g dry mass m−2 day−1 [3,10,11], which are far less than the theoretical maximum of 120~150 g m−2 day−1, or photosynthetic efficiency of 12.4% (based on total solar radiation spectrum) and 28% (based on visible light spectrum, 400~700 nm) [12,13]. The biofilm cultivation of microalgae, in which the algal cells are generally immobilized and fixed onto/into supporting materials in high density and fed with nutrient solutions, is a different cultivation method than conventional aqueous suspended cultures. Open ponds and PBRs have existed for a long time now; however, they often exhibited relatively low biomass productivity [14-21]. Recently, our research group proposed an improved cultivation method based on a reactor where both the immobilized biofilm technology and the light dilution theory work together in a patented design called 'attached cultivation' [22]. The basic principles of this 'attached cultivation' technique as well as the related photobioreactors include the following: (i) a highly dense wet algal paste attached onto artificial supporting materials to form a thin layer of the algal biofilm and (ii) many layers of these biofilms arranged in an array fashion to dilute the high light so that the light intensity impinging the algal biofilm is much lower than the light that reaches the photobioreactor footprint area (Figure 1A). The biomass productivity potential of this 'attached cultivation' method was evaluated both under indoor and outdoor conditions. Results showed that with indoor light of 700 μmol m−2 s−1 and a light dilution rate of 10, the maximum biomass productivity for Scenedesmus was close to 120 g m−2 day−1, with photosynthetic efficiency of ca. 18% (based on visible light). Under outdoor conditions, the maximum biomass productivity reached ca. 80 g m−2 day−1 corresponding to a photosynthetic efficiency of 17.3% (based on visible light) and 8.3% (based on total solar irradiation), which were both seven times higher than the data reported for a conventional open pond under the same environment conditions [22]. Similar biomass productivity and photosynthetic efficiency were also achieved with Botryococcus braunii which grows slowly with aqueous suspended cultivation but exhibits a high biomass productivity of ca. 50 g m−2 day−1 with attached cultivation, corresponding to a photosynthetic efficiency of ca. 15% (based on visible light) [23]. The schematic diagrams of the photobioreactors for attached cultivation. (A) The schematic diagram for the multiple-layer photobioreactor for attached cultivation (adapted from Liu et al. [22]). (B) The schematic diagram and the actual photograph for the single-layer photobioreactor for attached cultivation used in the research. As shown in these results, the 'attached cultivation' technique greatly improves the biomass productivity as well as photosynthetic efficiency of microalgae. However, the reason(s) and mechanism(s) for the superiorities of this method over suspended cultivation have not been fully understood yet. Maybe the secret of its superior performances over traditional suspended cultures lies in the differences in delivery of light and nutrients and in CO2 transfer between the two systems [24,25]. In this research, we especially focused on how the illumination worked, because light is always considered as the most important environmental factor that determines the behaviors of the microalgal cells [12,26]. A basic but important aspect to consider is how deep light penetrates during the microalgal cultivation. For open-pond suspended cultures, it is obvious that with the increases of biomass density, the light penetration depth decreases [12] and results in less and less percentage of algal cells being illuminated with light intensity which is higher than the compensation point (effectively illuminated) where the biomass accumulation rate due to photosynthesis equilibrates with the biomass loss rate due to respiration at a specific biomass density [11]. However, until to now, the dynamics of the 'light' vs. 'dark' for the open pond has not been quantitatively assessed. The information on illumination properties of the immobilized biofilm cultivation is scarce. How does the light transfer and penetrate inside the immobilized microalgal biofilm? Does light transfer and penetrate differently in the microalgal biofilm and in a suspended culture? How is the difference(s) related with the performances in photosynthetic efficiency? To answer these questions, the growth, photosynthetic oxygen evolution property, and specific growth rate for microalgal cells in open-pond and 'attached cultivation' systems were studied to determine the dynamics of effective light penetration (d E, depth from the surface where the light intensity is equal to the light compensation point) during cultivation. The light condition inside the outdoor open pond as well as the biofilm was revealed, and the effect of light distribution on the photosynthetic efficiency was discussed. The light distribution inside the suspended cultivation system The light intensity inside the open pond declined with the increase of broth depth as well as biomass concentration (Figure 2B). Similar results had been reported by Tredici [12]. According to the oxygen evolution properties, light intensity of 12.5 ± 3.9 μmol m−2 s−1 was considered as the light compensation point (LCP) for Scenedesmus dimorphus; under these circumstances, the oxygen consumption due to respiration already surpassed the oxygen evolution due to photosynthesis (Figure 2C). It should be noticed that the photosynthesis-light intensity (PI) curve of Figure 2C was measured on healthy vegetative algal cells that had been cultivated in the outdoor open pond for 5 days. According to our pilot experiment, from day 0 to day 30, there were non-significant differences among LCPs of algal cells from the outdoor open pond supplied with a non-nitrogen-deficient nutrient solution. The d E for any given biomass density can be estimated from the results of light attenuation and LCP (Figure 2D). Accordingly, the d E was only 0.57 cm at a biomass density of 5.0 g L−1, which means that only 40.4% of cells were effectively illuminated for a glass column having a diameter of 5 cm. In these conditions, the maximum biomass density would be 5.0 g L−1 after 10 days with continuous illumination of 100 μmol m−2 s−1 under indoor conditions. This d E value might also be affected by light conditions, for example, the d E might decrease in cloudy weather. To avoid this uncertainty, in this experiment, d E measurement was performed under outdoor conditions with natural sunlight intensity of 1,500~1,600 μmol m−2 s−1 so that this equation could be directly applied to outdoor cultivation. The light attenuation inside the culture broth of suspended cultivation of Scenedesmus dimorphus . (A) The schematic diagrams of light intensity measurements under different depths of the culture broth. (B) The dependence of light attenuation on culture density as well as depth. Seven different culture densities were tested, viz. 6.4 (black circle), 3.2 (white circle), 1.59 (black down-pointing triangle), 0.81 (white down-pointing triangle), 0.39 (black square), 0.18 (white square), and 0.07 (black diamond). The measurements were carried out outdoors with natural light. Data were mean ± standard deviation of three measurements. (C) The relationship of oxygen evolution rate versus light intensity (black triangle). Data were mean ± standard deviation of three measurements. The light compensation point (LCP) was indicated by arrows. (D) The effective illumination depth of aqueous suspended S. dimorphus culture broth at different biomass densities (white triangle). In this study, we proved that for a conventional 20-cm deep open pond, the maximum biomass density would reach 0.45 g L−1 in 10 days and remain stable in the following days (Figure 3A). This maximum biomass density of 0.45 g L−1 in the open pond, corresponding to 90 g m−2, was similar with the results of Chisti [11] where the biomass density for open pond could not exceed 0.5 g L−1. The pH during the cultivation slightly fluctuated in the range of 6.7 ~ 7.5, indicating that the carbon source was not limited during the experiment (Figure 3A). According to the equation in Figure 2D, thanks to the low biomass concentration, during the first 3 days of cultivation, light could easily penetrate the suspended culture delivering optimal light intensity to each algal cell. After 31 days, only 31.1% of the algal cells were effectively illuminated, with the effective d E being only 6.2 cm. In this research, the fastest biomass accumulation for the open pond was recorded from day 3 to day 7, corresponding to an increase of the biomass concentration from 0.1 to 0.3 g L−1 or areal biomass density from 20 to 60 g m−2. During the same time window, biomass productivity reached ca. 10 g m−2 day−1, and the effective light penetration continuously decreased from 100% to 45.5% (Figure 3B). The biomass productivity did not decrease with the reduced effective illumination depth during these times but remained at 10 g m−2 day−1. In microalgal suspended cultures, light distribution is not uniform at all. Light intensity decreases exponentially as we move farther away from the illuminated surface; for this reason, a thin layer on the water surface usually receives oversaturating light intensity while the bottom of the pond, as the algae cells reproduce and the biomass concentration increases, lies in total darkness. The algal cells traveling through the dark portion of the reactor consume biomass by dark respiration, if they spend a long enough time in this kind of environment. An appropriate depth of the water layer in an open pond is required to avoid biomass loss by cellular respiration. The optimal depth of any suspended culture of algae cells should be equal to the maximum depth at which a sufficient amount of solar light can penetrate. The aforementioned values for how 'light' and 'dark' regions expand and contract as the culture grows thicker are very likely to change depending on the different geographic locations considered other than Qingdao, China, where this experiment was conducted and the particular algal strain considered or the reactor design adopted. Identifying the upper limit of biomass density and the maximum depth at which algae cells are still effectively illuminated depending on the specific design of the reactor (different thicknesses of the water layer) has been an interesting task for our research team. Obtaining this information will definitely help to better understand how light travels through the water layer of an open pond, and it will also be useful knowledge for enhancing biomass productivity. The biomass increase, pH, and effective illumination depth of conventional aqueous suspended open ponds outdoors. (A) The biomass density (black circle) and pH (white circle) changes of the open pond outdoors. Data for biomass were mean ± standard deviation of six measurements for two independent open ponds (three measurements for each). Data for pH were mean ± standard deviation of two measurements for two independent open ponds (one measurement for each). (B) The effective illumination depth (black triangle) and percentage of effective illuminated algal cells (white triangle) at different days of cultivation in outdoor open ponds. The light distribution inside the biofilm of the attached cultivation system One of the methods to determine the d E of the attached microalgal biofilm is to directly measure the light intensity from underneath the biofilm. If the light intensity was equal to or higher than the LCP, then the corresponding depth could be effectively illuminated. However, we must also consider that the photosynthetically active wavelengths (red light) are more prone to be absorbed by chlorophyll than other less desired wavelengths (green light). The method we just described to measure light intensity might overestimate the d E if only the quantum flux intensity is measured because the undesired wavelengths travel farther away through layers of cells than the desired ones. To avoid the aforementioned problem in this research, we used a different method to estimate the d E by monitoring the specific growth rate of a thin layer of microalgal cells, which was called 'marker' layer, inserted just beneath the re-constructed biomass layer. An optimum 'marker' layer should have two main characteristics, viz. 1) thin enough to avoid the formation of light gradient inside the layer and 2) sensitive enough to the changes in light intensity. Algal cells from this thin layer should experience the same conditions as the upper layer. However, in pilot experiments, we found that such thin marker layers failed to meet requirement #2. In other words, the biomass changes for this kind of thin layers were too slight to be detected by regular gravimetric method, especially under lower light conditions. As an alternative, the algal cells used to inoculate the attached cultivation were chosen as the 'marker' layer because they were photosynthetically active even at low biomass densities. However, the risk of over- or underestimating the d E with this method was still present if the LCP of the inoculum algal cells was lower or higher than the LCPs of daughter cells that only experienced attached cultivation conditions. The LCP for the newly formed microalgal cells was firstly measured at different days. Typical light curves for oxygen evolution rate were obtained for all of these detached and re-suspended samples (Figure 4A). The LCPs of the daughter cells from day 0 to day 10 were in the range of 12.6 ± 2.2 ~ 19.8 ± 3.2 μmol m−2 s−1, without significant differences (P > 0.05) (Figure 4B). These results indicated that it is safe to determine the d E of the attached algal biofilm with this double-layer method under nutrient-replete condition without worrying about over- or underestimation. We also tested the LCP changes for the attached algal biofilm under nitrogen-depleted conditions and found that it increased dramatically to ca. 60 μmol m−2 s−1 at day 2. Because of this, a 'marker' layer consisting of the cells used to inoculate the attached cultivation surface cannot be used to estimate the d E of the nitrogen-starved attached cultivation. The oxygen evolution characters of Scenedesmus dimorphus under attached cultivation for different days of cultivation. (A) The oxygen evolution rate versus light intensity (light curve) of S. dimorphus under attached cultivation for 0 (black circle), 2 (white circle), 4 (black down-pointing triangle), 6 (white down-pointing triangle), 8 (black square), and 10 (white square) days of cultivation. Data were mean ± standard deviation of six measurements for three independent experiments (two measurements for each). (B) The light compensation point of S. dimorphus under attached cultivation for different days of cultivation. The ANOVA (SPSS, Chicago, IL, USA) result indicated that there were no differences for different days. The results of the double-layer experiments for attached cultivation of S. dimorphus under nitrogen-replete condition are shown in Figure 5. For the biofilm layer that has been attached cultivated for several days, the specific growth rate of the 'marker' layer (μ) decreased as the biomass of the upper layer increased, mainly because the light traveling through the upper layer became weaker and weaker (Figure 5A). Meanwhile, the rate of this decrease for μ gradually slowed down as the days went by, resulting in the increase of the thicknesses for the effectively illuminated upper layers (Figure 5A). According to the results in Figure 5A, the d E for attached cultivation under nitrogen-replete condition increased from 33.9 ± 6.5 μm for day 0 to 237.3 ± 11.8 μm for day 10, corresponding respectively to biomass densities of 14.6 ± 1.6 g m−2 and 111.5 ± 4.1 g m−2, whereas the actual biomass density increased from 8.8 ± 1.3 g m−2 to 107.6 ± 2.6 g m−2 in 10 days, corresponding to thicknesses of 21.7 ± 5.9 μm and 229.1 ± 8.6 μm (Figure 5B), respectively. The ratios of d E vs. actual biomass were almost equal to 1 throughout the experiments except for day 0 and day 2 which were 1.66 and 0.68, respectively. The fact that d E was similar to the actual biomass density indicated that nearly 100% of the algal cells inside the immobilized biofilm were effectively illuminated under attached cultivation. The estimation of effective illumination depth for the attached cultivation of S . dimorphus with twin-layer method. (A) The specific growth rate of lower layer versus the upper layer biomass density. Data were mean ± standard deviation of nine measurements for three independent experiments (three measurements for each). The upper layers were consisted with biomass grown with attached cultivated technique for 0 (black circle), 2 (white circle), 4 (black down-pointing triangle), 6 (white down-pointing triangle), 8 (black square), and 10 (white square) days. (B) The actual biomass density for the attached cultivation from day 0 to day 10 (closed bar), and the effective illumination depth for the attached cultivation from day 0 to day 10 (black triangle). Chlorophyll is the main light-harvesting molecule for photosynthetic organism, but some of the carotenoid molecules can also capture light and pass the excited energy to chlorophyll [27]. In this research, the content of chlorophyll as well as carotenoid was measured and values have been expressed as both a function of the dry biomass and the cultivated surface area. From day 0 to day 10, the dry mass-based chlorophyll content decreased from 3.3% to 0.4% and the carotenoid content decreased from 0.5% to 0.2%. The areal chlorophyll content increased from 0.29 to 0.70 g m−2 during the first 8 days of attached cultivation and then decreased to 0.45 g m−2 at day 10. The areal carotenoid content increased from 0.05 to 0.26 g m−2 from day 0 to day 8 and then remained constant. In general, during the attached cultivation, the increase of the d E was accompanied by the decrease in biomass-based chlorophyll and carotenoid contents as well as the increase in areal chlorophyll and carotenoid contents (Figure 6). Figure 7 suggests how the light transfer and distribution inside the immobilized microalgal biofilm are thought to be happening. In the earlier phase of attached cultivation, the pigment content in every single cell is relatively high, so that the light intensity decreased sharply when passing through each algal layer and can penetrate only a small depth of the algal biofilm. With the increase of the cultivation days, the algal layers as well as the pigment content along the light path increased but the pigment content in each single cell decreased, and only a moderate decrease in light intensity happens when passing through each algal layer. In the end, light can penetrate a bigger depth of the algal biofilm (Figure 7). The chlorophyll and carotenoid contents for the attached cultivation of S . dimorphus . (A) The areal (black circle) and dry mass-based (white circle) chlorophyll contents for attached cultivation of S. dimorphus for different days of cultivation. Data were mean ± standard deviation of six measurements for three independent experiments (two measurements for each). (B) The areal (black square) and dry mass-based (white square) carotenoid contents for attached cultivation of S. dimorphus for different days. Data were mean ± standard deviation of six measurements for three independent experiments (two measurements for each). The schematic model for the chlorophyll and light distribution inside the attached cultivation biofilm of S . dimorphus . The small oval represents the algal cells inside the biofilm. The green dots indicate the chlorophyll content of algal cells. The suggested light penetration path inside the biofilm is indicated by red arrows. Comparison of the illumination aspects of the suspended and attached cultivation The obvious difference between the two cultivation systems was that full illumination only happened for a short period of time (day 0 to day 3) in the open-pond system, when biomass concentration was very dilute (<20 g m−2), whereas in the attached cultivation system, the full illumination happened during a prolonged interval of 10 days as well as a wider range of biomass density (up to 100 g m−2 for a single layer). This high percentage of photosynthetic active biomass might explain the high biomass productivity and photosynthetic efficiency of attached cultivation under high light conditions. For example, in our previous research with a multiple-plate attached cultivation bioreactor with a light dilution rate of 10 (refer to Figure 1A), the land areal biomass density increased from 163.7 to 484.5 g m−2 in 5 days [22] under outdoor high light conditions (1,500 ~ 2,000 μmol m−2 s−1). Since the averaged light intensity impinging the cultivated surface was only about 100 ~ 150 μmol m−2 s−1, we can assume that nearly 100% of the biomass in this case was effectively illuminated. Similar results were also achieved under indoor conditions. The high photosynthetic efficiency of ca. 18% (based on visible light) was reached for Scenedesmus indoors with a light intensity of 700 μmol m−2 s−1 in the land areal biomass density range of 192.5 to 828.1 g m−2 [22]. Photosynthetic efficiency of ca. 15% (based on visible light) was also reached for Botryococcus braunii indoors with a light intensity of 500 μmol m−2 s−1 in the standing areal biomass density range of ca. 70 to ca. 580 g m−2 [23]. It is hard to imagine an open pond working with such high percentage of photosynthetic active cells and consequently such a condensed biomass concentration range. In general, for the attached cultivation, the high light intensity is diluted and absorbed by a cultivated area hosting an extremely dense biomass, where there is virtually no space left between adjacent cells, which is totally photosynthetically active, whereas in conventional open ponds, the high light intensity is only absorbed by a much smaller amount of algal cells so that the average light energy received by each photosynthetically active cell is higher in the open pond and a large portion of light is dissipated in a non-photosynthetic pathway before reaching other cells. In a conventional 20-cm deep open pond, all of the algal cells were fully illuminated for the first 3 days of outdoor cultivation. With the increase of the biomass concentration, the effective illuminated portion of the water layer decreased steeply to 45.5% from day 4 to day 10 and then gradually decreased to 31.1% at day 31, when the biomass density reached 0.45 g L−1 or 90 g m−2. In the attached cultivation system under nitrogen-replete condition, almost 100% of the immobilized algal cells inside the biofilm were effectively illuminated from day 0 through day 10, with biomass density increasing from 8.8 to 107.6 g m−2. Attached cultivation facilitates super high biomass density (up to ca. 800 g m−2) and super high efficient illumination (100%) when the reactor consists of multiple panels hosting the biofilm surface which might be the key to the high biomass productivity as well as high photosynthetic efficiency. Microalgae strain and medium The freshwater microalgal species S. dimorphus was locally screened in Qingdao, China. The algal cells were maintained in BG11 medium [28], each liter of which contains 1.5 g NaNO3, 0.075 g MgSO4·7H2O, 0.036 g CaCl2·2H2O, 0.04 g KH2PO4·H2O, 0.02 g Na2CO3, 6.0 × 10−3 g citric acid, 1.0 × 10−3 g Na2EDTA, 6.0 × 10−3 g ferric ammonium citrate, 2.22 × 10−4 g ZnSO4·7H2O, 6.9 × 10−5 g CuSO4·5H2O, 1.81 × 10−3 g MnCl2·4H2O, 3.9 × 10−4 g Na2MoO4·2H2O, 4.94 × 10−5 g Co(NO3)2·6H2O and 2.86 × 10−3 g H3BO3. Inoculum preparation The algal inoculum was prepared in glass columns (diameter = 0.05 m, working volume = 0.7 L) under indoor conditions of continuous illumination (100 μmol m−2 s−1; cold fluorescent lamp, white light; FSL T8 36W, Foshan Electrical Lighting Co., Ltd., Foshan, China) and continuous aeration (1.0 vvm; 98% air + 2% CO2, v/v). The temperature of the culture broth was 25°C ± 1°C. The algal biomass at late exponential phase (5~7 days after inoculation, 2~3 g L−1) was used to inoculate the experimental systems. The d E for the algal biofilm of the attached cultivation and conventional aqueous suspended open pond were determined under indoor and outdoor conditions, respectively. For the attached cultivation, the d E was measured with a single-layer attached cultivation system (Figure 1B) following the flow chart of Figure 8. Firstly, the algal cells were attached cultivated for 10 days with BG-11 (step 1; refer to the 'Cultivation systems' section for details). Every other day, some of the algal disks were sampled and the attached cells on it were detached and washed three times with de-ionized water (step 2). A 'marker' layer was built by filtering some fresh inoculum onto new cellulose acetate/nitrate membranes at the biomass density of s (g m−2). Then, some aliquots of the re-suspended algal culture from step 2 were gently filtered onto the 'marker' layers to form 'double-layer' attached algal disks. The corresponding 'single-layer' attached algal 'disks' were also prepared at the same time as a control using identical aliquots of re-suspended algal culture but without the 'marker' layer (step 3). Both of the newly prepared 'double-layer' and 'single-layer' algal disks were cultivated for 24 h under the same environmental conditions using the PBR depicted in Figure 1B. The biomass increase for the 'double-layer' algal disk was denoted by a (g m−2) and the biomass increase for the corresponding 'single-layer' algal disk was a' (g m−2), so that the net biomass gain for the 'marker' layer was calculated as δ = a − a' (g m−2), and μ was calculated as: The schematic flow chart of the experiment design to determine the effective illumination depth for the attached cultivation. $$ \mu = \ln \left(s+\delta \right)- \ln (s) $$ If μ > 0, it means that the 'marker' layer is receiving a light intensity higher than the LCP. In other words, the upper layer is not thick enough to fully attenuate the impinging light and the depth of the upper layer (d A) was only a fraction of the d E, d A < d E. If μ < 0, it means that the light intensity reaching the 'marker' layer is lower than the LCP and the upper layer is too thick to allow the 'marker' layer to accumulate biomass, d A > d E. If μ = 0, the light intensity at the 'marker' layer is equal to the LCP and the biomass increase of the biofilm due to photosynthesis is equal to the biomass loss due to cellular respiration. The thickness of the upper layer could be considered as the maximum distance the light can effectively penetrate, d A = d E (Figure 8, step 4). In actual experiments, however, considering the difficulties in precisely manipulating the amount of the upper-layer biomass to make d A = d E, the d E was generally approximately estimated as the intercept of the x-axis (Figure 8, step 5). The percentage of the effectively illuminated cells (P light, %) for attached cultivation was calculated as: $$ {P}_{\mathrm{light}}={d}_{\mathrm{E}}\times 100/{d}_{\mathrm{G}} $$ where d G represents the actual thickness of the attached algal 'disk' at the considered sampling day. The d E in the suspended cultivation system was determined with a conventional open-pond system and a set of homemade equipment with different light paths (Figure 2A; refer to the sections 'Equipments for determining the effective illumination depth for suspended cultures' and 'Cultivation systems' for detailed specifications and operations). Firstly, the relationships of biomass concentrations vs. penetrated light intensities were measured (Figure 2B). Secondly, according to the LCP determined by oxygen evolution rates at different light intensities (Figure 2C; refer to the 'Determination of LCP' section for details), the d E were calculated for each tested biomass densities. An equation was proposed to calculate the d E at any given biomass density in the range of 0.07 ~ 6.4 g L−1 (Figure 2D). Finally, the changes in biomass density of S. dimorphus in open ponds were studied outdoors (Figure 3A), and the d E was estimated according to the proposed equation. The percentage of the effectively illuminated cells (P light, %; Figure 3B) was calculated as: $$ {P}_{\mathrm{light}}={d}_{\mathrm{E}}\times 100/{d}_{\mathrm{P}} $$ where the d P represents the depth of the open pond, 0.2 m in this research. Equipments for determining the effective illumination depth for suspended cultures A series of white polyvinyl chloride (PVC) tubes (inner diameter = 0.03 m) with different lengths (2, 3, 5, 10, 20, 30, 40, and 50 cm; three tubes for each length) were prepared, and one of the open ends of these PVC tubes was covered with a piece of highly transparent polystyrene plate (thickness = 0.8 mm; >98% light transmittance). The outer wall of the PVC tubes and the outer circles of the polystyrene sheets were painted black to prevent the unwanted penetration of outside irradiance, whereas the inner wall and the inner circles (diameter = 1 cm) were not painted but thoroughly cleaned with water (Figure 2A). Algal suspensions with different biomass densities, that is, 0.07, 0.18, 0.39, 0.81, 1.59, 3.20 and 6.4 g L−1, were prepared in beakers and then used to fill the PVC tubes, so that the heights of the algal broth, viz. light paths, were equal to the lengths of the PVC tubes. The PVC pipes were placed vertically in open air, under direct sunlight with an intensity of approximately 1,500 ~ 1,600 μmol m−2 s−1. A photosynthetic quantum sensor (Li-250A, Li-Cor, Lincoln, NE, USA) was placed underneath each tube and tightly attached to the polystyrene sheets to measure the penetrated photon flux density. When low water layer thicknesses were to be tested, that is, 0.084 ~ 1.05 cm, the algal suspension was poured into a Petri dish made of highly transparent polystyrene (thickness = 0.8 mm; >98% light transmittance) and then the penetrated light intensity was measured. The thickness of the water layer in the Petri dish was calculated according to the volume of algal suspension poured into and the diameter of the Petri dish used. The external surface of the Petri dish was also painted with black paint with the exception of a small circular window (diameter = 1 cm) where the quantum sensor was to be attached. Cultivation systems Two different cultivation systems were adopted in this research to evaluate the active proportions for photosynthesis. The first one was a mini open-pond system which measured 2 m in length, 1 m in width, and 0.2 m in culture depth. The culture broth was propelled by a paddle wheel at a flow speed of 0.3 m s−1. A gas stream of pure CO2 (>99%, v/v) was injected into the bottom of the pond through a ventilation stone at an aeration speed of 4 L min−1 (0.01 vvm) from 8 am to 4 pm to supply the carbon source for the algal cells and maintain a pH of 7.5 ± 0.5. The measured maximum light intensity at the water surface was ca. 2,000 μmol m−2 s−1 during the daytime. The temperature of the culture broth was 30°C~35°C from 12 pm to 2 pm in the afternoon on a sunny day and 10°C~25°C for the rest of the time. Two identical ponds were operated synchronously in outdoor conditions at Qingdao, China (35°35′ N, 119°30′ E) from 17 September to 17 October 2012, with the same inoculated biomass density of 0.04 g L−1 (8 g m−2). The second adopted cultivation system was the single-layer attached biofilm cultivation system (Figure 1B) which has been introduced in detail in our previous research as a 'type 1' photobioreactor [22]. In brief, algal cells were evenly filtered onto a cellulose acetate/nitrate membrane (pore size 0.45 μm) to form algal disks with 10 ± 0.5 cm2 of footprint at an inoculated areal biomass density of 8 g m−2. These algal disks were then put onto a layer of regular filter paper and finally onto a 0.2 m × 0.4 m glass plate (3 mm thick). The glass plate as well as the filter paper and algal disks on it were sealed in a 0.5 m × 0.3 m × 0.05 m chamber made out of 5-mm-thick transparent glass plates. During the cultivation, the algal disks were continuously illuminated with 100 ± 10 μmol m−2 s−1 of white light from cold fluorescent lamps. A continuous air stream that contained 1.5% (v/v) of CO2 was injected into the chamber with a flow rate of 0.75 L min−1 (0.1 vvm). The BG11 medium was dripped onto the filter paper with a speed of 0.5 mL min−1 to keep the algal disks attached to the filter paper and provide nutrients for the growth of the algal cells. Only fresh medium was used, and any excess was immediately discarded. The temperature inside the glass chamber was kept at 25°C ± 2°C during the experiment. Growth analysis In this research, the growth of the algal culture was investigated by measuring the changes in dry biomass concentration (DW). Some aliquots of the culture broth (open pond) or detached biomass from the 'algal disk' (attached cultivation) were first washed with 50 mL of de-ionized water and then filtered onto a pre-weighted 0.45-μm GF/C filter membrane (Whatman, Little Chalfont, England; DW0). The membrane was oven dried at 105°C for 24 h and then weighted (DW1). The DW for open pond (g L−1) was calculated as: $$ \mathrm{D}\mathrm{W}=\left({\mathrm{DW}}_1-{\mathrm{DW}}_0\right)\times 1,000/\nu $$ where v represents the sample volume expressed in milliliters. The DW for attached cultivation was calculated as: $$ \mathrm{D}\mathrm{W}=\left({\mathrm{DW}}_1-{\mathrm{DW}}_0\right)/0.001 $$ where 0.001 represents the footprint area (m2) of attached algal cells. Determination of LCP The LCP was determined according to the photosynthetic oxygen evolution rate vs. light intensity curve (PI curve); this was measured with a Chlorolab-2 liquid-phase Clark-type oxygen electrode (Hansatech, Norfolk, England). The newly prepared inoculum for the algal disks (refer to the 'Inoculum preparation' section) was collected by centrifugation at 5,000 × g for 30 s (Allegra X-22R, Beckman Coulter, Brea, CA, USA). The obtained pellet was then re-suspended with BG-11 medium containing 100 mM NaHCO3 and adjusted to a proper biomass concentration so that the chlorophyll content was as close as possible to 10 μg per milliliter. One milliliter of the sample was added to the reaction cuvette and then bubbled with nitrogen gas for 30 s to expel the dissolved oxygen gas. Light intensities of 400, 200, 100, 80, 60, 40, 20 and 0 μmol m−2 s−1 were applied to drive the photosynthetic oxygen evolution. The measurement was repeated three times for each light intensity. The LCP was obtained as the intercept on the horizontal axis of the PI curve [29]. Determination of the thickness of the attached algal disk Prior to the experiments, the thickness of the algal disks at different cultivation days was measured by means of a surface profiler (Veeco Dektak150, Veeco Instruments Inc., Plainview, NY, USA) and the corresponding areal biomass density was also measured (refer to the 'Growth analysis' section). Accordingly, the correlation of the thickness (d, expressed in μm) of the algal film with the areal biomass density (DW, in g m−2) was obtained as follows: $$ d=2.1\times \mathrm{D}\mathrm{W}+3.2 $$ For later experiments, the thickness (μm) of the attached algal disks during cultivation was estimated by this equation. According to the pilot experiments, this equation could be applied in the biomass range of 0 ~ 120 g m−2 for S. dimorphus grown with the attached cultivation technique for up to 10 days under both nitrogen-replete or nitrogen-depleted conditions (R 2 = 0.99). Determination of the chlorophyll and carotenoid contents The chlorophyll and carotenoid contents were determined according to the methods described by Wellburn [30]. The algal biomass was harvested and washed with plenty of de-ionized water and then mixed with methanol under 60°C for 12 h in darkness until the algal cells whitened completely. The optical densities of the extraction were measured with a spectrophotometer at 666, 653, and 470 nm (Cary 50, Varian Inc., Palo Alto, CA, USA). The chlorophyll a (Chla), chlorophyll b (Chlb), and carotenoid (Car) concentrations in the extraction were calculated as follows (mg L−1): $$ {\mathrm{Chl}}_{\mathrm{a}}=15.65\times {\mathrm{OD}}_{666}-7.34\times {\mathrm{OD}}_{653} $$ $$ {\mathrm{Chl}}_{\mathrm{b}}=27.05\times {\mathrm{OD}}_{653}-11.21\times {\mathrm{OD}}_{666} $$ $$ \mathrm{Car}=\left(1,000\times {\mathrm{OD}}_{470}-2.86\times {\mathrm{Chl}}_{\mathrm{a}}-129.2\times {\mathrm{Chl}}_{\mathrm{b}}\right)/221 $$ biomass increase for the 'twin-layer' algal disk biomass increase for the corresponding 'single-layer' algal disk concentration of carotenoid Chla : concentration of chlorophyll a Chlb : concentration of chlorophyll b D : thickness of the algal film d A : the depth of the upper layer d E : effective illumination depth d G : actual thickness of the attached algal 'disk' at different sampling days DW: dry biomass concentration DW0 : dry weight of the filter membrane dry weight of the filter membrane after filtering the algal cells LCP: light compensation point PI curve: curve of photosynthetic oxygen evolution rate vs. light intensity P light : percentage of the effectively illuminated cells V : volume of the sample for biomass measurement δ : net biomass gain for the 'seed' layer μ : specific growth rate Chisti Y. 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Effect of initial biomass density on growth and astaxanthin production of Haematococcus pluvialis in an outdoor photobioreactor. J Appl Phycol. 2013;25:253–60. Green BR, Durnford DG. The chlorophyll-carotenoid proteins of oxygenic photosynthesis. Annu Rev Plant Physiol Plant Mol Biol. 1996;47:685–714. Stanier RY, Kunisawa R, Mandel M, Cohenbaz G. Purification and properties of unicellular blue-green algae (order Chroococcales). Bacteriol Rev. 1971;35:171–205. Ye Z. A new model for relationship between light intensity and the rate of photosynthesis in Oryza sativa. Photosynthetica. 2007;45:637–40. Wellburn AR. The spectral determination of chlorophylls a and b, as well as total carotenoids, using various solvents with spectrophotometers of different resolution. J Plant Physiol. 1994;144:307–13. This work was supported by the National Natural Science Foundation of China (41276144), the Director Innovation Foundation of Qingdao Institute of Bioenergy and Bioprocess Technology, CAS, and the Science and Technology Development Planning of Shandong Province (2013GHY11520). Key Laboratory of Biofuels, Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Sciences, Qingdao, Shandong, 266101, People's Republic of China Junfeng Wang & Tianzhong Liu Low-Carbon Energy Conversion Science and Engineering Center, Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, 201203, People's Republic of China Jinli Liu Junfeng Wang Tianzhong Liu Correspondence to Tianzhong Liu. JW and TL proposed the idea and hypothesis. JW carried out the experiment design and outdoor open-pond experiment and drafted the manuscript. JL carried out the biofilm experiment. TL performed the statistical analysis and helped to draft and revise the manuscript. All authors read and approved the final manuscript for publication. Wang, J., Liu, J. & Liu, T. The difference in effective light penetration may explain the superiority in photosynthetic efficiency of attached cultivation over the conventional open pond for microalgae. Biotechnol Biofuels 8, 49 (2015). https://doi.org/10.1186/s13068-015-0240-0 Photosynthetic efficiency Open pond Attached cultivation Effective light penetration	CommonCrawl
\begin{document} \begin{abstract} A flat of a matroid is cyclic if it is a union of circuits; such flats form a lattice under inclusion and, up to isomorphism, all lattices can be obtained this way. A lattice is a Tr-lattice if all matroids whose lattices of cyclic flats are isomorphic to it are transversal. We investigate some sufficient conditions for a lattice to be a Tr-lattice; a corollary is that distributive lattices of dimension at most two are Tr-lattices. We give a necessary condition: each element in a Tr-lattice has at most two covers. We also give constructions that produce new Tr-lattices from known Tr-lattices. \end{abstract} \maketitle \section{Introduction} A flat $X$ of a matroid $M$ is \emph{cyclic} if the restriction $M\|X$ has no isthmuses. Ordered by inclusion, the cyclic flats form a lattice, which we denote by $\mathcal{Z}(M)$. Every lattice is isomorphic to the lattice of cyclic flats of some (bi-transversal) matroid~\cite{cyclic,julie}. (All lattices considered in this paper are finite.) For certain lattices $L$, it is shown in~\cite{acketa,ac} that if $\mathcal{Z}(M)$ is isomorphic to $L$, then the matroid $M$ is transversal; lattices with this property are \emph{transversal lattices} or \emph{Tr-lattices}. In~\cite{cyclic}, lattices of width at most two are shown to be Tr-lattices. In this paper we treat a more general sufficient condition for a lattice to be a Tr-lattice, we prove a necessary condition, and we show that the class of Tr-lattices is closed under certain lattice operations. Following a section of background, Section~\ref{sec:sufficient} introduces MI-lattices and shows they are Tr-lattices. Special cases (e.g., distributive lattices of dimension at most two) are also treated. Section~\ref{sec:necessary} shows that each element of a Tr-lattice has at most two covers. Section~\ref{sec:examples} gives ways to construct new MI-lattices (resp., Tr-lattices) from known MI-lattices (resp., Tr-lattices). Some open problems suggested by this work are mentioned in the concluding section. \section{Background} We assume familiarity with basic matroid theory. Our notation and terminology for matroid theory follow~\cite{ox}; for ordered sets we mostly follow~\cite{tomt}. For a collection $\mathcal{F}$ of sets, we write $\bigcap(\mathcal{F})$ for the intersection $\bigcap_{X\in \mathcal{F}}X$ and $\bigcup(\mathcal{F})$ for $\bigcup_{X\in \mathcal{F}}X$. Recall that every ordered set $P$ can be embedded in a product of chains; the \emph{dimension} of $P$ is the least number of chains for which there is such an embedding. The lattices of dimension $2$ are the \emph{planar} lattices: their Hasse diagrams can be drawn in the plane without crossings (see, e.g.,~\cite[Chapter~3, Theorem~5.1]{tomt}). An \emph{antichain} in an ordered set is a collection of mutually incomparable elements. The \emph{width} of an ordered set is the maximal cardinality among its antichains. We say $y$ is a \emph{cover} of $x$ in an ordered set $P$ if $x<y$ and there is no $z$ in $P$ with $x<z<y$. The least and greatest elements in a lattice are denoted $\hat{0}$ and $\hat{1}$, respectively. The \emph{atoms} of a lattice are the elements that cover $\hat{0}$; dually, the \emph{coatoms} are the elements that $\hat{1}$ covers. An \emph{ideal} in an ordered set $P$ is a subset $I$ of $P$ such that if $x\in I$ and $y\leq x$, then $y\in I$. Dually, a \emph{filter} in $P$ is a subset $F$ such that if $x\in F$ and $y\geq x$, then $y\in F$. It is well known and easy to see that while nonisomorphic matroids can have the same cyclic flats, a matroid on a given set is determined by its collection of cyclic flats along with their ranks. In some cases we will want to ignore the cyclic flats and instead focus on the ranks assigned to the elements of an abstract lattice; this is justified by the following special case of~\cite[Theorem 1]{julie}. \begin{prop}\label{prop:sims} Let $L$ be a lattice. Given $\rho:L\rightarrow \mathbb{Z}$ with \begin{itemize} \item[(a)] $\rho(\hat{0})=0$, \item[(b)] $\rho(x)<\rho(y)$ whenever $x<y$, and \item[(c)] $\rho(x\lor y) + \rho(x\land y)\leq \rho(x) + \rho(y)$ whenever $x$ and $y$ are incomparable, \end{itemize} there is a matroid $M$ and an isomorphism $\phi:L\rightarrow\mathcal{Z}(M)$ with $\rho(x)=r\bigl(\phi(x)\bigr)$. \end{prop} A key result we use to prove that certain lattices are (or are not) Tr-lattices is the following characterization of transversal matroids, which was first formulated by Mason using cyclic sets and later refined to cyclic flats by Ingleton~\cite{ingleton}. (The statement in~\cite{ingleton} uses all nonempty collections of cyclic flats, but an elementary argument shows that it suffices to consider nonempty antichains of cyclic flats; see the discussion after~\cite[Lemma~5.6]{cyclic}.) \begin{prop}\label{prop:mi} A matroid $M$ is transversal if and only if for every nonempty antichain $\mathcal{A}$ in $\mathcal{Z}(M)$, \begin{equation}\label{mi} r\bigl(\bigcap(\mathcal{A})\bigr) \leq \sum_{\mathcal{F}\subseteq\mathcal{A}} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr).\tag{MI} \end{equation} \end{prop} The join in $\mathcal{Z}(M)$ (as in the lattice of flats) is given by $A\lor B=\hbox{\rm cl}(A\cup B)$, so one can replace the alternating sum in inequality~(MI) by the corresponding alternating sum of ranks of joins of cyclic flats. Unlike in the lattice of flats, the meet operation in $\mathcal{Z}(M)$ might not be intersection: $X\land Y$ is the union of the circuits that are contained in $X\cap Y$. Since the complements of the flats of a matroid are the unions of its cocircuits, $X$ is a cyclic flat of $M$ if and only if $E(M)-X$ is a cyclic flat of the dual, $M^$. Thus, $\mathcal{Z}(M^)$ is isomorphic to the order dual of $\mathcal{Z}(M)$. Let $S$ and $E$ be the least and greatest cyclic flats of $M$. Note that for $X\in \mathcal{Z}(M)$, the lattice $\mathcal{Z}(M\|X)$ is the interval $[S,X]$ in $\mathcal{Z}(M)$ and, dually, the lattice $\mathcal{Z}(M/X)$ is isomorphic to the interval $[X,E]$ in $\mathcal{Z}(M)$ via the isomorphism $Y\mapsto Y\cup X$. (The lattices of cyclic flats of other minors are not as simple to describe.) \section{Sufficient conditions for a lattice to be a Tr-lattices}\label{sec:sufficient} To convey the spirit of the main result of this section (Theorem~\ref{thm:primitive}) before defining the technical condition involved, we cite the following theorem, which, as we will show, is implied by the main result. \begin{thm}\label{thm:sublattice} If (a) $\mathcal{Z}(M)$ has dimension at most two and (b) for each antichain $\mathcal{A}$ of $\mathcal{Z}(M)$, the sublattice of $\mathcal{Z}(M)$ generated by $\mathcal{A}$ is distributive, then $M$ and all of its minors, as well as their duals, are transversal. \end{thm} \begin{cor} If $\mathcal{Z}(M)$ is distributive and has dimension at most two, then $M$ and all of its minors, as well as their duals, are transversal. \end{cor} The main result of this section uses the following notions. \begin{dfn}\label{def:mi} An \emph{MI-ordering} of an antichain $\mathcal{A}$ in a lattice $L$ is a permutation $a_1,a_2,\ldots,a_t$ of $\mathcal{A}$ so that \begin{itemize} \item[(i)] $a_i\lor a_{i+1}\lor \cdots\lor a_k = a_i\lor a_k$ for $1\leq i<k\leq t$ and \item[(ii)] $(a_1\land a_2\land\cdots\land a_k)\lor a_{k+1}=a_k\lor a_{k+1}$ for $1<k<t$. \end{itemize} An antichain is \emph{MI-orderable} if it has an MI-ordering. A lattice is \emph{MI-orderable}, or is an \emph{MI-lattice}, if each of its antichains is MI-orderable. \end{dfn} \begin{thm}\label{thm:primitive} Let $M$ be a matroid. \begin{enumerate} \item[(i)] Each MI-orderable antichain in $\mathcal{Z}(M)$ satisfies inequality~(MI). \item[(ii)] If $\mathcal{Z}(M)$ is MI-orderable, then $M$ and all of its minors are transversal. \end{enumerate} \end{thm} \begin{cor} MI-lattices are Tr-lattices. \end{cor} \begin{figure} \caption{The lattice of cyclic flats of a matroid $M$ and that of $M/x$.} \label{fig:minor} \end{figure} Before proving Theorem~\ref{thm:primitive}, we note a subtlety that explains the approach we take to prove part (ii): if $N$ is a minor of $M$ and $\mathcal{Z}(M)$ is MI-orderable, then $\mathcal{Z}(N)$ may or may not be MI-orderable. Indeed, $\mathcal{Z}(N)$ may not even be a Tr-lattice, and this is so even for deletions of $M$. (Recall that the class of transversal matroids is closed under deletions but not under contractions, so one might expect deletions to be somewhat more tame.) For example, for the matroid $M$ in Figure~\ref{fig:minor}, $\mathcal{Z}(M)$ is MI-orderable. Since this lattice is isomorphic to its order dual, $\mathcal{Z}(M^)$ is also MI-orderable. The lattice $\mathcal{Z}(M/x)$ is also shown; by checking directly or applying Theorem~\ref{thm:ideals}, we have that $\mathcal{Z}(M/x)$ is MI-orderable. However, by Theorem~\ref{thm:cover}, its order dual, which is $\mathcal{Z}(M^\backslash x)$, is not a Tr-lattice. This example also shows that the minor-closed, dual-closed class of matroids described in Theorem~\ref{thm:sublattice} is not determined by lattice-theoretic properties that apply to the lattices of cyclic flats of all matroids in the class. We prove Theorem~\ref{thm:primitive} via a sequence of lemmas. The first lemma gives a rank inequality associated with each MI-orderable antichain of $\mathcal{Z}(M)$. Note that for two-element antichains, this inequality is the semimodular inequality. (The meet and join operations in this and other results are in $\mathcal{Z}(M)$.) \begin{lemma}\label{lemma:in} Let $A_1,A_2,\ldots,A_t$ be an antichain of cyclic flats in a matroid $M$ such that $(A_1\land A_2\land\cdots\land A_k)\lor A_{k+1}=A_k\lor A_{k+1}$ whenever $1\leq k<t$. Then for $k$ with $k\leq t$, \begin{equation}\label{dkeyineq} r(A_1\cap A_2\cap \cdots \cap A_k)\leq \sum_{i=1}^k r(A_i)-\sum_{i=1}^{k-1}r(A_i\cup A_{i+1}). \end{equation} \end{lemma} \begin{proof} We prove the inequality by induction on $k$. Equality holds for $k=1$. Assume the result holds in case $k$. Semimodularity gives $$r(A_1\cap A_2\cap \cdots \cap A_{k+1})+ r\bigl((A_1\cap A_2\cap \cdots \cap A_k)\cup A_{k+1}\bigr)- r(A_1\cap A_2\cap \cdots \cap A_k)\leq r(A_{k+1}).$$ Adding this inequality to inequality~(\ref{dkeyineq}) gives $$r(A_1\cap A_2\cap \cdots \cap A_{k+1}) + r\bigl((A_1\cap A_2\cap \cdots \cap A_k)\cup A_{k+1}\bigr)\leq \sum_{i=1}^{k+1} r(A_i)-\sum_{i=1}^{k-1}r(A_i\cup A_{i+1}),$$ so if we show $r\bigl((A_1\cap A_2\cap \cdots \cap A_k)\cup A_{k+1}\bigr)=r(A_k\cup A_{k+1})$, then the inequality we want follows. This equality holds since $A_1\land A_2\land \cdots \land A_k\subseteq A_1\cap A_2\cap \cdots \cap A_k \subseteq A_k$ and $(A_1\land A_2\land \cdots \land A_k)\lor A_{k+1}=A_k\lor A_{k+1}$. \end{proof} \begin{lemma}\label{lemma:t} If an antichain $\mathcal{A}$ in $\mathcal{Z}(M)$ can be ordered as $A_1,A_2,\ldots,A_t$ so that \begin{itemize} \item[(i)] $A_i\lor A_{i+1}\lor \cdots\lor A_k = A_i\lor A_k$ whenever $1\leq i<k\leq t$ and \item[(ii)] $r(A_1\cap A_2\cap \cdots \cap A_t) \leq \sum_{i=1}^t r(A_i)-\sum_{i=1}^{t-1}r(A_i\cup A_{i+1}),$ \end{itemize} then $\mathcal{A}$ satisfies inequality~(MI). \end{lemma} \begin{proof} Assume properties (i) and (ii) hold. For $1\leq i\leq j\leq t$, set $$\mathcal{A}_{i,j}= \{\mathcal{F}\,:\,\mathcal{F}\subseteq\mathcal{A},\,\, i=\min(k\,:\,A_k\in\mathcal{F}), \text{ and } j=\max(k\,:\,A_k\in\mathcal{F}) \}.$$ Thus, if $\mathcal{F}\in \mathcal{A}_{i,j}$, then $\hbox{\rm cl}\bigl(\bigcup(\mathcal{F})\bigr)=A_i\lor A_j$. If $j>i+1$, then the terms on the right side of inequality~(MI) that arise from the sets in $\mathcal{A}_{i,j}$ cancel since there is a parity-switching involution $\phi$ of $\mathcal{A}_{i,j}$: fix $k$ with $i<k<j$ and let $$\phi(\mathcal{F})= \left\{ \begin{array}{ll} \mathcal{F}\cup\{A_k\}, &\mbox{if $A_k\not\in\mathcal{F}$;}\\ \mathcal{F}-\{A_k\}, &\mbox{if $A_k\in\mathcal{F}$.} \end{array}\right.$$ Thus, inequality~(MI) reduces to the inequality that is assumed in property (ii). \end{proof} The previous two lemmas show that MI-lattices are Tr-lattices. To prove the stronger assertion in part (ii) of Theorem~\ref{thm:primitive}, we show that if the antichains in $\mathcal{Z}(M)$ satisfy the hypotheses of Lemma~\ref{lemma:t}, then so do the antichains of single-element deletions and single-element contractions of $M$. (Note that unlike the hypotheses of Theorem~\ref{thm:primitive}, condition (ii) in Lemma~\ref{lemma:t} is not a lattice-theoretic property.) We start with a lemma about the cyclic flats of such minors. \begin{lemma}~\label{lem:cydel} For an element $x$ of $M$ and a cyclic flat $A$ of either $M\backslash x$ or $M/x$, the flat $\bar{A}=\hbox{\rm cl}_M(A)$ of $M$ is cyclic; furthermore, $\bar{A}$ is either $A$ or $A\cup x$, so $\bar{A}-x=A$. \end{lemma} \begin{proof} For a cyclic flat $A$ of $M\backslash x$, the assertions are transparent. Let $A$ be a cyclic flat of $M/x$ and let $S$ be the ground set of $M/x$. Thus, $S-A$ is a cyclic flat of the dual of $M/x$, that is, of $M^\backslash x$, so $\hbox{\rm cl}_{M^}(S-A)$, which is either $S-A$ or $(S-A)\cup x$, is a cyclic flat of $M^$. Therefore either $A\cup x$ or $A$ is a cyclic flat of $M$, from which the result follows. \end{proof} \begin{lemma} If each antichain in $\mathcal{Z}(M)$ can be ordered so that properties \emph{(i)} and \emph{(ii)} of Lemma~\emph{\ref{lemma:t}} hold, then the same is true for each antichain in $\mathcal{Z}(M\backslash x)$ and each antichain in $\mathcal{Z}(M/x)$. \end{lemma} \begin{proof} The proofs for $\mathcal{Z}(M\backslash x)$ and $\mathcal{Z}(M/x)$ are similar and, since each deletion of a transversal matroid is transversal, only the result about contractions is needed to prove Theorem~\ref{thm:primitive}, so we treat only $\mathcal{Z}(M/x)$. We use the notation $\bar{A}$ of Lemma~\ref{lem:cydel}. Let $\mathcal{A}$ be an antichain in $\mathcal{Z}(M/x)$. Note that $\{\bar{A}\,:\,A\in\mathcal{A}\}$ is an antichain in $\mathcal{Z}(M)$. By hypothesis, there is an ordering $A_1,A_2,\ldots,A_t$ of $\mathcal{A}$ so that in $M$ and $\mathcal{Z}(M)$, \begin{equation}\label{del1} \bar{A}_i\lor \bar{A}_{i+1}\lor \cdots\lor \bar{A}_k = \bar{A}_i\lor \bar{A}_k, \quad\text{ for } 1\leq i<k\leq t, \end{equation} and \begin{equation}\label{del2} r_M(\bar{A}_1\cap \bar{A}_2\cap \cdots \cap \bar{A}_t)+\sum_{i=1}^{t-1}r_M(\bar{A}_i\cup \bar{A}_{i+1})\leq \sum_{i=1}^t r_M(\bar{A}_i). \end{equation} Since $\bar{A}_j=\hbox{\rm cl}_M(A_j)$ and since $A\lor B$ in $\mathcal{Z}(M)$ is $\hbox{\rm cl}_M(A\cup B)$, by equation~(\ref{del1}) $A_i\cup A_{i+1}\cup\cdots\cup A_k$ and $A_i\cup A_k$ have the same closure in $M$, and so in $M/x$; thus, as needed, $A_i\lor A_{i+1}\lor \cdots\lor A_k = A_i\lor A_k$ in $\mathcal{Z}(M/x)$. The rank inequality in $M/x$ is immediate if $x$ is a loop of $M$, so assume this is not the case. Assume $x$ is in exactly $h$ of the cyclic flats $\bar{A}_1,\bar{A}_2,\ldots,\bar{A}_t$ of $M$. Thus, $$h+\sum_{i=1}^t r_{M/x}(A_i)=\sum_{i=1}^t r_M(\bar{A}_i).$$ Also, since $x$ is in at least $h$ of the sets $\bar{A}_1\cap \bar{A}_2\cap \cdots \cap \bar{A}_t$ and $\bar{A}_i\cup \bar{A}_{i+1}$, we have \begin{align} h+ r_{M/x}(A_1\cap A_2\cap& \cdots \cap A_t) +\sum_{i=1}^{t-1}r_{M/x}(A_i\cup A_{i+1}) \notag \\ & \leq r_M(\bar{A}_1\cap \bar{A}_2\cap \cdots \cap \bar{A}_t)+\sum_{i=1}^{t-1}r_M(\bar{A}_i\cup \bar{A}_{i+1}). \notag \end{align} The last two conclusions and inequality (\ref{del2}) give $$r_{M/x}(A_1\cap A_2\cap \cdots \cap A_t) +\sum_{i=1}^{t-1}r_{M/x}(A_i\cup A_{i+1})\leq \sum_{i=1}^t r_{M/x}(A_i),$$ as needed. \end{proof} The lemmas above complete the proof of Theorem~\ref{thm:primitive}. We now show that Theorem~\ref{thm:sublattice} follows. Recall that $\mathcal{Z}(M^)$ is isomorphic to the order dual of $\mathcal{Z}(M)$, so $M^$ satisfies the hypotheses of Theorem~\ref{thm:sublattice} if and only if $M$ does. Thus, the next lemma suffices to prove Theorem~\ref{thm:sublattice}. \begin{lemma}\label{lemma:o} If a lattice $L$ has dimension at most two and each of its antichains generates a distributive sublattice, then $L$ is MI-orderable. \end{lemma} \begin{proof} We may assume $L$ is a suborder of $\mathbb{N}^2$. List the elements of an antichain as $a_1,a_2,\ldots,a_t$ where $a_i=(x_i,y_i)$ with $x_1>x_2>\cdots>x_t$; thus, $y_1<y_2<\cdots<y_t$. Clearly $a_i\lor a_{i+1}\lor \cdots\lor a_k \geq a_i\lor a_k$. Let $a_i\lor a_k$ be $(p,q)$. Thus, $p\geq x_i$ and $q\geq y_k$, so $(p,q)\geq (x_j,y_j)$ for $i\leq j\leq k$, and so $a_i\lor a_{i+1}\lor \cdots\lor a_k = a_i\lor a_k$. One gets $a_i\land a_{i+1}\land \cdots\land a_k = a_i\land a_k$ similarly, so property (ii) of Definition~\ref{def:mi} can be rewritten as $(a_1\land a_k)\lor a_{k+1} = a_k\lor a_{k+1}$, or, by the distributive law, $(a_1\lor a_{k+1})\land (a_k\lor a_{k+1})=a_k\lor a_{k+1}$, that is, $a_1\lor a_{k+1} \geq a_k\lor a_{k+1}$. This property holds since $a_1\lor a_{k+1}=a_1\lor a_2\lor \cdots\lor a_k\lor a_{k+1}\geq a_k\lor a_{k+1}$. \end{proof} Antichains of at most two elements are trivially MI-orderable, so Theorem~\ref{thm:primitive} has the following corollary (as do Theorem~\ref{thm:sublattice} and Proposition~\ref{prop:mi}). \begin{cor}\label{cor:width2} \emph{\cite[Theorem 5.7.]{cyclic}} Lattices of width at most two are Tr-lattices. \end{cor} Section~\ref{sec:examples} includes examples of lattices to which Theorem~\ref{thm:primitive} but not Theorem~\ref{thm:sublattice} applies, as well as Tr-lattices that are not MI-lattices. \section{A necessary condition for a lattice to be a Tr-lattice}\label{sec:necessary} Condition (ii) of Definition~\ref{def:mi} is violated by any three covers of a given element, so each element of an MI-lattice has at most two covers. In this section, we show that the same is true of any Tr-lattice. (The examples in the next section show there is no bound on the number of elements that an element in a Tr-lattice covers.) \begin{thm}\label{thm:cover} Each element of a Tr-lattice has at most two covers. \end{thm} \begin{proof} Let the element $x$ of a lattice $L$ have at least three covers. We prove that $L$ is not a Tr-lattice by defining a function $\rho:L\rightarrow \mathbb{Z}$ so that properties (a)--(c) in Proposition~\ref{prop:sims} hold and inequality (MI) fails. For $y\in L$, let $F_y$ be the principal filter $\{u\,:u\geq y\}$ in $L$. Thus, the sublattice $F_x$ of $L$ has at least three atoms. Define $\rho':L\rightarrow \mathbb{Z}$ by $\rho'(y) = \bigl\|L-F_y\bigr\|$. It follows easily that $\rho'$ satisfies properties (a)--(c) in Proposition~\ref{prop:sims}. For $u,v,w\in F_x-\{x\}$, let $$m(u,v,w)=\rho'(u)+\rho'(v)+\rho'(w)-\rho'(u\lor v)-\rho'(u\lor w)-\rho'(v\lor w)+\rho'(u\lor v\lor w)-\rho'(x).$$ By inclusion-exclusion, $m(u,v,w)=\|F_x-(F_u\cup F_v\cup F_w)\|$. Set $$k=\,\min\{m(u,v,w)\,:\,u,v,w > x\} = \|F_x\|-\max\{\bigl\|F_u\cup F_v\cup F_w\bigr\| \,:\,u,v,w > x\}.$$ Thus, $k$ is the minimal size of the complement, in $F_x$, of the union of three proper principal filters in $F_x$. Note that if $k=m(u,v,w)$, then $u,v,w$ are distinct covers of $x$. Define $\rho:L\rightarrow\mathbb{Z}$ by \begin{eqnarray} \notag \rho(y) = \left\{ \begin{array}{ll} \rho'(y), &\mbox{if $y\leq x$,}\\ \rho'(x)-k-1, &\mbox{otherwise}. \end{array} \right. \end{eqnarray} Clearly $\rho$ satisfies property (a) of Proposition~\ref{prop:sims}. Properties (b) and (c) for $\rho$ follow from these properties for $\rho'$ except in two cases, which we address below: \begin{itemize} \item[(i)] $\rho(y)<\rho(z)$ if $y<z$, $y\leq x$, and $z\not\leq x$, and \item[(ii)] $\rho(y)+\rho(z)\geq \rho(y\lor z)+ \rho(y\land z)$ if $y\not\leq x$, $z\not\leq x$, and $y\land z\leq x$. \end{itemize} Assume $y<z$, $y\leq x$, and $z\not\leq x$. Thus, $F_x\subseteq F_y$. The inequality in statement (i) reduces to $k+2\leq \rho'(z) - \rho'(y) = \|F_y-F_z\|$. Note that $F_z\cap F_x$ is the principal filter $F_{z\lor x}$, which, since $z\not \leq x$, is properly contained in $F_x$; thus, there are at least $k+2$ elements in $F_x-F_z$, and so in $F_y-F_z$, which proves statement (i). Now assume $y\not\leq x$, $z\not\leq x$, and $y\land z\leq x$. The inequality in statement (ii) is $$\|L\|-\|F_y\|-k-1 + \|L\|-\|F_z\|-k-1\geq \|L\|-\|F_{y\lor z}\|-k-1 + \|L\|-\|F_{y\land z}\|,$$ that is, $\|F_{y\land z}-(F_y\cup F_z)\|\geq k+1.$ Note that $F_x\subseteq F_{y\land z}$ and $$(F_y\cup F_z)\cap F_x = (F_y\cap F_x)\cup(F_z\cap F_x)=F_{y\lor x}\cup F_{z\lor x},$$ which is the union of two principal filters that are properly contained in $F_x$; thus, there are at least $k+1$ elements in $F_x-(F_y\cup F_z)$ and so in $F_{y\land z}-(F_y\cup F_z)$, which proves statement (ii). Let $M$ be a matroid arising from $L$ and $\rho$ as in Proposition~\ref{prop:sims}. Fix $u,v,w$ with $k=m(u,v,w)$ and let $U$, $V$, and $W$ be the corresponding cyclic flats of $M$. The definitions of $m$ and $\rho$ give $$r(U)+r(V)+r(W)-r(U\cup V)-r(U\cup W)-r(V\cup W)+r(U\cup V\cup W)=r(X)-1.$$ Since $r(X)\leq r(U\cap V\cap W)$, it follows that the antichain $\{U,V,W\}$ of $\mathcal{Z}(M)$ does not satisfy inequality (MI). Thus, $M$ is not transversal, so $L$ is not a Tr-lattice. \end{proof} A matroid $M$ is \emph{nested} if $\mathcal{Z}(M)$ is a chain. These matroids have arisen many times in a variety of contexts (see~\cite[Section 4]{lpm2} for more information). That $\mathcal{Z}(M\oplus N)$ is isomorphic to the product $\mathcal{Z}(M)\times\mathcal{Z}(N)$ gives the following corollary. \begin{cor} If $\mathcal{Z}(M)$ is a Tr-lattice, then the matroid obtained from $M$ by deleting all loops and isthmuses is either a direct sum of at most two nested matroids or it is connected. \end{cor} \section{Examples and constructions}\label{sec:examples} This section gives examples of MI-lattices to which Theorem~\ref{thm:sublattice} does not apply and Tr-lattices that are not MI-lattices. We also show how to construct new MI-lattices from given MI-lattices, and likewise for Tr-lattices. \begin{figure} \caption{The lattices Acketa considered.} \label{fig:lattices} \end{figure} Acketa~\cite{acketa,ac} proved that chains and the lattices $L_1$, $L_2$, and $L_3$ of Figure~\ref{fig:lattices} are Tr-lattices (Corollary~\ref{cor:width2} applies); he noted that $L_4$ is not a Tr-lattice; he conjectured that $L_5$, $L_6$, and $L_7$ are Tr-lattices (Corollary~\ref{cor:width2} applies to $L_5$ and $L_6$; Theorem~\ref{thm:cover} shows that $L_7$ is not a Tr-lattice); he proved that $L_8$ is a Tr-lattice; he also showed that $L_9$ (the dual of $L_8$) is not a Tr-lattice. We note that $L_8$ is in an infinite family of MI-lattices; Figure~\ref{fig:examples1}.a gives another such lattice. The defining properties of these lattices are that the interval between $\hat{0}$ and any coatom is a chain, and for one of these chains (e.g., the left-most chain in Figure~\ref{fig:examples1}.a), all other such chains intersect it in different initial segments. \begin{figure} \caption{(a): A generic lattice like $L_8$. (b): A lattice $L_I$ obtained from an ideal in a product of two three-element chains.} \label{fig:examples1} \end{figure} Sublattices of MI-lattices are clearly MI-lattices. The next result gives another simple construction for MI-lattices. (See Figure~\ref{fig:examples1}.b.) \begin{thm}\label{thm:ideals} For any ideal $I$ in an MI-lattice $L$, the lattice $L_I$ induced on the set $I\cup\{\hat{1}\}$ by the same order is MI-orderable. \end{thm} \begin{proof} Each antichain $\mathcal{A}$ of $L_I$ is an antichain of $L$; order $\mathcal{A}$ so that properties (i) and (ii) of Definition~\ref{def:mi} hold in $L$. Let $z$ be the join of $\{a_i,a_{i+1},\ldots,a_k\}$ and of $\{a_i,a_k\}$ in $L$. If $z \in I$, then $z$ is the join of each of these sets in $L_I$, otherwise both sets have join $\hat{1}$ in $L_I$. Thus, property (i) holds in $L_I$. The same ideas show that property (ii) holds in $L_I$ since the meet operations are identical in $L$ and $L_I$. \end{proof} Recall that the linear sum (or ordinal sum) of partial orders $P$ and $Q$, where $P$ and $Q$ are disjoint, is the order on $P\cup Q$ in which the restriction to $P$ is the order on $P$, the restriction to $Q$ is the order on $Q$, and every element of $P$ is less than every element of $Q$. The following result is immediate. \begin{thm}\label{thm:linsum} The class of MI-lattices is closed under linear sums. \end{thm} The same result holds for the closely-related operation that, given lattices $L$ and $L'$, forms the Hasse diagram of the new lattice from those of $L$ and $L'$ by identifying the greatest element of $L$ with the least element of $L'$. It follows from Theorem~\ref{thm:lex} below that the same two results hold for Tr-lattices. By Theorem~\ref{thm:cover}, the class of MI-lattices and the class of Tr-lattices are not closed under direct products. We next treat three particular Tr-lattices of dimension $3$, only one of which is MI-orderable. These lattices, which are shown in Figure~\ref{fig:examples2}, are among the forbidden sublattices for planar lattices (see~\cite{kr}). (No other forbidden sublattices for planar lattices satisfy the necessary condition for Tr-lattices given in Theorem~\ref{thm:cover}.) \begin{figure} \caption{Three nonplanar Tr-lattices; only $D^d$ is MI-orderable.} \label{fig:examples2} \end{figure} \begin{thm}\label{thm:nonplanar} The lattice $D^d$ is MI-orderable. The lattices $F_0$ and $C$ are Tr-lattices that are not MI-orderable. \end{thm} \begin{proof} The sublattice of $D^d$ formed by removing $b$ is the linear sum of the lattice in Figure~\ref{fig:examples1}.b and a single-element lattice. Since this linear sum is MI-orderable, we need only check that the antichains in $D^d$ that contain $b$ are MI-orderable. Antichains of two elements are automatically MI-orderable; the only larger antichain in $D^d$ that includes $b$ is $\{a,b,c\}$, for which $b,a,c$ is an MI-ordering. In $F_0$, the antichains of more than two elements are $\{A,S,X\}$, $\{X,T,D\}$, and $\{X,A,D\}$. The first two are MI-orderable (ordered as written), so we need only show that in any matroid $M$ for which $\mathcal{Z}(M)$ is isomorphic to $F_0$, the flats corresponding to $X,A,D$ (for which we use the same notation) satisfy inequality (MI), which in this case is $r(X)+r(A)+r(D)-r(R)-r(E)\geq r(X\cap A\cap D)$. By semimodularity, $$r(A)+r(S)\geq r(E)+r(A\cap S).$$ The inclusions $T\subseteq A\cap S\subseteq S$ give $\hbox{\rm cl}\bigl((A\cap S)\cup X\bigr)=R$, so $$r(A\cap S)+r(X)\geq r(R)+r(A\cap S\cap X).$$ The inclusions $U\subseteq A\cap S\cap X\subseteq S$ give $\hbox{\rm cl}\bigl((A\cap S\cap X)\cup D\bigr)=S$, so $$r(A\cap S\cap X)+r(D)\geq r(S)+r(A\cap S\cap X\cap D).$$ Note that $A\cap S\cap X\cap D$ is $A\cap X\cap D$. Adding the three inequalities and simplifying yields the desired inequality. A similar argument applies to the lattice $C$, for which it suffices to consider the antichains $\{A,B,Y\}$, $\{A,W,Y\}$, $\{B,V,Y\}$, and $\{V,W,Y\}$. The last three are listed in MI-orderings. For $\{A,B,Y\}$, apply semimodularity to the pairs $\{A,S\}$, $\{A\cap S,B\}$, and $\{A\cap S\cap B,Y\}$; the inclusions $V\subseteq A\cap S\subseteq S$ and $T\subseteq A\cap S\cap B\subseteq S$ give $\hbox{\rm cl}\bigl((A\cap S)\cup B\bigr)=E$ and $\hbox{\rm cl}\bigl((A\cap S\cap B)\cup Y\bigr)=S$; add the resulting inequalities and cancel the common terms to get inequality (MI) for $\{A,B,Y\}$. \end{proof} We now consider two operations for producing new Tr-lattices. Given lattices $L_1$ and $L_2$, let $L_1L_2$ be the lattice on $(L_1\cup L_2\cup \{\hat{0},\hat{1}\})-\{\hat{1}_{L_1}, \hat{1}_{L_2}\}$ with $x\leq y$ if and only if (i) $y=\hat{1}$, or (ii) $x=\hat{0}$, or (iii) for some $i\in\{1,2\}$, both $x$ and $y$ are in $L_i$ and $x\leq y$ in $L_i$. Figure~\ref{fig:star}.a illustrates this operation; note that the unique four-element antichain in this lattice is not MI-orderable. \begin{thm}\label{thm:star} If $L_1$ and $L_2$ are Tr-lattices, then so is $L_1L_2$. \end{thm} The proofs of Theorems~\ref{thm:star} and~\ref{thm:lex} are similar, so we prove only the second result, which concerns lexicographic sums~\cite[Section~1.10]{tomt}. Let $L$ be a lattice and let $\mathcal{L}=(L_x\,:\,x\in L)$ be a family of lattices that is indexed by the elements of $L$. The lexicographic sum $L \oplus \mathcal{L}$ is defined on the set $\{(x,a)\,:\, x\in L,\, a\in L_x\}$; the order is given by $(x,a)\leq (y,b)$ if and only if either (i) $x<y$ in $L$ or (ii) $x=y$ and $a\leq b$ in $L_x$. Figure~\ref{fig:star}.b illustrates this operation. It is easy to see that $L \oplus \mathcal{L}$ is not necessarily MI-orderable even if all of the constituent lattices are. \begin{figure} \caption{(a) The lattice $L_1L_2$ where $L_1$ and $L_2$ are Boolean lattices on two elements. (b) A lexicographic sum; the indexing lattice is a Boolean lattice on two elements.} \label{fig:star} \end{figure} \begin{thm}\label{thm:lex} If $L$ has width at most two and if $\mathcal{L}=(L_x\,:\,x\in L)$ is a family of Tr-lattices, then $L\oplus \mathcal{L}$ is a Tr-lattice. \end{thm} \begin{proof} Let $\phi:\mathcal{Z}(M) \rightarrow L\oplus \mathcal{L}$ be an isomorphism. We must show that any antichain $\mathcal{A}$ in $\mathcal{Z}(M)$ satisfies inequality (MI). For $F\in \mathcal{Z}(M)$, let $\phi_1(F)$ be the first component of $\phi(F)$; thus, $\phi_1(F)\in L$. For $x\in \phi_1(\mathcal{A})$, set $\mathcal{A}_x= \{F\,:\, F\in \mathcal{A},\, \phi_1(F)=x\}$. Since $L$ has width at most two and $\mathcal{A}$ is an antichain in $\mathcal{Z}(M)$, there are at most two such sets; these sets partition $\mathcal{A}$. For $u\in L$, let $Z_u$ and $E_u$ be the least and greatest flats $F\in\mathcal{Z}(M)$ with $\phi_1(F)=u$. Thus, if $\phi_1(A)=u$ and $\phi_1(B)=v$ with $u\ne v$, then $A\lor B=Z_{u\lor v}$ and $A\land B=E_{u\land v}$ by the definition of $L\oplus \mathcal{L}$. Let $x$ be in $\phi_1(\mathcal{A})$. Note that $\mathcal{Z}(M\|E_x/Z_x)$ is isomorphic to $L_x$, so $M\|E_x/Z_x$ is transversal. Thus, by Proposition~\ref{prop:mi}, $$r_{M\|E_x/Z_x}\bigl(\bigcap(\mathcal{A}_x)-Z_x\bigr) \leq \sum_{\mathcal{F}\subseteq\mathcal{A}_x} (-1)^{\|\mathcal{F}\|+1} r_{M\|E_x/Z_x}\bigl(\bigcup(\mathcal{F})-Z_x\bigr),$$ which gives $$r\bigl(\bigcap(\mathcal{A}_x)\bigr) \leq \sum_{\mathcal{F}\subseteq\mathcal{A}_x} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr)$$ in $M$. If $\|\phi_1(\mathcal{A})\|=1$, then the last inequality is the required inequality (MI) for $\mathcal{A}$. If, instead, $\|\phi_1(\mathcal{A})\|=2$, let $\phi_1(\mathcal{A})=\{x,y\}$, so we also have $$r\bigl(\bigcap(\mathcal{A}_y)\bigr) \leq \sum_{\mathcal{F}\subseteq\mathcal{A}_y} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr).$$ The equality \begin{align} \sum_{\mathcal{F}\subseteq\mathcal{A}} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr) = \,& \sum_{\mathcal{F}\subseteq\mathcal{A}_x} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr) + \sum_{\mathcal{F}\subseteq\mathcal{A}_y} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr) \notag\\ & \qquad + \sum\limits_{\substack{\mathcal{F}_x\subseteq\mathcal{A}_x,\, \mathcal{F}_x\ne\emptyset \\ \mathcal{F}_y\subseteq\mathcal{A}_y,\, \mathcal{F}_y\ne\emptyset}} (-1)^{\|\mathcal{F}_x\|+\|\mathcal{F}_y\|+1} r\bigl(\bigcup(\mathcal{F}_x)\cup \bigcup(\mathcal{F}_y)\bigr)\notag \end{align} and that $r(X\cup Y)=r(Z_{x\lor y})$ for any $X\in L_x$ and $Y\in L_y$ give \begin{align} \sum_{\mathcal{F}\subseteq\mathcal{A}} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr) = \,& \sum_{\mathcal{F}\subseteq\mathcal{A}_x} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr) + \sum_{\mathcal{F}\subseteq\mathcal{A}_y} (-1)^{\|\mathcal{F}\|+1} r\bigl(\bigcup(\mathcal{F})\bigr) \notag\\ & \qquad -r(Z_{x\lor y}) \sum_{\mathcal{F}_x\subseteq\mathcal{A}_x,\, \mathcal{F}_x\ne\emptyset} (-1)^{\|\mathcal{F}_x\|} \sum_{\mathcal{F}_y\subseteq\mathcal{A}_y,\, \mathcal{F}_y\ne\emptyset} (-1)^{\|\mathcal{F}_y\|} \notag\\ \geq \,& r\bigl(\bigcap(\mathcal{A}_x)\bigr) + r\bigl(\bigcap(\mathcal{A}_y)\bigr) -r(Z_{x\lor y}) \notag\\ \geq \,& r\bigl(\bigcap(\mathcal{A})\bigr).\notag \end{align} (The last line uses semimodularity.) Thus, inequality (MI) holds, as needed. \end{proof} \section{Open problems}\label{sec:problems} The results in this paper suggest the following problems. \begin{enumerate} \item Is the converse of Theorem~\ref{thm:cover} true? \\ The following questions are of interest if the answer is negative. \begin{itemize} \item[(a)] Find a lattice-theoretic characterization of Tr-lattices, perhaps via a recursive description using operations such as those in Section~\ref{sec:examples}. \item[(b)] Is the converse of Theorem~\ref{thm:cover} true for planar lattices? \item[(c)] If $\mathcal{Z}(M)$ has the property of covers in Theorem~\ref{thm:cover}, is $M$ a gammoid? \item[(d)] Is every sublattice of a Tr-lattice also a Tr-lattice? Is this true at least for intervals, or upper intervals? \item[(e)] Is the counterpart of Theorem~\ref{thm:ideals} true for Tr-lattices? \end{itemize} \item Are there Tr-lattices, or MI-lattices, of all dimensions? \item Can one capture the minor-closed, dual-closed class of transversal matroids described in Theorem~\ref{thm:sublattice} by special presentations that such matroids have? What are the excluded minors for this class of matroids? \item Can one deduce any substantial properties of a matroid $M$ other than being a gammoid (or specializations, such as transversal or nested) from lattice-theoretic properties of $\mathcal{Z}(M)$? \item If $N$ is a minor of $M$ where $\mathcal{Z}(M)$ is a Tr-lattice, must $N$ be transversal? \item What can we say about $M$ (more particular than transversal) when $\mathcal{Z}(M)$ is a Tr-lattice? \end{enumerate} \begin{center} \textsc{Acknowledgements} \end{center} I am very grateful for Joseph Kung, whose questions and comments prompted me to pursue more deeply the implications of Proposition~\ref{prop:mi}. \end{document}	arXiv
\begin{document} \title{Limit laws for the diameter of a set of random points from a distribution supported by a smoothly bounded set } \author{Michael Schrempp } \institute{M. Schrempp \at Karlsruhe Institute of Technology, Institute of Stochastics, Englerstr. 2, 76131 Karlsruhe, Germany\\ Tel.: +49-721-60843264\\ \email{[email protected]} } \maketitle \begin{abstract} We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends to infinity, we consider the much more general setting in which the random points are the supports of appropriately defined Poisson processes. The main result covers the case of uniformly distributed points within a $d$-dimensional ellipsoid with a unique major axis. Moreover, several generalizations of the main result are established, for example a limit law for the maximum interpoint distance of random points from a Pearson type II distribution. \keywords{Maximum interpoint distance \and geometric extreme value theory \and Poisson process \and uniform distribution in an ellipsoid \and Pearson Type II distribution} \subclass{60D05 \and 60F05 \and 60G55 \and 60G70 \and 62E20} \end{abstract} \section{Introduction} \label{sec_introduction} For some fixed integer $d \ge 2$, let $Z,Z_1,Z_2,\ldots$ be independent and identically distributed (i.i.d.) $d$-dimensional random vectors, defined on a common probability space $(\Omega,{\cal A},\mathbb{P})$. We assume that the distribution $\mathbb{P}_Z$ of $Z$ is absolutely continuous with respect to Lebesgue measure. Writing $\|\cdot \|$ for the Euclidean norm on $\mathbb{R}^d$, the asymptotical behavior of the so-called maximum interpoint distance $$ M_n := \max\limits_{1 \le i , j \le n}\|Z_i - Z_j\| $$ as $n $ tends to infinity has been a topic of interest for more than 20 years. This behavior is closely related to the support $S\subset \mathbb{R}^d$ of $\mathbb{P}_{Z}$, which is the smallest closed set $C$ satisfying $\mathbb{P}_Z(C)=1$. Writing $$ \text{diam}(K):=\sup_{x,y \in K}\|x-y\| $$ for the diameter of a set $K \subset \mathbb{R}^d$, we obviously have $ M_n \overset{\text{a.s.}}{\longrightarrow } \text{diam}(S) $ as $n \to \infty $, but finding sequences $(a_n)_{n \in \mathbb{N}} $ and $(b_n)_{n \in \mathbb{N}}$ so that $a_n(b_n-M_n)$ has a non-degenerate limit distribution as $n \to \infty $ is a much more difficult problem, which has hitherto been solved only in a few special cases. We deliberately discard the case $d=1$ in what follows since then \begin{equation} M_n = \max_{1\le i \le n}Z_i - \min_{1\le i \le n}Z_i \end{equation} is the well-studied sample range. Results obtained so far mostly cover the case that $\mathbb{P}_{Z}$ is spherically symmetric, and they may roughly be classified according to whether $\mathbb{P}_{Z}$ has an unbounded or a bounded support. If $Z$ has a spherically symmetric normal distribution, \cite{Matthews1993} obtained a Gumbel limit distribution for $M_n$, and \cite{Henze1996} generalized this result to the case that $Z$ has a spherically symmetric Kotz type distribution. An even more general spherically symmetric setting with a Gumbel limit distribution has been studied by \cite{RaoJanson2015}. \cite{HenzeLao2010} studied unbounded distributions $\mathbb{P}_Z$, for which the norm $\|Z\|$ and the directional part $Z/\|Z\|$ of $Z$ are independent and the right tail of the distribution of $\|Z\|$ decays like a power law. In this case, they showed a (non-Gumbel) limit distribution of $M_n$ that can be described in terms of a suitably defined Poisson point process. Finally, \cite{Demichel2014} considered unbounded elliptical distributions of the form $ Z = TAW, $ where $T$ is a positive and unbounded random variable, $A$ is an invertible $(d \times d)$-dimensional matrix, and $W$ is uniformly distributed on the sphere $\mathcal{S}^{d-1} = \left\{ z \in \mathbb{R}^d: \|z\| = 1\right\}.$ In this case, the asymptotical behavior of $M_n$ depends on the right tail of the distribution function of $T$ and the multiplicity $k \in \left\{ 1,\ldots,d\right\}$ of the largest eigenvalue of $A.$ In that work, it was assumed that $T$ lies in the max-domain of attraction of the Gumbel law. If the matrix $A$ has a single largest eigenvalue, \cite{Demichel2014} derives a limit law for $M_n$ that can be represented in terms of two independent Poisson point processes on $\mathbb{R}^d$. On the other hand, if $A$ has a multiple largest eigenvalue and $T$ satisfies an additional technical assumption, $M_n$ has a Gumbel limit law. If $k=d$, the random vector $Z$ has a spherically symmetric distribution, and their result is the same as that stated by \cite{RaoJanson2015}. If $\mathbb{P}_{Z}$ has a bounded support, \cite{Lao2010} and \cite{MayerMol2007} deduced a Weibull limit distribution for $M_n$ in a very general setting if the distribution of $Z$ is supported by the $d$-dimensional unit ball $\mathbb{B}^d$ for $d \ge 2$. Furthermore, \cite{Lao2010} obtained limit laws for $M_n$ if $\mathbb{P}_{Z}$ is uniform or non-uniform in the unit square, uniform in regular polygons, or uniform in the $d$-dimensional unit cube, $d \ge 2$. \cite{Appel2002} obtained a convolution of two independent Weibull distributions as limit law of $M_n$ if $Z$ has a uniform distribution in a planar set with unique major axis and `sub-$\sqrt{x}$ decay' of its boundary at the endpoints. The latter property is \emph{not} fulfilled if $\mathbb{P}_Z$ is supported by a proper ellipse $E$. In that case, \cite{Appel2002} were able to derive bounds for the limit law of $M_n$ if $Z$ has a uniform distribution. The exact limit behavior of $M_n$ if $\mathbb{P}_{Z}$ is uniform in an ellipse has been an open problem for many years. Without giving a proof, \cite{RaoJanson2015} stated that $n^{2/3}(2-M_n)$ has a limit distribution (involving two independent Poisson processes) if $Z$ has a uniform distribution in a proper ellipse with major axis of length $2$. \cite{Schrempp2015} described this limit distribution in terms of two independent sequences of random variables, and \cite{Schrempp2016} generalized the result of \cite{RaoJanson2015} to the case that $\mathbb{P}_Z$ is uniform or non-uniform over a $d$-dimensional ellipsoid. Being more precise, the underlying set $E$ in \cite{Schrempp2016} is $$ E = \left\{z \in \mathbb{R}^d: \left( \frac{z_1}{a_1} \right)^2 + \left( \frac{z_2}{a_2} \right)^2 + \ldots + \left( \frac{z_d}{a_d} \right)^2 \le 1\right\}, $$ where $d\ge2$ and $a_1 > a_2 \ge a_3 \ge \ldots \ge a_d > 0$. Since $a_1 > a_2$, the ellipsoid $E$ has a unique major axis of length $2a_1$ with `poles' $(a_1,0,\ldots,0)$ and $(-a_1,0,\ldots,0)$. If the distribution $\mathbb{P}_Z$ is supported by such a set $E$ and $\mathbb{P}_Z(E \cap O)>0$ for each neighborhood $O$ of each of the two poles, the unique major axis makes sure that the asymptotical behavior of $M_n$ is determined solely by the shape of $\mathbb{P}_Z$ close to these poles. \cite{Schrempp2016} investigated distributions $\mathbb{P}_{Z}$ with a Lebesgue density $f$ on $E$, so that $f$ is continuous and bounded away from $0$ near the poles. Hence, the uniform distribution on $E$ was a special case of that work. It turned out that $2a_1 - M_n$ has to be scaled by the factor $n^{2/(d+1)}$ to obtain a non-degenerate limit distribution. In order to show this weak convergence, a related setting had been considered, in which the random points are the support of a specific series of Poisson point processes $\mathbf{Z}_n$ in $E$. Writing $\text{diam}(\mathbf{Z}_n)$ for the diameter of the support of $\mathbf{Z}_n$, it turned out that $n^{2/(d+1)}(2a_1-\text{diam}(\mathbf{Z}_n ))$ has a limiting distribution involving two independent Poisson processes that live on a subset $P$ of $\mathbb{R}^d$, the shape of which is determined by $a_1,\ldots,a_d$. By use of the so-called de-Poissonization technique, $n^{2/(d+1)}(2a_1-M_n)$ has the same limit distribution as $n$ tends to infinity.\\ From the proofs given in \cite{Schrempp2016}, it is quite obvious that only the values of the density at the poles and the curvature of the boundary $\partial E$ of $E$ at the poles determine the limiting distribution of $n^{2/(d+1)}(2a_1-M_n)$, but \emph{not} the fact that $E$ is an ellipsoid. The latter observation was the starting point for this work: Our main result is a generalization of the result stated in \cite{Schrempp2016} to distributions that are supported by a $d$-dimensional set $E$, $d \ge 2$, with `unique diameter' of length $2a>0$ between the poles $(-a,0,\ldots,0)$ and $(a,0,\ldots,0)$ and a smooth boundary at the poles. The formal assumptions on $E$ are stated in \autoref{sec_main_results}. If the density $f$ of $Z$ on $E$ is continuous and bounded away from $0$ close to the poles, $n^{2/(d+1)}(2a-\text{diam}(\mathbf{Z}_n ))$ has a non-degenerate limiting distribution also in this setting. Again, this limit law involves two independent Poisson processes that live on potentially different subsets $P_\ell$ and $P_r$ of $\mathbb{R}^d$. The shape of $P_\ell$ is only determined by the principal curvatures and the corresponding principal curvature directions of $\partial E$ at the left pole $(-a,0,\ldots,0)$. The same holds true for $P_r$ and the right pole $(a,0,\ldots,0)$. The paper is organized as follows. In \autoref{sec_fundamentals} we will fix our general notation, and \autoref{sec_main_results} contains our assumptions and our main result, which is \autoref{thm_main_result}, the proof of which will be given in \autoref{sec_proof_main_thm}. \autoref{sec_generalizations_unique_diameter} contains several generalizations of the main result for underlying sets with a `unique diameter'. These include more general distributions $\mathbb{P}_Z$, a limit theorem for the joint convergence of the $k$ largest distances among $Z_1,\ldots,Z_n$ and $p$-norms and so-called `$p$-superellipsoids', where $1 \le p < \infty $. \autoref{sec_generalizations_no_unique_diameter} deals with generalizations of our main result to settings where $E$ does not have a `unique diameter', and it concludes with a fundamental open problem concerning Pearson Type II distributions that are supported by an ellipsoid with at least two but less than $d$ major half-axes. \section{Fundamentals} \label{sec_fundamentals} Throughout, vectors are understood as column vectors, but if there is no danger of misunderstanding, we write them -- depending on the context -- either as row or as column vectors. We use the abbreviation $\widetilde z := (z_2,\ldots,z_d)$ for a point $z = (z_1,\ldots,z_d) \in \mathbb{R}^d$. Given a function $s:\mathbb{R}^{d-1} \to \mathbb{R}, \widetilde z \mapsto s(\widetilde z)$, let $s_j(\widetilde z)$ denote the partial derivative of $s$ with respect to the component $z_j$ for $j \in \left\{ 2,\ldots,d\right\}$. Notice that, for instance, $s_2$ stands for the partial derivative of $s$ with respect to $z_2$, \emph{not} with respect to the second component of $\widetilde z$. The gradient $\big(s_2(\widetilde z),\ldots,s_d(\widetilde z)\big)$ of $s$ at the point $\widetilde z$ will be denoted by $\nabla s(\widetilde z)$. Likewise $s_{ij}(\widetilde z)$ is the second-order partial derivative with respect to $z_i$ and $z_j$. Without stressing the dependence on the dimension, we write $\mathbf{0}$ for the origin in $\mathbb{R}^{i}$ and $\mathbf{e}_j$ for the $j$-th unit vector in $\mathbb{R}^{i}$ for $ i,j \in \mathbb{N} := \left\{ 1,2,\ldots\right\}$ with $j \le i$. The scalar product of $x,y \in \mathbb{R}^i$ will be denoted by $\langle x,y\rangle$, $i \in \mathbb{N}$. For a subset $ A \subset \mathbb{R}^{d}$ and $c > 0$ we write $c \cdot A := \left\{ c \cdot z: z \in A\right\}$, and we put $\mathbb{R}_{+}:= [0,\infty)$. Furthermore, $m_d$ stands for $d$-dimensional Lebesgue measure, and the $i$-dimensional identity matrix will be denoted by $\mathrm{I}_i$, $i \in \mathbb{N}$. Each unspecified limit refers to $n \to \infty$, and for two real-valued sequences $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$, where $b_n \ne 0$ for each $n \in \mathbb{N}$, we write $a_n \sim b_n$ if $a_n/b_n \to 1$. For a density $g$, a measure $\mu$ on $\mathbb{R}^d$ and a Borel set $A \in \mathcal{B}^d $ we put $g\big\|_A(z) := g(z)$ if $z \in A$ and $0$ otherwise, and write $\mu\big\|_A(B) := \mu(A \cap B)$ if $B \in \mathcal{B}^d $. Convergence in distribution and equality in distribution will be denoted by $\overset{\mathcal{D}}{\longrightarrow }$ and $\overset{\mathcal{D}}{= } $, respectively. The components of a random vector $Z_i$ are given by $Z_i = (Z_{i,1},\ldots,Z_{i,d})$ for $i \ge 1$. Finally, we write $N \overset{\mathcal{D}}{= } \text{Po}(\lambda) $ if the random variable $N$ has a Poisson distribution with parameter $\lambda >0 $. Regarding point processes, we mainly adopt the notation of~\cite{Resnick1987}, Chapter 3. A point process $\xi$ on some space $D$, equipped with a $\sigma$-field $\mathcal{D} $, is a measurable map from some probability space $(\Omega,\mathcal{A},\mathbb{P} )$ into $\big(M_p(D),\mathcal{M}_p(D)\big) $, where $M_p(D)$ is the set of all point measures $\chi$ on $D$, equipped with the smallest $\sigma$-field $\mathcal{M}_p(D)$ rendering the evaluation maps $\chi \mapsto \chi(A)$ from $M_p(D) \to [0,\infty]$ measurable for all $A \in \mathcal{D}.$ We call the point process $\xi$ simple if $ \mathbb{P}\big(\xi(\left\{ z\right\}) \in \left\{ 0,1\right\} \text{ for all } z \in D\big) = 1. $ A Poisson process with intensity measure $\mu$ is a point process $\xi$ satisfying \begin{equation} \label{eq_def_Poisson_process} \mathbb{P}\big(\xi(A) = k \big ) = \begin{cases} e^{-\mu(A)}\frac{\mu(A)^k}{k!}, & \text{if } \mu(A) < \infty,\\ 0, & \text{if } \mu(A) = \infty, \end{cases} \end{equation} for $A \in \mathcal{D} $ and $ k \in \mathbb{N}\cup \left\{ 0\right\}$. Moreover, $\xi(A_1),\ldots,\xi(A_i)$ are independent for any choice of $ i \ge 2$ and mutually disjoint sets $A_1,\ldots,A_i\in \mathcal{D}$. We briefly write $\xi \overset{\mathcal{D}}{= } \text{PRM}(\mu)$. If $\xi$ is a Poisson process with intensity measure $\mu$, \eqref{eq_def_Poisson_process} means $\xi(A) \overset{\mathcal{D}}{= } \text{Po}\big(\mu(A)\big)$ and hence $\mathbb{E}\big[\xi(A)\big] = \mu(A)$ for $A \in \mathcal{D}$. According to Corollary 6.5 in~\cite{Last2017}, for each Poisson process $\xi$ on $D$ there is a sequence $\mathcal{X}_1,\mathcal{X}_2,\ldots $ of random points in $D$ and a $\left\{ 0,1,\ldots,\infty \right\}$-valued random variable $N$ so that $$ \xi = \sum_{i=1}^{N}\varepsilon_{\mathcal{X}_i }, \quad \text{almost surely.} $$ Because of this property we use the notation $ \xi = \left\{ \mathcal{X}_i, i \ge 1\right\} $, whenever $\xi$ is a simple Poisson process and $\xi(D) = \infty$ almost surely. This terminology is motivated by the notion of a point process as a random set of points. We will use the bold letters $\mathbf{X},\mathbf{Y} $ and $\mathbf{Z} $ to denote point processes, and the convention will be as follows: Point processes supported by the whole underlying set $E$ will get a name involving the letter $\mathbf{Z}$. In contrast, the letter $\mathbf{X} $ always stands for processes that live only on the left half $E \cap \left\{ z_1 \le 0\right\}$ of $E$ and $\mathbf{Y} $ for those that are supported by the right half $E \cap \left\{ z_1 \ge 0\right\}$ of $E$. This distinction will be very useful to shorten the notation. If, for instance, $\mathbf{X} = \left\{ \mathcal{X}_i, i \ge 1 \right\}$ is a point process on $\mathbb{R}^d$, we write $\mathcal{X}_i = (\mathcal{X}_{i,1},\ldots,\mathcal{X}_{i,d} )$ to denote the coordinates of $\mathcal{X}_{i}. $ We finally introduce a very special sequence of Poisson processes: If $Z_1,Z_2,\ldots$ are i.i.d. with common distribution $\mathbb{P}_Z$, and $N_n$ is independent of this sequence and has a Poisson distribution with parameter $n$, then $$ \mathbf{Z}_n := \sum_{i=1}^{N_n}\varepsilon_{Z_{i}}, \quad n \in \mathbb{N}, $$ is a Poisson process in $\mathbb{R}^d$ with intensity measure $n\mathbb{P}_{Z}$, and we have $$ \text{diam}(\mathbf{Z}_n ) = M_{N_n} = \max_{1 \le i,j \le N_n} \big\|Z_i - Z_j\big\|. $$ \section{Conditions and main results} \label{sec_main_results} Our basic assumption on the shape and the orientation of the underlying set $E$ is that its finite diameter is attained by exactly one pair of points, both of which lie on the $z_1$-axis. Being more precise, we assume the following: \begin{condition} \label{cond_unique_diameter} Let $E \subset \mathbb{R}^d$ be a closed subset with $0 < 2a = \text{diam}(E) < \infty $ and $(-a,\mathbf{0} ), (a,\mathbf{0} )\in E$. Furthermore, we assume \begin{equation} \label{eq_cond_unique_diameter} \|x-y\| < 2a \qquad \text{for each} \qquad (x,y)\in \big( E\backslash \left\{ (-a,\mathbf{0} ), (a,\mathbf{0} )\right\} \big) \times E. \end{equation} \end{condition} Speaking of a `unique diameter', we will always mean that the underlying set satisfies \autoref{cond_unique_diameter}. The two points $(-a,\mathbf{0} ),(a,\mathbf{0} )\in E$ are henceforth called the `poles' of $E$. There is no loss of generality in assuming that the poles of $E$ are given by $(-a,\mathbf{0} )$ and $(a,\mathbf{0} )$. For every set having a diameter of length $2a >0$ we can find a suitable coordinate system so that this assumption is satisfied. Since we will consider distributions with $m_d$-densities supported by $E$, it will be no loss of generality either that we assume $E$ to be closed. So, condition~\eqref{eq_cond_unique_diameter} guarantees that $M_n$ will be determined by two points lying close to $(-a,\mathbf{0} )$ and $(a,\mathbf{0} )$, respectively, at least for large $n$ and a suitable distribution $\mathbb{P}_Z$. Our assumption on the shape of $E$ close to both poles is as follows: \begin{condition} \label{cond_shape_pole_caps} There are constants $\delta_\ell,\delta_r \in (0,a]$, open neighborhoods $O_\ell,O_r\subset \mathbb{R}^{d-1}$ of $\mathbf{0} \in \mathbb{R}^{d-1} $ and twice continuously differentiable functions $s^\ell : O_\ell \to \mathbb{R}_{+}$, $s^r : O_r \to \mathbb{R}_{+}$, so that \begin{align} E_\ell :=\ &E \cap \left\{ z_1 < -a + \delta_\ell\right\} =\ \left\{ (z_1,\widetilde z) \in \mathbb{R}^d: -a + s^\ell(\widetilde z) \le z_1 < -a + \delta_{\ell} , \widetilde z\in O_\ell\right\} \label{eq_def_E_l} \intertext{and} E_r :=\ & E \cap \left\{a - \delta_r < z_1\right\} =\ \left\{ (z_1,\widetilde z) \in \mathbb{R}^d: a - \delta_r < z_1 \le a - s^r(\widetilde z) , \widetilde z\in O_r\right\}. \label{eq_def_E_r} \end{align} \end{condition} Since $(-a,\mathbf{0} ),(a,\mathbf{0} )\in E$, we have $s^\ell(\mathbf{0} ) = s^r(\mathbf{0} ) = 0$, and we write $H_i$ for the Hessian of $s^{i}$ at the point $\mathbf{0} $. In view of the unique diameter of $E$ between $(-a,\mathbf{0} )$ and $(a,\mathbf{0} )$, we know the following facts about $\nabla s^{i}(\mathbf{0} )$ and $H_i$, $i \in \left\{ \ell,r\right\}$: \begin{lemma} \label{lem_first_part_deriv_are_0_and_Hessian_pos_def} For $i \in\left\{ \ell,r\right\}$ we have $\nabla s^i(\mathbf{0}) = \mathbf{0} $. Furthermore, the matrix $H_i$ is symmetric and positive definite, and all $d-1$ eigenvalues of $H_i$ are larger than $1/2a$. \end{lemma} The proof of this lemma can be found in \autoref{sec_appendix}. According to \autoref{lem_first_part_deriv_are_0_and_Hessian_pos_def}, the matrices $H_\ell$ and $H_r$ are orthogonally diagonalizable and all eigenvalues, denoted by $\kappa_2^i \le \ldots \le \kappa_{d}^i$, $i \in \left\{ \ell,r\right\}$, in ascending order, are real-valued and positive. The subscripts $2,\ldots,d$ instead of $1,\ldots,d-1$ are chosen deliberately. Because of the very close connection between these eigenvalues and the components $z_2,\ldots,z_d$ in our main theorem, this notation is much more intuitive for our purposes. See especially the end of this section for an illustration of the aforementioned connection. For $i \in \left\{ \ell,r\right\}$ we choose an orthonormal basis $\left\{ \mathbf{u}_2^i,\ldots,\mathbf{u}_d^i \right\}$ of $\mathbb{R}^{d-1}$, consisting of corresponding eigenvectors; namely $H_i \mathbf{u}_j^i =\kappa_j^i \mathbf{u}_j^i $ for $j \in \left\{ 2,\ldots,d\right\}$. Putting $U_i := (\mathbf{u}_2^i\ \|\ \ldots\ \|\ \mathbf{u}_d^i )$, we have $U_iU_i^\top = \mathrm{I}_{d-1}$ and $U_i^\top H_i U_i = \text{diag}(\kappa_2^i,\ldots,\kappa_d^i) =: D_i.$ \\ It is quite obvious that \autoref{cond_unique_diameter} restricts the possible Hessians $H_\ell$ and $H_r$. It would be desirable to find a one-to-one relation between the unique diameter of $E$ assumed in \autoref{cond_unique_diameter} on the one hand and all possible Hessians $H_\ell$ and $H_r$ on the other hand. But describing this relation in its whole generality would be technically very involved. Fortunately, we can state a simple but still very general condition on the Hessians to guarantee that $E \cap \left\{ \|z_1\| > a - \delta\right\}$ has a unique diameter between $(-a,\mathbf{0} )$ and $(a,\mathbf{0} )$ for $\delta >0$ sufficiently small. Unless otherwise stated we will always study sets fulfilling the following condition: \begin{condition} \label{cond_A_eta_pos_semi_definite} For some constant $\eta \in (0,1)$, the $2(d-1) \times 2(d-1)$-dimensional matrix $$ A(\eta) := \begin{pmatrix} 2a\eta D_\ell - \mathrm{I}_{d-1} & U_\ell^\top U_r \\ U_r^\top U_\ell & 2a\eta D_r - \mathrm{I}_{d-1} \end{pmatrix} $$ is positive semi-definite. \end{condition} We will briefly write $A(\eta) \ge 0$ to denote this property. Notice that $A(\eta_1) \ge 0 $ implies $A(\eta_2) \ge 0$ for each $\eta_2 > \eta_1$ since $D_\ell$ and $D_r$ are diagonal matrices with positive entries on their main diagonals. Due to the fact that $D_\ell,D_r,U_\ell$ and $U_r$ depend only on the curvature of $\partial E$ at the poles, \autoref{cond_A_eta_pos_semi_definite} is obviously not sufficient to ensure \eqref{eq_cond_unique_diameter} (figuring in \autoref{cond_unique_diameter}) for the whole set $E$. But \autoref{lem_suff_con_unique_diam} will show that \autoref{cond_A_eta_pos_semi_definite} guarantees that \eqref{eq_cond_unique_diameter} holds true for $E$ replaced with $E \cap \left\{ \|z_1\| > a - \delta\right\}$ and $\delta >0 $ sufficiently small. This assertion can be interpreted as `\autoref{cond_A_eta_pos_semi_definite} ensures the unique diameter of $E$ close to the poles'. Focussing on sets satisfying \autoref{cond_A_eta_pos_semi_definite} will be no strong limitation in the following sense: If $A(1)$ is \emph{not} positive semi-definite, then $E$ cannot have a unique diameter between the poles, see \autoref{lem_A_1_has_to_be_pos_def}. Hence, the only relevant case not covered by \autoref{cond_A_eta_pos_semi_definite} is given by $$ A(1)\ge0, \quad \text{ but } \quad A(\eta) \ngeq 0 \quad \text{ for each } \quad \eta \in (0,1). $$ At first sight, \autoref{cond_A_eta_pos_semi_definite} looks quite technical. A much more intuitive and sufficient, but \emph{not} necessary condition for \autoref{cond_A_eta_pos_semi_definite} to hold is \begin{equation} \label{eq_suff_cond_princ_curv} \frac{1}{\kappa_2^\ell}+ \frac{1}{\kappa_2^r} < 2a, \end{equation} see \autoref{lem_suff_cond_princ_curv}. We may thus check \autoref{cond_unique_diameter} (at least close to the poles) for many sets by merely looking at the smallest eigenvalues of $H_\ell$ and $H_r$. Now that we have stated our conditions on the underlying set $E$, we can focus on distributions supported by $E$. In this section we consider distributions $\mathbb{P}_{Z}$ with a Lebesgue density $f$ on $E$ satisfying the following property of continuity at the poles: \begin{condition} \label{cond_density_p_l_p_r} Let $f: E \to \mathbb{R}_{+}$ with $\int_{E}f(z)\,\mathrm{d}z =1 $. We further assume that $f$ is continuous at the poles $(-a,\mathbf{0} )$, $(a,\mathbf{0} )$ with $$ p_\ell := f(-a,\mathbf{0} ) > 0 \qquad \text{and} \qquad p_r := f(a,\mathbf{0} ) > 0. $$ \end{condition} Defining the `pole-caps of length $\delta$' via \begin{equation} \label{eq_def_E_l_delta_E_r_delta} E_{\ell,\delta} := E_\ell \cap \left\{ - a \le z_1 \le -a + \delta\right\} \quad \text{and} \quad E_{r,\delta} := E_r \cap \left\{ a - \delta \le z_1 \le a\right\} \end{equation} for $0<\delta < \min\left\{ \delta_\ell,\delta_r\right\}$, the property of continuity assumed in \autoref{cond_density_p_l_p_r} can be rewritten as $f(z) = p_i \big(1 + o(1)\big)$, where $o(1)$ is uniformly on $E_{i,\delta}$ as $\delta \to 0$, $i \in \left\{ \ell,r\right\}$. Now, we only need one more definition before we can formulate our main result. Putting \begin{equation} \label{eq_def_P_H} P(H) := \left\{ (z_1,\widetilde z) \in \mathbb{R}^d: \frac{1}{2} \widetilde z^\top H \widetilde z \le z_1 \right\} \end{equation} for some $(d-1)\times (d-1)$-dimensional matrix $H$, the set $P(H_\ell)$ (resp. $P(H_r)$) describes the shape of $E$ near the left (resp. right) pole if we `look through a suitably distorted magnifying glass', see \autoref{lemma_transformation_density} for details. The boundaries of $P(H_\ell)$ and $P(H_r)$ are elliptical paraboloids. Now we are prepared to state our main result. \begin{theorem} \label{thm_main_result} If Conditions~\ref{cond_unique_diameter} to \ref{cond_density_p_l_p_r} hold, then \begin{equation} \label{eq_theorem_main_result} n^{\frac{2}{d+1}}\big(2a - \mathrm{diam}(\mathbf{Z}_n )\big) \overset{\mathcal{D}}{\longrightarrow } \min_{i,j \ge 1} \left\{ \mathcal{X}_{i,1} + \mathcal{Y}_{j,1} - \frac{1}{4a}\big\|\widetilde {\mathcal{X}}_{i}- \widetilde {\mathcal{Y}}_{j}\big\|^2 \right\}, \end{equation} where $\left\{ \mathcal{X}_i, i \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(p_\ell\cdot m_d\big\|_{P(H_\ell)}\big) $ and $\left\{ \mathcal{Y}_j, j \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(p_r\cdot m_d\big\|_{P(H_r)}\big) $ are independent Poisson processes. The same holds true if we replace $ \mathrm{diam}(\mathbf{Z}_n )$ with $M_n$. \end{theorem} \begin{proof} See \autoref{sec_proof_main_thm}. \qed \end{proof} A special case of this result is given if we assume that $E$ is a proper ellipsoid. The following corollary illustrates that \autoref{thm_main_result} is a generalization of the main result in~\cite{Schrempp2016}: \begin{corollary} \label{cor_Ellipsoid} Let $a_1 > a_2 \ge a_3 \ge \ldots \ge a_d > 0$, and put $$ E := \left\{z \in \mathbb{R}^d: \sum_{j=1}^{d}\left(\frac{z_j}{a_j} \right)^2 \le 1\right\}. $$ The values $a_1,\ldots,a_d$ are called the `half-axes' of the ellipsoid $E$. Obviously, this set has a unique diameter of length $2a_1$ between the points $(-a_1,\mathbf{0} )$ and $(a_1,\mathbf{0} )$; i.e. \autoref{cond_unique_diameter} holds true with $a = a_1$. Putting $\delta_\ell := \delta_r := a_1$, $$ O_\ell := O_r := \left\{\widetilde z \in \mathbb{R}^{d-1}: \sum_{j=2}^{d}\left(\frac{z_j}{a_j} \right)^2 < 1\right\} \qquad \text{and} \qquad s^\ell(\widetilde z) := s^r(\widetilde z) := a_1 - a_1\sqrt{1 - \sum_{j=2}^{d}\left(\frac{z_j}{a_j} \right)^2 }, $$ \autoref{cond_shape_pole_caps} is fulfilled, too. Some easy calculations show that the Hessians $H_\ell$ and $H_r$ of $s^{\ell}$ and $s^{r}$ at $\mathbf{0}$ are given by \begin{align} H_\ell = H_r = \text{diag}\left( \frac{a_1}{a_2^2}\ ,\ \ldots\ ,\ \frac{a_1}{a_d^2}\right). \end{align} This means that the eigenvalues of both $H_\ell$ and $H_r$ are $\kappa_j^i = a_1/a_j^2$. Since $ a_2 < a_1$, we have $$ \frac{1}{\kappa_2^\ell}+ \frac{1}{\kappa_2^r} = \frac{a_2^2}{a_1} + \frac{a_2^2}{a_1} = 2\frac{a_2^2}{a_1} = 2a_1 \left( \frac{a_2}{a_1}\right)^2 < 2a_1 = 2a. $$ Hence, inequality \eqref{eq_suff_cond_princ_curv} holds true and thus \autoref{cond_A_eta_pos_semi_definite} is fulfilled. With \begin{align} P(H_\ell) = P(H_r) &= \left\{ z \in \mathbb{R}^d: \frac{1}{2} \widetilde z^\top H_{\ell}\widetilde z \le z_1\right\} = \left\{ z \in \mathbb{R}^d: \frac{1}{2} \sum_{j=2}^{d} \frac{a_1}{a_j^2}\cdot z_j^2 \le z_1\right\} = \left\{ z \in \mathbb{R}^d: \sum_{j=2}^{d} \left( \frac{z_j}{a_j} \right)^2 \le \frac{2z_1}{a_1}\right\}, \end{align} we can apply \autoref{thm_main_result} for distributions in $E$ satisfying \autoref{cond_density_p_l_p_r}, in accordance with Theorem 2.1 in \cite{Schrempp2016}. \end{corollary} \begin{corollary} \label{cor_ellipse_uniform} If $Z$ has a uniform distribution in the ellipsoid $E$ given in \autoref{cor_Ellipsoid}, \autoref{cond_density_p_l_p_r} holds true with $$ p_\ell := p_r := \frac{1}{m_d(E)} = \left( \frac{\pi^{\frac{d}{2}}}{\Gamma\left( \frac{d}{2}+1\right)}\prod_{i=1}^{d}a_i \right)^{-1} > 0. $$ Hence, \autoref{thm_main_result} is applicable. In the special case $d=2$ and $a_1 = 1$ we have $a_2 < 1$, $p_\ell = p_r = 1/(\pi a_2)$, $$ P:=P(H_\ell) = P(H_r)= \left\{ z \in \mathbb{R}^2: \left( \frac{z_2}{a_2} \right)^2 \le 2z_1\right\}, $$ and it follows that \begin{equation} \label{eq_Limit_Dist_Ellipse} n^{2/3}(2 - M_n ) \overset{\mathcal{D}}{\longrightarrow } \min_{i,j \ge 1} \left\{ \mathcal{X}_{i,1} + \mathcal{Y}_{j,1} - \frac{1}{4} (\mathcal{X}_{i,2}-\mathcal{Y}_{j,2})^2 \right\}, \end{equation} with two independent Poisson processes $\mathbf{X} = \left\{ \mathcal{X}_i, i \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(p_\ell\cdot m_2\big\|_P\big) $ and $\mathbf{Y} = \left\{ \mathcal{Y}_j, j \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(p_r\cdot m_2\big\|_P\big) $. \end{corollary} To illustrate the speed of convergence in \autoref{cor_ellipse_uniform}, we present the result of a simulation study. To this end, define $G(x,y) := x_1 + y_1 - (x_2 - y_2)^2/4$. In the proof of \autoref{lemma_continuity_of_G_hat} one can see that $G(x,y) \ge c(x_1+y_1)$, $(x,y) \in P(H_\ell) \times P(H_r)$, for some fixed $c \in (0,1)$. Therefore, the probability that a point $\mathcal{X}_i $ with a `large' first component $\mathcal{X}_{i,1} $ determines the minimum above is `small' (we omit details). The same holds for $\mathcal{Y}_j$. We can thus approximate the limiting distribution above by taking independent Poisson processes with intensity measures $p_\ell\cdot m_2\big\|_{\widetilde P}$ and $p_r\cdot m_2\big\|_{\widetilde P}$ where $\widetilde P := P(H_\ell) \cap \left\{ z \in \mathbb{R}^2: z_1 \le b\right\}$ for some fixed $b >0$. The larger the minor half-axis $a_2$ is (i.e. the more $E$ becomes `circlelike'), the larger $b$ has to be chosen in order to have a good approximation of the distributional limit in \eqref{eq_Limit_Dist_Ellipse} (we omit details). See \autoref{fig_plot_ellipse_simulation} for an illustration of the sets $E$ (left) and $P$ (right) and \autoref{fig_cdf_plot_ellipse_uniform} for the result of a simulation. Notice the different scalings between the left-~and the right-hand image in \autoref{fig_plot_ellipse_simulation}. \begin{figure} \caption{The sets $E$ (left) and $P$ (right) in the setting of \autoref{cor_ellipse_uniform} for $d=2$ with $a_1=1, a_2=1/2$.} \label{fig_plot_ellipse_simulation} \end{figure} \begin{figure} \caption{Empirical distribution function of $n^{2/3}(2-M_n)$ in the setting of \autoref{cor_ellipse_uniform} for $d=2$ with $a_1=1, a_2=1/2$, $n=1000$ (solid, 5000 replications). The limit distribution is approximated as described after \autoref{cor_ellipse_uniform} with $b = 10$ (dashed, 5000 replications). } \label{fig_cdf_plot_ellipse_uniform} \end{figure} \begin{remark} \label{rem_interpretation_as_hypersurfaces} The `outer boundaries' of $E_\ell$ and $E_r$, denoted by $$ M_\ell := \left\{(z_1,\widetilde z) \in \mathbb{R}^d: z_1 = -a+s^\ell(\widetilde z) , \widetilde z \in O_\ell\right\} \qquad \text{and} \qquad M_r := \left\{(z_1,\widetilde z) \in \mathbb{R}^d: z_1 = a-s^r(\widetilde z) , \widetilde z \in O_r\right\}, $$ can be interpreted as images of the two hypersurfaces $$ \mathbf{s}^\ell:\ O_\ell \to \mathbb{R}^d,\ \mathbf{s}^\ell(\widetilde z):=\big(-a + s^\ell(\widetilde z)\ ,\ \widetilde z \ \big) \qquad \text{and} \qquad \mathbf{s}^r:\ O_r \to \mathbb{R}^d,\ \mathbf{s}^r(\widetilde z):=\big(a -s^r(\widetilde z)\ ,\ \widetilde z \ \big). $$ For $ i \in \left\{ \ell,r\right\}$ the first partial derivatives of $\mathbf{s}^i $ are given by $$ \mathbf{s}_2^i(\widetilde z) = \Big( \ s_2^i(\widetilde z) \ , \ 1 \ , \ 0 \ , \ \ldots \ , \ 0 \ \Big) , \quad \ldots\quad , \ \mathbf{s}_d^i(\widetilde z) = \Big( \ s_d^i(\widetilde z) \ , \ 0 \ , \ \ldots\ , \ 0 \ , \ 1 \ \Big) . $$ These $d-1$ vectors are linearly independent for each $\widetilde z \in O_i$, which means that the hypersurfaces $\mathbf{s}^\ell $ and $\mathbf{s}^r $ are regular, see Definition 3.1.2 in~\cite{Csikos2014}. From \autoref{lem_first_part_deriv_are_0_and_Hessian_pos_def} we further know $\mathbf{s}_j^i(\mathbf{0} ) = \mathbf{e}_j$ for $i \in \left\{ \ell,r\right\}$ and each $j \in \left\{ 2,\ldots,d\right\}$. Hence, the two unit normal vectors of the hypersurface $\mathbf{s}^{i} $ at the pole $\mathbf{s}^{i}(\mathbf{0} ) $ are given by $\pm \mathbf{e}_1 $. Looking at Appendix A.2.2 in \cite{Schrempp2017} -- especially its ending -- we know (because of $\nabla s^i(\mathbf{0} )= \mathbf{0}$) that the eigenvalues $\kappa_j^i$ of the Hessian $H_i$ are exactly the principal curvatures of the hypersurface $\mathbf{s}^i $ at the pole $\mathbf{s}^i (\mathbf{0} )$ with respect to the unit normal vector $\mathbf{e}_1$ if $i = \ell$ and $-\mathbf{e}_1$ if $i = r$, respectively. We can further conclude that \begin{equation} \label{eq_Connection_v_and_u} \mathbf{v}_j^i := \begin{pmatrix} 0 \\ \mathbf{ u}_j^i \end{pmatrix} \in \mathbb{R}^d \end{equation} are the corresponding principal curvature directions. Using the notation introduced after \autoref{lem_first_part_deriv_are_0_and_Hessian_pos_def} and some easy transformations show \begin{align} P(H_i) &= \left\{ z_1 \mathbf{e}_1 + \sum_{j=2}^{d}z_j \mathbf{v}_j^i \in \mathbb{R}^d: \frac{1}{2} \sum_{j=2}^{d}\kappa_j^i z_j^2 \le z_1\right\}, \label{eq_represenatation_P_Hl_by_princ_dir_1} \end{align} see \cite[p. 30]{Schrempp2017} for more details. This representation is sometimes called the `normal representation of the osculating paraboloid $P(H_i)$', and it justifies the notation of the principal curvatures $\kappa_j^i$ with indices $2,\ldots,d$ instead of $1,\ldots,d-1$. If we have $\mathbf{v}_{j}^i = \mathbf{e}_j $ for each $ j \in \left\{ 2,\ldots,d\right\}$, we especially get \begin{equation} \label{eq_represenatation_P_Hl_by_princ_dir_2} P(H_i)= \left\{ z \in \mathbb{R}^d: \frac{1}{2} \sum_{j=2}^{d}\kappa_j^i z_j^2 \le z_1\right\}. \end{equation} \autoref{cor_Ellipsoid} is a special case of this situation. \end{remark} \section{Proof of \autoref{thm_main_result} } \label{sec_proof_main_thm} The proof of \autoref{thm_main_result} is divided into three subsections. The first one is mainly devoted to the study of some geometric properties of the set $E$ close to the poles. In \autoref{subsec_proof_main_thoerem_conv_point_processes} we will deal with the convergence of Poisson random measures, which will be crucial for the main part of the proof of \autoref{thm_main_result}, given in \autoref{subsubsec_main_part_proof_main_thm}. \subsection{Geometric considerations} \label{subsec_proof_main_thm_geom_considerations} First of all, note that \autoref{cond_A_eta_pos_semi_definite} holds true if, and only if, \begin{equation} \label{eq_cond_1_unique_diam} 0\le 2a\eta\left( \alpha^\top D_\ell\alpha + \beta^\top D_r \beta\right)+2\alpha^\top U_\ell^\top U_r\beta-\|\alpha\|^2 - \|\beta\|^2, \qquad \text{for all $\alpha,\beta \in \mathbb{R}^{d-1}$}, \end{equation} and since $\kappa_2^i$ and $\kappa_d^i$ are the smallest and the largest eigenvalues of $H_i$, the so-called min--max theorem by Courant--Fischer yields \begin{equation} \label{eq_Courant_Fischer} \kappa_2^{i}\| \widetilde z\|^2 \le \widetilde z^\top H_i \widetilde z \le \kappa_d^{i} \| \widetilde z\|^2 \end{equation} for each $\widetilde z \in \mathbb{R}^{d-1}$ and $i \in \left\{ \ell,r\right\}$. In view of \autoref{lem_first_part_deriv_are_0_and_Hessian_pos_def} it is clear that the second-order Taylor series expansions of $s^{i}$ at the point $\mathbf{0} $ is $$ s^i(\widetilde z) = \frac{1}{2}\widetilde z^\top H_i \widetilde z + R_i (\widetilde z), $$ where $R_i (\widetilde z) = o\big(\|\widetilde z\|^2 \big)$ and $i \in \left\{ \ell,r\right\}$. From \eqref{eq_def_E_l} and \eqref{eq_def_E_r} we obtain the representations \begin{align} E_\ell &= \left\{ (z_1,\widetilde z) \in \mathbb{R}^d: -a + \frac{1}{2}\widetilde z^\top H_\ell \widetilde z + R_\ell (\widetilde z) \le z_1 < -a + \delta_{\ell} , \widetilde z\in O_\ell\right\} \label{eq_representation_E_l_Taylor} \intertext{and} E_r &= \left\{ (z_1,\widetilde z) \in \mathbb{R}^d: a - \delta_r < z_1 \le a - \frac{1}{2}\widetilde z^\top H_r \widetilde z - R_r (\widetilde z) , \widetilde z\in O_r\right\}, \label{eq_representation_E_r_Taylor} \end{align} which will be widely used throughout this work. Now we need some additional definitions. We shift the set $E_\ell$ to the right by $a\cdot\mathbf{e}_1 $ along the $z_1$-axis and call this set $P_1(H_\ell)$. The set $E_r$ will be translated by $-a\cdot\mathbf{e}_1 $ along the $z_1$-axis to the left, and it will then be reflected at the plane $\left\{ z_1 = 0\right\}$. We call the resulting set $P_1(H_r)$. Looking at \eqref{eq_representation_E_l_Taylor} and~\eqref{eq_representation_E_r_Taylor}, we have \begin{equation} \label{eq_def_P_1_H_i} P_1(H_i) = \left\{ (z_1,\widetilde z) \in \mathbb{R}^d: \frac{1}{2}\widetilde z^\top H_i \widetilde z + R_i (\widetilde z) \le z_1 < \delta_i , \widetilde z\in O_i\right\} \end{equation} for $i \in \left\{ \ell,r\right\}$. The reason underlying this construction will be seen later in~\eqref{eq_reason_first_power_bigger}. In addition to $P_1(H_i)$, we introduce the constant \begin{equation} \label{eq_def_eta_hat} \widehat \eta := \frac{1+ \eta^{-1}}{2}, \end{equation} based on the constant $\eta \in (0,1)$ from \autoref{cond_A_eta_pos_semi_definite}. The subsequent remark will point out two very important properties of $\widehat \eta$, that will be essential for the proofs to follow: \begin{remark} \label{rem_properties_con_hat} Since $\eta \in (0,1)$, we have $\widehat \eta > 1$, and it follows that $P(H_i) \varsubsetneq \widehat \eta \cdot P(H_i)$ for $i \in \left\{ \ell,r\right\}$. Without this technical expansion of the limiting sets $P(H_i) $, several proofs would become much more complicated. The second important property is that $\widehat \eta$ is not `too large' in the sense that $$ 1 - \eta\widehat \eta = 1 - \eta\frac{1+ \eta^{-1}}{2} = 1 - \frac{\eta + 1}{2} = \frac{1 - \eta}{2} > 0. $$ This inequality will be crucial for the proofs of \autoref{lemma_R_is_little_o_of_G_tilde} and \autoref{lemma_continuity_of_G_hat}. \end{remark} As stated in \autoref{rem_properties_con_hat}, we will need the set $\widehat \eta \cdot P(H_i)$ for $i \in \left\{ \ell,r\right\}$. For later use, we give a more convenient representation of these sets: \begin{remark} \label{rem_representation_widehat_xi_P_H_i} For $i \in \left\{ \ell,r\right\}$ we obtain from \eqref{eq_def_P_H} \begin{align} \widehat \eta \cdot P(H_i) &= \left\{ \widehat \eta \cdot z \in \mathbb{R}^d: z \in P(H_i)\right\} \\ &= \left\{z\in \mathbb{R}^d: \widehat \eta ^{-1} z \in P(H_i)\right\}\\ &= \left\{z\in \mathbb{R}^d: \frac{1}{2}\left( \widehat \eta ^{-1} \widetilde z \right)^\top H_i \left( \widehat \eta ^{-1} \widetilde z \right) \le \widehat \eta ^{-1}z_1\right\} \\ &= \left\{z\in \mathbb{R}^d: \frac{1}{2}\widetilde z^\top H_i\widetilde z \le \widehat \eta z_1\right\}. \end{align} \end{remark} In the following, we have to consider simultaneously points $x$, that are lying close to the left pole, and points $y$, lying close to the right one. For this purpose, we use the definitions of the pole-caps $E_{\ell,\delta}$ and $E_{r,\delta}$ given in \eqref{eq_def_E_l_delta_E_r_delta} and put $E_{\delta} := E_{\ell,\delta} \times E_{r,\delta}$ to yield \begin{equation} \label{eq_Representation_E_delta} E_{\delta} = \big\{ (x,y) \in E_\ell \times E_r: -a \le x_1 \le -a + \delta, a - \delta \le y_1 \le a\big\}. \end{equation} The next lemma shows the reason for introducing the sets $\widehat \eta \cdot P(H_i)$, $i \in \left\{ \ell,r\right\}$. The inclusion stated there will be crucial for the proof of the subsequent \autoref{lemma_R_is_little_o_of_G_tilde} and for the main part of the proof of \autoref{thm_main_result} itself. \begin{lemma} \label{lem_shifted_poles_in_xi_hat_P_H_i} There is some constant $\delta^* \in \big(0, \min\left\{ \delta_\ell,\delta_r\right\}\big]$, so that the inclusion \begin{equation} \label{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_0} \big( P_1(H_\ell) \cap \left\{ z_1 \le \delta\right\}\big) \times \big( P_1(H_r) \cap \left\{ z_1 \le \delta\right\}\big) \subset \widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r) \end{equation} holds true for each $\delta \in (0,\delta^{}]$. In other words, we have \begin{align} \frac{1}{2}\widetilde x^\top H_\ell \widetilde x \le \widehat \eta (a+x_1) \qquad \text{and}\qquad \frac{1}{2}\widetilde y^\top H_r \widetilde y \le \widehat \eta (a-y_1) \label{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_1} \end{align} for all $(x,y) \in E_{\delta^{}}$. \end{lemma} \begin{proof} Observe \autoref{rem_representation_widehat_xi_P_H_i} and the construction of $P_1(H_\ell)$ and $P_1(H_r)$ at the beginning of this section for checking the equivalence between \eqref{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_0} and \eqref{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_1}. Without loss of generality we only show the first inequality of~\eqref{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_1} for $(x,y) \in E_{\delta}$ and $\delta > 0$ sufficiently small. If $\delta < \delta_\ell$, it follows from~\eqref{eq_representation_E_l_Taylor} and the definition of $E_\delta$ that $$ x \in \left\{ (z_1,\widetilde z) \in \mathbb{R}^d: -a + \frac{1}{2}\widetilde z^\top H_\ell \widetilde z + R_\ell (\widetilde z) \le z_1 \le -a + \delta , \widetilde z\in O_l\right\}, $$ whence \begin{equation} \label{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_2} \frac{1}{2}\widetilde x^\top H_\ell \widetilde x + R_\ell (\widetilde x) \le a + x_1 \le \delta. \end{equation} As $\delta \to 0$ we get $\|\widetilde x\| \to 0$ on $E_{\delta}$, and because of $R_\ell (\widetilde x) = o\left( \|\widetilde x\|^2\right)$ the relation $R_\ell (\widetilde x) = o\left( \widetilde x^\top H_\ell \widetilde x\right)$ holds true, too. Putting $\varepsilon := \frac{\eta^{-1}-1}{\eta^{-1}+1}>0$, we obtain for sufficiently small $\delta >0$ $$ \big\|R_\ell (\widetilde x)\big\| \le \frac{\varepsilon}{2}\widetilde x^\top H_\ell \widetilde x $$ for every $(x,y) \in E_{\delta}$. Combining this inequality with \eqref{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_2} shows that $$ \frac{1-\varepsilon}{2}\widetilde x^\top H_\ell \widetilde x \le a + x_1 $$ and hence, by the definition of $\widehat \eta$ given in \eqref{eq_def_eta_hat}, \begin{align} \frac{1}{2}\widetilde x^\top H_\ell \widetilde x &\le \frac{1}{1-\varepsilon}(a + x_1) = \widehat \eta (a+x_1). \end{align} Choosing $\delta^{}$ in such a way that both inequalities figuring in \eqref{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_1} hold true for each $(x,y) \in E_{\delta^{}}$ finishes the proof. \qed \end{proof} In the following, we will, without loss of generality, only investigate $E_{\delta}$ for $\delta \in (0,\delta^{}]$ to ensure the validity of \eqref{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_1}.\\ In the next step we examine the behavior of $\|x-y\|$ for $x$ close to the left pole of $E$ and $y$ close to the right one. For this purpose, we consider $\mathbb{R}^{2d}$ to describe the simultaneous convergence of $x$ to the left pole of $E$ and $y$ to the right pole. Some straightforward calculations show that the second-order Taylor polynomial of $(x,y) \mapsto \|x-y\|$ at the point $\mathbf{a} := (-a,\mathbf{0} , a,\mathbf{0} ) \in \mathbb{R}^{2d}$ is given by $ -x_1 + y_1 + \frac{1}{4a}\|\widetilde x - \widetilde y\|^2. $ As $(x,y )\to \mathbf{a} = (-a,\mathbf{0} , a,\mathbf{0} )$, we obtain \begin{equation} \label{eq_taylor_eukl_dist_at_poles} \|x-y\| = - x_1 + y_1 +\frac{1}{4a}\|\widetilde x- \widetilde y\|^2 + R(x,y), \end{equation} where $R(x,y) = o\big( \|(x,y) - \mathbf{a} \|^2\big)$, uniformly on the ball of radius $r$ and center $\mathbf{a} $ as $r \to 0$. This uniform convergence holds especially on $E_{\delta}$ (given in \eqref{eq_Representation_E_delta}) as $\delta \to 0$. Putting \begin{equation} \widetilde G(x,y) := (a + x_1) + (a - y_1) - \frac{1}{4a}\|\widetilde x- \widetilde y\|^2, \end{equation} we infer \begin{equation} \label{eq_2a_minus_norm_is_G_tilde_minus_R} 2a - \|x-y\| = \widetilde G(x,y) - R(x,y). \end{equation} \begin{lemma} \label{lemma_R_is_little_o_of_G_tilde} We have $R(x,y) = o\big( \widetilde G(x,y)\big)$, uniformly on $E_{\delta}$ as $\delta \to 0$. \end{lemma} \begin{proof} Notice that \begin{equation} \label{eq_proof_lemma_uniformly} \frac{R(x,y)}{\widetilde G(x,y)} = \frac{R(x,y)}{\|(x,y) - \mathbf{a} \|^2} \cdot\frac{\|(x,y) - \mathbf{a}\|^2}{\widetilde G(x,y)} = o(1)\frac{\|(x,y) - \mathbf{a}\|^2}{\widetilde G(x,y)} \end{equation} as $\delta \to 0$, where $o(1)$ is uniformly on $E_{\delta}$. It remains to show that $\|(x,y) - \mathbf{a}\|^2 / \widetilde G(x,y)$ is bounded on $E_{\delta}$ for small $\delta > 0$. Assume without loss of generality that $\|x_1\| \le \|y_1\| < a$. In view of $x \in E_\ell$ and $y \in E_r$, we get $0 < a - y_1 \le a + x_1$. Consider in a first step the numerator of the right-most fraction figuring in~\eqref{eq_proof_lemma_uniformly}. With \eqref{eq_Courant_Fischer} and \autoref{lem_shifted_poles_in_xi_hat_P_H_i} we obtain for $(x,y) \in E_\delta$ and sufficiently small $\delta>0$ \begin{align} \|(x,y) - \mathbf{a}\|^2 &= (a+x_1)^2 + (a-y_1)^2 + \|\widetilde x\|^2 + \|\widetilde y\|^2\\ &\le (a+x_1)^2 + (a-y_1)^2 + \frac{1}{\kappa_2^\ell}\widetilde x^\top H_\ell \widetilde x + \frac{1}{\kappa_2^r}\widetilde y^\top H_r \widetilde y\\ &\le (a+x_1)^2 + (a-y_1)^2 + \frac{2 \widehat \eta}{\kappa_2^\ell} (a+x_1) + \frac{2 \widehat \eta}{\kappa_2^r} (a-y_1)\\ &\le (a+x_1)^2 + (a+x_1)^2 + \frac{2 \widehat \eta}{\kappa_2^\ell} (a+x_1) + \frac{2 \widehat \eta}{\kappa_2^r} (a+x_1)\\ &= (a+x_1)\left( 2(a+x_1) + \frac{2 \widehat \eta}{\kappa_2^\ell} + \frac{2 \widehat \eta}{\kappa_2^r}\right). \end{align} As a consequence of $(x,y) \in E_{\delta}$ and $\delta \to 0$ we get $x_1 \to -a$, and thus the term inside the big brackets converges to $\frac{2 \widehat \eta}{\kappa_2^\ell}+\frac{2 \widehat \eta}{\kappa_2^r}$. We can conclude that there is a constant $c >0$ so that $ \|(x,y) - \mathbf{a} \|^2 < (a + x_1)\cdot c $ for every $(x,y) \in E_{\delta}$ and sufficiently small $\delta >0$. In a second step we look at the denominator of the right-most fraction figuring in~\eqref{eq_proof_lemma_uniformly}. Writing $\widetilde x = U_\ell \alpha$ and $\widetilde y = U_r \beta$, we deduce that \begin{align} \widetilde G(x,y) &= (a + x_1) + (a - y_1) - \frac{1}{4a}\big(\|\widetilde x\|^2 + \|\widetilde y\|^2 - 2\widetilde x^\top \widetilde y\big)\\ &= (a + x_1) + (a - y_1) - \frac{1}{4a}\big(\|\alpha\|^2 + \|\beta\|^2 - 2\alpha^\top U_\ell^\top U_r \beta\big) . \intertext{Inequality \eqref{eq_cond_1_unique_diam} now shows that} \widetilde G(x,y) &\ge (a + x_1) + (a - y_1) - \frac{1}{4a}2a\eta\left( \alpha^\top D_\ell\alpha + \beta^\top D_r \beta\right) \\ &= (a + x_1) + (a - y_1) - \frac{1}{2}\eta\left( \widetilde x^\top U_\ell D_\ell U_\ell^\top \widetilde x + \widetilde y^\top U_r D_r U_r^\top \widetilde y\right)\\ &= (a + x_1) + (a - y_1) - \frac{1}{2}\eta\left( \widetilde x^\top H_\ell\widetilde x + \widetilde y^\top H_r \widetilde y\right), \intertext{and by \autoref{lem_shifted_poles_in_xi_hat_P_H_i} we get for sufficiently small $\delta >0$} \widetilde G(x,y) &\ge (a + x_1) + (a - y_1) - \frac{1}{2}\eta\big(2\widehat \eta (a + x_1) + 2\widehat \eta (a-y_1)\big) \\ &= (a+x_1)\left( 1 + \frac{a-y_1}{a+x_1} - \eta \widehat \eta\left( 1 + \frac{a-y_1}{a+x_1}\right)\right)\\ &= (a+x_1)\left( 1 - \eta \widehat \eta\right)\left( 1 + \frac{a-y_1}{a+x_1}\right). \end{align} \autoref{rem_properties_con_hat} and $ \frac{a-y_1}{a+x_1} \ge 0$ now yield $$ \widetilde G(x,y) \ge (a+x_1) \frac{1-\eta}{2}\left( 1 + \frac{a-y_1}{a+x_1}\right) \ge (a+x_1) \frac{1-\eta}{2}, $$ where $\frac{1-\eta}{2}> 0$. Putting both parts together, we have \begin{align} \frac{\|(x,y) - \mathbf{a} \|^2 }{\widetilde G(x,y)} \le \frac{(a+x_1)\cdot c}{(a+x_1)\cdot \frac{1-\eta}{2}} = \frac{2c}{1 - \eta} \end{align} for every $(x,y) \in E_{\delta}$ and $\delta>0$ small enough, and the proof is finished. \qed \end{proof} \subsection{Convergence of Poisson random measures} \label{subsec_proof_main_thoerem_conv_point_processes} In this subsection we will focus on the convergence of Poisson processes inside the sets $P_1(H_i)$ for $i \in \left\{ \ell,r\right\}$. \autoref{lemma_Conv_of_V_n} will be the key to describe the asymptotical behavior of those points of $\mathbf{Z}_n $ lying close to one of the poles if we `look through a suitably distorted magnifying glass' and let $n$ tend to infinity. In what follows, put \begin{equation} \label{eq_def_of_nu} \nu := \frac{1}{d+1} \end{equation} and $$ T_n(z) := \left(\ n^{2\nu}z_1\ ,\ n ^{\nu}\widetilde z\ \right) $$ for $n \in \mathbb{N} $ and $ z = (z_1,\widetilde z) \in \mathbb{R}^d.$ \begin{lemma} \label{lemma_transformation_density} Suppose that, for $i \in \left\{ \ell,r\right\}$, the random vector $V = (V_1,\ldots,V_d)$ has a density $g$ on $P_1(H_i) \cap \left\{ z_1 \le \delta^\right\}$ with $g(z) = p\big(1+o(1)\big)$ uniformly on $P_1(H_i) \cap \left\{ z_1 \le \delta\right\}$ as $\delta \to 0$ for some $p >0$. Then, for every bounded Borel set $B \subset \mathbb{R}^d$, we have $\mathbb{P}\big( T_n(V) \in B \big)= \kappa_n(B)/n$ with $ \kappa_n(B) \to p \cdot m_d\big\|_{P(H_i)}(B)$ as $n \to \infty $. \end{lemma} \begin{proof} To emphasize the support of $g$, we write $g(z)\mathds{1} \big\{ z \in P_1(H_i)\cap \left\{ z_1 \le \delta^\right\}\big\} $ instead of $g(z)$. The Jacobian of $T_n$ is given by $$ \Delta T_n(x) = \text{det} \big( \text{diag}\left( n^{2\nu},n^{\nu},\ldots,n^{\nu}\right) \big) =n^{(d+1)\nu} = n, $$ and therefore the random vector $T_n(V)$ has the density \begin{align} g_n(z) =&\; \frac{g\left( T_n^{-1}(z)\right)}{n} = \; \frac{1}{n} g\left( \frac{z_1}{n^{2\nu}}, \frac{1}{n^{\nu}}\widetilde z\right) \mathds{1}\big\{ z \in P_n(H_i)\big\}, \end{align} where $P_n(H_i) := T_n\big(P_1(H_i)\cap \left\{ z_1 \le \delta^\right\}\big)$. In view of \eqref{eq_def_P_1_H_i} we get \begin{align} P_n(H_i)&= \left\{ z \in \mathbb{R}^d: T_n^{-1}(z) \in P_1(H_i)\cap \left\{ z_1 \le \delta^\right\}\right\}\\ &= \left\{ z \in \mathbb{R}^d: \frac{1}{2}\left( \frac{1}{n^{\nu}}\widetilde z\right)^\top H_i \left( \frac{1}{n^{\nu}} \widetilde z\right) + R_i \left( \frac{1}{n^{\nu}}\widetilde z\right) \le \frac{z_1}{n^{2\nu}} \le \delta^, \frac{1}{n^{\nu}}\widetilde z\in O_i\right\}\\ &= \left\{ z \in \mathbb{R}^d: \frac{1}{2}\widetilde z^\top H_i \widetilde z + n^{2\nu}R_i \left( \frac{1}{n^{\nu}}\widetilde z\right)\le z_1 \le n^{2\nu}\delta^, \widetilde z\in n^{\nu}O_i\right\}. \end{align} Since $O_i$ is an open neighborhood of $\mathbf{0} \in \mathbb{R}^{d-1} $ and $$ n^{2\nu}R_i \left(\frac{1}{n^{\nu}}\widetilde z\right) = \|\widetilde z\|^2\cdot \frac{R_i \left(\frac{1}{n^{\nu}}\widetilde z\right)}{\left\|\frac{1}{n^{\nu}}\widetilde z\right\|^2} \to 0 $$ as $n \to \infty $ for each fixed $\widetilde z \in \mathbb{R}^{d-1}$, we see that $\mathds{1}\big\{ z \in P_n(H_i)\big\} \to \mathds{1}\big\{ z \in P(H_i)\big\}$ for almost all $z \in \mathbb{R}^d$. Observe that this convergence does not hold true for $z = (z_1,\widetilde z) \in \mathbb{R}^d$ with $\frac{1}{2}\widetilde z^\top H_i \widetilde z = z_1$ and $R_i \big(\frac{1}{n^{\nu}}\widetilde z\big) > 0$ for infinitely many $n \in \mathbb{N}$. But, since $\left\{ z \in \mathbb{R}^d: \frac{1}{2}\widetilde z^\top H_i \widetilde z = z_1\right\}$ has Lebesgue measure $0$, these points will have no influence on the integrals to follow. For each Borel set $B \subset \mathbb{R}^d$, we have \begin{align} \mathbb{P}\big( T_n(V) \in B \big) &= \int_{B} g_n(z)\,\mathrm{d}z = \frac{1}{n} \int_{B} g\left( \frac{z_1}{n^{2\nu}}, \frac{1}{n^{\nu}}\widetilde z\right) \mathds{1}\left\{ z \in P_n(H_i)\right\}\,\mathrm{d}z. \end{align} If $B$ is bounded, $\sup\left\{ z_1 : (z_1,\widetilde z) \in B\right\} \le i_1$ for some $i_1 \in [0,\infty )$. Consequently, $\left( \frac{z_1}{n^{2\nu}}, \frac{1}{n^{\nu}}\widetilde z\right) \in \left\{t \in \mathbb{R}^d: t_1 \le \frac{i_1}{n^{2\nu}}\right\}$ for every $z \in B$. Since $g(z) = p\big(1+o(1)\big)$, uniformly on $P_1(H_i) \cap \left\{ z_1 \le \delta\right\}$ as $\delta \to 0$, we obtain $g\left( \frac{z_1}{n^{2\nu}}, \frac{1}{n^{\nu}}\widetilde z\right) = p\big(1+o(1)\big)$ uniformly on $B$ as $n \to \infty$, whence \begin{align} \mathbb{P}\big( T_n(V) \in B \big) &= \frac{1}{n} \cdot p\int_{B} \big(1 + o(1)\big) \mathds{1}\left\{ z \in P_n(H_i)\right\}\,\mathrm{d}z =: \frac{1}{n} \cdot \kappa_n(B). \end{align} Since $B$ is bounded and $\big(1 + o(1)\big)\mathds{1}\left\{ z \in P_n(H_i)\right\} \to \mathds{1}\left\{ z \in P(H_i)\right\}$ for almost all $z \in \mathbb{R}^d$, the dominated convergence theorem gives \begin{align} \lim_{n\to\infty} \kappa_n(B) &= p \int_{B} \lim_{n\to\infty} \big(1 + o(1)\big) \mathds{1}\left\{ z \in P_n(H_i)\right\}\,\mathrm{d}z = p \int_{B} \mathds{1}\left\{ z \in P(H_i)\right\}\,\mathrm{d}z = p\cdot m_d\big\|_{P(H_i)}(B). \end{align} \qed \end{proof} \begin{remark} \label{rem_inclusion_limiting_set} In the main part of the proof of \autoref{thm_main_result} in \autoref{subsubsec_main_part_proof_main_thm}, we will have to investigate point processes living inside the sets $P_1(H_i)$. But, contrary to the setting in~\cite{Schrempp2016}, the inclusion $P_1(H_i) \subset P(H_i)$ does \emph{not} hold in general, and hence especially \emph{not} $P_n(H_i) \subset P(H_i)$ for every $n \ge 1$. Therefore, the set $P(H_i)$ is in general not suitable as state space for our point processes. Letting $\mathbb{R}^d$ be the state space would rectify this problem, but then the proof of \autoref{lemma_continuity_of_G_hat} would fail. So, this is the point where it becomes crucial to slightly enlarge the sets $P(H_i)$ via $\widehat \eta \cdot P(H_i)$. According to \eqref{eq_shifted_poles_in_xi_hat_P_H_i_proof_help_0} and the choice of $\delta^{}$ we have \begin{equation} \label{eq_Inclusion_in_eta_hat_P_H_i} P_1(H_i) \cap \left\{ z_1 \le \delta^{}\right\} \subset \widehat \eta \cdot P(H_i) \end{equation} for $i \in \left\{ \ell,r\right\}$. If $z \in \widehat \eta \cdot P(H_i)$, then $T_n(z) = \left( n^{2\nu}z_1,n^{\nu}\widetilde z\right)$ and \autoref{rem_representation_widehat_xi_P_H_i} yield $$ \frac{1}{2}(n^{\nu}\widetilde z)^\top H_i (n^{\nu}\widetilde z) = n^{2\nu}\frac{1}{2}\widetilde z^\top H_i \widetilde z \le \widehat \eta n^{2\nu} z_1, $$ i.e, we have $T_n(z) \in \widehat\eta \cdot P(H_i)$ for every $n \ge 1$. We thus get the inclusion $$ T_n\big( \widehat\eta \cdot P(H_i)\big) \subset \widehat\eta \cdot P(H_i) $$ for each $n \ge 1$, and \eqref{eq_Inclusion_in_eta_hat_P_H_i} implies $$ T_n\big( P_1(H_i) \cap \left\{ z_1 \le \delta^{}\right\}\big) \subset \widehat\eta \cdot P(H_i). $$ Thus, we can use the state space $\widehat\eta \cdot P(H_\ell)$ for the point processes representing the random points near the left pole and $\widehat\eta \cdot P(H_r)$ for the corresponding processes near the right pole. In the proofs to follow, it will be very important to consider only the sets $E_\delta$ $($given in \eqref{eq_Representation_E_delta}$)$ with $\delta \in (0,\delta^{}]$. Without this restriction, the point processes could `leave' their state space, and the proof of \autoref{lemma_continuity_of_G_hat} would fail. Since the asymptotical behavior of the maximum distance will be determined close to the poles, this restriction does not mean any loss of generality. Without \autoref{cond_A_eta_pos_semi_definite} it could be very complicated to find state spaces that are large enough to include the processes $($close to the poles$)$ but are also small enough to allow an adapted version of \autoref{lemma_continuity_of_G_hat}. These state spaces would have to be defined depending on $($the signs of$)$ the error functions $R_i $ in every direction of $\mathbb{R}^{d-1}$, we omit details. \end{remark} As before, let $V = (V_1,\ldots,V_d)$ have a density $g$ on $P_1(H_i) \cap \left\{ z_1 \le \delta^{}\right\}$ with $g(z) = p\big(1+o(1)\big)$ uniformly on $P_1(H_i) \cap \left\{ z_1 \le \delta\right\}$ as $\delta \to 0$ for some $p >0$. For $n \in \mathbb{N}$ and some fixed $c > 0$ let $\widetilde {\mathbf{V}}_n$ be a Poisson process with intensity measure $nc\cdot \mathbb{P}_V$. With independently chosen $N_n \overset{\mathcal{D}}{= } \text{Po}(nc)$ and i.i.d. $\widetilde Z_1, \widetilde Z_2, \ldots$ with distribution $\mathbb{P}_V$, we have $ \widetilde {\mathbf{V}}_n \overset{\mathcal{D}}{= } \sum_{j=1}^{N_n}\varepsilon_{\widetilde Z_{j}}. $ According to the Mapping Theorem for Poisson processes, see \cite[p. 38]{Last2017}, $\mathbf{V}_n := \widetilde {\mathbf{V}}_n \circ T_n^{-1}$ is a Poisson process with intensity measure $ \mu_n := nc \cdot \mathbb{P}_V\circ T_n^{-1}$, and the representation above yields $ \mathbf{V}_n \overset{\mathcal{D}}{= } \sum_{j=1}^{N_n}\varepsilon_{T_n(\widetilde Z_{j})}. $ We have $\mathbf{V}_n \in M_p\Big( T_n\big(P_1(H_i)\cap \left\{ z_1 \le \delta^\right\}\big)\Big)$, $n \in \mathbb{N}$, and because of \autoref{rem_inclusion_limiting_set} it follows that $\mathbf{V}_n \in M_p\big( \widehat \eta \cdot P(H_i)\big)$. \begin{lemma} \label{lemma_Conv_of_V_n} Let $\mathbf{V}_n $ be defined as above. Then $\mathbf{V}_n \overset{\mathcal{D}}{\longrightarrow } \mathbf{V}$ with $\mathbf{V} \overset{\mathcal{D}}{= } \text{PRM}(\mu)$ and $\mu := p c \cdot m_d\big\|_{P(H_i)} $. \end{lemma} \begin{proof} We use Proposition 3.22 in~\cite{Resnick1987}. Writing $\mathcal{I}$ for the set of finite unions of bounded open rectangles, we have to show that $\mathbb{P}\left( \mathbf{V} (\partial I) = 0\right) = 1$, and that both the conditions (3.23) and~(3.24) in~\cite{Resnick1987} hold for every $I \in \mathcal{I} $. Because of $\mu(\partial I) = 0$, the first requirement obviously holds, and an application of \autoref{lemma_transformation_density} gives $$ \mu_n(I) = nc \cdot \left( \mathbb{P}_V\circ T_n^{-1} \right)(I) = nc \cdot \mathbb{P} \big( T_n(V) \in I \big) = c\kappa_n(I) \to \mu(I). $$ Since $\mathbf{V}_n $ and $\mathbf{V} $ are Poisson processes, we get $$ \mathbb{P}\big(\mathbf{V}_n(I) = 0 \big) = e^{-\mu_n(I)}\frac{\mu_n(I)^0}{0!} = e^{-\mu_n(I)} \to e^{-\mu(I)} = e^{-\mu(I)}\frac{\mu(I)^0}{0!} = \mathbb{P}\big(\mathbf{V}(I) = 0 \big) $$ and $$ \mathbb{E}\big[ \mathbf{V}_n(I)\big] = \mu_n(I) \to \mu(I) = \mathbb{E}\big[ \mathbf{V}(I)\big] < \infty . $$ \qed \end{proof} \subsection{Main part of the proof of \autoref{thm_main_result}} \label{subsubsec_main_part_proof_main_thm} \begin{proof} As stated before, we only consider $\delta \in (0,\delta^{}]$. Recall $$ E_{\delta} = \big\{ (x,y) \in E_\ell \times E_r: -a \le x_1 \le -a + \delta, a - \delta \le y_1 \le a\big\}, $$ $\delta >0$, and put $ I_{n}^{\delta} := \big\{ (i,j) : 1 \le i, j \le N_n, (Z_i,Z_j) \in E_{\delta} \big\}, $ $n \in \mathbb{N}$. Letting $ M_{n}^{\delta} := \max_{(i,j) \in I_n^{\delta}} \big\|Z_i - Z_j\big\|, $ we obtain $\mathbb{P}\big(M_{n}^{\delta} \neq \text{diam}(\mathbf{Z}_n )\big) \rightarrow 0$ for each $\delta >0$, since both \begin{align} \mathbb{P}\big( Z \in E \cap \left\{ -a \le z_1 \le - a + \delta\right\}\big) > 0 \qquad \text{and} \qquad \mathbb{P}\big( Z \in E \cap \left\{ a - \delta \le z_1 \le a \right\}\big) > 0 \end{align} hold true for each $\delta > 0$. Hence, it suffices to investigate $M_n^{\delta}$ for some fixed $\delta >0$ instead of $\text{diam}(\mathbf{Z}_n )$. According to \eqref{eq_2a_minus_norm_is_G_tilde_minus_R} and \autoref{lemma_R_is_little_o_of_G_tilde}, for each $\varepsilon >0$ there is some $\delta >0$ so that $$ \widetilde G(x,y)(1-\varepsilon) \le 2a - \|x-y\| \le \widetilde G(x,y)(1+\varepsilon) $$ for each $(x,y) \in E_{\delta}$. These inequalities imply \begin{align} (1-\varepsilon)\min_{(i,j) \in I_n^{\delta}}\Big\{ n^{2\nu}\widetilde G(Z_i,Z_j)\Big\} \le n^{2\nu}\big( 2a - M_n^{\delta} \big) &= \min_{(i,j) \in I_n^{\delta}}\Big\{ n^{2\nu}\big(2a - \|Z_i - Z_j\|\big)\Big\} \le (1+\varepsilon)\min_{(i,j) \in I_n^{\delta}}\Big\{ n^{2\nu}\widetilde G(Z_i,Z_j)\Big\}. \end{align} Putting $c_{\ell,\delta} := \int_{E_{\ell,\delta}} f(z)\,\mathrm{d}z $ and $c_{r,\delta} := \int_{E_{r,\delta}} f(z)\,\mathrm{d}z $, we define the independent random vectors $ X,Y $ with densities $c_{\ell,\delta}^{-1}f\big\|_{ E_{\ell,\delta}}$ and $c_{r,\delta}^{-1}f\big\|_{ E_{r,\delta}}$, respectively. Furthermore, for $n \in \mathbb{N}$, we introduce the independent Poisson processes $\widehat{\mathbf{X} }_n$ and $\widehat{\mathbf{Y} }_n$ with intensity measures $nc_{\ell,\delta} \cdot\mathbb{P}_{X}$ and $nc_{r,\delta} \cdot\mathbb{P}_{Y}$, respectively. With independent random elements $N_{\ell,n}, N_{r,n}$, $X_1,X_2,\ldots$, $Y_1,Y_2,\ldots$, where $N_{\ell,n} \overset{\mathcal{D}}{= } \text{Po}(nc_{\ell,\delta})$, $N_{r,n} \overset{\mathcal{D}}{= } \text{Po}(nc_{r,\delta})$, $X_1,X_2,\ldots $ are i.i.d. with distribution $\mathbb{P}_X$ and $Y_1,Y_2,\ldots $ are i.i.d. with distribution $\mathbb{P}_Y$, we get $$ \widehat{\mathbf{X} }_n \overset{\mathcal{D}}{= } \sum_{i=1}^{N_{\ell,n}}\varepsilon_{X_i} \qquad \text{and} \qquad \widehat{\mathbf{Y} }_n \overset{\mathcal{D}}{= } \sum_{j=1}^{N_{r,n}}\varepsilon_{Y_j}. $$ Letting $I_n := \big\{ (i,j) : 1 \le i\le N_{\ell,n}, 1 \le j \le N_{r,n}\big\}$, we obtain $ M_n^{\delta} \overset{\mathcal{D}}{= } \max_{(i,j) \in I_n} \big\|X_i - Y_j\big\|. $ As above, the inequalities \begin{align} (1-\varepsilon)\min_{(i,j) \in I_n}\Big\{ n^{2\nu}\widetilde G(X_i,Y_j)\Big\} &\le n^{2\nu}\left( 2a - \max_{(i,j) \in I_n} \big\|X_i - Y_j\big\| \right) \le (1+\varepsilon)\min_{(i,j) \in I_n}\Big\{ n^{2\nu}\widetilde G(X_i,Y_j)\Big\}\label{eq_proof_main_thm_inequalities} \end{align} hold, and since $\varepsilon >0$ can be chosen arbitrarily small, it suffices to examine $ \min_{(i,j) \in I_n}\big\{ n^{2\nu}\widetilde G(X_i,Y_j)\big\}. $ We get \begin{align} n^{2\nu}\widetilde G(X_i,Y_j) \ =\ & n^{2\nu}\left( (a + X_{i,1}) + (a - Y_{j,1}) - \frac{1}{4a}\big\|\widetilde X_{i} - \widetilde Y_{j}\big\|^2 \right) =\ G\Big ( n^{2\nu}\big(a + X_{i,1}\big) , n^{\nu}\widetilde X_{i} , n^{2\nu} \big(a - Y_{j,1}\big) , n^{\nu}\widetilde Y_{j} \Big),\label{eq_reason_first_power_bigger} \end{align} where $$ G: \begin{cases} \widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r) \to \mathbb{R}_{+}, \\ (x,y) \mapsto x_1 + y_1 - \frac{1}{4a}\|\widetilde x- \widetilde y\|^2. \end{cases} $$ The proof of \autoref{lemma_continuity_of_G_hat} will show that $G(x,y) \ge 0$ for every $(x,y) \in \widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r) $. It will be important that $G$ is only defined on $\widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r) $, not on $\mathbb{R}^{2d}$ (see the proof of \autoref{lemma_continuity_of_G_hat}). This will be no restriction: Because of \autoref{rem_inclusion_limiting_set} it suffices to use instead of $\mathbb{R}^d$ the state spaces $\widehat \eta \cdot P(H_\ell)$ and $\widehat \eta \cdot P(H_r) $ for the point processes $\mathbf{X}_n $ and $\mathbf{Y}_n $, respectively, where $\mathbf{X}_n $ and $\mathbf{Y}_n $ will be defined later. To this end, we introduce the Poisson processes $$ \widetilde{\mathbf{X} }_n := \sum_{i=1}^{N_{\ell,n}}\varepsilon_{( \ a+X_{i,1} \ , \ \widetilde X_{i} \ )} \qquad \text{and} \qquad \widetilde{\mathbf{Y} }_n := \sum_{j=1}^{N_{r,n}}\varepsilon_{( \ a-Y_{j,1} \ , \ \widetilde Y_{j} \ )} $$ on $\big(P_1(H_\ell) \cap \{z_1 \le \delta^{}\}\big) \subset \widehat \eta \cdot P(H_\ell)$ and $\big(P_1(H_r) \cap \{z_1 \le \delta^{}\}\big) \subset \widehat \eta \cdot P(H_r)$, respectively. In view of \autoref{cond_density_p_l_p_r}, we can apply \autoref{lemma_Conv_of_V_n}, and since $\widetilde{\mathbf{X} }_n$ and $\widetilde{\mathbf{Y} }_n$ are independent, we conclude that \begin{equation} \label{eq_conv_of_X_and_Y} \mathbf{X}_n := \widetilde {\mathbf{X}}_n \circ T_n^{-1} \overset{\mathcal{D}}{\longrightarrow } \mathbf{X} \qquad \text{and} \qquad \mathbf{Y}_n := \widetilde {\mathbf{Y}}_n \circ T_n^{-1} \overset{\mathcal{D}}{\longrightarrow } \mathbf{Y} \end{equation} on $M_p\big( \widehat \eta \cdot P(H_\ell)\big)$ and $M_p\big( \widehat \eta \cdot P(H_r)\big)$, respectively, with independent point processes $\mathbf{X} := \left\{ \mathcal{X}_i, i \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(p_\ell\cdot m_d\big\|_{P(H_\ell)}\big)$ and $ \mathbf{Y} := \left\{ \mathcal{Y}_j, j \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(p_r\cdot m_d\big\|_{P(H_r)}\big)$. Observe that an application of \autoref{lemma_Conv_of_V_n} to $\mathbf{X}_n$ yields $p = p_\ell/c_{\ell,\delta}$, $c = c_{\ell,\delta}$ and finally $\mu = pc \cdot m_d\big\|_{P(H_\ell)} = p_\ell \cdot m_d\big\|_{P(H_\ell)}$. By construction, we have the representations $$ \mathbf{X}_n = \sum_{i=1}^{N_{\ell,n}}\varepsilon_{T_n( \ a+X_{i,1} \ , \ \widetilde X_{i} \ )} = \sum_{i=1}^{N_{\ell,n}}\varepsilon_{\left( \ n^{2\nu}(a + X_{i,1}) \ , \ n^{\nu}\widetilde X_{i} \ \right)} \qquad \text{and} \qquad \mathbf{Y}_n = \sum_{j=1}^{N_{r,n}}\varepsilon_{T_n( \ a-Y_{j,1} \ , \ \widetilde Y_{j} \ )} = \sum_{j=1}^{N_{r,n}}\varepsilon_{\left( \ n^{2\nu}(a - Y_{j,1}) \ , \ n^{\nu}\widetilde Y_{j} \ \right)}. $$ According to Proposition 3.17 in~\cite{Resnick1987}, $M_p\big( \widehat \eta \cdot P(H_\ell)\big)$ and $M_p\big( \widehat \eta \cdot P(H_r)\big)$ are separable. By Appendix M10 in \cite{Billingsley1999} we know that $M_p\big( \widehat \eta \cdot P(H_\ell)\big)\times M_p\big( \widehat \eta \cdot P(H_r)\big)$ is separable, too, and invoking Theorem 2.8 of \cite{Billingsley1999} \eqref{eq_conv_of_X_and_Y} implies $\mathbf{X}_n \times \mathbf{Y}_n \overset{\mathcal{D}}{\longrightarrow } \mathbf{X} \times \mathbf{Y}.$ Define now \begin{equation} \label{eq_def_G_hat} \widehat G: \begin{cases} M_p\big( \widehat \eta \cdot P(H_\ell)\big)\times M_p\big( \widehat \eta \cdot P(H_r)\big) \to M_p(\mathbb{R}_{+}), \\ \mu \mapsto \mu \circ G^{-1}. \end{cases} \end{equation} By construction, we have the representations \begin{align} \widehat G(\mathbf{X}_n \times \mathbf{Y}_n) &= \sum_{i=1}^{N_{\ell,n}}\sum_{j=1}^{N_{r,n}} \varepsilon_{G\left( \ n^{2\nu}(a + X_{i,1}) \ , \ n^{\nu}\widetilde X_{i} \ , \ n^{2\nu}(a - Y_{j,1}) \ , \ n^{\nu}\widetilde Y_{j} \ \right) } \qquad \text{and} \qquad \widehat G(\mathbf{X} \times \mathbf{Y}) = \sum_{i,j \ge 1}\varepsilon_{G(\mathcal{X}_i,\mathcal{Y}_j )}. \end{align} Since the mapping $\widehat G$ is continuous (see \autoref{lemma_continuity_of_G_hat}), the continuous mapping theorem gives \begin{equation} \label{eq_conv_G_hat_X_times_Y} \widehat G(\mathbf{X}_n \times \mathbf{Y}_n) \overset{\mathcal{D}}{\longrightarrow } \widehat G(\mathbf{X} \times \mathbf{Y}). \end{equation} For a point process $\xi$ on $\mathbb{R}_+$ we define $t_1(\xi) := \min\left\{ t \ge 0: \xi\big([0,t]\big) \ge 1\right\}.$ The reason for introducing $t_1$ is the very useful relation $$ \min_{(i,j) \in I_n}\Big\{ n^{2\nu}\widetilde G(X_i,Y_j)\Big\} = t_1\big( \widehat G(\mathbf{X}_n \times \mathbf{Y}_n)\big). $$ \autoref{lemma_conv_of_t_1} says that $t_1\big( \widehat G(\mathbf{X}_n \times \mathbf{Y}_n) \big) \overset{\mathcal{D}}{\longrightarrow } t_1\big( \widehat G(\mathbf{X} \times \mathbf{Y}) \big)$ and, because of \begin{align} t_1\big( \widehat G(\mathbf{X} \times \mathbf{Y}) \big) = \min_{i,j \ge 1} \big\{ G(\mathcal{X}_i,\mathcal{Y}_j ) \big\} &= \min_{i,j \ge 1} \left\{ \mathcal{X}_{i,1} + \mathcal{Y}_{j,1} - \frac{1}{4a} \big\| \widetilde {\mathcal{X}}_{i}- \widetilde {\mathcal{Y}}_{j}\big\|^2\right\}, \end{align} the convergence stated in~\eqref{eq_theorem_main_result} follows from \eqref{eq_proof_main_thm_inequalities} as $\varepsilon \to 0$. Applying Theorem 3.2 in~\cite{MayerMol2007} to the functional $\Psi(\mathbf{Z}_n )= 2 - \text{diam}(\mathbf{Z}_n )$ shows that the same result holds true if we replace $\text{diam}(\mathbf{Z}_n )$ with $M_n$. \qed \end{proof} \begin{remark} \label{remark_reason_def_T_n} An explanation for the definition of the rescaling function $T_n(z) = (n^{2\nu}z_1,n^{\nu}\widetilde z)$ with $\nu = 1/(d+1)$ can be found in the proof of \autoref{lemma_Conv_of_V_n}: The $d$ powers of $n$ have to be chosen in such a way that their sum is $1$. This requirement implies $\Delta T_n(z) = n$ in the proof of \autoref{lemma_transformation_density}, whence $\mathbb{P}(T_n(V)\in B) = \kappa_n(B)/n$. As seen in the proof of \autoref{lemma_Conv_of_V_n}, the factors $1/n$ and $n$ cancel out, and only $c\kappa_n(B)$ remains. The reason why the first power is twice the other $d-1$ identical powers is due to the Taylor series expansion of $\|x-y\|$ in~\eqref{eq_taylor_eukl_dist_at_poles}. This fact fits exactly to the shape of $E$ near the poles, so that $P_n(H_i) = T_n\big(P_1(H_i) \cap \left\{ z_1 \le \delta^{}\right\}\big)$ can converge to the set $P(H_i)$, $i \in \left\{ \ell,r\right\}$ $($see the proof of \autoref{lemma_transformation_density}$)$. Finally, from~\eqref{eq_reason_first_power_bigger} it is clear that $n^{2\nu}$ is the correct scaling factor. \end{remark} We still have to verify the continuity of the function $\widehat G$: \begin{lemma} \label{lemma_continuity_of_G_hat} The function $\widehat G$ is continuous. \end{lemma} \begin{proof} This assertion may be proved in the same way as Proposition 3.18 in~\cite{Resnick1987}. We thus only have to demonstrate that $G^{-1}(K) \subset \widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r)$ is compact if $K \subset \mathbb{R}$ is compact. For this purpose, let $K \subset \mathbb{R}$ be compact. Since $G$ is continuous, $G^{-1}(K)$ is closed, and it remains to show that $G^{-1}(K)$ is bounded. From the specific form of $\widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r)$, $G^{-1}(K)$ can only be unbounded if it is unbounded in $x_1$- or $y_1$-direction (at this point it is important that our state spaces for the point processes are not $\mathbb{R}^d$, but only the subsets $\widehat \eta \cdot P(H_\ell)$ and $\widehat \eta \cdot P(H_r)$). For fixed $(x,y) \in \widehat\eta \cdot P(H_\ell) \times \widehat\eta \cdot P(H_r)$, let $\alpha,\beta \in \mathbb{R}^{d-1}$, so that $\widetilde x = U_\ell \alpha$ and $\widetilde y = U_r\beta$. Applying the same transformations as seen for $\widetilde G(x,y)$ in the proof of \autoref{lemma_R_is_little_o_of_G_tilde} to $G(x,y)$ yields $$ G(x,y) \ge x_1 + y_1 - \eta\left(\frac{1}{2} \widetilde x^\top H_\ell\widetilde x + \frac{1}{2}\widetilde y^\top H_r \widetilde y\right), $$ and using the representation of $\widehat \eta \cdot P(H_i)$ given in \autoref{rem_representation_widehat_xi_P_H_i} shows that $$ G(x,y) \ge x_1 + y_1 - \eta\left(\widehat \eta x_1 + \widehat \eta y_1\right) = \left( 1 - \eta\widehat \eta\right)(x_1 + y_1) = \frac{1-\eta}{2}(x_1 + y_1). $$ Since $\eta \in (0,1)$, we have $\frac{1-\eta}{2} > 0$ and the assumption $(x,y) \in \widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r)$ implies $(x_1,y_1) \in \mathbb{R}_+^2$, so that $G(x,y) \ge 0$ for each $(x,y) \in \widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r)$. If $x_1 \to \infty$ and/or $y_1 \to \infty$, the lower bound $ \frac{1-\eta}{2}(x_1+ y_1)$ for $G(x,y)$ also tends to infinity. From the boundedness of $K$ it follows that $G^{-1}(K)$ has to be bounded in $x_1$- and $y_1$-direction, too. This argument finishes the proof. \qed \end{proof} Finally, we have to prove the last lemma, used in the proof of \autoref{thm_main_result}: \begin{lemma} \label{lemma_conv_of_t_1} We have $t_1\big(\widehat G(\mathbf{X}_n \times \mathbf{Y}_n ) \big) \overset{\mathcal{D}}{\longrightarrow } t_1\big(\widehat G(\mathbf{X} \times \mathbf{Y} ) \big)$. \end{lemma} \begin{proof} In a first step we will show that $\widehat G(\mathbf{X} \times \mathbf{Y} ) \big(\left\{ t\right\}\big) = 0$ almost surely for each $t \ge 0$. For this purpose, we consider the set $$ G^{-1}\big( \left\{ t\right\}\big) = \left\{ (x,y) \in \widehat \eta \cdot P(H_\ell) \times \widehat \eta \cdot P(H_r): x_1 + y_1 - \frac{1}{4a}\|\widetilde x - \widetilde y\|^2 = t\right\}. $$ For some fixed $y^{} \in \widehat \eta \cdot P(H_r)$ we define $ A(y^{}):= \big\{ x \in \widehat \eta \cdot P(H_\ell): (x,y^{}) \in G^{-1}( \left\{ t\right\})\big\} $ and obtain $$ A(y^{}) \phantom{:}= \ \left\{ x \in \widehat \eta \cdot P(H_\ell): x_1 + y_1^{} - \frac{1}{4a}\|\widetilde x - \widetilde y^{}\|^2 = t\right\} = \ \left\{ x \in \widehat \eta \cdot P(H_\ell): \sqrt{4a\big(x_1 - (t- y_1^{})\big)} = \|\widetilde x - \widetilde y^{}\|\right\}. $$ Since the set $A(y^{})$ has Lebesgue-measure $0$, we can conclude that $\mathbf{X}\big(A(y^{})\big) = 0 $ almost surely for each $y^{} \in \widehat \eta \cdot P(H_r)$. This result implies $\widehat G(\mathbf{X} \times \mathbf{Y} ) \big(\left\{ t\right\}\big) = 0$ almost surely for each $t \ge 0$. In the following, we will write $\xi := \widehat G(\mathbf{X} \times \mathbf{Y} ) $ and $\xi_n :=\widehat G(\mathbf{X}_n \times \mathbf{Y}_n )$ for $n \in \mathbb{N}$. In view of \eqref{eq_conv_G_hat_X_times_Y}, the first part of this proof and Theorem 16.16 in~\cite{Kallenberg2002}, the convergence $\xi_n\big([0,t]\big) \overset{\mathcal{D}}{\longrightarrow } \xi\big([0,t]\big)$ holds true for each $ t >0$. Since $\xi_n$ and $\xi$ are point processes, $1/2$ is a point of continuity of the distribution functions of both $\xi_n\big([0,t]\big)$ and $\xi\big([0,t]\big)$, and we obtain $$ \mathbb{P}\Big( \xi_n\big([0,t]\big) = 0\Big) = \mathbb{P}\left( \xi_n\big([0,t]\big) \le \frac{1}{2}\right) \to \mathbb{P}\left( \xi\big([0,t]\big) \le \frac{1}{2}\right) = \mathbb{P}\Big( \xi\big([0,t]\big) = 0\Big) $$ for each $ t >0$. Thus, we have \begin{align} \mathbb{P}\big(t_1(\xi_n) \le t\big) =& \; 1 - \mathbb{P}\big(t_1(\xi_n) > t\big) \\ =& \; 1 - \mathbb{P}\Big( \xi_n\big([0,t]\big) = 0\Big)\\ \to& \; 1 - \mathbb{P}\Big( \xi\big([0,t]\big) = 0\Big)\\ =& \; 1 - \mathbb{P}\big(t_1(\xi) > t\big) \\ =& \; \mathbb{P}\big(t_1(\xi) \le t\big). \end{align} \qed \end{proof} \section{Generalizations 1 - Sets with unique diameter} \label{sec_generalizations_unique_diameter} This section deals with some obvious generalizations of \autoref{thm_main_result}. \autoref{subsec_more_general_densities_in_ellipsoids} is devoted to more general densities than those covered by \autoref{cond_density_p_l_p_r} in \autoref{sec_main_results}. Being more precise, we will investigate densities supported by ellipsoids that are allowed to tend to $0$ or $\infty $ close to the poles. It will turn out that the so-called Pearson Type II distributions are special distributions covered by this setting. \autoref{subsec_joint_convergence_k_largest_distances} establishes a limit theorem for the joint convergence of the $k $ largest distances among the random points in the settings of both \autoref{sec_main_results} and \autoref{subsec_more_general_densities_in_ellipsoids}. Moreover, \autoref{subsec_p_pllipsoids_p_norms} deals with $p$-superellipsoids and $p$-norms, where $1 \le p < \infty $. If the underlying $p$-superellipsoid has a unique diameter with respect to the $p$-norm and we use this norm to define the largest distance among the random points, we obtain very similar results as seen in \autoref{sec_main_results}. \subsection{More general densities supported by ellipsoids} \label{subsec_more_general_densities_in_ellipsoids} In this section we consider closed ellipsoids \begin{equation} \label{eq_def_open_ellipsoid_Pearson} E := \left\{z \in \mathbb{R}^d: \sum_{k=1}^{d}\left( \frac{z_k}{a_k} \right)^2 \le 1\right\}, \end{equation} with half axes $a_1 > a_2 \ge \ldots \ge a_d>0$, seen before in \autoref{cor_Ellipsoid}, and we define $\Sigma := \text{diag}(a_1^2,\ldots,a_d^2) \in \mathbb{R}^{d\times d}$. Inside of these ellipsoids we consider densities that satisfy the following condition: \begin{condition} \label{cond_f_gen_density_in_ellipsoid} We assume $f: E\to \mathbb{R}_{+}$, $\int_{E}f(z)\,\mathrm{d}z =1 $ and that there are constants $\alpha_\ell,\alpha_r > 0$ and $\beta_\ell,\beta_r >-1$ so that the function $$ z \mapsto \frac{f(z)}{\alpha_i\left(1- z^\top \Sigma^{-1} z\right)^{\beta_i}}, $$ that maps from $\text{int}(E)$ into $\mathbb{R}_{+}$, can be extended continuously at the poles $(-a,\mathbf{0} )$ and $(a,\mathbf{0} )$ with value $1$. Thereby, $\alpha_\ell, \beta_\ell$ correspond to the left pole $(-a,\mathbf{0} )$ and $\alpha_r, \beta_r$ to the right pole $(-a,\mathbf{0} )$, respectively. \end{condition} Notice that \autoref{cond_density_p_l_p_r} was a special case of this condition, namely for $\beta_i = 0$ and with $\alpha_i = p_i$, $i \in \left\{ \ell,r\right\}$ (observe that we can use $E$ instead of $\text{int}(E)$ in this case). The crucial difference to the setting of \autoref{thm_main_result} occurs in \autoref{lemma_transformation_density}. Before we state the main result of this section, which is \autoref{thm_main_result_gen_density_in_ellipsoid}, we will point out this essential difference. As already seen in \autoref{cor_Ellipsoid}, we have \begin{align} H_\ell = H_r = \text{diag}\left( \frac{a_1}{a_2^2}\ ,\ \ldots\ ,\ \frac{a_1}{a_d^2}\right), \end{align} and because of this symmetry, we briefly write $H := H_\ell = H_r$. Remember now the construction of $P_1(H)$ given at the beginning of \autoref{subsec_proof_main_thm_geom_considerations}. In this section, we use the same construction for $\text{int}(E)$ instead of $E$ to avoid divisions by $0$ for $\beta<0$, and we conclude that \begin{align} P_1(H)\ = &\ \left\{z \in \mathbb{R}^d: \left( \frac{z_1-a_1}{a_1} \right)^2 + \sum_{k=2}^{d}\left( \frac{z_k}{a_k} \right)^2 < 1, z_1 < a_1\right\} = \ \left\{z \in \mathbb{R}^d: \sum_{k=2}^{d}\left( \frac{z_k}{a_k} \right)^2 < \frac{2z_1}{a_1} - \left( \frac{z_1}{a_1}\right)^2, z_1 < a_1\right\}. \end{align} Since we only consider distributions of $Z$ that are absolutely continuous with respect to Lebesgue measure, this is no restriction at all. To show an adjusted version of \autoref{lemma_transformation_density}, we have, in generalization of~\eqref{eq_def_of_nu}, to define the constant $$ \nu := \frac{1}{d+1+2\beta}, $$ $\beta>-1$, the rescaling function $$ T_n(z) := \left( \ n^{2\nu}z_1\ ,\ n ^{\nu}\widetilde z\ \right) $$ for $n \in \mathbb{N}$, $ z = (z_1,\widetilde z) \in \mathbb{R}^d$ and the (now open) limiting set $$ P(H) := \ \left\{ z \in \mathbb{R}^d: \sum_{k=2}^{d}\left( \frac{z_k}{a_k}\right)^2 < \frac{2z_1}{a_1}\right\}. $$ Now we can state an adapted version of \autoref{lemma_transformation_density}. \begin{lemma} \label{lemma_transformation_density_gen_density} Suppose the random vector $V = (V_1,\ldots,V_d)$ has a density $g$ on $P_1(H)$ satisfying \begin{align} g(z) & = \left( 1+ o(1)\right)\cdot \alpha \cdot \left( 1 - \left( \frac{z_1-a_1}{a_1}\right)^2 - \sum_{k=2}^{d}\left( \frac{z_k}{a_k}\right)^2 \right)^{\beta}, \label{eq_Lemma_Trans_gen_density_Condition} \end{align} uniformly on $P_1(H) \cap \left\{ z_1 \le \delta\right\}$ as $\delta \to 0$, for some $\alpha>0$ and $ \beta > -1$. Then, for every bounded Borel set $B \subset \mathbb{R}^d$, we have \begin{equation} \label{eq_gen_density_P_T_n_V_in_B} \mathbb{P}\big( T_n(V) \in B \big)= \frac{\alpha}{n}\cdot\kappa_n(B) \end{equation} with $\kappa_n(B) \to \Lambda_{\beta}(B)$ and \begin{equation}\label{eq_Def_Lambda_beta} \Lambda_{\beta}(B) := \int_{B} \left( \frac{2z_1}{a_1} - \sum_{k=2}^{d}\left( \frac{z_k}{a_k}\right)^2 \right)^{\beta} \mathds{1}\left\{ z \in P(H)\right\}\,\mathrm{d}z. \end{equation} \end{lemma} The proof of this lemma is very technical since we can (in general) neither apply the dominated convergence theorem, nor the monotone convergence theorem to show $\kappa_n(B) \to \Lambda_{\beta}(B)$. Instead, an application of Scheff\'e's Lemma is necessary, see \cite{Schrempp2017} for more details and for the connection between this lemma and the following result. \begin{theorem} \label{thm_main_result_gen_density_in_ellipsoid} Let the density $f$ be supported by the ellipsoid $E$ with half-axes $a_1 > a_2 \ge \ldots \ge a_d>0$ and satisfy \autoref{cond_f_gen_density_in_ellipsoid} with $\beta_\ell = \beta_r =:\beta$. We then have \begin{equation} n^{\frac{2}{d+1+2\beta}}\big(2a_1 - \mathrm{diam}(\mathbf{Z}_n )\big) \overset{\mathcal{D}}{\longrightarrow } \min_{i,j \ge 1} \left\{ \mathcal{X}_{i,1} + \mathcal{Y}_{j,1} - \frac{1}{4a_1}\big\| \widetilde {\mathcal{X}}_{i}-\widetilde {\mathcal{Y}}_{j}\big\|^2 \right\}, \end{equation} where $\left\{ \mathcal{X}_i, i \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big( \alpha_\ell\cdot \Lambda_{\beta}\big) $ and $\left\{ \mathcal{Y}_j, j \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(\alpha_r\cdot \Lambda_{\beta}\big) $ are independent Poisson processes. If \autoref{cond_f_gen_density_in_ellipsoid} and -- without loss of generality -- the inequality $\beta_\ell > \beta_r$ hold true, we obtain \begin{equation} n^{\frac{2}{d+1+2\beta_{\ell}}}\big(2a_1 - \mathrm{diam}(\mathbf{Z}_n )\big) \overset{\mathcal{D}}{\longrightarrow } \min_{i \ge 1} \left\{ \mathcal{X}_{i,1} - \frac{1}{4a_1}\big\| \widetilde {\mathcal{X}}_{i}\big\|^2 \right\}, \end{equation} with $\left\{ \mathcal{X}_i, i \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big( \alpha_\ell\cdot \Lambda_{\beta_\ell}\big) $. The same results hold true if we replace $ \mathrm{diam}(\mathbf{Z}_n )$ with $M_n$. \end{theorem} \begin{proof} Under \autoref{cond_f_gen_density_in_ellipsoid} we have $f(z)>0$ for each $z$ arbitrarily close to one of the poles. In the case $\beta_\ell = \beta_r$, this inequality allows us to copy the proof of \autoref{thm_main_result} almost completely. The only difference is that we have to apply \autoref{lemma_transformation_density_gen_density} instead of \autoref{lemma_transformation_density} to show an adapted version of \autoref{lemma_Conv_of_V_n}. In the case $\beta_\ell>\beta_r$ we will observe a higher magnitude of points lying close to the right pole than to the left. This higher magnitude has far-reaching implications for the proof to follow. First of all, we define $$ \nu_\ell:=\frac{1}{d+1+2\beta_\ell}, \qquad T_n^{\ell}(z) := \left( \ n^{2\nu_\ell}z_1\ ,\ n ^{\nu_\ell}\widetilde z\ \right) \qquad \text{and} \qquad P_n^{\ell}(H) := T_n^{\ell}\big(P_1(H)\big). $$ The beginning of the main part of the proof of \autoref{thm_main_result} in \autoref{subsubsec_main_part_proof_main_thm} can be copied in this case, too. We will only point out the main difference. Let $N_{r,n}$ and $ Y_1,Y_2,\ldots $ be defined as in the proof of \autoref{thm_main_result} and write $V^r := (a-Y_{1,1},\widetilde Y_1)$. Then, $$ \widetilde{\mathbf{Y} }_n := \sum_{j=1}^{N_{r,n}}\varepsilon_{(a-Y_{j,1},\widetilde Y_{j})} $$ is a Poisson process with intensity measure $nc_{r,\delta}\cdot \mathbb{P}_{V^r}$, and $\mathbf{Y}_n^{\ell} := \mathbf{\widetilde Y}_n \circ (T_n^{\ell})^{-1}$ -- taking the part of $\mathbf{Y}_n $ in the proof of \autoref{thm_main_result} -- is a Poisson process with intensity measure $\widehat \mu_n := nc_{r,\delta} \cdot \mathbb{P}_{V^r} \circ (T_n^{\ell})^{-1}$. The density $f$ fulfills \autoref{cond_f_gen_density_in_ellipsoid} at the right pole with power $\beta_r$, but the shifted process $\widetilde{\mathbf{Y} }_n $ is scaled via $T_n^\ell$, which depends on $\beta_\ell$, \emph{not} on $ \beta_r$. Broadly speaking, this `wrong' (too slow) scaling has the effect, that $\mathbf{Y}_n^{\ell} $ will generate more and more points arbitrarily close to $\mathbf{0} $, see \cite{Schrempp2017} for technical details. \qed \end{proof} \begin{example} \label{ex_pearson_type_II} We now consider the so-called $d$-dimensional symmetric multivariate Pearson Type II distributions supported by an ellipsoid with half-axes $a_1 > a_2 \ge \ldots \ge a_d>0$, where $d \ge 2$. According to equation $($2.43$)$ in~\cite{Fang1990} and Example 2.11 in the same reference, we know that the corresponding densities are given by \begin{align} f_{\beta}(z) &=\frac{\Gamma\left( \frac{d}{2}+\beta + 1\right)}{\Gamma\left( \beta +1\right)\pi^{\frac{d}{2}}\prod_{i=1}^{d}a_i}\left(1-z^\top \Sigma^{-1}z\right)^{\beta}\cdot \mathds{1}\left\{ z \in \text{int}(E)\right\}. \end{align} Hence, \autoref{cond_f_gen_density_in_ellipsoid} holds true with $\beta_\ell = \beta_r = \beta$ and $ \alpha:=\alpha_\ell = \alpha_r = \frac{\Gamma\left( \frac{d}{2}+\beta + 1\right)}{\Gamma\left( \beta +1\right)\pi^{\frac{d}{2}}\prod_{i=1}^{d}a_i}, $ so that we can apply \autoref{thm_main_result_gen_density_in_ellipsoid}. \end{example} Figures~\ref{fig_pearson_beta_minus_0_5} and \ref{fig_pearson_beta_2} illustrate the densities $f_{\beta}$ and the corresponding densities of the intensity measures $\alpha \cdot \Lambda_\beta$ in the setting of \autoref{ex_pearson_type_II} for $d=2$, $a_1 = 1, a_2 = 1/2$ and $\beta \in \left\{ -1/2, 2\right\}$. See \cite{Schrempp2017} for the illustration of some more Pearson Type II densities in two dimensions and the results of a simulation study. \begin{figure} \caption{The density $f_{\beta}$ (left) and that of the intensity measure $\alpha \cdot \Lambda_\beta$ (right) in the setting of \autoref{ex_pearson_type_II} for $d=2$ with $a_1 =1, a_2 = 1/2$ and $\beta = -1/2$.} \label{fig_pearson_beta_minus_0_5} \end{figure} \begin{figure} \caption{The density $f_{\beta}$ (left) and that of the intensity measure $\alpha \cdot \Lambda_\beta$ (right) in the setting of \autoref{ex_pearson_type_II} for $d=2$ with $a_1 =1, a_2 = 1/2$ and $\beta = 2$.} \label{fig_pearson_beta_2} \end{figure} \subsection{Joint convergence of the \texorpdfstring{$k$}{k} largest distances} \label{subsec_joint_convergence_k_largest_distances} To state a result on the joint asymptotical behavior of the $k$ largest distances of the Poisson process $\mathbf{Z}_n = \sum_{i=1}^{N_n}\varepsilon_{Z_{i}}$, introduced in \autoref{sec_fundamentals}, we need some additional definitions. For $n \in \mathbb{N}$, let $D_n^{(1)} \ge D_n^{(2)} \ge \ldots \ge D_n^{\left( k\right)}$ be the $k$ largest distances in descending order between $Z_i$ and $Z_j$ for $1 \le i< j \le N_n$. So, we especially have $D_n^{(1)} = \text{diam}(\mathbf{Z}_n ).$ For a point process $\xi$ on $\mathbb{R}_+$ and $i \in \mathbb{N}$ we define $t_i(\xi) := \inf\left\{ t: \xi\big([0,t]\big) \ge i\right\}.$ According to Proposition 9.1.XII in~\cite{Daley2008}, each $t_i(\xi)$ is a well-defined random variable if $\xi$ is a simple point process. Since the point processes $ \widehat G(\mathbf{X}_n \times \mathbf{Y}_n)$ and $ \widehat G(\mathbf{X} \times \mathbf{Y})$ on $\mathbb{R}_+$ (introduced in the proof of \autoref{thm_main_result}) are simple, we conclude that the random variables $t_i\big( \widehat G(\mathbf{X}_n \times \mathbf{Y}_n) \big)$ and $t_i\big( \widehat G(\mathbf{X} \times \mathbf{Y}) \big)$ are well-defined for each fixed $ i \in \mathbb{N}$. Now we can state our result on the joint convergence of the $k$ largest distances in the setting of \autoref{sec_main_results}: \begin{theorem} \label{thm_main_result_joint_conv} If Conditions~\ref{cond_unique_diameter} to \ref{cond_density_p_l_p_r} hold true, we have for each $k \in \mathbb{N}$ the joint convergence \begin{equation} n^{\frac{2}{d+1}} \Big( 2a - D_n^{(1)}\ ,\ldots, \ 2a - D_n^{(k)} \Big) \overset{\mathcal{D}}{\longrightarrow } \Big( t_1\big( \widehat G(\mathbf{X} \times \mathbf{Y}) \big)\ ,\ldots, \ t_k\big( \widehat G(\mathbf{X} \times \mathbf{Y}) \big) \Big) , \end{equation} where $\mathbf{X} \overset{\mathcal{D}}{= } \text{PRM}\big(p_\ell\cdot m_d\big\|_{P(H_\ell)}\big) $ and $\mathbf{Y} \overset{\mathcal{D}}{= } \text{PRM}\big(p_r\cdot m_d\big\|_{P(H_r)}\big) $ are independent Poisson processes. \end{theorem} The proof of this theorem is a simple generalization of that of \autoref{thm_main_result}, see Section 5.3 in \cite{Schrempp2017} for more details. We can immediately generalize the results of \autoref{subsec_more_general_densities_in_ellipsoids}, too: \begin{theorem} \label{thm_joint_convergence_gen_density} Let the density $f$ be supported by the ellipsoid $E$ defined in~\eqref{eq_def_open_ellipsoid_Pearson} with half-axes $a_1 > a_2 \ge \ldots \ge a_d>0$, and put $a := a_1 $. If $f$ satisfies \autoref{cond_f_gen_density_in_ellipsoid} with $\beta_\ell = \beta_r =:\beta$ then, for each fixed $k \ge 1$, we have \begin{equation} n^{\frac{2}{d+1+2\beta}} \Big( 2a - D_n^{(1)}\ ,\ \ldots\ ,\ 2a - D_n^{(k)} \Big) \overset{\mathcal{D}}{\longrightarrow } \Big( t_1\big( \widehat G(\mathbf{X} \times \mathbf{Y}) \big)\ ,\ \ldots\ , \ t_k\big( \widehat G(\mathbf{X} \times \mathbf{Y}) \big) \Big), \end{equation} where $\mathbf{X} \overset{\mathcal{D}}{= } \text{PRM}\big( \alpha_\ell\cdot \Lambda_{\beta}\big) $ and $\mathbf{Y} \overset{\mathcal{D}}{= } \text{PRM}\big(\alpha_r\cdot \Lambda_{\beta}\big) $ are independent Poisson processes. \end{theorem} Notice that the definition $a := a_1$ in the theorem above is necessary, since the function $\widehat G$ has been defined in \autoref{subsubsec_main_part_proof_main_thm} in terms of $a$, not of $a_1$. See \cite{Schrempp2017} for the results of a simulation study. \subsection{\texorpdfstring{$p$}{p}-superellipsoids and \texorpdfstring{$p$}{p}-norms} \label{subsec_p_pllipsoids_p_norms} For $1 \le p < \infty$ and $a_1 > a_2 \ge a_3 \ge \ldots \ge a_d >0$ we define the so-called $p$-superellipsoid $$ E^p := \left\{ z \in \mathbb{R}^d: \sum_{k=1}^{d}\left( \frac{\|z_k\|}{a_k}\right)^p \le 1\right\} $$ and the corresponding $p$-norm $$ \|z\|_p := \left( \sum_{k=1}^{d}\|z_k\|^p \right)^{\frac{1}{p}}, \quad z \in \mathbb{R}^d. $$ Moreover, based on this norm, let $$ \text{diam}_p(A) := \sup_{x,y \in A} \|x-y\|_p $$ be the so-called $p$-diameter of a set $A \subset \mathbb{R}^d$. The definitions of $E^p$ and $\|\cdot\|_p$ yield $\big\|(-a_1,\mathbf{0} )-(a_1,\mathbf{0} )\big\|_p = 2a_1$, and in view of $a_1 > a_2 \ge a_3 \ge \ldots \ge a_d >0$ we have $\|z\|_p \le a_1$ for each $z \in E^p$, with equality only for $z \in \big\{ (-a_1,\mathbf{0} ),(a_1,\mathbf{0} )\big\}$. Together with $\|x-y\|_p \le \|x\|_p + \|y\|_p$ for all $x,y \in \mathbb{R}^d$ we can infer that the set $E^p$ has a unique diameter of length $2a_1$ with respect to the $p$-norm between the points $(-a_1,\mathbf{0} )$ and $(a_1,\mathbf{0} )$.\\ We assume that the random variables $Z_1,Z_2,\ldots$ are i.i.d. with a common density $f$, supported by the superellipsoid $E^p$. As in \autoref{sec_main_results}, we consider densities that are continuous and bounded away from 0 at the poles. In this subsection we will investigate the largest distance between these random points with respect to the corresponding $p$-norm, not with respect to the Euclidean norm, i.e. we consider $$ M_n^{p} := \max_{1 \le i , j \le n}\|Z_i - Z_j\|_p. $$ Using the Poisson process $\mathbf{Z}_n$ with intensity measure $n\mathbb{P}_{Z}$, defined in \autoref{sec_fundamentals}, we get $$ \text{diam}_p(\mathbf{Z}_n ) = \max_{1 \le i,j \le N_n} \big\|Z_i - Z_j\big\|_p. $$ Defining the new limiting set \begin{equation} \label{eq_p_ellips_def_limiting_set} P^p := \left\{ z \in \mathbb{R}^d: \sum_{k=2}^d \left( \frac{\|z_k\|}{a_k}\right)^p \le \frac{pz_1}{a_1}\right\} \end{equation} and using very similar techniques as seen before in \autoref{sec_proof_main_thm}, we can prove the following result: \begin{theorem} \label{thm_p_ellipsoid_p_norm} Under the standing assumptions of this section and if \autoref{cond_density_p_l_p_r} holds true for $E$ replaced with $E^p$ and $a = a_1$, then \begin{equation} n^{\frac{p}{d+p-1}}\big(2a_1 - \mathrm{diam}_p(\mathbf{Z}_n )\big) \overset{\mathcal{D}}{\longrightarrow } \min_{i,j \ge 1} \left\{ \mathcal{X}_{i,1} + \mathcal{Y}_{j,1} - \frac{1}{p(2a_1)^{p-1}} \big\|\widetilde {\mathcal{X}}_{i}- \widetilde {\mathcal{Y}}_{j}\|_p^p \right\}, \end{equation} where $\left\{ \mathcal{X}_i, i \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(p_\ell\cdot m_d\big\|_{P^p}\big) $ and $\left\{ \mathcal{Y}_j, j \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\big(p_r\cdot m_d\big\|_{P^p}\big) $ are independent Poisson processes. The same holds true if we replace $ \text{diam}_p(\mathbf{Z}_n )$ with $M_n^{p}$. \end{theorem} The proof of this theorem can be found in Section 5.5 in \cite{Schrempp2017}. \begin{corollary} \label{cor_p_norm} Given the uniform distribution on $E^p$, \autoref{cond_density_p_l_p_r} holds true for $E$ replaced with $E^p$, $a=a_1$ and $$ p_\ell = p_r = \frac{1}{m_d\big(E^p \big)} = \left( \frac{\left( 2 \Gamma\left( 1 + \frac{1}{p}\right) \right)^d\prod_{i=1}^d a_i}{\Gamma\left( 1 + \frac{d}{p}\right)} \right)^{-1} > 0, $$ see~\cite{Wang2005}. We can thus apply \autoref{thm_p_ellipsoid_p_norm}. Notice that \autoref{cor_ellipse_uniform} is a special case of this corollary, namely for $p = 2$. \end{corollary} Some more generalizations can be found in \cite{Schrempp2017}: Section 5.2 in that reference takes a look at more general densities supported by any set (not only ellipsoids), fulfilling the Conditions~\ref{cond_unique_diameter} to \ref{cond_A_eta_pos_semi_definite}. Furthermore, a different shape of $E$ close to the poles is considered in Section 5.4, and Section 5.6 illustrates that the smoothness of the boundary of $ E$ at the poles, as demanded in \autoref{cond_shape_pole_caps}, is by no means necessary to prove results similar to that of \autoref{thm_main_result}. \section{Generalizations 2 - Sets with no unique diameter} \label{sec_generalizations_no_unique_diameter} In this section we consider sets with no unique diameter, i.e. we no longer assume that \autoref{cond_unique_diameter} holds true. Basically, there are two different ways to modify this condition. The first is given by sets, having $ k $ pairs of poles, where $1 < k < \infty $, see \autoref{cond_several_axes_k_pole_pairs} below for a formal definition. Such sets will be studied in \autoref{subsec_several_major_axes}. An alternative modification of \autoref{cond_unique_diameter} is -- heuristically spoken in three dimensions -- given by sets with an equator, for example a three-dimensional ellipsoid with half-axes $1,1$ and $1/2$. For Pearson Type II distributed points in $d$-dimensional ellipsoids with at least two but less than $d$ major half-axes, we still do not know whether a limit distribution for $M_n$ exists, or not. However, at least for each of these Pearson Type II distributions, \autoref{subsec_ellipsoid_several_major_axes} exhibits bounds for the limit distribution of $M_n$, provided that such a limit law exists. \subsection{Several major axes} \label{subsec_several_major_axes} In this subsection we consider closed sets with more than one, but finitely many pairs of poles. To this end, we formulate a more general version of \autoref{cond_unique_diameter}: \begin{condition} \label{cond_several_axes_k_pole_pairs} Let $E \subset \mathbb{R}^d$ be closed, $a > 0 $ , $k \ge 2 $ and $x^{(1)},\ldots,x^{(k)},y^{(1)},\ldots, y^{(k)} \in E $ so that $$ \text{diam}(E) = \big\|x^{(1)}-y^{(1)}\big\| = \ldots = \big\|x^{(k)}-y^{(k)}\big\| = 2a $$ and \begin{equation} \label{eq_several_axes_pairs_of_poles_different} \left( x^{(i)},y^{(i)}\right) \ne \left( x^{(j)},y^{(j)}\right) \ne \left( y^{(i)},x^{(i)}\right) \end{equation} for $i \ne j$. Furthermore, we assume \begin{equation} \|x-y\| < 2a \qquad \text{for each} \qquad (x,y)\in \big( E\backslash \left\{ x^{(1)},\ldots,x^{(k)},y^{(1)},\ldots, y^{(k)} \right\} \big) \times E. \end{equation} \end{condition} Observe that \eqref{eq_several_axes_pairs_of_poles_different} makes sure that no pair of poles (points with distance $2a$) is considered twice. We want to emphasize the assumption $k < \infty $ in \autoref{cond_several_axes_k_pole_pairs}. Sets with an equator -- like an ellipsoid in $\mathbb{R}^{3}$ with half-axes $a_1 = a_2 > a_3$ -- are explicitly excluded by this condition, see \autoref{subsec_ellipsoid_several_major_axes} for some considerations in this setting.\\ For $m \in \left\{ 1,\ldots,k\right\}$, let $\phi^{(m)}$ be a rigid motion of $\mathbb{R}^d$ with $\phi^{(m)}\big( x^{(m)}\big) = (-a,\mathbf{0} )$ and $\phi^{(m)}\big( y^{(m)}\big) = (a,\mathbf{0} )$. If $f$ is a density with support $E$, we write $f^{(m)} := f \circ (\phi^{(m)})^{-1}$ for the transformed density supported by $\phi^{(m)}(E)$. Our basic assumption in this section will be that, for each $m \in \left\{ 1,\ldots,k\right\}$, the set $\phi^{(m)}(E)$ and the density $f^{(m)}$ fulfill all the requirements of \autoref{thm_main_result}, formally: \begin{condition} \label{cond_several_axes_polecaps} For each $m \in \left\{ 1,\ldots,k\right\}$, we assume that $\phi^{(m)}(E)$ satisfies Conditions~\ref{cond_shape_pole_caps} and \ref{cond_A_eta_pos_semi_definite}, and that the density $f^{(m)}$ fulfills \autoref{cond_density_p_l_p_r} with respect to some constants $p_\ell^{(m)},p_r^{(m)}>0$. \end{condition} \cite{Appel2002} investigated a similar setting in two dimensions for sets with boundary functions that -- in contrast to \autoref{cond_several_axes_polecaps} -- decay faster to zero at the poles than a square-root. In that setting, it was necessary to demand that any two different major axes have no vertex in common. Under \autoref{cond_several_axes_polecaps}, this requirement is given by definition: None of the points $x^{(1)},\ldots,x^{(k)},y^{(1)},\ldots,y^{(k)}$ can be part of more than one pair of points with distance $2a$, or, in other words, the set $E$ has exactly $2k$ poles, see Lemma 6.1 in \cite{Schrempp2017} for some more details. Writing $B_{\varepsilon}(z)$ for the closed ball with center $z \in \mathbb{R}^d$, we can infer that there exists an $\varepsilon >0$ so that the balls $ B_{\varepsilon}\big( x^{(1)}\big),\ldots,B_{\varepsilon}\big( x^{(k)}\big),B_{\varepsilon}\big( y^{(1)}\big),\ldots,B_{\varepsilon}\big( y^{(k)}\big) $ are pairwise disjoint. For $m \in \left\{ 1,\ldots,k\right\}$ we define the set $$ E^{(m)} := E \cap \left( B_{\varepsilon}\big( x^{(m)}\big) \cup B_{\varepsilon}\big( y^{(m)}\big)\right). $$ After moving $E^{(m)}$ via $\phi^{(m)}$ into the suitable position, \autoref{thm_main_result} is applicable for each $m \in \left\{ 1,\ldots,k\right\}$. We consider again the Poisson process $\mathbf{Z}_n= \sum_{i=1}^{N_n}\varepsilon_{Z_{i}} $, defined in \autoref{sec_fundamentals}. Since the sets $E^{(1)},\ldots,E^{(k)}$ are pairwise disjoint, the restrictions $\mathbf{Z}_n\big( \cdot \cap E^{(1)}\big),\ldots, \mathbf{Z}_n\big( \cdot \cap E^{(k)}\big)$ are independent Poisson processes. Consequently, for $m \in \left\{ 1,\ldots,k\right\}$, the maximum distances of points lying in $E^{(m)}$ are independent random variables. With \begin{align} I_{n}^{(m)} &:= \big\{ (i,j) : 1 \le i, j \le N_n, (Z_i,Z_j) \in E^{(m)} \times E^{(m)} \big\} \end{align} for $m \in \left\{ 1,\ldots,k\right\}$, we obtain $I_n^{(m)} \ne \emptyset$ for sufficiently large $n$ for each $ m \in \left\{ 1,\ldots,k\right\}$ almost surely and hence \begin{align} 2a - \max_{1\le i,j \le N_n}\|Z_i - Z_j\| = 2a - \max_{1 \le m \le k} \left\{ \max_{(i,j) \in I_n^{(m)}}\|Z_i - Z_j\|\right\} &= \min_{1 \le m \le k} \left\{2a -\max_{(i,j) \in I_n^{(m)}}\|Z_i - Z_j\|\right\}. \end{align} As mentioned before, we can apply \autoref{thm_main_result} to each of the random variables $\max_{(i,j) \in I_n^{(m)}}\|Z_i - Z_j\|$, and since these $k$ random variables are independent for each $n \in \mathbb{N}$, the $k$ limiting random variables inherit this property. Hence, we obtain as limiting distribution of the maximum distance of points within $E$ a minimum of $k$ independent random variables, each of which can be described as seen in \autoref{thm_main_result}. After stating one last definition we can formulate a generalized version of our main result \autoref{thm_main_result}. Instead of $H_\ell$ and $H_r$ we write $H_\ell^{(m)}$ and $H_r^{(m)}$ for the Hessian matrices of the corresponding boundary functions of $E^{(m)}$ at the poles, $m \in \{1,\ldots,k\}$. \begin{theorem} \label{thm_several_major_axes} Under \autoref{cond_several_axes_k_pole_pairs} and \autoref{cond_several_axes_polecaps} we have $$ n^{\frac{2}{d+1}}\big(2a - \mathrm{diam}(\mathbf{Z}_n )\big) \overset{\mathcal{D}}{\longrightarrow } \min_{1 \le m \le k} Z^{(m)}, $$ with independent random variables $Z^{(1)},\ldots,Z^{(k)}$, fulfilling \begin{equation} Z^{(m)}\overset{\mathcal{D}}{= } \min_{i,j \ge 1} \left\{ \mathcal{X}_{i,1}^{(m)} + \mathcal{Y}_{j,1}^{(m)} - \frac{1}{4a} \big\|\widetilde {\mathcal{X}}_{i}^{(m)}- \widetilde {\mathcal{Y}}_{j}^{(m)}\big\|^2\right\}, \end{equation} where all the Poisson processes $\left\{ \mathcal{X}_i^{(m)}, i \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\Big(p_\ell^{(m)}\cdot m_d\big\|_{P\big(H_\ell^{(m)}\big)}\Big) $, $\left\{ \mathcal{Y}_j^{(m)}, j \ge 1\right\} \overset{\mathcal{D}}{= } \text{PRM}\Big(p_r^{(m)}\cdot m_d\big\|_{P\big(H_r^{(m)}\big)}\Big) $, $m \in \left\{ 1,\ldots,k\right\}$, are independent. The same result holds true if we replace $ \mathrm{diam}(\mathbf{Z}_n )$ with $M_n$. \end{theorem} See \cite{Schrempp2017} for an application of this theorem to the ball of radius $r > 0$ with respect to the $p$-norm for $p > 2$. \subsection[Ellipsoids with no unique major half-axis]{Ellipsoids with no unique major half-axis } \label{subsec_ellipsoid_several_major_axes} In this subsection, we fix $d \ge 3$ and $e \in \left\{ 2,\ldots,d-1\right\}$ and consider the $d$-dimensional ellipsoid $E$ with half-axes $a_1 = \ldots =a_e = 1$ and $1 > a_{e+1} \ge \ldots \ge a_{d}$, formally: $$ E = \left\{ z \in \mathbb{R}^d: z_1^2 + \ldots +z_e^2 + \left( \frac{z_{e+1}}{a_{e+1}}\right)^2 + \ldots + \left( \frac{z_d}{a_d}\right)^2 \le 1\right\}. $$ There is no loss of generality in assuming that the $e$ major half-axes have length $1$. Otherwise, one would only have to scale $E$ and $M_n$ in a suitable way. We assume that the points $Z_1,Z_2,\ldots$ are independent and identically distributed according to a Pearson Type II distribution with parameter $\beta > -1$ on $\text{int}(E)$. This means that the density of $Z_1$ is given by $$ f(z) = c_1 \cdot \left(1-z^\top \Sigma^{-1}z\right)^{\beta} \cdot \mathds{1}\left\{ z \in \text{int}(E)\right\}, $$ where $\Sigma:= \text{diag}(1,\ldots,1,a_{e+1}^2,\ldots,a_d^2) \in \mathbb{R}^{d \times d}$ and $$ c_1 := \frac{\Gamma\left( \frac{d}{2}+\beta + 1\right)}{\Gamma\left( \beta +1\right)\pi^{\frac{d}{2}}\prod_{i=e+1}^{d}a_i}, $$ see \autoref{ex_pearson_type_II} and recall $a_1 = \ldots = a_e = 1$. Notice that we could use $E$ itself instead of $\text{int}(E)$ as support of $f$ for $\beta \ge 0$. But, since $\partial E$ has no influence at all on the limiting behavior of $M_n$ in our setting, the consideration of $\text{int}(E)$ instead of $E$ means no loss of generality. In this setting, we cannot state an exact limit theorem for $$ M_n = \max_{1\le i,j \le n}\| Z_i - Z_j\|. $$ However, by considering the projections $\overline Z_1, \overline Z_2,\ldots $ of $Z_1,Z_2,\ldots$ onto the first $e$ components and investigating $$ \overline M_n := \max_{1\le i,j \le n}\| \overline Z_i - \overline Z_j\|, $$ we can establish bounds for the unknown limit distribution, if it exists. To this end, we consider $\mathbb{R}^d$ as $\mathbb{R}^e \times \mathbb{R}^{d-e}$ and write $\overline z := (z_1,\ldots,z_e)$ for $z =(z_1,\ldots,z_d) \in \mathbb{R}^d$. In the same way, we put $\overline Z_n := (Z_{n,1},\ldots,Z_{n,e})$ for $Z_n = (Z_{n,1},\ldots,Z_{n,d})$ and $n\in\mathbb{N}$. Obviously, the random variables $\overline Z_1, \overline Z_2, \ldots $ are independent and identically distributed. Taking some orthogonal matrix $Q_{e} \in \mathbb{R}^{e \times e}$ and putting $Q := \text{diag}(Q_{e},\mathrm{I}_{d-e})$, the special form of $\Sigma$ yields $$ f(Qz) = c_1 \cdot \left(1-z^\top Q^\top \Sigma^{-1}Qz\right)^{\beta} = c_1 \cdot \left(1-z^\top \Sigma^{-1}z\right)^{\beta} = f(z) $$ for each $z \in \text{int}(E)$, and we can conclude that the distribution of $\overline{Z}_1,\overline{Z}_2,\ldots $ is spherically symmetric on the unit ball $\mathbb{B}^{e}$. In addition to that, the proof of \autoref{lem_several_major_axes_necessary_asymp_holds} (given in \cite{Schrempp2017}) reveals that this distribution solely depends on $d,e$ and $\beta$, \emph{not} on $a_{e+1},\ldots,a_d$. \\ The great advantage of assuming $a_1 = \ldots =a_e = 1$ is that we can directly apply Corollary 3.7 in~\cite{Lao2010} for the maximum distance of the random points $\overline Z_1,\overline Z_2,\ldots$ lying in the $e$-dimensional unit ball $\mathbb{B}^e$. For this purpose we write $\omega_e$ for its volume and obtain the following result: \begin{lemma} \label{lem_several_major_axes_necessary_asymp_holds} With \begin{align} a &:= \frac{\Gamma\left( \frac{d}{2}+\beta+1\right)}{\Gamma\left( \frac{d-e}{2}+\beta+2\right)}\cdot \pi^{-\frac{e}{2}} \cdot e \cdot \omega_{e} \cdot 2^{\frac{d-e}{2}+\beta} \qquad \qquad \text{and} \qquad \qquad \alpha := \frac{d-e}{2} + \beta + 1, \end{align} we have $$ \mathbb{P}\big( 1 - \|\overline Z_1\| \le s\big) \sim as^\alpha $$ as $s \downarrow 0$ \end{lemma} The proof of this lemma can be found in Subsection 6.2.2 of \cite{Schrempp2017}. In view of Corollary 3.7 in~\cite{Lao2010} we define $$ \sigma := \frac{2^{e-2}\Gamma\left( \frac{e}{2}\right)a^2\Gamma(\alpha+1)^2}{\sqrt{\pi}\Gamma\left( \frac{e+1}{2}+2\alpha\right)} $$ with $a$ and $\alpha$ given by \autoref{lem_several_major_axes_necessary_asymp_holds}, and we put \begin{align} b_n &:= \left( \frac{\sigma}{2} \right)^{\frac{2}{2d-e+4\beta+3}}\cdot n^{\frac{4}{2d-e+4\beta+3}}, \quad n \ge 1. \label{eq_several_maor_axes_ell_def_b_n} \intertext{Furthermore, we let} G(t) &:= 1 - \exp\left( -t^{\frac{2d - e + 4 \beta +3}{2}}\right) \label{eq_several_maor_axes_ell_def_G} \end{align} for $t \ge 0$. With Corollary 3.7 in~\cite{Lao2010} and \autoref{lem_several_major_axes_necessary_asymp_holds} we get \begin{equation} \label{eq_several_major_axes_conv_projections} \mathbb{P}\big(b_n(2 - \overline M_n ) \le t\big)\to G(t). \end{equation} But, since our focus lies on the asymptotic behavior of of $\max_{1\le i,j \le n}\| Z_i - Z_j\|$, \emph{not} on that of $\max_{1\le i,j \le n}\| \overline Z_i - \overline Z_j\|$, we have to find some useful relation between these two random variables. The key to success will be the following lemma, which provides bounds for $\|x-y\|$, $x,y \in E$, that depend merely on $\overline x$, $\overline y$ and the half-axis $a_{e+1}$. \begin{lemma} \label{lem_no_unique_major_axis_geom_ineq} Putting $$ g(\overline x, \overline y) := \sqrt{\big(\|\overline x\| + \|\overline y\|\big)^2 + 2a_{e+1}^2\big( 2- \|\overline x\|^2 - \|\overline y\|^2\big)}, $$ we have $$ \|\overline x - \overline y\| \le \|x-y\| \le g(\overline x, \overline y) $$ for all $x,y \in E$. \end{lemma} The proof of this lemma can be found in Subsection 6.2.2 of \cite{Schrempp2017}. Using the convergence given in~\eqref{eq_several_major_axes_conv_projections} and \autoref{lem_no_unique_major_axis_geom_ineq}, we can now state the main result of this section: \begin{theorem} \label{thm_several_major_axes_main_thm} Under the standing assumptions of this section we have \begin{align} G(t) \ &\le \ \liminf_{n \to \infty} \mathbb{P}\big( b_n( 2- M_n) \le t\big) \ \le \ \limsup_{n \to \infty} \mathbb{P}\big( b_n( 2- M_n) \le t\big)\label{eq_several_major_axes_assertion_main_theorem} \ \le \ G\left( \frac{t}{ 1-a_{e+1}^2}\right), \qquad t \ge 0, \end{align} where $b_n$ and $G$ are given in \eqref{eq_several_maor_axes_ell_def_b_n} and \eqref{eq_several_maor_axes_ell_def_G}, respectively. \end{theorem} See \cite{Schrempp2017} for the results of a simulation study. Before we give the proof of \autoref{thm_several_major_axes_main_thm}, we want to state an important corollary: \begin{corollary} From \autoref{thm_several_major_axes_main_thm} we immediately know that the sequence $$ \left( n^{\frac{4}{2d-e+4\beta+3}}(2-M_n) \right)_{n \in \mathbb{N}} $$ is tight. So, if there are a positive sequence $(a_n)_{n \in \mathbb{N}}$ and a non-degenerate distribution function $F$ with $\mathbb{P} \big(a_n(2-M_n)\le t\big) \to F(t)$, $t \ge 0$, we can conclude that $a_n \sim c \cdot n^{\frac{4}{2d-e+4\beta+3}}$ for some fixed $c \in \mathbb{R}$. \end{corollary} \begin{proof}[of \autoref{thm_several_major_axes_main_thm}] From \autoref{lem_no_unique_major_axis_geom_ineq} we have $$ \|\overline Z_i - \overline Z_j\| \le \|Z_i-Z_j\| \le g(\overline Z_i,\overline Z_j) $$ for all $i,j \in \mathbb{N}$. These inequalities imply $$ \max_{1\le i,j \le n} \|\overline Z_i - \overline Z_j\| \le \max_{1\le i,j \le n} \|Z_i-Z_j\| \le \max_{1\le i,j \le n} g(\overline Z_i,\overline Z_j) $$ and thus \begin{equation} \label{eq_no_unique_major_axis_ineq_minima} \min_{1\le i,j \le n}\left\{ 2 - g(\overline Z_i ,\overline Z_j) \right\} \le \min_{1\le i,j \le n}\left\{ 2 - \| Z_i - Z_j\| \right\} \le \min_{1\le i,j \le n}\left\{ 2 - \| \overline Z_i - \overline Z_j\| \right\}. \end{equation} Using~\eqref{eq_several_major_axes_conv_projections} and the upper inequality figuring in~\eqref{eq_no_unique_major_axis_ineq_minima} yields \begin{align} \mathbb{P}\left( b_n\Big( 2- \max_{1\le i,j \le n} \|Z_i-Z_j\|\Big) \le t\right) & = \mathbb{P}\left( 2- \max_{1\le i,j \le n} \|Z_i-Z_j\| \le \frac{t}{b_n}\right)\\ & = \mathbb{P}\left( \min_{1\le i,j \le n}\left\{ 2 - \| Z_i - Z_j\| \right\} \le \frac{t}{b_n}\right)\\ & \ge \mathbb{P}\left( \min_{1\le i,j \le n}\left\{ 2 - \| \overline Z_i - \overline Z_j\| \right\} \le \frac{t}{b_n}\right)\\ & = \mathbb{P}\left(b_n\Big(2 - \max_{1\le i,j \le n}\| \overline Z_i - \overline Z_j\| \Big) \le t\right)\\ & \to G(t). \end{align} Hence, the lower bound stated in~\eqref{eq_several_major_axes_assertion_main_theorem} has already been obtained. To establish the upper bound in~\eqref{eq_several_major_axes_assertion_main_theorem}, we consider $\mathbb{R}^e \times \mathbb{R}^{e}$. For $(\overline x, \overline y)\in \mathbb{B}^e \times \mathbb{B}^{e} $ close to $ \mathbf{a} :=(-1,\mathbf{0} ,1,\mathbf{0} ) \in \mathbb{R}^{2e}$ we have, putting $$ c := 1 - a_{e+1}^2, $$ the multivariate Taylor series expansions \begin{align} 2 - g(\overline x, \overline y) &= c \cdot\left( 2 + x_1 - y_1\right) + o\big(\|( \overline x, \overline y) - \mathbf{a} \|\big),\\ 2 - \|\overline x - \overline y\| &= \left( 2 + x_1 - y_1 \right) + o\big(\|( \overline x, \overline y) - \mathbf{a} \|\big), \intertext{and} \frac{2 - g(\overline x,\overline y)}{2 - \| \overline x - \overline y\|} &= c + o\big(\|( \overline x, \overline y) - \mathbf{a} \|\big). \end{align} By symmetry, we can conclude that $$ \frac{2 - g(\overline x,\overline y)}{2 - \| \overline x - \overline y\|} \to c $$ for $(\overline x, \overline y) \in \mathbb{B}^e \times \mathbb{B}^{e}$ with $(\overline x, \overline y) \to (\mathbf{a}^,-\mathbf{a}^ )$ and $\mathbf{a}^* \in \partial\mathbb{B}^{e} $. Furthermore, the symmetry guarantees that, for each $\delta \in (0,c)$, we can find a positive $\varepsilon$ so that $$ c - \delta \le \frac{2 - g(\overline x,\overline y)}{2 - \| \overline x - \overline y\|} $$ for all $(\overline x,\overline y) \in \mathbb{B}^{e} \times \mathbb{B}^{e}$ with $\|\overline x - \overline y\| \ge 2-\varepsilon$. For $n \in \mathbb{N}$, we write $\overline Z_{n}^{1}$ and $\overline Z_{n}^{2}$ for those elements of $\left\{ \overline Z_1,\ldots,\overline Z_n\right\}$ with $$ \max_{1 \le i,j\le n}\|\overline Z_i - \overline Z_j\| = \big\|\overline Z_{n}^{1} - \overline Z_{n}^{2}\big\|. $$ Based on these two random variables, we define for $\varepsilon$ given above the set $$ A_{n,\varepsilon} := \left\{ \big\|\overline Z_{n}^{1} - \overline Z_{n}^{2}\big\| > 2 - \varepsilon \right\}. $$ Obviously, $ \mathbb{P}(A_{n,\varepsilon}^c) \to 0$, and the event $A_{n,\varepsilon}$ entails $$ c - \delta \le \frac{2 - g\big(\overline Z_{n}^{1},\overline Z_{n}^{2}\big)}{2 - \big\| \overline Z_{n}^{1} - \overline Z_{n}^{2}\big\|}. $$ Together with the lower inequality given in~\eqref{eq_no_unique_major_axis_ineq_minima} we obtain \begin{align} \mathbb{P}\left( b_n\Big( 2- \max_{1\le i,j \le n} \|Z_i-Z_j\|\Big) \le t\right) \le\ &\mathbb{P}\left( b_n\min_{1\le i,j \le n}\left\{ 2 - \| Z_i - Z_j\| \right\} \le t , A_{n,\varepsilon}\right) + \mathbb{P}(A_{n,\varepsilon}^c) \\ \le\ & \mathbb{P}\left( b_n\min_{1\le i,j \le n}\left\{ 2 - g(\overline Z_i,\overline Z_j) \right\} \le t , A_{n,\varepsilon}\right) + \mathbb{P}(A_{n,\varepsilon}^c)\\ =\ &\mathbb{P}\left( b_n\min_{1\le i,j \le n}\left\{ \left( 2 - \| \overline Z_i - \overline Z_j\|\right) \cdot \frac{2 - g(\overline Z_i,\overline Z_j)}{2 - \| \overline Z_i - \overline Z_j\|} \right\} \le t , A_{n,\varepsilon}\right) + \mathbb{P}(A_{n,\varepsilon}^c) \\ \le \ &\mathbb{P}\left( b_n\min_{1\le i,j \le n}\left\{ \left( 2 - \| \overline Z_i - \overline Z_j\|\right) \cdot (c-\delta)\right\} \le t , A_{n,\varepsilon}\right) + \mathbb{P}(A_{n,\varepsilon}^c) \\ \le \ &\mathbb{P}\left(b_n\Big(2 - \max_{1\le i,j \le n}\| \overline Z_i - \overline Z_j\| \Big) \le \frac{t}{c-\delta}\right) + \mathbb{P}(A_{n,\varepsilon}^c) \\ \to \ & G\left( \frac{t}{c-\delta}\right). \end{align} Since $\delta$ can be chosen arbitrarily close to $0$, the continuity of $G$ implies $$ \limsup_{n\to \infty} \mathbb{P}\left( b_n\Big( 2- \max_{1\le i,j \le n} \|Z_i-Z_j\|\Big) \le t\right) \le G\left( \frac{t}{c} \right), $$ and the proof is finished. \qed \end{proof} \section{Appendix} \label{sec_appendix} \begin{proof}[of \autoref{lem_first_part_deriv_are_0_and_Hessian_pos_def}] We only consider $i= \ell$. It is clear that $H_\ell$ is symmetric, since $s^\ell$ is a twice continuously differentiable function. From \autoref{cond_unique_diameter} we know that \begin{equation} \label{eq_E_subset_ball_Radius_2a_at_right_pole} E \subset B_{2a}\big( (a,\mathbf{0} )\big) \qquad \text{and} \qquad E \cap \partial B_{2a}\big( (a,\mathbf{0} )\big) = \left\{ (-a,\mathbf{0} )\right\}. \end{equation} Writing $O_t := \left\{ \widetilde z \in \mathbb{R}^{d-1}: \|\widetilde z\|<2a \right\}$ and defining the mapping $t: O_t \to \mathbb{R}, \widetilde z \mapsto a - \sqrt{4a^2 - z_2^2- \ldots -z_d^2}$, the boundary of $B_{2a}\big( (a,\mathbf{0} )\big) $ in $ \left\{ z_1 < a\right\}$ can be parameterized as a hypersurface via $$ \mathbf{t}: \begin{cases} O_t \to \mathbb{R}^d,\\ \widetilde z \mapsto \big(\ t( \widetilde z)\ ,\ \widetilde z\ \big). \end{cases} $$ For $j,k \in \left\{ 2,\ldots,d\right\}$, we obtain \begin{align} t_j(\widetilde z) &= (4a^2 - z_2^2 - \ldots - z_d^2)^{-\frac{1}{2}}\cdot z_j,\\ t_{jk}(\widetilde z) &= (4a^2 - z_2^2 - \ldots - z_d^2)^{-\frac{3}{2}}\cdot z_jz_k + (4a^2 - z_2^2 - \ldots - z_d^2)^{-\frac{1}{2}}\cdot \delta_{jk}. \end{align} Hence, $\nabla t(\mathbf{0} ) = \mathbf{0} $, and the Hessian of $t$ at $\mathbf{0} $ is given by $H_t := \frac{1}{2a}\mathrm{I}_{d-1} $. So, the second-order Taylor series expansion of $t$ at this point has the form \begin{equation} \label{eq_Taylor_function_t} t(\widetilde z) = -a + \mathbf{0}^\top\widetilde z + \frac{1}{2}\widetilde z^\top H_t \widetilde z + R_t(\widetilde z), \end{equation} where $R_t (\widetilde z) = o\big(\|\widetilde z\|^2 \big)$. Furthermore, we have \begin{equation} \label{eq_Taylor_s_l} s^{\ell}(\widetilde z) = -a + \nabla s^{\ell}(\mathbf{0} )^\top \widetilde z + \frac{1}{2}\widetilde z^\top H_\ell \widetilde z + R_\ell(\widetilde z), \end{equation} where $R_\ell (\widetilde z) = o\big(\|\widetilde z\|^2 \big)$. In view of \eqref{eq_E_subset_ball_Radius_2a_at_right_pole} and \autoref{cond_shape_pole_caps}, we have $t(\widetilde z) <-a + s^{\ell}(\widetilde z)$ for each $\widetilde z \in O_\ell \backslash\left\{ \mathbf{0} \right\}$ (observe that \eqref{eq_E_subset_ball_Radius_2a_at_right_pole} ensures $O_\ell \subset O_t$). Using \eqref{eq_Taylor_function_t} and\eqref{eq_Taylor_s_l}, this inequality can be rewritten as $$ -a + \frac{1}{2}\widetilde z^\top H_t \widetilde z + R_t(\widetilde z)< -a + \nabla s^{\ell}(\mathbf{0} )^\top \widetilde z + \frac{1}{2}\widetilde z^\top H_\ell \widetilde z + R_\ell(\widetilde z), $$ and hence $$ 0< \nabla s^{\ell}(\mathbf{0} )^\top \widetilde z + \frac{1}{2}\widetilde z^\top (H_\ell - H_t) \widetilde z + \big(R_\ell(\widetilde z) -R_t(\widetilde z) \big) $$ for each $\widetilde z \in O_\ell\backslash\left\{ \mathbf{0} \right\}$. Since $R_\ell (\widetilde z)-R_t (\widetilde z) = o\big(\|\widetilde z\|^2 \big)$, this inequality shows $\nabla s^{\ell}(\mathbf{0} ) = \mathbf{0} $ and that the matrix $H_\ell - H_t$ is positive definite. Remembering $H_t = \frac{1}{2a}\mathrm{I}_{d-1} $, $H_\ell$ has to be positive definite, too, and all eigenvalues of $H_\ell$ have to be larger than $1/2a$. \qed \end{proof} Now we will show that \autoref{cond_A_eta_pos_semi_definite} really ensures the unique diameter of $E$ `close to the poles': \begin{lemma} \label{lem_suff_con_unique_diam} Under Conditions~\ref{cond_shape_pole_caps} and \ref{cond_A_eta_pos_semi_definite}, \eqref{eq_cond_unique_diameter} holds true for $E$ replaced with $E \cap \left\{ \|z_1\| > a - \delta\right\}$ and $\delta >0 $ sufficiently small. \end{lemma} \begin{proof} Since the diameter of $E$ cannot be determined by interior points, it suffices to investigate points on the boundaries $M_\ell$ and $M_r$ of the pole-caps of $E$. To this end, let $(\widetilde x ,\widetilde y)\in O_\ell \times O_r\backslash \left\{ \mathbf{0}\right\}$. Invoking \eqref{eq_representation_E_l_Taylor} and \eqref{eq_representation_E_r_Taylor} and putting $$ \Xi:= \frac{1}{2}\left(\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) + R_\ell (\widetilde x) + R_r (\widetilde y), $$ we get \begin{align} \big\| (-a + s^\ell(\widetilde x),\widetilde x ) - \big( a -s^r(\widetilde y),\widetilde y\big) \big\|^2\ = \ &\Big\| \Big(-a + \frac{1}{2}\widetilde x^\top H_\ell \widetilde x + R_\ell (\widetilde x),\widetilde x \Big) - \Big( a - \frac{1}{2}\widetilde y^\top H_r \widetilde y - R_r (\widetilde y),\widetilde y\Big) \Big\|^2\\ = \ &\left( -2a + \Xi\right)^2 + \|\widetilde x- \widetilde y\|^2\\ = \ &4a^2 - 4a\Xi + \Xi^2 + \|\widetilde x- \widetilde y\|^2\\ = \ &4a^2 - \left( 4a\Xi -\Xi^2 \right) +\|\widetilde x\|^2 + \| \widetilde y\|^2 - 2\widetilde x^\top \widetilde y. \end{align} \autoref{lem_inequality_for_Xi} will show that \begin{equation} \label{eq_inequality_for_Xi} 4a\Xi - \Xi^2 > 2a\eta\left( \widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) \end{equation} for every $(\widetilde x,\widetilde y) \ne \mathbf{0}$ sufficiently close to $\mathbf{0} $. Representing the points $\widetilde x$ and $\widetilde y$ in terms of the bases $\left\{ \mathbf{u}_2^\ell,\ldots,\mathbf{u}_d^\ell \right\}$ and $\left\{ \mathbf{u}_2^r,\ldots,\mathbf{u}_d^r \right\}$, namely $\widetilde x = U_\ell \alpha$ and $\widetilde y = U_r \beta$, \eqref{eq_cond_1_unique_diam} gives \begin{align} \big\| (-a + s^\ell(\widetilde x),\widetilde x ) - \big( a -s^r(\widetilde y),\widetilde y\big) \big\|^2\ < \ &4a^2 - 2a\eta\left( \widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) + \|\widetilde x\|^2 + \| \widetilde y\|^2 - 2\widetilde x^\top \widetilde y \\ = \ &4a^2 - 2a\eta\left( \alpha^\top U_\ell^\top H_\ell U_\ell \alpha + \beta^\top U_r^\top H_r U_r \beta \right) + \|U_\ell\alpha\|^2 + \| U_r \beta\|^2 - 2\alpha^\top U_\ell^\top U_r \beta\\ = \ &4a^2 - 2a\eta\left( \alpha^\top D_\ell \alpha + \beta^\top D_r \beta \right) - 2\alpha^\top U_\ell^\top U_r \beta+ \|\alpha\|^2 + \|\beta\|^2\\ \le \ &4a^2. \end{align} Thus, for $\delta>0$ sufficiently small, the only pair of points in $E \cap \left\{ \|z_1\| > a - \delta\right\}$ with distance $2a$ is given by $(-a,\mathbf{0} )$ and $(a,\mathbf{0} )$, and the proof is finished. \qed \end{proof} It remains to prove the validity of \eqref{eq_inequality_for_Xi}. \begin{lemma} \label{lem_inequality_for_Xi} For $\eta \in \left( 0,1\right)$ and $(\widetilde x,\widetilde y) \ne \mathbf{0}$ sufficiently close to $\mathbf{0} $ we have \begin{align} 4a\Xi - \Xi^2 > 2a\eta\left( \widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right). \end{align} \end{lemma} \begin{proof} Let $\varepsilon := \frac{1-\eta}{2}>0$. Without loss of generality we assume $\widetilde x \ne \mathbf{0} $. For $\widetilde x$ sufficiently close to $\mathbf{0} $, \eqref{eq_Courant_Fischer} and $R_\ell (\widetilde x) = o\big(\|\widetilde x\|^2 \big)$ lead to $$ \big\| R_\ell (\widetilde x)\big\| < \frac{\varepsilon}{2}\kappa_2^\ell\|\widetilde x\|^2 \le \frac{\varepsilon}{2}\widetilde x^\top H_\ell \widetilde x, $$ whence $$ \frac{1}{2}\widetilde x^\top H_\ell \widetilde x + R_\ell (\widetilde x) > \frac{1}{2}\widetilde x^\top H_\ell \widetilde x - \frac{\varepsilon}{2}\widetilde x^\top H_\ell \widetilde x = \frac{1-\varepsilon}{2}\widetilde x^\top H_\ell \widetilde x . $$ By the same reasoning for $\widetilde y$ we get $$ \frac{1}{2}\widetilde y^\top H_r \widetilde y + R_r (\widetilde y) \ge \frac{1-\varepsilon}{2}\widetilde y^\top H_r \widetilde y. $$ Observe that, in the line above, equality holds if $\widetilde y = \mathbf{0} $. Putting both inequalities together yields $$ \Xi > \frac{1-\varepsilon}{2}\left(\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) $$ and thus \begin{equation} \label{eq_inequality_Xi_1} 4a\Xi > 2a(1-\varepsilon)\left(\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right). \end{equation} Since close to $\mathbf{0}$ both $\big\|R_\ell (\widetilde x)\big\| \le \frac{\kappa_d^\ell}{2} \|\widetilde x\|^2$ and $\big\|R_r (\widetilde y) \big\|\le \frac{\kappa_d^r }{2}\|\widetilde y\|^2$ hold true, \eqref{eq_Courant_Fischer} gives \begin{align} \Xi^2 &= \left( \frac{1}{2}\left(\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) + R_\ell (\widetilde x) + R_r (\widetilde y)\right)^2\\ &\le \left( \frac{1}{2}\big( \kappa_d^\ell\|\widetilde x\|^2 + \kappa_d^r\|\widetilde y\|^2\big) + \frac{\kappa_d^\ell }{2}\|\widetilde x\|^2 + \frac{\kappa_d^r }{2}\|\widetilde y\|^2 \right)^2 \\ &= \left( \kappa_d^\ell\|\widetilde x\|^2 + \kappa_d^r\|\widetilde y\|^2 \right)^2 \\ &\le \max\left\{ \kappa_d^\ell,\kappa_d^r\right\}^2 \left( \|\widetilde x\|^2 +\|\widetilde y\|^2 \right)^2. \end{align} Using \eqref{eq_Courant_Fischer} again yields $$ 0 \le \frac{\left( \|\widetilde x\|^2 +\|\widetilde y\|^2 \right)^2}{\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y } \le \frac{\left( \|\widetilde x\|^2 +\|\widetilde y\|^2 \right)^2}{\kappa_2^\ell \|\widetilde x\|^2 + \kappa_2^r \|\widetilde y\|^2}. $$ Since the fraction on the right-hand side tends to $0$ as $(\widetilde x,\widetilde y) \to \mathbf{0}$ we infer \begin{equation} \label{eq_inequality_Xi_2} \Xi^2 \le 2 a \varepsilon \left(\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) \end{equation} for all $(\widetilde x,\widetilde y)$ sufficiently close to $\mathbf{0} .$ From \eqref{eq_inequality_Xi_1} and \eqref{eq_inequality_Xi_2} we deduce that \begin{align} 4a\Xi - \Xi^2 &> 2a(1-\varepsilon)\left(\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) - 2 a \varepsilon \left(\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) \\ &= 2a(1-2\varepsilon)\left(\widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right), \end{align} and since $1-2\varepsilon= 1- 2\frac{1-\eta}{2}= \eta$, the proof is finished. \qed \end{proof} Now we want to show that the matrix $A(1)$ is necessarily positive semi-definite. Otherwise, we would obtain a contradiction to \autoref{cond_unique_diameter}. \begin{lemma} \label{lem_A_1_has_to_be_pos_def} Under Conditions~\ref{cond_unique_diameter} and \ref{cond_shape_pole_caps} we have $A(1) \ge 0$. \end{lemma} \begin{proof} Assuming $A(1) \ngeq 0$, there exists $z \in \mathbb{R}^{2(d-1)}$ with $ z^\top A(1)z <0$. Then, we can also find an $\eta^>1$ with \begin{align} z^\top A(\eta^)z &= z^\top \big( A(1) + 2a(\eta^{}-1) \text{diag}(D_\ell,D_r) \big)z\\ &= z^\top A(1)z + (\eta^{}-1) 2a z^\top \text{diag}(D_\ell,D_r)z \\ &< 0, \end{align} which entails $A(\eta^) \ngeq 0$. Notice that $(\eta^{}-1)2az^\top \text{diag}(D_\ell,D_r)z > 0$ can be made arbitrarily small by choosing $\eta^{}$ sufficiently close to $1$. In a similar way as in the proof of \autoref{lem_inequality_for_Xi}, one can show $$ 4a\Xi - \Xi^2 < 2a\eta^\left( \widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) $$ for all $(\widetilde x,\widetilde y) \ne \mathbf{0}$ sufficiently close to $\mathbf{0} $. As in the proof of \autoref{lem_suff_con_unique_diam} we obtain \begin{align} &\big\| (-a + s^\ell(\widetilde x),\widetilde x ) - \big( a -s^r(\widetilde y),\widetilde y\big) \big\|^2 > \ 4a^2 - 2a\eta^\left( \widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) + \|\widetilde x\|^2 + \| \widetilde y\|^2 - 2\widetilde x^\top \widetilde y. \label{eq_proof_A_1_pos_definit_help_1} \end{align} Because of $A(\eta^) \ngeq 0$ we can find $\alpha,\beta \in \mathbb{R}^{d-1}$ arbitrarily close to $\mathbf{0} $ with \begin{align} \Big( \alpha^\top \ , \ \beta^\top \Big) A(\eta^) \begin{pmatrix} \alpha \\ \beta \end{pmatrix} &< 0. \end{align} This inequality can be rewritten to \begin{equation} \label{eq_proof_A_1_pos_definit_help_2} -2a\eta^\left( \alpha^\top D_\ell\alpha + \beta^\top D_r \beta\right)-2\alpha^\top U_\ell^\top U_r\beta+\|\alpha\|^2 + \|\beta\|^2 > 0. \end{equation} If we choose $\|\alpha\|$ and $\|\beta\|$ small enough, we have $\widetilde x := U_\ell\alpha \in O_\ell$ and $\widetilde y := U_r \beta \in O_r$. Putting \eqref{eq_proof_A_1_pos_definit_help_1} and \eqref{eq_proof_A_1_pos_definit_help_2} together yields \begin{align} \big\| (-a + s^\ell(\widetilde x),\widetilde x ) - \big( a -s^r(\widetilde y),\widetilde y\big) \big\|^2\ > \ &4a^2 - 2a\eta^\left( \widetilde x^\top H_\ell \widetilde x + \widetilde y^\top H_r \widetilde y \right) + \|\widetilde x\|^2 + \| \widetilde y\|^2 - 2\widetilde x^\top \widetilde y \\ = \ &4a^2 - 2a\eta^\left( \alpha^\top U_\ell^\top H_\ell U_\ell \alpha + \beta^\top U_r^\top H_r U_r \beta \right)- 2\alpha^\top U_\ell^\top U_r \beta + \|U_\ell\alpha\|^2 + \| U_r \beta\|^2\\ = \ &4a^2 - 2a\eta^\left( \alpha^\top D_\ell \alpha + \beta^\top D_r \beta \right)- 2\alpha^\top U_\ell^\top U_r \beta + \|\alpha\|^2 + \|\beta\|^2\\ > \ &4a^2. \end{align} This inequality contradicts \autoref{cond_unique_diameter}, and the proof is finished. \qed \end{proof} The following lemma shows that inequality~\eqref{eq_suff_cond_princ_curv} is sufficient for \autoref{cond_A_eta_pos_semi_definite}: \begin{lemma} \label{lem_suff_cond_princ_curv} If \eqref{eq_suff_cond_princ_curv} holds true, then \autoref{cond_A_eta_pos_semi_definite} is fulfilled. \end{lemma} \begin{proof} Inequality~\eqref{eq_suff_cond_princ_curv} ensures the existence of an $\eta^{} \in (0,1)$ with \begin{equation} \label{eq_suff_cond_princ_curv_help} \frac{1}{\kappa_2^\ell} + \frac{1}{\kappa_2^r} = 2a\eta^{}. \end{equation} Applying \eqref{eq_Courant_Fischer} (with $H_\ell$ and $H_r$ replaced with the matrices $D_\ell$ and $D_r$, respectively), using \eqref{eq_suff_cond_princ_curv_help} and some obvious transformations yield \begin{align} &2a\eta^{}\left( \alpha^\top D_\ell\alpha + \beta^\top D_r \beta\right)+2\alpha^\top U_\ell^\top U_r\beta-\|\alpha\|^2 - \|\beta\|^2 \ge \ \left\| \sqrt{\frac{\kappa_2^\ell}{\kappa_2^r}}U_\ell\alpha + \sqrt{\frac{\kappa_2^r}{\kappa_2^\ell}}U_r\beta \right\|^2 \ge\ 0. \end{align} Consequently, \autoref{cond_A_eta_pos_semi_definite} holds with $\eta = \eta^{*}$, see \eqref{eq_cond_1_unique_diam}. \qed \end{proof} As mentioned before, \eqref{eq_suff_cond_princ_curv} is only sufficient for the unique diameter close to the poles, \emph{not} necessary. See Example 3.13 in \cite{Schrempp2017} for an illustration of a set with unique diameter between $(-a,\mathbf{0} )$ and $(a,\mathbf{0} )$ for which inequality~\eqref{eq_suff_cond_princ_curv} is \emph{not} fulfilled. \end{document}	arXiv
Pencil (geometry) In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane. "Bundle (geometry)" redirects here. Not to be confused with Bundle (mathematics). Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle.[1] Thus, the set of all lines through a point in three-space is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two-dimensional nature of such a pencil, it is sometimes referred to as a flat pencil.[2] Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line.[1] A more common term for this set is a range of points. Pencil of lines In a plane, let u and v be two distinct intersecting lines. For concreteness, suppose that u has the equation, aX + bY + c = 0 and v has the equation a'X + b'Y + c′ = 0. Then λu + μv = 0, represents, for suitable scalars λ and μ, any line passing through the intersection of u = 0 and v = 0. This set of lines passing through a common point is called a pencil of lines.[3] The common point of a pencil of lines is called the vertex of the pencil. In an affine plane with the reflexive variant of parallelism, a set of parallel lines forms an equivalence class called a pencil of parallel lines.[4] This terminology is consistent with the above definition since in the unique projective extension of the affine plane to a projective plane a single point (point at infinity) is added to each line in the pencil of parallel lines, thus making it a pencil in the above sense in the projective plane. Pencil of planes A pencil of planes, is the set of planes through a given straight line in three-space, called the axis of the pencil. The pencil is sometimes referred to as a axial-pencil[5] or fan of planes or a sheaf of planes.[6] For example, the meridians of the globe are defined by the pencil of planes on the axis of Earth's rotation. Two intersecting planes meet in a line in three-space, and so, determine the axis and hence all of the planes in the pencil. The four-space of quaternions can be seen as an axial pencil of complex planes all sharing the same real line. In fact, quaternions contain a sphere of imaginary units, and a pair of antipodal points on this sphere, together with the real axis, generate a complex plane. The union of all these complex planes constitutes the 4-algebra of quaternions. Pencil of circles Any two circles in the plane have a common radical axis, which is the line consisting of all the points that have the same power with respect to the two circles. A pencil of circles (or coaxial system) is the set of all circles in the plane with the same radical axis.[7] To be inclusive, concentric circles are said to have the line at infinity as a radical axis. There are five types of pencils of circles,[8] the two families of Apollonian circles in the illustration above represent two of them. Each type is determined by two circles called the generators of the pencil. When described algebraically, it is possible that the equations may admit imaginary solutions. The types are: • An elliptic pencil (red family of circles in the figure) is defined by two generators that pass through each other in exactly two points. Every circle of an elliptic pencil passes through the same two points. An elliptic pencil does not include any imaginary circles. • A hyperbolic pencil (blue family of circles in the figure) is defined by two generators that do not intersect each other at any point. It includes real circles, imaginary circles, and two degenerate point circles called the Poncelet points of the pencil. Each point in the plane belongs to exactly one circle of the pencil. • A parabolic pencil (as a limiting case) is defined where two generating circles are tangent to each other at a single point. It consists of a family of real circles, all tangent to each other at a single common point. The degenerate circle with radius zero at that point also belongs to the pencil. • A family of concentric circles centered at a common center (may be considered a special case of a hyperbolic pencil where the other point is the point at infinity). • The family of straight lines through a common point; these should be interpreted as circles that all pass through the point at infinity (may be considered a special case of an elliptic pencil).[9][10] Properties A circle that is orthogonal to two fixed circles is orthogonal to every circle in the pencil they determine.[11] The circles orthogonal to two fixed circles form a pencil of circles.[11] Two circles determine two pencils, the unique pencil that contains them and the pencil of circles orthogonal to them. The radical axis of one pencil consists of the centers of the circles of the other pencil. If one pencil is of elliptic type, the other is of hyperbolic type and vice versa.[11] The radical axis of any pencil of circles, interpreted as an infinite-radius circle, belongs to the pencil. Any three circles belong to a common pencil whenever all three pairs share the same radical axis and their centers are collinear. Projective space of circles There is a natural correspondence between circles in the plane and points in three-dimensional projective space; a line in this space corresponds to a one-dimensional continuous family of circles, hence a pencil of points in this space is a pencil of circles in the plane. Specifically, the equation of a circle of radius r centered at a point (p,q), $(x-p)^{2}+(y-q)^{2}=r^{2},$ may be rewritten as $\alpha (x^{2}+y^{2})-2\beta x-2\gamma y+\delta =0,$ where α = 1, β = p, γ = q, and δ = p2 + q2 − r2. In this form, multiplying the quadruple (α,β,γ,δ) by a scalar produces a different quadruple that represents the same circle; thus, these quadruples may be considered to be homogeneous coordinates for the space of circles.[12] Straight lines may also be represented with an equation of this type in which α = 0 and should be thought of as being a degenerate form of a circle. When α ≠ 0, we may solve for p = β/α, q = γ/α, and r =√(p2 + q2 − δ/α); the latter formula may give r = 0 (in which case the circle degenerates to a point) or r equal to an imaginary number (in which case the quadruple (α,β,γ,δ) is said to represent an imaginary circle). The set of affine combinations of two circles (α1,β1,γ1,δ1), (α2,β2,γ2,δ2), that is, the set of circles represented by the quadruple $z(\alpha _{1},\beta _{1},\gamma _{1},\delta _{1})+(1-z)(\alpha _{2},\beta _{2},\gamma _{2},\delta _{2})$ for some value of the parameter z, forms a pencil; the two circles being the generators of the pencil. Cardioid as envelope of a pencil of circles Main article: Cardioid § Cardioid as envelope of a pencil of circles Another type of pencil of circles can be obtained as follows. Consider a given circle (called the generator circle) and a distinguished point P on the generator circle. The set of all circles that pass through P and have their centers on the generator circle form a pencil of circles. The envelope of this pencil is a cardioid. Pencil of spheres A sphere is uniquely determined by four points that are not coplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc.[13] This property is analogous to the property that three non-collinear points determine a unique circle in a plane. Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle. By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres.[14] Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).[15] If f(x, y, z) = 0 and g(x, y, z) = 0 are the equations of two distinct spheres then $\lambda f(x,y,z)+\mu g(x,y,z)=0$ is also the equation of a sphere for arbitrary values of the parameters λ and μ. The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.[16] If the pencil of spheres does not consist of all planes, then there are three types of pencils:[15] • If the spheres intersect in a real circle C, then the pencil consists of all the spheres containing C, including the radical plane. The centers of all the ordinary spheres in the pencil lie on a line passing through the center of C and perpendicular to the radical plane. • If the spheres intersect in an imaginary circle, all the spheres of the pencil also pass through this imaginary circle but as ordinary spheres they are disjoint (have no real points in common). The line of centers is perpendicular to the radical plane, which is a real plane in the pencil containing the imaginary circle. • If the spheres intersect in a point A, all the spheres in the pencil are tangent at A and the radical plane is the common tangent plane of all these spheres. The line of centers is perpendicular to the radical plane at A. All the tangent lines from a fixed point of the radical plane to the spheres of a pencil have the same length.[15] The radical plane is the locus of the centers of all the spheres that are orthogonal to all the spheres in a pencil. Moreover, a sphere orthogonal to any two spheres of a pencil of spheres is orthogonal to all of them and its center lies in the radical plane of the pencil.[15] Pencil of conics A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.[17] The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles. In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.[18] A pencil of conics can be represented algebraically in the following way. Let C1 and C2 be two distinct conics in a projective plane defined over an algebraically closed field K. For every pair λ, μ of elements of K, not both zero, the expression: $\lambda C_{1}+\mu C_{2}$ represents a conic in the pencil determined by C1 and C2. This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.) Thinking of C1, say, as a ternary quadratic form, then C1 = 0 is the equation of the "conic C1". Another concrete realization would be obtained by thinking of C1 as the 3×3 symmetric matrix which represents it. If C1 and C2 have such concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant. Pencil of plane curves More generally, a pencil is the special case of a linear system of divisors in which the parameter space is a projective line. Typical pencils of curves in the projective plane, for example, are written as $\lambda C+\mu C'=0\,$ where C = 0, C′ = 0 are plane curves. History Desargues is credited with inventing the term "pencil of lines" (ordonnance de lignes).[19] An early author of modern projective geometry G. B. Halsted introduced the terms copunctal and flat-pencil to define angle: "Straights with the same cross are copunctal." Also "The aggregate of all coplanar, copunctal straights is called a flat-pencil" and "A piece of a flat-pencil bounded by two of the straights as sides, is called an angle."[20] See also • Bundle adjustment • Lefschetz pencil • Matrix pencil • Pencil beam • Fibration • Locus Notes 1. Young 1971, p. 40 2. Halsted 1906, p. 9 3. Pedoe 1988, p. 106 4. Artin 1957, p. 53 5. Halsted 1906, p. 9 6. Woods 1961, p. 12 7. Johnson 2007, p. 34 8. Some authors combine types and reduce the list to three. Schwerdtfeger (1979, pp. 8–10) 9. Johnson 2007, p. 36 10. Schwerdtfeger 1979, pp. 8–10 11. Johnson 2007, p. 37 12. Pfeifer & Van Hook 1993. 13. Albert 2016, p. 55. 14. Albert 2016, p. 57. 15. Woods 1961, p. 267. 16. Woods 1961, p. 266 17. Faulkner 1952, pg. 64. 18. Samuel 1988, pg. 50. 19. Earliest Known Uses of Some Words of Mathematics, retrieved July 14, 2020 20. Halsted 1906, p. 9 References • Albert, Abraham Adrian (2016) [1949], Solid Analytic Geometry, Dover, ISBN 978-0-486-81026-3 • Artin, E. (1957), Geometric Algebra, Interscience Publishers • Faulkner, T. E. (1952), Projective Geometry (2nd ed.), Edinburgh: Oliver and Boyd, ISBN 9780486154893 • Halsted, George Bruce (1906), Synthetic Projective Geometry, New York Wiley • Johnson, Roger A. (2007) [1929], Advanced Euclidean Geometry, Dover, ISBN 978-0-486-46237-0 • Pedoe, Dan (1988) [1970], Geometry /A Comprehensive Course, Dover, ISBN 0-486-65812-0 • Pfeifer, Richard E.; Van Hook, Cathleen (1993), "Circles, Vectors, and Linear Algebra", Mathematics Magazine, 66 (2): 75–86, doi:10.2307/2691113, JSTOR 2691113 • Samuel, Pierre (1988), Projective Geometry, Undergraduate Texts in Mathematics (Readings in Mathematics), New York: Springer-Verlag, ISBN 0-387-96752-4 • Schwerdtfeger, Hans (1979) [1962], Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry, Dover, pp. 8–10. • Young, John Wesley (1971) [1930], Projective Geometry, Carus Monograph #4, Mathematical Association of America • Woods, Frederick S. (1961) [1922], Higher Geometry / An introduction to advanced methods in analytic geometry, Dover External links • Weisstein, Eric W., "Pencil", MathWorld	Wikipedia
\begin{definition}[Definition:Reflexive Relation] Let $\RR \subseteq S \times S$ be a relation in $S$. \end{definition}	ProofWiki
Hyperbolic set In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under f, with respect to some Riemannian metric on M. An analogous definition applies to the case of flows. In the special case when the entire manifold M is hyperbolic, the map f is called an Anosov diffeomorphism. The dynamics of f on a hyperbolic set, or hyperbolic dynamics, exhibits features of local structural stability and has been much studied, cf. Axiom A. Definition Let M be a compact smooth manifold, f: M → M a diffeomorphism, and Df: TM → TM the differential of f. An f-invariant subset Λ of M is said to be hyperbolic, or to have a hyperbolic structure, if the restriction to Λ of the tangent bundle of M admits a splitting into a Whitney sum of two Df-invariant subbundles, called the stable bundle and the unstable bundle and denoted Es and Eu. With respect to some Riemannian metric on M, the restriction of Df to Es must be a contraction and the restriction of Df to Eu must be an expansion. Thus, there exist constants 0<λ<1 and c>0 such that $T_{\Lambda }M=E^{s}\oplus E^{u}$ and $(Df)_{x}E_{x}^{s}=E_{f(x)}^{s}$ and $(Df)_{x}E_{x}^{u}=E_{f(x)}^{u}$ for all $x\in \Lambda $ and $\\|Df^{n}v\\|\leq c\lambda ^{n}\\|v\\|$ for all $v\in E^{s}$ and $n>0$ and $\\|Df^{-n}v\\|\leq c\lambda ^{n}\\|v\\|$ for all $v\in E^{u}$ and $n>0$. If Λ is hyperbolic then there exists a Riemannian metric for which c = 1 — such a metric is called adapted. Examples • Hyperbolic equilibrium point p is a fixed point, or equilibrium point, of f, such that (Df)p has no eigenvalue with absolute value 1. In this case, Λ = {p}. • More generally, a periodic orbit of f with period n is hyperbolic if and only if Dfn at any point of the orbit has no eigenvalue with absolute value 1, and it is enough to check this condition at a single point of the orbit. References • Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. Reading Mass.: Benjamin/Cummings. ISBN 0-8053-0102-X. • Brin, Michael; Garrett, Stuck (2002). Introduction to Dynamical Systems. Cambridge University Press. ISBN 0-521-80841-3. This article incorporates material from Hyperbolic Set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.	Wikipedia
Does black hole formation contradict the Pauli exclusion principle? A star's collapse can be halted by the degeneracy pressure of electrons or neutrons due to the Pauli exclusion principle. In extreme relativistic conditions, a star will continue to collapse regardless of the degeneracy pressure to form a black hole. Does this violate the Pauli exclusion principle? If so, are theorists ok with that? And if it doesn't violate the Pauli exclusion principle, why not? general-relativity black-holes fermions pauli-exclusion-principle kd88kd88 $\begingroup$ Take the case of a star -> neutron star first. There is enough pressure (energy density) to convert protons into neutrons and radiate the leptons away as neutrinos. I'm not sure what happens in a collapse to a black hole but my guess is that something similar happens with the quarks. Good question! $\endgroup$ – Brandon Enright Jan 16 '14 at 21:03 $\begingroup$ Then perhaps analogously to your example of $e + p \rightarrow n + \nu$ there could be some unknown process that allows $u + d \rightarrow X$ where $X$ is either bosonic or somehow free to propagate away. $\endgroup$ – kd88 Jan 16 '14 at 21:17 $\begingroup$ There are probably lots of other options besides that. I really don't know. $\endgroup$ – Brandon Enright Jan 16 '14 at 21:21 $\begingroup$ related: physics.stackexchange.com/questions/354614/… $\endgroup$ – Ben Crowell Aug 31 '17 at 3:21 I don't have a very satisfactory description of the microscopic picture, but let me share my thoughts. The Pauli exclusion doesn't quite say that fermions can't be squeezed together in space. It says that two fermions can't share the same quantum state (spin included). A black hole has an enormous amount of entropy (proportional to its area, from the famous Bekenstein-Hawking formula $S = \frac{A}{4}$) and hence, its state count is $\sim e^A$. Now, this might not seem like a big deal since usual matter has entropy proportional to volume. However, volume of such collections is also proportional to the mass. This means that a counting of the number of states goes as $e^M$ For a black hole, it's Schwarzschild radius is proportional to the mass, hence $A \sim M^2$. So, the number of states scales as $e^{M^2}$ which is much much more than ordinary matter, especially if the mass is "not small". So there seem to be a lot of quantum states into which one can shove the fermions. So it seems like the fermions should have an easier time in a black hole than in (say) a neutron star. SivaSiva $\begingroup$ There may be two different reasons why people would imagine the exclusion principle to be violated in gravitational collapse. One is the one given in the question, and other has to do with what happens below the Planck scale, at the formation of the singularity. This answer is interesting, but it seems to have more to do with the latter question, not the one asked by the OP. $\endgroup$ – Ben Crowell Aug 31 '17 at 3:25 $\begingroup$ So it seems like the fermions should have an easier time in a black hole than in (say) a neutron star. The Bekenstein-Hawking entropy doesn't represent the microstates of the infalling matter, since, e.g., an eternal black hole has the same entropy. AFAICT semiclassical gravity doesn't give a satisfactory answer as to what microscopic degrees of freedom are represented by black hole entropy, but presumably they would be gravitational in nature. $\endgroup$ – Ben Crowell Aug 31 '17 at 4:03 $\begingroup$ @ Ben Crowell-I had thought the NON-eternality of ANY black hole was why Hawking radiation (whose existence seems to be very widely accepted, in spite of the trans-Planckian problem you mention) had been hypothesized, so please correct me if I'm wrong. Among cosmologies, the trans-Planckian problem is dodged by Nikodem J. Poplawski's "Cosmology with torsion" (the subject of many papers by him between 2010 and 2019), because it uses fermionic interactions of Einstein-Cartan gravity, whose Cartan radius is much larger than the Planck length. $\endgroup$ – Edouard Jun 4 at 3:12 All the known laws of physics, including the exclusion principle, are believed to be valid at all times during the collapse, up until the matter that you're talking is just about to hit the singularity. ("Just about to hit" may mean when the density reaches the Planck density, so that quantum gravity effects become important, or it may be a little earlier, if there is other physics beyond the standard model that we don't know about.) Note that just because an event horizon has formed, and some matter has fallen past the event horizon, that doesn't mean that the conditions experienced by that matter are extreme. They needn't be extreme at all. The equivalence principle says that the laws of physics are always locally the same, because spacetime can always locally be approximated as flat, so that special relativity applies. Actually the conditions at the formation of the event horizon are pretty solidly within the range of conditions (temperature, pressure) that can be described by the standard model of particle physics. Degenerate matter is simply matter in a state where the degeneracy pressure is significant. In a white dwarf or neutron star, for example, the degeneracy pressure happens to be in equilibrium with gravity. The matter can be compressed further without violating the exclusion principle. The exclusion principle essentially has the effect of imposing a maximum wavelength on each particle, which goes roughly like $\sim(V/n)^{-1/3}$, where $V$ is the total volume and $n$ is the number of identical fermions. If $V$ gets smaller, you get a smaller wavelength, therefore bigger momenta and higher pressure. In stars that aren't massive enough to form black holes, you reach a point where this higher pressure gets big enough to give an equilibrium with gravity. If the star is more massive, so that we are going to form a black hole, then we just don't reach such an equilibrium. The wavelengths of the fermions simply get very short, and their momenta very high, as we approach the formation of the singularity. We don't try to say anything about the singularity itself using the presently known laws of physics. In pure classical general relativity, the singularity is not even considered to be part of the spacetime. Jus12 Ben CrowellBen Crowell $\begingroup$ Does it mean that Pauli's principle does not prohibit particles occupying the same state, but rather it says that there will be a resistance when this is forced upon the particles (a kind of repulsion/pressure). $\endgroup$ – Jus12 May 15 '15 at 11:13 Does this violate the Pauli exclusion principle? If so, are theorists ok with that? The short answers are "yes" and "yes". Recall that we are talking about what happens inside the event horizon ... Perhaps the density of states diverges as volume decreases. However iirc most thinking is around the idea that there is a quark degeneracy limit that has to be overcome like the neutron degeneracy limit. i.e. those people speculating some process that combines quarks into some boson can pat themselves on the back. The bottom line is that we don't know enough about how matter/energy behaves in such extreme conditions to be able to do more than speculate. Also see: http://www.physicsforums.com/showthread.php?t=600360 Simon BridgeSimon Bridge $\begingroup$ +1 for pointing out that "what happens in the event horizon stays in the event horizon" $\endgroup$ – kd88 Jan 17 '14 at 10:40 $\begingroup$ Recall that we are talking about what happens inside the event horizon ... That's irrelevant. The equivalence principle says that the laws of physics are always locally the same, because spacetime can always locally be approximated as flat, so that special relativity applies. The bottom line is that we don't know enough about how matter/energy behaves in such extreme conditions to be able to do more than speculate. What extreme conditions do you have in mind? The conditions are not necessarily extreme just because you're inside the event horizon. Do you mean at the singularity? $\endgroup$ – Ben Crowell Oct 17 '14 at 21:35 Not the answer you're looking for? Browse other questions tagged general-relativity black-holes fermions pauli-exclusion-principle or ask your own question. Violation of Pauli exclusion principle How does the infinite density of the early universe or black holes not violate Fermi-Dirac statistics? The trouble with a singularity Why can a neutron star implode? What happens when electron degeneracy pressure is being overcome (right before an object turns into a black hole)? How does gravity acts on the electrons to collapse a neutrons star or white dwarfs against the degeneracy pressure? What causes Paulis Exclusion Principle? What prevents a star from collapsing after stellar death? How does the Pauli exclusion principle create a force in degenerate matter? Nature of the quantum degeneracy pressure Are protons and neutrons affected by the Pauli Exclusion Principle? How does a quark star collapse into a black hole? Pauli Exclusion and Black Holes	CommonCrawl
Structural Basis for the Energetics of Jacalin-Sugar Interactions: Promiscuity Versus Specificity Jeyaprakash, Arockia A and Jayashree, G and Mahanta, SK and Swaminathan, CP and Sekar, K and Surolia, A and Vijayan, M (2005) Structural Basis for the Energetics of Jacalin-Sugar Interactions: Promiscuity Versus Specificity. In: Journal of Molecular Biology, 347 (1). pp. 181-188. Structural_Basis.pdf Jacalin, a tetrameric lectin, is one of the two lectins present in jackfruit (Artocarpus integrifolia) seeds. Its crystal structure revealed, for the first time, the occurrence of the $\beta$-prism I fold in lectins. The structure led to the elucidation of the crucial role of a new N terminus generated by post-translational proteolysis for the lectin's specificity for galactose. Subsequent X-ray studies on other carbohydrate complexes showed that the extended binding site of jacalin consisted of, in addition to the primary binding site, a hydrophobic secondary site A composed of aromatic residues and a secondary site B involved mainly in water-bridges. A recent investigation involving surface plasmon resonance and the X-ray analysis of a methyl-$\alpha$-mannose complex, had led to a suggestion of promiscuity in the lectin's sugar specificity. To explore this suggestion further, detailed isothermal titration calorimetric studies on the interaction of galactose (Gal), mannose (Man), glucose (Glc), Me-$\alpha$-Gal, Me-$\alpha$-Man, Me-$\alpha$-Glc and other mono- and oligosaccharides of biological relevance and crystallographic studies on the jacalin-Me-$\alpha$-Glc complex and a new form of the jacalin-Me-$\alpha$-Man complex, have been carried out. The binding affinity of Me-$\alpha$-Man is 20 times weaker than that of Me-$\alpha$-Gal. The corresponding number is 27, when the binding affinities of Gal and Me-$\alpha$-Gal, and those of Man and Me-$\alpha$-Man are compared. Glucose (Glc) shows no measurable binding, while the binding affinity of Me-$\alpha$-Glc is slightly less than that of Me-$\alpha$-Man. The available crystal structures of jacalin-sugar complexes provide a convincing explanation for the energetics of binding in terms of interactions at the primary binding site and secondary site A. The other sugars used in calorimetric studies show no detectable binding to jacalin. These results and other available evidence suggest that jacalin is specific to O-glycans and its affinity to N-glycans is extremely weak or non-existent and therefore of limited value in processes involving biological recognition. The Copyright belongs to Elsevier. Moraceae lectin;b-prism I fold;sugar specificity;thermodynamic parameters;glycans recognition Division of Biological Sciences > Molecular Biophysics Unit http://eprints.iisc.ac.in/id/eprint/5673	CommonCrawl
\begin{document} \selectlanguage{english} \title{Higher dimensional formal loop spaces} \begin{abstract} If $M$ is a symplectic manifold then the space of smooth loops $\mathrm C^{\infty}(\mathrm S^1,M)$ inherits of a quasi-symplectic form. We will focus in this article on an algebraic analogue of that result. In their article \cite{kapranovvasserot:loop1}, Kapranov and Vasserot introduced and studied the formal loop space of a scheme $X$. We generalize their construction to higher dimensional loops. To any scheme $X$ -- not necessarily smooth -- we associate $\mathcal L^d(X)$, the space of loops of dimension $d$. We prove it has a structure of (derived) Tate scheme -- ie its tangent is a Tate module: it is infinite dimensional but behaves nicely enough regarding duality. We also define the bubble space $\operatorname{\mathfrak{B}}^d(X)$, a variation of the loop space. We prove that $\operatorname{\mathfrak{B}}^d(X)$ is endowed with a natural symplectic form as soon as $X$ has one (in the sense of \cite{ptvv:dersymp}). Throughout this paper, we will use the tools of $(\infty,1)$-categories and symplectic derived algebraic geometry. \end{abstract} \selectlanguage{french} \begin{abstract} {\bf Espaces des lacets formels de dimension supérieure : }L'espace des lacets $\mathrm C^{\infty}(\mathrm S^1,M)$ associé à une variété symplectique $M$ se voit doté d'une structure (quasi-)symplectique induite par celle de $M$. Nous traiterons dans cet article d'un analogue algébrique de cet énoncé. Dans leur article \cite{kapranovvasserot:loop1}, Kapranov et Vasserot ont introduit l'espace des lacets formels associé à un schéma. Nous généralisons leur construction à des lacets de dimension supérieure. Nous associons à tout schéma $X$ -- pas forcément lisse -- l'espace $\mathcal L^d(X)$ de ses lacets formels de dimension $d$. Nous démontrerons que ce dernier admet une structure de schéma (dérivé) de Tate : son espace tangent est de Tate : de dimension infinie mais suffisamment structuré pour se soumettre à la dualité. Nous définirons également l'espace $\operatorname{\mathfrak{B}}^d(X)$ des bulles de $X$, une variante de l'espace des lacets, et nous montrerons que le cas échéant, il hérite de la structure symplectique de $X$. \end{abstract} \selectlanguage{english} \tableofcontents \section{Introduction} \addcontentsline{toc}{section}{Introduction} Considering a differential manifold $M$, one can build the space of smooth loops $\operatorname{L}(M)$ in $M$. It is a central object of string theory. Moreover, if $M$ is symplectic then so is $\operatorname{L}(M)$ -- more precisely quasi-symplectic since it is not of finite dimension -- see for instance \cite{munozpresas:symp}. We will be interested here in an algebraic analogue of that result. The first question is then the following: what is an algebraic analogue of the space of smooth loops? An answer appeared in 1994 in Carlos Contou-Carrère's work (see \cite{contoucarrere:jacobienne}). He studies there $\mathbb{G}_m(\mathbb{C}(\!(t)\!))$, some sort of holomorphic functions in the multiplicative group scheme, and defines the famous Contou-Carrère symbol. This is the first occurrence of a \emph{formal loop space} known by the author. This idea was then generalised to algebraic groups as the affine Grassmannian $\mathfrak{Gr}_G = \quot{G(\mathbb C (\!(t)\!))}{G(\mathbb C[\![t]\!])}$ showed up and got involved in the geometric Langlands program. In their paper \cite{kapranovvasserot:loop1}, Mikhail Kapranov and Éric Vasserot introduced and studied the formal loop space of a smooth scheme $X$. It is an ind-scheme $\mathcal L(X)$ which we can think of as the space of maps $\operatorname{Spec} \mathbb C(\!(t)\!) \to X$. This construction strongly inspired the one presented in this article. There are at least two ways to build higher dimensional formal loops. The most studied one consists in using higher dimensional local fields $k(\!( t_1 )\!) \dots (\!( t_d )\!)$ and is linked to Beilinson's adèles. There is also a generalisation of Contou-Carrère symbol in higher dimensions using those higher dimensional local fields -- see \cite{osipovzhu:contoucarrere} and \cite{bgw:contoucarrere}. If we had adopted this angle, we would have considered maps from some torus\footnote{The variable $\el{t}{d}$ are actually ordered. The author likes to think of $\operatorname{Spec}(k(\!( t_1 )\!) \dots (\!( t_d )\!))$ as a formal torus equipped with a flag representing this order.} $\operatorname{Spec}(k(\!( t_1 )\!) \dots (\!( t_d )\!))$ to $X$. The approach we will follow in this work is different. We generalize here the definition of Kapranov and Vasserot to higher dimensional loops in the following way. For $X$ a scheme of finite presentation, not necessarily smooth, we define $\mathcal L^d(X)$, the space of formal loops of dimension $d$ in $X$. We define $\mathcal L^d_V(X)$ the space of maps from the formal neighbourhood of $0$ in $\mathbb{A}^d$ to $X$. This is a higher dimensional version of the space of germs of arcs as studied by Jan Denef and François Loeser in \cite{denefloeser:germs}. Let also $\mathcal L_U^d(X)$ denote the space of maps from a \emph{punctured} formal neighbourhood of $0$ in $\mathbb{A}^d$ to $X$. The formal loop space $\mathcal L^d(X)$ is the formal completion of $\mathcal L_V^d(X)$ in $\mathcal L_U^d(X)$. Understanding those three items is the main goal of this work. The problem is mainly to give a meaningful definition of the punctured formal neighbourhood of dimension $d$. We can describe what its cohomology should be: \[ \mathrm H^n(\hat \mathbb{A}^d\smallsetminus \{0\}) = \begin{cases} k[\![\el{X}{d}]\!] & \text{ if } n = 0 \\ (\el{X}{d}[])^{-1} k[\el{X^{-1}}{d}] & \text{ if } n=d-1 \\ 0 & \text{ otherwise} \end{cases} \] but defining this punctured formal neighbourhood with all its structure is actually not an easy task. Nevertheless, we can describe what maps out of it are, hence the definition of $\mathcal L^d_U(X)$ and the formal loop space. This geometric object is of infinite dimension, and part of this study is aimed at identifying some structure. Here comes the first result in that direction. \begin{thmintro}[see \autoref{L-indpro}]\label{intro-kaploop} The formal loop space of dimension $d$ in a scheme $X$ is represented by a derived ind-pro-scheme. Moreover, the functor $X \mapsto \mathcal L^d(X)$ satisfies the étale descent condition. \end{thmintro} We use here methods from derived algebraic geometry as developed by Bertrand Toën and Gabriele Vezzosi in \cite{toen:hagii}. The author would like to emphasize here that the derived structure is necessary since, when $X$ is a scheme, the underlying schemes of $\mathcal L^d(X)$, $\mathcal L^d_U(X)$ and $\mathcal L^d_V(X)$ are isomorphic as soon as $d \geq 2$. Let us also note that derived algebraic geometry allowed us to define $\mathcal L^d(X)$ for more general $X$'s, namely any derived stack. In this case, the formal loop space $\mathcal L^d(X)$ is no longer a derived ind-pro-scheme but an ind-pro-stack. The case $d = 1$ and $X$ is a smooth scheme gives a derived enhancement of Kapranov and Vasserot's definition. This derived enhancement is conjectured to be trivial when $X$ is a smooth affine scheme in \cite[9.2.10]{gaitsgoryrozenblyum:dgindschemes}. Gaitsgory and Rozenblyum also prove in \emph{loc. cit.} their conjecture holds when $X$ is an algebraic group. The proof of \autoref{intro-kaploop} is based on an important lemma. We identify a full sub-category $\mathcal{C}$ of the category of ind-pro-stacks such that the realisation functor $\mathcal{C} \to \mathbf{dSt}_k$ is fully faithful. We then prove that whenever $X$ is a derived affine scheme, the stack $\mathcal L^d(X)$ is in the essential image of $\mathcal{C}$ and is thus endowed with an \emph{essentially unique} ind-pro-structure satisfying some properties. The generalisation to any $X$ is made using a descent argument. Note that for general $X$'s, the ind-pro-structure is not known to satisfy nice properties one could want to have, for instance on the transition maps of the diagrams. We then focus on the following problem: can we build a symplectic form on $\mathcal L^d(X)$ when $X$ is symplectic? Again, this question requires the tools of derived algebraic geometry and \emph{shifted symplectic structures} as in \cite{ptvv:dersymp}. A key feature of derived algebraic geometry is the cotangent complex $\mathbb{L}_X$ of any geometric object $X$. A ($n$-shifted) symplectic structure on $X$ is a closed $2$-form $\mathcal{O}_X[-n] \to \mathbb{L}_X \wedge \mathbb{L}_X$ which is non degenerate -- ie induces an equivalence \[ \mathbb{T}_X \to \mathbb{L}_X[n] \] Because $\mathcal L^d(X)$ is not finite, linking its cotangent to its dual -- through an alleged symplectic form -- requires to identify once more some structure. We already know that it is an ind-pro-scheme but the proper context seems to be what we call Tate stacks. Before saying what a Tate stack is, let us talk about Tate modules. They define a convenient context for infinite dimensional vector spaces. They where studied by Lefschetz, Beilinson and Drinfeld, among others, and more recently by Bräunling, Gröchenig and Wolfson \cite{bgw:tate}. We will use here the notion of Tate objects in the context of stable $(\infty,1)$-categories as developed in \cite{hennion:tate}. If $\mathcal{C}$ is a stable $(\infty,1)$-category -- playing the role of the category of finite dimensional vector spaces, the category $\operatorname{\mathbf{Tate}}(\mathcal{C})$ is the full subcategory of the $(\infty,1)$-category of pro-ind-objects $\operatorname{\mathbf{Pro}} \operatorname{\mathbf{Ind}}(\mathcal{C})$ in $\mathcal{C}$ containing both $\operatorname{\mathbf{Ind}}(\mathcal{C})$ and $\operatorname{\mathbf{Pro}}(\mathcal{C})$ and stable by extensions and retracts. We will define the derived category of Tate modules on a scheme -- and more generally on a derived ind-pro-stack. An Artin ind-pro-stack $X$ -- meaning an ind-pro-object in derived Artin stacks -- is then gifted with a cotangent complex $\mathbb{L}_X$. This cotangent complex inherits a natural structure of pro-ind-module on $X$. This allows us to define a Tate stack as an Artin ind-pro-stack whose cotangent complex is a Tate module. The formal loop space $\mathcal L^d(X)$ is then a Tate stack as soon as $X$ is a finitely presented derived affine scheme. For a more general $X$, what precedes makes $\mathcal L^d(X)$ some kind of \emph{locally} Tate stack. This structure suffices to define a determinantal anomaly \[ \mathopen{}\mathclose\bgroup\originalleft[\mathrm{Det}_{\mathcal L^d(X)}\aftergroup\egroup\originalright] \in \mathrm H^2\mathopen{}\mathclose\bgroup\originalleft(\mathcal L^d(X), \mathcal{O}_{\mathcal L^d(X)}^{\times}\aftergroup\egroup\originalright) \] for any quasi-compact quasi-separated (derived) scheme $X$ -- this construction also works for slightly more general $X$'s, namely Deligne-Mumford stacks with algebraisable diagonal, see \autoref{determinantalanomaly}. Kapranov and Vasserot proved in \cite{kapranovvasserot:loop4} that in dimension $1$, the determinantal anomaly governs the existence of sheaves of chiral differential operators on $X$. One could expect to have a similar result in higher dimensions, with higher dimensional analogues of chiral operators and vertex algebras. The author plans on studying this in a future work. Another feature of Tate modules is duality. It makes perfect sense and behaves properly. Using the theory of symplectic derived stacks developed by Pantev, Toën, Vaquié and Vezzosi in \cite{ptvv:dersymp}, we are then able to build a notion of symplectic Tate stack: a Tate stack $Z$ equipped with a ($n$-shifted) closed $2$-form which induces an equivalence \[ \mathbb{T}_Z \to^\sim \mathbb{L}_Z[n] \] of Tate modules over $Z$ between the tangent and (shifted) cotangent complexes of $Z$. To make a step toward proving that $\mathcal L^d(X)$ is a symplectic Tate stack, we actually study the bubble space $\operatorname{\mathfrak{B}}^d(X)$ -- see \autoref{dfbubble}. When $X$ is affine, we get an equivalence \[ \operatorname{\mathfrak{B}}^d(X) \simeq \mathcal L_V^d(X) \times_{\mathcal L_U^d(X)} \mathcal L^d_V(X) \] Note that the fibre product above is a \emph{derived} intersection. We then prove the following result \begin{thmintro}[see \autoref{B-symplectic}]\label{intro-bubble} If $X$ is an $n$-shifted symplectic stack then the bubble space $\operatorname{\mathfrak{B}}^d(X)$ is endowed with a structure of $(n-d)$-shifted symplectic Tate stack. \end{thmintro} The proof of this result is based on a classical method. The bubble space is in fact, as an ind-pro-stack, the mapping stack from what we call the formal sphere $\hat S^d$ of dimension $d$ to $X$. There are therefore two maps \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{\mathfrak{B}}^d(X) & \operatorname{\mathfrak{B}}^d(X) \times \hat S^d \ar[r]^-{\operatorname{ev}} \ar[l]_-{\operatorname{pr}} & X } \] The symplectic form on $\operatorname{\mathfrak{B}}^d(X)$ is then $\int_{\hat S^d} \operatorname{ev}^ \omega_X$, where $\omega_X$ is the symplectic form on $X$. The key argument is the construction of this integration on the formal sphere, ie on an oriented pro-ind-stack of dimension $d$. The orientation is given by a residue map. On the level of cohomology, it is the morphism \[ \mathrm H^d(\hat S^d) \simeq (\el{X}{d}[])^{-1} k[\el{X^{-1}}{d}] \to k \] mapping $(\el{X}{d}[])^{-1}$ to $1$. This integration method would not work on $\mathcal L^d(X)$, since the punctured formal neighbourhood does not have as much structure as the formal sphere: it is not known to be a pro-ind-scheme. Nevertheless, \autoref{intro-bubble} is a first step toward proving that $\mathcal L^d(X)$ is symplectic. We can consider the nerve $Z_\bullet$ of the map $\mathcal L^d_V(X) \to \mathcal L^d_U(X)$. It is a groupoid object in ind-pro-stacks whose space of maps is $\operatorname{\mathfrak{B}}^d(X)$. The author expects that this groupoid is compatible in some sense with the symplectic structure so that $\mathcal L^d_U(X)$ would inherit a symplectic form from realising this groupoid. One the other hand, if $\mathcal L^d_U(X)$ was proven to be symplectic, then the fibre product defining $\operatorname{\mathfrak{B}}^d(X)$ should be a Lagrangian intersection. The bubble space would then inherit a symplectic structure from that on $\mathcal L^d(X)$. \subsubsection{Techniques and conventions} Throughout this work, we will use the techniques of $(\infty,1)$-category theory. We will once in a while use explicitly the model of quasi-categories developed by Joyal and Lurie (see \cite{lurie:htt}). That being said, the results should be true with any equivalent model. Let us fix now two universes $\mathbb U \in \mathbb V$ to deal with size issues. Every algebra, module or so will implicitly be $\mathbb U$-small. The first part will consist of reminders about $(\infty,1)$-categories. We will fix there some notations. Note that we will often refer to \cite{hennion:these} for some $(\infty,1)$-categorical results needed in this article. We will also use derived algebraic geometry, as introduced in \cite{toen:hagii}. We refer to \cite{toen:dagems} for a recent survey of this theory. We will denote by $k$ a base field and by $\mathbf{dSt}_k$ the $(\infty,1)$-category of ($\mathbb U$-small) derived stacks over $k$. In the first section, we will dedicate a few page to introduce derived algebraic geometry. \subsubsection{Outline} This article begins with a few paragraphs, recalling some notions we will use. Among them are $(\infty,1)$-categories and derived algebraic geometry. In \autoref{chapterIP}, we set up a theory of geometric ind-pro-stacks. We then define in \autoref{chapterSymptate} symplectic Tate stacks and give a few properties, including the construction of the determinantal anomaly (see \autoref{determinantalanomaly}). Comes \autoref{chapterfloops} where we finally define higher dimensional loop spaces and prove \autoref{intro-kaploop} (see \autoref{L-indpro}). We finally introduce the bubble space and prove \autoref{intro-bubble} (see \autoref{B-symplectic}) in \autoref{chapterBubbles}. \subsubsection{Aknowledgements} I would like to thank Bertrand Toën, Damien Calaque and Marco Robalo for the many discussions we had about the content of this work. I am grateful to Mikhail Kapranov, James Wallbridge and Giovanni Faonte for inviting me at the IPMU. My stay there was very fruitful and the discussions we had were very interesting. I learned after writing down this article that Kapranov had an unpublished document in which higher dimensional formal loops are studied. I am very grateful to Kapranov for letting me read those notes, both inspired and inspiring. This work is extracted from my PhD thesis \cite{hennion:these} under the advisement of Bertrand Toën. I am very grateful to him for those amazing few years. \section{Preliminaries} \addcontentsline{toc}{section}{Preliminaries} In this part, we recall some results and definitions from $(\infty,1)$-category theory and derived algebraic geometry. \subsection{A few tools from higher category theory} In the last decades, theory of $(\infty,1)$-categories has tremendously grown. The core idea is to consider categories enriched over spaces, so that every object or morphism is considered up to higher homotopy. The typical example of such a category is the category of topological spaces itself: for any topological spaces $X$ and $Y$, the set of maps $X \to Y$ inherits a topology. It is often useful to talk about topological spaces up to homotopy equivalences. Doing so, one must also consider maps up to homotopy. To do so, one can of course formally invert every homotopy equivalence and get a set of morphisms $[X,Y]$. This process loses information and mathematicians tried to keep trace of the space of morphisms. The first fully equipped theory handy enough to work with such examples, called model categories, was introduced by Quillen. A model category is a category with three collections of maps -- weak equivalences (typically homotopy equivalences), fibrations and cofibrations -- satisfying a bunch of conditions. The datum of such collections allows us to compute limits and colimits up to homotopy. We refer to \cite{hovey:modcats} for a comprehensive review of the subject. Using model categories, several mathematicians developed theories of $(\infty,1)$-categories. Let us name here Joyal's quasi-categories, complete Segal spaces or simplicial categories. Each one of those theories is actually a model category and they are all equivalent one to another -- see \cite{bergner:oneinfty} for a review. In \cite{lurie:htt}, Lurie developed the theory of quasi-categories. In this book, he builds everything necessary so that we can think of $(\infty,1)$-categories as we do usual categories. To prove something in this context still requires extra care though. We will use throughout this work the language as developed by Lurie, but we will try to keep in mind the $1$-categorical intuition. In this section, we will fix a few notations and recall some results to which we will often refer. \paragraph{Notations:} Let us first fix a few notations, borrowed from \cite{lurie:htt}. \begin{itemize} \item We will denote by $\inftyCatu U$ the $(\infty,1)$-category of $\mathbb U$-small $(\infty,1)$-categories -- see \cite[3.0.0.1]{lurie:htt}; \item Let $\PresLeftu U$ denote the $(\infty,1)$-category of $\mathbb U$-presentable (and thus $\mathbb V$-small) $(\infty,1)$-categories with left adjoint functors -- see \cite[5.5.3.1]{lurie:htt}; \item The symbol $\mathbf{sSets}$ will denote the $(\infty,1)$-category of $\mathbb U$-small simplicial sets up to homotopy equivalences (this is equivalent to the category of (nice) topological spaces up to homotopy); \item For any $(\infty,1)$-categories $\mathcal{C}$ and $\mathcal{D}$ we will write $\operatorname{Fct}(\mathcal{C},\mathcal{D})$ for the $(\infty,1)$-category of functors from $\mathcal{C}$ to $\mathcal{D}$ (see \cite[1.2.7.3]{lurie:htt}). The category of presheaves will be denoted $\operatorname{\mathcal{P}}(\mathcal{C}) = \operatorname{Fct}(\mathcal{C}^{\mathrm{op}}, \mathbf{sSets})$. \item For any $(\infty,1)$-category $\mathcal{C}$ and any objects $c$ and $d$ in $\mathcal{C}$, we will denote by $\operatorname{Map}_{\mathcal{C}}(c,d)$ the space of maps from $c$ to $d$. \item For any simplicial set $K$, we will denote by $K^{\triangleright}$ the simplicial set obtained from $K$ by formally adding a final object. This final object will be called the cone point of $K^\triangleright$. \end{itemize} The following theorem is a concatenation of results from Lurie. \begin{thm}[Lurie]\label{indu-thm} Let $\mathcal{C}$ be a $\mathbb V$-small $(\infty,1)$-category. There is an $(\infty,1)$-category $\Indu U(\mathcal{C})$ and a functor $j \colon \mathcal{C} \to \Indu U(\mathcal{C})$ such that \begin{enumerate} \item The $(\infty,1)$-category $\Indu U(\mathcal{C})$ is $\mathbb V$-small; \item The $(\infty,1)$-category $\Indu U(\mathcal{C})$ admits $\mathbb U$-small filtered colimits and is generated by $\mathbb U$-small filtered colimits of objects in $j(\mathcal{C})$; \item The functor $j$ is fully faithful and preserves finite limits and finite colimits which exist in $\mathcal{C}$; \item For any $c \in \mathcal{C}$, its image $j(c)$ is $\mathbb U$-small compact in $\Indu U (\mathcal{C})$; \item For every $(\infty,1)$-category $\mathcal{D}$ with every $\mathbb U$-small filtered colimits, the functor $j$ induces an equivalence \[ \operatorname{Fct}^{\mathbb U\mathrm{-c}}(\Indu U(\mathcal{C}), \mathcal{D}) \to^\sim \operatorname{Fct}(\mathcal{C},\mathcal{D}) \] where $\operatorname{Fct}^{\mathbb U \mathrm{-c}}(\Indu U(\mathcal{C}), \mathcal{D})$ denote the full subcategory of $\operatorname{Fct}(\Indu U(\mathcal{C}),\mathcal{D})$ spanned by functors preserving $\mathbb U$-small filtered colimits. \item If $\mathcal{C}$ is $\mathbb U$-small and admits all finite colimits then $\Indu U(\mathcal{C})$ is $\mathbb U$-presentable; \item If $\mathcal{C}$ is endowed with a symmetric monoidal structure then there exists such a structure on $\Indu U(\mathcal{C})$ such that the monoidal product preserves $\mathbb U$-small filtered colimits in each variable.\label{indmonoidal} \end{enumerate} \end{thm} \begin{proof} Let us use the notations of \cite[5.3.6.2]{lurie:htt}. Let $\mathcal K$ denote the collection of $\mathbb U$-small filtered simplicial sets. We then set $\Indu U (\mathcal{C}) = \operatorname{\mathcal{P}}^\mathcal K _\emptyset(\mathcal{C})$. It satisfies the required properties because of \emph{loc. cit.} 5.3.6.2 and 5.5.1.1. We also need tiny modifications of the proofs of \emph{loc. cit.} 5.3.5.14 and 5.3.5.5. The last item is proved in \cite[6.3.1.10]{lurie:halg}. \end{proof} \begin{rmq} Note that when $\mathcal{C}$ admits finite colimits then the category $\Indu U(\mathcal{C})$ embeds in the $\mathbb V$-presentable category $\Indu V(\mathcal{C})$. \end{rmq} \begin{df} Let $\mathcal{C}$ be a $\mathbb V$-small $\infty$-category. We define $\Prou U(\mathcal{C})$ as the $(\infty,1)$-category \[ \Prou U(\mathcal{C}) = \mathopen{}\mathclose\bgroup\originalleft( \Indu U(\mathcal{C}^{\mathrm{op}}) \aftergroup\egroup\originalright) ^{\mathrm{op}} \] It satisfies properties dual to those of $\Indu U(\mathcal{C})$. \end{df} \begin{df} \label{indext} Let $\mathcal{C}$ be a $\mathbb V$-small $(\infty,1)$-category. Let \[ i \colon \operatorname{Fct}(\mathcal{C}, \inftyCatu V) \to \operatorname{Fct}(\Indu U(\mathcal{C}), \inftyCatu V) \] denote the left Kan extension functor. We will denote by $\operatorname{\underline{\mathbf{Ind}}}^\mathbb U_\mathcal{C}$ the composite functor \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{Fct}(\mathcal{C}, \inftyCatu V) \ar[r]^-i & \operatorname{Fct}(\Indu U(\mathcal{C}), \inftyCatu V) \ar[rr]^-{\Indu U \circ -} && \operatorname{Fct}(\Indu U(\mathcal{C}), \inftyCatu V) } \] We will denote by $\operatorname{\underline{\mathbf{Pro}}}^\mathbb U_\mathcal{C}$ the composite functor \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{Fct}(\mathcal{C}, \inftyCatu V) \ar[rr]^{\Prou U \circ -} && \operatorname{Fct}(\mathcal{C}, \inftyCatu V) \ar[r] & \operatorname{Fct}(\Prou U(\mathcal{C}), \inftyCatu V) } \] We define the same way \begin{align} &\Indextu V_C \colon \operatorname{Fct}(\mathcal{C}, \inftyCatu V) \to \operatorname{Fct}(\Indu V(\mathcal{C}), \mathbf{Cat}_\infty) \\ &\Proextu V_C \colon \operatorname{Fct}(\mathcal{C}, \inftyCatu V) \to \operatorname{Fct}(\Prou V(\mathcal{C}), \mathbf{Cat}_\infty) \\ \end{align} \end{df} \begin{rmq} The \autoref{indext} can be expanded as follows. To any functor $f \colon \mathcal{C} \to \inftyCatu V$ and any ind-object $c$ colimit of a diagram \[ \shorthandoff{;:!?} \xymatrix{ K \ar[r]^-{\bar c} & \mathcal{C} \ar[r] & \Indu U(\mathcal{C}) } \] we construct an $(\infty,1)$-category \[ \operatorname{\underline{\mathbf{Ind}}}^\mathbb U_\mathcal{C}(f)(c) \simeq \Indu U\mathopen{}\mathclose\bgroup\originalleft(\colim f(\bar c)\aftergroup\egroup\originalright) \] To any pro-object $d$ limit of a diagram \[ \shorthandoff{;:!?} \xymatrix{ K^{\mathrm{op}} \ar[r]^-{\bar d} & \mathcal{C} \ar[r] & \Prou U(\mathcal{C}) } \] we associate an $(\infty,1)$-category \[ \operatorname{\underline{\mathbf{Pro}}}^\mathbb U_\mathcal{C}(f)(d) \simeq \lim \Prou U(f(\bar d)) \] \end{rmq} \begin{df} Let $\mathbf{Cat}_\infty^{\mathbb V\mathrm{,st}}$ denote the subcategory of $\inftyCatu V$ spanned by stable categories with exact functors between them -- see \cite[1.1.4]{lurie:halg}. Let $\mathbf{Cat}_\infty^{\mathbb V\mathrm{,st,id}}$ denote the full subcategory of $\mathbf{Cat}_\infty^{\mathbb V\mathrm{,st}}$ spanned by idempotent complete stable categories. \end{df} \begin{rmq} It follows from \cite[1.1.4.6, 1.1.3.6, 1.1.1.13 and 1.1.4.4]{lurie:halg} that the functors $\operatorname{\underline{\mathbf{Ind}}}^\mathbb U_\mathcal{C}$ and $\operatorname{\underline{\mathbf{Pro}}}^\mathbb U_\mathcal{C}$ restricts to functors \begin{align} & \operatorname{\underline{\mathbf{Ind}}}^\mathbb U_\mathcal{C} \colon \operatorname{Fct}(\mathcal{C}, \mathbf{Cat}_\infty^{\mathbb V\mathrm{,st}}) \to \operatorname{Fct}(\Indu U (\mathcal{C}), \mathbf{Cat}_\infty^{\mathbb V\mathrm{,st}}) \\ & \operatorname{\underline{\mathbf{Pro}}}^\mathbb U_\mathcal{C} \colon \operatorname{Fct}(\mathcal{C}, \mathbf{Cat}_\infty^{\mathbb V\mathrm{,st}}) \to \operatorname{Fct}(\Prou U (\mathcal{C}), \mathbf{Cat}_\infty^{\mathbb V\mathrm{,st}}) \end{align} \end{rmq} \paragraph{Symmetric monoidal $(\infty,1)$-categories:} We will make use in the last chapter of the theory of symmetric monoidal $(\infty,1)$-categories as developed in \cite{lurie:halg}. Let us give a (very) quick review of those objects. \begin{df}\label{ptfin} Let $\mathrm{Fin}^\pt$ denote the category of pointed finite sets. For any $n \in \mathbb{N}$, we will denote by $\langle n \rangle$ the set $\{{}, 1, \dots ,n\}$ pointed at ${}$. For any $n$ and $i \leq n$, the Segal map $\delta^i \colon \langle n \rangle \to \langle 1 \rangle$ is defined by $\delta^i(j) = 1$ if $j = i$ and $\delta^i(j) = {}$ otherwise. \end{df} \begin{df}(see \cite[2.0.0.7]{lurie:halg})\label{monoidalcats} Let $\mathcal{C}$ be an $(\infty,1)$-category. A symmetric monoidal structure on $\mathcal{C}$ is the datum of a coCartesian fibration $p \colon \mathcal{C}^{\otimes} \to \mathrm{Fin}^\pt$ such that \begin{itemize} \item The fibre category $\mathcal{C}^{\otimes}_{\langle 1 \rangle}$ is equivalent to $\mathcal{C}$ and \item For any $n$, the Segal maps induce an equivalence $\mathcal{C}^{\otimes}_{\langle n \rangle} \to (\mathcal{C}^{\otimes}_{\langle 1 \rangle})^n \simeq \mathcal{C}^n$. \end{itemize} where $\mathcal{C}^{\otimes}_{\langle n \rangle}$ denote the fibre of $p$ at $\langle n \rangle$. We will denote by $\monoidalinftyCatu V$ the $(\infty,1)$-category of $\mathbb V$-small symmetric monoidal $(\infty,1)$-categories -- see \cite[2.1.4.13]{lurie:halg}. \end{df} Such a coCartesian fibration is classified by a functor $\phi \colon \mathrm{Fin}^\pt \to \inftyCatu V$ -- see \cite[3.3.2.2]{lurie:htt} -- such that $\phi(\langle n \rangle) \simeq \mathcal{C}^n$. The tensor product on $\mathcal{C}$ is induced by the map of pointed finite sets $\mu \colon \langle 2 \rangle \to \langle 1 \rangle$ mapping both $1$ and $2$ to $1$ \[ \otimes = \phi(\mu) \colon \mathcal{C}^2 \to \mathcal{C} \] \begin{rmq}\label{indext-monoidal} The forgetful functor $\monoidalinftyCatu V \to \inftyCatu V$ preserves all limits as well as filtered colimits -- see \cite[3.2.2.4 and 3.2.3.2]{lurie:halg}. Moreover, it follows from \autoref{indu-thm} - \ref{indmonoidal} that the functor $\Indu U$ induces a functor \[ \Indu U \colon \monoidalinftyCatu V \to \monoidalinftyCatu V \] The same holds for $\Prou U$. The constructions $\operatorname{\underline{\mathbf{Ind}}}^\mathbb U$ and $\operatorname{\underline{\mathbf{Pro}}}^\mathbb U$ therefore restrict to \begin{align} &\operatorname{\underline{\mathbf{Ind}}}^\mathbb U_\mathcal{C} \colon \operatorname{Fct}(\mathcal{C},\monoidalinftyCatu V) \to \operatorname{Fct}(\Indu U(\mathcal{C}), \monoidalinftyCatu V) \\ &\operatorname{\underline{\mathbf{Pro}}}^\mathbb U_\mathcal{C} \colon \operatorname{Fct}(\mathcal{C},\monoidalinftyCatu V) \to \operatorname{Fct}(\Prou U(\mathcal{C}), \monoidalinftyCatu V) \end{align} \end{rmq} \paragraph{Tate objects:} We now recall the definition and a few properties of Tate objects in a stable and idempotent complete $(\infty,1)$-category. The content of this paragraph comes from \cite{hennion:tate}. See also \cite{hennion:these}. \begin{df} Let $\mathcal{C}$ be a stable and idempotent complete $(\infty,1)$-category. Let $\Tateu U(\mathcal{C})$ denote the smallest full subcategory of $\Prou U \Indu U(\mathcal{C})$ containing $\Indu U(\mathcal{C})$ and $\Prou U(\mathcal{C})$, and both stable and idempotent complete. \end{df} The category $\Tateu U(\mathcal{C})$ naturally embeds into $\Indu U\Prou U(\mathcal{C})$ as well. \begin{prop}\label{tate-dual} If moreover $\mathcal{C}$ is endowed with a duality equivalence $\mathcal{C}^{\mathrm{op}} \to^\sim \mathcal{C}$ then the induced functor \[ \Prou U \Indu U(\mathcal{C}) \to \mathopen{}\mathclose\bgroup\originalleft(\Prou U \Indu U(\mathcal{C})\aftergroup\egroup\originalright)^{\mathrm{op}} \simeq \Indu U\Prou U(\mathcal{C}) \] preserves Tate objects and induces an equivalence $\Tateu U(\mathcal{C}) \simeq \Tateu U(\mathcal{C})^{\mathrm{op}}$. \end{prop} \begin{df} Let $\mathcal{C}$ be a $\mathbb V$-small $(\infty,1)$-category. We define the functor \[ \Tateextu U \colon \shorthandoff{;:!?} \xymatrix@1{\operatorname{Fct}(\mathcal{C},\mathbf{Cat}_\infty^{\mathbb V\mathrm{,st}}) \ar[r]^-i & \operatorname{Fct}(\Indu U(\mathcal{C}),\mathbf{Cat}_\infty^{\mathbb V\mathrm{,st}}) \ar[rr]^-{\Tateu U \circ -} && \operatorname{Fct}(\Indu U(\mathcal{C}),\mathbf{Cat}_\infty^{\mathbb V\mathrm{,st,id}})} \] \end{df} \subsection{Derived algebraic geometry} We present here some background results about derived algebraic geometry. Let us assume $k$ is a field of characteristic $0$. First introduced by Toën and Vezzosi in \cite{toen:hagii}, derived algebraic geometry is a generalisation of algebraic geometry in which we replace commutative algebras over $k$ by commutative differential graded algebras (or cdga's). We refer to \cite{toen:dagems} for a recent survey of this theory. \paragraph{Generalities on derived stacks:}We will denote by $\cdgaunbounded^{\leq 0}_k$ the $(\infty,1)$-category of cdga's over $k$ concentrated in non-positive cohomological degree. It is the $(\infty,1)$-localisation of a model category along weak equivalences. Let us denote $\mathbf{dAff}_k$ the opposite $(\infty,1)$-category of $\cdgaunbounded^{\leq 0}_k$. It is the category of derived affine schemes over $k$. In this work, we will adopt a cohomological convention for cdga's. A derived prestack is a presheaf $\mathbf{dAff}_k^{\mathrm{op}} \simeq \cdgaunbounded^{\leq 0}_k \to \mathbf{sSets}$. We will thus write $\operatorname{\mathcal{P}}(\mathbf{dAff}_k)$ for the $(\infty,1)$-category of derived prestacks. A derived stack is a prestack satisfying the étale descent condition. We will denote by $\mathbf{dSt}_k$ the $(\infty,1)$-category of derived stacks. It comes with an adjunction \[ (-)^+ \colon \operatorname{\mathcal{P}}(\mathbf{dAff}_k) \rightleftarrows \mathbf{dSt}_k \] where the left adjoint $(-)^+$ is called the stackification functor. \begin{rmq} The categories of varieties, schemes or (non derived) stacks embed into $\mathbf{dSt}_k$. \end{rmq} \begin{df} The $(\infty,1)$-category of derived stacks admits an internal hom $\operatorname{\underline{Ma}p}(X,Y)$ between two stacks $X$ and $Y$. It is the functor $\cdgaunbounded^{\leq 0}_k \to \mathbf{sSets}$ defined by \[ A \mapsto \operatorname{Map}_{\mathbf{dSt}_k}(X \times \operatorname{Spec} A, Y) \] We will call it the mapping stack from $X$ to $Y$. \end{df} There is a derived version of Artin stacks of which we first give a recursive definition. \begin{df}(see for instance \cite[5.2.2]{toenmoerdijk:crm})\label{artinstacks} Let $X$ be a derived stack. \begin{itemize} \item We say that $X$ is a derived $0$-Artin stack if it is a derived affine scheme ; \item We say that $X$ is a derived $n$-Artin stack if there is a family $(T_\alpha)$ of derived affine schemes and a smooth atlas \[ u \colon \coprod T_\alpha \to X \] such that the nerve of $u$ has values in derived $(n-1)$-Artin stacks ; \item We say that $X$ is a derived Artin (or algebraic) stack if it is an $n$-Artin stack for some $n$. \item We say that $X$ is locally of finite presentation if there exists a smooth atlas $\bigsqcup T_\alpha \to X$ as above, such that the derived affine schemes $T_\alpha$ are all finitely presented (ie their cdga of functions is finitely presented, or equivalently compact is the category of cdga's). We also say that $X$ is finitely presented if there is such an atlas with a finite number of $T_\alpha$'s. \end{itemize} We will denote by $\dSt^{\mathrm{Art}}_k$ the full subcategory of $\mathbf{dSt}_k$ spanned by derived Artin stacks. \end{df} \begin{df} A morphism $X \to Y$ of derived stacks is called algebraic if for any cdga $A$ and any map $\operatorname{Spec} A \to Y$, the derived intersection $X \times_Y \operatorname{Spec} A$ is an algebraic stack. \end{df} To any cdga $A$ we associate the category $\mathbf{dgMod}_A$ of dg-modules over $A$. Similarly, to any derived stack $X$ we can associate a derived category $\mathbf{Qcoh}(X)$ of quasicoherent sheaves. It is a $\mathbb U$-presentable $(\infty,1)$-category given by the formula \[ \mathbf{Qcoh}(X) \simeq \lim_{\operatorname{Spec} A \to X} \mathbf{dgMod}_A \] Moreover, for any map $f \colon X \to Y$, there is a natural pull back functor $f^* \colon \mathbf{Qcoh}(Y) \to \mathbf{Qcoh}(X)$. This functor admits a right adjoint, which we will denote by $f_$. This construction is actually a functor of $(\infty,1)$-categories. \begin{df} Let us denote by $\mathbf{Qcoh}$ the functor \[ \mathbf{Qcoh} \colon \mathbf{dSt}_k^{\mathrm{op}} \to \PresLeftu U \] For any $X$ we can identify a full subcategory $\mathbf{Perf}(X) \subset \mathbf{Qcoh}(X)$ of perfect complexes. This defines a functor \[ \mathbf{Perf} \colon \mathbf{dSt}_k^{\mathrm{op}} \to \inftyCatu U \] \end{df} \begin{rmq} For any derived stack $X$ the categories $\mathbf{Qcoh}(X)$ and $\mathbf{Perf}(X)$ are actually stable and idempotent complete $(\infty,1)$-categories. The inclusion $\mathbf{Perf}(X) \to \mathbf{Qcoh}(X)$ is exact. Moreover, for any map $f \colon X \to Y$ the pull back functor $f^$ preserves perfect modules and is also exact. \end{rmq} \begin{df}[see for instance the appendix of \cite{halpernleistern-preygel:properness}] Let $X$ be a derived stack and let $\pi \colon X \to {}$ denote the projection. We say that $X$ is of finite cohomological dimension if there is a non-negative integer $d$ such that the complex $\pi_ \mathcal{O}_X = \mathbb{R} \Gamma(X,\mathcal{O}_X) \in \mathbf{dgMod}_k$ is concentrated in degree lower or equal to $d$. \end{df} \begin{ex} Any derived affine scheme is of finite cohomological dimension (take $d = 0$). Any quasi-compact quasi-separated derived stack (ie a finite colimit of derived affine schemes) is of finite cohomological dimension. \end{ex} Any derived Artin stack $X$ over a basis $S$ admits a cotangent complex $\mathbb{L}_{X/S} \in \mathbf{Qcoh}(X)$. If $X$ is locally of finite presentation, then the its cotangent complex is perfect \[ \mathbb{L}_{X/S} \in \mathbf{Perf}(X) \] \paragraph{Symplectic structures:} Following \cite{ptvv:dersymp}, to any derived stack $X$ we associate two complexes $\forms p(X)$ and $\closedforms p(X)$ in $\mathbf{dgMod}_k$, respectively of $p$-forms and closed $p$-forms on $X$. They come with a natural morphism $\closedforms p(X) \to \forms p(X)$ forgetting the lock closing the forms\footnote{This lock is a structure on the form: being closed in not a property in this context.}. This actually glues into a natural transformation \[ \shorthandoff{;:!?} \xymatrix{ \mathbf{dSt}_k \dcell[r][\closedforms p][\forms p] & \mathbf{dgMod}_k } \] Let us emphasize that, as soon as $X$ is Artin, the complex $\forms 2(X)$ is canonically equivalent to the global sections complex of $\mathbb{L}_X \wedge \mathbb{L}_X$. In particular, any $n$-shifted $2$-forms $k[-n] \to \forms p(X)$ induces a morphism $\mathcal{O}_X[-n] \to \mathbb{L}_X \wedge \mathbb{L}_X$ in $\mathbf{Qcoh}(X)$. If $X$ is locally of finite presentation, the cotangent $\mathbb{L}_X$ is perfect and we then get a map \[ \mathbb{T}_X[-n] \to \mathbb{L}_X \] \begin{df} Let $X$ be a derived Artin stack locally of finite presentation. \begin{itemize} \item An $n$-shifted $2$-form $\omega_X \colon k[-n] \to \forms 2(X)$ is called non-degenerated if the induced morphism $\mathbb{T}_X[-n] \to \mathbb{L}_X$ is an equivalence; \item An $n$-shifted symplectic form on $X$ is a $n$-shifted closed $2$-form $\omega_X \colon k[-n] \to \closedforms 2(X)$ such that underlying $2$-form $k[-n] \to \closedforms 2(X) \to \forms 2(X)$ is non degenerate. \end{itemize} \end{df} \paragraph{Obstruction theory:} Let $A \in \cdgaunbounded^{\leq 0}_k$ and let $M \in \mathbf{dgMod}_A^{\leq -1}$ be an $A$-module concentrated in negative cohomological degrees. Let $d$ be a derivation $A \to A \oplus M$ and $s \colon A \to A \oplus M$ be the trivial derivation. The square zero extension of $A$ by $M[-1]$ twisted by $d$ is the fibre product \[ \shorthandoff{;:!?} \xymatrix{ A \oplus_d M[-1] \cart \ar[r]^-p \ar[d] & A \ar[d]^d \\ A \ar[r]^-s & A \oplus M } \] Let now $X$ be a derived stack and $M \in \mathbf{Qcoh}(X)^{\leq -1}$. We will denote by $X[M]$ the trivial square zero extension of $X$ by $M$. Let also $d \colon X[M] \to X$ be a derivation -- ie a retract of the natural map $X \to X[M]$. We define the square zero extension of $X$ by $M[-1]$ twisted by $d$ as the colimit \[ X_d[M[-1]] = \colim_{f \colon \operatorname{Spec} A \to X} \operatorname{Spec}(A \oplus_{f^d} f^ M[-1]) \] It is endowed with a natural morphism $X \to X_d[M[-1]]$ induced by the projections $p$ as above. \begin{prop}[Obstruction theory on stacks]\label{obstruction} Let $F \to G$ be an algebraic morphism of derived stacks. Let $X$ be a derived stack and let $M \in \mathbf{Qcoh}(X)^{\leq -1}$. Let $d$ be a derivation \[ d \in \operatorname{Map}_{X/-}(X[M], X) \] We consider the map of simplicial sets \[ \psi \colon \operatorname{Map}(X_d[M[-1]],F) \to \operatorname{Map}(X,F) \times_{\operatorname{Map}(X,G)} \operatorname{Map}(X_d[M[-1]],G) \] Let $y \in \operatorname{Map}(X,F) \times_{\operatorname{Map}(X,G)} \operatorname{Map}(X_d[M[-1]],G)$ and let $x \in \operatorname{Map}(X,F)$ be the induced map. There exists a point $\alpha(y) \in \operatorname{Map}(x^* \mathbb{L}_{F/G}, M)$ such that the fibre $\psi_y$ of $\psi$ at $y$ is equivalent to the space of paths from $0$ to $\alpha(y)$ in $\operatorname{Map}(x^* \mathbb{L}_{F/G}, M)$ \[ \psi_y \simeq \Omega_{0,\alpha(y)}\operatorname{Map}(x^* \mathbb{L}_{F/G}, M) \] \end{prop} \begin{proof} This is a simple generalisation of \cite[1.4.2.6]{toen:hagii}. The proof is very similar. We have a natural commutative square \[ \shorthandoff{;:!?} \xymatrix{ X[M] \ar[r]^-d \ar[d] & X \ar[d] \\ X \ar[r] & X_d[M[-1]] } \] It induces a map \[ \alpha \colon \operatorname{Map}(X,F) \times_{\operatorname{Map}(X,G)} \operatorname{Map}(X_d[M[-1]],G) \to \operatorname{Map}_{X/-/G}(X[M],F) \simeq \operatorname{Map}(x^* \mathbb{L}_{F/G},M) \] Let $\Omega_{0,\alpha(y)}\operatorname{Map}_{X/-/G}(X[M],F)$ denote the space of paths from $0$ to $\alpha(y)$. It is the fibre product \[ \shorthandoff{;:!?} \xymatrix{ \Omega_{0,\alpha(y)}\operatorname{Map}_{X/-/G}(X[M],F) \cart[][20] \ar[r] \ar[d] & {} \ar[d]^{\alpha(y)} \\ {} \ar[r]^-0 & \operatorname{Map}_{X/-/G}(X[M],F) } \] The composite map $\alpha \psi$ is by definition homotopic to the $0$ map. This defines a morphism \[ f \colon \Omega_{0,\alpha(y)}\operatorname{Map}_{X/-/G}(X[M],F) \to \psi_y \] It now suffices to see that the category of $X$'s for which $f$ is an equivalence contains every derived affine scheme and is stable by colimits. The first assertion is exactly \cite[1.4.2.6]{toen:hagii} and the second one is trivial. \end{proof} \paragraph{Postnikov towers:} To any cdga $A$, one can associate its $n$-truncation $A_{\leq n}$ for some $n$. It is, by definition, the universal cdga with vanishing cohomology $\mathrm H^p(A_{\leq n})$ for $p < -n$ associated to $A$. The truncation comes with a canonical map $A \to A_{\leq n}$ so that one can form the diagram \[ A_{\leq 0} \from A_{\leq 1} \from \dots \] This induces a canonical morphism $A \to \lim_n A_{\leq n}$ which is an equivalence. This phenomenon has a counterpart when dealing with derived stacks. We denote by $\mathbf{cdga}_k^{[-n,0]}$ the category of cdga's with cohomology concentrated in degrees $-n$ to $0$. It comes with the fully faithful embedding $i_n \colon \mathbf{cdga}_k^{[-n,0]} \to \cdgaunbounded^{\leq 0}_k$. For any prestack $X \colon \cdgaunbounded^{\leq 0}_k \to \mathbf{sSets}$, we define its truncation $\tau_{\leq n} X$ as the restriction of $X$ to the category $\mathbf{cdga}_k^{[-n,0]}$. We will abuse notations and also denote by $\tau_{\leq n} X$ the functor obtained after left Kan extending along $i_n$. The prestack $\tau_{\leq n} X$ comes with a canonical morphism $\tau_{\leq n} X \to X$ which, as $n$ varies, assembles to define a canonical map \[ \colim_n \tau_{\leq n} X \to X \] Remark that this morphism is not necessarily an equivalence. We will study it in \autoref{coconnective}. \paragraph{Algebraisable stacks:} Let $X$ be a derived stack and $A$ be a cdga. Let $a = (\el{a}{p})$ be a sequence of elements of $A^0$ forming a regular sequence in $\mathrm H^0(A)$. Let $\quot{A}{\el{a^n}{p}}$ denote the Kozsul complex associated with the regular sequence $(\el{a^n}{p})$. It is endowed with a cdga structure. There is a canonical map \[ \psi(A)_a \colon \colim_n X\mathopen{}\mathclose\bgroup\originalleft( \textstyle \quot{A}{\el{a^n}{p}} \aftergroup\egroup\originalright) \to X\mathopen{}\mathclose\bgroup\originalleft(\lim_n \textstyle \quot{A}{\el{a^n}{p}} \aftergroup\egroup\originalright) \] This map is usually not an equivalence. \begin{df}\label{alg-diag} A derived stack $X$ is called algebraisable if for any $A$ and any regular sequence $a$ the map $\psi(A)_a$ is an equivalence. A map $f \colon X \to Y$ is called algebraisable if for any derived affine scheme $T$ and any map $T \to Y$, the fibre product $X \times_Y T$ is algebraisable. We will say that a derived stack $X$ has algebraisable diagonal if the diagonal morphism $X \to X \times X$ is algebraisable. \end{df} \begin{rmq} A derived stack $X$ has algebraisable diagonal if for any $A$ and $a$ the map $\psi(A)_a$ is fully faithful. One could also rephrase the definition of being algebraisable as follows. A stack is algebraisable if it does not detect the difference between \[ \colim_n \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( \textstyle \quot{A}{\el{a^n}{p}} \aftergroup\egroup\originalright) \text{~~~and~~~} \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( \lim_n \textstyle \quot{A}{\el{a^n}{p}} \aftergroup\egroup\originalright) \] \end{rmq} \begin{ex} Any derived affine scheme is algebraisable. Another important example of algebraisable stack is the stack of perfect complexes. In \cite{bhatt:algebraisable}, Bhargav Bhatt gives some more examples of algebraisable (non-derived) stacks -- although our definition slightly differs from his. He proves that any quasi-compact quasi-separated algebraic space is algebraisable and also provides with examples of non-algebraisable stacks. Let us name $\mathrm K(\mathbb{G}_m,2)$ -- the Eilenberg-Maclane classifying stack of $\mathbb{G}_m$ -- as an example of non-algebraisable stack. Algebraisability of Deligne-Mumford stacks is also look at in \cite{lurie:dagxii}. \end{ex} \section{Ind-pro-stacks}\label{chapterIP} Throughout this section, we will denote by $S$ a derived stack over some base field $k$ and by $\mathbf{dSt}_S$ the category of derived stack over the base $S$. \subsection{Cotangent complex of a pro-stack} \begin{df} A pro-stack over $S$ an object of $\Prou U \mathbf{dSt}_S$. \end{df} \begin{rmq} Note that the category $\Prou U \mathbf{dSt}_S$ is equivalent to the category of pro-stacks over $k$ with a morphism to $S$. \end{rmq} \begin{df}\label{iperf-df} Let $\mathbf{Perf} \colon \mathbf{dSt}_S^{\mathrm{op}} \to \inftyCatu U$ denote the functor mapping a stack to its category of perfect complexes. We will denote by $\operatorname{\mathbf{IPerf}}$ the functor \[ \operatorname{\mathbf{IPerf}} = \operatorname{\underline{\mathbf{Ind}}}^\mathbb U_{\mathbf{dSt}_S^{\mathrm{op}}}(\mathbf{Perf}) \colon (\Prou U\mathbf{dSt}_S)^{\mathrm{op}} \to \mathbf{Pr}^{\mathrm{L}}_\infty \] where $\operatorname{\underline{\mathbf{Ind}}}^\mathbb U$ was defined in \autoref{indext}. Whenever $X$ is a pro-stack, we will call $\operatorname{\mathbf{IPerf}}(X)$ the derived category of ind-complexes on $X$. It is $\mathbb U$-presentable. If $f \colon X \to Y$ is a map of pro-stacks, then the functor \[ \operatorname{\mathbf{IPerf}}(f) \colon \operatorname{\mathbf{IPerf}}(Y) \to \operatorname{\mathbf{IPerf}}(X) \] admits a right adjoint (as both the involved categories are presentable and the functor preserves all colimits). We will denote $f_\mathbf{I}^* = \operatorname{\mathbf{IPerf}}(f)$ and $f^\mathbf{I}_$ its right adjoint. \end{df} \begin{rmq} Let $X$ be a pro-stack and let $\bar X \colon K^{\mathrm{op}} \to \mathbf{dSt}_S$ denote a $\mathbb U$-small cofiltered diagram of whom $X$ is a limit in $\Prou U\mathbf{dSt}_S$. The derived category of ind-perfect complexes on $X$ is by definition the category \[ \operatorname{\mathbf{IPerf}}(X) = \Indu U(\colim \mathbf{Perf}(\bar X)) \] It thus follows from \cite[1.1.4.6 and 1.1.3.6]{lurie:halg} that $\operatorname{\mathbf{IPerf}}(X)$ is stable. Note that it is also equivalent to the colimit \[ \operatorname{\mathbf{IPerf}}(X) = \colim \operatorname{\mathbf{IPerf}}(\bar X) \in \PresLeftu V \] It is therefore equivalent to the limit of the diagram \[ \operatorname{\mathbf{IPerf}}_(\bar X) \colon K \to \mathbf{dSt}_S^{\mathrm{op}} \to \PresLeftu V \simeq (\PresRightu V)^{\mathrm{op}} \] An object $E$ in $\operatorname{\mathbf{IPerf}}(X)$ is therefore the datum of an object $E_k$ of $\operatorname{\mathbf{IPerf}}(\bar X(k))$ for each $k \in K$ and of some compatibilities between them. We will then have $E_k \simeq {p_k}_* E$ where $p_k \colon X \to \bar X(k)$ is the natural projection. \end{rmq} \begin{df} Let $X$ be a pro-stack. We define its derived category of pro-perfect complexes \[ \operatorname{\mathbf{PPerf}}(X) = \mathopen{}\mathclose\bgroup\originalleft( \operatorname{\mathbf{IPerf}}(X) \aftergroup\egroup\originalright) ^{\mathrm{op}} \] Recall that perfect complexes are precisely the dualizable objects in the category of quasi-coherent complexes. They therefore come with a duality equivalence $\mathbf{Perf}(-) \to^\sim (\mathbf{Perf}(-))^{\mathrm{op}}$. This gives rise to the equivalence \[ \operatorname{\mathbf{PPerf}}(X) \simeq \Prou U(\colim \mathbf{Perf}(\bar X)) \] whenever $\bar X \colon \operatorname{K}^{\mathrm{op}} \to \mathbf{dSt}_S$ is a cofiltered diagram of whom $X$ is a limit in $\Prou U\mathbf{dSt}_S$. \end{df} \begin{df} Let us define the functor $\Tateu U_\mathbf P \colon (\Prou U \mathbf{dSt}_S)^{\mathrm{op}} \to \mathbf{Cat}_\infty^{\mathbb V\mathrm{,st,id}}$ \[ \Tateu U_\mathbf P = \Tateextu U_{\mathbf{dSt}_S^{\mathrm{op}}}(\mathbf{Perf}) \] \end{df} \begin{rmq} The functor $\Tateu U_\mathbf P$ maps a pro-stack $X$ given by a diagram $\bar X \colon K^{\mathrm{op}} \to \mathbf{dSt}_S$ to the stable $(\infty,1)$-category \[ \Tateu U_\mathbf P(X) = \Tateu U(\colim \mathbf{Perf}(\bar X)) \] There is a canonical fully faithful natural transformation \[ \Tateu U_\mathbf P \to \Prou U \circ \operatorname{\mathbf{IPerf}} \] We also get a fully faithful \[ \Tateu U_\mathbf P \to \Indu U \circ \operatorname{\mathbf{PPerf}} \] \end{rmq} \begin{df} Let $\mathbf{Qcoh} \colon \mathbf{dSt}_S^{\mathrm{op}} \to \inftyCatu V$ denote the functor mapping a derived stack to its derived category of quasi-coherent sheaves. It maps a morphism between stacks to the appropriate pullback functor. We will denote by $\operatorname{\mathbf{IQcoh}}$ the functor \[ \operatorname{\mathbf{IQcoh}} = \operatorname{\underline{\mathbf{Ind}}}^\mathbb U_{\mathbf{dSt}_S^{\mathrm{op}}}(\mathbf{Qcoh}) \colon (\Prou U\mathbf{dSt}_S)^{\mathrm{op}} \to \inftyCatu V \] If $f \colon X \to Y$ is a map of pro-stacks, we will denote by $f^_\mathbf{I}$ the functor $\operatorname{\mathbf{IQcoh}}(f)$. We also define \[ \operatorname{\mathbf{IQcoh}}^{\leq 0} = \operatorname{\underline{\mathbf{Ind}}}^\mathbb U_{\mathbf{dSt}_S^{\mathrm{op}}}(\mathbf{Qcoh}^{\leq 0}) \] the functor of connective modules. \end{df} \begin{rmq} There is a fully faithful natural transformation $\operatorname{\mathbf{IPerf}} \to \operatorname{\mathbf{IQcoh}}$ ; for any map $f \colon X \to Y$ of pro-stacks, there is therefore a commutative diagram \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{\mathbf{IPerf}}(Y) \ar[r] \ar[d]_{f_\mathbf I^} & \operatorname{\mathbf{IQcoh}}(Y) \ar[d]^{f_\mathbf I^}\\ \operatorname{\mathbf{IPerf}}(X) \ar[r] & \operatorname{\mathbf{IQcoh}}(X) } \] The two functors denoted by $f_\mathbf I^$ are thus compatible. Let us also say that the functor \[ f_\mathbf I^* \colon \operatorname{\mathbf{IQcoh}}(Y) \to \operatorname{\mathbf{IQcoh}}(X) \] does \emph{not} need to have a right adjoint. We next show that it sometimes has one. \end{rmq} \begin{prop}\label{prop-iqcoh} Let $f \colon X \to Y$ be a map of pro-stacks. If $Y$ is actually a stack then the functor $f_\mathbf I^* \colon \operatorname{\mathbf{IQcoh}}(Y) \to \operatorname{\mathbf{IQcoh}}(X)$ admits a right adjoint. \end{prop} \begin{proof}[sketch of] For a complete proof, we refer to \cite[1.2.0.8]{hennion:these}. Let us denote by $\bar X \colon K^{\mathrm{op}} \to \mathbf{dSt}_S$ a cofiltered diagram of whom $X$ is a limit in $\Prou U \mathbf{dSt}_S$. The map $X \to Y$ factors through the projection $X \to \bar X(k)$ for some $k \in K$. The right adjoint of $f_\mathbf I^$ is then (informally) given by the limit \[ \lim_{k \to l} \bar f(l)_ \] where $\bar f(l)_$ is the right adjoint to the induced functor $\bar f(l)^ \colon \operatorname{\mathbf{IQcoh}}(Y) \to \operatorname{\mathbf{IQcoh}}(\bar X(l))$. \end{proof} \newcommand{\mathbf{IQ}}{\mathbf{IQ}} \begin{df} Let $f \colon X \to Y$ be a map of pro-stacks. We will denote by $f_^\mathbf{IQ}$ the right adjoint to $f_\mathbf I^ \colon \operatorname{\mathbf{IQcoh}}(Y) \to \operatorname{\mathbf{IQcoh}}(X)$ \emph{if it exists}. \end{df} \begin{rmq} In the situation of \autoref{prop-iqcoh}, there is a natural transformation \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{\mathbf{IPerf}}(X) \ar[r] \ar[d]_{f_^\mathbf I} & \operatorname{\mathbf{IQcoh}}(X) \ar[d]^{f_^\mathbf{IQ}} \\ \operatorname{\mathbf{IPerf}}(Y) \ar@{=>}[ur] \ar[r] & \operatorname{\mathbf{IQcoh}}(Y) } \] It does not need to be an equivalence. \end{rmq} \begin{df} Let $X$ be a pro-stack over $S$. The structural sheaf $\mathcal{O}_X$ of $X$ is the pull-back of $\mathcal{O}_S$ along the structural map $X \to S$. \end{df} \begin{ex} Let $X$ be a pro-stack over $S$ and $\bar X \colon K^{\mathrm{op}} \to \mathbf{dSt}_S$ be a $\mathbb U$-small cofiltered diagram of whom $X$ is a limit in $\Prou U\mathbf{dSt}_S$. Let $k$ be a vertex of $K$, let $X_k$ denote $\bar X(k)$ and let $p_k$ denote the induced map of pro-stacks $X \to X_k$. If $f \colon k \to l$ is an arrow in $K$, we will also denote by $f$ the map of stacks $\bar X(f)$. We have \[ (p_k)_^\mathbf{IQ} (\mathcal{O}_X) \simeq \colim_{f \colon k \to l} f_ \mathcal{O}_{X_l} \] One can see this using \cite[1.2.0.7]{hennion:these}. \end{ex} \begin{df} For any category $\mathcal{C}$ with finite colimits, we will denote by $\operatorname{\mathrm{B}}^{\amalg}_\mathcal{C}$ the functor $\mathcal{C}^{\Delta^1} \to \inftyCatu V$ mapping a morphism $\phi \colon c \to d$ to the category of factorizations $c \to e \to d$ of $\phi$. For a formal definition in the context of $(\infty,1)$-categories, we refer to \cite[1.3.0.14]{hennion:these}. \end{df} \newcommand{\operatorname{Ex}}{\operatorname{Ex}} \begin{df}\label{derivation-pdst} Let $T$ be a stack over $S$. Let us consider the functor \[ \mathbf{Qcoh}(T)^{\leq 0} \to \operatorname{\mathrm{B}}_{\mathbf{dSt}_S^{\mathrm{op}}}^{\amalg}(\operatorname{id}_T) \simeq \mathopen{}\mathclose\bgroup\originalleft(\comma{T}{\mathbf{dSt}_T}\aftergroup\egroup\originalright)^{\mathrm{op}} \] mapping a quasi-coherent sheaf $E$ to the square zero extension $T \to T[E] \to T$. This construction is functorial in $T$ and actually comes from a natural transformation \[ \operatorname{Ex} \colon \mathbf{Qcoh}^{\leq 0} \to \operatorname{\mathrm{B}}_{\mathbf{dSt}_S^{\mathrm{op}}}^{\amalg}(\operatorname{id}_-) \] of functors $\mathbf{dSt}_S^{\mathrm{op}} \to \inftyCatu V$. We will denote by $\operatorname{Ex}^{\operatorname{\mathbf{Pro}}}$ the natural transformation \[ \operatorname{Ex}^{\operatorname{\mathbf{Pro}}} = \operatorname{\underline{\mathbf{Ind}}}^\mathbb U_{\mathbf{dSt}_S^{\mathrm{op}}}(\operatorname{Ex}) \colon \operatorname{\mathbf{IQcoh}}^{\leq 0} \to \Indextu U_{\mathbf{dSt}_S^{\mathrm{op}}}(\operatorname{\mathrm{B}}_{\mathbf{dSt}_S^{\mathrm{op}}}^{\amalg}(\operatorname{id}_-)) \simeq \operatorname{\mathrm{B}}_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}}^{\amalg}(\operatorname{id}_-) \] between functors $(\Prou U\mathbf{dSt}_S)^{\mathrm{op}} \to \mathbf{Cat}_\infty$. The equivalence on the right is the one from \cite[1.3.0.18]{hennion:these}. If $X$ is a pro-stack and $E \in \operatorname{\mathbf{IQcoh}}(X)^{\leq 0}$ then we will denote by $X \to X[E] \to X$ the image of $E$ by the functor $\operatorname{Ex}^{\operatorname{\mathbf{Pro}}}(X)$. \end{df} \begin{rmq} Let us give a description of this functor. Let $X$ be a pro-stack and let $\bar X \colon K^{\mathrm{op}} \to \mathbf{dSt}_S$ denote a $\mathbb U$-small cofiltered diagram of whom $X$ is a limit in $\Prou U\mathbf{dSt}_S$. For every $k \in K$ we can compose the functor mentioned above with the base change functor \[ \shorthandoff{;:!?} \xymatrix{ (\mathbf{Qcoh}(X_k))^{\mathrm{op}} \ar[r]^-{X_k[-]} & \comma{X_k}{\mathbf{dSt}_{X_k}} \ar[r]^-{- \times_{X_k} X} & \comma{X}{\Prou U\mathbf{dSt}_X} } \] This is functorial in $k$ and we get a functor $\mathopen{}\mathclose\bgroup\originalleft(\colim \mathbf{Qcoh}(\bar X) \aftergroup\egroup\originalright)^{\mathrm{op}} \to \comma{X}{\Prou U\mathbf{dSt}_X}$ which we extend and obtain a more explicit description of the square zero extension functor \[ X[-] \colon (\operatorname{\mathbf{IQcoh}}(X))^{\mathrm{op}} \to \comma{X}{\Prou U\mathbf{dSt}_X} \] \end{rmq} \begin{df} Let $X$ be a pro-stack. \begin{itemize} \item We finally define the functor of derivations over $X$ : \[ \operatorname{Der}(X,-) = \operatorname{Map}_{X/-/S}(X[-],X) \colon \operatorname{\mathbf{IQcoh}}(X)^{\leq 0} \to \mathbf{sSets} \] \item We say that $X$ admits a cotangent complex if the functor $\operatorname{Der}(X,-)$ is corepresentable -- ie there exists a $\mathbb{L}_{X/S} \in \operatorname{\mathbf{IQcoh}}(X)$ such that for any $E \in \operatorname{\mathbf{IQcoh}}(X)^{\leq 0}$ \[ \operatorname{Der}(X,E) \simeq \operatorname{Map}(\mathbb{L}_{X/S},E) \] \end{itemize} \end{df} \begin{df} Let $\dSt^{\mathrm{Art}}_S$ be the full sub-category of $\mathbf{dSt}_S$ spanned by derived Artin stacks over $S$. An Artin pro-stack is an object of $\Prou U \dSt^{\mathrm{Art}}_S$. Let $\dSt^{\mathrm{Art,lfp}}_S$ be the full sub-category of $\dSt^{\mathrm{Art}}_S$ spanned by derived Artin stacks locally of finite presentation over $S$. An Artin pro-stack locally of finite presentation is an object of $\Prou U\dSt^{\mathrm{Art,lfp}}_S$ \end{df} \begin{prop}\label{cotangent-pdst} Any Artin pro-stack $X$ over $S$ admits a cotangent complex $\mathbb{L}_{X/S}$. Let us assume that $\bar X \colon K^{\mathrm{op}} \to \dSt^{\mathrm{Art}}_S$ is a $\mathbb U$-small cofiltered diagram of whom $X$ is a limit in $\Prou U\dSt^{\mathrm{Art}}_S$. When $k$ is a vertex of $K$, let us denote by $X_k$ the derived Artin stack $\bar X(k)$. If $f \colon k \to l$ is an arrow in $K$, we will also denote by $f \colon X_l \to X_k$ the map of stacks $\bar X(f)$. The cotangent complex is given by the formula \[ \mathbb{L}_{X/S} = \colim_k p_k^* \mathbb{L}_{X_k/S} \in \Indu U\mathopen{}\mathclose\bgroup\originalleft(\colim \mathbf{Qcoh}(\bar X) \aftergroup\egroup\originalright) \simeq \operatorname{\mathbf{IQcoh}}(X) \] where $p_k$ is the canonical map $X \to X_k$. The following formula stands \[ {p_k}_^\mathbf{IQ} \mathbb{L}_{X/S} \simeq \colim_{f \colon k \to l} f_ \mathbb{L}_{X_l/S} \] If $X$ is moreover locally of finite presentation over $S$, then its cotangent complex belongs to $\operatorname{\mathbf{IPerf}}(X)$. \end{prop} Before proving this proposition, let us fix the following notation \begin{df} Let $\mathcal{C}$ be a full sub-category of an $\infty$-category $\mathcal{D}$. There is a natural transformation from $\operatorname{O}_\mathcal{D} \colon d \mapsto \quot{\mathcal{D}}{d}$ to the constant functor $\mathcal{D} \colon \mathcal{D} \to \mathbf{Cat}_\infty$. We denote by $\operatorname{O}_\mathcal{D}^\mathcal{C}$ the fiber product \[ \operatorname{O}_\mathcal{D}^\mathcal{C} = \operatorname{O}_\mathcal{D} \times_\mathcal{D} \mathcal{C} \colon \mathcal{D} \to \mathbf{Cat}_\infty \] \end{df} \begin{rmq} The functor $\operatorname{O}_\mathcal{D}^\mathcal{C} \colon \mathcal{D} \to \mathbf{Cat}_\infty$ maps an object $d \in \mathcal{D}$ to the comma category of objects in $\mathcal{C}$ over $d$ \[ \quot{\mathcal{C}}{d} = (\mathcal{C} \times \{d\} ) \timesunder[\mathcal{D} \times \mathcal{D}] \mathcal{D}^{\Delta^1} \] \end{rmq} \begin{proof}[of the proposition] The cotangent complex defines a natural transformation \[ \lambda \colon \operatorname{O}_{\mathbf{dSt}_S^{\mathrm{op}}}^{(\dSt^{\mathrm{Art}}_S)^{\mathrm{op}}} \to \mathbf{Qcoh}(-) \] To any stack $T$ and any Artin stack $U$ over $S$ with a map $f \colon T \to U$, it associates the quasi-coherent complex $f^* \mathbb{L}_{U/S}$ on $T$. Applying the functor $\operatorname{\underline{\mathbf{Ind}}}^\mathbb U_{\mathbf{dSt}_S^{\mathrm{op}}}$ we get a natural transformation $\lambda^{\operatorname{\mathbf{Pro}}}$ \[ \lambda^{\operatorname{\mathbf{Pro}}} = \operatorname{\underline{\mathbf{Ind}}}^\mathbb U_{\mathbf{dSt}_S^{\mathrm{op}}}(\lambda) \colon \operatorname{O}^{(\Prou U\dSt^{\mathrm{Art}}_S)^{\mathrm{op}}}_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}} \to \operatorname{\mathbf{IQcoh}}(-) \] Specifying it to $X$ we get a functor \[ \lambda^{\operatorname{\mathbf{Pro}}}_X \colon \mathopen{}\mathclose\bgroup\originalleft( \comma{X}{\Prou U\dSt^{\mathrm{Art}}_S} \aftergroup\egroup\originalright)^{\mathrm{op}} \to \operatorname{\mathbf{IQcoh}}(X) \] Let us set $\mathbb{L}_{X/S} =\lambda^{\operatorname{\mathbf{Pro}}}_X(X) \in \operatorname{\mathbf{IQcoh}}(X)$. We have by definition the equivalence \[ \mathbb{L}_{X/S} \simeq \colim_k p_k^* \mathbb{L}_{X_k/S} \] Let us now check that it satisfies the required universal property. The functor $\operatorname{Der}(X,-)$ is the limit of the diagram $K^{\mathrm{op}} \to \operatorname{Fct}(\operatorname{\mathbf{IQcoh}}(X)^{\leq 0}, \mathbf{sSets})$ \[ \operatorname{Map}_{X/-/S}(X[-], \bar X) \] Fixing $k \in K$, the functor $\operatorname{Map}_{X/-/S}(X[-], X_k) \colon \operatorname{\mathbf{IQcoh}}(X)^{\leq 0} \simeq \Indu U(\colim \mathbf{Qcoh}(\bar X)^{\leq 0}) \to \mathbf{sSets}$ preserves filtered colimits. It is hence induced by its restriction to $\colim \mathbf{Qcoh}(\bar X)^{\leq 0}$. It follows that the diagram $\operatorname{Map}_{X/-/S}(X[-], \bar X)$ factors through a diagram \[ \delta \colon K^{\mathrm{op}} \to \operatorname{Fct}\mathopen{}\mathclose\bgroup\originalleft(\colim \mathbf{Qcoh}(\bar X)^{\leq 0}, \mathbf{sSets}\aftergroup\egroup\originalright) \simeq \lim \operatorname{Fct}(\mathbf{Qcoh}(\bar X)^{\leq 0}, \mathbf{sSets}) \] Similarly, the functor $\operatorname{Map}(\mathbb{L}_{X/S},-)$ is the limit of a diagram \[ \shorthandoff{;:!?} \xymatrix{K^{\mathrm{op}} \ar[r]^-\mu & \lim \operatorname{Fct}(\mathbf{Qcoh}(\bar X)^{\leq 0}, \mathbf{sSets}) \ar[r] & \operatorname{Fct}(\operatorname{\mathbf{IQcoh}}(X)^{\leq 0}, \mathbf{sSets})} \] The universal property of the usual cotangent complex defines an equivalence between $\delta$ and $\mu$. To get the formula for ${p_k}_^\mathbf{IQ} \mathbb{L}_{X/S}$, one uses \cite[1.2.0.7]{hennion:these} and the last statement is obvious. \end{proof} \begin{rmq} The definition of the derived category of ind-quasi-coherent modules on a pro-stack is build for the above proposition and remark to hold. \end{rmq} \begin{rmq}\label{cotangent-prostacks} We have actually proven that for any pro-stack $X$, the two functors \[ \operatorname{\mathbf{IQcoh}}(X)^{\leq 0} \times \comma{X}{\dSt^{\mathrm{Art}}_S} \to \mathbf{sSets} \] defined by \begin{align} (E,Y) &\mapsto \operatorname{Map}_{X/-/S}(X[E], Y) \\ (E,Y) &\mapsto \operatorname{Map}_{\operatorname{\mathbf{IQcoh}}(X)}(\lambda_X^{\operatorname{\mathbf{Pro}}}(Y), E) \end{align} are equivalent. \end{rmq} \subsection{Cotangent complex of an ind-pro-stack} \begin{df} An ind-pro-stack is an object of the category \[ \mathbf{IP}\mathbf{dSt}_S = \Indu U \Prou U\mathbf{dSt}_S \] \end{df} \begin{df} Let us define the functor $\operatorname{\mathbf{PIPerf}} \colon (\mathbf{IP}\mathbf{dSt}_S)^{\mathrm{op}} \to \inftyCatu V$ as \[ \operatorname{\mathbf{PIPerf}} = \operatorname{\underline{\mathbf{Pro}}}^\mathbb U_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}}(\operatorname{\mathbf{IPerf}}) \] where $\operatorname{\underline{\mathbf{Pro}}}^\mathbb U$ was defined in \autoref{indext}. Whenever we have a morphism $f \colon X \to Y$ of ind-pro-stacks, we will denote by $f^_\mathbf{PI}$ the functor \[f^_\mathbf{PI} = \operatorname{\mathbf{PIPerf}}(f) \colon \operatorname{\mathbf{PIPerf}}(Y) \to \operatorname{\mathbf{PIPerf}}(X) \] \end{df} \begin{rmq} Let $X$ be an ind-pro-stack. Let $\bar X \colon K \to \Prou U\mathbf{dSt}_S$ denote a $\mathbb U$-small filtered diagram of whom $X$ is a colimit in $\mathbf{IP}\mathbf{dSt}_S$. We have by definition \[ \operatorname{\mathbf{PIPerf}}(X) \simeq \lim \Prou U(\operatorname{\mathbf{IPerf}}(\bar X)) \] \end{rmq} \begin{prop}\label{prop-piperf-right-adjoint} Let $f \colon X \to Y$ be a map of ind-pro-stacks. If $Y$ is a pro-stack then the functor $f^_\mathbf{PI} \colon \operatorname{\mathbf{PIPerf}}(Y) \to \operatorname{\mathbf{PIPerf}}(X)$ admits a right adjoint. \end{prop} \begin{df} Let $f \colon X \to Y$ be a map of ind-pro-stacks. If the functor \[ f^_\mathbf{PI} \colon \operatorname{\mathbf{PIPerf}}(Y) \to \operatorname{\mathbf{PIPerf}}(X) \] admits a right adjoint, we will denote it by $f^\mathbf{PI}_$. \end{df} \begin{proof}[of the proposition] If both $X$ and $Y$ are pro-stacks, then $f^\mathbf{PI}_* = \Prou U(f^\mathbf I_)$ is right adjoint to $f_\mathbf{PI}^ = \Prou U(f_\mathbf I^)$. Let now $X$ be an ind-pro-stack and let $\bar X \colon K \to \Prou U\mathbf{dSt}_S$ denote a $\mathbb U$-small filtered diagram of whom $X$ is a colimit in $\mathbf{IP}\mathbf{dSt}_S$. We then have \[ f_\mathbf{PI}^ \colon \operatorname{\mathbf{PIPerf}}(Y) \to \operatorname{\mathbf{PIPerf}}(X) \simeq \lim \operatorname{\mathbf{PIPerf}}(\bar X) \] The right adjoint is the informally given by the formula \[ f^\mathbf{PI}_* = \lim_k \bar f(k)^\mathbf{PI}_* \] where $\bar f(k)$ is the induced map $\bar X(k) \to Y$. For a formal proof, we refer to \cite[1.2.0.5]{hennion:these}. \end{proof} \begin{df} Let $X \in \mathbf{IP}\mathbf{dSt}_S$. We define $\operatorname{\mathbf{IPPerf}}(X) = (\operatorname{\mathbf{PIPerf}}(X))^{\mathrm{op}}$. If $X$ is the colimit in $\mathbf{IP}\mathbf{dSt}_S$ of a filtered diagram $K \to \Prou U\mathbf{dSt}_S$ then we have \[ \operatorname{\mathbf{IPPerf}}(X) \simeq \lim (\Indu U \circ \operatorname{\mathbf{PPerf}} \circ \bar X) \] We will denote by $\dual{(-)} \colon \operatorname{\mathbf{IPPerf}}(X) \to (\operatorname{\mathbf{PIPerf}}(X))^{\mathrm{op}}$ the duality functor. \end{df} \begin{df} Let us define the functor $\Tateu U_\mathbf{IP} \colon (\mathbf{IP}\mathbf{dSt}_S)^{\mathrm{op}} \to \mathbf{Cat}_\infty^{\mathbb V\mathrm{,st,id}}$ as the right Kan extension of $\Tateu U_\mathbf P$ along the inclusion $(\Prou U \mathbf{dSt}_S)^{\mathrm{op}} \to (\mathbf{IP}\mathbf{dSt}_S)^{\mathrm{op}}$. It is by definition endowed with a canonical fully faithful natural transformation \[ \Tateu U_\mathbf{IP} \to \operatorname{\mathbf{PIPerf}} \] For any $X \in \mathbf{IP}\mathbf{dSt}_S$, an object of $\Tateu U_\mathbf{IP}(X)$ will be called a Tate module on $X$. \end{df} \begin{rmq} We can characterise Tate objects: a module $E \in \operatorname{\mathbf{PIPerf}}(X)$ is a Tate module if and only if for any pro-stack $U$ and any morphism $f \colon U \to X \in \mathbf{IP}\mathbf{dSt}_S$, the pullback $f^_\mathbf{IP}(E)$ is in $\Tateu U_\mathbf P(U)$. Let us also remark here that \end{rmq} \begin{lem}\label{ipdst-tate-in-ipp} Let $X$ be an ind-pro-stack over $S$. The fully faithful functors \[ \shorthandoff{;:!?} \xymatrix{ \Tateu U_\mathbf{IP}(X) \ar[r] & \operatorname{\mathbf{PIPerf}}(X) \ar@{=}[r]^-{\dual{(-)}}& (\operatorname{\mathbf{IPPerf}}(X))^{\mathrm{op}} & \mathopen{}\mathclose\bgroup\originalleft(\Tateu U_\mathbf{IP}(X)\aftergroup\egroup\originalright)^{\mathrm{op}} \ar[l] } \] have the same essential image. We thus have an equivalence \[ \dual{(-)} \colon \Tateu U_\mathbf{IP}(X) \simeq \mathopen{}\mathclose\bgroup\originalleft(\Tateu U_\mathbf{IP}(X)\aftergroup\egroup\originalright)^{\mathrm{op}} \] \end{lem} \begin{proof} This is a corollary of \autoref{tate-dual}. \end{proof} \begin{df} Let us define $\operatorname{\mathbf{PIQcoh}} \colon (\mathbf{IP}\mathbf{dSt}_S)^{\mathrm{op}} \to \inftyCatu V$ to be the functor \[ \operatorname{\mathbf{PIQcoh}} = \operatorname{\underline{\mathbf{Pro}}}^\mathbb U_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}}(\operatorname{\mathbf{IQcoh}}) \] From \autoref{indext-monoidal}, for any ind-pro-stack $X$, the category $\operatorname{\mathbf{PIQcoh}}(X)$ admits a natural monoidal structure. We also define the subfunctor \[ \operatorname{\mathbf{PIQcoh}}^{\leq 0} = \operatorname{\underline{\mathbf{Pro}}}^\mathbb U_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}}(\operatorname{\mathbf{IQcoh}}^{\leq 0}) \] \end{df} \begin{rmq} Let us give an informal description of the above definition. To an ind-pro-stack $X = \colim_\alpha \lim_\beta X_{\alpha\beta}$ we associate the category \[ \operatorname{\mathbf{PIQcoh}}(X) = \lim_\alpha \Prou U \Indu U\mathopen{}\mathclose\bgroup\originalleft(\colim_\beta \mathbf{Perf}(X_{\alpha\beta})\aftergroup\egroup\originalright) \] \end{rmq} \begin{df} Let $f \colon X \to Y$ be a map of ind-pro-stacks. We will denote by $f_\mathbf{PI}^$ the functor $\operatorname{\mathbf{PIQcoh}}(f)$. Whenever it exists, we will denote by $f^\mathbf{PIQ}_$ the right adjoint to $f_\mathbf{PI}^$. \end{df} \begin{prop}\label{prop-piqcoh-right-adjoint} Let $f \colon X \to Y$ be a map of ind-pro-stacks. If $Y$ is actually a stack, then the induced functor $f^_\mathbf{PI}$ admits a right adjoint. \end{prop} \begin{proof} This is very similar to the proof of \autoref{prop-piperf-right-adjoint} but using \autoref{prop-iqcoh}. \end{proof} \begin{rmq} There is a fully faithful natural transformation $\operatorname{\mathbf{PIPerf}} \to \operatorname{\mathbf{PIQcoh}}$. Using the same notation $f_\mathbf{PI}^$ for the images of a map $f \colon X \to Y$ is therefore only a small abuse. Moreover, for any such map $f \colon X \to Y$, for which the right adjoints drawn below exist, there is a natural tranformation \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{\mathbf{PIPerf}}(Y) \ar[r] \ar[d]_{f^\mathbf{PI}_} & \operatorname{\mathbf{PIQcoh}}(Y) \ar[d]^{f^\mathbf{PIQ}_} \\ \operatorname{\mathbf{PIPerf}}(X) \ar[r] \ar@{=>}[ur] & \operatorname{\mathbf{PIQcoh}}(X) } \] It is generally not an equivalence. \end{rmq} \begin{df} Let $\operatorname{Ex}^{\mathbf{IP}}$ denote the natural transformation $\operatorname{\underline{\mathbf{Pro}}}^\mathbb U_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}}(\operatorname{Ex}^{\operatorname{\mathbf{Pro}}})$ \[ \operatorname{Ex}^{\mathbf{IP}} \colon \operatorname{\mathbf{PIQcoh}}^{\leq 0} \to \operatorname{\underline{\mathbf{Pro}}}^\mathbb U_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}} \mathopen{}\mathclose\bgroup\originalleft( \operatorname{\mathrm{B}}_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}}^{\amalg}(\operatorname{id}_-) \aftergroup\egroup\originalright) \simeq \operatorname{\mathrm{B}}_{(\mathbf{IP}\mathbf{dSt}_S)^{\mathrm{op}}}^{\amalg}(\operatorname{id}_-) \] of functors $(\mathbf{IP}\mathbf{dSt}_S)^{\mathrm{op}} \to \mathbf{Cat}_\infty$. If $X$ is an ind-pro-stack and $E \in \operatorname{\mathbf{PIQcoh}}(X)^{\leq 0}$ then we will denote by $X \to X[E] \to X$ the image of $E$ by the functor \[ \operatorname{Ex}^{\mathbf{IP}}(X) \colon \operatorname{\mathbf{PIQcoh}}(X)^{\leq 0} \to \mathopen{}\mathclose\bgroup\originalleft(\comma{X}{\mathbf{IP}\mathbf{dSt}_X}\aftergroup\egroup\originalright)^{\mathrm{op}} \] \end{df} \begin{rmq} Let us decipher the above definition. Let $X = \colim_\alpha \lim_\beta X_{\alpha\beta}$ be an ind-pro-stack and let $E$ be a pro-ind-module over it. By definition $E$ is the datum, for every $\alpha$, of a pro-ind-object $E^\alpha$ in the category $\Prou U \Indu U (\colim_\beta \mathbf{Qcoh}^{\leq 0}(X_{\alpha\beta}))$. Let us denote $E^\alpha = \lim_\gamma \colim_\delta E^\alpha_{\gamma\delta}$. For any $\gamma$ and $\delta$, there is a $\beta_0(\gamma,\delta)$ such that $E^\alpha_{\gamma\delta}$ is in the essential image of $\mathbf{Qcoh}^{\leq 0} (X_{\alpha\beta_0(\gamma,\delta)})$. We then have \[ X[E] = \colim_{\alpha,\gamma} \lim_{\delta} \lim_{\beta \geq \beta_0(\gamma,\delta)} X_{\alpha\beta}[E_{\gamma\delta}] \in \mathbf{IP}\mathbf{dSt}_S \] \end{rmq} \begin{df}\label{derivation-ipdst} Let $X$ be an ind-pro-stack. \begin{itemize} \item We define the functor of derivations on $X$ \[ \operatorname{Der}(X,-) = \operatorname{Map}_{X/-/S}(X[-],X) \] \item We say that $X$ admits a cotangent complex if there exists $\mathbb{L}_{X/S} \in \operatorname{\mathbf{PIQcoh}}(X)$ such that for any $E \in \operatorname{\mathbf{PIQcoh}}(X)^{\leq 0}$ \[ \operatorname{Der}(X,E) \simeq \operatorname{Map}(\mathbb{L}_{X/S},E) \] \item Let us assume that $f \colon X \to Y$ is a map of ind-pro-stacks and that $Y$ admits a cotangent complex. We say that $f$ is formally étale if $X$ admits a cotangent complex and the natural map $f^* \mathbb{L}_{Y/S} \to \mathbb{L}_{X/S}$ is an equivalence. \end{itemize} \end{df} \begin{df} An Artin ind-pro-stack over $S$ is an object in the category \[ \mathbf{IP}\dSt^{\mathrm{Art}}_S = \Indu U \Prou U\dSt^{\mathrm{Art}}_S \] An Artin ind-pro-stack locally of finite presentation is an object of \[ \mathbf{IP}\dSt^{\mathrm{Art,lfp}}_S = \Indu U \Prou U\dSt^{\mathrm{Art,lfp}}_S \] \end{df} \begin{prop}\label{ipcotangent} Any Artin ind-pro-stack $X$ admits a cotangent complex \[ \mathbb{L}_{X/S} \in \operatorname{\mathbf{PIQcoh}}(X) \] Let us assume that $\bar X \colon K \to \operatorname{\mathbf{Pro}}\dSt^{\mathrm{Art}}_S$ is a $\mathbb U$-small filtered diagram of whom $X$ is a colimit in $\mathbf{IP}\dSt^{\mathrm{Art}}_S$. For any vertex $k \in K$ we will denote by $X_k$ the pro-stack $\bar X(k)$ and by $i_k$ the structural map $X_k \to X$. For any $f \colon k \to l$ in $K$, let us also denote by $f$ the induced map $X_k \to X_l$. We have for all $k \in K$ \[ i_{k,\mathbf{PI}}^* \mathbb{L}_{X/S} \simeq \lim_{f \colon k \to l} f^_\mathbf I \mathbb{L}_{X_l/S} \in \operatorname{\mathbf{PIQcoh}}(X_k) \] If moreover $X$ is locally of finite presentation then $\mathbb{L}_{X/S}$ belongs to $\operatorname{\mathbf{PIPerf}}(X)$. \end{prop} \begin{proof} Let us recall the natural transformation $\lambda^{\operatorname{\mathbf{Pro}}}$ from the proof of \autoref{cotangent-pdst} \[ \lambda^{\operatorname{\mathbf{Pro}}} = \operatorname{\underline{\mathbf{Ind}}}^\mathbb U_{\mathbf{dSt}_S^{\mathrm{op}}}(\lambda) \colon \operatorname{O}^{(\Prou U\dSt^{\mathrm{Art}}_S)^{\mathrm{op}}}_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}} \to \operatorname{\mathbf{IQcoh}}(-) \] of functors $(\Prou U\mathbf{dSt}_S)^{\mathrm{op}} \to \mathbf{Cat}_\infty$. Applying the functor $\operatorname{\underline{\mathbf{Pro}}}^\mathbb U_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}}$ we define the natural transformation $\lambda^{\mathbf{IP}}$ \[ \lambda^{\mathbf{IP}} = \operatorname{\underline{\mathbf{Pro}}}^\mathbb U_{(\Prou U\mathbf{dSt}_S)^{\mathrm{op}}} \mathopen{}\mathclose\bgroup\originalleft( \lambda^{\operatorname{\mathbf{Pro}}} \aftergroup\egroup\originalright) \colon \operatorname{O}_{(\mathbf{IP}\mathbf{dSt}_S)^{\mathrm{op}}}^{(\mathbf{IP}\dSt^{\mathrm{Art}}_S)^{\mathrm{op}}} \to \operatorname{\mathbf{PIQcoh}}(-) \] between functors $(\mathbf{IP}\mathbf{dSt}_S)^{\mathrm{op}} \to \mathbf{Cat}_\infty$. Specifying to $X$ we get a functor \[ \lambda^{\mathbf{IP}}_X \colon \mathopen{}\mathclose\bgroup\originalleft(\comma{X}{\mathbf{IP}\dSt^{\mathrm{Art}}_S}\aftergroup\egroup\originalright) ^{\mathrm{op}} \to \operatorname{\mathbf{PIQcoh}}(X) \] We now define $\mathbb{L}_{X/S} = \lambda^{\mathbf{IP}}_X(X)$. By definition we have \[ i_{k,\mathbf{PI}}^ \mathbb{L}_{X/S} \simeq \lim \lambda^{\operatorname{\mathbf{Pro}}}_{X_k}(\bar X) \simeq \lim_{f \colon k \to l} f^_\mathbf I \mathbb{L}_{X_l/S} \] for every $k \in K$. Let us now prove that it satisfies the expected universal property. It suffices to compare for every $k \in K$ the functors \[ \operatorname{Map}_{X_k/-/S}(X_k[-], X) \hspace{1cm} \text{and} \hspace{1cm} \operatorname{Map}_{\operatorname{\mathbf{PIQcoh}}(X_k)}(i_{k,\mathbf{PI}}^\mathbb{L}_{X/S}, -) \] defined on $\operatorname{\mathbf{PIQcoh}}(X_k)^{\leq 0}$. They are both pro-extensions to $\operatorname{\mathbf{PIQcoh}}(X_k)^{\leq 0}$ of their restrictions $\operatorname{\mathbf{IQcoh}}(X_k)^{\leq 0} \to \mathbf{sSets}$. The restricted functor $\operatorname{Map}_{X_k/-/S}(X_k[-], X)$ is a colimit of the diagram \[ \operatorname{Map}_{X_k/-/S}(X_k[-], \bar X) \colon \mathopen{}\mathclose\bgroup\originalleft(\comma{k}{K}\aftergroup\egroup\originalright)^{\mathrm{op}} \to \operatorname{Fct}(\operatorname{\mathbf{IQcoh}}(X_k)^{\leq 0}, \mathbf{sSets}) \] while $\operatorname{Map}_{\operatorname{\mathbf{PIQcoh}}(X_k)}(i_{k,\mathbf{PI}}^\mathbb{L}_{X/S}, -)$ is a colimit to the diagram \[ \operatorname{Map}_{\operatorname{\mathbf{IQcoh}}(X_k)}(\lambda^{\operatorname{\mathbf{Pro}}}_{X_k}(\bar X), -) \colon \mathopen{}\mathclose\bgroup\originalleft(\comma{k}{K}\aftergroup\egroup\originalright)^{\mathrm{op}} \to \operatorname{Fct}(\operatorname{\mathbf{IQcoh}}(X_k)^{\leq 0},\mathbf{sSets}) \] We finish the proof with \autoref{cotangent-prostacks}. \end{proof} Let us record here a technical result we will use later on. For a proof, we refer to \cite[2.1.2.20]{hennion:these}. \begin{prop}\label{IPcotangent-underlying} Let $X \in \mathbf{IP}\dSt^{\mathrm{Art}}_S$. Let us denote by $\pi \colon X \to S$ the structural map. Let also $\tilde \mathbb{L}^\mathbf{IP}$ denote the functor \[ \mathopen{}\mathclose\bgroup\originalleft(\mathbf{IP}\dSt^{\mathrm{Art}}_S\aftergroup\egroup\originalright)^{\mathrm{op}} \to \Prou U \Indu U \mathbf{Qcoh}(S) \] obtained by extending the functor $(\dSt^{\mathrm{Art}}_S)^{\mathrm{op}} \to \mathbf{Qcoh}(S)$ mapping $f \colon T \to S$ to $f_ \mathbb{L}_{T/S}$. Then we have $\pi_^\mathbf{PIQ} \mathbb{L}_{X/S} \simeq \tilde \mathbb{L}^{\mathbf{IP}} (X)$ \end{prop} \begin{df} Let $X$ by an Artin ind-pro-stack locally of finite presentation over $S$. We will call the tangent complex of $X$ the ind-pro-perfect complex on $X$ \[ \mathbb{T}_{X/S} = \dual \mathbb{L}_{X/S} \in \operatorname{\mathbf{IPPerf}}(X) \] \end{df} \subsection{Uniqueness of pro-structure} \begin{lem}\label{coconnective} Let $Y$ and $Z$ be derived Artin stacks. The following is true \begin{enumerate} \item The canonical map \[ \operatorname{Map}(Z,Y) \to \lim_n \operatorname{Map}(\tau_{\leq n} Z,Y) \] is an equivalence; \item If $Y$ is $q$-Artin and $Z$ is $m$-truncated then the mapping space $\operatorname{Map}(Z,Y)$ is $(m + q)$-truncated. \end{enumerate} \end{lem} \begin{proof} We prove both items recursively on the Artin degree of $Z$. The case of $Z$ affine is proved in \cite[C.0.10 and 2.2.4.6]{toen:hagii}. We assume that the result is true for $n$-Artin stacks. Let $Z$ be $(n+1)$-Artin. There is an atlas $u \colon U \to Z$. Let us remark that for $k \in\mathbb{N}$ the truncation $\tau_{\leq k} u \colon \tau_{\leq k} U \to \tau_{\leq k} Z$ is also a smooth atlas --- indeed we have $\tau_{\leq k} U \simeq U \times_Z \tau_{\leq k} Z$. Let us denote by $U_\bullet$ the nerve of $u$ and by $\tau_{\leq k}U_\bullet$ the nerve of $\tau_{\leq k} u$. Because $k$-truncated stacks are stable by flat pullbacks, the groupoid $\tau_{\leq k}U_\bullet$ is equivalent to $\tau_{\leq k}(U_\bullet)$. We have \[ \operatorname{Map}(Z,Y) \simeq \lim_{[p] \in \Delta} \operatorname{Map}(U_p,Y) \simeq \lim_{[p] \in \Delta} \lim_k \operatorname{Map}(\tau_{\leq k}U_p,Y) \simeq \lim_k \operatorname{Map}(\tau_{\leq k} Z,Y) \] That proves item \emph{(i)}. If moreover $Z$ is $m$-truncated, then we can replace $U$ by $\tau_{\leq m} U$. If follows that $\operatorname{Map}(Z,Y)$ is a limit of $(m + q)$-truncated spaces. This finishes the proof of \emph{(ii)}. \end{proof} We will use this well known lemma: \begin{lem}[see {\cite[Chapter XI]{bousfieldkan:holim}}]\label{finite-groupoids} Let $S \colon \Delta \to \mathbf{sSets}$ be a cosimplicial object in simplicial sets. Let us assume that for any $[p] \in \Delta$ the simplicial set $S_p$ is $n$-coconnective. Then the natural morphism \[ \lim_{[p] \in \Delta} S_p \to \underset{p \leq n+1}{\lim_{[p] \in \Delta}} S_p \] is an equivalence. \end{lem} \begin{lem}\label{truncatedcompact} Let $\bar X \colon \mathbb{N}^{\mathrm{op}} \to \mathbf{dSt}_S$ be a diagram such that \begin{enumerate} \item There are two integers $m$ and $n$ such that for any $k \in \mathbb{N}$ the stack $\bar X(k)$ is $n$-Artin, $m$-truncated and of finite presentation; \item There exists a diagram $\bar u \colon \mathbb{N}^{\mathrm{op}} \times \Delta^1 \to \mathbf{dSt}_S$ such that the restriction of $\bar u$ to $\mathbb{N}^{\mathrm{op}} \times \{1\}$ is equivalent to $\bar X$, every map $\bar u(k) \colon \bar u(k)(0) \to \bar u(k)(1) \simeq \bar X(k)$ is a smooth atlas and the limit $\lim_k \bar u(k)$ is an epimorphism. \label{truncatedcompact:atlas} \end{enumerate} If $Y$ is an algebraic derived stack of finite presentation then the canonical morphism \[ \colim \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft( \bar X,Y\aftergroup\egroup\originalright) \to \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(\lim \bar X, Y \aftergroup\egroup\originalright) \] is an equivalence. \end{lem} \begin{proof} Let us prove the statement recursively on the Artin degree $n$. If $n$ equals $0$, this is a simple reformulation of the finite presentation of $Y$. Let us assume that the statement at hand is true for some $n$ and let $\bar X(0)$ be $(n+1)$-Artin. Considering the nerves of the epimorphisms $\bar u(k)$, we get a diagram \[ \bar Z \colon \mathbb{N}^{\mathrm{op}} \times \Delta^{\mathrm{op}} \to \mathbf{dSt}_S \] Note that $\bar Z$ has values in $n$-Artin stacks. We observe that the diagram $\lim_k \bar Z(k)_\bullet \colon \Delta^{\mathrm{op}} \to \mathbf{dSt}_S$ is the nerve of the map $\lim_k \bar u(k)$. Since $\lim_k \bar u(k)$ is by assumption an epimorphism (whose target is $\lim \bar X$), the natural map \[ \colim_{[p] \in \Delta} \lim_{k \in \mathbb{N}} \bar Z(k)_p \to \lim \bar X \simeq \lim_{k \in \mathbb{N}} \colim_{[p] \in \Delta} \bar Z(k)_p \] is an equivalence. We now write \begin{align} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(\lim \bar X,Y \aftergroup\egroup\originalright) &\simeq \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(\colim_{[p] \in \Delta} \lim_{k \in \mathbb{N}} \bar Z(k)_p,Y \aftergroup\egroup\originalright) \\ &\simeq \lim_{[p] \in \Delta} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(\lim_{k \in \mathbb{N}} \bar Z(k)_p,Y \aftergroup\egroup\originalright) \\ &\simeq \lim_{[p] \in \Delta} \colim_{k \in \mathbb{N}} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(\bar Z(k)_p,Y \aftergroup\egroup\originalright) \end{align} We also have \[ \colim \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft( \bar X,Y \aftergroup\egroup\originalright) \simeq \colim_{k \in \mathbb{N}} \lim_{[p] \in \Delta} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft( \bar Z(k)_p, Y \aftergroup\egroup\originalright) \] It thus suffices to prove that the canonical morphism of simplicial sets \[ \colim_{k \in \mathbb{N}} \lim_{[p] \in \Delta} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft( \bar Z(k)_p, Y \aftergroup\egroup\originalright) \to \lim_{[p] \in \Delta} \colim_{k \in \mathbb{N}} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(\bar Z(k)_p,Y \aftergroup\egroup\originalright) \] is an equivalence. Let us notice that each $\bar Z(k)_p$ is $m$-truncated. It is indeed a fibre product of $m$-truncated derived stacks along flat maps. Let $q$ be an integer such that $Y$ is $q$-Artin. The simplicial set $\operatorname{Map}(\bar Z(k)_p,Y)$ is then $(m + q)$-coconnective (\autoref{coconnective}). It follows from \autoref{finite-groupoids} that the limit at hand is in fact finite and we have the required equivalence. \end{proof} \begin{lem}\label{exact-seq} Let $\bar M \colon \mathbb{N}^{\mathrm{op}} \to \mathbf{sSets}$ be a diagram. For any $i \in \mathbb{N}$ and any point $x = (x_n) \in \lim \bar M$, we have the following exact sequence \[ \shorthandoff{;:!?} \xymatrix{ 0 \ar[r] & \displaystyle {\lim_n}^1 \pi_{i+1}(\bar M(n),x_n) \ar[r] & \displaystyle \pi_i\mathopen{}\mathclose\bgroup\originalleft(\lim_n \bar M(n), x\aftergroup\egroup\originalright) \ar[r] & \displaystyle \lim_n \pi_i(\bar M(n), x_n) \ar[r] & 0 } \] \end{lem} A proof of that lemma can be found for instance in \cite{hirschhorn:limfibrations}. \begin{lem}\label{colim-lim-swap} Let $M \colon \mathbb{N}^{\mathrm{op}} \times K \to \mathbf{sSets}$ denote a diagram, where $K$ is a filtered simplicial set. If for any $i \in \mathbb{N}$ there exists $N_i$ such that for any $n \geq N_i$ and any $k \in K$ the induced morphism $M(n,k) \to M(n-1,k)$ is an $i$-equivalence then the canonical map \[ \phi \colon \colim_{k \in K} \lim_{n \in \mathbb{N}} M(n,k) \to \lim_{n \in \mathbb{N}} \colim_{k \in K} M(n,k) \] is an equivalence. We recall that an $i$-equivalence of simplicial sets is a morphism which induces isomorphisms on the homotopy groups of dimension lower or equal to $i$. \end{lem} \begin{proof} We can assume that $K$ admits an initial object $k_0$. Let us write $M_{nk}$ instead of $M(n,k)$. Let us fix $i \in \mathbb{N}$. If $i \geq 1$, we also fix a base point $x \in \lim_n M_{nk_0}$. Every homotopy group below is computed at $x$ or at the natural point induced by $x$. We will omit the reference to the base point. We have a morphism of short exact sequences \[ \shorthandoff{;:!?} \xymatrix{ 0 \ar[r] & \displaystyle \colim_k {\lim_{n}}^1 \pi_{i+1}(M_{nk}) \ar[r] \ar[d] & \displaystyle \colim_k \pi_{i}\mathopen{}\mathclose\bgroup\originalleft(\lim_{n} M_{nk}\aftergroup\egroup\originalright) \ar[r] \ar[d] & \displaystyle \colim_k \lim_{n} \pi_i(M_{nk}) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \displaystyle {\lim_{n}}^1 \colim_k \pi_{i+1}(M_{nk}) \ar[r] & \displaystyle \pi_{i}\mathopen{}\mathclose\bgroup\originalleft(\lim_{n} \colim_k M_{nk}\aftergroup\egroup\originalright) \ar[r] & \displaystyle \lim_{n} \colim_k \pi_i(M_{nk}) \ar[r] & 0 \\ } \] We can restrict every limit to $n \geq N_{i+1}$. Using the assumption we see that the limits on the right hand side are then constant and so are the $1$-limits on the left. If follows that the vertical maps on the sides are isomorphisms, and so is the middle map. This begin true for any $i$, we conclude that $\phi$ is an equivalence. \end{proof} \begin{df}\label{shy} Let $\bar X \colon \mathbb{N}^{\mathrm{op}} \to \mathbf{dSt}_S$ be a diagram. We say that $\bar X$ is a shy diagram if \begin{enumerate} \item For any $k \in \mathbb{N}$ the stack $\bar X(k)$ is algebraic and of finite presentation; \item For any $k \in \mathbb{N}$ the map $\bar X(k \to k+1) \colon \bar X(k+1) \to \bar X(k)$ is affine; \item The stack $\bar X(0)$ is of finite cohomological dimension. \end{enumerate} If $X$ is the limit of $\bar X$ in the category of prostacks, we will also say that $\bar X$ is a shy diagram for $X$. \end{df} \begin{prop}\label{prop-cocompact} Let $\bar X \colon \mathbb{N}^{\mathrm{op}} \to \mathbf{dSt}_S$ be a shy diagram. If $Y$ is an algebraic derived stack of finite presentation then the canonical morphism \[ \colim \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft( \bar X,Y\aftergroup\egroup\originalright) \to \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(\lim \bar X, Y \aftergroup\egroup\originalright) \] is an equivalence. \end{prop} \begin{proof} Since for any $n$, the truncation functor $\tau_{\leq n}$ preserves shy diagrams, let us use \autoref{coconnective} and \autoref{truncatedcompact} \begin{align} \operatorname{Map}(\lim \bar X,Y) \simeq & \lim_n \operatorname{Map}(\tau_{\leq n}(\lim \bar X),Y)\\ & \simeq \lim_n \operatorname{Map}(\lim \tau_{\leq n}\bar X,Y) \simeq \lim_n \colim \operatorname{Map}(\tau_{\leq n} \bar X,Y) \end{align} On the other hand we have \[ \colim \operatorname{Map}(\bar X,Y) \simeq \colim \lim_n \operatorname{Map}(\tau_{\leq n} \bar X,Y) \] and we are to study the canonical map \[ \phi \colon \colim \lim_n \operatorname{Map}(\tau_{\leq n} \bar X,Y) \to \lim_n \colim \operatorname{Map}(\tau_{\leq n} \bar X,Y) \] Because of \autoref{colim-lim-swap}, it suffices to prove the assertion \begin{enumerate}[label=(\arabic)] \item For any $i \in \mathbb{N}$ there exists $N_i \in \mathbb{N}$ such that for any $n \geq N_i$ and any $k \in \mathbb{N}$ the map \[ p_{n,k} \colon \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft( \tau_{\leq n} \bar X(k), Y \aftergroup\egroup\originalright) \to \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft( \tau_{\leq n -1} \bar X(k), Y \aftergroup\egroup\originalright) \] induces an equivalence on the $\pi_j$'s for any $j \leq i$. \end{enumerate} For any map $f \colon \tau_{\leq n-1} \bar X(k) \to Y$ we will denote by $F_{n,k}(f)$ the fibre of $p_{n,k}$ at $f$. We have to prove that for any such $f$ the simplicial set $F_{n,k}(f)$ is $i$-connective. Let thus $f$ be one of those maps. The derived stack $\tau_{\leq n} \bar X(k)$ is a square zero extension of $\tau_{\leq n-1} \bar X(k)$ by a module $M[n]$, where \[ M = \ker \mathopen{}\mathclose\bgroup\originalleft(\mathcal{O}_{\tau_{\leq n} \bar X(k)} \to \mathcal{O}_{\tau_{\leq n-1} \bar X(k)} \aftergroup\egroup\originalright) [-n] \] Note that $M$ is concentrated in degree $0$. It follows from the obstruction theory of $Y$ --see \autoref{obstruction} -- that $F_{n,k}(f)$ is not empty if and only if the obstruction class \[ \alpha(f) \in G_{n,k}(f) = \operatorname{Map}_{\mathcal{O}_{\tau_{\leq n-1} \bar X(k)}}(f^* \mathbb{L}_Y, M[n+1]) \] of $f$ vanishes. Moreover, if $\alpha(f)$ vanishes, then we have an equivalence \[ F_{n,k}(f) \simeq \operatorname{Map}_{\mathcal{O}_{\tau_{\leq n-1} \bar X(k)}}(f^* \mathbb{L}_Y, M[n]) \] Using assumptions \emph{(iii)} and \emph{(ii)} we have that $\bar X(k)$ --- and therefore its truncation too --- is of finite cohomological dimension $d$. Let us denote by $[a,b]$ the Tor-amplitude of $\mathbb{L}_Y$. We get that $G_{n,k}(f)$ is $(s+1)$-connective for $s = a + n - d$ and that $F_{n,k}(f)$ is $s$-connective if $\alpha(f)$ vanishes. Let us remark here that $d$ and $a$ do not depend on either $k$ or $f$ and thus neither does $N_i = i + d - a$ (we set $N_i = 0$ if this quantity is negative). For any $n \geq N_i$ and any $f$ as above, the simplicial set $G_{n,k}(f)$ is at least $1$-connective. The obstruction class $\alpha(f)$ therefore vanishes and $F_{n,k}(f)$ is indeed $i$-connective. This proves (1) and concludes this proof. \end{proof} \newcommand{\mathbf P\dSt^{\mathrm{shy}}}{\mathbf P\mathbf{dSt}^{\mathrm{shy}}} \begin{df}\label{prochamps} Let $\mathbf P\dSt^{\mathrm{shy}}_S$ denote the full subcategory of $\Prou U\mathbf{dSt}_S$ spanned by the prostacks which admit shy diagrams. Every object $X$ in $\mathbf P\dSt^{\mathrm{shy}}_S$ is thus the limit of a shy diagram $\bar X \colon \mathbb{N}^{\mathrm{op}} \to \mathbf{dSt}_S$. We will say that $X$ is of cotangent tor-amplitude in $[a,b]$ if there exists a shy diagram $\bar X \colon \mathbb{N}^{\mathrm{op}} \to \mathbf{dSt}_S$ for $X$ such that every cotangent $\mathbb{L}_{\bar X(n)}$ is of tor-amplitude in $[a,b]$. We will also say that $X$ is of cohomological dimension at most $d$ if there is a shy diagram $\bar X$ with values in derived stacks of cohomological dimension at most $d$. The pro-stack $X$ will be called $q$-Artin if there is a shy diagram for it, with values in $q$-Artin derived stacks. Let us denote by $\mathcal{C}^{[a,b]}_{d,q}$ the full subcategory of $\mathbf P\dSt^{\mathrm{shy}}_S$ spanned by objects of cotangent tor-amplitude in $[a,b]$, of cohomological dimension at most $d$ and $q$-Artin. \end{df} \begin{thm} The limit functor $i_\mathrm{shy} \colon \mathbf P\dSt^{\mathrm{shy}}_S \to \mathbf{dSt}_S$ is fully faithful and has values in Artin stacks. \end{thm} \begin{proof} This follows directly from \autoref{prop-cocompact}. \end{proof} \begin{df} A map of pro-stacks $f \colon X \to Y$ if an open immersion if there exists a diagram \[ \bar f \colon \mathbb{N}^{\mathrm{op}} \times \Delta^1 \to \mathbf{dSt}_k \] such that \begin{itemize} \item The limit of $\bar f$ in maps of pro-stacks is $f$; \item The restriction $\mathbb{N}^{\mathrm{op}} \times \{0\} \to \mathbf{dSt}_k$ of $\bar f$ is a shy diagram for $X$ and the restriction $\mathbb{N}^{\mathrm{op}} \times \{1\} \to \mathbf{dSt}_k$ is a shy diagram for $Y$; \item For any $n$, the induced map of stacks $\{n\} \times \Delta^1 \to \mathbf{dSt}_k$ is an open immersion. \end{itemize} \end{df} \subsection{Uniqueness of ind-pro-structures} \label{unique-ipstructure} \begin{df}\label{indprochamps} Let $\mathbf{IP} \dSt^{\mathrm{shy,b}}_S$ denote the full subcategory of $\Indu U(\mathbf P\dSt^{\mathrm{shy}}_S)$ spanned by colimits of $\mathbb U$-small filtered diagrams $K \to \mathbf P\dSt^{\mathrm{shy}}_S$ which factors through $\mathcal{C}^{[a,b]}_{d,q}$ for some 4-uplet $a,b,d,q$. For any $X \in \mathbf{IP} \dSt^{\mathrm{shy,b}}_S$ we will say that $X$ is of cotangent tor-amplitude in $[a,b]$ and of cohomological dimension at most $d$ if it is the colimit (in $\Indu U(\mathbf P\dSt^{\mathrm{shy}}_S)$) of a diagram $K \to \mathcal{C}^{[a,b]}_{d,q}$. \end{df} \begin{thm}\label{ff-realisation} The colimit functor $\Indu U(\mathbf P\dSt^{\mathrm{shy}}_S) \to \mathbf{dSt}_S$ restricts to a full faithful embedding $\mathbf{IP} \dSt^{\mathrm{shy,b}}_S \to \mathbf{dSt}_S$. \end{thm} \begin{lem}\label{iequi} Let $a,b,d,q$ be integers with $a \leq b$. Let $T \in \mathbf P\dSt^{\mathrm{shy}}_S$ and $\bar X \colon K \to \mathcal{C}^{[a,b]}_{d,q}$ be a $\mathbb U$-small filtered diagram. For any $i \in \mathbb{N}$ there exists $N_i$ such that for any $n \geq N_i$ and any $k \in K$, the induced map \[ \operatorname{Map}(\tau_{\leq n}T, \bar X(k)) \to \operatorname{Map}(\tau_{\leq n-1} T, \bar X(k)) \] is an $i$-equivalence. We recall that an $i$-equivalence of simplicial sets is a morphism which induces isomorphisms on the homotopy groups of dimension lower or equal to $i$. \end{lem} \begin{rmq} For the proof of this lemma, we actually do not need the integer $q$. \end{rmq} \begin{proof} Let us fix $i \in \mathbb{N}$. Let $k \in K$ and $\bar T \colon \mathbb{N} \to \mathbf{dSt}_S$ be a shy diagram for $T$. We observe here that $\tau_{\leq n} \bar T$ is a shy diagram whose limit is $\tau_{\leq n} T$. Let also $\bar Y_k \colon \mathbb{N} \to \mathbf{dSt}_S$ be a shy diagram for $\bar X(k)$. The map at hand \[ \psi_{nk} \colon \operatorname{Map}(\tau_{\leq n}T, \bar X(k)) \to \operatorname{Map}(\tau_{\leq n-1} T, \bar X(k)) \] is then the limit of the colimits \[ \lim_{p \in \mathbb{N}} \colim_{q \in \mathbb{N}} \operatorname{Map}(\tau_{\leq n} \bar T(q), \bar Y_k(p)) \to \lim_{p \in \mathbb{N}} \colim_{q \in \mathbb{N}} \operatorname{Map}(\tau_{\leq n-1} \bar T(q), \bar Y_k(p)) \] Let now $f$ be a map $\tau_{\leq n-1} T \to \bar X(k)$. It corresponds to a family of morphisms \[ f_p \colon {} \to \colim_{q \in \mathbb{N}} \operatorname{Map}(\tau_{\leq n-1} \bar T(q), \bar Y_k(p)) \] For any $p$, let $F_{nk}^p(f)$ denote the fibre of the map \[ \psi_{nk}^p \colon \colim_{q \in \mathbb{N}} \operatorname{Map}(\tau_{\leq n} \bar T(q), \bar Y_k(p)) \to \colim_{q \in \mathbb{N}} \operatorname{Map}(\tau_{\leq n-1} \bar T(q), \bar Y_k(p)) \] over the point $f_p$. We also set $F_{nk}(f) = \lim_p F_{nk}^p(f)$ and observe that $F_{nk}(f)$ is nothing but the fibre of $\psi_{nk}$ over $f$. To prove the result, it suffices to show that for any such $f$, the fibre $F_{nk}(f)$ is $i$-connective. Using the exact sequence of \autoref{exact-seq}, it suffices to prove that $F_{nk}^p(f)$ is $(i+1)$-connective for any $f$ and any $p$. Fixing such an $f$ and such a $p$, there exists $q_0 \in \mathbb{N}$ such that the map $f_p$ factors through the canonical map \[ \operatorname{Map}(\tau_{\leq n-1} \bar T(q_0), \bar Y_k(p)) \to \colim_{q \in \mathbb{N}} \operatorname{Map}(\tau_{\leq n-1} \bar T(q), \bar Y_k(p)) \] We deduce that $F_{nk}^p(f)$ is equivalent to the colimit \[ F_{nk}^p(f) \simeq \colim_{q \geq q_0} G_{nk}^{pq}(f) \] where $G_{nk}^{pq}(f)$ is the fibre at the point induced by $f_p$ of the map \[ \operatorname{Map}(\tau_{\leq n} \bar T(q), \bar Y_k(p)) \to \operatorname{Map}(\tau_{\leq n-1} \bar T(q), \bar Y_k(p)) \] The interval $[a,b]$ contains the tor-amplitude of $\mathbb{L}_{\bar Y_k(p)}$ and $d$ is an integer greater than the cohomological dimension of $\bar T(q)$. We saw in the proof of \autoref{prop-cocompact} that $G_{nk}^{pq}(f)$ is then $(a + n - d)$-connective. We set $N_i = i + d - a +1$. \end{proof} \begin{proof}[of \autoref{ff-realisation}] We will prove the sufficient following assertions \begin{enumerate}[label=(\arabic)] \item The colimit functor $\Indu U(\mathbf P\dSt^{\mathrm{shy}}_S) \to \operatorname{\mathcal{P}}(\mathbf{dAff}_S)$ restricts to a fully faithful functor \[ \eta \colon \mathbf{IP} \dSt^{\mathrm{shy,b}}_S \to \operatorname{\mathcal{P}}(\mathbf{dAff}_S) \] \item The functor $\eta$ has values in the full subcategory of stacks. \end{enumerate} Let us focus on assertion (1) first. We consider two $\mathbb U$-small filtered diagrams $\bar X \colon K \to \mathbf P\dSt^{\mathrm{shy}}_S$ and $\bar Y \colon L \to \mathbf P\dSt^{\mathrm{shy}}_S$. We have \[ \operatorname{Map}_{\Indu U(\mathbf P\dSt^{\mathrm{shy}}_S)}\mathopen{}\mathclose\bgroup\originalleft(\colim \bar X, \colim \bar Y\aftergroup\egroup\originalright) \simeq \lim_k \operatorname{Map}_{\Indu U(\mathbf P\dSt^{\mathrm{shy}}_S)}(\bar X(k), \colim \bar Y) \] and \[ \operatorname{Map}_{\operatorname{\mathcal{P}}(\mathbf{dAff})}\mathopen{}\mathclose\bgroup\originalleft( \colim i_\mathrm{shy} \bar X, \colim i_\mathrm{shy} \bar Y \aftergroup\egroup\originalright) \simeq \lim_k \operatorname{Map}_{\operatorname{\mathcal{P}}(\mathbf{dAff})}\mathopen{}\mathclose\bgroup\originalleft( i_\mathrm{shy} \bar X(k), \colim i_\mathrm{shy} \bar Y \aftergroup\egroup\originalright) \] We can thus replace the diagram $\bar X$ in $\mathbf P\dSt^{\mathrm{shy}}_S$ by a simple object $X \in \mathbf P\dSt^{\mathrm{shy}}_S$. We now assume that $\bar Y$ factors through $\mathcal{C}^{[a,b]}_{d,q}$ for some $a,b,d,q$. We have to prove that the following canonical morphism is an equivalence \[ \phi \colon \colim_{l \in L} \operatorname{Map}(i_\mathrm{shy} X,i_\mathrm{shy} \bar Y(l)) \to \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(i_\mathrm{shy} X, \colim i_\mathrm{shy} \bar Y\aftergroup\egroup\originalright) \] where the mapping spaces are computed in prestacks. If $i_\mathrm{shy} X$ is affine then $\phi$ is an equivalence because colimits in $\operatorname{\mathcal{P}}(\mathbf{dAff}_S)$ are computed pointwise. Let us assume that $\phi$ is an equivalence whenever $i_\mathrm{shy} X$ is $(q-1)$-Artin and let us assume that $i_\mathrm{shy} X$ is $q$-Artin. Let $u \colon U \to i_\mathrm{shy} X$ be an atlas of $i_\mathrm{shy} X$ and let $Z_\bullet$ be the nerve of $u$ in $\mathbf{dSt}_S$. We saw in the proof of \autoref{truncatedcompact} that $Z_\bullet$ factors through $\mathbf P\dSt^{\mathrm{shy}}_S$. The map $\phi$ is now equivalent to the natural map \begin{align} \colim_{l \in L} \operatorname{Map}(i_\mathrm{shy} X,i_\mathrm{shy} \bar Y(l)) \to \lim_{[p]\in \Delta} & \colim_{l \in L} \operatorname{Map}(Z_p,i_\mathrm{shy} \bar Y(l)) \\ & \simeq \lim_{[p]\in \Delta} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(Z_p,\colim i_\mathrm{shy} \bar Y\aftergroup\egroup\originalright) \simeq \operatorname{Map}(i_\mathrm{shy} X,\colim i_\mathrm{shy} \bar Y) \end{align} Remembering \autoref{coconnective}, it suffices to study the map \[ \colim_{l \in L} \lim_n \operatorname{Map}(\tau_{\leq n} i_\mathrm{shy} X,i_\mathrm{shy} \bar Y(l)) \to \lim_{[p]\in \Delta} \colim_{l \in L} \lim_n \operatorname{Map}(\tau_{\leq n}Z_p,i_\mathrm{shy} \bar Y(l)) \] Applying \autoref{iequi} and then \autoref{colim-lim-swap}, we see that $\phi$ is an equivalence if the natural morphism \[ \lim_n \colim_{l \in L} \lim_{[p] \in \Delta} \operatorname{Map}(\tau_{\leq n}Z_p,i_\mathrm{shy} \bar Y(l)) \to \lim_n \lim_{[p] \in \Delta} \colim_{l \in L} \operatorname{Map}(\tau_{\leq n}Z_p, i_\mathrm{shy} \bar Y(l)) \] is an equivalence. The stack $i_\mathrm{shy} \bar Y(l)$ is by assumption $q$-Artin, where $q$ does not depend on $l$. Now using \autoref{coconnective} and \autoref{finite-groupoids}, we conclude that $\phi$ is an equivalence. This proves (1). We now focus on assertion (2). If suffices to see that the colimit in $\operatorname{\mathcal{P}}(\mathbf{dAff}_S)$ of the diagram $i_\mathrm{shy} \bar Y$ as above is actually a stack. Let $H_\bullet \colon \Delta^{\mathrm{op}} \cup \{-1\} \to \mathbf{dAff}_S$ be an hypercovering of an affine $\operatorname{Spec}(A) = H_{-1}$. We have to prove the following equivalence \[ \colim_l \lim_{[p] \in \Delta} \operatorname{Map}(H_p,i_\mathrm{shy} \bar Y(l)) \to \lim_{[p] \in \Delta} \colim_l \operatorname{Map}(H_p,i_\mathrm{shy} \bar Y(l)) \] Using the same arguments as for the proof of (1), we have \begin{align} \colim_l \lim_{[p] \in \Delta} \operatorname{Map}(H_p,i_\mathrm{shy} \bar Y(l)) &\simeq \colim_l \lim_{[p] \in \Delta} \lim_n \operatorname{Map}(\tau_{\leq n}H_p,i_\mathrm{shy} \bar Y(l)) \\ &\simeq \lim_n \colim_l \lim_{[p] \in \Delta} \operatorname{Map}(\tau_{\leq n}H_p,i_\mathrm{shy} \bar Y(l)) \\ &\simeq \lim_n \lim_{[p] \in \Delta} \colim_l \operatorname{Map}(\tau_{\leq n}H_p,i_\mathrm{shy} \bar Y(l)) \\ &\simeq \lim_{[p] \in \Delta} \colim_l \lim_n \operatorname{Map}(\tau_{\leq n}H_p,i_\mathrm{shy} \bar Y(l)) \\ &\simeq \lim_{[p] \in \Delta} \colim_l \operatorname{Map}(H_p,i_\mathrm{shy} \bar Y(l)) \end{align} \end{proof} We will need one last lemma about that category $\mathbf{IP} \dSt^{\mathrm{shy,b}}_S$. \begin{lem}\label{shydaff-limits} The fully faithful functor $\mathbf{IP} \dSt^{\mathrm{shy,b}}_S \cap \mathbf{IP}\mathbf{dAff}_S \to \mathbf{IP}\mathbf{dSt}_S \to \mathbf{dSt}_S$ preserves finite limits. \end{lem} \begin{proof} The case of an empty limit is obvious. Let then $X \to Y \from Z$ be a diagram in $\mathbf{IP} \dSt^{\mathrm{shy,b}}_S \cap \mathbf{IP}\mathbf{dAff}_S$. There exist $a$ and $b$ and a diagram \[ \sigma \colon K \to \operatorname{Fct}\mathopen{}\mathclose\bgroup\originalleft( \Lambda^2_1, \mathcal{C}^{[a,b]}_{0,0} \aftergroup\egroup\originalright) \] such that $K$ is a $\mathbb U$-small filtered simplicial set and the colimit in $\mathbf{IP}\mathbf{dSt}_S$ is $X \to Y \from Z$. We can moreover assume that $\sigma$ has values in $\operatorname{Fct}(\Lambda^2_1, \Prou U(\mathbf{dAff}_S)) \simeq \Prou U(\operatorname{Fct}(\Lambda^2_1, \mathbf{dAff}_S))$. We deduce that the fibre product $X \times_Y Z$ is the realisation of the ind-pro-diagram in derived affine stacks with cotangent complex of tor amplitude in $[a-1,b+1]$. It follows that $X \times_Y Z$ is again in $\mathbf{IP} \dSt^{\mathrm{shy,b}}_S \cap \mathbf{IP}\mathbf{dAff}_S$. \end{proof} \section{Symplectic Tate stacks}\label{chapterSymptate} \subsection{Tate stacks: definition and first properties} \label{tatestacks}We can now define what a Tate stack is. \begin{df} A Tate stack is a derived Artin ind-pro-stack locally of finite presentation whose cotangent complex -- see \autoref{ipcotangent} -- is a Tate module. Equivalently, an Artin ind-pro-stack locally of finite presentation is Tate if its tangent complex is a Tate module. We will denote by $\dSt^{\mathrm{Tate}}_k$ the full subcategory of $\mathbf{IP}\mathbf{dSt}_k$ spanned by Tate stacks. \end{df} This notion has several good properties. For instance, using \autoref{ipdst-tate-in-ipp}, if a $X$ is a Tate stack then comparing its tangent $\mathbb{T}_X$ and its cotangent $\mathbb{L}_X$ makes sense, in the category of Tate modules over $X$. We will explore that path below, defining symplectic Tate stacks. Another consequence of Tatity\footnote{or Tateness or Tatitude} is the existence of a determinantal anomaly as defined in \cite{kapranovvasserot:loop2}. Let us consider the natural morphism of prestacks \[ \theta \colon \Tateu U \to \mathrm K^{\operatorname{\mathbf{Tate}}} \] where $\Tateu U$ denote the prestack $A \mapsto \Tateu U(\mathbf{Perf}(A))$ and $\mathrm K^{\operatorname{\mathbf{Tate}}} \colon A \mapsto \operatorname{K}(\Tateu U(\mathbf{Perf}(A)))$ -- $\mathrm K$ denoting the connective $K$-theory functor. From \cite[Section 5]{hennion:tate} we have a determinant \[ \mathrm K^{\operatorname{\mathbf{Tate}}} \to \mathrm K(\mathbb{G}_m,2) \] where $\mathrm K(\mathbb{G}_m,2)$ is the Eilenberg-Maclane classifying stack. \begin{df} We define the Tate determinantal map as the composite map \[ \Tateu U \to \mathrm K(\mathbb{G}_m,2) \] To any derived stack $X$ with a Tate module $E$, we associate the determinantal anomaly $[\det_E] \in \mathrm H^2(X,\mathcal{O}_X^{\times})$, image of $E$ by the morphism \[ \operatorname{Map}(X,\Tateu U) \to \operatorname{Map}(X,\mathrm K(\mathbb{G}_m,2)) \] \end{df} Let now $X$ be an ind-pro-stack. Let also $R$ denote the realisation functor $\Prou U\mathbf{dSt}_k \to \mathbf{dSt}_k$ mapping a pro-stack to its limit in $\mathbf{dSt}_k$. Let finally $\bar X \colon K \to \Prou U\mathbf{dSt}_k$ denote a $\mathbb U$-small filtered diagram whose colimit in $\mathbf{IP}\mathbf{dSt}_k$ is $X$. We have a canonical functor \[ F_X \colon \lim \Tateu U_\mathbf P(\bar X) \simeq \Tateu U_\mathbf{IP}(X) \to \lim \Tateu U(R \bar X) \] \begin{df}\label{determinantalanomaly} Let $X$ be an ind-pro-stack and $E$ be a Tate module on $X$. Let $X'$ be the realisation of $X$ in $\Indu U\mathbf{dSt}_k$ and $X''$ be its image in $\mathbf{dSt}_k$. We define the determinantal anomaly of $E$ the image of $F_X(E)$ by the map \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{Map}_{\Indu U\mathbf{dSt}_k}(X',\Tateu U) \to \operatorname{Map}_{\Indu U\mathbf{dSt}_k}(X',\mathrm K(\mathbb{G}_m,2)) \simeq \operatorname{Map}_{\mathbf{dSt}_k}(X'',\mathrm K(\mathbb{G}_m,2)) } \] In particular if $X$ is a Tate stack, we will denote by $[\det_X] \in \mathrm H^2(X'',\mathcal{O}_{X''}^{\times})$ the determinantal anomaly associated to its tangent $\mathbb{T}_X \in \Tateu U_\mathbf{IP}(X)$. \end{df} The author plans on studying more deeply this determinantal class in future work. For now, let us conclude this section with following \begin{lem}\label{tate-limits} The inclusion $\dSt^{\mathrm{Tate}}_k \to \mathbf{IP}\mathbf{dSt}_k$ preserves finite limits. \end{lem} \begin{proof} Let us first notice that a finite limit of Artin ind-pro-stacks is again an Artin ind-pro-stack. Let now $X \to Y \from Z$ be a diagram of Tate stacks. The fibre product \[ \shorthandoff{;:!?} \xymatrix{ X \times_Y Z \cart \ar[d]_{p_Z} \ar[r]^-{p_X} & X \ar[d]^g \\ Z \ar[r] & Y } \] is an Artin ind-pro-stack. It thus suffices to test if its tangent $\mathbb{T}_{X \times_Y Z}$ is a Tate module. The following cartesian square concludes \[ \shorthandoff{;:!?} \xymatrix{ \mathbb{T}_{X \times_Y Z} \cart \ar[r] \ar[d] & p_X^* \mathbb{T}_X \ar[d] \\ p_Z^* \mathbb{T}_Z \ar[r] & p_X^* g^* \mathbb{T}_Y } \] \end{proof} \subsection{Shifted symplectic Tate stacks} \newcommand{\operatorname{\mathbf{DR}}}{\operatorname{\mathbf{DR}}} \newcommand{\operatorname{\mathrm{NC^w}}}{\operatorname{\mathrm{NC^w}}} \newcommand{\operatorname{C}}{\operatorname{C}} We assume now that the basis $S$ is the spectrum of a ring $k$ of characteristic zero. Recall from \cite{ptvv:dersymp} the stack in graded complexes $\operatorname{\mathbf{DR}}$ mapping a cdga over $k$ to its graded complex of forms. It actually comes with a mixed structure induced by the de Rham differential. The authors also defined there the stack in graded complexes $\operatorname{\mathrm{NC^w}}$ mapping a cdga to its graded complex of closed forms. Those two stacks are linked by a morphism $\operatorname{\mathrm{NC^w}} \to \operatorname{\mathbf{DR}}$ forgetting the closure. We will denote by $\forms p, \closedforms p \colon \cdgaunbounded^{\leq 0}_k \to \mathbf{dgMod}_k$ the complexes of weight $p$ in $\operatorname{\mathbf{DR}}[-p]$ and $\operatorname{\mathrm{NC^w}}[-p]$ respectively. The stack $\forms p$ will therefore map a cdga to its complexes of $p$-forms while $\closedforms p$ will map it to its closed $p$-forms. For any cdga $A$, a cocycle of degree $n$ of $\forms p (A)$ is an $n$-shifted $p$-forms on $\operatorname{Spec} A$. The functors $\closedforms p$ and $\forms p$ extend to functors \[ \closedforms p,~ \forms p \colon \mathbf{dSt}_k^{\mathrm{op}} \to \mathbf{dgMod}_k \] \begin{df} Let us denote by $\IPforms p$ and $\IPclosedforms p$ the extensions \[ (\mathbf{IP}\mathbf{dSt}_k)^{\mathrm{op}} \to \Prou U\Indu U\mathbf{dgMod}_k \] of $\forms p$ and $\closedforms p$, respectively. They come with a natural projection $\IPclosedforms p \to \IPforms p$. Let $X \in \mathbf{IP}\mathbf{dSt}_k$. An $n$-shifted (closed) $p$-form on $X$ is a morphism $k[-n] \to \IPforms p(X)$ (resp. $\IPclosedforms p(X)$). For any closed form $\omega \colon k[-n] \to \IPclosedforms p(X)$, the induced map $k[-n] \to \IPclosedforms p(X) \to \IPforms p(X)$ is called the underlying form of $\omega$. \end{df} \begin{rmq} In the above definition, we associate to any ind-pro-stack $X = \colim_\alpha \lim_\beta X_{\alpha\beta}$ its complex of forms \[ \IPforms p(X) = \lim_\alpha \colim_\beta \forms p(X_{\alpha\beta}) \in \Prou U \Indu U \mathbf{dgMod}_k \] \end{rmq} For any ind-pro-stack $X$, the derived category $\operatorname{\mathbf{PIQcoh}}(X)$ is endowed with a canonical monoidal structure. In particular, one defines a symmetric product $E \mapsto \operatorname{Sym}^2_\mathbf{PI}(E)$ as well as an antisymmetric product \[ E \wedge_\mathbf{PI} E = \operatorname{Sym}_\mathbf{PI}^2(E[-1])[2] \] \begin{thm}\label{prop-formsareforms} Let $X$ be an Artin ind-pro-stack over $k$ and $\pi \colon X \to {}$ the projection. The push-forward functor \[ \pi^\mathbf{PIQ}_ \colon \operatorname{\mathbf{PIQcoh}}(X) \to \Prou V \Indu V(\mathbf{dgMod}_k) \] exists (see \autoref{prop-piqcoh-right-adjoint}) and maps $\mathbb{L}_X \wedge_\mathbf{PI} \mathbb{L}_X$ to $\IPforms 2(X)$. In particular, any $2$-form $k[-n] \to \IPforms 2(X)$ corresponds to a morphism $\mathcal{O}_X[-n] \to \mathbb{L}_X \wedge_\mathbf{PI} \mathbb{L}_X$ in $\operatorname{\mathbf{PIQcoh}}(X)$. \end{thm} \begin{proof} This follows from \cite[1.14]{ptvv:dersymp}, from \autoref{IPcotangent-underlying} and from the equivalence \[ \lambda^\mathbf{IP} \wedge_\mathbf{PI} \lambda^\mathbf{IP} = \Proextu U \Indextu U (\lambda) \wedge_\mathbf{PI} \Proextu U \Indextu U(\lambda) \simeq \Proextu U \Indextu U (\lambda \wedge \lambda) \] where $\lambda^\mathbf{IP}$ is defined in the proof of \autoref{ipcotangent}. \end{proof} \begin{df} Let $X$ be a Tate stack. Let $\omega \colon k[-n] \to \IPforms 2(X)$ be an $n$-shifted $2$-form on $X$. It induces a map in the category of Tate modules on $X$ \[ \underline \omega \colon \mathbb{T}_X \to \mathbb{L}_X[n] \] We say that $\omega$ is non-degenerate if the map $\underline \omega$ is an equivalence. A closed $2$-form is non-degenerate if the underlying form is. \end{df} \begin{df}\index{Symplectic Tate stack} A symplectic form on a Tate stack is a non-degenerate closed $2$-form. A symplectic Tate stack is a Tate stack equipped with a symplectic form. \end{df} \subsection{Mapping stacks admit closed forms} \newcommand{\operatorname{EV}}{\operatorname{EV}} In this section, we will extend the proof from \cite{ptvv:dersymp} to ind-pro-stacks. Note that if $X$ is a pro-ind-stack and $Y$ is a stack, then $\operatorname{\underline{Ma}p}(X,Y)$ is an ind-pro-stack. We will then need an evaluation functor $\operatorname{\underline{Ma}p}(X,Y) \times X \to Y$. It appears that this evaluation map only lives in the category of ind-pro-ind-pro-stacks \[ \colim_\alpha \lim_{\beta} \colim_{\xi} \lim_{\zeta} \operatorname{\underline{Ma}p}(X_{\alpha\zeta},Y) \times X_{\beta\xi} \to Y \] To use this map properly, we will need the following remark. \begin{df} Let $\mathcal{C}$ be a category. There is one natural fully faithful functor \[ \phi \colon \mathbf{PI}(\mathcal{C}) \to (\mathbf{IP})^2(\mathcal{C}) \] but three $\mathbf{IP}(\mathcal{C}) \to (\mathbf{IP})^2(\mathcal{C})$. The first one is given by applying $\mathbf{IP}$ to the canonical embedding functor $\mathcal{C} \to \mathbf{IP}(\mathcal{C})$. The second one by considering the canonical embedding functor $\mathcal{D} \to \mathbf{IP}(\mathcal{D})$ for $\mathcal{D} = \mathbf{IP}(\mathcal{C})$. In this work, we will only consider the third functor \[ \psi \colon \mathbf{IP}(\mathcal{C}) \to (\mathbf{IP})^2(\mathcal{C}) \] given by applying $\Indu U$ to the canonical embedding $\mathcal{D} \to \mathbf{PI}(\mathcal{D})$ for $\mathcal{D} = \Prou U(\mathcal{C})$. Let us also denote by $\xi$ the natural fully faithful functor $\mathcal{C} \to (\mathbf{IP})^2 (\mathcal{C})$. \end{df} \begin{df} Let $Y$ be a stack and $X$ be a pro-ind-stack. Let us denote the evaluation map in $\mathbf{IP}^2\mathbf{dSt}_S$ \[ \operatorname{ev}^{X,Y} \colon \shorthandoff{;:!?} \xymatrix@1{\displaystyle \psi \operatorname{\underline{Ma}p}_S(X,Y) \times_S \phi X \ar[r] & \xi Y} \] For a formal definition of this map, we refer to \cite[2.2.3.2]{hennion:these}. \end{df} We assume now that $S = \operatorname{Spec} k$. Let us recall the following definition from \cite[2.1]{ptvv:dersymp} \begin{df} A derived stack $X$ is $\mathcal{O}$-compact if for any derived affine scheme $T$ the following conditions hold \begin{itemize} \item The quasi-coherent sheaf $\mathcal{O}_{X \times T}$ is compact in $\mathbf{Qcoh}(X \times T)$ ; \item Pushing forward along the projection $X \times T \to T$ preserves perfect complexes. \end{itemize} Let us denote by $\mathbf{dSt}_k^{\mathcal{O}}$ the full subcategory of $\mathbf{dSt}_k$ spanned by $\mathcal{O}$-compact derived stacks. \end{df} \begin{df} An $\mathcal{O}$-compact pro-ind-stack is a pro-ind-object in the category of $\mathcal{O}$-compact derived stacks. We will denote by $\mathbf{PI}\mathbf{dSt}^{\mathcal{O}}_k$ their category. \end{df} \begin{lem} There is a functor \[ \mathbf{PI} \mathbf{dSt}_k^\mathcal{O} \to \operatorname{Fct}\mathopen{}\mathclose\bgroup\originalleft( \mathbf{IP}\mathbf{dSt}_k \times \Delta^1 \times \Delta^1, (\mathbf{IP})^2 (\mathbf{dgMod}_k)^{\mathrm{op}} \aftergroup\egroup\originalright) \] defining for any $\mathcal{O}$-compact pro-ind-stack $X$ and any ind-pro-stack $F$ a commutative square \[ \shorthandoff{;:!?} \xymatrix{ \closedforms p_{\mathbf{IP}^2} (\psi F \times \phi X) \ar[r] \ar[d] & \IPclosedforms p(\psi F) \otimes_k \phi \mathcal{O}_X \ar[d] \\ \forms p_{\mathbf{IP}^2} (\psi F \times \phi X) \ar[r] & \IPforms p(\psi F) \otimes_k \phi \mathcal{O}_X } \] where $\closedforms p_{\mathbf{IP}^2}$ and $\forms p_{\mathbf{IP}^2}$ are the extensions of $\IPclosedforms p$ and $\IPforms p$ to \[ (\mathbf{IP})^2\mathbf{dSt}_k \to (\mathbf{IP})^2 (\mathbf{dgMod}_k^{\mathrm{op}}) \] \end{lem} \begin{proof} Recall in \cite[part 2.1]{ptvv:dersymp} the construction for any $\mathcal{O}$-compact stack $X$ and any stack $F$ of a commutative diagram (of graded complexes): \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{\mathrm{NC^w}}(F \times X) \ar[r] \ar[d] & \operatorname{\mathrm{NC^w}}(F) \otimes_k \pi_* \mathcal{O}_X \ar[d] \\ \operatorname{\mathbf{DR}}(F \times X) \ar[r] & \operatorname{\mathbf{DR}}(F) \otimes_k \pi_* \mathcal{O}_X } \] where $\pi \colon X \to {}$. Taking the part of weight $p$ and shifting, we get \[ \shorthandoff{;:!?} \xymatrix{ \closedforms p(F \times X) \ar[r] \ar[d] & \closedforms p(F) \otimes_k \pi_ \mathcal{O}_X \ar[d] \\ \forms p(F \times X) \ar[r] & \forms p(F) \otimes_k \pi_* \mathcal{O}_X } \] This construction is functorial in both $F$ and $X$ so it corresponds to a functor \[ \mathbf{dSt}^\mathcal{O}_k \to \operatorname{Fct}(\mathbf{dSt}_k \times \Delta^1 \times \Delta^1, \mathbf{dgMod}_k^{\mathrm{op}}) \] We can now form the functor \begin{align} \mathbf{PI}\mathbf{dSt}^\mathcal{O}_k &\to \mathbf{PI} \operatorname{Fct}\mathopen{}\mathclose\bgroup\originalleft(\operatorname{\mathbf{Pro}} \mathbf{dSt}_k \times \Delta^1 \times \Delta^1, \operatorname{\mathbf{Pro}} (\mathbf{dgMod}_k^{\mathrm{op}}) \aftergroup\egroup\originalright) \\ &\to \operatorname{Fct}\mathopen{}\mathclose\bgroup\originalleft(\operatorname{\mathbf{Pro}} \mathbf{dSt}_k \times \Delta^1 \times \Delta^1, \mathbf{PI} \operatorname{\mathbf{Pro}} (\mathbf{dgMod}_k^{\mathrm{op}}) \aftergroup\egroup\originalright) \\ &\to \operatorname{Fct}\mathopen{}\mathclose\bgroup\originalleft(\mathbf{IP} \mathbf{dSt}_k \times \Delta^1 \times \Delta^1, (\mathbf{IP})^2 (\mathbf{dgMod}_k^{\mathrm{op}}) \aftergroup\egroup\originalright) \\ \end{align} By construction, for any ind-pro-stack $F$ and any $\mathcal{O}$-compact pro-ind-stack, it induces the commutative diagram \[ \shorthandoff{;:!?} \xymatrix{ \closedforms p_{\mathbf{IP}^2}(\psi F \times \phi X) \ar[r] \ar[d] & \psi\IPclosedforms p(F) \otimes_k \phi\mathcal{O}_X \ar[d] \\ \forms p_{\mathbf{IP}^2}(\psi F \times \phi X) \ar[r] & \psi\IPforms p(F) \otimes_k \phi\mathcal{O}_X } \] \end{proof} \begin{rmq} Let us remark that we can informally describe the horizontal maps using the maps from \cite{ptvv:dersymp}: \begin{align} \Theta_{\mathbf{IP}^2}(\psi F \times \phi X) = \lim_\alpha & \colim_\beta \lim_\gamma \colim_\delta \Theta(F_{\alpha\delta} \times X_{\beta\gamma})\\ & \to \lim_\alpha \colim_\beta \lim_\gamma \colim_\delta \Theta(F_{\alpha\delta}) \otimes (\mathcal{O}_{X_{\beta\gamma}}) = \psi \Theta_{\mathbf{IP}}(F) \otimes \phi \mathcal{O}_X \\ \end{align} where $\Theta$ is either $\closedforms p$ or $\forms p$. \end{rmq} \begin{df} Let $F$ be an ind-pro-stack and let $X$ be an $\mathcal{O}$-compact pro-ind-stack. Let $\eta \colon \mathcal{O}_X \to k[-d]$ be a map of ind-pro-$k$-modules. Let finally $\Theta$ be either $\closedforms p$ or $\forms p$. We define the integration map \[ \int_\eta \colon \shorthandoff{;:!?} \xymatrix@1{\Theta_{\mathbf{IP}^2}(\psi F \times \phi X) \ar[r] & \psi \Theta_{\mathbf{IP}}(F) \otimes \phi \mathcal{O}_X \ar[r]^-{\operatorname{id} \otimes \phi \eta} & \psi\Theta_{\mathbf{IP}}(F)[-d]} \] \end{df} \begin{thm}\label{ipdst-form} Let $Y$ be a derived stack and $\omega_Y$ be an $n$-shifted closed $2$-form on $Y$. Let $X$ be an $\mathcal{O}$-compact pro-ind-stack, let $\pi \colon X \to {}$ be the projecton, and let $\eta \colon \pi_ \mathcal{O}_X \to k[-d]$ be a map. The mapping ind-pro-stack $\operatorname{\underline{Ma}p}(X,Y)$ admits an $(n-d)$-shifted closed $2$-form. \end{thm} \begin{proof} Let us denote by $Z$ the mapping ind-pro-stack $\operatorname{Map}(X,Y)$. We consider the diagram \[ \shorthandoff{;:!?} \xymatrix@1{ \chi k[-n] \ar[r]^-{\omega_Y} & \chi \closedforms 2 (Y) \ar[r]^-{\operatorname{ev}^} & \closedforms 2_{\mathbf{IP}^2} (X \times Z) \ar[r]^-{\int_\eta} & \psi\closedforms 2_{\mathbf{IP}} (Z)[-d] } \] where $\chi \colon \mathbf{dgMod}_k^{\mathrm{op}} \to^\xi \mathbf{IP}(\mathbf{dgMod}_k^{\mathrm{op}}) \to^\psi (\mathbf{IP})^2(\mathbf{dgMod}_k^{\mathrm{op}})$ is the canonical inclusion. Note that since the functor $\psi$ is fully faithful, this induces a map in $\mathbf{IP}(\mathbf{dgMod}_k^{\mathrm{op}})$ \[ \shorthandoff{;:!?} \xymatrix@1{ \xi k \ar[r] & \IPclosedforms 2 (Z)[n-d]} \] and therefore a $(n-d)$-shifted closed $2$-form on $Z = \operatorname{Map}(X,Y)$. The underlying form is given by the composition \[ \shorthandoff{;:!?} \xymatrix@1{ \chi k[-n] \ar[r]^-{\omega_Y} & \chi \closedforms 2 (Y) \ar[r] & \chi \forms 2 (Y) \ar[r]^-{\operatorname{ev}^} & \forms 2_{\mathbf{IP}^2} (X \times Z) \ar[r]^-{\int_\eta} & \psi\forms 2_{\mathbf{IP}} (Z)[-d] } \] \end{proof} \begin{rmq} \label{describe-form} Let us describe the form issued by \autoref{ipdst-form}. We set the notations $X = \lim_\alpha \colim_\beta X_{\alpha\beta}$ and $Z_{\alpha\beta} = \operatorname{Map}(X_{\alpha\beta},Y)$. By assumption, we have a map \[ \eta \colon \colim_\alpha \lim_\beta \mathcal{O}_{X_{\alpha\beta}} \to k[-d] \] For any $\alpha$, there exists therefore $\beta(\alpha)$ and a map $\eta_{\alpha\beta(\alpha)} \colon \mathcal{O}_{X_{\alpha\beta(\alpha)}} \to k[-d]$ in $\mathbf{dgMod}(k)$. Unwinding the definitions, we see that the induced form $\int_\eta \omega_Y$ \[ \shorthandoff{;:!?} \xymatrix@1{ \xi k \ar[r] & \IPforms 2 (\operatorname{\underline{Ma}p}(X,Y))[n-d] \simeq \lim_\alpha \colim_\beta \forms 2 (Z_{\alpha\beta})[n-d] } \] is the universal map obtained from the maps \[ \shorthandoff{;:!?} \xymatrix@1{ k \ar[r]^-{\omega_{\alpha\beta(\alpha)}} & \forms 2 (Z_{\alpha\beta(\alpha)})[n-d] \ar[r] & \colim_\beta \forms 2 (Z_{\alpha\beta})[n-d] } \] where $\omega_{\alpha\beta(\alpha)}$ is built using $\eta_{\alpha\beta(\alpha)}$ and the procedure of \cite{ptvv:dersymp}. Note that $\omega_{\alpha\beta(\alpha)}$ can be seen as a map $\mathbb{T}_{X_{\alpha\beta(\alpha)}} \otimes \mathbb{T}_{X_{\alpha\beta(\alpha)}} \to \mathcal{O}_{X_{\alpha\beta(\alpha)}}$. We also know from \autoref{prop-formsareforms} that the form $\int_\eta \omega_Y$ induces a map \[ \mathbb{T}_Z \otimes \mathbb{T}_Z \to \mathcal{O}_Z[n-d] \] in $\operatorname{\mathbf{IPP}}(Z)$. Let us fix $\alpha_0$ and pull back the map above to $Z_{\alpha_0}$. We get \[ \colim_{\alpha \geq \alpha_0} \lim_\beta g_{\alpha_0\alpha}^* p_{\alpha\beta}^* ( \mathbb{T}_{Z_{\alpha\beta}} \otimes \mathbb{T}_{Z_{\alpha\beta}}) \simeq i_{\alpha_0}^* (\mathbb{T}_Z \otimes \mathbb{T}_Z) \to \mathcal{O}_{Z_{\alpha_0}}[n-d] \] This map is the universal map obtained from the maps \begin{align} \lim_\beta g_{\alpha_0\alpha}^ p_{\alpha\beta}^* ( \mathbb{T}_{Z_{\alpha\beta}} \otimes \mathbb{T}_{Z_{\alpha\beta}}) \to{} & g_{\alpha_0\alpha}^* p_{\alpha\beta(\alpha)}^* ( \mathbb{T}_{Z_{\alpha\beta(\alpha)}} \otimes \mathbb{T}_{Z_{\alpha\beta(\alpha)}}) \\ &\to g_{\alpha_0\alpha}^* p_{\alpha\beta(\alpha)}^* (\mathcal{O}_{X_{\alpha\beta(\alpha)}})[n-d] \simeq \mathcal{O}_{X_{\alpha_0}}[n-d] \end{align} where $g_{\alpha_0\alpha}$ is the structural map $Z_{\alpha_0} \to Z_\alpha$ and $p_{\alpha\beta}$ is the projection $Z_\alpha = \lim_\beta Z_{\alpha\beta} \to Z_{\alpha\beta}$. \end{rmq} \subsection{Mapping stacks have a Tate structure} \newcommand{\operatorname{coker}}{\operatorname{coker}} \begin{df}\label{map-cotate} Let $S$ be an $\mathcal{O}$-compact pro-ind-stack. We say that $S$ is an $\mathcal{O}$-Tate stack if there exist a poset $K$ and a diagram $\bar S \colon K^{\mathrm{op}} \to \Indu U \mathbf{dSt}_k$ such that \begin{enumerate} \item The limit of $\bar S$ in $\mathbf{PI}\mathbf{dSt}_k$ is equivalent to $S$ ;\label{map-diagproj} \item For any $i \leq j \in K$ the pro-module over $\bar S(i)$ \label{map-structuralring} \[ \operatorname{coker}\mathopen{}\mathclose\bgroup\originalleft(\mathcal{O}_{\bar S(i)} \to \bar S(i \leq j)_ \mathcal{O}_{\bar S(j)} \aftergroup\egroup\originalright) \] is trivial in the pro-direction -- ie belong to $\mathbf{Qcoh}(\bar S(i))$. \item For any $i \leq j \in K$ the induced map $\bar S(i \leq j)$ is represented by a diagram \[ \bar f \colon L \times \Delta^1 \to \mathbf{dSt}_k \] such that \begin{itemize} \item For any $l \in L$ the projections $\bar f(l,0) \to {}$ and $\bar f(l,1) \to {}$ satisfy the base change formula ; \item For any $l \in L$ the map $\bar f(l)$ satisfies the base change and projection formulae ; \item For any $m \leq l \in L$ the induced map $\bar f(m \leq l, 0)$ satisfies the base change and projection formulae. \end{itemize} \end{enumerate} \end{df} \begin{rmq} We will usually work with pro-ind-stacks $S$ given by an explicit diagram already satisfying those assumptions. \end{rmq} \begin{prop}\label{map-tate} Let us assume that $Y$ is a derived Artin stack locally of finite presentation. Let $S$ be an $\mathcal O$-compact pro-ind-stack. If $S$ is an $\mathcal{O}$-Tate stack then the ind-pro-stack $\operatorname{\underline{Ma}p}(S,Y)$ is a Tate stack. \end{prop} \begin{proof} Let $Z = \operatorname{Map}(S,Y)$ as an ind-pro-stack. Let $\bar S \colon K^{\mathrm{op}} \to \Indu U \mathbf{dSt}_k$ be as in \autoref{map-cotate}. We will denote by $\bar Z \colon K \to \Prou U\mathbf{dSt}_k$ the induced diagram and for any $i \in K$ by $s_i \colon \bar Z(i) \to \bar Z$ the induced map. Let us first remark that $Z$ is an Artin ind-pro-stack locally of finite presentation. It suffices to prove that $s_i^* \mathbb{L}_{Z}$ is a Tate module on $\bar Z(i)$, for any $i \in K$. Let us fix such an $i$ and denote by $Z_i$ the pro-stack $\bar Z(i)$. We consider the differential map \[ s_i^* \mathbb{L}_{Z} \to \mathbb{L}_{Z_i} \] It is by definition equivalent to the natural map \[ \lim \lambda^{\operatorname{\mathbf{Pro}}}_{Z_i}(\bar Z\|_{K^{\geq i}}) \to^f \lambda^{\operatorname{\mathbf{Pro}}}_{Z_i}(Z_i) \] where $K^{\geq i}$ is the comma category $\comma{i}{K}$ and $\bar Z\|_{K^{\geq i}}$ is the induced diagram \[ K^{\geq i} \to \comma{Z_i}{\Prou U\mathbf{dSt}_S} \] Let $\phi_i$ denote the diagram \[ \phi_i \colon \mathopen{}\mathclose\bgroup\originalleft(K^{\geq i}\aftergroup\egroup\originalright)^{\mathrm{op}} \to \operatorname{\mathbf{IPerf}}(Z_i) \] obtained as the kernel of $f$. It is now enough to prove that $\phi_i$ factors through $\mathbf{Perf}(Z_i)$. Let $j \geq i$ in $K$ and let us denote by $g_{ij}$ the induced map $Z_i \to Z_j$ of pro-stacks. Let $\bar f \colon L \times \Delta^1 \to \mathbf{dSt}_k$ represents the map $\bar S(i \leq j) \colon \bar S(j) \to \bar S(i) \in \Indu U \mathbf{dSt}_k$ as in assumption \ref{map-diagproj} in \autoref{map-cotate}. Up to a change of $L$ through a cofinal map, we can assume that the induced diagram \[ \operatorname{coker}\mathopen{}\mathclose\bgroup\originalleft(\mathcal{O}_{\bar S(i)} \to \bar S(i \leq j)_* \mathcal{O}_{\bar S(j)}\aftergroup\egroup\originalright) \] is essentially constant -- see assumption \ref{map-structuralring}. We denote by $\bar h \colon L^{\mathrm{op}} \times \Delta^1 \to \mathbf{dSt}_k$ the induced diagram, so that $g_{ij}$ is the limit of $\bar h$ in $\Prou U \mathbf{dSt}_k$. For any $l \in L$ we will denote by $h_l \colon Z_{il} \to Z_{jl}$ the map $\bar h(l)$. Let us denote by $\bar Z_i$ the induced diagram $l \mapsto Z_{il}$ and by $\bar Z_j$ the diagram $l \mapsto Z_{jl}$. Let also $p_l$ denote the projection $Z_i \to Z_{il}$ We have an exact sequence \[ \phi_i(j) \to \colim_l p_l^* h_l^* \mathbb{L}_{Z_{jl}} \to \colim_l p_l^* \mathbb{L}_{Z_{il}} \] Let us denote by $\psi_{ij}$ the diagram obtained as the kernel \[ \psi_{ij} \to \lambda^{\operatorname{\mathbf{Pro}}}_{Z_i}(\bar Z_j) \to \lambda^{\operatorname{\mathbf{Pro}}}_{Z_i}(\bar Z_i) \] so that $\phi_i(j)$ is the colimit $\colim \psi_{ij}$ in $\operatorname{\mathbf{IPerf}}(Z_i)$. It suffices to prove that the diagram $\psi_{ij} \colon L \to \mathbf{Perf}(Z_i)$ is essentially constant (up to a cofinal change of posets). By definition, we have \[ \psi_{ij}(l) \simeq p_l^* \mathbb{L}_{Z_{il}/Z_{jl}} [-1] \] Let $m \to l$ be a map in $L$ and $t$ the induced map $Z_{il} \to Z_{im}$. The map $\psi_{ij}(m \to l)$ is equivalent to the map $p_l^* \xi$ where $\xi$ fits in the fibre sequence in $\mathbf{Perf}(Z_{il})$ \[ \shorthandoff{;:!?} \xymatrix{ t^* \mathbb{L}_{Z_{im}/Z_{jm}} [-1] \ar[r] \ar[d]_\xi & t^* h_m^* \mathbb{L}_{Z_{jm}} \ar[d] \ar[r] & t^* \mathbb{L}_{Z_{im}} \ar[d] \\ \mathbb{L}_{Z_{il}/Z_{jl}} [-1] \ar[r] & h_l^* \mathbb{L}_{Z_{jl}} \ar[r] & \mathbb{L}_{Z_{il}} } \] We consider the dual diagram \[ \shorthandoff{;:!?} \xymatrix{ t^* \mathbb{T}_{Z_{im}/Z_{jm}} [1] \ar@{<-}[r] \ar@{<-}[d] & t^* h_m^* \mathbb{T}_{Z_{jm}} \ar@{<-}[d] \ar@{<-}[r] & t^* \mathbb{T}_{Z_{im}} \ar@{<-}[d] \\ \mathbb{T}_{Z_{il}/Z_{jl}} [1] \ar@{<-}[r] & h_l^* \mathbb{T}_{Z_{jl}} \ar@{<-}[r] & \mathbb{T}_{Z_{il}} \ar@{}[ul]\|{(\sigma)} } \] Using base change along the maps from $S_{im}$, $S_{jm}$ and $S_{jl}$ to the point, we get that the square $(\sigma)$ is equivalent to \[ \shorthandoff{;:!?} \xymatrix{ \pi_* (\operatorname{id} \times s f_m)_* (\operatorname{id} \times s f_m)^* E & \ar[l] \pi_* (\operatorname{id} \times s)_* (\operatorname{id} \times s)^* E \\ \pi_* (\operatorname{id} \times f_l)_* (\operatorname{id} \times f_l)^* E \ar[u] & \pi_* E \ar[l] \ar[u] } \] where $\pi \colon Z_{il} \times S_{il} \to Z_{il}$ is the projection, where $s \colon S_{im} \to S_{il}$ is the map induced by $m \to l$ and where $E \simeq \operatorname{ev}^* \mathbb{T}_Y$ with $\operatorname{ev} \colon Z_{il} \times S_{il} \to Y$ the evaluation map. Note that we use here the well known fact $\mathbb{T}_{\operatorname{Map}(X,Y)} \simeq \operatorname{pr}_* \operatorname{ev}^* \mathbb{T}_Y$ where \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{Map}(X,Y) & \operatorname{Map}(X,Y) \times X \ar[r]^-\operatorname{ev} \ar[l]_\operatorname{pr} & Y } \] are the canonical maps. Now using the projection and base change formulae along the morphisms $s$, $f_l$ and $f_m$ we get that $(\sigma)$ is equivalent to the image by $\pi_$ of the square \[ \shorthandoff{;:!?} \xymatrix{ E \otimes p^ s_* {f_m}_* \mathcal{O}_{S_{jm}} & E \otimes p^* s_* \mathcal{O}_{S_{im}} \ar[l] \\ E \otimes p^* {f_l}_* \mathcal{O}_{S_{jl}} \ar[u] & E \otimes p^* \mathcal{O}_{S_{il}} \ar[u] \ar[l] } \] We therefore focus on the diagram \[ \shorthandoff{;:!?} \xymatrix{ s_* {f_{m}}_* \mathcal{O}_{S_{jm}} & s_* \mathcal{O}_{S_{im}} \ar[l] \\ {f_l}_* \mathcal{O}_{S_{jl}} \ar[u] & \mathcal{O}_{S_{il}} \ar[l] \ar[u] } \] The map induced between the cofibres is an equivalence, using assumption \ref{map-structuralring}. It follows that the diagram $\psi_{ij}$ is essentially constant, and thus that $Z$ is a Tate stack. \end{proof} \section{Formal loops}\label{chapterfloops} In this part, we will at last define and study the higher dimensional formal loop spaces. We will prove it admits a local Tate structure. \subsection{Dehydrated algebras and de Rham stacks} In this part, we define a refinement of the reduced algebra associated to a cdga. This allows us to define a well behaved de Rham stack associated to an infinite stack. Indeed, without any noetherian assumption, the nilradical of a ring -- the ideal of nilpotent elements -- is a priori not nilpotent itself. The construction below gives an alternative definition of the reduced algebra -- which we call the dehydrated algebra -- associated to any cdga $A$, so that $A$ is, in some sense, a nilpotent extension of its dehydrated algebra. Whenever $A$ is finitely presented, this construction coincides with the usual reduced algebra. \begin{df} Let $A \in \cdgaunbounded^{\leq 0}_k$. We define its dehydrated algebra as the ind-algebra $A_\mathrm{deh} = \colim_{I} \quot{\mathrm H^0(A)}{I}$ where the colimit is taken over the filtered poset of nilpotent ideals of $\mathrm H^0(A)$. The case $I = 0$ gives a canonical map $A \to A_\mathrm{deh}$ in ind-cdga's. This construction is functorial in $A$. \end{df} \begin{rmq} Whenever $A$ is of finite presentation, then $A_\mathrm{deh}$ is equivalent to the reduced algebra associated to $A$. In that case, the nilradical $\sqrt{A}$ of $A$ is nilpotent. Moreover, if $A$ is any cdga, it is a filtered colimits of cdga's $A_\alpha$ of finite presentation. We then have $A_\mathrm{deh} \simeq \colim (A_\alpha)_\mathrm{red}$ in ind-algebras. \end{rmq} \begin{lem} The colimit $B$ of the ind-algebra $A_\mathrm{deh}$ in the category of algebras is equivalent to the reduced algebra $A_\mathrm{red}$. \end{lem} \begin{proof} Let us first remark that $B$ is reduced. Indeed any nilpotent element $x$ of $B$ comes from a nilpotent element of $A$. It therefore belongs to a nilpotent ideal $(x)$. This define a natural map of algebras $A_\mathrm{red} \to B$. To see that it is an isomorphism, it suffices to say that $\sqrt{A}$ is the union of all nilpotent ideals. \end{proof} \begin{df} Let $X$ be a prestack. We define its de Rham prestack $X_\mathrm{dR}$ as the composition \[ \shorthandoff{;:!?} \xymatrix{ \cdgaunbounded^{\leq 0}_k \ar[r]^-{(-)_\mathrm{deh}} & \Indu U(\cdgaunbounded^{\leq 0}_k) \ar[r]^-{\Indu U(X)} & \Indu U(\mathbf{sSets}) \ar[r]^-{\colim} & \mathbf{sSets} } \] This defines an endofunctor of $(\infty,1)$-category $\operatorname{\mathcal{P}}(\mathbf{dAff}_k)$. We have by definition \[ X_\mathrm{dR}(A) = \colim_{I} X\mathopen{}\mathclose\bgroup\originalleft( \quot{\mathrm H^0(A)}{I} \aftergroup\egroup\originalright) \] \end{df} \begin{rmq} If $X$ is a stack of finite presentation, then it is determined by the images of the cdga's of finite presentation. The prestack $X_\mathrm{dR}$ is then the left Kan extension of the functor \[ \app{\mathbf{cdga}_k^{\leq 0\mathrm{,fp}}}{\mathbf{sSets}}{A}{X(A_\mathrm{red})} \] \end{rmq} \begin{df} Let $f \colon X \to Y$ be a functor of prestacks. We define the formal completion $\hat X_Y$ of $X$ in $Y$ as the fibre product \[ \shorthandoff{;:!?} \xymatrix{ \hat X_Y \cart \ar[r] \ar[d] & X_\mathrm{dR} \ar[d] \\ Y \ar[r] & Y_\mathrm{dR} } \] This construction obviously defines a functor $\mathrm{FC} \colon \operatorname{\mathcal{P}}(\mathbf{dAff}_k)^{\Delta^1} \to \operatorname{\mathcal{P}}(\mathbf{dAff}_k)$. \end{df} \begin{rmq} The natural map $\hat X_Y \to Y$ is formally étale, in the sense that for any $A \in \cdgaunbounded^{\leq 0}_k$ and any nilpotent ideal $I \subset \mathrm H^0(A)$ the morphism \[ \hat X_Y (A) \to \hat X_Y \mathopen{}\mathclose\bgroup\originalleft(\textstyle \quot{\mathrm H^0(A)}{I} \aftergroup\egroup\originalright) \timesunder[Y\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I} \aftergroup\egroup\originalright)][][-4pt] Y(A) \] is an equivalence. \end{rmq} \subsection{Higher dimensional formal loop spaces} Here we finally define the higher dimensional formal loop spaces. To any cdga $A$ we associate the formal completion $V_A^d$ of $0$ in $\mathbb{A}^d_A$. We see it as a derived affine scheme whose ring of functions $A[\![X_{1\dots d}]\!]$ is the algebra of formal series in $d$ variables $\el{X}{d}$. Let us denote by $U_A^d$ the open subscheme of $V_A^d$ complementary of the point $0$. We then consider the functors $\mathbf{dSt}_k \times \cdgaunbounded^{\leq 0}_k \to \mathbf{sSets}$ \begin{align} & \tilde{\mathcal L}^d_V \colon (X,A) \mapsto \operatorname{Map}_{\mathbf{dSt}_k}(V_A^d, X) \\ & \tilde{\mathcal L}^d_U \colon (X,A) \mapsto \operatorname{Map}_{\mathbf{dSt}_k}(U_A^d, X) \end{align} \begin{df} Let us consider the functors $\tilde{\mathcal L}_U^d$ and $\tilde{\mathcal L}_V^d$ as functors $\mathbf{dSt}_k \to \operatorname{\mathcal{P}}(\mathbf{dAff})$. They come with a natural morphism $\tilde{\mathcal L}_V^d \to \tilde{\mathcal L}_U^d$. We define $\tilde{\mathcal L}^d$ to be the pointwise formal completion of $\tilde{\mathcal L}_V^d$ into $\tilde{\mathcal L}_U^d$ : \[ \tilde{\mathcal L}^d(X) = \mathrm{FC}\mathopen{}\mathclose\bgroup\originalleft(\tilde{\mathcal L}^d_V(X) \to \tilde{\mathcal L}^d_U(X)\aftergroup\egroup\originalright) \] We also define $\mathcal L^d$, $\mathcal L^d_U$ and $\mathcal L^d_V$ as the stackified version of $\tilde{\mathcal L}^d$, $\tilde{\mathcal L}^d_U$ and $\tilde{\mathcal L}^d_V$ respectively. We will call $\mathcal L^d(X)$ the formal loop stack in $X$. \end{df} \begin{rmq} The stack $\mathcal L^d_V(X)$ is a higher dimensional analogue to the stack of germs in $X$, as studied for instance by Denef and Loeser in \cite{denefloeser:germs}. \end{rmq} \begin{rmq} By definition, the derived scheme $U_A^d$ is the (finite) colimit in derived stacks \[ U_A^d = \colim_q \colim_{\el{i}{q}} \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( A[\![X_{1\dots d}]\!][X^{-1}_{i_1\dots i_q}] \aftergroup\egroup\originalright) \] where $A[\![X_{1\dots d}]\!][X^{-1}_{i_1\dots i_q}]$ denote the algebra of formal series localized at the generators $\iel{X^{-1}}{q}$. It follows that the space of $A$-points of $\mathcal L^d(X)$ is equivalent to the simplicial set \[ \mathcal L^d(X)(A) \simeq \colim _{I\subset \mathrm H^0(A)} \lim_q \lim_{\el{i}{q}} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft( \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft(A[\![X_{1\dots d}]\!][X^{-1}_{i_1\dots i_q}]^{\sqrt{I}}\aftergroup\egroup\originalright), X \aftergroup\egroup\originalright) \] where $A[\![X_{1 \dots d}]\!][X^{-1}_{i_1\dots i_q}]^{\sqrt{I}}$ is the sub-cdga of $A[\![X_{1 \dots d}]\!][X^{-1}_{i_1\dots i_q}]$ consisting of series \[ \sum_{\el{n}{d}} a_{\el{n}{d}} X_1^{n_1} \dots X_d^{n_d} \] where $a_{\el{n}{d}}$ is in the kernel of the map $A \to \quot{\mathrm H^0(A)}{I}$ as soon as at least one of the $n_i$'s is negative. Recall that in the colimit above, the symbol $I$ denotes a nilpotent ideal of $\mathrm H^0(A)$. \end{rmq} \begin{lem}\label{LV-epi} Let $X$ be a derived Artin stack of finite presentation with algebraisable diagonal (see \autoref{alg-diag}) and let $t \colon T = \operatorname{Spec}(A) \to X$ be a smooth atlas. The induced map $\mathcal L_V^d(T) \to \mathcal L_V^d(X)$ is an epimorphism of stacks. \end{lem} \begin{proof} It suffices to study the map $\tilde{\mathcal L}_V^d(T) \to \tilde{\mathcal L}_V^d(X)$. Let $B$ be a cdga. Let us consider a $B$-point $x \colon \operatorname{Spec} B \to \tilde{\mathcal L}_V^d(X)$. It induces a $B$-point of $X$ \[ \operatorname{Spec} B \to \operatorname{Spec}(B[\![X_{1\dots d}]\!]) \to^x X \] Because $t$ is an epimorphism, there exists an étale map $f \colon \operatorname{Spec} C \to \operatorname{Spec} B$ and a commutative diagram \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{Spec} C \ar[r]^-c \ar[d]_f & T \ar[d]^t \\ \operatorname{Spec} B \ar[r] & X } \] It corresponds to a $C$-point of $\operatorname{Spec} B \times_X T$. For any $n \in \mathbb{N}$, let us denote by $C_n$ the cdga \[ C_n := \quot{C[\el{x}{d}]}{(\el{x^n}{d})} \] and by $S_n$ the spectrum $\operatorname{Spec} C_n$ We also set $B_n = \quot{B[\el{x}{d}]}{(\el{x^n}{d})}$ and $X_n = \operatorname{Spec} B_n$. Finally, we define $T_n$ as the pullback $T \times_X X_n$. We will also consider the natural fully faithful functor $\Delta^n \simeq \{0,\dots,n\} \to \mathbb{N}$. We have a natural diagram \[ \alpha_0 \colon \Lambda^{2,2} \times \mathbb{N} \amalg_{\Lambda^{2,2} \times \Delta^0} \Delta^2 \times \Delta^0 \to \mathbf{dSt}_k \] informally drown has a commutative diagram \[ \shorthandoff{;:!?} \xymatrix@R=0pt{ S_0 \ar[d] \ar@/_20pt/[dd] \ar[r] & \dots \ar[r] & S_n \ar[r] \ar[d] & \dots \\ X_0 \ar[r] & \dots \ar[r] & X_n \ar[r] & \dots \\ T_0 \ar[u] \ar[r] & \dots \ar[r] & T_n \ar[r] \ar[u] & \dots } \] Let $n \in \mathbb{N}$ and let us assume we have built a diagram \[ \alpha_n \colon (\Lambda^{2,2} \times \mathbb{N}) \amalg_{\Lambda^{2,2} \times \Delta^n} \Delta^2 \times \Delta^n \to \mathbf{dSt}_k \] extending $\alpha_{n-1}$. There is a sub-diagram of $\alpha_n$ \[ \shorthandoff{;:!?} \xymatrix{ S_n \ar[r] \ar[d] & S_{n+1} \\ T_n \ar[r] & T_{n+1} \ar[d]^{t_{n+1}} \\ & X_{n+1} } \] Since the map $t_{n+1}$ is smooth (it is a pullback of $t$), we can complete this diagram with a map $S_{n+1} \to T_{n+1}$ and a commutative square. Using the composition in $\mathbf{dSt}_k$, we get a diagram $\alpha_{n+1}$ extending $\alpha_n$. We get recursively a diagram $\alpha \colon \Delta^2 \times \mathbb{N} \to \mathbf{dSt}_k$. Taking the colimit along $\mathbb{N}$, we get a commutative diagram \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{Spec} C \ar[d]_f \ar[r] & \colim_n \operatorname{Spec} C_n \ar[d] \ar[rr] && T \ar[d]^t \\ \operatorname{Spec} B \ar[r] & \colim_n \operatorname{Spec} B_n \ar[r] & \operatorname{Spec}(B[\![X_{1\dots d}]\!]) \ar[r] & X } \] This defines a map $\phi \colon \colim \operatorname{Spec}(C_n) \to \operatorname{Spec}(B[\![X_{1\dots d}]\!]) \times_X T$. We have the cartesian diagram \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{Spec}(B[\![X_{1\dots d}]\!]) \times_X T \ar[r] \ar[d] \cart & X \ar[d] \\ \operatorname{Spec}(B[\![X_{1 \dots d}]\!]) \times T \ar[r] & X \times X } \] The diagonal of $X$ is algebraisable and thus so is the stack $\operatorname{Spec}(B[\![X_{1\dots d}]\!]) \times_X T$. The morphism $\phi$ therefore defines the required map \[ \operatorname{Spec}(C[\![X_{1 \dots d}]\!]) \to \operatorname{Spec}(B[\![X_{1\dots d}]\!]) \times_X T \] \end{proof} \begin{rmq} Let us remark here that if $X$ is an algebraisable stack, then $\tilde{\mathcal L}_V^d(X)$ is a stack, hence the natural map is an equivalence \[ \tilde{\mathcal L}_V^d(X) \simeq \mathcal L_V^d(X) \] \end{rmq} \begin{lem}\label{LU-fetale} Let $f \colon X \to Y$ be an étale map of derived Artin stacks. For any cdga $A \in \cdgaunbounded^{\leq 0}_k$ and any nilpotent ideal $I \subset \mathrm H^0(A)$, the induced map \[ \theta \colon \shorthandoff{;:!?} \xymatrix{ \tilde{\mathcal L}^d_U(X)(A) \ar[r] & \tilde{\mathcal L}^d_U(X)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \displaystyle \timesunder[\tilde{\mathcal L}^d_U(Y)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright)][][-4pt] \tilde{\mathcal L}^d_U(Y)(A) } \] is an equivalence. \end{lem} \begin{proof} The map $\theta$ is a finite limit of maps \[ \mu \colon \shorthandoff{;:!?} \xymatrix{ X(\xi A) \ar[r] & X\mathopen{}\mathclose\bgroup\originalleft(\xi \mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \aftergroup\egroup\originalright) \displaystyle \timesunder[Y\mathopen{}\mathclose\bgroup\originalleft(\xi\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \aftergroup\egroup\originalright)][][-4pt] Y (\xi A ) } \] where $\xi A = A[\![X_{1 \dots d}]\!][X^{-1}_{i_1\dots i_p}]$ is obtained from the cdga of formal power series in $A$ with $d$ variables by inverting the variables $x_{i_j}$. Let also $\xi (\quot{\mathrm H^0(A)}{I})$ be defined similarly. The natural map $\xi(\mathrm H^0(A)) \to \xi (\quot{\mathrm H^0(A)}{I})$ is also a nilpotent extension. We deduce from the étaleness of $f$ that the map \[ \shorthandoff{;:!?} \xymatrix{ X(\xi(\mathrm H^0(A))) \ar[r] & X\mathopen{}\mathclose\bgroup\originalleft(\xi \mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \aftergroup\egroup\originalright) \displaystyle \timesunder[Y\mathopen{}\mathclose\bgroup\originalleft(\xi\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \aftergroup\egroup\originalright)][][-4pt] Y (\xi( \mathrm H^0(A)) ) } \] is an equivalence. Let now $n \in \mathbb{N}$. We assume that the natural map \[ \shorthandoff{;:!?} \xymatrix{ X(\xi(A_{\leq n})) \ar[r] & X\mathopen{}\mathclose\bgroup\originalleft(\xi \mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \aftergroup\egroup\originalright) \displaystyle \timesunder[Y\mathopen{}\mathclose\bgroup\originalleft(\xi\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \aftergroup\egroup\originalright)][][-4pt] Y (\xi(A_{\leq n})) } \] is an equivalence. The cdga $\xi(A_{\leq n+1}) \simeq (\xi A)_{\leq n+1}$ is a square zero extension of $\xi(A_{\leq n})$ by $\mathrm H^{-n-1}(\xi A)$. We thus have the equivalence \[ \shorthandoff{;:!?} \xymatrix{ X(\xi(A_{\leq n+1})) \ar[r]^-\sim & X(\xi(A_{\leq n})) \displaystyle \timesunder[Y(\xi(A_{\leq n}))][][-3pt] Y(\xi(A_{\leq n+1})) } \] The natural map \[ \shorthandoff{;:!?} \xymatrix{ X(\xi(A_{\leq n+1})) \ar[r] & X\mathopen{}\mathclose\bgroup\originalleft(\xi \mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \aftergroup\egroup\originalright) \displaystyle \timesunder[Y\mathopen{}\mathclose\bgroup\originalleft(\xi\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \aftergroup\egroup\originalright)][][-4pt] Y (\xi(A_{\leq n+1})) } \] is thus an equivalence too. The stacks $X$ and $Y$ are nilcomplete, hence $\mu$ is also an equivalence -- recall that a derived stack $X$ is nilcomplete if for any cdga $B$ we have \[ X(B) \simeq \lim_n X(B_{\leq n}) \] Also recall that any Artin stack is nilcomplete. It follows that $\theta$ is an equivalence. \end{proof} \begin{cor}\label{L-fetale} Let $f \colon X \to Y$ be an étale map of derived Artin stacks. For any cdga $A \in \cdgaunbounded^{\leq 0}_k$ and any nilpotent ideal $I \subset \mathrm H^0(A)$, the induced map \[ \theta \colon \shorthandoff{;:!?} \xymatrix{ \tilde{\mathcal L}^d(X)(A) \ar[r] & \tilde{\mathcal L}^d(X)\mathopen{}\mathclose\bgroup\originalleft(\textstyle \quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \displaystyle \timesunder[\tilde{\mathcal L}^d(Y)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright)][][-4pt] \tilde{\mathcal L}^d(Y)(A) } \] is an equivalence. \end{cor} \begin{prop}\label{L-epi} Let $X$ be a derived Deligne-Mumford stack of finite presentation with algebraisable diagonal. Let $t \colon T \to X$ be an étale atlas. The induced map $\mathcal L^d(T) \to \mathcal L^d(X)$ is an epimorphism of stacks. \end{prop} \begin{proof} We can work on the map of prestacks $\tilde{\mathcal L}^d(T) \to \tilde{\mathcal L}^d(X)$. Let $A \in \cdgaunbounded^{\leq 0}_k$. Let $x$ be an $A$-point of $\tilde{\mathcal L}^d(X)$. It corresponds to a vertex in the simplicial set \[ \shorthandoff{;:!?} \xymatrix{ \displaystyle \colim_I \textstyle {\tilde{\mathcal L}_V^d(X)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright)} \displaystyle \timesunder[\tilde{\mathcal L}_U^d(X)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright)][][-4pt] \tilde{\mathcal L}_U^d(X)(A) } \] There exists therefore a nilpotent ideal $I$ such that $x$ comes from a commutative diagram \[ \shorthandoff{;:!?} \xymatrix{ U_{\quot{\mathrm H^0(A)}{I}}^d \ar[d] \ar[r] & U_A^d \ar[d] \\ V_{\quot{\mathrm H^0(A)}{I}} \ar[r]_-v & X } \] Using \autoref{LV-epi} we get an étale morphism $\psi \colon A \to B$ such that the map $v$ lifts to a map $u \colon V_{\quot{B}{J}} \to T$ where $J$ is the image of $I$ by $\psi$. This defines a point in \[ \textstyle \tilde{\mathcal L}^d_U(T)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(B)}{J} \aftergroup\egroup\originalright) \displaystyle \timesunder[\tilde{\mathcal L}^d_U(X)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(B)}{J}\aftergroup\egroup\originalright)][][-4pt] \tilde{\mathcal L}^d_U(X)(B) \] Because of \autoref{LU-fetale}, we get a point of $\tilde{\mathcal L}^d(T)(B)$. We now observe that this point is compatible with $x$. \end{proof} In the case of dimension $d=1$, \autoref{LU-fetale} can be modified in the following way. Let $f \colon X \to Y$ be a smooth map of derived Artin stacks. For any cdga $A \in \cdgaunbounded^{\leq 0}_k$ and any nilpotent ideal $I \subset \mathrm H^0(A)$, the induced map \[ \theta \colon \shorthandoff{;:!?} \xymatrix{ \tilde{\mathcal L}^1_U(X)(A) \ar[r] & \tilde{\mathcal L}^1_U(X)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright) \displaystyle \timesunder[\tilde{\mathcal L}^1_U(Y)\mathopen{}\mathclose\bgroup\originalleft(\quot{\mathrm H^0(A)}{I}\aftergroup\egroup\originalright)][][-4pt] \tilde{\mathcal L}^1_U(Y)(A) } \] is essentially surjective. The following proposition follows. \begin{prop} Let $X$ be an Artin derived stack of finite presentation and with algebraisable diagonal. Let $t \colon T \to X$ be a smooth atlas. The induced map $\mathcal L^1(T) \to \mathcal L^1(X)$ is an epimorphism of stacks. \end{prop} \begin{ex} The proposition above implies for instance that $\mathcal L^1(\mathrm{B} G) \simeq \mathrm{B} \mathcal L^1(G)$ for any algebraic group $G$ -- where $\mathrm{B} G$ is the classifying stack of $G$-bundles. \end{ex} \subsection{Tate structure and determinantal anomaly}\label{determinantalclass} We saw in \autoref{tatestacks} that to any Tate stack $X$, we can associate a determinantal anomaly. It a class in $\mathrm H^2(X,\mathcal{O}_X^{\times})$. We will prove in this subsection that the stack $\mathcal L^d(X)$ is endowed with a structure of Tate stack as soon as $X$ is affine. We will moreover build a determinantal anomaly on $\mathcal L^d(X)$ for any quasi-compact and separated scheme $X$. \begin{lem}\label{L-shy} For any $B \in \cdgaunbounded^{\leq 0}_k$ of finite presentation, the functors \[ \tilde{\mathcal L}^d_U(\operatorname{Spec} B), \tilde{\mathcal L}^d(\operatorname{Spec} B) \colon \cdgaunbounded^{\leq 0}_k \to \mathbf{sSets} \] are in the essential image of the fully faithful functor \[ \mathbf{IP} \dSt^{\mathrm{shy,b}}_k \cap \mathbf{IP}\mathbf{dAff}_k \to \mathbf{IP}\mathbf{dSt}_k \to \mathbf{dSt}_k \to \operatorname{\mathcal{P}}(\mathbf{dAff}) \] (see \autoref{indprochamps}). It follows that $\tilde{\mathcal L}^d_U(\operatorname{Spec} B) \simeq \mathcal L^d_U(\operatorname{Spec} B)$ and $\tilde{\mathcal L}^d(\operatorname{Spec} B) \simeq \mathcal L^d(\operatorname{Spec} B)$. \end{lem} \begin{proof} Let us first remark that $\operatorname{Spec} B$ is a retract of a \emph{finite} limit of copies of the affine line $\mathbb{A}^1$. It follows that the functor $\tilde{\mathcal L}_U^d(\operatorname{Spec} B)$ is, up to a retract, a finite limit of functors \[ Z_E^d \colon A \mapsto \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(k[Y], A[\![X_{1 \dots d}]\!][X^{-1}_{i_1\dots i_q}]\aftergroup\egroup\originalright) \] where $E = \{ \el{i}{q} \} \subset F = \{1,\dots,d\}$. The functor $Z^d_E$ is the realisation of an affine ind-pro-scheme \[ Z^d_E \simeq \colim_n \lim_p \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( k[ a_{\el{\alpha}{d}}, -n \delta_i \leq \alpha_i \leq p] \aftergroup\egroup\originalright) \] where $\delta_i = 1$ if $i \in E$ and $\delta_i = 0$ otherwise. The variable $a_{\el{\alpha}{d}}$ corresponds to the coefficient of $X_1^{\alpha_1} \dots X_d^{\alpha_d}$. The functor $Z^d_E$ is thus in the category $\mathbf{IP} \dSt^{\mathrm{shy,b}} \cap \mathbf{IP}\mathbf{dAff}_k$. The result about $\tilde{\mathcal L}^d_U(\operatorname{Spec} B)$ then follows from \autoref{shydaff-limits}. The case of $\tilde{\mathcal L}^d(\operatorname{Spec} B)$ is similar: we decompose it into a finite limit of functors \[ G_E^d \colon A \mapsto \colim_{I \subset \mathrm H^0(A)} \operatorname{Map}\mathopen{}\mathclose\bgroup\originalleft(k[Y], A[\![X_{1 \dots d}]\!][X^{-1}_{i_1\dots i_q}]^{\sqrt{I}}\aftergroup\egroup\originalright) \] where $I$ is a nilpotent ideal of $\mathrm H^0(A)$. We then observe that $G_E^d$ is the realisation of the ind-pro-scheme \[ G^d_E \simeq \colim_{n,m} \lim_p \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( \quot{k[ a_{\el{\alpha}{d}}, -n \delta_i \leq \alpha_i \leq p]}{J} \aftergroup\egroup\originalright) \] where $J$ is the ideal generated by the symbols $a_{\el{\alpha}{d}}^m$ with at least one of the $\alpha_i$'s negative. \end{proof} \begin{rmq} Let $n$ and $p$ be integers and let $k(E,n,p)$ denote the number of families $(\el{\alpha}{d})$ such that $-n \delta_i \leq \alpha_i \leq p$ for all $i$. We have \[ Z^d_E \simeq \colim_n \lim_p (\mathbb{A}^1)^{k(E,n,p)} \] \end{rmq} \begin{df} From \autoref{L-shy}, we get a functor $\underline \mathcal L^d \colon \mathbf{dAff}_k^\mathrm{fp} \to \mathbf{IP}\mathbf{dSt}_k$. It follows from \autoref{L-epi} that $\underline \mathcal L^d$ is a costack in ind-pro-stacks. We thus define \[ \underline \mathcal L^d \colon \mathbf{dSt}_k^{\mathrm{lfp}} \to \mathbf{IP}\mathbf{dSt}_k \] to be its left Kan extension along the inclusion $\mathbf{dAff}_k^\mathrm{fp} \to \mathbf{dSt}_k^\mathrm{lfp}$ -- where $\mathbf{dSt}_k^\mathrm{lfp}$ is $(\infty,1)$-category of derived stacks locally of finite presentation. This new functor $\underline \mathcal L^d$ preserves small colimits by definition. \end{df} \begin{prop}\label{L-indpro} There is a natural transformation $\theta$ from the composite functor \[ \shorthandoff{;:!?} \xymatrix{ \mathbf{dSt}_k^\mathrm{lfp} \ar[r]^-{\underline \mathcal L^d} & \mathbf{IP}\mathbf{dSt}_k \ar[r]^{\|-\|^\mathbf{IP}} & \mathbf{dSt}_k } \] to the functor $\mathcal L^d$. Moreover, the restriction of $\theta$ to derived Deligne-Mumford stacks of finite presentation with algebraisable diagonal is an equivalence. \end{prop} \begin{proof} There is by definition a natural transformation \[ \theta \colon \| \underline \mathcal L^d(-) \|^\mathbf{IP} \to \mathcal L^d(-) \] Moreover, the restriction of $\theta$ to affine derived scheme of finite presentation is an equivalence -- see \autoref{L-shy}. The fact that $\theta_X$ is an equivalence for any Deligne-Mumford stack $X$ follows from \autoref{L-epi}. \end{proof} \begin{lem}\label{subsets-colim} Let $F$ be a non-empty finite set. For any family $(M_D)$ of complexes over $k$ indexed by subsets $D$ of $F$, we have \[ \colim_{\emptyset \neq E \subset F} \bigoplus_{\emptyset \neq D \subset E} M_D \simeq M_F[d-1] \] where $d$ is the cardinal of $F$ (the maps in the colimit diagram are the canonical projections). \end{lem} \begin{proof} We can and do assume that $F$ is the finite set $\{1, \dots, d\}$ and we proceed recursively on $d$. The case $d = 1$ is obvious. Let now $d \geq 2$ and let us assume the statement is true for $F \smallsetminus \{d\}$. Let $(M_D)$ be a family as above. We have a cocartesian diagram \[ \shorthandoff{;:!?} \xymatrix{ \displaystyle \colim_{\{d\} \subsetneq E \subset F} \bigoplus_{\emptyset \neq D \subset E} M_D \ar[r] \ar[d] & \displaystyle \colim_{ \emptyset \neq E \subset F\smallsetminus\{d\}} \bigoplus_{\emptyset \neq D \subset E} M_D \ar[d] \\ \displaystyle M_{\{d\}} \ar[r] & \displaystyle \colim_{\emptyset \neq E \subset F} \bigoplus_{\emptyset \neq D \subset E} M_D \cocart } \] We have by assumption \[ \colim_{ \emptyset \neq E \subset F\smallsetminus\{d\}} \bigoplus_{\emptyset \neq D \subset E} M_D \simeq M_{F \smallsetminus \{d\}} [d-2] \] and \begin{align} \colim_{\{d\} \subsetneq E \subset F} \bigoplus_{\emptyset \neq D \subset E} M_D &\simeq M_{\{d\}} \oplus \mathopen{}\mathclose\bgroup\originalleft(\colim_{\{d\} \subsetneq E \subset F} \bigoplus_{\{d\} \subsetneq D \subset E} M_D \aftergroup\egroup\originalright) \oplus \mathopen{}\mathclose\bgroup\originalleft(\colim_{\{d\} \subsetneq E \subset F} \bigoplus_{\emptyset \neq D \subset E \smallsetminus \{d\}} M_D\aftergroup\egroup\originalright) \\ &\simeq M_{\{d\}} \oplus M_F[d-2] \oplus M_{F \smallsetminus \{d\}}[d-2] \end{align} The result follows. \end{proof} \begin{lem}\label{LU-affine-tate} For any $B \in \cdgaunbounded^{\leq 0}_k$ of \emph{finite presentation}, the ind-pro-stack $\underline \mathcal L^d_U(\operatorname{Spec} B)$ is a Tate stack. \end{lem} \begin{proof} Let us first focus on the case of the affine line $\mathbb{A}^1$. We have to prove that the cotangent complex $\mathbb{L}_{\underline \mathcal L^d_U(\mathbb{A}^1)}$ is a Tate module. For any subset $D \subset F$ we define $M_D^{p,n}$ to be the free $k$-complex generated by the symbols \[ \{a_{\el{\alpha}{d}}, -n \leq \alpha_i < 0 \text{ if } i \in D, 0 \leq \alpha_i \leq p \text{ otherwise}\} \] in degree $0$. From the proof of \autoref{L-shy}, we have \[ Z^d_E \simeq \colim_n \lim_p \operatorname{Spec} \mathopen{}\mathclose\bgroup\originalleft( k\mathopen{}\mathclose\bgroup\originalleft[ \textstyle \bigoplus_{D \subset E} M_D^{p,n} \aftergroup\egroup\originalright] \aftergroup\egroup\originalright) \text{~~~ and ~~~} \underline \mathcal L^d_U(\mathbb{A}^1) \simeq \lim_{\emptyset \neq E \subset F} Z^d_E \] where $F = \{1 , \dots , d\}$. If we denote by $\pi$ the projection $\underline \mathcal L_U^d(\mathbb{A}^1) \to \operatorname{Spec} k$, we get \[ \mathbb{L}_{\underline \mathcal L^d_U(\mathbb{A}^1)} \simeq \pi^* \mathopen{}\mathclose\bgroup\originalleft(\colim_{\emptyset \neq E \subset F} \lim_n \colim_p \bigoplus_{D \subset E} M_D^{p,n}\aftergroup\egroup\originalright) \simeq \pi^* \mathopen{}\mathclose\bgroup\originalleft(\lim_n \colim_p \colim_{\emptyset \neq E \subset F} \bigoplus_{D \subset E} M_D^{p,n}\aftergroup\egroup\originalright) \] Using \autoref{subsets-colim} we have \[ \mathbb{L}_{\underline \mathcal L^d_U(\mathbb{A}^1)} \simeq \pi^* \mathopen{}\mathclose\bgroup\originalleft(\lim_n \colim_p M_\emptyset^{p,n} \oplus M_F^{p,n}[d-1] \aftergroup\egroup\originalright) \] Moreover, we have $M_\emptyset^{p,n} \simeq M_\emptyset^{p,0}$ and $M_F^{p,n} \simeq M_F^{0,n}$. It follows that $\mathbb{L}_{\underline \mathcal L^d_U(\mathbb{A}^1)}$ is a Tate module on the ind-pro-stack $\underline \mathcal L^d_U(\mathbb{A}^1)$. The case of $\underline \mathcal L^d_U(\operatorname{Spec} B)$ then follows from \autoref{shydaff-limits} and from \autoref{tate-limits}. \end{proof} \begin{lem}\label{LU-ip-fetale} Let $B \to C$ be an étale map between cdga's of finite presentation. The induced map $f \colon \underline \mathcal L^d_U(\operatorname{Spec} C) \to \underline \mathcal L^d_U(\operatorname{Spec} B)$ is formally étale -- see \autoref{derivation-ipdst}. \end{lem} \begin{proof} Let us denote $X = \operatorname{Spec} B$ and $Y = \operatorname{Spec} C$. We have to prove that the induced map \[ j \colon \operatorname{Map}_{\underline \mathcal L^d_U(Y)/-}\mathopen{}\mathclose\bgroup\originalleft(\underline \mathcal L^d_U(Y)[-], \underline \mathcal L^d_U(Y)\aftergroup\egroup\originalright) \to \operatorname{Map}_{\underline \mathcal L^d_U(Y)/-}\mathopen{}\mathclose\bgroup\originalleft(\underline \mathcal L^d_U(Y)[-], \underline \mathcal L_U^d(X)\aftergroup\egroup\originalright) \] is an equivalence of functors $\operatorname{\mathbf{PIQcoh}}( \underline \mathcal L^d(Y) )^{\leq 0} \to \mathbf{sSets}$. Since $\underline \mathcal L^d_U(Y)$ is ind-pro-affine, we can restrict to the study of the morphism \[ j_Z \colon \operatorname{Map}_{Z/-}\mathopen{}\mathclose\bgroup\originalleft(Z[-], \underline \mathcal L^d_U(Y)\aftergroup\egroup\originalright) \to \operatorname{Map}_{Z/-}\mathopen{}\mathclose\bgroup\originalleft(Z[-], \underline \mathcal L_U^d(X)\aftergroup\egroup\originalright) \] of functors $\operatorname{\mathbf{IQcoh}}(Z)^{\leq 0} \to \mathbf{sSets}$, for any pro-affine scheme $Z$ and any map $Z \to \underline \mathcal L^d_U(Y)$. Let us fix $E \in \operatorname{\mathbf{IQcoh}}(Z)^{\leq 0}$. The pro-stack $Z[E]$ is in fact an affine pro-scheme. Recall that both $\underline \mathcal L^d_U(Y)$ and $\underline \mathcal L^d_U(X)$ belong to $\mathbf{IP} \dSt^{\mathrm{shy,b}}_k$. It follows from the proof of \autoref{ff-realisation} that the morphism $j_Z(E)$ is equivalent to \[ \|j_Z(E)\| \colon \operatorname{Map}_{\|Z\|/-}\mathopen{}\mathclose\bgroup\originalleft( \|Z[E]\|, \mathcal L^d_U(Y) \aftergroup\egroup\originalright) \to \operatorname{Map}_{\|Z\|/-}\mathopen{}\mathclose\bgroup\originalleft( \|Z[E]\|, \mathcal L^d_U(X) \aftergroup\egroup\originalright) \] where $\|-\|$ is the realisation functor and the mapping spaces are computed in $\mathbf{dSt}_k$. It now suffices to see that $\|Z[E]\|$ is a trivial square zero extension of the derived affine scheme $\|Z\|$ and to use \autoref{LU-fetale}. \end{proof} \begin{prop}\label{L-affine-tate} Let $\operatorname{Spec} B$ be a derived affine scheme of finite presentation. The ind-pro-stack $\underline \mathcal L^d(\operatorname{Spec} B)$ admits a cotangent complex. This cotangent complex is moreover a Tate module. For any étale map $B \to C$ the induced map $f \colon \underline \mathcal L^d(\operatorname{Spec} C) \to \underline \mathcal L^d(\operatorname{Spec} B)$ is formally étale -- see \autoref{derivation-ipdst}. \end{prop} \begin{proof} Let us write $Y = \operatorname{Spec} B$. Let us denote by $i \colon \underline \mathcal L^d(Y) \to \underline \mathcal L^d_U(Y)$ the natural map. We will prove that the map $i$ is formally étale, the result will then follow from \autoref{LU-affine-tate} and \autoref{LU-ip-fetale}. To do so, we consider the natural map \[ j \colon \operatorname{Map}_{\underline \mathcal L^d(Y)/-}\mathopen{}\mathclose\bgroup\originalleft(\underline \mathcal L^d(Y)[-], \underline \mathcal L^d(Y)\aftergroup\egroup\originalright) \to \operatorname{Map}_{\underline \mathcal L^d(Y)/-}\mathopen{}\mathclose\bgroup\originalleft(\underline \mathcal L^d(Y)[-], \underline \mathcal L_U^d(Y)\aftergroup\egroup\originalright) \] of functors $\operatorname{\mathbf{PIQcoh}}( \underline \mathcal L^d(Y) )^{\leq 0} \to \mathbf{sSets}$. To prove that $j$ is an equivalence, we can consider for every affine pro-scheme $X \to \underline \mathcal L^d(Y)$ the morphism of functors $\operatorname{\mathbf{IQcoh}}(X)^{\leq 0} \to \mathbf{sSets}$ \[ j_X \colon \operatorname{Map}_{X/-}\mathopen{}\mathclose\bgroup\originalleft( X[-], \underline \mathcal L^d(Y) \aftergroup\egroup\originalright) \to \operatorname{Map}_{X/-}\mathopen{}\mathclose\bgroup\originalleft( X[-], \underline \mathcal L^d_U(Y) \aftergroup\egroup\originalright) \] Let us fix $E \in \operatorname{\mathbf{IQcoh}}(X)^{\leq 0}$. The morphism $j_X(E)$ is equivalent to \[ \|j_X(E)\| \colon \operatorname{Map}_{\|X\|/-}\mathopen{}\mathclose\bgroup\originalleft( \|X[E]\|, \mathcal L^d(Y) \aftergroup\egroup\originalright) \to \operatorname{Map}_{\|X\|/-}\mathopen{}\mathclose\bgroup\originalleft( \|X[E]\|, \mathcal L^d_U(Y) \aftergroup\egroup\originalright) \] where the mapping space are computed in $\mathbf{dSt}_k$. The map $\|j_X(E)\|$ is a pullback of the map \[ f \colon \operatorname{Map}_{\|X\|/-}\mathopen{}\mathclose\bgroup\originalleft( \|X[E]\|, \mathcal L^d_V(Y)_\mathrm{dR} \aftergroup\egroup\originalright) \to \operatorname{Map}_{\|X\|/-}\mathopen{}\mathclose\bgroup\originalleft( \|X[E]\|, \mathcal L^d_U(Y)_\mathrm{dR} \aftergroup\egroup\originalright) \] It now suffices to see that $\|X[E]\|$ is a trivial square zero extension of the derived affine scheme $\|X\|$ and thus $f$ is an equivalence (both of its ends are actually contractible). \end{proof} Let us recall from \autoref{determinantalanomaly} the determinantal anomaly \[ [\mathrm{Det}_{\underline \mathcal L^d(\operatorname{Spec} A)}] \in \mathrm H^2\mathopen{}\mathclose\bgroup\originalleft(\mathcal L^d(\operatorname{Spec} A), \mathcal{O}_{\mathcal L^d(\operatorname{Spec} A)}^{\times}\aftergroup\egroup\originalright) \] It is associated to the tangent $\mathbb{T}_{\underline \mathcal L^d(\operatorname{Spec} A)} \in \Tateu U_\mathbf{IP}(\underline \mathcal L^d(\operatorname{Spec} A))$ through the determinant map. Using \autoref{L-affine-tate}, we see that this construction is functorial in $A$, and from \autoref{L-epi} we get that it satisfies étale descent. Thus, for any quasi-compact and quasi-separated (derived) scheme (or Deligne-Mumford stack with algebraisable diagonal), we have a well-defined determinantal anomaly \[ [\mathrm{Det}_{\underline \mathcal L^d(X)}] \in \mathrm H^2\mathopen{}\mathclose\bgroup\originalleft(\mathcal L^d(X), \mathcal{O}_{\mathcal L^d(X)}^{\times}\aftergroup\egroup\originalright) \] \begin{rmq} It is known since \cite{kapranovvasserot:loop4} that in dimension $d=1$, if $[\mathrm{Det}_{\mathcal L^1(X)}]$ vanishes, then there are essentially no non-trivial automorphisms of sheaves of chiral differential operators on $X$. \end{rmq} \section{Bubble spaces}\label{chapterBubbles} In this section, we study the bubble space, an object closely related to the formal loop space. We will then prove the bubble space to admit a symplectic structure. \subsection{Two lemmas} In this subsection, we will develop two duality results we will need afterwards. Let $A \in \cdgaunbounded^{\leq 0}_k$ be a cdga over a field $k$. Let $(\el{f}{p})$ be points of $A^0$ whose images in $\mathrm H^0(A)$ form a regular sequence. Let us denote by $A_{n,k}$ the Kozsul complex associated to the regular sequence $(\el{f^n}{k})$ for $k \leq p$. We set $A_{n,0} = A$ and $A_n = A_{n,p}$ for any $n$. If $k<p$, the multiplication by $f^n_{k+1}$ induces an endomorphism $\varphi^n_{k+1}$ of $A_{n,k}$. Recall that $A_{n,k+1}$ is isomorphic to the cone of $\varphi^n_{k+1}$: \[ \shorthandoff{;:!?} \xymatrix{ A_{n,k} \ar[r]^{\varphi^n_{k+1}} \ar[d] & A_{n,k} \ar[d] \\ 0 \ar[r] & A_{n,k+1} \cocart } \] Let us now remark that for any couple $(n,k)$, the $A$-module $A_{n,k}$ is perfect. \begin{lem} \label{dual-over-A} Let $k \leq p$. The $A$-linear dual $A_{n,k}^{\quot{\vee}{A}} = \operatorname{\mathbb{R}\underline{Hom}}_A(A_{n,k},A)$ of $A_{n,k}$ is equivalent to $A_{n,k}[-k]$; \end{lem} \begin{proof} We will prove the statement recursively on the number $k$. When $k = 0$, the result is trivial. Let $k \geq 0$ and let us assume that $A_{n,k}^{\quot{\vee}{A}}$ is equivalent to $A_{n,k}[-k]$. Let us also assume that for any $a \in A$, the diagram induced by multiplication by $a$ commutes \[ \shorthandoff{;:!?} \xymatrix{ A_{n,k}^{\quot{\vee}{A}} \ar@{-}[r]^-\sim \ar[d]_{\dual a} & A_{n,k}[-k] \ar[d]^a \\ A_{n,k}^{\quot{\vee}{A}} \ar@{-}[r]^-\sim & A_{n,k}[-k] } \] We obtain the following equivalence of exact sequences \[ \shorthandoff{;:!?} \xymatrix{ A_{n,k+1}[-k-1] \ar[r] \ar@{-}[d]^\sim & A_{n,k}[-k] \ar@{-}[d]^\sim \ar[r]^{\varphi^n_{k+1}} & A_{n,k}[-k] \ar@{-}[d]^\sim \\ A_{n,k+1}^{\quot{\vee}{A}} \ar[r] & A_{n,k}^{\quot{\vee}{A}} \ar[r]^{\dual{\mathopen{}\mathclose\bgroup\originalleft(\varphi^n_{k+1}\aftergroup\egroup\originalright)}} & A_{n,k}^{\quot{\vee}{A}} } \] The statement about multiplication is straightforward. \end{proof} \begin{lem} \label{dual-over-k} Let us assume $A$ is a formal series ring over $A_1$: \[ A = A_1[\![\el{f}{p}]\!] \] It follows that for any $n$, the $A_1$-module $A_n$ is free of finite type and that there is map $r_n \colon A_n \to A_1$ mapping $\el{f^n}{p}[]$ to $1$ and any other generator to zero. We deduce an equivalence \[ A_n \to^\sim A_n^{\quot{\vee}{A_1}} = \operatorname{\mathbb{R}\underline{Hom}}_{A_1}(A_n, A_1) \] given by the pairing \[ \shorthandoff{;:!?} \xymatrix{ A_n \otimes_{A_1} A_n \ar[r]^-{\times} & A_n \ar[r]^{r_n} & A_1 } \] \end{lem} \begin{rmq} Note that we can express the inverse $A_n^{\quot{\vee}{A_1}} \to A_n$ of the equivalence above: it maps a function $\alpha \colon A_n \to A_1$ to the serie \[ \sum_{\underline i} \alpha(f^{\underline i}) f^{n-1-\underline i} \] where $\underline i$ varies through the uplets $(\el{i}{p})$ and where $f^{\underline i} = f_1^{i_1} \dots f_p^{i_p}$. \end{rmq} \subsection{Definition and properties} We define here the bubble space, obtained from the formal loop space. We will prove in the next sections it admits a structure of symplectic Tate stack. \begin{df} The formal sphere of dimension $d$ is the pro-ind-stack \[ {\hat{\mathrm{S}}}^d = \lim_n \colim_{p \geq n} \operatorname{Spec}(A_p \oplus \operatorname{\underline{Hom}}_A(A_n,A)) \simeq \lim_n \colim_{p \geq n} \operatorname{Spec}(A_p \oplus A_n[-d]) \] where $A = k[\el{x}{d}]$ and $A_n = \quot{A}{(\el{x^n}{d})}$. \end{df} \begin{rmq} The notation $\operatorname{Spec}(A_p \oplus A_n[-d])$ is slightly abusive. The cdga $A_p \oplus A_n[-d]$ is not concentrated in non positive degrees. In particular, the derived stack $\operatorname{Spec}(A_p \oplus A_n[-d])$ is not a derived affine scheme. It behaves like one though, regarding its derived category: \[ \mathbf{Qcoh}(\operatorname{Spec}(A_p \oplus A_n[-d])) \simeq \mathbf{dgMod}_{A_p \oplus A_n[-d]} \] \end{rmq} Let us define the ind-pro-algebra \[ \mathcal{O}_{{\hat{\mathrm{S}}}^d} = \colim_n \lim_{p \geq n} A_p \oplus A_n[-d] \] where $A_p \oplus A_n[-d]$ is the trivial square zero extension of $A_p$ by the module $A_n[-d]$. For any $m \in \mathbb{N}$, let us denote by ${\hat{\mathrm{S}}}^d_m$ the ind-stack \[ {\hat{\mathrm{S}}}^d_m = \colim_{p \geq m} \operatorname{Spec}(A_p \oplus A_m[-d]) \] \begin{df}\label{dfbubble} Let $T$ be a derived Artin stack. We define the $d$-bubble stack of $T$ as the mapping ind-pro-stack \[ \operatorname{\underline{\mathfrak{B}}}(T) = \operatorname{\underline{Ma}p}({\hat{\mathrm{S}}}^d, T) \colon \operatorname{Spec} B \mapsto \colim_n \lim_{p \geq n} T \mathopen{}\mathclose\bgroup\originalleft(B \otimes (A_p \oplus A_n[-d]) \aftergroup\egroup\originalright) \] Again, the cdga $A_p \oplus A_n[-d]$ is not concentrated in non positive degree. This notation is thus slightly abusive and by $T(B \otimes (A_p \oplus A_n[-d]))$ we mean \[ \operatorname{Map}(\operatorname{Spec}(A_p \oplus A_n[-d]) \times \operatorname{Spec} B,T) \] We will denote by $\bar \operatorname{\underline{\mathfrak{B}}}(T)$ the diagram $\mathbb{N} \to \Prou U \mathbf{dSt}_k$ of whom $\operatorname{\underline{\mathfrak{B}}}(T)$ is a colimit in $\mathbf{IP}\mathbf{dSt}_k$. Let us also denote by $\operatorname{\underline{\mathfrak{B}}}_m(T)$ the mapping pro-stack \[ \operatorname{\underline{\mathfrak{B}}}_m(T) = \operatorname{Map}({\hat{\mathrm{S}}}^d_m, T) \colon \operatorname{Spec} B \mapsto \lim_{p \geq m} T \mathopen{}\mathclose\bgroup\originalleft(B \otimes (A_p \oplus A_m[-d]) \aftergroup\egroup\originalright) \] and $\bar \operatorname{\underline{\mathfrak{B}}}_m(T) \colon \{ p \in \mathbb{N} \| p \geq m\}^{\mathrm{op}} \to \mathbf{dSt}_S$ the corresponding diagram. In particular \[ \operatorname{\underline{\mathfrak{B}}}_0(T) = \operatorname{Map}({\hat{\mathrm{S}}}^d_0, T) \colon \operatorname{Spec} B \mapsto \lim_{p} T \mathopen{}\mathclose\bgroup\originalleft(B \otimes A_p\aftergroup\egroup\originalright) \] Those stacks come with natural maps \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{\underline{\mathfrak{B}}}_0(T) \ar[r]^-{s_0} & \operatorname{\underline{\mathfrak{B}}}(T) \ar[r]^-r & \operatorname{\underline{\mathfrak{B}}}_0(T) } \] \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{\underline{\mathfrak{B}}}_m(T) \ar[r]^-{s_m} & \operatorname{\underline{\mathfrak{B}}}(T) } \] \end{df} \begin{prop}\label{B-and-L-IP} If $T$ is an affine scheme of finite type, the bubble stack $\operatorname{\underline{\mathfrak{B}}}(T)$ is the product in ind-pro-stacks \[ \shorthandoff{;:!?} \xymatrix{ \operatorname{\underline{\mathfrak{B}}}(T) \ar[r] \ar[d] \cart[3] & {} \underline{\mathcal L}_V^d(T) \ar[d] \\ {} \underline \mathcal L_V^d(T) \ar[r] & {} \underline \mathcal L_U^d(T) } \] \end{prop} \begin{proof} There is a natural map $V_k^d \to {\hat{\mathrm{S}}}^d$ induced by the morphism \[ \colim_n \lim_{p \geq n} A_p \oplus A_n[-d] \to \lim_p A_p \] Because $T$ is algebraisable, it induces a map $\operatorname{\underline{\mathfrak{B}}}(T) \to \underline \mathcal L_V^d(T)$ and thus a diagonal morphism \[ \delta \colon \operatorname{\underline{\mathfrak{B}}}(T) \to \underline \mathcal L_V^d(T) \timesunder[\underline \mathcal L_U^d(T)] \underline \mathcal L_V^d(T) \] We will prove that $\delta$ is an equivalence. Note that because $T$ is a (retract of a) finite limit of copies of $\mathbb{A}^1$, we can restrict to the case $T = \mathbb{A}^1$. Let us first compute the fibre product $Z = \underline \mathcal L^d_V(\mathbb{A}^1) \times_{\underline \mathcal L_U^d(\mathbb{A}^1)} \underline \mathcal L_V^d(\mathbb{A}^1)$. It is the pullback of ind-pro-stacks \[ \shorthandoff{;:!?} \xymatrix{ Z \ar[r] \ar[d] \cart & \displaystyle \lim_p \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( k[ a_{\el{\alpha}{d}}, 0 \leq \alpha_i \leq p] \aftergroup\egroup\originalright) \ar[d] \\ \displaystyle \lim_p \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( k[ a_{\el{\alpha}{d}}, 0 \leq \alpha_i \leq p] \aftergroup\egroup\originalright) \ar[r] & \displaystyle \colim_n \lim_p \lim_{I \subset J} \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( k[ a_{\el{\alpha}{d}}, -n \delta_{i \in I} \leq \alpha_i \leq p] \aftergroup\egroup\originalright) } \] where $J = \{1, \dots , d\}$ and $\delta_{i \in I} = 1$ if $i \in I$ and $0$ otherwise. For any subset $K \subset J$ we define $M_K^{p,n}$ to be the free complex generated by the symbols \[ \{a_{\el{\alpha}{d}}, -n \leq \alpha_i < 0 \text{ if } i \in K, 0 \leq \alpha_i \leq p \text{ otherwise}\} \] We then have the cartesian diagram \[ \shorthandoff{;:!?} \xymatrix{ Z \ar[r] \ar[d] \cart & \lim_p \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( k[ M^{p,0}_\emptyset ] \aftergroup\egroup\originalright) \ar[d] \\ \lim_p \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( k[ M^{p,0}_\emptyset ] \aftergroup\egroup\originalright) \ar[r] & \colim_n \lim_p \lim_{I \subset J} \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft(k\mathopen{}\mathclose\bgroup\originalleft[ \bigoplus_{K \subset I} M_K^{p,n} \aftergroup\egroup\originalright] \aftergroup\egroup\originalright) } \] Using \autoref{subsets-colim} we get \[ Z \simeq \colim_n \lim_p \operatorname{Spec}\mathopen{}\mathclose\bgroup\originalleft( k\mathopen{}\mathclose\bgroup\originalleft[M^{p,0}_\emptyset \oplus M^{0,n}_J[d]\aftergroup\egroup\originalright] \aftergroup\egroup\originalright) \] \end{proof} \begin{rmq} Let us consider the map $\lim_p A_p \to A_0 \simeq k$ mapping a formal serie to its coefficient of degree $0$. The $(\lim A_p)$-ind-module $\colim A_n[-d]$ is endowed with a natural map to $k[-d]$. This induces a morphism $\mathcal{O}_{{\hat{\mathrm{S}}}^d} \to k \oplus k[-d]$ and hence a map $\mathrm S^d \to {\hat{\mathrm{S}}}^d$, where $\mathrm S^d$ is the topological sphere of dimension $d$. We then have a rather natural morphism \[ \operatorname{\underline{\mathfrak{B}}}^d(X) \to \operatorname{\underline{Ma}p}(\mathrm S^d,X) \] \end{rmq} \subsection{Its tangent is a Tate module} We will prove in this subsection that the bubble stack is a Tate stack. To do so, we could bluntly apply \autoref{map-tate} but we will give here a direct proof of that statement. We will get another decomposition of its tangent complex that will be needed when proving $\operatorname{\underline{\mathfrak{B}}}^d(T)$ is symplectic. \begin{prop}\label{prop-formal-tate} Let us assume that the Artin stack $T$ is locally of finite presentation. For any $m \in \mathbb{N}$ we have an exact sequence \[ \shorthandoff{;:!?} \xymatrix{ s_m^r^ \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \ar[r] & s_m^\mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \ar[r] & s_m^\mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)/\operatorname{\underline{\mathfrak{B}}}^d(T)_0} } \] where the left hand side is an ind-perfect module and the right hand side is a pro-perfect module. In particular, the middle term is a Tate module, and the ind-pro-stack $\operatorname{\underline{\mathfrak{B}}}^d(T)$ is a Tate stack. \end{prop} \begin{proof} Throughout this proof, we will write $\operatorname{\underline{\mathfrak{B}}}$ instead of $\operatorname{\underline{\mathfrak{B}}}^d(T)$ and $\operatorname{\underline{\mathfrak{B}}}_m$ instead of $\operatorname{\underline{\mathfrak{B}}}^d(T)_m$ for any $m$. Let us first remark that $\operatorname{\underline{\mathfrak{B}}}$ is an Artin ind-pro-stack locally of finite presentation. It suffices to prove that $s_m^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}}$ is a Tate module on $\operatorname{\underline{\mathfrak{B}}}_m$, for any $m \in \mathbb{N}$. We will actually prove that it is an elementary Tate module. We consider the map \[ s_m^* r^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_0} \to s_m^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}} \] It is by definition equivalent to the natural map \[ \lambda^{\operatorname{\mathbf{Pro}}}_{\operatorname{\underline{\mathfrak{B}}}_m}(\operatorname{\underline{\mathfrak{B}}}_0) \to^f \lim \lambda^{\operatorname{\mathbf{Pro}}}_{\operatorname{\underline{\mathfrak{B}}}_m}(\bar \operatorname{\underline{\mathfrak{B}}}_{\geq m}(T)) \] where $\bar \operatorname{\underline{\mathfrak{B}}}_{\geq m}(T)$ is the restriction of $\bar \operatorname{\underline{\mathfrak{B}}}(T)$ to $\{ n \geq m \} \subset \mathbb{N}$. Let $\phi$ denote the diagram \[ \phi \colon \{ n \in \mathbb{N} \| n \geq m\}^{\mathrm{op}} \to \operatorname{\mathbf{IPerf}}(\operatorname{\underline{\mathfrak{B}}}_m(T)) \] obtained as the cokernel of $f$. It is now enough to prove that $\phi$ factors through $\mathbf{Perf}(\operatorname{\underline{\mathfrak{B}}}_m(T))$. Let $n \geq m$ be an integer and let $g_{mn}$ denote the induced map $\operatorname{\underline{\mathfrak{B}}}_m(T) \to \operatorname{\underline{\mathfrak{B}}}_n(T)$. We have an exact sequence \[ s_m^* r^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_0(T)} \simeq g_{mn}^* s_n^* r^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_0(T)} \to g_{m,n}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_n(T)} \to \phi(n) \] Let us denote by $\psi(n)$ the cofiber \[ s_n^* r^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_0(T)} \to \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_n(T)} \to \psi(n) \] so that $\phi(n) \simeq g_{mn}^* \psi(n)$. This sequence is equivalent to the colimit (in $\operatorname{\mathbf{IPerf}}(\operatorname{\underline{\mathfrak{B}}}_n(T))$) of a cofiber sequence of diagrams $\{ p \in \mathbb{N} \| p \geq n \}^{\mathrm{op}} \to \mathbf{Perf}(\operatorname{\underline{\mathfrak{B}}}_n(T))$ \[ \lambda^{\operatorname{\mathbf{Pro}}}_{\operatorname{\underline{\mathfrak{B}}}_n(T)}(\bar \operatorname{\underline{\mathfrak{B}}}_0(T)) \to \lambda^{\operatorname{\mathbf{Pro}}}_{\operatorname{\underline{\mathfrak{B}}}_n(T)}(\bar \operatorname{\underline{\mathfrak{B}}}_n(T) ) \to \bar \psi(n) \] It suffices to prove that the diagram $\bar \psi(n) \colon \{ p \in \mathbb{N} \| p \geq n \}^{\mathrm{op}} \to \mathbf{Perf}(\operatorname{\underline{\mathfrak{B}}}_n(T))$ is (essentially) constant. Let $p \in \mathbb{N}$, $p \geq n$. The perfect complex $\bar \psi(n)(p)$ fits in the exact sequence \[ t_{np}^* \varepsilon_{np}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{0,p}(T)} \to \pi_{n,p}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{n,p}(T)} \to \bar \psi(n)(p) \] where $t_{np} \colon \operatorname{\underline{\mathfrak{B}}}_n(T) \to \operatorname{\underline{\mathfrak{B}}}_{n,p}(T)$ is the canonical projection and $\varepsilon_{np} \colon \operatorname{\underline{\mathfrak{B}}}_{n,p}(T) \to \operatorname{\underline{\mathfrak{B}}}_{0,p}(T)$ is induced by the augmentation $\mathcal{O}_{S_{n,p}} \to \mathcal{O}_{S_{0,p}}$. It follows that $\bar \psi(n)(p)$ is equivalent to \[ t_{np}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{n,p}(T)/\operatorname{\underline{\mathfrak{B}}}_{0,p}(T)} \] Moreover, for any $q \geq p \geq n$, the induced map $\bar \psi(n)(p) \to \bar \psi(n)(q)$ is obtained (through $t_{nq}^$) from the cofiber, in $\mathbf{Perf}(\operatorname{\underline{\mathfrak{B}}}_{n,q}(T))$ \[ \shorthandoff{;:!?} \xymatrix@R=2mm{ \alpha_{npq}^ \varepsilon_{np}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{0,p}(T)} \ar[r] \ar@{}[rdddd]\|{(\sigma)} & \alpha_{npq}^ \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{n,p}(T)} \ar[dddd] \ar[r] & \alpha_{npq}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{n,p}(T)/\operatorname{\underline{\mathfrak{B}}}_{0,p}(T)} \ar[dddd] \\ \varepsilon_{nq}^* \alpha_{0pq}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{0,p}(T)} \ar@{=}[u] \ar[ddd] \\ \\ \\ \varepsilon_{nq}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{0,q}(T)} \ar[r] & \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{n,q}(T)} \ar[r] & \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}_{n,q}(T)/\operatorname{\underline{\mathfrak{B}}}_{0,q}(T)} } \] where $\alpha_{npq}$ is the map $\operatorname{\underline{\mathfrak{B}}}_{n,q}(T) \to \operatorname{\underline{\mathfrak{B}}}_{n,p}(T)$. Let us denote by $(\sigma)$ the square on the left hand side above. Let us fix a few more notations \[ \shorthandoff{;:!?} \xymatrix@R=7mm@C=5mm{ & \operatorname{\underline{\mathfrak{B}}}_{n,p}(T) \times S_{0,p} \ar[dd]_{\varphi_{np}} \ar[dl]_(0.6){a_{0p}} & & \operatorname{\underline{\mathfrak{B}}}_{n,q}(T) \times S_{0,p} \ar[dd]_{\psi_{npq}} \ar[ll] \ar[rr] & & \operatorname{\underline{\mathfrak{B}}}_{n,q}(T) \times S_{0,q} \ar[dd]^{\varphi_{nq}} \\ S_{0,p} \ar[dd]_{\xi_{np}} \\ & \operatorname{\underline{\mathfrak{B}}}_{n,p}(T) \times S_{n,p} \ar[dl]^{a_{np}} \ar@{-}[d] & & \operatorname{\underline{\mathfrak{B}}}_{n,q}(T) \times S_{n,p} \ar[ll] \ar[rr]^{b_{npq}} \ar@{-}[d] \ar[ld] & & \operatorname{\underline{\mathfrak{B}}}_{n,q}(T) \times S_{n,q} \ar[dl]^{a_{nq}} \ar[dd]^{\varpi_{nq}} \ar[dr]^(0.7){\operatorname{ev}_{nq}} \\ S_{n,p} & \ar[d]^(0.35){\varpi_{np}} & S_{n,p} \ar@{-}[ll]_(0.4)= & \ar[d] & S_{n,q} \ar[ll]_(0.3){\beta_{npq}} & & T\\ & \operatorname{\underline{\mathfrak{B}}}_{n,p}(T) & & \operatorname{\underline{\mathfrak{B}}}_{n,q}(T) \ar[ll]_{\alpha_{npq}} \ar@{-}[rr]_{=} & & \operatorname{\underline{\mathfrak{B}}}_{n,q}(T) } \] The diagram $(\sigma)$ is then dual to the diagram \[ \shorthandoff{;:!?} \xymatrix{ \alpha_{npq}^* \varepsilon_{np}^* {\varpi_{0p}}_* \operatorname{ev}_{0p}^* \mathbb{T}_T & \alpha_{npq}^* {\varpi_{np}}_* \operatorname{ev}_{np}^* \mathbb{T}_T \ar[l] \\ \varepsilon_{nq}^* {\varpi_{0q}}_* \operatorname{ev}_{0q}^* \mathbb{T}_T \ar[u] & {\varpi_{nq}}_* \operatorname{ev}_{nq}^* \mathbb{T}_T \ar[l] \ar[u] } \] Moreover, the functor $\varpi_{np}$ (for any $n$ and $p$) satisfies the base change formula. This square is thus equivalent to the image by ${\varpi_{nq}}_$ of the square \[ \shorthandoff{;:!?} \xymatrix{ {\psi_{npq}}_ {b_{npq}}_* b_{npq}^* \psi_{npq}^* \operatorname{ev}_{nq}^* \mathbb{T}_T & {b_{npq}}_* b_{npq}^* \operatorname{ev}_{nq}^* \mathbb{T}_T \ar[l] \\ {\varphi_{nq}}_* \varphi_{nq}^* \operatorname{ev}_{nq}^* \mathbb{T}_T \ar[u] & \operatorname{ev}_{nq}^* \mathbb{T}_T \ar[l] \ar[u] } \] Using now the projection and base change formulae along the morphisms $\varphi_{nq}$, $b_{npq}$ and $\psi_{npq}$, we see that this last square is again equivalent to \[ \shorthandoff{;:!?} \xymatrix{ (a_{nq}^* {\beta_{npq}}_* {\xi_{np}}_* \mathcal{O}_{S_{0,p}}) \otimes (\operatorname{ev}_{nq}^* \mathbb{T}_T) & (a_{nq}^* {\beta_{npq}}_* \mathcal{O}_{S_{n,p}}) \otimes (\operatorname{ev}_{nq}^* \mathbb{T}_T) \ar[l] \\ (a_{nq}^* {\xi_{nq}}_* \mathcal{O}_{S_{0,q}}) \otimes (\operatorname{ev}_{nq}^* \mathbb{T}_T) \ar[u] & (a_{nq}^* \mathcal{O}_{S_{n,q}}) \otimes (\operatorname{ev}_{nq}^* \mathbb{T}_T) \ar[l] \ar[u] } \] We therefore focus on the diagram \[ \shorthandoff{;:!?} \xymatrix{ \mathcal{O}_{S_{n,q}} \ar[r] \ar[d] & {\xi_{nq}}_* \mathcal{O}_{S_{0,q}} \ar[d] \\ {\beta_{npq}}_* \mathcal{O}_{S_{n,p}} \ar[r] & {\beta_{npq}}_* {\xi_{np}}_* \mathcal{O}_{S_{0,p}} } \] By definition, the fibres of the horizontal maps are both equivalent to $A_n[-d]$ and the map induced by the diagram above is an equivalence. We have proven that for any $q \geq p \geq n$ the induced map $\bar \psi(n)(p) \to \bar \psi(n)(q)$ is an equivalence. It implies that $\mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}(T)}$ is a Tate module. \end{proof} \subsection{A symplectic structure (shifted by \texorpdfstring{$d$}{d})} In this subsection, we will prove the following \begin{thm}\label{B-symplectic} Assume $T$ is $q$-shifted symplectic. The ind-pro-stack $\operatorname{\underline{\mathfrak{B}}}^d(T)$ admits a symplectic Tate structure shifted by $q-d$. Moreover, for any $m \in \mathbb{N}$ we have an exact sequence \[ s_m^* r^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \to s_m^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \to s_m^* r^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_0}[q-d] \] \end{thm} \begin{proof} Let us start with the following remark: the residue map $r_n \colon A_n \to k = A_1$ defined in \autoref{dual-over-k} defines a map $\mathcal{O}_{{\hat{\mathrm{S}}}^d} \to k[-d]$. From \autoref{ipdst-form}, we have a $(q-d)$-shifted closed $2$-form on $\operatorname{\underline{\mathfrak{B}}}^d(T)$. We have a morphism from \autoref{prop-formsareforms} \[ \mathcal{O}_{\operatorname{\underline{\mathfrak{B}}}^d(T)}[q-d] \to \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \otimes \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \] in $\operatorname{\mathbf{PIPerf}}(\operatorname{\underline{\mathfrak{B}}}^d(T))$. Let $m \in \mathbb{N}$. We get a map \[ \mathcal{O}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_m}[q-d] \to s_m^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \otimes s_m^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \] and then \[ s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \otimes s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \to \mathcal{O}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_m}[q-d] \] in $\operatorname{\mathbf{IPPerf}}(\operatorname{\underline{\mathfrak{B}}}^d(T)_m)$. We consider the composite map \[ \theta \colon s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)/\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \otimes s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)/\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \to s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \otimes s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \to \mathcal{O}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_m}[q-d] \] Using the \autoref{describe-form} and the proof of \autoref{prop-formal-tate} we see that $\theta$ is induced by the morphisms (varying $n$ and $p$) \[ \shorthandoff{;:!?} \xymatrix{ {\varpi_{np}}_* \mathopen{}\mathclose\bgroup\originalleft( E \otimes E \otimes \operatorname{ev}_{np}^* \mathopen{}\mathclose\bgroup\originalleft( \mathbb{T}_T \otimes \mathbb{T}_T \aftergroup\egroup\originalright) \aftergroup\egroup\originalright) \ar[r]^-A & {\varpi_{np}}_* \mathopen{}\mathclose\bgroup\originalleft( E \otimes E [q] \aftergroup\egroup\originalright) \ar[r]^-B & {\varpi_{np}}_* \mathopen{}\mathclose\bgroup\originalleft( \mathcal{O}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{np} \times S_{n,p}} [q] \aftergroup\egroup\originalright) } \] where $E = a_{np}^* {\xi_{np}}_* {h_{np}}_* \gamma_n^! \mathcal{O}_{\mathbb{A}^d}$ and the map $A$ is induced by the symplectic form on $T$. The map $B$ is induced by the multiplication in $\mathcal{O}_{S_{n,p}}$. This sheaf of functions is a trivial square zero extension of augmentation ideal ${\xi_{np}}_* {h_{np}}_* \gamma_n^! \mathcal{O}_{\mathbb{A}^d}$ and $B$ therefore vanishes. It follows that the morphism \[ s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \otimes s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)/\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \to s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \otimes s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \to \mathcal{O}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_m}[q-d] \] factors through $s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \otimes s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)/\operatorname{\underline{\mathfrak{B}}}^d(T)_0}$. Now using \autoref{prop-formal-tate} we get a map of exact sequences in the category of Tate modules over $\operatorname{\underline{\mathfrak{B}}}^d(T)_m$ \[ \shorthandoff{;:!?} \xymatrix{ s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)/\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \ar[r] \ar[d]_{\tau_m} & s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)} \ar[r] \ar[d] & s_m^* r^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \ar[d] \\ s_m^* r^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_0}[d-q] \ar[r] & s_m^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)}[d-q] \ar[r] & s_m^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)/\operatorname{\underline{\mathfrak{B}}}^d(T)_0}[d-q] } \] where the maps on the sides are dual one to another. It therefore suffices to see that the map $\tau_m \colon s_m^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)/\operatorname{\underline{\mathfrak{B}}}^d(T)_0} \to s_m^* r^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_0}[d-q]$ is an equivalence. We now observe that $\tau_m$ is a colimit indexed by $p \geq m$ of maps \[ g_{pm}^* t_{pp}^* \mathopen{}\mathclose\bgroup\originalleft( \varepsilon_{pp}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{0p}} \to \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{pp}/\operatorname{\underline{\mathfrak{B}}}^d(T)_{0p}} \aftergroup\egroup\originalright) \] Let us fix $p \geq m$ and $G = a_{pp}^* {\xi_{pp}}_* \mathcal{O}_{S_{0p}}$. The map $ F_p \colon \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{pp}/\operatorname{\underline{\mathfrak{B}}}^d(T)_{0p}} \to \varepsilon_{pp}^* \mathbb{L}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{0p}}$ at hand is induced by the pairing \[ \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{pp}/\operatorname{\underline{\mathfrak{B}}}^d(T)_{0p}} \otimes \varepsilon_{pp}^* \mathbb{T}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{0p}} \simeq \shorthandoff{;:!?} \xymatrix{ {\varpi_{pp}}_* \mathopen{}\mathclose\bgroup\originalleft(E \otimes \operatorname{ev}_{pp}^* \mathbb{T}_T \aftergroup\egroup\originalright) \otimes {\varpi_{pp}}_* \mathopen{}\mathclose\bgroup\originalleft( G \otimes \operatorname{ev}_{pp}^* \mathbb{T}_T \aftergroup\egroup\originalright) \ar[d] \\ {\varpi_{pp}}_* \mathopen{}\mathclose\bgroup\originalleft( E \otimes \operatorname{ev}_{pp}^* \mathbb{T}_T \otimes G \otimes \operatorname{ev}_{pp}^* \mathbb{T}_T \aftergroup\egroup\originalright) \ar[d] \\ {\varpi_{pp}}_* \mathopen{}\mathclose\bgroup\originalleft( E \otimes G \aftergroup\egroup\originalright)[q] \ar[d] \\ {\varpi_{pp}}_* \mathopen{}\mathclose\bgroup\originalleft( \mathcal{O}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{pp} \times S_{pp}} \aftergroup\egroup\originalright) [q] \ar[d] \\ \mathcal{O}_{\operatorname{\underline{\mathfrak{B}}}^d(T)_{pp}} [q-d] } \] We can now conclude using \autoref{dual-over-k}. \end{proof} \phantomsection \addcontentsline{toc}{section}{References} \end{document}	arXiv
\begin{document} \raggedbottom \title{Quantum coherence and speed limit in the mean-field Dicke model of superradiance} \author{D. Z. Rossatto} \thanks{These authors contributed equally to this work.} \affiliation{Universidade Estadual Paulista (Unesp), Campus Experimental de Itapeva, 18409-010 Itapeva, S\~{a}o Paulo, Brazil} \author{D. P. Pires} \thanks{These authors contributed equally to this work.} \affiliation{Departamento de F\'{i}sica Te\'{o}rica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, Rio Grande do Norte, Brazil} \author{F. M. de Paula} \affiliation{Centro de Ci\^{e}ncias Naturais e Humanas, Universidade Federal do ABC, 09606-070 S\~{a}o Bernardo do Campo, S\~{a}o Paulo, Brazil} \author{O. P. de S\'{a} Neto} \affiliation{Coordena\c{c}\~{a}o de Ci\^{e}ncia da Computa\c{c}\~{a}o, Universidade Estadual do Piau\'{i}, 64202-220 Parna\'{i}ba, Piau\'{i}, Brazil} \affiliation{PPGQ, Universidade Estadual do Piau\'{i}, Rua Jo\~{a}o Cabral 2231, 64002-150 Teresina, Piau\'{i}, Brazil} \begin{abstract} Dicke superrandiance is a cooperative phenomenon which arises from the collective coupling of an ensemble of atoms to the electromagnetic radiation. Here we discuss the quantifying of quantum coherence for the Dicke model of superradiance in the mean-field approximation. We found the single-atom $l_1$-norm of coherence is given by the square root of the normalized average intensity of radiation emitted by the superradiant system. This validates quantum coherence as a useful figure of merit towards the understanding of superradiance phenomenon in the mean-field approach. In particular, this result suggests probing the single-atom coherence through the radiation intensity in superradiant systems, which might be useful in experimental realizations where is unfeasible to address atoms individually. Furthermore, given the nonlinear unitary dynamics of the time-dependent single-atom state that effectively describes the system of $N$ atoms, we analyze the quantum speed limit time and its interplay with the $l_1$-norm of coherence. We verify the quantum coherence speeds up the evolution of the superradiant system, i.e., the more coherence stored on the single-atom state, the faster the evolution. These findings unveil the role played by quantum coherence in superradiant systems, which in turn could be of interest for communities of both condensed matter physics and quantum optics. \end{abstract} \maketitle \section{Introduction} \label{section0001} Light-matter interaction plays a striking role in the understanding of several physical phenomena~\cite{PhysRevA.8.2517}, thus being a subject of significant interest to research on laser cooling and atomic trapping~\cite{RevModPhys.58.699,PhysRevLett.61.826,Chu:Wieman:89,RevModPhys.70.721}, cavity quantum electrodynamics~\cite{RevModPhys.73.565,RevModPhys.75.281}, and more recently quantum computing~\cite{PhysRevLett.92.127902}. It is noteworthy that the Dicke model, which des\-cri\-bes the coupling of a single mode of the radiation field with an ensemble of two-level systems, stands as a paradigmatic toy model from quantum optics~\cite{Dicke1954,PhysRevA.7.831,Garraway2011,Kirton2018}. In turn, the collective and coherent interaction can promote the well-known Dicke superradiance, in which the system spontaneously emits radiation at high intensity in a short time window~\cite{Nature_285_70_1980,Gross1982}. Remarkably, experimental rea\-li\-za\-tion of superradiance has been performed in a variety of quantum platforms~\cite{PhysRevLett.30.309,PhysRevLett.36.1035,PhysRevLett.76.2049,Inouye1999,NaturePhys_3_2,Rohlsberger2010,Mlynek2014,PhysRevLett.115.063601,PhysRevLett.114.023601,PhysRevLett.114.023602,PhysRevLett.117.210503,PhysRevLett.124.013601,Kim2017}. Moreover, exploiting the physical richness of the superradiant phase~\cite{Hepp1973,Nature_464_7293,PhysRevA.96.033633,PhysRevLett.125.050402}, superradiance has applications in ultra-narrow-linewidth lasers~\cite{PhysRevLett.71.995,Bohnet2012}, quantum communication~\cite{Duan2001,Kimble2008}, sensitive gravimeters~\cite{Liao2015}, and quantum batteries~\cite{PhysRevLett.120.117702,PhysRevLett.122.047702,PhysRevB.99.205437}. Motivated by the ubiquitous role of radiation intensity in the superradiance phenomenon, one can ask if Dicke superradiance somehow accelerates the evolution of the quantum system, the latter remaining as a challenge for the design of faster quantum information-processing devices~\cite{PhysRevLett.103.240501,LARS_2018}. In particular, this quantum signature could be captured by the so-called quantum speed limit (QSL), i.e., the minimum time of evolution required for a quantum system evolve between two given states~\cite{1945_JPhysURSS_9_249,1992_PhysicaD_120_188,2009_PhysRevLett_103_160502}. Nowadays, QSL find applications in quantum computation and quantum communication~\cite{PhysRevA.82.022318}, quantum metrology~\cite{PhysRevA.97.022109}, and quantum thermodynamics~\cite{PhysRevLett.118.150601}. Furthermore, Dicke superradiance has been recently addressed under the viewpoint of local quantum uncertainty quantifiers~\cite{PhysRevA.94.023819}, as well as quantum correlations~\cite{PhysRevLett.112.140402,PhysRevA.101.052310}. This motivates an investigation of the superradiance employing another figure of merit as quantum coherence, particularly focusing on its resource-theoretical approach~\cite{PhysRevLett.113.140401,RevModPhys.89.041003}. Quantum co\-he\-ren\-ce is a remar\-ka\-ble fingerprint of non-classical systems linked to the quantum superposition principle, which plays an essential role in quantum optics~\cite{PhysRev.130.2529}, quantum thermodyna\-mics~\cite{PhysRevA.93.052335}, condensed matter physics~\cite{PhysRevB.93.184428}, and bio\-lo\-gi\-cal systems~\cite{2013_829687}. In fact, it has been shown that coherence and superradiance are mutually interconvertible resources~\cite{PhysRevA.97.052304}. Moreover, one can investigate the interplay between the many-body coherences of Dicke superradiance and the QSL in multiparticle systems~\cite{PhysRevResearch.2.023125}. In this paper, we discuss the quantifying of quantum coherence and quantum speed limit time, and also their interplay, for the Dicke model of superradiance in the mean-field approximation. We find quantum coherence is suddenly suppressed for a large number of atoms, while exhibits a maximum value at the time delay of superradiance (time of maximum intensity), at which in turn is robust to the number of atom increasing. Importantly, we show that $\ell_1$-norm of coherence is related to the intensity of radiation emitted by the superradiant system. This result suggests that probing the single-atom coherence through the radiation intensity in superradiant systems might be useful in experimental setups where is unfeasible to address atoms individually. It is noteworthy that the QSL bound saturates as one increases the number of atoms $N$ in the system, thus suggesting that quantum information-processing devices based on Dicke superradiance could operate at maximum speed in the limit $N \gg 1$. Moreover, we explore the relation of quantum coherence and QSL time, thus analyzing the former as a resource capable to speed up the unitary dynamics of each single two-level atom. The paper is organized as follows. In Sec.~\ref{sec:model} we briefly review the basic features of the Dicke model of superradiance in view of mean-field approximation. In Sec.~\ref{sec:quantumcoherence001} we discuss the role of quantum coherence in the referred model. In Sec.~\ref{sec:QSLsection0001} we study the quantum speed limit time with regard to the effective nonlinear unitary evolution of two non-orthogonal single-atom pure states. In addition, we investigate the interplay between the QSL bound and quantum coherence. Finally, we sumarize our main results and present the conclusions. \section{Physical System} \label{sec:model} Let us consider the Dicke model of superradiance, a system of $N$ identical two-level atoms with transition frequency $\omega$, which interacts collectively with their surrounding electromagnetic field in the vacuum state (zero temperature)~\cite{Benedict1996,Garraway2011}. For transitions between Dicke states~\cite{MandelWolf}, considering $N \gg 1$ and the system weakly coupled to its environment, the dynamics of this system can be described by the Lindblad-type mean-field master equation ($\hbar=1$)~\cite{Gross1982,BreuerPet} \begin{equation} \label{mecoll} \frac{d{\rho_N}}{dt} = -i\omega[{J^z},{\rho_N}] - \frac{\gamma_0}{2}\left(\{ {J^+}{J^-},{\rho_N}\} - 2\, {J^-}{\rho_N}{J^+} \right) ~, \end{equation} where ${J^z} = (1/2)\, {\sum_{j=1}^N} {\sigma_j^z}$ and ${J^{\pm}} = {\sum_{j=1}^N} {\sigma^{\pm}_j}$ are collective operators, with ${\sigma_j^z}$ and ${\sigma^{\pm}_j}$ denoting the Pauli matrices associated with the $j$-th atom, $[X,Y] =XY - YX$ is the commutator, and $\{ X,Y \} = XY + YX$ is the anticommutator. In particular, this mean-field master equation can be mapped onto a nonlinear Schr\"{o}dinger-type equation by embedding the dy\-na\-mics of the $N$-atom mean-field state ${\rho_N} \approx {(\|{\psi_t}\rangle\langle{\psi_t}\|)^{\otimes N}}$ into an effective unitary dynamics of each two-level atom, described by $\ket{\psi_t}$, which in turn evolves accor\-ding to the nonlinear Hamiltonian $H_t = (\omega/2)\,{\sigma_z} + i \, ({N\gamma_0}/{2}) (\bra{\psi_t}\sigma_{+}\ket{\psi_t}\sigma_{-} - \bra{\psi_t}\sigma_{-}\ket{\psi_t}\sigma_{+})$, where ${\sigma_+}$ (${\sigma_-}$) is the raising (lowering) operator, and $\gamma_0$ is the spontaneous emission rate of each single two-level atom~\cite{BreuerPet}. The solution for the nonlinear Schr\"{o}dinger equation $({d}/{dt})\ket{\psi_t} = -i {H_t}\ket{\psi_t}$ is described by the time-dependent single-atom state~\cite{BreuerPet} \begin{equation} \label{hamilteffective0002} \ket{\psi_t} = \sqrt{1 - {p_t}} \, {e^{i\frac{\omega}{2} t}} \ket{g} + \sqrt{{p_t}} \, {e^{-i\frac{\omega}{2} t}} \ket{e} ~, \end{equation} where $\|g\rangle$ ($\|e\rangle$) stands for the ground (excited) state of the two-level atom. Here \begin{equation} \label{hamilteffective0002001} {p_t} = [{e^{\gamma_0 N (t - t_D)}+1}]^{-1} \end{equation} denotes the probability of finding a single atom in the excited state at time $t$, with $0 \leq {p_t} \leq 1$, while ${t_D} = {(\gamma_0} N)^{-1} \ln (N)$ stands for the time delay of the superradiance (time of maximum intensity)~\cite{Gross1982,Benedict1996}. Finally, with $\ket{\psi_t}$ given by Eq.~\eqref{hamilteffective0002}, the nonlinear Hamiltonian can be written as \begin{equation} \label{hamilteffective0001} {H_t} = \frac{\omega}{2}\sigma_{z} -i \frac{N\gamma_0}{2}\sqrt{{p_t}(1 - {p_t})}(\sigma_{+} e^{-i\omega t}- \sigma_{-} e^{i\omega t}) ~. \end{equation} It is noteworthy that, given the probability distribution $p_t$ in Eq.~\eqref{hamilteffective0002001}, the average intensity of radiation emitted by the system of $N$ atoms can be written as $I(t) = -N\omega (dp_t/dt) = ({N^2}\omega{\gamma_0}/4) \, {\text{sech}^2}[(N{\gamma_0}/2)(t - {t_D}) ]$, which is proportional to $N^2$, characterizing the superradiance~\cite{BreuerPet}. For $t = {t_D}$, the system emits radiation with the maximum intensity value ${I_\text{max}} = I( t = {t_D}) = {N^2}\omega{\gamma_0}/4$. In general, Eq.~(1) describes a system of two-level atoms in free space in the mean-field approximation, which in turn is valid in the limit $N \to \infty$~\cite{BreuerPet}. Nevertheless, it is worth mention that Eq.~(1) also describes an atomic cloud with a finite number $N$ of noninteracting two-level atoms coupled to a leaking cavity~\cite{Rossatto2011,PhysRevA.4.302,Delanty2011,Mlynek2014,PhysRevLett.124.013601} in the so-called bad-cavity limit~\cite{haroche2006exploring}. Just to clarify, in this case the cavity dissipation surpasses the effective coupling between the cavity mode and the atomic cloud. Therefore, our results might embody the case of a finite number of atoms. \section{Quantum Coherence} \label{sec:quantumcoherence001} In this section we briefly introduce the main concepts of quantum coherence and discuss its role in the superradiance phenomenon addressed in Sec.~\ref{sec:model}. Over half decade ago, the seminal work by Baumgratz {\it et al.}~\cite{PhysRevLett.113.140401} introduced the minimal theoretical framework for the quantification of quantum coherence. This approach opened an avenue for the characterization of quantum coherence under the scope of resource theories, which currently still is a matter of intense debate~\cite{PhysRevA.94.052336,PhysRevA.95.019902,PhysRevLett.117.030401,PhysRevLett.116.120404,PhysRevX.6.041028}. Here we will briefly review the key aspects of quantifying quantum coherence discussed in Ref.~\cite{PhysRevLett.113.140401}. We shall consider a physical system defined on a $d$-dimensional Hilbert space $\mathcal{H}$ endowed with some re\-fe\-ren\-ce basis ${\{ \| {j} \rangle \}_{j = 0}^{d - 1}}$. In turn, the state of the system is described by a density matrix $\rho \in \mathcal{D}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H}) = \{ {\rho^{\dagger}} = \rho,~\rho\geq 0,~\text{Tr}(\rho) = 1\}$ stands for the convex space of po\-si\-ti\-ve semi-definite density operators. In particular, the subset $\mathcal{I}\in \mathcal{D}(\mathcal{H})$ of incoherent states encompass the family of density matrices as $\delta = {\sum_j}\, {q_j}\|{j}\rangle\langle{j}\|$ that are diagonal in the reference basis, with $0 \leq {q_j} \leq 1$ and ${\sum_j}\, {q_j} = 1$. In summary, a {\it bona fide} quantum cohe\-ren\-ce quantifier $C(\rho)$ must satisfy the following properties~\cite{PhysRevLett.113.140401,RevModPhys.89.041003} (i) non-negativity, i.e., $C(\rho) \geq 0$ for all state $\rho$, with $C(\rho) = 0$ iff $\rho \in \mathcal{I}$; (ii) con\-ve\-xi\-ty under mixing, i.e., $C({\sum_n}\, {q_n} {\rho_n}) \leq {\sum_n}\, {q_n} C({\rho_n})$, with ${\rho_n} \in \mathcal{D}(\mathcal{H})$, $0 \leq {q_n} \leq 1$, and ${\sum_n}\, {q_n} =1$; (iii) mo\-no\-to\-ni\-city under in\-co\-he\-rent completely positive and trace-preserving (ICPTP) maps, i.e., $C(\mathcal{E}[\rho]) \leq C(\rho)$, for all ICPTP map $\mathcal{E}[\bullet]$; (iv) strong monotonicity, i.e., $C(\rho) \geq {\sum_n} {q_n} C({\rho_n})$, where ${\rho_n} = {q_n^{-1}}{K_n} \rho {K^{\dagger}_n}$ sets the post-measured states for arbitrary Kraus o\-pe\-rators $\{ {K_n} \}$ satisfying ${\sum_n} {K^{\dagger}_n} {K_n} = \mathbb{I}$ and $K_n\mathcal{I}K^{\dagger}_{n}\subset \mathcal{I}$, with ${q_n} = \mathrm{Tr}({K_n} \rho {K^{\dagger}_n})$. Some widely known quantum coherence measures include relative entropy of coherence~\cite{PhysRevLett.113.140401}, geometric co\-he\-ren\-ce~\cite{PhysRevLett.115.020403}, and robustness of coherence~\cite{PhysRevLett.116.150502}. In addition, the $l_1$-norm of coherence, which is defined in terms of $l_1$-distance between $\rho$ and its closest incoherent state, is of fundamental interest since its exact calculation is readily given by ${C}(\rho) = {\sum_{j,l (j\neq l)}}{\| {\rho_{jl}} \|}$, where $\rho_{jl}$ sets the off-diagonal ele\-ments of $\rho$ eva\-lua\-ted with respect to the re\-fe\-ren\-ce basis ${\{ \| {j} \rangle \}_{j = 0}^{d - 1}}$~\cite{PhysRevLett.113.140401}. Importantly, for a pure single-qubit state (i.e., $d = 2$ and $\rho = \|\phi\rangle\langle{\phi}\|$), the referred quantum coherence measures are monotonically related to each other, and thus they shall exhibit the same qua\-li\-ta\-ti\-ve behavior~\cite{RevModPhys.89.041003}. In this context, without any loss of generality, from now on we will address the $l_1$-norm of co\-he\-ren\-ce to characterize the role of quantum coherence in the superradiant system previously discussed. From Sec.~\ref{sec:model}, fixing the reference basis $\{ \|{e}\rangle, \|{g}\rangle \}$, Eq.~\eqref{hamilteffective0002} implies the single-atom density operator \begin{align} {\rho_t} &= (1 - {p_t})\|{g}\rangle\langle{g}\| + {p_t}\|{e}\rangle\langle{e}\| \nonumber\\ &+ \sqrt{{p_t}(1 - {p_t})}({e^{i\omega t}}\|{g}\rangle\langle{e}\| + {e^{- i\omega t}}\|{e}\rangle\langle{g}\|) ~, \end{align} with $p_t$ given in Eq.~\eqref{hamilteffective0002001}. In this case, $l_1$-norm of co\-he\-ren\-ce is given by \begin{equation} \label{eq:coherence00001} {C}({\rho_t}) = \text{sech}\left( \frac{N{\gamma_0}}{2}\, (t - {t_D}) \right) ~. \end{equation} Note that ${C}({\rho_t})$ is a bell-shaped symmetric function over time $t$, centered at $t = t_D$, with a full width at half maximum scaling as $(\gamma_0 N)^{-1}$. For $t = 0$, Eq.~\eqref{eq:coherence00001} reduces to ${C}({\rho_0}) = \text{sech}\left( N{\gamma_0}{t_D}/{2}\right)$, which in turn approaches to zero in the limit $N \rightarrow \infty$ for ${t_D} \neq 0$. Figure~\ref{figure}(a) shows the density plot of ${C}({\rho_t})$ as a function of dimensionless parameters $\omega(t - {t_D})$ and $N{\gamma_0}/(2\omega)$. For $0 < t \le {t_D}$, ${C}({\rho_t})$ starts increasing monotonically and reaches its maximum value ${C}({\rho_{t_D}}) = 1$ at the time delay $t = {t_D}$. In fact, ${C}({\rho_{t_D}})$ is always maximum regardless the value of $N{\gamma_0}/(2\omega)$. For $t > {t_D}$, ${C}({\rho_t})$ decreases mono\-to\-nically and asymptotically approaches zero in time. In the very underdamped regime, i.e., $N{\gamma_0}/(2\omega) \ll 1$, the quantum co\-he\-rence remains appro\-xi\-mately constant around its maximum value ${C}({\rho_t}) \sim 1$, while for the very overdamped regime, $N{\gamma_0}/(2\omega) \gg 1$, it follows that ${C}({\rho_t}) \sim 0$ for all $t \neq {t_D}$. In other words, the more atoms the system has, the less quantum coherence is stored on each single-atom state and therefore on the $N$-atom state [see Eq.~\eqref{eq:coherence00001002}], except at time $t = t_D$. Roughly speaking, for $t = {t_D}$ the system of two-level atoms undergoes a constructive quantum interference yielding the cooperative effect of superradiance. In fact, the system will radiate with maximum intensity, in a short time window. However, for all $t \neq {t_D}$, the single-particle coherences of the atomic ensemble will be suppressed as the system becomes larger, and thus the quantum properties tend to be negligible for $N$ sufficiently large. Quite interestingly, Eq.~\eqref{eq:coherence00001} can be written in terms of the average intensity of radiation, $I(t) = ({N^2}\omega{\gamma_0}/4) \, {\text{sech}^2}[(N{\gamma_0}/2)(t - {t_D})]$. Indeed, one readily concludes \begin{equation} \label{eq:coherence00002} {C}({\rho_t}) = \sqrt{\frac{{I(t)} }{I_\text{max}}} ~, \end{equation} with $I_\text{max} = {N^2}\omega{\gamma_0}/4$ the maximum intensity value. From Eq.~\eqref{eq:coherence00002}, note the single-atom co\-he\-ren\-ce depends on the square root of the normalized average intensity coo\-pe\-ratively emitted by the whole system. We shall stress the intensity $I(t)$ immediately vanishes for the case in which quantum coherence $C(t)$ is identically zero. Therefore, quantum coherence, instead of quantum entanglement, stands as a figure of merit towards the understanding of the superradiance phenomenon in the mean-field approach. More fundamentally, the relation $C({\rho_t}) \propto \sqrt{I(t)}$ between coherence and radiation intensity becomes clearer when one rea\-li\-zes that for two-level atoms the $l_1$-norm of co\-he\-ren\-ce is given by the normalized microscopic dipole moment, i.e., $\moy{\sigma_x}$, which in turn stand as a natural measure of coherence in the process of collective spontaneous emission~\cite{PhysRevA.97.052304}. Indeed, remembering the average intensity of radiation can be written in terms of the coherence of the normalized total electric dipole moment, i.e., $I(t) \propto \moy{J^{+}J^{-}}$~\cite{Gross1982,Benedict1996}, one may readily verify that ${[C({\rho_t})]^2} \propto \moy{J^{+}J^{-}}$, thus validating the $l_1$-norm of coherence as a figure of merit of the superradiance phenomenon in the mean-field approach. Finally, one may prove the quantum coherence for the uncorrelated $N$-particle state ${\rho_N} = {\rho_t^{\otimes N}}$, i.e., the coherence of the entire system, is given by \begin{equation} \label{eq:coherence00001002} {C}({\rho_t^{\otimes N}}) = {\left[1 + {C}({\rho_t}) \right]^N} - 1 ~. \end{equation} In particular, for the case in which the single-particle quantum coherence is much smaller than one, i.e., ${C}({\rho_t}) \ll 1$, the $N$-particle quantum coherence in Eq.~\eqref{eq:coherence00001002} approximately becomes ${C}({\rho_t^{\otimes N}}) \approx N {C}({\rho_t})$. In principle, the result given by Eq.~\eqref{eq:coherence00002} suggests that, when experimentally measuring the intensity $I(t)$, one could infer the quantum coherence of a single two-level atom of the system. For example, this link would be of particularly interest for experimental setups in which it is unfeasible to address atoms individually. It is important to notice that our results are based on Eq.~\eqref{mecoll} following the mean-field approximation ($N \gg 1$), which refers to a toy model. However, it is straightforward to show that our results are also achieved, still for $N\gg1$, even when we consider individual atomic decay and dephasing as small perturbations in the collective decay. Importantly, it is possible that such constraints do not hold in certain experimental implementations of the referred physical setting, thus entailing some additional complications to obtain a direct relation between the single-atom coherence and the intensity of the emitted radiation. We point out that this deserves further investigation. \section{Quantum Speed Limit} \label{sec:QSLsection0001} In this section we briefly introduce the quantum speed limit (QSL) time and relate it to the superradiance phenomenon. Quantum mechanics imposes a thre\-shold on the minimum evolution time required for a system evolve between two given quantum states, which in turn is certified by the QSL~\cite{1945_JPhysURSS_9_249,1992_PhysicaD_120_188,2009_PhysRevLett_103_160502}. Given a unitary evolution of pure states $\|{\psi_0}\rangle$ and $\|{\psi_{\tau}}\rangle$ generated by a time-independent Hamiltonian $H$, Mandelstam and Tamm (MT)~\cite{1945_JPhysURSS_9_249} have proved the QSL bound $\tau \geq \hbar\arccos(\|\langle{\psi_0}\|{\psi_{\tau}}\rangle\|)/\Delta E$, in which ${(\Delta E)^2} = \langle{\psi_0}\|{H^2}\|{\psi_0}\rangle - {\langle{\psi_0}\|{H}\|{\psi_0}\rangle^2}$ stands for the variance of $H$. Later, Margolous and Le\-vi\-tin (ML)~\cite{1992_PhysicaD_120_188} derived a novel QSL bound for closed quantum systems evolving between two ortho\-go\-nal states, with time-independent Hamiltonian $H$, which reads $\tau \geq \hbar\pi/(2E)$, in which $E = {\langle{\psi_0}\|{H}\|{\psi_0}\rangle} - {E_0}$ is the mean energy, and $E_0$ the ground state energy of the system. More than a decade after this result, Levitin and Toffoli~\cite{2009_PhysRevLett_103_160502} showed that, by focusing on the case of orthogonal pure states evolving unitarily, the tightest QSL sets ${\tau_{\text{QSL}}} = \max \{\hbar \pi/(2\Delta{E}),\hbar\pi/(2E)\}$. Giovannetti {\it et al.}~\cite{2003_PhysRevA_67_052109} addressed the case of QSLs for mixed states undergoing unitary evolutions, also concluding that entanglement is able to speed up the evolution of composite systems. For more details on QSLs for closed quantum systems, see Refs.~\cite{1983_JPhysAMathGen_16_2993,Aharonov1990,1992_AmJPhys_60_182_Vaidman,1993_PhysRevLett_70_3365,1999_PhysLettA_262_296,2003_36_5587_JPhysAMathGen_Dorje_Brody,2004_PhysicaD_1_189_Luo,2010_PhysRevA_82_022107,2013_JMathPhys_46_335302,Deffner2017,PhysRevA.90.012303_Russell,1506.03199_Mondal_Datta_Sk,PhysRevA.93.052331,Okuyama2018,Modi2018}. \begin{figure}\label{figure} \end{figure} QSL has been also largely investigated for the dynamics of open quantum systems. Indeed, Taddei {\it et al.}~\cite{2013_PhysRevLett_110_050402} and del Campo {\it et al.}~\cite{2013_PhysRevLett_110_050403} have derived the MT bound for arbitrary physical processes, which can be either unitary or nonunitary. Furthermore, Deffner and Lutz~\cite{Deffner2013} derived another class of MT and ML bounds, also showing that non-Markovian signatures can speed up the nonunitary dynamics. Nevertheless, it has been proved the link between speeding up the evolution and non-Markovianity exists only for a certain class of dynamical maps and initial states~\cite{Teittinen_2019}. For completeness, we refer to Refs.~\cite{PhysRevA.89.012307,PhysRevA.91.022102_LiuXuZhu,2015_JPhysAMathTheor_48_045301,PhysRevLett.115.210402,Uzdin_2016,Pires2016,Volkoff2018distinguishability,PhysRevLett.120.070402,PhysRevLett.121.070601,Funo_2019,Garc_a_Pintos_2019,Campaioli2019tightrobust} for other derivations and applications of QSLs for open quantum systems. In Sec.~\ref{sec:model} we have shown that each single two-level atom of the system undergoes an effective unitary evolution governed by the time-dependent non\-li\-near Hamiltonian $H_t$. The effective two-level system is initialized in the pure state $\|{\psi_0}\rangle$, and thus the evolved state $\|{\psi_t}\rangle$ will also be pure during the unitary dynamics for any $t \in [0,\tau]$. Therefore, here we will deal with a quantum system undergoing a nonlinear physical process, but still unitary. In this case, the lower bound on $\tau$, which holds for initial and final pure states undergoing a unitary physical process, is obtained from the inequality $\tau \geq {\tau_{\rm QSL}}$, with the QSL time given by~\cite{2013_JMathPhys_46_335302,Deffner2017} \begin{equation} \label{tQSL} {\tau_{\rm QSL}} = \frac{\mathcal{L}(\ket{\psi_0}, \ket{\psi_\tau})}{\overline{\Delta E}_{\tau}} ~, \end{equation} where $\mathcal{L}(\ket{\psi_0}, \ket{\psi_\tau}) = \arccos{(\|\braket{\psi_0}{\psi_\tau}\|)}$ is the Bures angle, i.e., a distance measure between quantum states, while $\overline{\Delta E}_{\tau} = {\tau^{-1}} {\int_{0}^{\tau}} dt \, \Delta {E_t}$ is the time-average of the variance $\Delta {E_t} = \sqrt{\bra{\psi_t}H_t^2 \ket{\psi_t} - \bra{\psi_t}H_t \ket{\psi_t}^2}$ of the time-dependent Hamiltonian $H_t$. In the particular case where $\|{\psi_0}\rangle$ is orthogonal to $\|{\psi_{\tau}}\rangle$, Eq.~\eqref{tQSL} reduces to ${\tau_{\rm QSL}} = \pi/(2{\overline{\Delta E}_{\tau}})$. Physically, ${\tau_{\rm QSL}}$ sets the minimal time the system requires to evolve between states $\|{\psi_0}\rangle$ and $\|{\psi_{\tau}}\rangle$, also presenting a geometric interpretation discussed as follows. On the one hand, the unitary evolution of $\|{\psi_t}\rangle$ des\-cribes an arbitrary path in the manifold of pure states for $t \in [0 ,\tau ]$, thus connecting states $\|{\psi_0}\rangle$ and $\|{\psi_{\tau}}\rangle$. The length of this path, which generally is not the shor\-test one with respect to the set of paths drawn by $\|{\psi_t}\rangle$, is written as ${\int_{0}^{\tau}} dt \, \Delta {E_t}$ and depends on the variance of the Hamiltonian $H_t$, which in turn is nothing but the quantum Fisher information metric for the case of pure states. On the other hand, Bures angle describes the length of the geodesic path connecting states $\|{\psi_0}\rangle$ and $\|{\psi_{\tau}}\rangle$, and is a function of the overlap of both states. It is quite re\-mar\-ka\-ble that the Bures angle plays the role of a distinguishability measure of quantum states, and stands as the geodesic distance regarding to the quantum Fisher information metric. For a detailed discussion of geometric QSLs, by exploiting the family of Riemannian information metrics defined on the space of quantum states, which in turn encompasses open and closed quantum systems and pure and mixed states, see Ref.~\cite{Pires2016}. Now we will discuss the role played by the superradiance phenomenon and the collective excitations of Dicke states into the QSL time. In order to see this, we will first proceed with the analytical calculation of QSL ratio ${\tau_{\rm QSL}}/\tau$ in Eq.~\eqref{tQSL}. Given that $\bra{\psi_t}H_t^2 \ket{\psi_t} = \left({\omega}/{2}\right)^2 + {\left({N\gamma_0}/{2}\right)^2} {p_t}(1 - {p_t})$, and $\bra{\psi_t}H_t \ket{\psi_t} = ({\omega}/{2})(2{p_t}-1)$, one obtains the time-average of variance as \begin{widetext} \begin{equation} \label{lhs} {\overline{\Delta E}_{\tau}} = \frac{1}{2\tau}\sqrt{1 + {\alpha^{-2}}} \, \arccos{\left[(1-2p_\tau)(1-2p_0) + 4 \sqrt{p_0 p_\tau (1-p_0)(1-p_\tau)} \, \right]} ~, \end{equation} with $\alpha := N{\gamma_0}/(2\omega)$, while the Bures angle becomes \begin{equation} \label{rhs} \mathcal{L}(\ket{\psi_0}, \ket{\psi_\tau}) = \frac{1}{2} \arccos{\left[(1-2p_\tau)(1-2p_0) + 4 \sqrt{p_0 p_\tau (1-p_0)(1-p_\tau)} \cos(\omega \tau)\right] } ~. \end{equation} \end{widetext} Next, we discuss some remarkable features involving the QSL of the evolution due to superradiant transitions between Dicke states. The state $\|{\psi_{\tau}}\rangle$ becomes maximally distinguishable, i.e., orthogonal, to the initial state $\|{\psi_0}\rangle$, in the limit $\alpha \to \infty$ (for $\tau > t_D \neq 0$) or $\tau \to \infty$. This means the maximum distinguishability is attainable if the time of evolution exceeds the time delay $t_D$, i.e., when the system populate its superradiant state. Moreover, the bound $\tau \geq {\tau_{\text{QSL}}}$ saturates in the limit $\alpha \to \infty$, as long as $t_D \ne0$ ($p_0 \to 1$). To summarize, QSL time saturates (i) in the so-called overdamped regime, i.e., $N\gamma_0/2\omega \gtrsim 1$; or (ii) if one increases the number of atoms $N$ in the system, maintaining the ratio $\gamma_0/\omega$ fixed. Physically, the system evolves along the geodesic path connecting $\|{\psi_0}\rangle$ and $\|{\psi_{\tau}}\rangle$ in the mani\-fold of pure quantum states. For instance, this result suggests that quantum information-processing devices based on Dicke superradiance could operate at maximum speed as long as $N \gamma_0/2\omega \gg 1$. Let us now discuss the role played by quantum coherence into the QSL time. Without loss of generality, here we will set the number of atoms as $N = {10^6}$. It has been proved that entanglement~\cite{2003_PhysRevA_67_052109} can promote a speed up in the time evolution of a quantum system, while the quantum coherence also plays a non-trivial role on the time evolution~\cite{1506.03199_Mondal_Datta_Sk,PhysRevA.93.052331,Pires2016}. Figure~\ref{figure}(b) shows the density plot for the ratio ${\tau_{\text{QSL}}}/\tau$ as a function of dimensionless parameters $\omega(\tau - {t_D})$ and $N{\gamma_0}/(2\omega)$. On the one hand, the inequality $\tau \geq {\tau_{\rm QSL}}$ suddenly saturates in the overdamped regime $N{\gamma_0}/(2\omega) \gtrsim 1$, region in which ${C}({\rho_t}) \to 0$ when $t \ne t_D$, as depicted in Fig.~\ref{figure}(a). On the other hand, in the very underdamped regime $N{\gamma_0}/(2\omega) \ll 1$ where ${C}({\rho_t}) \sim 1$, the ratio ${\tau_{\text{QSL}}}/\tau$ approaches zero, which in turn implies a speed up into the evolution of the two-level system. Therefore, for a fixed difference $\tau - {t_D}$, the ratio ${\tau_{\rm QSL}}/\tau$ decreases as the coherence increases [see Fig.~\ref{figure}(c)], i.e., quantum coherence speeds up the dy\-na\-mics. In other words, the more coherence, the faster the evolution in the mean-field Dicke model of superradiance. Note that the QSL time saturates whenever $C({\rho_{\tau}}) \approx 0$, or close to the peak of the superradiance intensity, which is marked by the delaying time $\tau \approx {t_D}$. It is worth mention that the aforementioned relation between QSL and quantum coherence can be directly observed by rewriting Eqs.~\eqref{lhs} and~\eqref{rhs} in terms of ${C}({\rho_\tau})$, such that \begin{widetext} \begin{equation} \label{eq:QSLcoherence000xxx001} \frac{\tau_{\text{QSL}}}{\tau} = \frac{\frac{1}{2}\arccos{\left({C}({\rho_0}){C}({\rho_{\tau}})\cos{(\omega \tau)} - \text{sgn}(\tau - {t_D})\sqrt{(1 - {{C}({\rho_0})^2})(1 - {{C}({\rho_{\tau}})^2})} \, \right)}}{\frac{1}{2}\sqrt{1 + {\alpha^{-2}}}\, \arccos{\left({C}({\rho_0}){C}({\rho_{\tau}}) - \text{sgn}(\tau - {t_D})\sqrt{(1 - {{C}({\rho_0})^2})(1 - {{C}({\rho_{\tau}})^2})} \, \right)}} \,. \end{equation} \end{widetext} \section{Conclusions} \label{sec:conclusions0001} Dicke superradiance, a phenomenon triggered by the collective coupling of atomic levels with the electromagnetic field, is a subject of wide interest in quantum optics and condensed matter physics, not only from the viewpoint of fundamentals of physics, but also for its promising application in the devising of quantum devices. In this work we discussed the quantifying of quantum coherence in the Dicke model of superradiance, under the mean-field approximation description, particularly focusing on the $l_1$-norm of coherence. We found that, for all $t \neq {t_D}$, the more particles in the atomic ensemble, the less quantum coherence is stored in the state of the system. It is noteworthy that quantum coherence exhibits its maximum value at time delay $t_D$, for any number $N$ of atoms. Furthermore, we show that quantum coherence is related to the radiation intensity, and thus the former stand as a useful figure of merit to investigate the superradiance phenomenon in the mean-field approach. Due to the aforementioned link, we point out the intensity of radiation emitted by the system vanishes when quantum coherence is identically zero. Remarkably, this result suggests probing the single-atom coherence through the radiation intensity in superradiant systems, which might be useful in experimental setups where is unfeasible to address atoms individually. In addition, given the time-dependent single-atom state describing the effective dynamics of each two-level atom [see Eq.~\eqref{hamilteffective0002}], we address the quantum speed limit (QSL) time ${\tau_{\text{QSL}}}$ regarding the unitary evolution between two non-orthogonal pure states. We observed that the in\-crea\-sing of the number of atoms $N$ in the system implies the saturation of the QSL time, thus indicating that Dicke-superradiance-based quantum devices could operate at maximum speed as long as $N \gg 1$. Finally, since QSL time can be recast in terms of the $l_1$-norm of co\-he\-ren\-ce [see Eq.~\eqref{eq:QSLcoherence000xxx001}], we have seen the QSL ratio ${\tau_{\rm QSL}}/\tau$ decreases as the quantum coherence increases, thus concluding that the quantum coherence is a resource that speeds up the o\-verall dy\-na\-mics of the superradiant system. Our findings unveil the role played by quantum coherence, quantum speed limit, and their interplay in superradiant systems, which in turn could be of interest for the communities of condensed matter physics and quantum optics, particularly for further investigation of Dicke-superradiance-based quantum devices, and for an analysis which goes beyond the mean-field approximation~\cite{PhysRevA.97.052304,PhysRevResearch.2.023125}. \begin{thebibliography}{104} \makeatletter \providecommand \@ifxundefined [1]{ \@ifx{#1\undefined} } \providecommand \@ifnum [1]{ \ifnum #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi } \providecommand \@ifx [1]{ \ifx #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi } \providecommand \natexlab [1]{#1} \providecommand \enquote [1]{``#1''} \providecommand \bibnamefont [1]{#1} \providecommand \bibfnamefont [1]{#1} \providecommand \citenamefont [1]{#1} \providecommand \href@noop [0]{\@secondoftwo} \providecommand \href [0]{\begingroup \@sanitize@url \@href} \providecommand \@href[1]{\@@startlink{#1}\@@href} \providecommand \@@href[1]{\endgroup#1\@@endlink} \providecommand \@sanitize@url [0]{\catcode `\\12\catcode `\$12\catcode `\&12\catcode `\#12\catcode `\^12\catcode `\_12\catcode `\%12\relax} \providecommand \@@startlink[1]{} \providecommand \@@endlink[0]{} \providecommand \url [0]{\begingroup\@sanitize@url \@url } \providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix }} \providecommand \urlprefix [0]{URL } \providecommand \Eprint [0]{\href } \providecommand \doibase [0]{http://dx.doi.org/} \providecommand \selectlanguage [0]{\@gobble} \providecommand \bibinfo [0]{\@secondoftwo} \providecommand \bibfield [0]{\@secondoftwo} \providecommand \translation [1]{[#1]} \providecommand \BibitemOpen [0]{} \providecommand \bibitemStop [0]{} \providecommand \bibitemNoStop [0]{.\EOS\space} \providecommand \EOS [0]{\spacefactor3000\relax} \providecommand \BibitemShut [1]{\csname bibitem#1\endcsname} \let\auto@bib@innerbib\@empty \bibitem [{\citenamefont {Hepp}\ and\ \citenamefont {Lieb}(1973{\natexlab{a}})}]{PhysRevA.8.2517} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Hepp}}\ and\ \bibinfo {author} {\bibfnamefont {E.~H.}\ \bibnamefont {Lieb}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Equilibrium {S}tatistical {M}echanics of {M}atter {I}nteracting with the {Q}uantized {R}adiation {F}ield},}\ }\href {\doibase 10.1103/PhysRevA.8.2517} {\bibfield {journal} {\bibinfo {journal} {Phys. 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Rev. A}\ }\textbf {\bibinfo {volume} {96}},\ \bibinfo {pages} {033633} (\bibinfo {year} {2017})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Zhu}\ \emph {et~al.}(2020)\citenamefont {Zhu}, \citenamefont {Xu}, \citenamefont {Zhang},\ and\ \citenamefont {Liu}}]{PhysRevLett.125.050402} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont {Zhu}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Xu}}, \bibinfo {author} {\bibfnamefont {G.-F.}\ \bibnamefont {Zhang}}, \ and\ \bibinfo {author} {\bibfnamefont {W.-M.}\ \bibnamefont {Liu}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{F}inite-{C}omponent {M}ulticriticality at the {S}uperradiant {Q}uantum {P}hase {T}ransition},}\ }\href {\doibase 10.1103/PhysRevLett.125.050402} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. 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Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo {pages} {047702} (\bibinfo {year} {2019}{\natexlab{a}})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Andolina}\ \emph {et~al.}(2019{\natexlab{b}})\citenamefont {Andolina}, \citenamefont {Keck}, \citenamefont {Mari}, \citenamefont {Giovannetti},\ and\ \citenamefont {Polini}}]{PhysRevB.99.205437} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~M.}\ \bibnamefont {Andolina}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Keck}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Mari}}, \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Giovannetti}}, \ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Polini}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum versus classical many-body batteries},}\ }\href {\doibase 10.1103/PhysRevB.99.205437} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. 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Rev. Lett.}\ }\textbf {\bibinfo {volume} {103}},\ \bibinfo {pages} {240501} (\bibinfo {year} {2009})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Jaeger}(2018)}]{LARS_2018} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Jaeger}},\ }\href {\doibase 10.1007/978-3-319-98824-5} {\emph {\bibinfo {title} {The {S}econd {Q}uantum {R}evolution}}}\ (\bibinfo {publisher} {Springer},\ \bibinfo {address} {New York},\ \bibinfo {year} {2018})\BibitemShut {NoStop} \bibitem [{\citenamefont {Mandelstam}\ and\ \citenamefont {Tamm}(1945)}]{1945_JPhysURSS_9_249} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Mandelstam}}\ and\ \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Tamm}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The uncertainty relation between energy and time in non-relativistic quantum mechanics},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Phys. 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Rev. Lett.}\ }\textbf {\bibinfo {volume} {103}},\ \bibinfo {pages} {160502} (\bibinfo {year} {2009})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Murphy}\ \emph {et~al.}(2010)\citenamefont {Murphy}, \citenamefont {Montangero}, \citenamefont {Giovannetti},\ and\ \citenamefont {Calarco}}]{PhysRevA.82.022318} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Murphy}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Montangero}}, \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Giovannetti}}, \ and\ \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Calarco}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Communication at the quantum speed limit along a spin chain},}\ }\href {\doibase 10.1103/PhysRevA.82.022318} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. 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Rev. Lett.}\ }\textbf {\bibinfo {volume} {118}},\ \bibinfo {pages} {150601} (\bibinfo {year} {2017})}\BibitemShut {NoStop} \bibitem [{\citenamefont {dos Santos}\ and\ \citenamefont {Duzzioni}(2016)}]{PhysRevA.94.023819} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~M.}\ \bibnamefont {dos Santos}}\ and\ \bibinfo {author} {\bibfnamefont {E.~I.}\ \bibnamefont {Duzzioni}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Elucidating {D}icke superradiance by quantum uncertainty},}\ }\href {\doibase 10.1103/PhysRevA.94.023819} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. 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\begin{document} \title{Diagnosability of labeled max-plus automata} \thispagestyle{empty} \pagestyle{empty} \begin{abstract} In this paper, \emph{diagnosability} is characterized for a labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ over a dioid $\mathcal{D}$ as a real-time system. In order to represent time elapsing, a special class of dioids called \emph{progressive} are considered, in which there is a total canonical order, there is at least one element greater than ${\bf1}$, the product of sufficiently many elements greater than ${\bf1}$ is arbitrarily large, and the cancellative law is satisfied. Then a notion of diagnosability is formulated for $\mathcal{A}^{\mathcal{D}}$ over a progressive dioid $\mathcal{D}$. By developing a notion of \emph{concurrent composition}, a sufficient and necessary condition is given for diagnosability of automaton $\mathcal{A}^{\mathcal{D}}$. It is also proven that the problem of verifying diagnosability of $\mathcal{A}^{\underline{\mathbb{Q}}}$ is $\mathsf{coNP}$-complete, where $\mathsf{coNP}$-hardness even holds for deterministic, deadlock-free, and divergence-free $\mathcal{A}^{\underline{\mathbb{N}_0}}$, where $\underline{\mathbb{Q}}$ and $\underline{\mathbb{N}_0}$ are the max-plus dioids having elements in $\mathbb{Q}\cup\{-\infty\}$ and $\mathbb{N}_0\cup\{-\infty\}$, respectively. \end{abstract} \section{Introduction} \subsection{Background} The study of \emph{diagnosability} of discrete-event systems (DESs) modeled by finite-state machines originates from \cite{Lin1994DiagnosabilityDES,Sampath1995DiagnosabilityDES}, where in the former state-based diagnosis was studied, and in the latter event-based diagnosis was studied and a \emph{formal definition} of diagnosability was given. A DES usually consists of discrete states and transitions between states caused by spontaneous occurrences of partially-observed (labeled) events. Intuitively, a DES is called diagnosable if the occurrence of a faulty event can be determined after a finite number of subsequent events occur by observing the generated output sequence. Widely studied DES models have finitely many events, but may have infinitely many states, e.g., Petri nets, timed automata, max-plus systems, etc. The event-based notion of diagnosability can be naturally extended to these models, although the characterization of diagnosability is much more difficult than in finite-state machines. \subsection{Literature review} \label{subsec:LiterRev} In the following, we briefly review diagnosability results in different models. \emph{Finite-state automata.} In finite-state automata (FSAs), a notion of diagnosability was formulated in \cite{Sampath1995DiagnosabilityDES}, and a \emph{diagnoser} method was proposed to verify diagnosability. The diagnoser of an FSA records state estimates along observed output sequences and also records fault propagation along transitions of states of the FSA. The diagnoser has exponential complexity, and diagnosability is verifiable by a relatively simple cycle condition on the diagnoser. Hence diagnosability can be verified in $\mathsf{EXPTIME}$. Later, a \emph{twin-plant} structure with polynomial complexity was proposed in \cite{Jiang2001PolyAlgorithmDiagnosabilityDES, Yoo2002DiagnosabiliyDESPTime} so that a polynomial-time algorithm for verifying diagnosability was given. The verification algorithms in the above three papers all depend on two fundamental assumptions of deadlock-freeness (an FSA will always run) and divergence-freeness (the running of an FSA will always be eventually observed, more precisely, for every infinitely long transition sequence, its output sequence is also of infinite length). Later on, in many papers, verification of all kinds of variants of diagnosability depends on the two assumptions. The two assumptions were removed in \cite{Cassez2008FaultDiagnosisStDyObser} by using a generalized version of the twin-plant structure to verify \emph{negation} of diagnosability, in polynomial time. Results on diagnosability of probabilistic FSAs can be found in \cite{Thorsley2005DiagnosabilityStochDES,Thorsley2017EquivConditionDiagnosabilityStochDES,Keroglou2019AA-DiagnosabilityProbAutomata}, etc. Results on decentralized settings of diagnosability can be found in \cite{Debouk2000CodiagnosabilityAutomata,Cassez2012ComplexityCodiagnosability,Zhang2021UnifyingDetDiagPred}, etc. \emph{Labeled Petri nets.} For results on diagnosability of labeled Petri nets, we refer the reader to \cite{Cabasino2012DiagnosabilityPetriNet,Berard2018DiagnosabilityPetriNet,Yin2017DiagnosabilityLabeledPetriNets}, etc. In \cite{Cabasino2012DiagnosabilityPetriNet}, a new technique called \emph{verifier} (which can be regarded as the twin-plant structure extended to labeled Petri nets) was developed to verify diagnosability, and two notions of diagnosability were verified by using the technique under several assumptions. It was also pointed out that the two notions are equivalent in FSAs but not in labeled Petri nets. In \cite{Yin2017DiagnosabilityLabeledPetriNets}, the weaker notion of diagnosability studied in \cite{Cabasino2012DiagnosabilityPetriNet} was proven to be decidable with an $\mathsf{EXPSPACE}$ lower bound under the first of the two previously mentioned assumptions, by using the verifier and Yen's path formulae \cite{Yen1992YenPathLogicPetriNet,Atig2009YenPathLogicPetriNet}; in \cite{Berard2018DiagnosabilityPetriNet}, an even weaker notion of diagnosability called trace diagnosability was proven to be decidable in $\mathsf{EXPSPACE}$ with an $\mathsf{EXPSPACE}$ lower bound without any assumption, by using the verifier and linear temporal logic. Results on diagnosability in special classes (e.g., bounded, unobservable-event-induced-subnets being acyclic) of labeled Petri nets can be found in \cite{Giua2005FaultDetectionPetriNets,Cabasino2005FaultDetectionPetriNets, Basile2009DiagnosisPetriNetsILP}, etc. \emph{Labeled timed automata.} Diagnosability was firstly defined for (labeled) timed automata in \cite{Tripakis2002DiagnosisTimedAutomata}, and its decision problem was proven to be $\mathsf{PSPACE}$-complete, where $\mathsf{PSPACE}$-membership was proved by computing a \emph{parallel composition} (which falls back on the generalized twin-plant structure of FSAs used in \cite{Cassez2008FaultDiagnosisStDyObser}) in time polynomial in the size of the timed automaton and finding Zeno runs of the parallel composition in $\mathsf{PSPACE}$ in the size of the parallel composition. The diagnoser of a timed automaton was defined (which may be a Turing machine) and fault diagnosis was done in $2$-$\mathsf{EXPTIME}$ in the size of the timed automaton and in the size of the observation. In \cite{Bouyer2005FaultDiagnosisTimedAutomata}, it was studied whether a timed automaton has a diagnoser that can be realized by a deterministic timed automaton (or particularly an event recording automaton as a subclass of the former). The former was proven to be $2$-$\mathsf{EXPTIME}$-complete and the latter $\mathsf{PSPACE}$-complete, provided that a bound on the resources available to the diagnoser is given. \emph{Labeled max-plus automata.} In \cite{Lai2019FaultDiagnosisMax-PlusAutomata}, a notion of diagnoser was defined for a (labeled) max-plus automaton over the max-plus dioid $\underline{\mathbb{R}}=(\mathbb{R}\cup\{-\infty\},\max,+,-\infty,0)$, where $\mathbb{R}$ denotes the set of real numbers, and computed under the second of the previously mentioned two assumptions and an additional strong assumption that for every two different states $p,q$, there is at most one unobservable transition from $p$ to $q$. In the paper, a max-plus automaton was not treated as a real-time system as above but a max-plus system, because the diagnoser does not record state estimates along observed timed output sequences. Given a timed output sequence $\gamma=(\sigma_1,t_1)\dots(\sigma_n,t_n)$, when the diagnoser receives $\gamma$, it does not necessarily return the ending states of the runs that produce output $\sigma_i$ at time $t_i$, where $1\le i\le n$, but returns the ending states of the runs who produce output sequence $\sigma_1\dots\sigma_n$ and have the maximal time consumption equal to $t_i$ for generating $\sigma_1\dots \sigma_i$. In addition, it was not pointed out that generally for the above max-plus dioid $\underline{\mathbb{R}}$, a diagnoser is usually uncomputable. Finally, a notion of diagnosability was not formulated. \subsection{Contribution of the paper} In this paper, we regard a labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ over a dioid $\mathcal{D}$ as a real-time system and characterize a notion of diagnosability. \begin{enumerate} \item In order to represent time elapsing, we consider a special class of dioids which we call \emph{progressive}. Intuitively in this class, there is a total canonical order, there is at least one element greater than $\bf1$, the product of sufficiently many elements greater than ${\bf1}$ is arbitrarily large, and the cancellative law is satisfied. The classical max-plus dioids such as the tropical semiring $\underline{\mathbb{R}}$, $\underline{\mathbb{Q}}=(\mathbb{Q}\cup\{-\infty\},\max,+,-\infty,0)$, and $\underline{\mathbb{N}_0}=(\mathbb{N}_0\cup\{-\infty\},\max,+,-\infty,0)$ are all progressive. \item For a labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ over a progressive dioid $\mathcal{D}$, we give a formal definition of diagnosability (Definition~\ref{def_diag_LMPautomata}). In order to conveniently represent the definition, we choose runs but not traces (i.e., generated event sequences) as in almost all results in the literature (see Section~\ref{subsec:LiterRev}). \item By developing a new notion of \emph{concurrent composition} (Definition~\ref{def_CC_diag_LMPautomata}) for automaton $\mathcal{A}^{\mathcal{D}}$, we give a sufficient and necessary condition for negation of diagnosability of $\mathcal{A}^{\mathcal{D}}$ (Theorem~\ref{thm1_diag_LMPautomata}). \item Particularly, we prove that the problem of verifying diagnosability of automaton $\mathcal{A}^{\underline{\mathbb{Q}}}$ over dioid $\underline{\mathbb{Q}}$ is $\mathsf{coNP}$-complete (Theorem~\ref{thm3_diag_LMPautomata}), where $\mathsf{coNP}$-hardness even holds for deterministic, deadlock-free, and divergence-free automaton $\mathcal{A}^{\underline{\mathbb{N}_0}}$. $\mathsf{coNP}$-membership is obtained by proving that a concurrent composition can be computed in $\mathsf{NP}$ in the size of $\mathcal{A}^{\underline{\mathbb{Q}}}$ by connecting $\mathcal{A}^{\underline{\mathbb{Q}}}$ and the $\mathsf{NP}$-complete \emph{exact path length problem} \cite{Nykanen2002ExactPathLength}, and negation of diagnosability can be verified in time polynomial in the size of the concurrent composition. $\mathsf{coNP}$-hardness is obtained by constructing a polynomial-time reduction from the $\mathsf{NP}$-complete \emph{subset sum problem} \cite{Garey1990ComputerIntractability} to negation of diagnosability. \end{enumerate} Compared with the results given in \cite{Lai2019FaultDiagnosisMax-PlusAutomata}, the results in our paper have several advantages: (1) We formulate a notion of diagnosability. (2) We regard a max-plus automaton as a real-time system, so that the occurrence of any faulty event will be diagnosed if an automaton is diagnosable. (3) The results in our paper do not depend any of the two fundamental assumptions as mentioned before, but the results in \cite{Lai2019FaultDiagnosisMax-PlusAutomata} depend on the second of the two assumptions and an additional strong assumption. The remainder of the paper is structured as follows. In Section~\ref{sec:pre}, we introduce notation, basic facts, the exact path length problem, the subset sum problem, and labeled max-plus automata over dioids. We also propose a notion of progressive dioid and prove several useful properties in such dioids. In Section~\ref{sec:mainresult}, we show the main results of the paper. Section~\ref{sec:conc} ends the paper with a short conclusion. \section{Preliminaries} \label{sec:pre} \subsection{Notation} Symbols $\mathbb{N}_0$, $\mathbb{Z}$, $\mathbb{Z}_{+}$, $\mathbb{Q}$, and $\mathbb{Q}_{+}$ denote the sets of nonnegative integers (natural numbers), integers, positive integers, rational numbers, and positive rational numbers, respectively. For a finite \emph{alphabet} $\Sigma$, $\Sigma^$ denotes the set of \emph{words} (i.e., finite sequences) over $\Sigma$ including the empty word $\epsilon$. Elements of $\Sigma$ are called \emph{letters}. For $s\in \Sigma^$ and $\sigma \in\Sigma$, we write $\sigma\in s$ if $\sigma$ appears in $s$. $\Sigma^{+}:=\Sigma^\setminus\{\epsilon\}$. For a word $s\in \Sigma^$, $\|s\|$ stands for its length. For $s\in \Sigma^+$ and natural number $k$, $\last(s)$ denotes its last letter, $s^k$ denotes the concatenation of $k$ copies of $s$. For a word $s\in \Sigma^$, a word $s'\in \Sigma^$ is called a \emph{prefix} of $s$, denoted as $s'\sqsubset s$, if there exists another word $s''\in \Sigma^$ such that $s=s's''$. For $s\in\Sigma^$ and $s'\in\Sigma^$, we use $s\sqsubsetneq s'$ to denote $s\sqsubset s'$ and $s\ne s'$. For a word $s\in\Sigma^$, a word $s'\in\Sigma^$ is called a \emph{suffix} of $s$ if $s''s'=s$ for some $s''\in\Sigma^$. For two real numbers $a$ and $b$, $[a,b]$ denotes the closed interval with lower and upper endpoints being $a$ and $b$, respectively; for two integers $i\le j$, $\llbracket i,j\rrbracket$ denotes the set of all integers no less than $i$ and no greater than $j$; and for a set $S$, $\|S\|$ denotes its cardinality and $2^S$ its power set. $\subset$ and $\subsetneq$ denote the subset and strict subset relations. We will use the $\mathsf{NP}$-complete \emph{exact path length} (EPL) problem and \emph{subset sum} (SS) problem in the literature to prove the main results. \subsection{The exact path length problem} \label{subsec:EPLProblem} Consider a $k$-dimensional weighted directed graph $G=(\mathbb{Q}^k,V,A)$, where $k\in\mathbb{Z}_{+}$, $\mathbb{Q}^k=\underbrace{\mathbb{Q}\times\cdots\times\mathbb{Q}}_{k}$, $V$ is a finite set of vertices, $A\subset V\times\mathbb{Q}^k\times V$ a finite set of weighted edges with weights in $\mathbb{Q}^k$. For a path $v_1\xrightarrow[]{z_1}\cdots\xrightarrow[]{z_{n-1}}v_n$, its weight is defined by $\sum_{i=1}^{n-1}z_i$. The EPL problem \cite{Nykanen2002ExactPathLength} is stated as follows. \begin{problem}[EPL]\label{prob1_det_MPautomata} Given a positive integer $k$, a $k$-dimensional weighted directed graph $G=(\mathbb{Q}^k,V,A)$, two vertices $v_1,v_2\in V$, and a vector $z\in \mathbb{Q}^k$, determine whether there is a path from $v_1$ to $v_2$ with weight $z$. \end{problem} We set that for a positive integer $n$, the size $\size(n)$ of $n$ to be the length of its binary representation, $\size(0)=1$; for a negative integer $-n$, $\size(-n)=\size(n)+1$ (here $1$ is used to denote the size of ``$-$''); for a rational number $m/n$, where $m,n$ are relatively prime integers, $\size(m/n)=\size(m)+\size(n)$; then for a vector $z\in\mathbb{Q}^{k}$, its size is the sum of the sizes of its components. The size of an instance $(k,G,v_1,v_2,z)$ of the EPL problem is defined by $\size(k)+\size(G)+2+\size(z)$, where $\size(G)=\|V\|+\size(A)$, $\size(A)=\sum_{(v_1,z',v_2)\in A}(2+\size(z'))$. \begin{lemma}[\cite{Nykanen2002ExactPathLength}]\label{lem1_det_MPautomata} The EPL problem belongs to $\mathsf{NP}$. The EPL problem is $\mathsf{NP}$-hard already for graph $(\mathbb{Z},V,A)$. \end{lemma} Note that EPL problem studied in \cite{Nykanen2002ExactPathLength} is on graph $(\mathbb{Z}^k,V,A)$. However, the proof in \cite{Nykanen2002ExactPathLength} also applies to the more general case for graph $(\mathbb{Q}^k,V,A)$, resulting in Lemma~\ref{lem1_det_MPautomata}. \subsection{The subset sum problem} The SS problem \cite{Garey1990ComputerIntractability} is as follows. \begin{problem}[SS]\label{prob2_det_MPautomata} Given positive integers $n_1,\dots,n_m$, and $N$, determine whether $N=\sum_{i\in I}n_i$ for some $I\subset\llbracket 1,m\rrbracket$. \end{problem} \begin{lemma}[\cite{Garey1990ComputerIntractability}]\label{lem2_det_MPautomata} The SS problem is $\mathsf{NP}$-complete. \end{lemma} \subsection{Dioids} \begin{definition}\label{dioid_diag_LMPautomata} An \emph{idempotent semiring} (or \emph{dioid}) is a set $T$ with two binary operations $\oplus$ and $\otimes$, called \emph{addition} and \emph{multiplication}, such that $(T,\oplus)$ is an \emph{idempotent commutative monoid} with \emph{identity element} ${\bf0}\in T$ (also called \emph{zero}), $(T,\otimes)$ is a monoid with \emph{identity element} ${\bf1}\in T$ (also called \emph{one}), $\bf0$ is absorbing, and $\otimes$ distributes over $\oplus$ on both sides. A dioid is denoted by $\mathcal{D}=(T,\oplus,\otimes,{\bf0},{\bf1})$. \end{definition} For $a\in \mathcal{D}$ (i.e., $a\in T$), we write $a^n=\underbrace{a\otimes\cdots\otimes a}_{n}$ for all $n\in\mathbb{N}_0$, and denote $a^0={\bf1}$. An \emph{order} over a set $T$ is a relation $\preceq\subset T\times T$ that is reflexive, anti-symmetric, and transitive. For all $a,b\in T$ such that $a\preceq b$, we also write $b\succeq a$. If additionally $a\ne b$, then we write $a\prec b$ or $b\succ a$. In dioid $\mathcal{D}$, the relation $\preceq\subset T\times T$ such that $a\preceq b$ if and only if $a\oplus b=b$ for all $a,b\in T$ is an order (called the \emph{canonical order}). In the sequel, $\preceq$ always means the canonical order. Order $\preceq$ is \emph{total} if for all $a,b\in \mathcal{D}$, either $a\preceq b$ or $b\preceq a$. We have some direct properties as follows. \begin{lemma}\label{lem1_diag_LMPautomata} \begin{enumerate}[(1)] \item\label{item4_diag_LMPautomata} Let $a,b\in \mathcal{D}$ be such that ${\bf 1}\preceq b$. Then for all $a\in \mathcal{D}$, $a\preceq a\otimes b$, $a\preceq b\otimes a$. \item\label{item5_diag_LMPautomata} Let $a,b\in \mathcal{D}$ be such that ${\bf 1}\prec a$ and ${\bf 1}\preceq b$. Then ${\bf 1}\prec a\otimes b$ and ${\bf 1}\prec b\otimes a$. \item\label{item15_diag_LMPautomata} Let $a,b,c,d\in\mathcal{D}$ be such that $a\preceq b$ and $c\preceq d$. Then $a\otimes c\preceq b\otimes d$. \item\label{item14_diag_LMPautomata} Let $a,b\in \mathcal{D}$ be such that ${\bf 1}\prec a\prec b$. Then $a\prec a\otimes b$ and $a\prec b\otimes a$. \item\label{item14'_diag_LMPautomata} Let $a,b\in \mathcal{D}$ be such that ${\bf 1}\succ a\succ b$. Then $a\succ a\otimes b$ and $a\succ b\otimes a$. \end{enumerate} \end{lemma} The results in Lemma~\ref{lem1_diag_LMPautomata} show that a dioid has some features of \emph{time elapsing}, if we consider an element greater than ${\bf1}$ as a positive time elapsing (${\bf1}$ means no time elapses), and also consider the product of two elements greater than ${\bf1}$ as the total time elapsing (see \eqref{item14_diag_LMPautomata}). In order to see whether a dioid has more features of time elapsing, one may ask given $a,b\in\mathcal{D}$ such that ${\bf 1}\prec a\prec b$, is it true that (a) $b\prec b\otimes a$ and (b) $a\prec a^2$? Generally not. Consider dioid \begin{equation}\label{eqn5_diag_LMPautomata} (\mathbb{Z}\cup\{-\infty,+\infty\},\max,+,-\infty,0), \end{equation} where $+\infty$ is the maximal element. For $0<1<+\infty$, one does not have $(+\infty)<(+\infty)+1$ or $+\infty<(+\infty)+(+\infty)$, but one has $1<1+1$. In this paper, we consider dioids that have as more features of time elapsing as possible, because we want to use a dioid to describe the time in a max-plus automaton as a real-time system. To this end, we consider dioids with the above properties (a) and (b) and additionally with the \emph{cancellative law}: for all $a,b,c\in T$ with $a\ne{\bf0}$, $a\otimes b=a\otimes c$ $\implies$ $b=c$ and $b\otimes a=c\otimes a$ $\implies$ $b=c$. A dioid with the cancellative law is called \emph{cancellative}. \begin{lemma}\label{lem6_diag_LMPautomata} Consider a cancellative dioid $\mathcal{D}$. For all $a,b,c,d\in\mathcal{D}$, the following hold. \begin{enumerate}[(1)] \item\label{item19_diag_LMPautomata} If $a\otimes b\succeq a\otimes c$ and $a\ne{\bf0}$, then $b\succeq c$. \item\label{item20_diag_LMPautomata} If $c\otimes d\succeq a\otimes b$, $c\preceq a$, and $a\ne{\bf0}$, then $d\succeq b$. \item\label{item29_diag_LMPautomata} If $a\ne{\bf0}$ and $b\succ c$ then $a\otimes b\succ a\otimes c$ and $b\otimes a\succ c\otimes a$. \end{enumerate} \end{lemma} \begin{proof} We only prove \eqref{item20_diag_LMPautomata}. By $c\preceq a$ we have $c\oplus a=a$. Then $a\otimes d=(c\oplus a)\otimes d =c\otimes d\oplus a\otimes d\succeq a\otimes b\oplus a\otimes d=a\otimes (b\oplus d)$. By the cancellative law, $d\succeq b\oplus d$. Furthermore, we have $d=b\oplus d\oplus d=b\oplus d$ (idempotency). Hence $d\succeq b$. \end{proof} Now we give the notion of \emph{progressive} dioid. \begin{definition}\label{progressive_dioid_diag_LMPautomata} A dioid $\mathcal{D}=(T,\oplus,\otimes,{\bf0},{\bf1})$ is called \emph{progressive} if (1) $\mathcal{D}$ has a total canonical order, (2) $\mathcal{D}$ has at least one element greater than ${\bf1}$, (3) for all $a,b\in\mathcal{D}$ with $a\succ {\bf1}$ and $b\succ {\bf1}$, there exists $n\in\mathbb{Z}_+$ such that $a^{n}\succ b$, (4) for all $a,b\in\mathcal{D}$ with $a\prec {\bf1}$ and $b\prec{\bf1}$, there exists $n\in\mathbb{Z}_+$ such that $a^n\prec b$, (5) $\mathcal{D}$ is cancellative, (6) for all $a,b\in\mathcal{D}$ with $a\prec{\bf1}\prec b$, there exists $n,m\in\mathbb{Z}_+$ such that $a\otimes b^n\succ {\bf1}$ and $b^m\otimes a\succ {\bf1}$, (7) for all $a,b\in\mathcal{D}$ with $a\prec{\bf1}\prec b$, there exists $n,m\in\mathbb{Z}_+$ such that $a^n\otimes b\prec{\bf1}$, $b\otimes a^{m}\prec{\bf1}$, and (8) $\mathcal{D}$ has no zero divisor (i.e., for all nonzero $a,b$ in $\mathcal{D}$, $a\otimes b\ne{\bf0}$). \end{definition} \begin{remark} If in $\mathcal{D}$, every nonzero element $a$ has a (unique) multiplicative inverse $a^{-1}$, then (3) (resp., (6)) is equivalent to (4) (resp., (7)), because in this case if $a\otimes b={\bf1}$, then $a\succ{\bf1}\iff b\prec{\bf1}$, $a\prec b\iff a^{-1} \succ b^{-1}$. \end{remark} \begin{lemma}\label{lem2_diag_LMPautomata} Let $\mathcal{D}$ be a progressive dioid. \begin{enumerate}[(1)] \item\label{item25_diag_LMPautomata} Let $a,b\in\mathcal{D}$ be such that ${\bf1}\prec a$ and ${\bf1}\prec b$. Then $a\prec a\otimes b$ and $a\prec b\otimes a$. \item\label{item26_diag_LMPautomata} Let $a,b\in\mathcal{D}$ be such that ${\bf1}\succ a$ and ${\bf1}\succ b$. Then $a\succ a\otimes b$ and $a\succ b\otimes a$. \end{enumerate} \end{lemma} \begin{proof} We only need to prove \eqref{item25_diag_LMPautomata}. \eqref{item26_diag_LMPautomata} is a symmetric form of \eqref{item25_diag_LMPautomata}. By \eqref{item4_diag_LMPautomata} of Lemma~\ref{lem1_diag_LMPautomata}, we have $a\preceq a\otimes b$ and $a\preceq b\otimes a$. Suppose $a=a\otimes b$. Then we have $a\otimes b=a\otimes b^2=a$. Analogously we have $a\otimes b^n=a$ for all $n\in\mathbb{Z}_{+}$. Because $\mathcal{D}$ is progressive, we have $b^m\succ a$ for some $m\in\mathbb{Z}_{+}$. By \eqref{item14_diag_LMPautomata} of Lemma~\ref{lem1_diag_LMPautomata}, we have $a\prec a\otimes b^m$, which is a contradiction. Hence $a\prec a\otimes b$. Similarly $a\prec b\otimes a$. \end{proof} \begin{corollary}\label{cor1_diag_LMPautomata} Let $\mathcal{D}$ be a progressive dioid. Let $a,b\in\mathcal{D}$ be such that $b\prec {\bf1}\prec a$. Then \begin{align} &a\prec a^2\prec a^3 \prec \cdots,\\ &b\succ b^2\succ b^3 \succ \cdots. \end{align} \end{corollary} \begin{remark} By Corollary~\ref{cor1_diag_LMPautomata}, one sees in a progressive dioid $\mathcal{D}$, the maximal element must not exists, that is, $\oplus_{t\in T}t\notin T$. For example, dioid \eqref{eqn5_diag_LMPautomata} is not progressive. \end{remark} One sees that the dioids \begin{align} \underline{\mathbb{Q}} &:= (\mathbb{Q}\cup\{-\infty\},\max,+,-\infty,0),\\ \underline{\mathbb{Q}_{\ge0}} &:= (\mathbb{Q}_{\ge 0}\cup\{-\infty\},\max,+,-\infty,0),\\ \underline{\mathbb{N}_0} &:= (\mathbb{N}_0\cup\{-\infty\},\max,+,-\infty,0) \end{align} are progressive and have a total canonical order $\le$. \subsection{Labeled max-plus automata over dioids} A \emph{max-plus automaton} is a tuple $\mathfrak{G}=(Q,E,\Delta,Q_0,\alpha,\mu)$ over dioid $\mathcal{D}=(T,\oplus,\otimes,{\bf0},{\bf1})$, denoted by $(\mathcal{D},\mathfrak{G})$ for short, where $Q$ is a nonempty finite set of \emph{states}, $E$ a nonempty finite \emph{alphabet} (elements of $E$ are called \emph{events}), $\Delta\subset Q\times E\times Q$ a \emph{transition relation} (elements of $\Delta$ are called \emph{transitions}), $Q_0\subset Q$ is a nonempty set of initial states, $\alpha$ is a map $Q_0\to T\setminus\{ {\bf0}\}$, map $\mu:\Delta\to T\setminus\{ {\bf0}\}$ assigns to each transition $(q,e,q')\in\Delta$ a nonzero \emph{weight} $\mu(e)_{qq'}$ in $T$, where this transition is also denoted by $q\xrightarrow[]{e/\mu(e)_{qq'}}q'$. Automaton $(\mathcal{D},\mathfrak{G})$ is called \emph{deterministic} if $\|Q_0\|=1$ and for all $q,q',q''\in Q$ and $e\in E$, $(q,e,q')\in \Delta$ and $(q,e,q'')\in\Delta$ $\implies$ $q'=q''$. For all $q\in Q$, we also regard $q\xrightarrow[]{\epsilon/{\bf1}}q$ as a transition (which is called an \emph{$\epsilon$-transition}). A transition $q\xrightarrow[]{e/\mu(e)_{qq'}}q'$ is called \emph{instantaneous} if $\mu(e)_{qq'}={\bf1}$, and called \emph{noninstantaneous} if $\mu(e)_{qq'}\ne{\bf1}$. Call a state $q\in Q$ \emph{dead} if for all $q'\in Q$ and $e\in E$, $(q,e,q')\not\in\Delta$, i.e., there exists no transition starting at $q$ (apart from the $\epsilon$-transition). Call an automaton $(\mathcal{D},\mathfrak{G})$ \emph{deadlock-free} if it has no reachable dead state. Particularly for $(\underline{\mathbb{Q}_{\ge0}},\mathfrak{G})$, for an initial state $q\in Q_0$, $\alpha(q)$ denotes its initial time delay, and in a transition $q\xrightarrow[]{e/\mu(e)_{qq'}}q'$, $\mu(e)_{qq'}$ denotes its time delay, i.e., the time consumption of the execution of the transition. Hence the execution of an instantaneous transition requires zero time, while the execution of a noninstantaneous transition requires a positive rational number $\mu(e)_{qq'}$ as time. For $q_0,q_1,\dots,q_n\in Q$ (where $q_0$ is not necessarily initial), and $e_1,\dots,e_n\in E$, where $n\in \mathbb{N}_0$, we call \begin{align}\label{path_det_MPautomaton} \pi:=q_0\xrightarrow[]{e_1}q_1\xrightarrow[]{e_2}\cdots\xrightarrow[]{e_n}q_n \end{align} a \emph{path} if for all $i\in\llbracket 0,n-1\rrbracket$, $(q_i,e_{i+1},q_{i+1})\in\Delta$. We write $\init(\pi)=q_0$, $\last(\pi)=q_n$. A path $\pi$ is called a \emph{cycle} if $q_0=q_n$. A cycle $\pi$ is called \emph{simple} if it has no repetitive states except for $q_0$ and $q_n$. For paths $\pi_1$ and $\pi_2$ with $\last(\pi_1)=\init(\pi_2)$, we use $\pi_1\pi_2$ to denote the concatenation of $\pi_1$ and $\pi_2$ after removing one of $\last(\pi_1)$ or $\init(\pi_2)$, and write $(\pi_1\pi_2)\setminus \pi_1=\pi_2$. The set of paths starting at $q_0\in Q$ and ending at $q\in Q$ (under event sequence $s:=e_1\dots e_n\in E^+$) is denoted by $q_0\rightsquigarrow q$ ($q_0\xrsquigarrow{s}q$), where $q_0\xrsquigarrow{s}q=\{q_0\xrightarrow[]{e_1}q_1\xrightarrow[]{e_2}\cdots\xrightarrow[]{e_{n-1}} q_{n-1} \xrightarrow[]{e_n}q\|q_1,\dots,q_{n-1}\in Q\}$. Then for $q\in Q$ and $s\in E^$, we denote $Q_0\rightsquigarrow q:=\bigcup_{q_0\in Q_0}q_0\rightsquigarrow{}q$ and $Q_0\xrsquigarrow{s}q:= \bigcup_{q_0\in Q_0}q_0\xrsquigarrow{s}q$. The \emph{timed word} of path $\pi$ is defined by \begin{align}\label{timedword_det_MPautomaton} \tau(\pi):=(e_1,t_1)(e_2,t_2)\dots(e_n,t_n), \end{align} where for all $i\in\llbracket 1,n\rrbracket$, $t_i=\bigotimes_{j=1}^{i}\mu(e_j)_{q_{j-1}q_j}$. The \emph{weight} $\WEI(\pi)$ (also denoted by $\WEI_{\pi}$) of path $\pi$ and the \emph{weight} $\WEI(\tau(\pi))$ (also denoted by $\WEI_{\tau(\pi)}$) of timed word $\tau(\pi)$ are both defined by $t_n$. Particularly, we set $\WEI_{\epsilon}={\bf1}$ for the empty path $\epsilon$. A path always has nonzero weight if $\mathcal{D}$ has no zero divisor. A path $\pi$ is called \emph{instantaneous} if $t_1=\cdots=t_n={\bf1}$, and called \emph{noninstantaneous} otherwise. Call a state $q\in Q$ \emph{stuck} if either $q$ is dead or starting at $q$ there exist only instantaneous paths. Automaton $(\mathcal{D},\mathfrak{G})$ is called \emph{stuck-free} if it has no reachable stuck state. Particularly for $(\underline{\mathbb{Q}_{\ge0}},\mathfrak{G})$, one has $t_i=\sum_{j=1}^{i}{\mu(e_j)_{q_{j-1}q_j}}$, hence $t_i$ denotes the time needed for the first $i$ transitions in path $\pi$, $i\in\llbracket 1,n\rrbracket$. We define a labeling function $\ell:E\to\Sigma\cup\{\epsilon\}$, where $\Sigma$ is a finite alphabet, to distinguish \emph{observable} and \emph{unobservable} events. The set of observable events and the set of unobservable events are denoted by $E_o=\{e\in E\|\ell(e)\in\Sigma\}$ and $E_{uo}=\{e\in E\|\ell(e)=\epsilon\}$, respectively. When an observable event $e$ occurs, one observes $\ell(e)$; while an unobservable event occurs, one observes nothing. A transition $q\xrightarrow[]{e/\mu(e)_{qq'}}q'$ is called \emph{observable} (resp., \emph{unobservable}) if $e$ is observable (resp., unobservable). Labeling function $\ell$ is recursively extended to $E^\to \Sigma^$ as $\ell(e_1e_2\dots e_n)=\ell(e_1)\ell(e_2)\dots\ell(e_n)$. A path $\pi$ \eqref{path_det_MPautomaton} is called \emph{unobservable} if $\ell(e_1\dots e_n)=\epsilon$, and called \emph{observable} otherwise. A \emph{labeled max-plus automaton} is formulated as \begin{align}\label{LMPA_diag_opa_MPautomata} \mathcal{A}^{\mathcal{D}}:=(\mathcal{D},\mathfrak{G},\Sigma,\ell). \end{align} Labeling function $\ell$ is extended as follows: for all $(e,t)\in E\times T$, $\ell((e,t))=(\ell(e),t)$ if $\ell(e)\ne\epsilon$, and $\ell((e,t))=\epsilon$ otherwise. Hence $\ell$ is also recursively extended to $(E\times T)^\to (\Sigma\times T)^$. For a path $\pi$, $\ell(\tau(\pi))$ is called a \emph{timed label/output sequence}. We also extend the previously defined function $\tau$ as follows: for all $\gamma=(\sigma_1,t_1)\dots(\sigma_n,t_n)\in(\Sigma\times T)^{}$, \begin{equation}\label{eqn11_det_MPautomata} \tau(\gamma)=(\sigma_1,t_1')\dots(\sigma_n,t_n'), \end{equation} where $t_j'=\bigotimes_{i=1}^{j}t_i$ for all $j\in\llbracket 1,n\rrbracket$. Particularly, $\tau(\epsilon)=\epsilon$. \begin{remark} A \emph{finite-state automaton} (studied in \cite{Shu2007Detectability_DES,Shu2011GDetectabilityDES,Masopust2018ComplexityDetectabilityDES,Zhang2017PSPACEHardnessWeakDetectabilityDES}, etc.) can be regarded as a labeled max-plus automaton $\mathcal{A}^{\underline{\mathbb{N}_0}}$ such that all unobservable transitions have weight $0$ and every two observable transitions with the same label have the same positive weight. \end{remark} The size of a given automaton $\mathcal{A}^{\underline{\mathbb{Q}}}$ is defined by $\|Q\|+\|\Delta\|+\|Q_0\|+\size(\alpha)+\size(\mu) +\size(\ell)$, where the size of a rational number has already been defined before, $\size(\alpha)=\sum_{q_0\in Q_0}\size(\alpha(q_0))$, $\size(\mu)=\sum_{(q,e,q')\in\Delta} \size(\mu(e)_{qq'})$, $\size(\ell)=\|\{(e,\ell(e))\|e\in E\}\|$. From now on, without loss of generality, we assume for each initial state $q_0\in Q_0$, $\alpha(q_0)={\bf1}$, because otherwise we can add a new initial state $q_0'$ not in $Q_0$ and set $\alpha(q_0')={\bf1}$, and for each initial state $q_0\in Q_0$ such that $\alpha(q_0)\ne{\bf1}$ we add a new transition $q_0'\xrightarrow[]{\varepsilon/\alpha(q_0)}q_0$ and set $q_0$ to be not initial any more, where $\varepsilon$ is a new event not in $E$. For consistency we also assume that $\ell(\varepsilon)=\epsilon$. Particularly for $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$, if it generates a path $\pi$ as in \eqref{path_det_MPautomaton} such that $q_0\in Q_0$, consider its timed word $\tau(\pi)$ as in \eqref{timedword_det_MPautomaton}, then at time $t_i$, one will observe $\ell(e_i)$ if $\ell(e_i)\ne\epsilon$; and observe nothing otherwise, where $i\in\llbracket 1,n\rrbracket$. \begin{example}\label{exam2_diag_LMPautomata} A stuck-free labeled max-plus automaton $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ is shown in Fig.~\ref{fig3_diag_LMPautomata}. \begin{figure}\label{fig3_diag_LMPautomata} \end{figure} \end{example} \section{Main results} \label{sec:mainresult} \subsection{The definition of diagnosability} In this subsection, we formulate the definition of diagnosability for labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ \eqref{LMPA_diag_opa_MPautomata}. The set of \emph{faulty} events is denoted by a subset $E_f$ of $E$. Usually, one assumes that $E_f\subset E_{uo}$ \cite{Sampath1995DiagnosabilityDES,Tripakis2002DiagnosisTimedAutomata,Jiang2001PolyAlgorithmDiagnosabilityDES}, because the occurrence of observable events can be directly seen. However, technically, this assumption is usually not needed. We do not make this assumption in this paper, because different observable events may have the same label (i.e., labeling function $\ell$ is not necessarily injective on $E_{o}$), so the occurrences of observable faulty events may also need to be distinguished. As usual, with slight abuse of notion, for $s\in E^$, we use $E_f\in s$ to denote that some faulty event $e_f\in E_f$ appears in $s$. Similarly for $w\in (E\times T)^$, we use $E_f\in w$ to denote that some $(e_f,t)$ appears in $w$, where $e_f\in E_f$, $t\in T$. In $\mathcal{A}^{\mathcal{D}}$, all transitions of the form $q_1\xrightarrow[]{e_f/\mu(e_f)_{q_1q_2}} q_2$ for some $e_f\in E_f$ are called \emph{faulty transitions}, the other transitions are called \emph{normal transitions}. We denote by $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$ the \emph{normal subautomaton} of $\mathcal{A}^{\mathcal{D}}$ obtained by removing all faulty transitions of $\mathcal{A}^{\mathcal{D}}$. We also denote by $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$ the \emph{faulty subautomaton} of $\mathcal{A}^{\mathcal{D}}$ by only keeping all faulty transitions and their predecessors and successors, where predecessors mean the transitions from which some faulty transition is reachable, and successors mean the transitions that are reachable from some faulty transition. \begin{definition}\label{def_diag_LMPautomata} Let $\mathcal{D}$ be a progressive dioid, $\mathcal{A}^{\mathcal{D}}$ a labeled max-plus automaton as in \eqref{LMPA_diag_opa_MPautomata} and $E_f\subset E$ a set of faulty events. $\mathcal{A}^{\mathcal{D}}$ is called \emph{$E_f$-diagnosable} (see Fig.~\ref{fig2_diag_LMPautomata} for illustration) if \begin{equation}\label{eqn17_diag_LMPautomata} \begin{split} &(\exists t\in \mathcal{D}:t\succ {\bf1})(\forall \pi\in Q_0 \xrsquigarrow{se_f} q\text{ with }e_f\in E_f)\\ &(\forall \pi' \in q\xrsquigarrow{s'} q')[ ( (\WEI_{\pi'} \succ t) \vee (q'\text{ is stuck}) ) \implies {\bf D}], \end{split} \end{equation} where ${\bf D}=(\forall \pi''\in Q_0\xrsquigarrow{s''}q''\text{ with }\ell(\tau(\pi''))=\ell(\tau(\pi\pi'))) [((\WEI_{\pi''}\succeq \WEI_{\pi}\otimes t)\vee(q''\text{ is stuck}))\implies (E_f\in s'')].$ \end{definition} \begin{figure} \caption{Illustration of diagnosability. Each line represents a path starting from initial states, where the timed label sequences of all dashed lines are the same as that of the solid line at the instant when the last event of the solid line occurs. Bullets denote faulty events. The circle denotes a stuck state.} \label{fig2_diag_LMPautomata} \end{figure} Definition~\ref{def_diag_LMPautomata} can be interpreted as follows. For a progressive dioid $\mathcal{D}$, automaton $\mathcal{A}^{\mathcal{D}}$ is $E_f$-diagnosable if and only if there exists an element $t\succ{\bf1}$ in $\mathcal{D}$, for every path $\pi$ whose last event is faulty, for every path $\pi'$ as a continuation of $\pi$, if either $\WEI_{\pi'}\succ t$ or $\pi'$ ends at a stuck state, then one can make sure that a faulty event (although not necessarily $e_f$) must have occurred when the timed label sequence $\ell(\tau(\pi\pi'))$ has been generated after at least time $t$ since $e_f$ occurs. Here we consider progressive dioids because in such dioids, for a path $\pi'$ as in \eqref{eqn17_diag_LMPautomata}, in a large extent we have $\WEI_{\pi'}\succ t$, i.e., automaton $\mathcal{A}^{\mathcal{D}}$ can run for a long time. Particularly, the $E_f$-diagnosability of $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$ has the following practical meaning: if $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$ is $E_f$-diagnosable, then once a faulty event (e.g., $e_f$) occurs at some instant (e.g., $\WEI_\pi$), then after a sufficiently long time (e.g., $t\in \mathbb{Q}_+$) that only depends on $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$, one can make sure that some faulty event must have occurred by observing the generated timed label sequence (e.g., $\ell(\tau(\pi\pi'))$) at every instant $\ge t$ since $e_f$ occurs, no matter whether $q'$ is a stuck state. In order to reduce complicated discussions when characterizing $E_f$-diagnosability of automaton $\mathcal{A}^{\mathcal{D}}$, we can equivalently simplify its definition shown in \eqref{eqn17_diag_LMPautomata} by adding at each stuck state a normal, noninstantaneous, and unobservable transition to a \emph{sink} state on which there is also a normal, noninstantaneous, and unobservable self-loop. Let the labeled stuck-free max-plus automaton \begin{equation}\label{SF_LMPA_diag_opa_LMPautomata} \bar\mathcal{A}^{\mathcal{D}}=(\mathcal{D},\bar\mathfrak{G},\Sigma,\bar\ell) \end{equation} be obtained from $\mathcal{A}^{\mathcal{D}}$ by doing the following modifications on $\mathcal{A}^{\mathcal{D}}$: at each stuck state $q\in Q$, add a $\mathfrak{u_n}$-transition $q\xrightarrow[]{\mathfrak{u_n}/\mathfrak{t}}\mathfrak{q}$, where $\mathfrak{u_n}$ is a normal, unobservable event not in $E$, $\mathfrak{q}$ is a fresh state not in $Q$, ${\bf1}\succ \mathfrak{t}\in \mathcal{D}$, and add a self-loop $\mathfrak{q}\xrightarrow[]{\mathfrak{u_n}/\mathfrak{t}}\mathfrak{q}$. \begin{lemma}\label{lem5_diag_LMPautomata} Let $\mathcal{D}$ be a progressive dioid. Let $\mathcal{A}^{\mathcal{D}}$ be a labeled max-plus automaton as in \eqref{LMPA_diag_opa_MPautomata} and $E_f\subset E$ a set of faulty events. Let $\bar\mathcal{A}^{\mathcal{D}}$ be its stuck-free automaton as in \eqref{SF_LMPA_diag_opa_LMPautomata}. Then $\mathcal{A}^{\mathcal{D}}$ is $E_f$-diagnosable if and only if $\bar\mathcal{A}^{\mathcal{D}}$ satisfies \begin{equation}\label{eqn20_diag_LMPautomata} \begin{split} &(\exists t\in \mathcal{D}:t\succ {\bf1})(\forall \pi\in Q_0 \xrsquigarrow{se_f} q\text{ with }e_f\in E_f)\\ &(\forall \pi' \in q\xrsquigarrow{s'} Q)[ (\WEI_{\pi'} \succ t) \implies {\bf D}], \end{split} \end{equation} where ${\bf D}=(\forall \pi''\in Q_0\xrsquigarrow{s''}q''\text{ with }\bar\ell(\tau(\pi''))=\bar\ell (\tau(\pi\pi')))[(\WEI_{\pi''}\succeq \WEI_{\pi}\otimes t)\implies (E_f\in s'')]$. \end{lemma} \begin{proof} Observe that in a path $\pi$ of $\bar\mathcal{A}^{\mathcal{D}}$, if $\mathfrak{u_n}$ appears, then after the $\mathfrak{u_n}$, only $\mathfrak{q}$ can be visited, hence all successors are $\mathfrak{q}\xrightarrow[]{\mathfrak{u_n}/\mathfrak{t}}\mathfrak{q}$. Without loss of generality, we assume that $\mathcal{A}^{\mathcal{D}}$ is not stuck-free, otherwise $\mathcal{A}^{\mathcal{D}}$ is the same as its stuck-free automaton $\bar\mathcal{A}^{\mathcal{D}}$. ``only if'': Suppose that $\mathcal{A}^{\mathcal{D}}$ is $E_f$-diagnosable, i.e., \eqref{eqn17_diag_LMPautomata} holds. Choose a $t\succ {\bf1}$ as in \eqref{eqn17_diag_LMPautomata}. In $\bar\mathcal{A}^{\mathcal{D}}$, arbitrarily choose paths $\pi\in Q_0\xrsquigarrow{se_f}q$, $\pi'\in q\xrsquigarrow{s'}q'$, and $\pi''\in Q_0\xrsquigarrow{s''}q''$ such that $e_f\in E_f$, $\WEI_{\pi'}\succ t$, $\bar\ell(\tau(\pi''))=\bar\ell(\tau(\pi\pi'))$, and $\WEI_{\pi''}\succeq\WEI_{\pi}\otimes t$. Let $\bar \pi''\in Q_0\xrsquigarrow{\bar s''}\bar q''$ be the longest prefix of $\pi''$ that contains no $\mathfrak{u_n}$. Then $\bar\pi''$ is a path of $\mathcal{A}^{\mathcal{D}}$ and $\ell(\tau(\bar\pi''))= \bar\ell(\tau(\pi''))$. In addition, $\last(\bar\pi'')$ is a stuck state of $\mathcal{A}^{\mathcal{D}}$ if $\bar\pi''$ is not the same as $\pi''$, (\romannumeral1) Assume $\mathfrak{u_n}\notin se_fs'$. Then $\pi\pi'$ is also a path of $\mathcal{A}^{\mathcal{D}}$, and hence $E_f\in\bar s''$ by \eqref{eqn17_diag_LMPautomata}. It follows that $E_f\in s''$. (\romannumeral2) Assume $\mathfrak{u_n}\notin se_f$ and $\mathfrak{u_n}\in s'$. Then $\pi$ is also a path of $\mathcal{A}^{\mathcal{D}}$. Let $\bar \pi'$ be the longest prefix of $\pi'$ that contains no $\mathfrak{u_n}$, then $\pi\bar\pi'$ is a path of $\mathcal{A}^{\mathcal{D}}$, $\ell(\tau(\pi\bar\pi')) = \bar\ell(\tau(\pi\pi'))$, and $\last(\bar \pi')$ is a stuck state of $\mathcal{A}^{\mathcal{D}}$ if $\bar\pi'$ is not the same as $\pi'$. We then have $\ell(\tau(\bar\pi''))=\ell(\tau(\pi\bar\pi'))$. Then by \eqref{eqn17_diag_LMPautomata}, we have $E_f\in \bar s''$. It also follows that $E_f\in s''$. (\romannumeral3) By definition of $\bar \mathcal{A}^{\mathcal{D}}$, $\mathfrak{u_n}\notin se_f$. Based on the above (\romannumeral1), (\romannumeral2), and (\romannumeral3), no matter whether $\mathfrak{u_n}\in se_fs'$ or not, one has $E_f\in s''$, then \eqref{eqn20_diag_LMPautomata} holds. ``if'': Suppose $\bar\mathcal{A}^{\mathcal{D}}$ satisfies \eqref{eqn20_diag_LMPautomata}. Choose a $t\succ{\bf1}$ as in \eqref{eqn20_diag_LMPautomata}. In $\mathcal{A}^{\mathcal{D}}$, arbitrarily choose paths $\pi\in Q_0\xrsquigarrow{se_f}q$, $\pi'\in q\xrsquigarrow{s'}q'$, and $\pi''\in Q_0\xrsquigarrow{s''}q''$ such that $e_f\in E_f$, $\WEI_{\pi'}\succ t$ or $q'$ is stuck, $\ell(\tau(\pi''))=\ell(\tau(\pi\pi'))$, $\WEI_{\pi''}\succeq\WEI_{\pi}\otimes t$ or $q''$ is stuck. One directly sees that $\pi\pi'$ and $\pi''$ are paths of $\bar\mathcal{A}^{\mathcal{D}}$. If $\WEI_{\pi'}\nsucc t$, then $q'$ is stuck, we denote $\bar\pi':=\pi'\left( \xrightarrow[]{\mathfrak{u_n}}\mathfrak{q} \right)^{2n}$, where $n\in\mathbb{Z}_+$ is such that $\WEI_{\pi'}\otimes \mathfrak{t}^n\succ {\bf1}$ and $\mathfrak{t}^n\succ t$ ($n$ exists because $\mathcal{D}$ is progressive), $\mathfrak{t}=\mu(\mathfrak{u_n})_{\mathfrak{q} \mathfrak{q}}\succ{\bf1}$ (see \eqref{SF_LMPA_diag_opa_LMPautomata}), then $\WEI_{\bar\pi'}=\WEI_{\pi'}\otimes \mathfrak{t}^{2n}\succeq \mathfrak{t}^n \succ t$ by Lemma~\ref{lem1_diag_LMPautomata}. Similarly, if $\WEI_{\pi''}\nsucceq\WEI_{\pi}\otimes t$ then $q''$ is stuck, we can add sufficiently many $\mathfrak{u_n}$-transitions after $q''$ so that the updated $\pi''$ satisfies $\WEI_{\pi''}\succeq \WEI_{\pi}\otimes t$. Then by \eqref{eqn20_diag_LMPautomata}, $e_f\in s''$. \end{proof} By Lemma~\ref{lem5_diag_LMPautomata}, in order to investigate whether a labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ over a progressive dioid $\mathcal{D}$ is $E_f$-diagnosable (Definition~\ref{def_diag_LMPautomata}), we only need to investigate whether its stuck-free automaton $\bar\mathcal{A}^{\mathcal{D}}$ satisfies the following Definition~\ref{def'_diag_LMPautomata}. In the sequel, $E_f$-diagnosability always means Definition~\ref{def'_diag_LMPautomata}. Apparently, Definition~\ref{def'_diag_LMPautomata} is simpler than Definition~\ref{def_diag_LMPautomata}. \begin{definition}\label{def'_diag_LMPautomata} Let $\mathcal{D}$ be a progressive dioid, $\mathcal{A}^{\mathcal{D}}$ a labeled stuck-free max-plus automaton as in \eqref{LMPA_diag_opa_MPautomata}, and $E_f\subset E$ a set of faulty events. $\mathcal{A}^{\mathcal{D}}$ is called \emph{$E_f$-diagnosable} if \begin{equation}\label{eqn20'_diag_LMPautomata} \begin{split} &(\exists t\in \mathcal{D}:t\succ {\bf1})(\forall \pi\in Q_0 \xrsquigarrow{se_f} q\text{ with }e_f\in E_f)\\ &(\forall \pi' \in q\xrsquigarrow{s'} Q)[ (\WEI_{\pi'} \succ t) \implies {\bf D}], \end{split} \end{equation} where ${\bf D}=(\forall \pi''\in Q_0\xrsquigarrow{s''}q''\text{ with }\ell(\tau(\pi''))=\ell(\tau(\pi\pi'))) [(\WEI_{\pi''}\succeq \WEI_{\pi}\otimes t)\implies (E_f\in s'')]$. \end{definition} By Definition~\ref{def'_diag_LMPautomata}, one directly has the following result. \begin{proposition}\label{prop2_diag_LMPautomata} Let $\mathcal{D}$ be a progressive dioid. A labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ \eqref{LMPA_diag_opa_MPautomata} is not $E_f$-diagnosable if and only if \begin{equation}\label{eqn10_diag_LMPautomata} \begin{split} &(\forall t\in \mathcal{D}:t\succ {\bf1})(\exists \pi_t\in Q_0 \xrsquigarrow{se_f} q\text{ with }e_f\in E_f)\\ &(\exists \pi_t' \in q\xrsquigarrow{s'} Q) (\exists \pi_t''\in Q_0\xrsquigarrow{s''}q'')[(\WEI_{\pi_t'}\succ t) \wedge \\ &(\ell(\tau(\pi_t''))=\ell(\tau(\pi_t\pi_t'))) \wedge (\WEI_{\pi_t''}\succeq\WEI_{\pi}\otimes t)\\ &\wedge (E_f\notin s'')]. \end{split} \end{equation} \end{proposition} \begin{example}\label{exam3_diag_LMPautomata} Recall the automaton $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ in Example~\ref{exam2_diag_LMPautomata} (shown in Fig.~\ref{fig3_diag_LMPautomata}). We choose paths \begin{align} {\color{red}\pi_t} &:= q_0 \xrightarrow[]{{\color{cyan} f}/3} q_1, \quad {\color{green}\pi_t'} := q_1 \xrightarrow[]{a/1}q_3\left( \xrightarrow[]{a/1}q_3 \right)^t,\\ {\color{blue}\pi_t''} &:= q_0 \left( \xrightarrow[]{u/1}q_2 \right) ^3 \left( \xrightarrow[]{a/1}q_4 \right)^{t+1}, \end{align} where $t\in\mathbb{Z}_+$. It holds that \begin{align} &\ell(\tau({\color{blue}\pi_t''})) = \ell(\tau({\color{red}\pi_t}{\color{green}\pi_t'})) = (a,4)(a,5)\dots(a,t+4). \end{align} Then $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ is not $\color{cyan}\{f\}$-diagnosable by Proposition.~\ref{prop2_diag_LMPautomata}. \end{example} \subsection{The notion of concurrent composition} In order to give a sufficient and necessary condition for $E_f$-diagnosability of a labeled (stuck-free) max-plus automaton $\mathcal{A}^{\mathcal{D}}$, we define a notion of \emph{concurrent composition} (similar but technically different structures have been used to study diagnosability of finite-state automata \cite{Cassez2008FaultDiagnosisStDyObser} and timed automata \cite{Tripakis2002DiagnosisTimedAutomata}) of its faulty subautomaton $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$ and its normal subautomaton $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$. From now on, $\mathcal{A}^{\mathcal{D}}$ always means a stuck-free automaton. \begin{definition}\label{def_CC_diag_LMPautomata} Consider a labeled max-plus automaton $\mathcal{A}^\mathcal{D}$ \eqref{LMPA_diag_opa_MPautomata} and a faulty event set $E_f\subset E$. We define the \emph{concurrent composition} of $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$ and $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$ by a labeled finite-state automaton \begin{align}\label{CC_diag_LMPautomata} \CCa(\mathcal{A}^\mathcal{D}_{\mathsf{f}},\mathcal{A}^\mathcal{D}_{\mathsf{n}})=(Q',E',\delta',Q_0',\Sigma,\ell'), \end{align} where $Q'=Q\times Q$; $E'=E'_{o}\cup E'_{uo}$, where $E'_{o}=\{(e_1,e_2)\in E_o\times (E_o\setminus E_f)\|\ell(e_1)=\ell(e_2)\}$; $E'_{uo}=\{(e_{uo},\epsilon)\|e_{uo}\in E_{uo}\}\cup\{(\epsilon,e_{uo})\|e_{uo}\in E_{uo}\setminus E_f\}$; $Q_0'=Q_0\times Q_0$; $\delta'\subset Q'\times E'\times Q'$ is the transition relation, for all states $(q_1,q_2),(q_3,q_4)\in Q'$, events $(e_1,e_2)\in E_o'$, $(e_{uo}^1,\epsilon),(\epsilon,e_{uo}^2)\in E_{uo}'$, \begin{enumerate}[(1)] \item\label{item1_diag_LMPautomata} (observable transition) $((q_1,q_2),(e_1,e_2),(q_3,q_4))\in\delta'$ if and only if in $\mathcal{A}^\mathcal{D}$, there exist states $q_5,q_6\in Q$, event sequences $s_1\in (E_{uo})^$, $s_2\in (E_{uo}\setminus E_f)^$, and paths \begin{equation}\label{eqn6_diag_LMPautomata} \begin{split} \pi_1 &:= q_1\xrightarrow[]{s_1}q_5\xrightarrow[]{e_1}q_3,\\ \pi_2 &:= q_2\xrightarrow[]{s_2}q_6\xrightarrow[]{e_2}q_4, \end{split} \end{equation} in $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$, $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$, respectively, such that $\WEI_{\pi_1}=\WEI_{\pi_2}$ (particularly when $\mathcal{D}=\underline{\mathbb{Q}}$, the \emph{weight} of this transition is defined by $0$ because $\WEI_{\pi_1}=\WEI_{\pi_2}$), \item\label{item2_diag_LMPautomata} (unobservable transition) $((q_1,q_2),(e_{uo}^1,\epsilon),(q_3,q_4))\in\delta'$ if and only if $(q_1,e_{uo}^1,q_3)\in\Delta$, $q_2=q_4$ (particularly when $\mathcal{D}=\underline{\mathbb{Q}}$, $\mu(e_{uo}^1)_{q_1q_3}$ is the \emph{weight} of the transition), \item\label{item3_diag_LMPautomata} (unobservable transition) $((q_1,q_2),(\epsilon,e_{uo}^2),(q_3,q_4))\in\delta'$ if and only if $q_1=q_3$, $(q_2,e_{uo}^2,q_4)\in\Delta$ (particularly when $\mathcal{D}=\underline{\mathbb{Q}}$, $-\mu(e_{uo}^2)_{q_2q_4}$ is the \emph{weight} of the transition); \end{enumerate} for all $(e_1,e_2)\in E_o'$, $\ell'((e_1,e_2))=\ell(e_1)$; for all $e'\in E_{uo}'$, $\ell'(e')=\epsilon$. $\ell'$ is recursively extended to $(E')^\to \Sigma^$. For a state $q'$ of $\CCa(\mathcal{A}^\mathcal{D}_{\mathsf{f}},\mathcal{A}^\mathcal{D}_{\mathsf{n}})$, we write $q'=(q'(L),q'(R))$\footnote{Note that $\CCa(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$ is a labeled max-plus automaton over dioid $\underline{\mathbb{Q}}$. We did not define weights for transitions of $\CCa(\mathcal{A}^\mathcal{D}_{\mathsf{f}},\mathcal{A}^\mathcal{D}_{\mathsf{n}})$ because generally in $\mathcal{D}$, an element does not necessarily have a multiplicative inverse.}. \end{definition} One sees that in $\CCa(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$, an observable transition is obtained by merging two paths of $\mathcal{A}^{\mathcal{D}}$, where the two paths contain exactly one observable event each, have the same weight and both end at the occurrences of the observable events; in addition, the second path contains only normal transitions. Hence the two paths are consistent with observations. Particularly for $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$, in both paths, the observable events occur at the same instant of time if $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$ starts at the starting states of the two paths at the same instant. However, an unobservable transition of $\CCa(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ is obtained by merging an unobservable transition and an $\epsilon$-transition of $\mathcal{A}^{\mathcal{D}}$. The two transitions are not necessarily consistent with observations, e.g., an unobservable transition may have weight $\ne{\bf1}$, but an $\epsilon$-transition must have weight equal to $\bf1$. A sequence $q_0'\xrightarrow[]{s_1'}\cdots\xrightarrow[]{s_n'}q_n'$ of transitions of $\CCa(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ ($n\in\mathbb{N}_0$) is called a \emph{run}. An \emph{unobservable run} is a run consisting of unobservable transitions. In an unobservable run $\pi'$, its left component $\pi'(L)$ is an unobservable path of $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$, its right component $\pi'(R)$ is an unobservable path of $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$. We denote by $\CCao(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ and $\CCauo(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ the subautomata of $\CCa(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ consisting of all observable transitions and all unobservable transitions, respectively. Computing all unobservable transitions of $\CCa(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$ takes time polynomial in the size of $\mathcal{A}^{\underline{\mathbb{Q}}}$. Now for states $(q_1,q_2),(q_3,q_4)\in Q'$ and observable event $(e_1,e_2)\in E_o'$, we check whether there exists an observable transition $((q_1,q_2),(e_1,e_2),(q_3,q_4))$ in $\CCa(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$: \begin{myenumerate} \item\label{item16_diag_LMPautomata} Nondeterministically choose $q_5,q_6\in Q$ such that $q_5\xrightarrow[]{e_1} q_3$ and $q_6\xrightarrow[]{e_2} q_4$ are two observable transitions of $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}}$ and $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}}$, respectively. \item\label{item17_diag_LMPautomata} Regard $\CCauo(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$ as a $1$-dimensional weighted directed graph $(\mathbb{Q},V,A)$ as in Subsection~\ref{subsec:EPLProblem}, where $V=Q'$, $A=\{((p_1,p_2), \mu(e_{uo})_{p_1p_3},(p_3,p_2))\|p_1,p_2,p_3\in Q,e_{uo}\in E_{uo},(p_1,e_{uo},p_3)\in\Delta \} \cup \{((p_1,p_2),-\mu(e_{uo})_{p_2p_3},(p_1,\\p_3))\|p_1,p_2,p_3\in Q,e_{uo}\in E_{uo},(p_2,e_{uo}, p_3)\in\Delta\}$. \item\label{item18_diag_LMPautomata} If there exists a path from $(q_1,q_2)$ to $(q_5,q_6)$ with weight $\mu(e_2)_{q_6q_4}-\mu(e_1) _{q_5q_3}$ in $\CCauo(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$, then $((q_1,q_2),(e_1,e_2),(q_3,q_4))$ is an observable transition of $\CCa(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$. \end{myenumerate} By Lemma~\ref{lem1_det_MPautomata}, the existence of such a path can be checked in $\mathsf{NP}$. Hence the following result holds. \begin{theorem}\label{thm4_diag_LMPautomata} Consider a labeled max-plus automaton $\mathcal{A}^{\underline{\mathbb{Q}}}$, a faulty event set $E_f\subset E$, and the corresponding faulty subautomaton $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}}$ and normal subautomaton $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}}$. The concurrent composition $\CCa(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$ can be computed in time nondeterministically polynomial in the size of $\mathcal{A}^{\underline{\mathbb{Q}}}$. \end{theorem} In order to characterize $E_f$-diagnosability, we define several special kinds of transitions and runs in $\CCa(\mathcal{A}^\mathcal{D}_{\mathsf{f}},\mathcal{A}^\mathcal{D}_{\mathsf{n}})$. \begin{definition}\label{def1_diag_LMPautomata} Consider an observable transition $(q_1,q_2)\xrightarrow[]{(e_1,e_2)}(q_3,q_4)=:\pi_o$ of $\CCa(\mathcal{A}^ \mathcal{D}_{\mathsf{f}},\mathcal{A}^\mathcal{D}_{\mathsf{n}})$ as in Definition~\ref{def_CC_diag_LMPautomata} \eqref{item1_diag_LMPautomata}, a path as $\pi_1$ or $\pi_2$ in \eqref{eqn6_diag_LMPautomata} is called an \emph{admissible path} of $\pi_o$. Transition $\pi_o$ is called \emph{faulty} if it has an admissible path $\pi_1$ containing a faulty event, called \emph{positive} if it has an admissible path $\pi_1$ such that $\WEI_{\pi_1}\succ{\bf1}$. A run $q_0'\xrightarrow[]{e_1'}\cdots\xrightarrow[]{e_n'}q_n'$ of $\CCao(\mathcal{A}^\mathcal{D}_{\mathsf{f}},\mathcal{A}^\mathcal{D}_{\mathsf{n}})$ is called \emph{positive} if each transition $q_i'\xrightarrow[]{e_{i+1}'}q_{i+1}'$ has an admissible path $\pi_i$ such that $\bigotimes_{i=0}^{n-1}\WEI_{\pi_i}\succ{\bf1}$. (Hence a positive observable transition is a positive run, a run consisting of positive observable transitions is a positive run by Lemma~\ref{lem1_diag_LMPautomata}.) An unobservable transition of $\CCa(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ is \emph{faulty} if its event belongs to $E_f\times\{\epsilon\}$. \end{definition} In order to check in $\CCao(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$, whether an observable transition $\pi_o$ is faulty, we only need to add a condition $E_f\in s_1e_1$ into \eqref{item18_diag_LMPautomata} so that we need to solve a subproblem of the EPL problem. Hence whether $\pi_o$ is faulty can be checked in $\mathsf{NP}$ in the size of $\mathcal{A}^{\underline{\mathbb{Q}}}$. Similarly, whether $\pi_o$ is positive can also be checked in $\mathsf{NP}$. Consider a simple cycle $q_0'\xrightarrow[]{e_1'}\cdots\xrightarrow[]{e_n'}q_n'$ of $\CCao(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$, where $q_0'=q_n'$. In order to check whether the cycle is positive, we can make $n$ copies of $\CCauo(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$, and use these copies and $\mathcal{A}^{\underline{\mathbb{Q}}}$ to check for each $i\in\llbracket 0,n-1 \rrbracket$, whether $q_i'\xrightarrow[] {e_{i+1}'}q_{i+1}'$ has an admissible run $\pi_1^i$ such that $\sum_{i=0}^{n-1} \WEI_{\pi_1^i}>0$. This can be done also in $\mathsf{NP}$, because $n$ must be less than or equal to $\|Q\|^2$. \begin{proposition}\label{prop3_diag_LMPautomata} In $\CCao(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$, whether a transition is faulty, whether a transition is positive, whether a simple cycle is positive, can be verified in $\mathsf{NP}$ in the size of $\mathcal{A}^{\underline{\mathbb{Q}}}$. \end{proposition} \begin{definition}\label{def2_diag_LMPautomata} Given $k\in\mathcal{D}$, in $\mathcal{A}^{\mathcal{D}}$, a state $q$ is called \emph{$k$-crucial} (resp., \emph{anti-$k$-crucial}) if there exists an unobservable cycle $q\xrightarrow[]{s}q=:\pi$ such that $\WEI_{\pi}\succ k$ (resp., $\WEI_{\pi}\prec k$). Such a cycle is called a \emph{$k$-crucial cycle} (resp., \emph{anti-$k$-crucial cycle}). A state $q$ is called \emph{eventually $k$-crucial} if either $q$ is $k$-crucial or some $k$-crucial state $q'$ is reachable from $q$ through an unobservable path. \end{definition} \begin{remark} When $\mathcal{D}$ is progressive, if a state is ${\bf1}$-crucial, then it is $k$-crucial for any $k\in\mathcal{D}$ greater than ${\bf1}$, because a cycle can be repeated arbitrarily often. \end{remark} For $\mathcal{A}^{\underline{\mathbb{Q}}}$, we next show that all $0$-crucial states can be computed in time polynomial in the size of $\mathcal{A}^{\underline{\mathbb{Q}}}$: (1) Compute all reachable unobservable transitions of $\mathcal{A}^{\underline{\mathbb{Q}}}$, denote the obtained subautomaton by $\Acc(\mathcal{A}^{\underline{\mathbb{Q}}}_{uo})$, which is considered as a weighted directed graph $(\mathbb{Q},V,A)$. (2) Compute all strongly connected components of $(\mathbb{Q},V,A)$ by using Tarjan algorithm, which takes time linear in the size of $(\mathbb{Q},V,A)$. (3) Choose a vertex $q$ of $(\mathbb{Q},V,A)$ and the strongly connected component $SCC_{q}$ containing $q$. In $SCC_{q}$, replace the weight $w$ of any transition by $-w$, denote the currently updated $SCC_{q}$ by $\overline{SCC_{q}}$. Then $q$ is $0$-crucial if and only if in $\overline{SCC_{q}}$ there is a cycle with negative weight. One can use Bellman-Ford algorithm \cite[Chap. 24]{Cormen2009Algorithms} to check whether there is a cycle with negative weight reachable from $q$ in time $O(\|V\|\|A\|)$. Similarly, all anti-$0$-crucial states can also be computed in polynomial time. \begin{proposition}\label{prop4_diag_LMPautomata} In $\mathcal{A}^{\underline{\mathbb{Q}}}$, whether a state is $0$-crucial (resp., anti-$0$-crucial) can be verified in time polynomial in the size of $\mathcal{A}^{\underline{\mathbb{Q}}}$. \end{proposition} Furthermore, we need one property of $\CCauo(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}, \mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ in order to characterize $E_f$-diagnosability. \begin{proposition}\label{prop1_diag_LMPautomata} Consider a labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$. In the corresponding $\CCauo(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}, \mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$, for every run \begin{equation}\label{eqn1_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}\cdots\xrightarrow[]{s_n'}q_n', \end{equation} where $n\in\mathbb{Z}_{+}$, $s_1',\dots,s_n'\in (E'_{uo})^+$, there exists a run \begin{equation}\label{eqn2_diag_LMPautomata} q_0'\xrightarrow[]{\bar s_1'}\bar q_1'\xrightarrow[]{\bar s_2'}q_n' \end{equation} such that $\bar s_1'\in (E_{uo}\times\{\epsilon\})^$, $\bar s_2'\in (\{\epsilon\}\times(E_{uo}\setminus E_f))^$, $q_0'(R)=\bar q_1'(R)$, $\bar q_1'(L)=q_n'(L)$, and the left (resp., right) component of \eqref{eqn1_diag_LMPautomata} is the same as the left (resp., right) component of \eqref{eqn2_diag_LMPautomata}. \end{proposition} \begin{proof} One sees in $\CCauo(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$, all runs are unobservable. Then for every run $r$, we can swap any two consecutive transitions whose events have $\epsilon$ components in different positions but do not change the components of $r$. Without loss of generality, we consider the following run \begin{equation}\label{eqn4_diag_LMPautomata} \begin{split} &(q_1,q_2)\xrightarrow[]{(e_1,\epsilon)}(q_3,q_2)\xrightarrow[]{(\epsilon,e_2)}(q_3,q_4)\xrightarrow[]{(e_3,\epsilon)}\\ &(q_5,q_4)\xrightarrow[]{(e_4,\epsilon)}(q_6,q_4)\xrightarrow[]{(\epsilon,e_5)}(q_6,q_7), \end{split} \end{equation} and then change it as follows: \begin{align} &(q_1,q_2)\xrightarrow[]{(e_1,\epsilon)}(q_3,q_2)\xrightarrow[]{\bf(e_3,\epsilon)}{\bf(q_5,q_2)}\xrightarrow[]{\bf(\epsilon,e_2)}\\ &(q_5,q_4)\xrightarrow[]{(e_4,\epsilon)}(q_6,q_4)\xrightarrow[]{(\epsilon,e_5)}(q_6,q_7), \end{align} \begin{subequations}\label{eqn3_diag_LMPautomata} \begin{align} &(q_1,q_2)\xrightarrow[]{(e_1,\epsilon)}(q_3,q_2)\xrightarrow[]{(e_3,\epsilon)}(q_5,q_2)\xrightarrow[]{\bf(e_4,\epsilon)}\label{eqn3_1_diag_LMPautomata}\\ &{\bf(q_6,q_2)}\xrightarrow[]{\bf(\epsilon,e_2)}(q_6,q_4)\xrightarrow[]{(\epsilon,e_5)}(q_6,q_7),\label{eqn3_2_diag_LMPautomata} \end{align} \end{subequations} that is, in \eqref{eqn3_1_diag_LMPautomata}, $(e_1,\epsilon)(e_3,\epsilon)(e_4,\epsilon)\in (E_{uo}\times\{\epsilon\})^$, in \eqref{eqn3_2_diag_LMPautomata}, $(\epsilon,e_2)(\epsilon,e_5)\in (\{\epsilon\}\times(E_{uo}\setminus E_f))^$; and the left (resp., right) component of \eqref{eqn4_diag_LMPautomata} is the same as the left (resp., right) component of \eqref{eqn3_diag_LMPautomata}. \end{proof} \begin{example} Reconsider the automaton $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ in Example~\ref{exam2_diag_LMPautomata} (shown in Fig.~\ref{fig3_diag_LMPautomata}). Its faulty subautomaton $\mathcal{A}_{1\mathsf{f}}^{\underline{\mathbb{N}_0}}$ and normal subautomaton $\mathcal{A}_{1\mathsf{n}}^{\underline{\mathbb{N}_0}}$ are shown in Fig.~\ref{fig4_diag_LMPautomata}. Its concurrent composition $\CCa(\mathcal{A}_{1\mathsf{f}}^{\underline{\mathbb{N}_0}},\mathcal{A}_{1\mathsf{n}}^{\underline{\mathbb{N}_0}})$ is shown in Fig.~\ref{fig5_diag_LMPautomata}. \begin{figure}\label{fig4_diag_LMPautomata} \end{figure} \begin{figure}\label{fig5_diag_LMPautomata} \end{figure} \end{example} \subsection{A sufficient and necessary condition for $E_f$-diagnosability of $\mathcal{A}^{\mathcal{D}}$} In this subsection, for a labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ over a progressive dioid $\mathcal{D}$, we use the corresponding concurrent composition $\CCa(\mathcal{A}^\mathcal{D}_{\mathsf{f}},\mathcal{A}^\mathcal{D}_{\mathsf{n}})$ to give a sufficient and necessary condition for negation of its $E_f$-diagnosability (as in Definition~\ref{def'_diag_LMPautomata}). By Lemma~\ref{lem5_diag_LMPautomata}, we assume $\mathcal{A}^{\mathcal{D}}$ is stuck-free without loss of generality. \begin{theorem}\label{thm1_diag_LMPautomata} Consider a labeled max-plus automaton $\mathcal{A}^{\mathcal{D}}$ \eqref{LMPA_diag_opa_MPautomata}, where $\mathcal{D}$ is progressive. $\mathcal{A}^{\mathcal{D}}$ is not $E_f$-diagnosable if and only if in the corresponding concurrent composition $\CCa(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ \eqref{CC_diag_LMPautomata}, at least one of the following four conditions holds. \begin{myenumerate} \item\label{item6_diag_LMPautomata} There exists a path \begin{align}\label{eqn7_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}q_1'\xrightarrow[]{e_{2}'}q_2'\xrightarrow[]{s_{3}'}q_{3}', \end{align} where $q_0'\in Q_0'$, $s_1',s_3'\in(E_o')^$, $e_2'\in E_o'$, $q_1'\xrightarrow[] {e_{2}'}q_{2}'$ is faulty, and $q_{3}'$ belongs to a positive transition cycle. \item\label{item7_diag_LMPautomata} There exists a path \begin{align}\label{eqn8_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}q_1'\xrightarrow[]{e_{2}'}q_2'\xrightarrow[]{s_{3}'}q_{3}', \end{align} where $q_0'\in Q_0'$, $s_1',s_3'\in(E_o')^$, $e_2'\in E_o'$, $q_1'\xrightarrow[] {e_{2}'}q_{2}'$ is faulty, $q_{3}'(L)$ is eventually ${\bf1}$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$, and $q_3'(R)$ is eventually ${\bf1}$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$. \item\label{item8_diag_LMPautomata} There exists a path \begin{align}\label{eqn9_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}q_1'\xrightarrow[]{s_{2}'}q_2'\xrightarrow[]{e_{3}'}q_{3}', \end{align} where $q_0'\in Q_0'$, $s_1'\in(E_o')^$, $s_2'\in(E_{uo}')^$, $e_3'\in E_f\times\{\epsilon\}$, $q_{3}'(L)$ is eventually ${\bf1}$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$, and $q_3'(R)$ is eventually ${\bf1}$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$. \item\label{item27_diag_LMPautomata} There exists a path \begin{align}\label{eqn22_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}q_1'\xrightarrow[]{s_{2}'}q_2'\xrightarrow[]{e_{3}'}q_{3}', \end{align} where $q_0'\in Q_0'$, $s_1'\in(E_o')^$, $s_2'\in(E_{uo}')^$, $e_3'\in E_f\times\{\epsilon\}$, $q_{3}'(L)$ is eventually ${\bf1}$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$, and some state in $q_1'(L)\xrightarrow[]{s_{2}'(L)}q_2'(L)$ is anti-$\bf1$-crucial in $\mathcal{A}_{\mathsf{f}}^{\mathcal{D}}$. \end{myenumerate} \end{theorem} \begin{proof} The proof of this result is put into Appendix in order not to delay the statement at this point. \end{proof} Because dioid $\underline{\mathbb{Q}}$ is progressive, by Theorem~\ref{thm1_diag_LMPautomata} we directly have the following result. \begin{theorem}\label{thm2_diag_LMPautomata} A labeled max-plus automaton $\mathcal{A}^{\underline{\mathbb{Q}}}$ is not $E_f$-diagnosable if and only if in the corresponding concurrent composition $\CCa(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$, at least one of the following four conditions holds. \begin{myenumerate} \item\label{item11_diag_LMPautomata} There exists a path \begin{align}\label{eqn13_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}q_1'\xrightarrow[]{e_{2}'}q_2'\xrightarrow[]{s_{3}'}q_{3}', \end{align} where $q_0'\in Q_0'$, $s_1',s_3'\in(E_o')^$, $e_2'\in E_o'$, $q_1'\xrightarrow[] {e_{2}'}q_{2}'$ is faulty, and $q_{3}'$ belongs to a positive transition cycle. \item\label{item12_diag_LMPautomata} There exists a path \begin{align}\label{eqn14_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}q_1'\xrightarrow[]{e_{2}'}q_2'\xrightarrow[]{s_{3}'}q_{3}', \end{align} where $q_0'\in Q_0'$, $s_1',s_3'\in(E_o')^$, $e_2'\in E_o'$, $q_1'\xrightarrow[] {e_{2}'}q_{2}'$ is faulty, $q_{3}'(L)$ is eventually $0$-crucial in $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}}$, and $q_3'(R)$ is eventually $0$-crucial in $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}}$. \item\label{item13_diag_LMPautomata} There exists a path \begin{align}\label{eqn15_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}q_1'\xrightarrow[]{s_{2}'}q_2'\xrightarrow[]{e_{3}'}q_{3}', \end{align} where $q_0'\in Q_0'$, $s_1'\in(E_o')^$, $s_2'\in(E_{uo}')^$, $e_3'\in E_f\times\{\epsilon\}$, $q_{3}'(L)$ is eventually $0$-crucial in $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}}$, and $q_3'(R)$ is eventually $0$-crucial in $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}}$. \item\label{item28_diag_LMPautomata} There exists a path \begin{align}\label{eqn23_diag_LMPautomata} q_0'\xrightarrow[]{s_1'}q_1'\xrightarrow[]{s_{2}'}q_2'\xrightarrow[]{e_{3}'}q_{3}', \end{align} where $q_0'\in Q_0'$, $s_1'\in(E_o')^$, $s_2'\in(E_{uo}')^$, $e_3'\in E_f\times\{\epsilon\}$, $q_{3}'(L)$ is eventually $0$-crucial in $\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}}$, and some state in $q_1'(L) \xrightarrow[]{s_{2}'(L)}q_2'(L)$ is anti-$0$-crucial in $\mathcal{A}_{\mathsf{f}}^{\underline{\mathbb{Q}}}$. \end{myenumerate} \end{theorem} \begin{corollary} A labeled max-plus automaton $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$ is not $E_f$-diagnosable if and only if in the corresponding concurrent composition $\CCa(\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}_{\mathsf{n}})$, at least one of \eqref{item11_diag_LMPautomata}, \eqref{item12_diag_LMPautomata}, and \eqref{item13_diag_LMPautomata} adapted to $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$ to holds (\eqref{item28_diag_LMPautomata} must not hold because $\mathcal{A}^{\underline{\mathbb{Q}_{\ge0}}}$ contains no transition with negative weight). \end{corollary} \begin{example}\label{exam4_diag_LMPautomata} Reconsider the automaton $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ in Example~\ref{exam2_diag_LMPautomata} (shown in Fig.~\ref{fig3_diag_LMPautomata}). In its concurrent composition $\CCa(\mathcal{A}_{1\mathsf{f}}^{\underline{\mathbb{N}_0}}, \mathcal{A}_{1\mathsf{n}}^{\underline{\mathbb{N}_0}})$ (in Fig.~\ref{fig5_diag_LMPautomata}), the observable transition $(q_0,q_0) \xrightarrow[]{(a,a)}(q_3,q_4)$ is faulty, because it has two admissible paths $q_0\xrightarrow[]{\color{cyan} f} q_1\xrightarrow[]{a}q_3$ and $q_0\left( \xrightarrow[]{u}q_2 \right)^3\xrightarrow[]{a}q_4$ both with weight $4$; moreover, the observable self-loop $(q_3,q_4)\xrightarrow[]{(a,a)}(q_3,q_4)$ is positive, because it has two admissible paths $q_3\xrightarrow[]{a}q_3$ and $q_4\xrightarrow[]{a}q_4$ both with positive weight $1$. Hence \eqref{item11_diag_LMPautomata} of Theorem~\ref{thm2_diag_LMPautomata} is satisfied, $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ is not $\color{cyan}\{f\}$-diagnosable, which is consistent with the result obtained in Example~\ref{exam3_diag_LMPautomata}. \eqref{item12_diag_LMPautomata} is not satisfied. \eqref{item13_diag_LMPautomata} is satisfied, because in path $(q_0,q_0)\xrightarrow[]{({\color{cyan} f},\epsilon)} (q_1,q_0)$, $q_1$ is $0$-crucial in $\mathcal{A}_{1\mathsf{f}}^{\underline{\mathbb{N}_0}}$ (because of the unobservable self-loop on $q_1$ with positive weight) and $q_0$ is eventually $0$-crucial in $\mathcal{A}_{1\mathsf{n}}^{\underline{\mathbb{N}_0}}$ (because of the unobservable path $q_0\xrightarrow[]{u}q_2\xrightarrow[]{u}q_2$). \eqref{item28_diag_LMPautomata} is not be satisfied. \end{example} Next we give an example for which only \eqref{item28_diag_LMPautomata} of Theorem~\ref{thm2_diag_LMPautomata} is satisfied. \begin{example}\label{exam5_diag_LMPautomata} Consider a second stuck-free labeled max-plus automaton $\mathcal{A}_2^{\underline{\mathbb{Q}}}$ shown in Fig.~\ref{fig6_diag_LMPautomata}. One easily sees that in $\CCa(\mathcal{A}_{2\mathsf{f}}^{\underline{\mathbb{Q}}},\mathcal{A}_{2\mathsf{n}}^{\underline{\mathbb{Q}}})$, there is no observable transition. Then neither \eqref{item11_diag_LMPautomata} nor \eqref{item12_diag_LMPautomata} is satisfied. \eqref{item13_diag_LMPautomata} is not satisfied either, because in $\mathcal{A}_{2\mathsf{n}}^{\underline{\mathbb{Q}}}$ there is no reachable eventually $0$-crucial state. \eqref{item28_diag_LMPautomata} is satisfied, because in path $q_0\xrightarrow[]{u}q_1 \xrightarrow[]{u}q_1 \xrightarrow[]{{\color{cyan} f}}q_2$, $q_1 \xrightarrow[]{u}q_1$ is an anti-$0$-crucial cycle, $q_2$ is $0$-crucial. Hence $\mathcal{A}_2^{\underline{\mathbb{Q}}}$ is not $\color{cyan} \{f\}$-diagnosable. Moreover, choose paths \begin{align} q_0\xrightarrow[]{u}q_1\left( \xrightarrow[]{u}q_1 \right)^{t+2} \xrightarrow[]{{\color{cyan} f}}q_2 =: {\color{red}\pi_t},\\ q_2\left( \xrightarrow[]{u}q_2 \right)^{t+1} =: {\color{green}\pi_t'},\\ q_0\xrightarrow[]{u}q_1 =: {\color{blue}\pi_t''}, \end{align} where $t\in \mathbb{Z}_+$. Then one has \begin{align} &\ell(\tau({\color{red}\pi_t}{\color{green}\pi_t'})) = \ell(\tau({\color{blue}\pi_t''})) = \epsilon,\\ &\WEI_{{\color{green}\pi_t'}} = t+1 > t,\\ &\WEI_{{\color{blue}\pi_t''}} = 0 > \WEI_{{\color{red}\pi_t}} + t = -2. \end{align} By Proposition~\ref{prop2_diag_LMPautomata}, one also has $\mathcal{A}_2^{\underline{\mathbb{Q}}}$ is not $\color{cyan} \{f\}$-diagnosable. \begin{figure}\label{fig6_diag_LMPautomata} \end{figure} \end{example} \subsection{The complexity of verifying diagnosability of labeled max-plus automaton $\mathcal{A}^{\underline{\mathbb{Q}}}$} We give the following complexity result on verifying $E_f$-diagnosability of $\mathcal{A}^{\underline{\mathbb{Q}}}$. \begin{theorem}\label{thm3_diag_LMPautomata} The problem of verifying $E_f$-diagnosability of a labeled max-plus automaton $\mathcal{A}^{\underline{\mathbb{Q}}}$ is $\mathsf{coNP}$-complete, where $\mathsf{coNP}$-hardness even holds for deterministic, deadlock-free, and divergence-free $\mathcal{A}^{\underline{\mathbb{N}_0}}$. \end{theorem} \begin{proof} ``$\mathsf{coNP}$-membership'': Recall that the concurrent composition $\CCa(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$ can be computed in $\mathsf{NP}$ (Theorem~\ref{thm4_diag_LMPautomata}). In \eqref{item11_diag_LMPautomata}, by definition, we can assume $q_3'$ belongs to a positive simple transition cycle without loss of generality. By Proposition~\ref{prop3_diag_LMPautomata}, in $\CCa(\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{f}},\mathcal{A}^{\underline{\mathbb{Q}}}_{\mathsf{n}})$, whether an observable transition is faulty can be verified in $\mathsf{NP}$, whether a simple observable transition cycle is positive can also be verified in $\mathsf{NP}$, so \eqref{item11_diag_LMPautomata} can be verified in $\mathsf{NP}$. By Proposition~\ref{prop4_diag_LMPautomata}, whether a state of $\mathcal{A}^{\underline{\mathbb{Q}}}$ is $0$-crucial (resp., anti-$0$-crucial) can be verified in polynomial time. Hence in \eqref{item12_diag_LMPautomata}, \eqref{item13_diag_LMPautomata}, and \eqref{item28_diag_LMPautomata}, whether $q_{3}'(L)$ is eventually $0$-crucial in $\mathcal{A}^{\mathbb{Q}}_{\mathsf{f}}$, whether $q_3'(R)$ is eventually $0$-crucial in $\mathcal{A}^{\mathbb{Q}}_{\mathsf{n}}$, and whether some state in $q_1'(L) \xrightarrow[]{s_{2}'(L)}q_2'(L)$ is anti-$0$-crucial in $\mathcal{A}_{\mathsf{f}}^{\underline{\mathbb{Q}}}$ can be verified in polynomial time. Hence \eqref{item12_diag_LMPautomata}, \eqref{item13_diag_LMPautomata}, and \eqref{item28_diag_LMPautomata} can be verified in $\mathsf{NP}$. ``$\mathsf{coNP}$-hardness'': We reduce the $\mathsf{NP}$-complete SS problem (Problem~\ref{prob2_det_MPautomata}) to negation of $E_f$-diagnosability of $\mathcal{A}^{\underline{\mathbb{N}_0}}$ in polynomial time. Given positive integers $n_1,\dots,n_m$, and $N$, next we construct in polynomial time a labeled max-plus automaton $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ as illustrated in Fig.~\ref{fig1_diag_LMPautomata}. $q_0$ is the unique initial state and has initial time delay $0$. Events $u,u_1,u_2$, and $e_f$ are unobservable. Events $a$ and $b$ are observable, $\ell(a)=a$, $\ell(b)=b$. Only $e_f$ is faulty. For all $i\in\llbracket 0,m-1\rrbracket$, there exist two unobservable transitions $q_{i}\xrightarrow[]{u_1/n_{i+1}}q_{i+1}$ and $q_{i}\xrightarrow[]{u_2/0}q_{i+1}$. The other unobservable transitions are $q_{m+1}^1\xrightarrow[]{e_f/1}q_{m+2}^1$ and $q_{m+1}^2\xrightarrow[]{u/1}q_{m+2}^2$. The observable transitions are $q_{m}\xrightarrow[]{a/1}q_{m+1}^1$, $q_{0}\xrightarrow[]{a/N+1}q_{m+1}^2$, and two self-loops $q_{m+2}^1\xrightarrow[]{b/1}q_{m+2}^1$ and $q_{m+2}^2\xrightarrow[]{b/1}q_{m+2}^2$. Denote $\{u,u_1,u_2,e_f,a,b\}=:E$. Apparently $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ is deterministic, deadlock-free, and divergence-free (having no reachable unobservable cycle). \begin{figure} \caption{Sketch of the reduction in the proof of Theorem~\ref{thm3_diag_LMPautomata}.} \label{fig1_diag_LMPautomata} \end{figure} Suppose there exists $I\subset\llbracket 1,m\rrbracket$ such that $N=\sum_{i\in I}n_i$. Then there is an unobservable path $\pi\in q_0\rightsquigarrow q_m $ whose weight is equal to $N$. Then for an arbitrary $t\in\mathbb{N}_0$, we choose \begin{align} \pi_t =& \pi\xrightarrow[]{a} q_{m+1}^1\xrightarrow[]{e_f} q_{m+2}^1,\\ \pi_t' =& q_{m+2}^1 \left( \xrightarrow[]{b} q_{m+2}^1 \right)^t ,\\ \pi_t'' =& q_0 \xrightarrow[]{a} q_{m+1}^2\xrightarrow[]{u} q_{m+2}^2 \left( \xrightarrow[]{b} q_{m+2}^2 \right)^t. \end{align} Then we have \begin{align} e_f &\notin aub^t,\\ \WEI_{\pi_t'} &= t,\\ \WEI_{\pi_t''} &= N+2+t = \WEI_{\pi_t}+t,\\ \ell(\tau(\pi_t\pi_t')) &= \ell(\tau(\pi_t'')) \\ &= (a,N+1)(b,N+3)\dots (b,N+t+2). \end{align} By definition, $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ is not $\{e_f\}$-diagnosable. Assume for all $I\subset\llbracket 1,m\rrbracket$, $N\ne\sum_{i\in I}n_i$. Choose $t=1$, and choose an arbitrary path starting from $q_0$ whose last event is faulty, then the path must be of the form \begin{align} \pi\xrightarrow[]{a} q_{m+1}^1 \xrightarrow[]{e_f} q_{m+2}^1, \end{align} where $\pi\in q_0\rightsquigarrow q_m$. We have $\WEI_\pi\ne N$. Choose an arbitrary continuation $q_{m+2}^1\left( \xrightarrow[]{b} q_{m+2}^1 \right) ^{t'}$ with $t'\ge t$, then if we observe $\ell\left(\tau\left(\pi\xrightarrow[]{a} q_{m+1}^1 \xrightarrow[]{e_f} q_{m+2}^1 \left( \xrightarrow[]{b} q_{m+2}^1 \right) ^{t'}\right)\right)$, we know that $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ must be in state $q_{m+2}^1$. Hence $e_f$ has occurred. That is, $\mathcal{A}_1^{\underline{\mathbb{N}_0}}$ is $\{e_f\}$-diagnosable. \end{proof} \section{conclusion} \label{sec:conc} In this paper, we formulated a notion of diagnosability for labeled max-plus automata over a subclass of dioids called progressive which could represent time elapsing. We gave a notion of concurrent composition for such automata and used the notion to give a sufficient and necessary condition for diagnosability. We also proved that for the dioid $(\mathbb{Q}\cup\{-\infty\},\max,+,-\infty,0)$ (which is progressive), a concurrent composition can be computed in $\mathsf{NP}$ and verifying diagnosability is $\mathsf{coNP}$-complete, where $\mathsf{coNP}$-hardness even holds for deterministic, deadlock-free, and divergence-free automata. Note that particularly when the weights of the transitions of the considered automata are nonnegative rational numbers, the automata become a subclass of (labeled) timed automata. Recall that for timed automata, the diagnosability verification problem is $\mathsf{PSPACE}$-complete \cite{Tripakis2002DiagnosisTimedAutomata}, hence in this paper a subclass of timed automata for which the diagnosability problem belongs to $\mathsf{coNP}$ was found. \appendix \begin{proof}[of Theorem~\ref{thm1_diag_LMPautomata}] ``if'': Assume \eqref{item6_diag_LMPautomata} holds. Then in $\mathcal{A}^{\mathcal{D}}$ there exist paths \begin{align} &q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)\xrightarrow[]{\bar s_2} q_2'(L)\xrightarrow[]{\bar s_3} q_3'(L)\xrightarrow[]{\bar s_4} q_3'(L),\\ &q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)\xrightarrow[]{\hat s_2} q_2'(R)\xrightarrow[]{\hat s_3} q_3'(R)\xrightarrow[]{\hat s_4} q_3'(R) \end{align} such that \begin{align} &\bar s_i\in E^E_o, \hat s_i\in(E\setminus E_f)^(E_o\setminus E_f), \ell(\bar s_i)=\ell(\hat s_i)\\ &\text{for }i= 1,3,4,\\ &\bar s_2\in(E_{uo})^E_o, \hat s_2\in(E_{uo}\setminus E_f)^(E_o\setminus E_f),\ell(\bar s_2)=\ell(\hat s_2),\\ &\ell(\bar s_1)=\ell(s_1'), \ell(\bar s_2)=\ell(e_2'), \ell(\bar s_3)=\ell(s_3'),\\ &\ell(\tau(q_{i-1}'(L)\xrightarrow[]{\bar s_{i}} q_{i}'(L)))= \ell(\tau(q_{i-1}'(R)\xrightarrow[]{\hat s_{i}} q_{i}'(R)))\\ &\text{for all }i\in\llbracket 1,3\rrbracket,\\ &\ell(\tau(q_{3}'(L)\xrightarrow[]{\bar s_{4}} q_{3}'(L)))= \ell(\tau(q_{3}'(R)\xrightarrow[]{\hat s_{4}} q_{3}'(R))),\\ &E_f\in\bar s_2,\\ &\WEI(q_{3}'(L)\xrightarrow[]{\bar s_{4}} q_{3}'(L))=\WEI(q_{3}'(R)\xrightarrow[]{\hat s_{4}} q_{3}'(R))=:l\succ{\bf1}\\ &(\text{because }q_3'\text{ belongs to a positive cycle}). \end{align} Let $t\in\mathcal{D}$ be such that $t\succ{\bf1}$. Choose $n\in\mathbb{N}_0$ such that $l^n\succ t$ and $\WEI(q_2'(L)\xrightarrow[]{\bar s_3}q_3'(L))\otimes t^n \succ{\bf1}$ ($n$ exists because $\mathcal{D}$ is progressive). Choose \begin{align} q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)\xrightarrow[]{\bar s_2} q_2'(L) &=: \pi_t,\\ q_2'(L)\xrightarrow[]{\bar s_3}q_3'(L)\left(\xrightarrow[] {\bar s_4} q_3'(L)\right)^{2n} &=: \pi_t',\\ q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)\xrightarrow[]{\hat s_2} q_2'(R)\xrightarrow[]{\hat s_3} q_3'(R)\left(\xrightarrow[]{\hat s_4} q_3'(R)\right)^{2n} &=: \pi_t'', \end{align} one then has $\ell(\tau(\pi_t\pi_t'))=\ell(\tau(\pi_t''))$, $E_f\in \bar s_2$, $E_f\notin \hat s_1\hat s_2\hat s_3(\hat s_4)^{2n}$, $\WEI_{\pi_t'} = \WEI(q_2'(L)\xrightarrow[]{\bar s_3} q_3'(L))\otimes l^{2n} \succeq l^n\succ t$, and $\WEI_{\pi_t''}=\WEI_{\pi_t}\otimes \WEI_{\pi_t'}\succ \WEI_{\pi_t}\otimes t$ (by $\WEI_{\pi_t}\ne{\bf0}$ and Lemma~\ref{lem6_diag_LMPautomata}). Then By Proposition~\ref{prop2_diag_LMPautomata}, $\mathcal{A}^{\mathcal{D}}$ is not $E_f$-diagnosable. Assume \eqref{item7_diag_LMPautomata} holds. Then there exist unobservable paths \begin{subequations}\label{eqn21_diag_LMPautomata} \begin{align} &q_3'(L) \xrightarrow[]{s_4} q_4 \xrightarrow[]{s_5} q_4 \text{ in }\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\\ &q_3'(R) \xrightarrow[]{s_6} q_5 \xrightarrow[]{s_7} q_5 \text{ in }\mathcal{A}^{\mathcal{D}}_{\mathsf{n}} \end{align} \end{subequations} such that $s_4\in(E_{uo})^$, $s_5\in(E_{uo})^+$, $s_6\in(E_{uo}\setminus E_f)^$, $s_7\in(E_{uo}\setminus E_f)^+$, $\WEI(q_4 \xrightarrow[]{s_5} q_4)=:l_1\succ{\bf1}$, and $\WEI(q_5 \xrightarrow[]{s_7} q_5)=:l_2\succ{\bf1}$. By \eqref{eqn8_diag_LMPautomata}, in $\mathcal{A}^{\mathcal{D}}$ there exist paths \begin{align} &q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)\xrightarrow[]{\bar s_2} q_2'(L) \xrightarrow[]{\bar s_3}q_3'(L) \xrightarrow[]{s_4} q_4 \xrightarrow[]{s_5} q_4,\\ &q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)\xrightarrow[]{\hat s_2} q_2'(R)\xrightarrow[]{\hat s_3} q_3'(R) \xrightarrow[]{s_6} q_5 \xrightarrow[]{s_7} q_5 \end{align} such that \begin{align} &\bar s_i\in E^E_o, \hat s_i\in(E\setminus E_f)^(E_o\setminus E_f)\text{ for }i=1,3,\\ &\bar s_2\in (E_{uo})^E_o, \hat s_2\in (E_{uo}\setminus E_f)^(E_o\setminus E_f),\\ &\ell(\bar s_i)=\ell(\hat s_i)=\ell(s_i')\text{ for }i=1,3,\\ &\ell(\bar s_2)=\ell(\hat s_2)=\ell(e_2'),\\ &\ell(\tau(q_{i-1}'(L)\xrightarrow[]{\bar s_{i}} q_{i}'(L)))=\ell(\tau(q_{i-1}'(R)\xrightarrow[]{\hat s_{i}} q_{i}'(R)))\\ &\text{for }i=1,2,3,\\ &E_f\in\bar s_2. \end{align} Let $t\in\mathcal{D}$ be such that $t\succ{\bf1}$. Choose $n\in\mathbb{N}_0$ such that $(l_1)^n\succ t$, $(l_2)^n\succ t$, $\WEI(q_2'(L)\xrightarrow[]{\bar s_3}q_3'(L) \xrightarrow[]{s_4}q_4)\otimes (l_1)^n\succ {\bf1}$, and $\WEI(q_2'(R)\xrightarrow[]{\hat s_3}q_3'(R) \xrightarrow[]{s_6}q_5)\otimes (l_2)^n\succ{\bf1}$. Denote \begin{align} q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)\xrightarrow[]{\bar s_2} q_2'(L) &=: \pi_t,\\ q_2'(L)\xrightarrow[]{\bar s_3}q_3'(L) \xrightarrow[]{s_4}q_4 \left(\xrightarrow[] {s_5}q_4\right)^{2n} &=: \pi_t',\\ q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)\xrightarrow[]{\hat s_2} q_2'(R) &\\ \xrightarrow[]{\hat s_3} q_3'(R) \xrightarrow[]{s_6}q_5 \left(\xrightarrow[]{s_7} q_5\right)^{2n} &=: \pi_t'', \end{align} One then has $E_f\in\bar s_2$, $E_f\notin\hat s_1\hat s_2\hat s_3s_6 (s_7)^{2n}$, $\ell(\tau(\pi_t\pi_t'))=\ell(\tau(\pi_t''))$, $\WEI_{\pi_t'} = \WEI(q_2'(L)\xrightarrow[]{\bar s_3}q_3'(L) \xrightarrow[]{s_4}q_4)\otimes (l_1)^{2n} \succeq (l_1)^n\succ t$, and $\WEI_{\pi_{t}''} = \WEI_{\pi_t}\otimes \WEI(q_2'(R)\xrightarrow[] {\hat s_3}q_3'(R) \xrightarrow[]{s_6}q_5)\otimes (l_2)^{2n} \succeq \WEI_{\pi_t}\otimes (l_2)^n \succ \WEI_{\pi_t}\otimes t$ (by Lemma~\ref{lem6_diag_LMPautomata}). Then also by Proposition~\ref{prop2_diag_LMPautomata}, $\mathcal{A}^{\mathcal{D}}$ is not $E_f$-diagnosable. Assume \eqref{item8_diag_LMPautomata} holds. Then similarly to \eqref{item7_diag_LMPautomata}, there also exist unobservable paths in $\mathcal{A}^{\mathcal{D}}$ as in \eqref{eqn21_diag_LMPautomata}. By \eqref{eqn9_diag_LMPautomata}, in $\mathcal{A}^{\mathcal{D}}$ there exist paths \begin{align} &q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)\xrightarrow[]{s_2'(L)} q_2'(L) \xrightarrow[]{e_f}q_3'(L)\xrightarrow[]{s_4} q_4 \xrightarrow[]{s_5} q_4, \\ &q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)\xrightarrow[]{s_2'(R)} q_2'(R)\xrightarrow[]{\epsilon} q_3'(R) \xrightarrow[]{s_6} q_5 \xrightarrow[]{s_7} q_5\\ \end{align} such that \begin{align} &\bar s_1\in E^E_o, \hat s_1\in(E\setminus E_f)^(E_o\setminus E_f),\\ &\ell(\bar s_1)=\ell(\hat s_1)=\ell(s_1'),\\ &\ell(\tau(q_{0}'(L)\xrightarrow[]{\bar s_{1}} q_{1}'(L)))= \ell(\tau(q_{0}'(R)\xrightarrow[]{\hat s_{1}} q_{1}'(R))),\\ &\WEI(q_{0}'(L)\xrightarrow[]{\bar s_{1}} q_{1}'(L))= \WEI(q_{0}'(R)\xrightarrow[]{\hat s_{1}} q_{1}'(R)),\\ &q_2'(R)=q_3'(R),\quad (e_f,\epsilon)= e_3'. \end{align} Let $t\in\mathcal{D}$ be such that $t\succ{\bf1}$. Choose a sufficiently large $n\in\mathbb{N}_0$ such that $(l_1)^n\succ t$, $(l_2)^n\succ t$, $\WEI(q_3'(L) \xrightarrow[]{s_4}q_4)\otimes (l_1)^n\succ {\bf1}$, $\WEI(q_1'(R)\xrightarrow[]{s_2'(R)} q_2'(R)\xrightarrow[]{s_6}q_5 )\otimes (l_2)^n\succ{\bf1}$, and $(l_2)^n\succ \WEI(q_1'(L)\xrightarrow[]{s_2'(L)} q_2'(L)\xrightarrow[]{e_f}q_3'(L))$. Denote \begin{align} q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)\xrightarrow[]{s_2'(L)} q_2'(L)\xrightarrow[] {e_f}q_3'(L) &=: \pi_t,\\ q_3'(L) \xrightarrow[]{s_4}q_4 \left(\xrightarrow[] {s_5}q_4 \right)^{2n} &=: \pi_t',\\ q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)\xrightarrow[]{s_2'(R)} q_2'(R) \xrightarrow[]{s_6}q_5\left(\xrightarrow[]{s_7} q_5\right)^{3n} &=: \pi_t''. \end{align} One then has $E_f\notin\hat s_1s_2'(R)s_6(s_7)^{3n}$, $\ell(\tau(\pi_t\pi_t'))=\ell(\tau(\pi_t''))= \ell(\tau(q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)))$, $\WEI_{\pi_t'} = \WEI(q_3'(L) \xrightarrow[]{s_4}q_4)\otimes (l_1)^{2n} \succeq (l_1)^n \succ t$, and $\WEI_{\pi_{t}''}= \WEI(q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)) \otimes \WEI(q_1'(R)\xrightarrow[]{s_2'(R)}q_2'(R) \xrightarrow[]{s_6}q_5)\otimes (l_2)^{3n} = \WEI(q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)) \otimes \WEI(q_1'(R)\xrightarrow[]{s_2'(R)}q_2'(R) \xrightarrow[]{s_6}q_5)\otimes (l_2)^{3n} \succeq \WEI(q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L))\otimes (l_2)^{2n} \succeq \WEI(q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L))\otimes \WEI(q_1'(L)\xrightarrow[]{s_2'(L)} q_2'(L)\xrightarrow[]{e_f}q_3'(L)) \otimes (l_2)^n \succ \WEI_{\pi_t}\otimes t $ (by Lemma~\ref{lem6_diag_LMPautomata}). Also by Proposition~\ref{prop2_diag_LMPautomata}, $\mathcal{A}^{\mathcal{D}}$ is not $E_f$-diagnosable. Assume \eqref{item27_diag_LMPautomata} holds. Then there exist unobservable paths \begin{subequations}\label{} \begin{align} &q_3'(L) \xrightarrow[]{s_4} q_4 \xrightarrow[]{s_5} q_4 \text{ in }\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\\ &q_5 \xrightarrow[]{s_6} q_5 \text{ in }\mathcal{A}^{\mathcal{D}}_{\mathsf{f}} \end{align} \end{subequations} such that $q_5$ appears in $q_1'(L)\xrightarrow[]{s_2'(L)}q_2'(L)$, $s_4\in(E_{uo})^$, $s_5,s_6\in(E_{uo})^+$, $\WEI(q_4 \xrightarrow[]{s_5} q_4)=:l_1\succ{\bf1}$, and $\WEI(q_5 \xrightarrow[]{s_6} q_5)=:l_2\prec{\bf1}$. By \eqref{eqn22_diag_LMPautomata}, in $\mathcal{A}^{\mathcal{D}}$ there exist paths \begin{align} &q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)\xrightarrow[]{s_2'(L)_1} q_5 \xrightarrow[]{s_6} q_5\xrightarrow[]{s_2'(L)_2} q_2'(L) \xrightarrow[]{e_f}q_3'(L)\\ &\xrightarrow[]{s_4} q_4 \xrightarrow[]{s_5} q_4, \\ &q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)\xrightarrow[]{s_2'(R)} q_2'(R)\xrightarrow[]{\epsilon} q_3'(R)\\ \end{align} such that \begin{align} &\bar s_1\in E^E_o, \hat s_1\in(E\setminus E_f)^(E_o\setminus E_f),\\ &\ell(\bar s_1)=\ell(\hat s_1)=\ell(s_1'),\\ &\ell(\tau(q_{0}'(L)\xrightarrow[]{\bar s_{1}} q_{1}'(L)))= \ell(\tau(q_{0}'(R)\xrightarrow[]{\hat s_{1}} q_{1}'(R))),\\ &\WEI(q_{0}'(L)\xrightarrow[]{\bar s_{1}} q_{1}'(L))= \WEI(q_{0}'(R)\xrightarrow[]{\hat s_{1}} q_{1}'(R)),\\ &q_2'(R)=q_3'(R),\quad (e_f,\epsilon)= e_3',\\ &q_1'(L)\xrightarrow[]{s_2'(L)_1} q_5 \xrightarrow[]{s_2'(L)_2} q_2'(L) = q_1'(L)\xrightarrow[]{s_2'(L)} q_2'(L). \end{align} Let $t\in\mathcal{D}$ be such that $t\succ{\bf1}$. Choose sufficiently large $n\in\mathbb{N}_0$ such that $(l_1)^n\succ t$ and $\WEI(q_3'(L) \xrightarrow[]{s_4}q_4)\otimes (l_1)^n\succ {\bf1}$. Then choose sufficiently large $m\in\mathbb{N}_0$ such that $\WEI(q_1'(L)\xrightarrow[]{s_2'(L)_1} q_5 )\otimes (l_2)^m\prec{\bf1}$, $(l_2)^m\otimes\WEI( q_5\xrightarrow[]{s_2'(L)_2}q_2'(L) \xrightarrow[]{e_f}q_3'(L))\prec{\bf1}$, and $(l_2)^m\otimes \WEI\left(q_3'(L) \xrightarrow[]{s_4} q_4\left( \xrightarrow[]{s_5} q_4 \right)^{2n}\right)\prec \WEI\left( q_1'(R)\xrightarrow[]{s_2'(R)} q_2'(R) \right)$. Denote \begin{align} q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)\xrightarrow[]{s_2'(L)_1} q_5 \left( \xrightarrow[]{s_6} q_5 \right) ^{3m}&\\ \xrightarrow[]{s_2'(L)_2} q_2'(L)\xrightarrow[] {e_f}q_3'(L) &=: \pi_t,\\ q_3'(L) \xrightarrow[]{s_4}q_4 \left(\xrightarrow[] {s_5}q_4 \right)^{2n} &=: \pi_t',\\ q_0'(R)\xrightarrow[]{\hat s_1} q_1'(R)\xrightarrow[]{s_2'(R)} q_2'(R) &=: \pi_t''. \end{align} One then has $E_f\notin\hat s_1s_2'(R)$, $\ell(\tau(\pi_t\pi_t'))=\ell(\tau(\pi_t''))= \ell(\tau(q_0'(L)\xrightarrow[]{\bar s_1} q_1'(L)))$, $\WEI_{\pi_t'} = \WEI(q_3'(L) \xrightarrow[]{s_4}q_4)\otimes (l_1)^{2n} \succeq (l_1)^n \succ t$, $\WEI(q_1'(L)\xrightarrow[]{s_2'(L)_1} q_5 \left( \xrightarrow[]{s_6} q_5 \right) ^{3m} \xrightarrow[]{s_2'(L)_2} q_2'(L)\xrightarrow[] {e_f}q_3'(L))\otimes\WEI_{\pi_t'} \preceq (l_2)^m\otimes \WEI_{\pi_t'}\preceq \WEI\left( q_1'(R)\xrightarrow[]{s_2'(R)} q_2'(R) \right)$, $\WEI_{\pi_t''}= \WEI(q_{0}'(R)\xrightarrow[]{\hat s_{1}} q_{1}'(R))\otimes \WEI\left( q_1'(R)\xrightarrow[]{s_2'(R)} q_2'(R) \right)\succeq \WEI(q_{0}'(L)\xrightarrow[]{\bar s_{1}} q_{1}'(L))\otimes \WEI(q_1'(L)\xrightarrow[]{s_2'(L)_1} q_5 \left( \xrightarrow[]{s_6} q_5 \right) ^{3m} \xrightarrow[]{s_2'(L)_2} q_2'(L)\xrightarrow[] {e_f}q_3'(L)) \otimes\WEI_{\pi_t'} =\WEI_{\pi_t}\otimes\WEI_{\pi_t'} \succ \WEI_{\pi_t}\otimes t$. Also by Proposition~\ref{prop2_diag_LMPautomata}, $\mathcal{A}^{\mathcal{D}}$ is not $E_f$-diagnosable. ``only if'': Assume that $\mathcal{A}^{\mathcal{D}}$ is not $E_f$-diagnosable. We choose a sufficiently large ${\bf1}\prec t\in\mathcal{D}$. This can be done because there is an element $t'$ in $\mathcal{D}$ greater than ${\bf1}$ and $t'\prec (t')^2 \prec (t')^3 \prec \cdots$ (by Corollary~\ref{cor1_diag_LMPautomata}). By Proposition~\ref{prop2_diag_LMPautomata}, there exist paths $\pi_t= q_0^1\xrightarrow[]{s}q_1^1$, $\pi_t'= q_1^1\xrightarrow[]{s'}q_2^1$, and $\pi_t''= q_0^2\xrightarrow[]{s''}q_2^2$, such that $q_0^1,q_0^2\in Q_0$, $\last(s)\in E_f$, $\ell(\tau(\pi_t''))=\ell(\tau(\pi_t\pi_t'))$, $E_f\notin s''$, $\WEI_{\pi_t'}\succ t$, and $\WEI_{\pi_t''}\succeq\WEI_{\pi_t}\otimes t$. By definition of $\CCa(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$, the two paths $\pi_t\pi_t'$ and $\pi_t''$ generate a run $\pi'$ of $\CCa(\mathcal{A}^{\mathcal{D}}_{\mathsf{f}},\mathcal{A}^{\mathcal{D}}_{\mathsf{n}})$ consisting of a sequence of observable transitions followed by a sequence of unobservable transitions. Note that particularly, $\pi'$ may contain only observable transitions or only unobservable transitions. Hence $\pi_t\pi_t'$ and $\pi_t''$ can be rewritten as \begin{subequations}\label{eqn18_diag_LMPautomata} \begin{align} &\bar q_0\xrightarrow[]{\bar s_1} \cdots \xrightarrow[]{\bar s_n}\bar q_n \xrightarrow[]{\bar s_{n+1}}\bar q_{n+1},\\ &\hat q_0\xrightarrow[]{\hat s_1} \cdots \xrightarrow[]{\hat s_n}\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}, \end{align} \end{subequations} respectively, where $\bar q_0=q_0^1$, $\bar q_{n+1}=q_2^1$, $\hat q_0=q_0^2$, $\hat q_{n+1}=q_2^2$, $n\in\mathbb{N}_0$, for all $i\in\llbracket 1,n\rrbracket$, $\bar s_i\in(E_{uo})^E_o$, $\hat s_i\in(E_{uo}\setminus E_f)^* (E_o\setminus E_f)$, $\WEI(\bar q_{i-1}\xrightarrow[]{\bar s_i}\bar q_i)=\WEI(\hat q_{i-1}\xrightarrow[]{\hat s_i}\hat q_i)$, $\bar s_{n+1}\in(E_{uo})^$, $\hat s_{n+1}\in(E_{uo}\setminus E_f)^$. Then \begin{align} ss' &= \bar s_1 \dots \bar s_n \bar s_{n+1},\\ s'' &= \hat s_1 \dots \hat s_n \hat s_{n+1}, \end{align} and \begin{align} \pi' =& (\bar q_0,\hat q_0)\xrightarrow[]{(\last(\bar s_1),\last(\hat s_1))} \cdots \xrightarrow[]{(\last(\bar s_n),\last(\hat s_n))}\\ &(\bar q_n,\hat q_n) \xrightarrow[]{s_{n+1}'}(\bar q_{n+1},\hat q_{n+1}), \end{align} where $s_{n+1}'\in(E_{uo}')^$, $s_{n+1}'(L)=\bar s_{n+1}$, $s_{n+1}'(R)=\hat s_{n+1}$, for every $i\in\llbracket 1,n\rrbracket$, $(\bar q_{i-1},\hat q_{i-1})\xrightarrow[]{(\last(\bar s_i),\last(\hat s_i))} (\bar q_{i},\hat q_{i})$ is an observable transition, $(\bar q_n,\hat q_n) \xrightarrow[]{s_{n+1}'} (\bar q_{n+1},\hat q_{n+1})$ is a sequence of unobservable transitions. The subsequent argument is divided into four cases. \begin{enumerate}[(a)] \item\label{item21_diag_LMPautomata} Case $\bar s_1\dots \bar s_n= s$: In this case, $\last(\bar s_n)\in E_f$, $\pi_t=\bar q_0\xrightarrow[]{\bar s_1} \cdots \xrightarrow[]{\bar s_n}\bar q_n$, $\pi_t'=\bar q_n\xrightarrow[]{\bar s_{n+1}}\bar q_{n+1}$, $\WEI_{\pi_t}=\WEI(\hat q_0\xrightarrow[]{\hat s_1} \cdots \xrightarrow[]{\hat s_n}\hat q_n)$, $\WEI_{\pi_t'}\succ t$, $\WEI_{\pi_t''}=\WEI_{\pi_t}\otimes \WEI(\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1})\succeq \WEI_{\pi_t}\otimes t$. Then we have $\WEI(\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}) \succeq t$ because a path always has nonzero weight and $\mathcal{D}$ is cancellative. Recall that $t$ is sufficiently large. Divide $\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}$ to paths $\hat\pi_1\dots\hat\pi_m$ such that $\last(\hat\pi_i)=\init(\hat\pi_{i+1})$ for all $i\in\llbracket 1,m-1 \rrbracket$, where $\hat\pi_1$ is the shortest prefix of $\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}$ having weight greater than $\bf1$, $\hat\pi_2$ is the shortest prefix of $(\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1})\setminus \pi_1$ having weight greater than $\bf1$, \dots, $\hat\pi_{m-1}$ is the shortest prefix of $(\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1})\setminus (\pi_1\dots\pi_{m-2})$ having weight greater than $\bf1$. Then we have $\WEI_{\hat\pi_1}\succ{\bf1}$, \dots, $\WEI_{\hat\pi_{m-1}}\succ{\bf1}$, but $\WEI_{\hat\pi_m}$ is not necessarily greater than $\bf1$. We also have $m$ is sufficiently large, because $t$ is sufficiently large. Then there exist repetitive states among $\init(\hat\pi_1)$, $\init(\hat\pi_2)$, \dots, $\init(\hat\pi_{m-1})$, and all these repetitive states are $\bf1$-crucial by Lemma~\ref{lem1_diag_LMPautomata}. That is, $\init(\hat\pi_1)=\hat q_n$ is eventually $\bf1$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$. Similarly, we have $\init(\pi_t')=\bar q_n$ is eventually $\bf1$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$. Then \eqref{item7_diag_LMPautomata} holds, and the $s_3'$ in \eqref{eqn8_diag_LMPautomata} is equal to $\epsilon$. \item\label{item22_diag_LMPautomata} Case $\bar s_1\dots \bar s_n\sqsubsetneq s$: In this case, denote $\pi_t:=\bar q_0\xrightarrow[]{\bar s_1} \cdots \xrightarrow[]{\bar s_n}\bar q_n\pi_1'$, then $\pi_t'=(\bar q_n\xrightarrow[]{\bar s_{n+1}}\bar q_{n+1}) \setminus \pi_1'$. We have $\WEI_{\pi_t'}\succ t$, $\WEI(\bar q_0\xrightarrow[]{\bar s_1} \cdots \xrightarrow[]{\bar s_n}\bar q_n) = \WEI(\hat q_0\xrightarrow[]{\hat s_1} \cdots \xrightarrow[]{\hat s_n} \hat q_n)$, $\WEI_{\pi_t''}=\WEI(\hat q_0\xrightarrow[]{\hat s_1} \cdots \xrightarrow[]{\hat s_n} \hat q_n)\otimes \WEI(\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}) \succeq \WEI_{\pi_t}\otimes t = \WEI(\bar q_0\xrightarrow[]{\bar s_1} \cdots \xrightarrow[]{\bar s_n}\bar q_n)\otimes \WEI_{\pi_1'} \otimes t$. Using the argument as in \eqref{item21_diag_LMPautomata}, we know $\last(\pi_1')$ is eventually ${\bf1}$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{f}}$. By Lemma~\ref{lem6_diag_LMPautomata}, we have $\WEI(\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}) \succeq \WEI_{\pi_1'} \otimes t$. \begin{itemize} \item Assume $\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}$ has a prefix $\pi_2'$ such that $\WEI_{\pi_2'} \preceq\WEI_{\pi_1'}$. Then we have $\WEI(\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}) = \WEI_{\pi_2'} \otimes \WEI((\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1})\setminus\pi_2') \succeq \WEI_{\pi_1'}\otimes t$. By $\WEI_{\pi_2'} \preceq \otimes \WEI_{\pi_1'}$ and Lemma~\ref{lem6_diag_LMPautomata}, we have $\WEI((\hat q_n \xrightarrow[]{\hat s_{n+1}} \hat q_{n+1})\setminus \pi_2') \succeq t$. Also by using the argument as in \eqref{item21_diag_LMPautomata}, $\last(\pi_2')$ is eventually ${\bf1}$-crucial in $\mathcal{A}^{\mathcal{D}}_{\mathsf{n}}$. That is, \eqref{item8_diag_LMPautomata} holds. \item Assume $\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}$ has no prefix with weight no greater than $\WEI_{\pi_1'}$. Then $\WEI_{\pi_1'}\prec{\bf1}$. If $\hat q_n \xrightarrow[]{\hat s_{n+1}}\hat q_{n+1}$ contains a $\bf1$-crucial cycle in $\mathcal{A}_{\mathsf{n}}^{\mathcal{D}}$, then \eqref{item8_diag_LMPautomata} holds; otherwise, we must have $\WEI_{\pi_1'}$ is sufficiently small, then also using the argument as in \eqref{item21_diag_LMPautomata} we have $\pi_1'$ contains an anti-$\bf1$-crucial cycle in $\mathcal{A}_{\mathsf{f}}^{\mathcal{D}}$, i.e., \eqref{item27_diag_LMPautomata} holds. \end{itemize} \item\label{item23_diag_LMPautomata} Case $s\sqsubsetneq \bar s_1\dots \bar s_n$ and $\bar s_{n+1}=\epsilon$: Choose $m\in\llbracket 1,n\rrbracket$ such that $\bar s_1\dots \bar s_{m-1}\sqsubset s\sqsubsetneq \bar s_1\dots \bar s_m$. \begin{itemize} \item Assume $\bar s_1\dots \bar s_{m-1}=s$. If there exists $l\in\llbracket m,n\rrbracket$ such that $\WEI(\bar q_{l-1}\xrightarrow[]{\bar s_l} \bar q _l)=\WEI(\hat q_{l-1}\xrightarrow[]{\hat s_l} \hat q_l)$ is sufficiently large, then $\bar q_ {l-1}\xrightarrow[]{\bar s_l} \bar q_l$ contains a ${\bf1}$-crucial cycle in $\mathcal{A}_{\mathsf{f}}^{\mathcal{D}}$, $\hat q_{l-1}\xrightarrow[]{\hat s_l} \hat q_l$ contains a ${\bf1}$-crucial cycle in $\mathcal{A}_{\mathsf{n}}^{\mathcal{D}}$, then \eqref{item7_diag_LMPautomata} holds. Otherwise if such $l$ does not exist, then the weights of $\bar q_{m-1}\xrightarrow[] {\bar s_m}\bar q_m$, $\dots$, $\bar q_{n-1} \xrightarrow[] {\bar s_n}\bar q_n$ cannot be arbitrarily large. Because $t$ is sufficiently large and $t\prec\WEI(\bar q_{m-1}\xrightarrow[] {\bar s_m}\cdots\xrightarrow[] {\bar s_n}\bar q_n)$, by using the argument as in \eqref{item21_diag_LMPautomata}, in $(\bar q_{m-1},\hat q_{m-1})\xrightarrow[]{(\last(\bar s_m), \last(\hat s_m))} \cdots \xrightarrow[]{(\last(\bar s_n),\last(\hat s_n))}(\bar q_n,\hat q_n)$ there exists a positive cycle. Hence \eqref{item6_diag_LMPautomata} holds. \item Assume $\bar s_1\dots \bar s_{m-1}\sqsubsetneq s$. From $\bar q_{m-1}\xrightarrow[]{\bar s_m} \bar q_m$ and $\hat q_{m-1}\xrightarrow[]{\hat s_m} \hat q_m$, by Proposition~\ref{prop1_diag_LMPautomata}, we obtain an unobservable sequence $(\bar q_{m-1},\hat q_{m-1})\xrightarrow[]{s_m^1}\bar v \xrightarrow[]{s_m^2}(\bar q_m',\hat q_m')=:\bar \pi$, where $s_m^1(L)$ ($\ne\epsilon$) is a suffix of $s$, $s_m^1(R)=\epsilon$, and \begin{align} &\bar \pi(L)\xrightarrow[]{\last(\bar s_m)}\bar q_m = \bar q_{m-1}\xrightarrow[]{\bar s_m}\bar q_m,\\ &\bar \pi(R)\xrightarrow[]{\last(\hat s_m)}\hat q_m=\hat q_{m-1}\xrightarrow[]{\hat s_m}\hat q_m.\\ &\text{(the path before $=$ is the same as the one after $=$)} \end{align*} If there exist $\bf1$-crucial cycles in both components of $\bar v\xrightarrow[]{s_m^2}(\bar q_m',\hat q_m')$, \eqref{item8_diag_LMPautomata} holds; if there exists a $\bf1$-crucial cycle in $\bar v(L)\xrightarrow[]{s_m^2(L)} \bar q_m'$ and there exists an anti-$\bf1$-crucial cycle in $\bar q_{m-1}\xrightarrow[]{s_m^1(L)}\bar v(L)$, \eqref{item27_diag_LMPautomata} holds; otherwise, also by argument similar as above, either \eqref{item6_diag_LMPautomata} or \eqref{item7_diag_LMPautomata} holds. \end{itemize} \item\label{item24_diag_LMPautomata} Case $s\sqsubsetneq \bar s_1\dots \bar s_n$ and $\bar s_{n+1}\ne\epsilon$: Using argument similar as above, one also has at least one of \eqref{item6_diag_LMPautomata}, \eqref{item7_diag_LMPautomata}, \eqref{item8_diag_LMPautomata}, and \eqref{item27_diag_LMPautomata} holds. This finishes the proof. \end{enumerate} \end{proof} \end{document}	arXiv
\begin{document} \flushbottom \title{A quantum walk simulation of extra dimensions with warped geometry} \thispagestyle{empty} \section{Introduction} Quantum walks (QWs) constitute an interesting possibility for simulating physical phenomena from many fields. The discrete time version describes the motion of a spin $1/2$ particle on a lattice. For instance, by simply incorporating suitable position-dependent phases on the unitary operator that implements the time evolution, one can mimic the effects of an external electromagnetic field \cite{PhysRevLett.111.160601,Arnault2016,di2014quantum,PhysRevA.92.042324,PhysRevA.93.032333,PhysRevA.93.052301,PhysRevA.98.032333,Cedzich2018}. In the continuum limit (when both the time step and the lattice spacing tend to zero), the Dirac equation in presence of such fields is recovered. In an analogous way, the motion of a Dirac particle in presence of a gravitational field can be simulated by an appropriate choice of the operator that drives the evolution, either on a rectangular or other types of lattices \cite{di2014quantum,DebbaschWaves,Arrighi2019}. Other scenarios include vacuum or matter neutrino oscillations \cite{Molfetta2016,Mallick2017,Jha2020}, and one can even establish some connections to lattice field theories \cite{PhysRevA.99.032110}. There is also a different connection of QWs with quantum field theories, namely the possibility to explore some models which include extra dimensions, which are only manifested at very high energies. The possibility of extra dimensions of space was first suggested by Theodor Kaluza and Oscar Klein \cite{Kaluza1921,Klein1926} seeking an unified theory of electromagnetic and gravitational fields into a higher dimensional field, with one of the dimensions compactified. Experimental data from particle colliders restrict the compactification radius to such small scales that it becomes virtually impossible to explore these extra dimensions. Different ideas have been proposed to overcome this difficulty, for example the domain wall model introduced by Rubakov and Shaposhnikov \cite{Rubakov1983}, in which the particle couples to an external scalar field. The motion of a spin $1/2$ particle moving inside such a geometry was analyzed in \cite{PhysRevA.95.042112}. In addition to recovering the corresponding Dirac equation in the continuum limit, the QW shows, at finite spacetime spacing, localization of the particle within the brane due to the coupling to the field. Spatial localization is an important phenomenon in physics, which appears within the context of diffusion processes in lattices. It can arise from random noise on the lattice sites, giving rise to Anderson localization \cite{Lattices1956} and causing a metal-insulator transition, but it can also be the consequence of the action of an external periodic potential (see e.g. \cite{aubry1980analyticity,PhysRevLett.49.833,PhysRevLett.103.013901}). Similarly, one obtains localization for the 1-dimensional QW when spatial disorder is included \cite{Joye2010,PhysRevLett.106.180403,Crespi2013a}, non-linear effects \cite{Navarrete-Benlloch2007}, or by the use of a spatially periodic coin \cite{PhysRevE.82.031122}. The results in \cite{PhysRevA.95.042112} show, however, that localization can also appear as a consequence of the interaction with a\textit{ smooth} external potential, instead of a random, or even periodic, perturbation. In this paper, we investigate localization effects that arise within a different context, which is also inspired on high energy physics, and was originally proposed to address the \textit{hierarchy problem} (the observed difference between the Higgs mass, and the Planck scale, in many orders of magnitude), and is commonly referred to as the Randall-Sundrum model \cite{RSoriginal}. This model assumes an extra dimension which extends between two \textit{branes} (with a topology that will be discussed later). Here we consider a simplified version with one ordinary spatial dimension and one extra dimension, and define a QW that reproduces the dynamics of a spin $1/2$ particle in the continuum spacetime limit. Unlike the Rubakov and Shaposhnikov model, there is no coupling to an external scalar field. Instead, this model presents a warped geometry along the extra dimension. As we will show, this curvature is at the root of a localization effect of the QW towards the second (low energy) brane. The stationary states of the model in the continuum limit become concentrated close to the low energy brane for high values of the warp coefficient, which quantifies the strength of the localization. The localization of the QW can be analyzed by quantifying its overlap with these stationary states. This allows us to tailor the dynamics of the QW, showing a different behavior as the value of the warp coefficient is changed. In this way, we arrive at a QW model with a rich phenomenology, where some properties are inherited from the continuum field theoretic model. There is, in this sense, a mutual multidisciplinary benefit: one can design a QW which simulates an important high energy physics model. In exchange, the knowledge of the continuum properties is useful to understand, and to control, the dynamics of the QW in different regimes. This paper is organized as follows. We first define the Randall-Sundrum model in $1+1$ spatial dimensions, along with its main properties. We pay special attention to the stationary states of the Hamiltonian, which play a crucial role in understanding the dynamics of the proposed QW. Next, we define a QW which allows to recover the dynamics of the Randall-Sundrum model for a spin $1/2$ particle, and we study its phenomenology. Namely, we show that the distribution probability, as well as the expected value of the position along the extra dimension, approaches the lower brane at large time, and that this approaching proceeds more slowly for larger values of the warp coefficient, which turns out to be the main parameter in controlling the dynamics. We also analyze the entanglement entropy between spatial and internal degrees of freedom, exhibiting a complex behavior as a function of that parameter, which can be attributed to the different sharpness of the probability distribution. We finally conclude by collecting and discussing our main results. \section{The Model} \subsection{Orbifold $S^{1}/\mathbb{Z}_{2}$ and Background Geometry} As described in the Introduction, we consider the Randall-Sundrum model (RSM) \cite{RSoriginal} with a single extra dimension $y$, together with a 2-dimensional ordinary spacetime, whose coordinates are denoted by $x^{\mu}=\{t,x\}$. The total spacetime possesses $D=3$ dimensions. The extra dimension $y$ is compactified on a circle of radius $R$, and subject to a $\mathbb{Z}_{2}$ symmetry. These features are captured by the equivalences \begin{align} S^{1}:\;y & \sim y+2\pi R~,\label{eqn:PeriodicCond}\\ \mathbb{Z}_{2}:\;y & \sim-y~,\label{eqn:Z2Cond} \end{align} which define the orbifold $S^{1}/\mathbb{Z}_{2}$ describing this extra dimension. Along the $y$ dimension, the orbifold is a finite segment with two fixed points at $y=0$ and $y=\pi R\equiv L$. The RSM assumes that there is a $(D-1)$-brane of ordinary dimensions at each fixed point, see Fig.~\ref{fig:Branes} for a sketch of the space configuration and the orbifold symmetries. \begin{figure} \caption{Schematic representation of the extra dimension in the Randall-Sundrum model. } \label{fig:Branes} \end{figure} The matter fields are supposed to reside on the brane at $y=L$, which is referred to as the ``visible brane'', while the brane at $y=0$ is the ``hidden brane''. Both branes contribute to the bulk background geometry through their tensions, or vacuum energies, $T_{\mathrm{vis}}$ and $T_{\mathrm{hid}}$ respectively \cite{RSoriginal,BraneTensions}. The total background action is \begin{equation}\label{eqn:ActionBackgroundTotal} S=\int_{-L}^{L}dy\int dx^{\mu}\sqrt{\|g\|}\left(2\alpha\mathcal{R}-\Lambda\right)+S_{\mathrm{vis}}+S_{\mathrm{hid}}~, \end{equation} where the first term is the usual Einstein-Hilbert action of the total space, with $\Lambda$ the bulk cosmological constant, $\alpha$ a constant and $\|g\|$ the absolute value of the metric determinant, while \begin{align} S_{\mathrm{vis}} & =-\int dx^{\mu}\sqrt{\|g_{\mathrm{vis}}\|}T_{\mathrm{vis}}~,\\ S_{\mathrm{hid}} & =-\int dx^{\mu}\sqrt{\|g_{\mathrm{hid}}\|}T_{\mathrm{hid}}~, \end{align} are the action contributions of the branes tensions, with the induced metrics $g_{\mathrm{vis}}(x^{\mu})=g(x^{\mu},y=L)$ and $g_{\mathrm{hid}}(x^{\mu})=g(x^{\mu},y=0)$. To address the hierarchy problem, the following metric was proposed \begin{equation} ds^{2}=e^{-2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}~,\label{eqn:Metric} \end{equation} where $e^{-2A(y)}$ is a \textit{warp factor}, a rapidly changing function along the additional dimension, and $\eta_{\mu\nu}$ is the Minkowski metric with signature $(+,-)$. The metric in Eq.~(\ref{eqn:Metric}) obeys Einstein's equations that are obtained from the action (\ref{eqn:ActionBackgroundTotal}): We refer the reader to the Supplementary Material for the standard computation particularized to this lower-dimensional spacetime. We also show that, as a consequence of these equations, the function in the exponent is given by \begin{equation} A(y)=k\|y\| \end{equation} where $k$ is the so called \emph{warp coefficient}. \subsection{Fermions in the Randall-Sundrum model} We now focus on the study of spin $1/2$ fermions, whose evolution equation is the Dirac equation in curved spacetime \begin{equation} (i\gamma^{a}e_{a}^{\mu}D_{\mu}-m)\Psi=0~.\label{eqn:DiracCurved} \end{equation} The $\gamma^{a}$ are the Dirac gamma matrices in a local rest frame, and the covariant derivative is \begin{equation} D_{\mu}=\partial_{\mu}-\frac{i}{4}\omega_{\mu}^{ab}\sigma_{ab}~,\quad\text{with}\quad\sigma_{ab}=\frac{i}{2}[\gamma_{a},\gamma_{b}]~, \end{equation} where $\omega_{\mu}^{ab}$ is the spin connection. The \emph{vierbeins} $e_{a}^{\mu}$ allow to express the Dirac matrices in a rest frame, that is, they perform a change of basis to a non-coordinate system in which the metric becomes the Minkowski metric \begin{equation} g_{\mu\nu}e_{a}^{\mu}e_{b}^{\nu}=\eta_{ab}~.\label{eqn:VierbeinsDef} \end{equation} Equation~(\ref{eqn:DiracCurved}) defines the vector current \begin{equation} j^{\mu}=\sqrt{\|g\|}e_{a}^{\mu}\overline{\Psi}\gamma_{a}\Psi~, \end{equation} whose conservation $\partial_{\mu}j^{\mu}=0$ imposes the normalization condition \begin{equation} \int dx^{\mu}\sqrt{\|g\|}e_{0}^{0}\Psi^{\dagger}\Psi=1~. \end{equation} In the case of 2 spatial dimensions, the Dirac equation (\ref{eqn:DiracCurved}) can be reduced, after some algebra, to \begin{equation} i\gamma^{a}\left[e_{a}^{\mu}\partial_{\mu}\Psi+\frac{1}{2\sqrt{\|g\|}}\partial_{\mu}(e_{a}^{\mu}\sqrt{\|g\|})\Psi\right]-m\Psi=0~,\label{eqn:Dirac2+1} \end{equation} where the $\gamma_{a}$ matrices become Pauli matrices. A simple choice of the vierbeins obeying relation (\ref{eqn:VierbeinsDef}) is \begin{equation} e_{0}=(e^{A(y)},0,0)\quad e_{1}=(0,e^{A(y)},0)\quad e_{2}=(0,0,1)~, \end{equation} which yields the following expression for the Dirac equation \begin{equation} i\partial_{t}\Psi=-i\gamma^{0}\gamma^{1}\partial_{x}\Psi-i\gamma^{0}\gamma^{2}\partial_{y}(e^{-A(y)}\Psi)+\gamma_{0}e^{-A(y)}m\Psi~.\label{eqn:DiracExplicit} \end{equation} This expression can be rewritten in Hamiltonian form as \begin{equation} i\partial_{t}\chi=\mathcal{H}\chi,\label{eqn:DiracSimp} \end{equation} with \begin{equation} \mathcal{H}=-\frac{i}{2}\{B^{x},\partial_{x}\}-\frac{i}{2}\{B^{y},\partial_{y}\}+\gamma_{0}e^{-A(y)}m~, \label{eq:HamiltonianDirac} \end{equation} where the change of variable $\chi=e^{-A(y)/2}\Psi$ was performed, and we defined \begin{equation} B^{x}=\gamma^{0}\gamma^{1}~,\quad B^{y}=e^{-A(y)}\gamma^{0}\gamma^{2}~. \end{equation} The symbol $\{\cdot,\cdot\}$ represents the anticommutator of two operators. There is some freedom in the choice of the gamma matrices. For convenience, we choose \begin{equation} \gamma^{0}=\sigma_{x}~,\quad\gamma^{1}=i\sigma_{y}~,\quad\gamma^{2}=i\sigma_{z}~. \end{equation} \subsection{Boundary Conditions for Fermionic Fields} \label{sec:Boundary} The periodic condition (\ref{eqn:PeriodicCond}) simply implies that the fermionic fields need also to be periodic \begin{equation} \chi(x^{\mu},y+2L)=\chi(x^{\mu},y)~,\label{eqn:BoundaryPeriodic} \end{equation} but the $\mathbb{Z}_{2}$ needs a deeper consideration, since it has to leave the fermionic action invariant. We can write the fermionic action as \begin{equation} S_{F}= \int dx^{\mu}\int dy\overline{\chi}(x^{\mu},y) \left(i\gamma^{\mu}\partial_{\mu}+i\gamma^{2}\partial_{y}e^{-A(y)}-e^{-A(y)m}\right)\chi(x^{\mu},y)~. \label{eqn:ActionFermion} \end{equation} which is extremized by the Dirac equation (\ref{eqn:DiracExplicit}). Under the action of $\mathbb{Z}_{2}$ it becomes \begin{equation} S_{F}= \int dx^{\mu}\int dy\overline{\chi}(x^{\mu},-y) \left(i\gamma^{\mu}\partial_{\mu}-i\gamma^{2}\partial_{y}e^{-A(-y)}-e^{-A(-y)}m\right)\chi(x^{\mu},-y)~. \label{eqn:ActionFermionZ2} \end{equation} We have to find an operator $M$, defined as $\chi(x^{\mu},-y)=M\chi(x^{\mu},y)$, so as to allow the action to remain invariant. Action (\ref{eqn:ActionFermionZ2}) then becomes \begin{equation} S_{F}= \int dx^{\mu}\int dy\overline{\chi}(x^{\mu},y)\gamma^{0}M\gamma^{0} \left(i\gamma^{\mu}\partial_{\mu}-i\gamma^{2}\partial_{y}e^{-A(y)}-e^{-A(y)}m\right)M^{\dagger}\chi(x^{\mu},y)~, \end{equation} and establishes the following restrictions for $M$ to keep the action (\ref{eqn:ActionFermion}) invariant, \begin{align} & \gamma^{0}M\gamma^{0}\gamma^{\mu}M^{\dagger}=\gamma^{\mu}~,\\ & \gamma^{0}M\gamma^{0}\gamma^{2}M^{\dagger}=-\gamma^{2}~,\\ & \gamma^{0}M\gamma^{0}M^{\dagger}=\mathbb{I}~, \end{align} where the first 2 conditions come from the kinetic terms of the action, and the last one arises from the mass term. There does not exist a solution for $M$ that solves all conditions simultaneously, although $M=\eta\sigma_{z}$ is a solution for the first 2, with $\eta=\pm1$. This means that a constant mass term is forbidden. In the following we restrict ourselves to the case where the ``bulk mass'' $m$ vanishes. The action of the fermionic field is therefore \begin{equation} \begin{split}S_{F}= & \int dx^{\mu}dy\overline{\chi}(x^{\mu},y)\left(i\gamma^{\mu}\partial_{\mu}+i\gamma^{2}\partial_{y}e^{-A(y)}\right)\chi(x^{\mu},y)~,\end{split} \label{eqn:ActionFermionNoMass} \end{equation} and the fermionic field has to obey the boundary condition \begin{equation} \chi(x^{\mu},-y)=\eta\sigma_{z}\chi(x^{\mu},y)~,\label{eqn:BoundaryZ2} \end{equation} with $\eta=\pm1$. \begin{figure} \caption{Plots of the probability distribution for the first four stationary states, with positive energy and a value of $q=10$, for $kL=3$ on the left and for $kL=7$ on the right.} \label{fig:Confinement} \end{figure} \subsection{Stationary Solutions} \label{sect:StationaryStates} In this model, the Dirac field satisfies a complicated equation, Eq.~(\ref{eqn:DiracExplicit}), which is difficult to address even numerically. In order to obtain some insight, we first look for stationary solutions, which are defined as the eigenstates of the Hamiltonian. For $m=0$, and with our choice of the gamma matrices, the Hamiltonian takes the form \begin{equation} \mathcal{H}=-\sigma_{z}\hat{p}_{x}+\frac{\sigma_{y}}{2}\left(e^{-A(y)}\hat{p}_{y}+\hat{p}_{y}e^{-A(y)}\right)~,\label{eqn:DiracMassless} \end{equation} where $p_{k}=-i\partial_{k}$ is the momentum operator along the $k$ direction ($k=x,y$). The stationary states $\phi_{n}(x,y)$ corresponding to energy $E_{n}$ satisfy \begin{equation} \mathcal{H}\phi_{n}(x,y)=E_{n}\phi_{n}(x,y)~.\label{eq:Eigenproblem} \end{equation} It is convenient to introduce a Fourier transform on the ordinary dimension $x$: \begin{equation} \tilde{\phi}_{n}(q,y)=\int dxe^{-iqx}\phi_{n}(x,y)~, \end{equation} since the field is free to move along this direction. We found the energies \begin{equation} E_{n}=\pm\sqrt{q^{2}+\left(k\alpha_{n}\right)^{2}}~,\label{eqn:SpectrumSS} \end{equation} where \begin{equation} \alpha_{n}=\frac{n\pi}{e^{kL}-1}~,~n=0,1,\;\dots \end{equation} The eigenfunctions associated with this spectrum that satisfy the boundary condition (\ref{eqn:BoundaryZ2}), for the particular case with $\eta=1$, are \begin{align} \tilde{\phi}_{n}^{\uparrow}(q,y)&=\sqrt{\frac{2k}{e^{kL}-1}} \frac{E_{n}+q}{\sqrt{(E_{n}+q)^{2}+(k\alpha_{n})^{2}}} e^{\frac{k\|y\|}{2}}\cos\left[\alpha_{n}\left(e^{k\|y\|}-1\right)\right]~,\label{eqn:EigenStateUpEta+}\\ \tilde{\phi}_{n}^{\downarrow}(q,y)&=\sqrt{\frac{2k}{e^{kL}-1}} \frac{k\alpha_{n}}{\sqrt{(E_{n}+q)^{2}+(k\alpha_{n})^{2}}} e^{\frac{k\|y\|}{2}}\sin\left[\alpha_{n}\left(e^{k\|y\|}-1\right)\right]\mathrm{sign}(y)~,\label{eqn:EigenStateDownEta+} \end{align} where the components of the spinor field are $\tilde{\phi}_{n}=(\tilde{\phi}_{n}^{\uparrow},\tilde{\phi}_{n}^{\downarrow})^{T}$. The particular case $n=0$ only has an upper component, which is given by \begin{equation} \tilde{\phi}_{0}^{\uparrow}(q,y)=\sqrt{\frac{k}{e^{kL}-1}}e^{\frac{k\|y\|}{2}}\mathrm{sign}(E_{n}+q)~,\label{eq:n0mode} \end{equation} and is undefined for energy and momentum with different sign. The procedure to obtain the eigenfunctions is detailed in the Supplementary Material, as well as the solution for $\eta = -1$. The probability distribution associated to these wavefunctions is concentrated around $y=L$ for high values of the warp coefficient $k$. We illustrate this behavior in Fig.~\ref{fig:Confinement}, where we have plotted the probability density for the first modes with positive energy, and momentum $q=10$, for a value of the warp coefficient $kL=3$ and $kL=7$, respectively. \section{A Quantum Walk for the Randall-Sundrum model} Once we have discussed the main properties of the RSM in the continuum spacetime, we focus on the main goal of our work, which consists in constructing a QW that is able to simulate the dynamics of a spin $1/2$ particle subject to the geometric effects and symmetries of the model. To incorporate the metric, we adapt the scheme introduced in \cite{DebbaschWaves}, which allows to reproduce (in the continuum limit) a Dirac equation of the form Eq.~(\ref{eqn:DiracSimp}). The QW is defined on a 2-dimensional discrete grid with $x$ and $y$ axis, with discrete positions labeled by $r$ and $s$, respectively. The grid points are equally spaced by $\epsilon$, so that the spatial coordinates can be related to the grid points by $x=\epsilon r$ and $y=\epsilon s$. The Hilbert space that corresponds to these spatial degrees of freedom, $\mathcal{H}_{spatial}$ is spanned by the basis $\{\|x=\epsilon r,y=\epsilon s\rangle\}/r,s\in\mathbb{Z}$. Time steps are labeled by $j\in\mathbb{Z}$, and are also equally spaced by $\epsilon$. The coin (or internal) space is a 2 dimensional Hilbert space $\mathcal{H}_{\mathrm{coin}}$, so that the total Hilbert space is $\mathcal{H}_{tot}=\mathcal{H}_{spatial}\otimes\mathcal{H}_{\mathrm{coin}}$. At a given time step, the state of the walker will be represented by a two component spinor $\|\chi_{j}\rangle\in\mathcal{H}_{tot}$. The one step evolution of the QW is given by \begin{equation} \|\chi_{j+1}\rangle=U\|\chi_{j}\rangle~,\label{eqn:QWevolution} \end{equation} where $U$ is a unitary operator which consists on rotations in the components of $\|\chi_{j}\rangle$, and translations by $\epsilon$ in the two directions of physical space $x$ and $y$. The angles of rotation can, in general, be dependent on the spacetime coordinates of the walker. Following \cite{DebbaschWaves}, we adopt \begin{equation} U=R^{-1}(y)\left[\Theta(y)S_{y}(\epsilon/2)\right]^{2}R(y)S_{x}(\epsilon)~,\label{eq:Uoperator} \end{equation} where $S_{k}(\epsilon)=\exp(-i\sigma_{z}p_{k}\epsilon)$ are spin-dependent shift operators in the direction $\pm k$ (with $k=x,y$), \begin{equation} \Theta(y)=\begin{pmatrix}c(y) & is(y)\\ -is(y) & c(y) \end{pmatrix}~,\label{eq:Thetarotation} \end{equation} with $c(y)=e^{-A(y)}$, $s(y)=\sqrt{1-e^{-2A(y)}}$, and \begin{equation} R(y)=\frac{1}{\sqrt{2}}\begin{pmatrix}f^{}(y) & if(y)\\ -f^{}(y) & -if(y) \end{pmatrix}~,\label{eq:RRotation} \end{equation} where $f(y)=\sqrt{\frac{1+c(y)}{2}}+i\sqrt{\frac{1-c(y)}{2}}$. At each position $(r,s)$ we introduce \begin{equation} \chi_{j,r,s}\equiv\langle x=\epsilon r,y=\epsilon s\|\chi_{j}\rangle=\begin{pmatrix}\chi_{j,r,s}^{\uparrow}\\ \chi_{j,r,s}^{\downarrow} \end{pmatrix}~,\label{eq:Xicomponents} \end{equation} which represents the amplitude (given a component of the spin) for the particle to be localized at the position labeled by $(r,s)$ and time step $j$. In this way, the time step defined by (\ref{eqn:QWevolution}) can be recast as a recursive formula for $\chi_{j,r,s}$, which is provided in the Supplementary Material. In order to implement this QW to simulate fermions in the RSM, appropriate conditions have to be set to comply with the boundary conditions \eqref{eqn:BoundaryPeriodic} and \eqref{eqn:BoundaryZ2}. It can be explicitly shown, from the recursive formula for $\chi_{j,r,s}$, that this QW dynamics respects (\ref{eqn:BoundaryZ2}), in the sense that, if the walker obeys the condition \begin{equation} \chi_{j,r,-s}=\eta\sigma_{z}\chi_{j,r,s}~,\label{eqn:BoundaryQW} \end{equation} at time $j$, it is also obeyed at time $j+1$. For the simulations, we discretize the $y$ coordinate along the segment $[-L,L]$ with a spacing $\epsilon$, and impose an initial condition which satisfies Eq.~(\ref{eqn:BoundaryQW}). We use the same lattice spacing in the $x$ direction, together with an strategy that adapts its effective extension to the time step. We also impose periodic boundary conditions on the grid to respect condition (\ref{eqn:BoundaryPeriodic}), taking into account that functions evaluated at $y=L+\epsilon$ should be identified with functions at $y=-L+\epsilon$ to respect the periodicity in the range $[-L,L]$. \section{Results} The QW defined in the previous section is guaranteed to reproduce (in the continuum limit) a Dirac equation of the form \eqref{eqn:DiracSimp}, such as the one corresponding to the RSM. The question that arises concerns the dynamics appearing at a finite lattice and time step spacing. Of course, one does not expect the QW to behave exactly as the continuum field but, to what extent do they differ? Are there any new features that appear in the discrete case? In particular, we are interested in looking for some kind of probability concentration towards the visible brane, for a given initial condition. In this Section we explore all these features. \subsection{Stationarity of the Eigenstates Solutions on the Quantum Walk} \label{sec:StationaryQW} As an initial comparison, we start by considering the discretized version of the eigenstates corresponding to the continuum limit Hamiltonian, obtained before. Such states remain stationary within this limit (i.e. they just evolve by adopting a trivial phase). How do they evolve under the action of the QW? We consider an initial state which corresponds to an eigenstate of the continuum, with fixed momentum $q$, and check whether the QW evolution of this state is stationary. The initial condition of the walker is therefore \begin{equation} \chi_{0,r,s}=\tilde{\phi}_{n}(q,\epsilon s)e^{iq\epsilon r}~,\label{eqn:InitialPlaneWave} \end{equation} which represents a constant probability density along the ordinary dimension $x$. As expected, the QW evolution does not remain stationary, although it keeps a close resemblance to the initial state. This can be observed from Fig.~\ref{fig:SS_evolution}, where we represented the normalized marginal probability along the $y$ direction of the walker (after summing over $x$) at different time steps, for an initial stationary state solution with $n=2$, and warp coefficient $kL=3$. \begin{figure} \caption{Snapshots of the probability density starting from an initial eigenstate with $n=2$ and positive energy, for a value of $kL=3$, and $q=10$. The simulation grid has 100 points along the $y$ direction, and enough points have been taken in the $x$ direction to ensure that the total probability density does not leak outside the boundaries.} \label{fig:SS_evolution} \end{figure} \subsection{Localization in the QW\label{subsec:Confining-in-the}} We now investigate the localization capability of the above defined QW, i.e., whether it shows a tendency to concentrate the walker towards the visible brane at $y=L$. We consider an initial walker which is fully localized \begin{equation} \chi_{0,r,s}^ {}=\delta_{x,0}\delta_{y,y_{0}}C_{0}~,\label{eqn:InitialDelta} \end{equation} where $C_{0}$ is the initial coin state, and we recall that $x=\epsilon r$ and $y=\epsilon s$. We explore the evolution of a walker which is initially localized at the center of the extra dimension, that is at $y_{0}=\frac{L}{2}$, and we study the probability distribution for different values of the warp coefficient, at a given time step. In Fig.~\ref{fig:panell_confinament} we show the surface plot of the probability density with the above initial conditions, and $C_{0}=\frac{1}{\sqrt{2}}(1,i)^{T}$, which induces a symmetric evolution in the ordinary dimension. The blue (red) color of the surface represents dominance of the upper (lower) coin component, while yellow stands for a superposition of both components. \begin{figure}\label{fig:panell_confinament} \end{figure} \begin{figure} \caption{Expected value of the probability distribution along the extra dimension $y$, as calculated from the FPD, for different values of the warp coefficient $kL$. The initial condition is the same as in Fig.~\ref{fig:panell_confinament}.} \label{fig:Expected value} \end{figure} We notice that most of the probability distribution in the $x$ direction is concentrated along a freely propagating front which moves at the maximum speed ($x=\pm t$), consistently with the fact that the QW simulates massless fermions. We also notice that most of the right propagating distribution (positive values of $x$) is dominated by the upper coin component, while the part propagating to the left (negative values of $x$) mainly contains the lower coin component, a fact that can also be inferred from the explicit evolution of the QW (see Supplementary Material for details). The propagation of the walker along the extra dimension $y$ strongly depends on the value of the warp coefficient. At $t=5L$, the distribution with the lowest value of $kL$ possesses non-zero values on the visible brane $y=L$, while the other two do not. In fact, the displacement of the probability distribution towards $y=L$ is slower for the highest $kL$. In other words, a larger value of the warp coefficient dramatically increases the time scale of the dynamics along the extra dimension, and makes it prohibitively expensive (in terms of computational cost) to explore larger values of $kL$ than those considered here. In order to investigate whether the QW exhibits the same behavior as the stationary states, in the sense that a higher value of the warp coefficient induces a stronger localization near the visible brane, we study the distribution of the freely propagated parts of the walker (the regions around $x=\pm t$), where most of the probability density is concentrated, as can be readily seen in Fig.~\ref{fig:panell_confinament}. The probability distribution associated to these two zones will be referred to as the ``freely propagating distribution'' (FPD). In terms of the spinor components, those are the probability density distributions obtained from $\chi_{j,s}^{\mathrm{R}}\equiv\chi_{j,j,s}$ and $\chi_{j,s}^{\mathrm{L}}\equiv\chi_{j,-j,s}$, where $r=\pm j$ restricts the wavefunction to the two freely propagating peaks. In Fig.~\ref{fig:Expected value} we represent the expected value for these distributions along the $y$ dimension, which can be defined as \begin{equation} \langle y_{R(L)}(t)\rangle=\sum_{s}\epsilon s\;\chi_{j,s}^{\mathrm{R(L)}\dagger}\chi_{j,s}^{\mathrm{R(L)}}~,\label{eqn:ExpectedVal} \end{equation} where $t=\epsilon j$, for different values of $kL$. First of all we notice that this quantity reaches an asymptotic value, which is closer to $L$ for higher warp coefficients. Secondly, as discussed above, the warp coefficient induces a change in the time scale of the dynamics, so that lower values of the warp coefficient show a faster convergence towards the asymptotic state, consistently with the features already observed in Fig.~\ref{fig:panell_confinament}. \begin{figure} \caption{Probability distributions of the FPDs along the extra dimension $y$, for the value $kL=3$. The inset is a histogram showing the value of the $B_{n}(t)$ coefficients, as defined by Eq. (\ref{eqn:IntegratedModeCoeff}): see the text for an explanation. The left (right) panels show the left (right) FPD. The top panels are calculated at a shorter time $t=50 L$ and the bottom ones at a longer time $t=1000 L$. The initial condition is the same as in Fig.~\ref{fig:Expected value}, and the simulation grid has 200 points along the $y$ direction.} \label{fig:ModeDecomposition} \end{figure} \subsection{Mode decomposition of the freely propagating distribution} Our simulations indicate that the FPD reaches a steady state along the extra dimension, in a similar fashion as the expected value (\ref{eqn:ExpectedVal}). This evolution can be appreciated from the plots of Fig.~\ref{fig:ModeDecomposition}. Al late times (lower row), the probability distribution resembles the probability density of a stationary state with positive energy and momentum in one of the lowest modes: $n=0$ for the right FPD, and $n=1$ for the left FPD. It is important to recall that, as discussed above, the right (left) FPD is predominantly composed by the upper (lower) component of the spinor, and that $n=0$ has no lower component: see Eq. (\ref{eq:n0mode}). This causes a fundamental difference when comparing the left and right contributions. In order to investigate these features on the time evolution, we introduce a decomposition on the wavefunction of the walker as a combination of the stationary states basis. This allows us to write \begin{equation}\label{eqn:ModeDecomposition} \chi_{j,r,s}=\int_{-\pi/\epsilon}^{\pi/\epsilon}\frac{dq}{2\pi}\sum_{n}\beta_{n}(q,t)\tilde{\phi}_{n}(q,\epsilon s)e^{-iq\epsilon r}~, \end{equation} where the temporal dependence is included on the $\beta_{n}(q,t)$ coefficients. In the Supplementary Material we detail how these factors can be computed, and define their normalization conditions. In particular, we are interested on the contribution of each value $n$, therefore we integrate out the dependence in the quasi-momentum $q$. In other words, we are interested on the following (time-dependent) coefficients: \begin{equation} B_{n}(t)=\int_{-\pi/\epsilon}^{\pi/\epsilon}\frac{dq}{2\pi}\left\|\beta_{n}(q,t)\right\|^{2}~.\label{eqn:IntegratedModeCoeff} \end{equation} The different mode components $B_{n}(t)$ of Fig.~\ref{fig:ModeDecomposition} have been included as an inset in those plots. On the one hand, it can be observed that, at long times, when a steady state has been reached, the FPDs are mostly composed by the lowest possible mode ($n=0$ or $n=1$, as discussed above). On the other hand, at short times, the FPDs contain additional higher modes. \subsection{Entanglement Entropy \label{subsec:Entanglement-Entropy}} Finally, we study the entanglement properties that the QW exhibits between the coin and position degrees of freedom for the already considered, initially localized state. The entanglement can be quantified using the von Neumann entropy of the reduced density matrix in the coin space \begin{equation} S(t)=-\mathrm{Tr}\left\{ \rho_{c}(t)\log_{2}\rho_{c}(t)\right\} ~, \end{equation} where $\rho_{c}(t=\epsilon j)=\sum_{r,s}\chi_{j,r,s}\chi_{j,r,s}^{\dagger}$ is the reduced density matrix in the coin space, i.e. after tracing out the spatial degrees of freedom. In Fig.~\ref{fig:entropy} we plot the evolution of the entanglement entropy of a fully localized initial state for different values of the warp coefficient, with a coin state $C_{0}=\frac{1}{\sqrt{5}}(1,2i)^{T}$. Notice that this choice is different from that one used in the previous section, for reasons that are explained below. It can be seen that the entanglement entropy reaches lower values as $kL$ increases, an effect that can probably be due to the fact that the probability density in between the FPDs becomes more spread (and therefore ``less ordered'') at lower values of $kL$. This can be observed in Fig.~\ref{fig:panell_confinament_zoom}, where we plotted a zoomed version of Fig.~\ref{fig:panell_confinament}, but obtained with the above initial coin components $C_{0}=\frac{1}{\sqrt{5}}(1,2i)^{T}$. One can see that, for lower values of the warp coefficient, a significant part of the probability distribution is scrambled in the intermediate region between both parts of the FPD. This diffusion effect can be totally mitigated for extreme values of the warp coefficient, leading to a minimum value of the entropy which is completely dominated by the FPD, and can be obtained from the initial coin components. In the Supplementary Material we show this limiting situation, and how the corresponding entropy can be computed. The initial coin state $C_{0}=\frac{1}{\sqrt{2}}(1,i)^{T}$ previously used produces values of the entropy which are very close to unity in all cases, making it difficult to appreciate the effects that are discussed above. \begin{figure}\label{fig:entropy} \end{figure} \begin{figure}\label{fig:panell_confinament_zoom} \end{figure} \section{Conclusions} We have investigated a quantum walk which allows to simulate the Randall-Sundrum model of extra dimensions, while satisfying the constrains imposed by the symmetries of that model. This model has played an important role in high energy physics, aiming to solve the hierarchy problem, by introducing one finite extra dimension that possesses two branes at its extremes. The matter fields are confined in the visible brane, while gravity is allowed to span along this whole dimension. We worked it out for the case of spin $1/2$ fermions in a two dimensional space, composed by an ordinary dimension and an orbifolded one, apart from time, and obtained the Dirac equation in this spacetime configuration. The boundary conditions of the orbifold on the fermionic field forced it to be massless on the bulk. In this lower dimensional space we were able to obtain the eigenenergies of the fermionic field, as well as the corresponding eigenstates, showing a probability density which is concentrated near the visible brane, a phenomenon that bears an analogy with the localization effect that can be found in many scenarios \cite{Lattices1956,aubry1980analyticity,PhysRevLett.49.833,Joye2010,PhysRevLett.106.180403,Navarrete-Benlloch2007,PhysRevE.82.031122}. This analogy motivated us to seek localization effects on the QW that we introduced to simulate the RSM. The QW is defined in such a way that, in the continuum limit, the Dirac equation of the fermionic field for the RSM metric is recovered. We investigated the confining capabilities of the QW, by considering an initially localized walker away from the visible brane. We concluded that the freely propagating parts of the probability distribution, where the probability is mostly concentrated, reach an asymptotic value of the expected position along the extra dimension. Moreover, the asymptotic value gets closer to the visible brane for higher values of the warp coefficient, which therefore drives the strength of localization, and also noticed that it had an effect on the timescale of the dynamics, by delaying them for higher values of the coefficient. At long time steps, the probability densities show an asymptotic shape, with a resemblance with the eigenstates that were obtained in the continuous model, which suggested a study based on the decomposition of the wavefunction in terms of these stationary states. We found that the freely propagating parts of the QW are dominated, in the asymptotic regime, by the lowest possible (i.e., compatible with the symmetries of the model) modes. At intermediate time steps, the same decomposition manifests a combination of multiple modes with higher energy. Finally, we found that the entanglement between coin and spatial degrees of freedom is reduced for stronger warp coefficients. We associated this result to the higher spreading of the density distribution for the lower values of the warp coefficient. We conclude that quantum walks are suitable candidates for simulating models of field theories with extra dimensions that rely on the curvature of the spacetime. Not only the model is interesting from the point of view of the field theory: It allows to design a quantum process that can be tailored to exhibit very rich dynamics, showing free propagation in one dimension, and an asymptotic confining behavior on the other one, with rates that can be tuned by an appropriate choice of the parameters. In this way, the interplay between high energy physics and quantum simulations can be of mutual benefit. \section{Supplementary Material} \subsection{Metric Solution} \label{append:WrapFactor} Here we prove that the metric (\ref{eqn:Metric}) extremizes the background action (\ref{eqn:ActionBackgroundTotal}), which can be rewritten as \begin{equation} S=\int dy\int dx^{\mu}\sqrt{\|g\|}\Big(2\alpha\mathcal{R}-\Lambda-\delta(y)T_{\mathrm{hid}}-\delta(y-L)T_{\mathrm{vis}}\Big)~. \end{equation} The extrema of this action gives the following Einstein equations \begin{equation} \sqrt{\|g\|}\bigg(R_{MN} -\frac{1}{2}g_{MN}\mathcal{R}+\frac{1}{4\alpha}\Lambda g_{MN}\bigg)= -\frac{\sqrt{\|g\|}}{4\alpha} \bigg(T_{\mathrm{hid}}\delta(y)g_{\mu\nu}\delta_{M}^{\mu}\delta_{N}^{\nu}+T_{\mathrm{hid}}\delta(y-L)g_{\mu\nu}\delta_{M}^{\mu}\delta_{N}^{\nu}\bigg)~, \end{equation} with indices $M,N=\{t,x,y\}$, while $\mu,\nu=\{t,x\}$ only account for ordinary dimensions. After computing the curvature tensor, we obtain the following equation for the $yy$ component \begin{equation} A'(y)^{2}+\frac{\Lambda}{4\alpha}=0~,\label{eqn:Einstein55} \end{equation} which yields the solution \begin{equation} A(y)=k\|y\|~,\quad\mathrm{with}\quad k\equiv\sqrt{-\frac{\Lambda}{4\alpha}}~,\label{eqn:WrapFactorSol} \end{equation} and has to be consistent with the orbifold symmetry (\ref{eqn:Z2Cond}). There are no $\mu y$ components, as the metric and tensors with these components vanish. The $\mu\nu$ components of the Einstein equations are \begin{equation} (A'(y)^{2}-A'' (y))e^{-2A(y)}\eta_{\mu\nu}+\frac{1}{4\alpha}\Lambda e^{-2A(y)}\eta_{\mu\nu}= - \frac{1}{4\alpha}e^{-2A(y)}\eta_{\mu\nu}\left[T_{\mathrm{hid}}\delta(y)+T_{\mathrm{vis}}\delta(y-L)\right]~, \end{equation} which, making use of equation (\ref{eqn:Einstein55}), can be simplified to \begin{equation} A''(y)=\frac{1}{4\alpha}\left[T_{\mathrm{hid}}\delta(y)+T_{\mathrm{vis}}\delta(y-L)\right]~.\label{eqn:Einstein_munu} \end{equation} After computing the second derivative of $A(y)$ from Eq. (\ref{eqn:WrapFactorSol}), and taking into account the periodicity of the metric (\ref{eqn:PeriodicCond}), yields \begin{equation} A''(y)=2k(\delta(y)-\delta(y-L))~, \end{equation} which allows us to identify, from Eq. (\ref{eqn:Einstein_munu}), the values of the tensions \begin{equation} -T_{\mathrm{vis}}=T_{\mathrm{hid}}=8\alpha k=\sqrt{-16\Lambda\alpha}~. \end{equation} The results obtained in this section indicate that the bulk geometry has to be Anti-de Sitter, with a negative bulk cosmological constant, and that the visible brane has negative tension, while the hidden one is positive. These results differ, from standard works on Randall-Sundrum, on the constant coefficient appearing in the expression for $k$, Eq.~(\ref{eqn:WrapFactorSol}) because we are considering a one dimensional ordinary space, so that the computations of the curvature tensor yield different constant factors. \subsection{Hamiltonian eigenstates} \label{append:SteadyState} In order to solve the eigenvalue problem \eqref{eq:Eigenproblem}, it is convenient to perform the change of basis $\xi(q,y)=H\tilde{\phi}(q,y)$, with \begin{equation} H=\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \end{equation} the Hadamard matrix, so that the eigenvalue equation becomes \begin{align} \left[q+\frac{i}{2}\left(p_{y}e^{-A(y)}+e^{-A(y)}p_{y}\right)\right]\xi_{n}^{-}=E_{n}\xi_{n}^{+}~,\label{eqn:Differential1stOrd-Pos}\\ \left[q-\frac{i}{2}\left(p_{y}e^{-A(y)}+e^{-A(y)}p_{y}\right)\right]\xi_{n}^{+}=E_{n}\xi_{n}^{-}~,\label{eqn:Differential1stOrd-Neg} \end{align} where $\xi^{\pm}$ are the components of $\xi=(\xi^{+},\xi^{-})^{T}$. This system of equations can be decoupled, giving \begin{equation} \left[q^{2}+\frac{1}{4}\left(p_{y}e^{-A(y)}+e^{-A(y)}p_{y}\right)^{2}\right]\xi_{n}^{\pm}=E_{n}^{2}\xi_{n}^{\pm}~,\label{eqn:Differential2ndOrd} \end{equation} which is a second order differential equation that can be solved for the appropriate boundary conditions. We solve this equation both in the positive $\left[\xi^{\pm}(0<y<L)\right]_{P}$ and negative domain $\left[\xi^{\pm}(-L<y<0)\right]_{N}$, delivering \begin{align} \left[\xi_{n}^{\pm}(y)\right]_{P}= & Ae^{\frac{ky}{2}}\cos\left(e^{ky}\alpha_{n}\right)+Be^{\frac{ky}{2}}\sin\left(e^{ky}\alpha_{n}\right)~,\label{eqn:GenericSolution}\\ \left[\xi_{n}^{\pm}(y)\right]_{N}= & Ce^{-\frac{ky}{2}}\cos\left(e^{-ky}\alpha_{n}\right)+De^{\frac{ky}{2}}\sin\left(e^{-ky}\alpha_{n}\right)~, \end{align} where we defined \begin{equation} \alpha_{n}=\frac{\sqrt{E_{n}^{2}-q^{2}}}{k}~. \end{equation} These solutions are related by the continuity conditions \begin{equation} \left[\xi_{n}^{\pm}(0)\right]_{P}=\left[\xi_{n}^{\pm}(0)\right]_{N}~,\;\text{and}\;\left[\xi_{n}^{\pm}(L)\right]_{P}=\left[\xi_{n}^{\pm}(-L)\right]_{N}~, \end{equation} where in the last one the periodicity of the wavefunctions, Eq.~(\ref{eqn:BoundaryPeriodic}), has been used, and imply that the solutions are related by $A=C$ and $B=D$. The discontinuity introduced by the delta terms at $y=0$ and $y=\pm L$, coming from $A''(y)$, imposes $B=A\tan\alpha_{n}$, and the following restrictions to the energies \begin{equation} \tan\alpha_{n}=\tan\left(e^{kL}\alpha_{n}\right)~, \end{equation} which yields the spectrum in Eq.~(\ref{eqn:SpectrumSS}). After taking into account these conditions, the eigenstates become \begin{equation} \xi_{n}^{\pm}(y)=Ae^{\frac{k\|y\|}{2}}\left[\cos\left(e^{k\|y\|}\alpha_{n}\right)+\tan\alpha_{n}\sin\left(e^{k\|y\|}\alpha_{n}\right)\right]~. \end{equation} However, these solutions come from the second order differential equation (\ref{eqn:Differential2ndOrd}), whereas the original equations were first order, and relate $\xi^{+}(y)$ to $\xi^{-}(y)$. To find the appropriate solution of the eigenfunctions, we need to take into account these relations. Since any lineal combination of solutions is also a solution of the equations, we consider the solution $\left[\xi_{n}^{+}(y)\right]_{2}$ of Eq.~(\ref{eqn:Differential2ndOrd}) to obtain $\left[\xi_{n}^{-}(y)\right]_{1}$ from Eq.~(\ref{eqn:Differential1stOrd-Neg}), where $[~\cdot~]_{i}$ denotes whether the solution comes from a first ($i=1$) or second ($i=2$) order differential equation. Similarly, from $\left[\xi_{n}^{-}(y)\right]_{2}$ we obtain $\left[\xi_{n}^{+}(y)\right]_{1}$, so that \begin{equation} \left[\xi_{n}^{\pm}(y)\right]_{1}=\bigg[\sin\left(\alpha_{n}e^{k\|y\|}\right)\left(\frac{q}{E}\tan\alpha_{n}\pm\frac{k\alpha_{n}}{E}\mathrm{sign}(y)\right)+ \cos\left(\alpha_{n}e^{k\|y\|}\right)\left(\frac{q}{E}\mp\frac{k\alpha_{n}}{E}\tan\alpha_{n}\mathrm{sign}(y)\right)\bigg]Ae^{\frac{k\|y\|}{2}}~. \end{equation} The general solution for the eigenstates is a lineal combination of this pair of solutions \begin{align} \xi_{n}^{+}(y)=K_{1}\left[\xi_{n}^{+}(y)\right]_{2}+K_{2}\left[\xi_{n}^{+}(y)\right]_{1}~,\\ \xi_{n}^{-}(y)=K_{1}\left[\xi_{n}^{-}(y)\right]_{1}+K_{2}\left[\xi_{n}^{-}(y)\right]_{2}~, \end{align} where the relation between the constants $K_{1}$ and $K_{2}$ is set by Eq.~(\ref{eqn:BoundaryZ2}), which, depending on the possible values of $\eta$, implies the restrictions \begin{align} \eta=+1 & \implies K_{1}=K_{2}~,\\ \eta=-1 & \implies K_{1}=-K_{2}~. \end{align} Finally, undoing the change of basis, we recover the original eigenstate components of Eqs.~(\ref{eqn:EigenStateUpEta+},\ref{eqn:EigenStateDownEta+}) for $\eta=+1$, while \begin{align} \phi_{n}^{\uparrow}(y)&=\sqrt{\frac{2k}{e^{kL}-1}} \frac{k\alpha_{n}}{\sqrt{(E_{n}+q)^{2}+(k\alpha_{n})^{2}}} e^{\frac{k\|y\|}{2}}\sin\left[\alpha_{n}\left(1-e^{k\|y\|}\right)\right]\mathrm{sign}(y)~,\label{eqn:EigenStateUpEta-}\\ \phi_{n}^{\downarrow}(y)&=\sqrt{\frac{2k}{e^{kL}-1}} \frac{E_{n}+q}{\sqrt{(E_{n}+q)^{2}+(k\alpha_{n})^{2}}} e^{\frac{k\|y\|}{2}}\cos\left[\alpha_{n}\left(1-e^{k\|y\|}\right)\right]~,\label{eqn:EigenStateDownEta-} \end{align} are obtained for $\eta=-1$, and where the remaining constant was set by the normalization of the wavefunction \begin{equation} \int_{0}^{L}dy\tilde{\phi}_{n}(q,y)^{\dagger}\tilde{\phi}_{n}(q,y)=1~.\label{eqn:SS_Norm_Cont} \end{equation} The solution for the particular case of $n=0$ has only a lower component, and is given by \begin{equation} \phi_{0}^{\downarrow}(y)=\sqrt{\frac{k}{e^{kL}-1}}e^{\frac{k\|y\|}{2}}\mathrm{sign}(E_{n}+q)~. \end{equation} \subsection{QW explicit time step} \label{append:QW} Making use of the equations that define the QW, Eqs. (\ref{eqn:QWevolution}, \ref{eq:Uoperator}, \ref{eq:Thetarotation}) and (\ref{eq:RRotation}), one can recast the evolution of $\|\chi_{j}\rangle$ as a recurrence relation relating the spinor components Eq. (\ref{eq:Xicomponents}) at two consecutive time steps. We arrive at \begin{align} \chi_{j+1,r,s}^{\uparrow}= & -\frac{i}{2}e^{i\theta(y)}\left[s\left(y+\frac{\epsilon}{2}\right)+s\left(y-\frac{\epsilon}{2}\right)\right]\chi_{j,r+1,s}^{\uparrow} -\frac{1}{2}\left[s\left(y+\frac{\epsilon}{2}\right)-s\left(y-\frac{\epsilon}{2}\right)\right]\chi_{j,r-1,s}^{\downarrow}\nonumber \\ & +\frac{1}{2}f(y)f(y+\epsilon)c\left(y+\frac{\epsilon}{2}\right)\chi_{j,r+1,s+1}^{\uparrow} +\frac{1}{2}f(y)f(y-\epsilon)c\left(y-\frac{\epsilon}{2}\right)\chi_{j,r+1,s-1}^{\uparrow}\nonumber \\ & +\frac{i}{2}f(y)f^{}(y+\epsilon)c\left(y+\frac{\epsilon}{2}\right)\chi_{j,r-1,s+1}^{\downarrow} -\frac{i}{2}f(y)f^{}(y-\epsilon)c\left(y-\frac{\epsilon}{2}\right)\chi_{j,r-1,s-1}^{\downarrow}~,\label{eq:mapup} \end{align} for the upper component, where we recall that $y=\epsilon s$, and we defined $e^{\pm i\theta(y)}=c(y)\pm is(y)$. For the lower component one finds \begin{align} \chi_{j+1,r,s}^{\downarrow}= & \frac{i}{2}e^{-i\theta(y)}\left[s\left(y+\frac{\epsilon}{2}\right)+s\left(y-\frac{\epsilon}{2}\right)\right]\chi_{j,r-1,s}^{\downarrow} -\frac{1}{2}\left[s\left(y+\frac{\epsilon}{2}\right)-s\left(y-\frac{\epsilon}{2}\right)\right]\chi_{j,r+1,s}^{\uparrow}\nonumber \\ & +\frac{1}{2}f^{}(y)f^{}(y+\epsilon)c\left(y+\frac{\epsilon}{2}\right)\chi_{j,r-1,s+1}^{\downarrow} +\frac{1}{2}f^{}(y)f^{}(y-\epsilon)c\left(y-\frac{\epsilon}{2}\right)\chi_{j,r-1,s-1}^{\downarrow}\nonumber \\ & -\frac{i}{2}f^{}(y)f(y+\epsilon)c\left(y+\frac{\epsilon}{2}\right)\chi_{j,r+1,s+1}^{\uparrow} +\frac{i}{2}f^{}(y)f(y-\epsilon)c\left(y-\frac{\epsilon}{2}\right)\chi_{j,r+1,s-1}^{\uparrow}~.\label{eq:mapdown} \end{align} We notice that the upper components are displaced in one direction along the $x$ dimension, while the lower components are displaced in the opposite direction. \subsection{Mode decomposition of the freely propagating distribution} \label{append:ModeDecomposition} The stationary states found above form an orthonormal basis, in the continuum limit, that allow for a decomposition of any function along the $y$ coordinate, for a given value of $q$. They can also be used, after a proprer discretization, in the lattice on which the QW is defined. Following this idea, we introduced the decomposition in Eq.~\eqref{eqn:ModeDecomposition}, which is a function in the space of $q$, the lattice quasimomentum along the $x$ coordinate. For this quasimomentum space, the spinor components are related to Eq. (\ref{eq:Xicomponents}) via a discrete Fourier transform \begin{equation} \tilde{\chi}_{j,s}(q)=\sum_{r}e^{-iq\epsilon r}\chi_{j,r,s}~. \end{equation} Making use of \begin{equation} \sum_{r}e^{ix(q-q')}=\frac{2\pi}{\epsilon}\delta(q-q')~, \end{equation} and the orthonormality condition (\ref{eqn:SS_Norm_Cont}) on the grid \begin{equation} \epsilon\sum_{s}\tilde{\phi}_{n}(q,\epsilon s)^{\dagger}\tilde{\phi}_{m}(q,\epsilon s)=\delta_{n,m}~, \end{equation} the coefficients can be obtained as \begin{equation} \beta_{n}(q,t)=\epsilon^{2}\sum_{s}\tilde{\phi}_{n}(q,\epsilon s)\tilde{\chi}_{j,s}(q)~. \end{equation} The coefficients of the freely propagating distribution with $x=t$ are \begin{equation} \beta_{n}(q,t)=\epsilon^{2}\sum_{s}\tilde{\phi}_{n}(q,\epsilon s)e^{-iqt}\chi_{j,j,s}~, \end{equation} while, for $x=-t$, they read as \begin{equation} \beta_{n}(q,t)=\epsilon^{2}\sum_{s}\tilde{\phi}_{n}(q,\epsilon s)e^{iqt}\chi_{j,-j,s}~. \end{equation} From the normalization condition of the spinor on the grid \begin{equation} \epsilon^{2}\sum_{r,s}\chi_{j,r,s}^{\dagger}\chi_{j,r,s}=1~, \end{equation} and making use of the definition \eqref{eqn:ModeDecomposition}, it can be shown that the mode coefficients satisfy \begin{equation} \sum_{n}\int_{-\pi/\epsilon}^{\pi/\epsilon}\frac{dq}{2\pi}\left\|\beta_{n}(q,t)\right\|^{2}=1~, \end{equation} which can be expressed in terms of the integrated coefficients (\ref{eqn:IntegratedModeCoeff}) as \begin{equation} \sum_{n}B_{n}(t)=1~. \end{equation} \subsection*{High $kL$ limit of the QW time step and limiting entropy} \label{append:QWlimitK} In the limit of a high warp factor $kL$, the exponential $e^{-A(L)}$ becomes very small, so that the QW discrete time recursive evolution Eqs. (\ref{eq:mapup},\ref{eq:mapdown}) can be expanded up to the lowest order in this factor, giving \begin{align} \chi_{j+1,r,s}^{\uparrow}= & \chi_{j,r+1,s}^{\uparrow}~,\nonumber \\ \chi_{j+1,r,s}^{\downarrow}= & \chi_{j,r-1,s}^{\downarrow} \label{eqn:QWLimit}~. \end{align} Although this expansion is only valid for values of $y$ close to $L$, it is still accurate enough for the initial condition located at $y=L/2$. As discussed in the main text, the asymptotic value of the entanglement entropy decreases as $kL$ is increased. Therefore, the minimum value of the entropy is reached in the limit $e^{-A(L)}\approx0$. The initial condition $\chi_{0,r,s} = \delta_{r,0}\delta_{s,s_0}C_0$ can be iterated with the help of Eqs.~\eqref{eqn:QWLimit} to produce the explicit time evolution \begin{align} \chi_{j,r,s}^{\uparrow}= & C_{0}^{\uparrow}\delta_{r,j}\delta_{s,s_0}~,\nonumber \\ \chi_{j,r,s}^{\downarrow}= & C_{0}^{\downarrow}\delta_{r,-j}\delta_{s,s_0}~. \end{align} The corresponding reduced density matrix becomes time-independent and diagonal: \begin{equation} \rho_{c}(t)=diag(\|C_{0}^{\uparrow}\|^{2},\|C_{0}^{\downarrow}\|^{2})\label{eq:SLlarge} \end{equation} from which the minimum value of the entropy can finally be obtained: \begin{equation} S_\mathrm{min}=-\|C_{0}^{\uparrow}\|^{2} \log_2 \|C_{0}^{\uparrow}\|^{2} - \|C_{0}^{\downarrow}\|^{2} \log_2 \|C_{0}^{\downarrow}\|^{2}~. \end{equation} \end{document}	arXiv
JEE Main 2021 (Online) 16th March Morning Shift Numerical +4 -1 Let f : R $$ \to $$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$$\in$$R. If $${I_1} = \int\limits_0^8 {f(x)dx} $$ and $${I_2} = \int\limits_{ - 1}^3 {f(x)dx} $$, then the value of I1 + 2I2 is equal to ____________. Your input ____ If the normal to the curve y(x) = $$\int\limits_0^x {(2{t^2} - 15t + 10)dt} $$ at a point (a, b) is parallel to the line x + 3y = $$-$$5, a > 1, then the value of \| a + 6b \| is equal to ___________. JEE Main 2021 (Online) 26th February Evening Shift Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, $$-$$3) and (4, $$-$$2$$\sqrt 2 $$), and given that a $$-$$ 2$$\sqrt 2 $$ b = 3, then (a2 + b2 + ab) is equal to __________. If $${I_{m,n}} = \int\limits_0^1 {{x^{m - 1}}{{(1 - x)}^{n - 1}}dx} $$, for m, $$n \ge 1$$, and $$\int\limits_0^1 {{{{x^{m - 1}} + {x^{n - 1}}} \over {{{(1 + x)}^{m + 1}}}}} dx = \alpha {I_{m,n}}\alpha \in R$$, then $$\alpha$$ equals ___________. Questions Asked from Definite Integrals and Applications of Integrals (Numerical) JEE Main 2022 (Online) 28th July Evening Shift (1) JEE Main 2022 (Online) 28th July Morning Shift (1) JEE Main 2022 (Online) 27th July Evening Shift (2) JEE Main 2022 (Online) 26th July Morning Shift (2) JEE Main 2022 (Online) 25th July Evening Shift (2) JEE Main 2022 (Online) 25th July Morning Shift (1) JEE Main 2022 (Online) 30th June Morning Shift (2) JEE Main 2022 (Online) 29th June Evening Shift (1) JEE Main 2022 (Online) 27th June Evening Shift (1) JEE Main 2022 (Online) 27th June Morning Shift (1) JEE Main 2022 (Online) 26th June Evening Shift (1) JEE Main 2022 (Online) 26th June Morning Shift (2) JEE Main 2022 (Online) 25th June Evening Shift (1) JEE Main 2022 (Online) 24th June Evening Shift (1) JEE Main 2022 (Online) 24th June Morning Shift (3) JEE Main 2021 (Online) 31st August Evening Shift (1) JEE Main 2021 (Online) 31st August Morning Shift (2) JEE Main 2021 (Online) 26th August Evening Shift (1) JEE Main 2021 (Online) 26th August Morning Shift (1) JEE Main 2021 (Online) 27th July Evening Shift (1) JEE Main 2021 (Online) 27th July Morning Shift (1) JEE Main 2021 (Online) 25th July Evening Shift (1) JEE Main 2021 (Online) 22th July Evening Shift (1) JEE Main 2021 (Online) 20th July Morning Shift (1) JEE Main 2021 (Online) 18th March Evening Shift (1) JEE Main 2021 (Online) 18th March Morning Shift (1) JEE Main 2021 (Online) 17th March Evening Shift (2) JEE Main 2021 (Online) 17th March Morning Shift (1) JEE Main 2021 (Online) 16th March Morning Shift (3) JEE Main 2021 (Online) 26th February Evening Shift (2) JEE Main 2021 (Online) 26th February Morning Shift (2) JEE Main 2021 (Online) 25th February Evening Shift (1) JEE Main 2021 (Online) 25th February Morning Shift (1) JEE Main 2021 (Online) 24th February Morning Shift (1) JEE Main 2020 (Online) 4th September Evening Slot (1) JEE Main 2020 (Online) 2nd September Evening Slot (1) JEE Main 2020 (Online) 2nd September Morning Slot (1) JEE Main 2020 (Online) 8th January Evening Slot (1)	CommonCrawl
Abel's theorem In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. This article is about Abel's theorem on power series. For Abel's theorem on algebraic curves, see Abel–Jacobi map. For Abel's theorem on the insolubility of the quintic equation, see Abel–Ruffini theorem. For Abel's theorem on linear differential equations, see Abel's identity. For Abel's theorem on irreducible polynomials, see Abel's irreducibility theorem. For Abel's formula for summation of a series, using an integral, see Abel's summation formula. Theorem Let the Taylor series $G(x)=\sum _{k=0}^{\infty }a_{k}x^{k}$ be a power series with real coefficients $a_{k}$ with radius of convergence $1.$ Suppose that the series $\sum _{k=0}^{\infty }a_{k}$ converges. Then $G(x)$ is continuous from the left at $x=1,$ that is, $\lim _{x\to 1^{-}}G(x)=\sum _{k=0}^{\infty }a_{k}.$ The same theorem holds for complex power series $G(z)=\sum _{k=0}^{\infty }a_{k}z^{k},$ provided that $z\to 1$ entirely within a single Stolz sector, that is, a region of the open unit disk where $\|1-z\|\leq M(1-\|z\|)$ for some fixed finite $M>1$. Without this restriction, the limit may fail to exist: for example, the power series $\sum _{n>0}{\frac {z^{3^{n}}-z^{2\cdot 3^{n}}}{n}}$ converges to $0$ at $z=1,$ but is unbounded near any point of the form $e^{\pi i/3^{n}},$ so the value at $z=1$ is not the limit as $z$ tends to 1 in the whole open disk. Note that $G(z)$ is continuous on the real closed interval $[0,t]$ for $t<1,$ by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that $G(z)$ is continuous on $[0,1].$ Stolz sector The Stolz sector $\|1-z\|\leq M(1-\|z\|)$ has explicit formula $y^{2}=-{\frac {M^{4}(x^{2}-1)-2M^{2}((x-1)x+1)+2{\sqrt {M^{4}(-2M^{2}(x-1)+2x-1)}}+(x-1)^{2}}{(M^{2}-1)^{2}}}$ and is plotted on the right for various values. The left end of the sector is $x={\frac {1-M}{1+M}}$, and the right end is $x=1$. On the right end, it becomes a cone with angle $2\theta $, where $\cos \theta ={\frac {1}{M}}$. Remarks As an immediate consequence of this theorem, if $z$ is any nonzero complex number for which the series $\sum _{k=0}^{\infty }a_{k}z^{k}$ converges, then it follows that $\lim _{t\to 1^{-}}G(tz)=\sum _{k=0}^{\infty }a_{k}z^{k}$ in which the limit is taken from below. The theorem can also be generalized to account for sums which diverge to infinity. If $\sum _{k=0}^{\infty }a_{k}=\infty $ then $\lim _{z\to 1^{-}}G(z)\to \infty .$ However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for ${\frac {1}{1+z}}.$ At $z=1$ the series is equal to $1-1+1-1+\cdots ,$ but ${\tfrac {1}{1+1}}={\tfrac {1}{2}}.$ We also remark the theorem holds for radii of convergence other than $R=1$: let $G(x)=\sum _{k=0}^{\infty }a_{k}x^{k}$ be a power series with radius of convergence $R,$ and suppose the series converges at $x=R.$ Then $G(x)$ is continuous from the left at $x=R,$ that is, $\lim _{x\to R^{-}}G(x)=G(R).$ Applications The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, $z$) approaches $1$ from below, even in cases where the radius of convergence, $R,$ of the power series is equal to $1$ and we cannot be sure whether the limit should be finite or not. See for example, the binomial series. Abel's theorem allows us to evaluate many series in closed form. For example, when $a_{k}={\frac {(-1)^{k}}{k+1}},$ we obtain $G_{a}(z)={\frac {\ln(1+z)}{z}},\qquad 0<z<1,$ by integrating the uniformly convergent geometric power series term by term on $[-z,0]$; thus the series $\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}$ converges to $\ln(2)$ by Abel's theorem. Similarly, $\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}$ converges to $\arctan(1)={\tfrac {\pi }{4}}.$ $G_{a}(z)$ is called the generating function of the sequence $a.$ Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative sequences, such as probability-generating functions. In particular, it is useful in the theory of Galton–Watson processes. Outline of proof After subtracting a constant from $a_{0},$ we may assume that $\sum _{k=0}^{\infty }a_{k}=0.$ Let $s_{n}=\sum _{k=0}^{n}a_{k}\!.$ Then substituting $a_{k}=s_{k}-s_{k-1}$ and performing a simple manipulation of the series (summation by parts) results in $G_{a}(z)=(1-z)\sum _{k=0}^{\infty }s_{k}z^{k}.$ Given $\varepsilon >0,$ pick $n$ large enough so that $\|s_{k}\|<\varepsilon $ for all $k\geq n$ and note that $\left\|(1-z)\sum _{k=n}^{\infty }s_{k}z^{k}\right\|\leq \varepsilon \|1-z\|\sum _{k=n}^{\infty }\|z\|^{k}=\varepsilon \|1-z\|{\frac {\|z\|^{n}}{1-\|z\|}}<\varepsilon M$ when $z$ lies within the given Stolz angle. Whenever $z$ is sufficiently close to $1$ we have $\left\|(1-z)\sum _{k=0}^{n-1}s_{k}z^{k}\right\|<\varepsilon ,$ so that $\left\|G_{a}(z)\right\|<(M+1)\varepsilon $ when $z$ is both sufficiently close to $1$ and within the Stolz angle. Related concepts Converses to a theorem like Abel's are called Tauberian theorems: There is no exact converse, but results conditional on some hypothesis. The field of divergent series, and their summation methods, contains many theorems of abelian type and of tauberian type. See also • Abel's summation formula – Integration by parts version of Abel's method for summation by parts • Nachbin resummation – Theorem bounding the growth rate of analytic functionsPages displaying short descriptions of redirect targets • Summation by parts – Theorem to simplify sums of products of sequences Further reading • Ahlfors, Lars Valerian (September 1, 1980). Complex Analysis (Third ed.). McGraw Hill Higher Education. pp. 41–42. ISBN 0-07-085008-9. - Ahlfors called it Abel's limit theorem. External links • Abel summability at PlanetMath. (a more general look at Abelian theorems of this type) • A.A. Zakharov (2001) [1994], "Abel summation method", Encyclopedia of Mathematics, EMS Press • Weisstein, Eric W. "Abel's Convergence Theorem". MathWorld.	Wikipedia
\begin{document} \title{Trimming of metric spaces and the tight span} \author[Vladimir Turaev]{Vladimir Turaev} \address{ Department of Mathematics, \newline \indent Indiana University \newline \indent Bloomington IN47405 \newline \indent USA \newline \indent e-mail: [email protected]} \subjclass[2010]{54E35} \begin{abstract} We use the trimming transformations to study the tight span of a metric space. \end{abstract} \maketitle \section{Introduction}\label{intro} The theory of tight spans due to J.~Isbell \cite{Is} and A.~Dress \cite{Dr} embeds any metric space~$X$ in a hyperconvex metric space $T(X)$ called the tight span of~$X$. In this paper we split $T(X)$ as a union of two metric subspaces $\tau= \tau(X) $ and $\overline {C}=\overline {C(X)} $. The space~$\tau $ is the tight span of a certain quotient $\overline{X_\infty}$ of~$X$. The space~$\overline {C } $ is a disjoint union of metric trees which either do not meet~$\tau $ or meet~$\tau $ at their roots lying in $\overline{X_\infty}\subset \tau$. In this picture, the original metric space $X\subset T(X)$ consists of the tips of the branches of the trees. The construction of $\tau $ and~$\overline {C } $ uses the trimming transformations of metric spaces studied in \cite{Tu} for finite metric spaces. In the present paper - essentially independent of~\cite{Tu} - we discuss trimming for all metric spaces and introduce related objects including the subspaces $ \tau $ and~$ \overline C$ of $T(X)$. Our main theorem says that $\tau \cup \overline C=T(X)$ and $\tau \cap{\overline C }$ is the set of the roots of the trees forming~$\overline C$. This work was partially supported by the NSF grant DMS-1664358. \section{Trim pseudometric spaces}\label{sectionBIS0} \subsection{Pseudometrics}\label{PreliminariesBIS01} We recall basics on metric and pseudometric spaces. Set $\mathbb{R}_+=\{r\in \mathbb{R} \, \vert \, r\geq 0\}$. A \emph{pseudometric space} is a pair consisting of a set~$X$ and a mapping $d: X\times X \to \mathbb{R}_+ $ (the \emph{pseudometric}) such that for all $x,y,z\in X$, $$d(x,x)=0, \quad d(x,y)=d(y,x) , \quad d(x,y)+ d(y,z)\geq d(x,z).$$ A pseudometric space $(X,d)$ is a \emph{metric space} (and~$d$ is a \emph{metric}) if $d(x,y) >0$ for all distinct $x, y\in X $. A map $f:X\to X'$ between pseudometric spaces $(X,d)$ and $(X',d')$ is \emph{distance preserving} if $d(x,y) = d'(f(x), f(y))$ for all $x,y \in X$ and is \emph{non-expansive} if $d(x,y) \geq d'(f(x), f(y))$ for all $x,y \in X$. Pseudometric spaces $X,X'$ are \emph{isometric} if there is a distance preserving bijection $X \to X'$. We call distance preserving maps between metric spaces \emph{metric embeddings}; they are always injective. Each pseudometric space $(X,d)$ carries an equivalence relation $\sim_d$ defined by $x\sim_d y$ if $d(x,y)=0$ for $x,y\in X$. The quotient set $ \overline X = X/{\sim_d }$ carries a metric~$\overline d$ defined by $ {\overline d}(\overline x, \overline y)=d(x,y)$ where $x,y$ are any points of~$X$ and $\overline x, \overline y \in \overline X$ are their equivalence classes. The metric space $(\overline X, \overline d)$ is the \emph{metric quotient} of $(X,d)$. Any distance preserving map from $X$ to a metric space $Y$ expands uniquely as the composition of the projection $X\to \overline X$ and a metric embedding $\overline X\hookrightarrow Y$. \subsection{Trim spaces}\label{Preliminaries} Given a set~$X$ and a map $d:X\times X\to \mathbb{R}$, we use the same symbol~$d$ for the map $X\times X\times X \to \mathbb{R} $ carrying any triple $x,y,z \in X$ to $$d(x,y,z)= \frac{ d(x, y)+ d(x, z)-d(y,z)}{2} . $$ The right-hand side is called the Gromov product, see, for instance, \cite{BCK}. A pseudometric~$d$ in a set~$X$ determines a function $\underline d: X \to \mathbb{R}_+ $ as follows: if $\operatorname{card} (X)=1$, then $\underline d =0$; if $X$ has two points $x,y$, then $\underline d(x)=\underline d(y)= d(x,y)/2$; if $\operatorname{card} (X)\geq 3$, then for all $x \in X$, $$ {\underline d} (x)= \inf_{y,z \in X\setminus \{x\}, y \neq z } d(x, y,z) . $$ If $\underline d(x)=0$ for all $x\in X$, then we say that the pseudometric space $(X,d)$ is \emph{trim}. Following K.~Menger \cite{Me}, we say that a point~$x$ of a pseudometric space $(X,d)$ \emph{lies between} $y\in X$ and $z\in X$ if $d(y,z)=d(x,y)+d(x,z)$. A simple sufficient condition for $(X,d)$ to be trim says that each point $x\in X$ lies between two distinct points of $ X\setminus \{x\}$. \subsection{Examples} A pseudometric space having only one point is trim. A pseudometric space having two points is trim iff the distance between these points is equal to zero. More generally, any set with zero pseudometric is trim. We give two examples of trim metric spaces: (a) the set of words of a fixed finite length in a given finite alphabet with the Hamming distance between words defined as the number of positions at which the letters of the words differ; (b) a subset of a Euclidean circle $C\subset \mathbb{R}^2 $ meeting each half-circle in $C$ in at least three points; here the distance between two points is the length of the shorter arc in $C$ connecting these points. \section{Drift and trimming} \subsection{The drift}\label{Drift} Given a function $\delta: X\to \mathbb{R}$ on a metric space $(X,d)$, we define a map $d_\delta:X\times X\to \mathbb{R}$ as follows: for $x,y \in X$, \begin{equation} d_\delta(x,y)=\left\{ \begin{array}{ll} 0 \quad {\text {if}} \,\, x=y , \\ d(x,y)-\delta(x)- \delta(y) \quad {\text {if}} \,\, x \neq y . \end{array} \right. \end{equation} We say that $d_\delta$ is obtained from~$d$ by a \emph{drift}. The idea behind the definition of $d_\delta$ is that each point $x\in X$ is pulled towards all other points of~$X$ by $\delta(x)$. \begin{lemma}\label{le49} If $\delta \leq {\underline d}$ (i.e., if $\delta(x)\leq {\underline d}(x)$ for all $x\in X$), then $ {d_\delta} $ is a pseudometric in~$X$. Moreover, if $\operatorname{card}(X) \geq 3$ or $\operatorname{card}(X) =2, \delta=const$, then $\underline{d_\delta}= \underline d-\delta$. \end{lemma} \begin{proof} We first prove that for any distinct $x,y \in X$, \begin{equation}\label{eqsimple} {\underline d} (x) + {\underline d} (y) \leq d(x,y). \end{equation} If $X=\{x,y\}$, then \eqref{eqsimple} follows from the definition of~$\underline d$. If $\operatorname{card}(X)\geq 3$, pick any $z\in X\setminus \{x,y\}$. Then ${\underline d} (x) \leq { d(x, y ,z)} $ and ${\underline d} (y) \leq d(y,x ,z) $. So, $${\underline d} (x) + {\underline d} (y) \leq d(x,y,z) + d(y,x,z)=d(x,y).$$ We now check that ${d_\delta}:X \times X\to \mathbb{R}$ is a pseudometric. Clearly, ${d_\delta}$ is symmetric and, by definition, ${d_\delta}(x,x)=0$ for all $x\in X$. Formula \eqref{eqsimple} and the assumption $\delta \leq {\underline d}$ imply that for any distinct $x,y\in X$, $${d_\delta}(x,y)=d(x,y)-\delta(x)- \delta(y)\geq {\underline d} (x) -\delta(x)+ {\underline d} (y) - \delta(y) \geq 0.$$ To prove the triangle inequality for $d_\delta$ we rewrite it as $d_\delta(x,y,z)\geq 0$ for any $x,y,z\in X$. If $x=y$ or $x=z$, then $d_\delta(x,y,z) =0$; if $y=z$, then $d_\delta(x,y,z)=d_\delta(x,y) \geq 0$. Finally, if $x,y,z$ are pairwise distinct, then \begin{equation}\label{ert} d_\delta(x,y,z) = d(x,y,z) - \delta(x) \geq {\underline d} (x) - \delta(x) \geq 0 . \end{equation} The equality $\underline{d_\delta} (x)= \underline d (x) -\delta (x)$ for all $x\in X$ follows from \eqref{ert} if $\operatorname{card}(X)\geq 3$ and from the definitions if $\operatorname{card}(X)=2$ and $\delta=const$. \end{proof} \subsection{Trimming}\label{Trimming} Let $(X,d)$ be a metric space. Applying Lemma~\ref{le49} to $\delta= \underline d:X\to \mathbb{R}$, we obtain a pseudometric $d^{-}=d_\delta$ in~$X$. For $x,y \in X$, \begin{equation}\label{dbulletdef} d^{-}(x,y)=\left\{ \begin{array}{ll} 0 \quad {\text {if}} \,\, x=y , \\ d(x,y)- \underline d(x)- \underline d(y) \quad {\text {if}} \,\, x \neq y. \end{array} \right. \end{equation} Lemma~\ref{le49} implies that $\underline{d^{-}}=0$, i.e., that $(X, d^{-})$ is a trim pseudometric space. Let $t(X)$ be the metric quotient of $(X, d^{-})$. By definition, $t(X)=X/{\sim_{d^{-}}}$ and the metric in $t(X)$ is induced by the pseudometric $d^{-}$ in~$X$. We say that $t(X)$ is obtained from $(X,d)$ by \emph{trimming} and call the projection $p_X:X \to t(X) $ the \emph{trimming projection}. Note that $p_X$ is a non-expansive surjection of metric spaces. It is bijective if and only if ${d^{-}}$ is a metric in~$X$ and then $t(X) =(X, d^{-}) $ is a trim metric space. In general, $t(X)$ may be non-trim. Starting from $ (X_0,d_0)=(X,d)$ and iterating the trimming, we obtain metric spaces $\{ t^k(X)=(X_k, d_k)\}_{k\geq 1}$ and non-expansive surjections \begin{equation}\label{line} X =X_0 \stackrel{p_0}{\longrightarrow} X_1 \stackrel{p_1}{\longrightarrow} X_2 \stackrel{p_2}{\longrightarrow} \cdots \end{equation} where $p_0=p_X$ and $p_k=p_{t^k(X)}: X_k \to X_{k+1} $ for $k\geq 1$. We call the sequence \eqref{line} the \emph{trimming sequence} of~$X$. Each point $x \in X $ determines points $(x_{(k)} \in X_k)_{k\geq 0}$ by $x_{(0)}=x$ and $x_{(k+1)}=p_k(x_{(k)})$ for all~$k$. The sequence $(x_{(k)} )_{k\geq 0}$ is the \emph{trimming sequence} of~$x$. For $x,y \in X$, we write $x \cong y$ if there is an integer $k\geq 0$ such that $x_{(k)}=y_{(k)}$. We call the smallest such~$k$ the \emph{meeting index} of $x, y$ and denote it by $m(x,y)$. Note that $x_{(k)} =y_{(k)} $ for all $k \geq m(x,y)$ which easily implies that $\cong$ is an equivalence relation in~$X$. Clearly, $m(x,x)=0$ and $m(x,y) \geq 1$ for $x\neq y$. Set $X_\infty=X/\cong$ and for $x\in X$, denote its projection to $X_\infty$ by $x_{(\infty)}$. We have $$d(x,y) \geq d_1(x_{(1)} , y_{(1)} ) \geq d_2(x_{(2)} , y_{(2)} ) \geq \cdots \geq 0 $$ for any $x,y \in X$. As a consequence, the formula $$d_\infty (x_{(\infty)},y_{(\infty)})= \lim_{k\to \infty} d_k(x_{(k)}, y_{(k)}) $$ defines a pseudometric $d_\infty$ in $X_\infty$. Clearly, the projection $ X\to X_\infty, x\mapsto x_{(\infty)}$ is a non-expansive surjection. \subsection{Remark} One can take the metric quotient $\overline {X_\infty}$ of $X_\infty$ and then iterate the construction $X\mapsto \overline {X_\infty}$. We will not do it in this paper. \section{Leaf spaces of metric trees}\label{examplesfmt} We compute the trimming for the leaf spaces of metric trees. \subsection{Metric graphs and trees}\label{Trees} A \emph{metric graph} is a graph without loops whose every edge~$e$ is endowed with a real number $\ell_e > 0$, the \emph{length}, and with a homeomorphism $e \approx [0, \ell_e] \subset \mathbb{R}_+ $. We use this homeomorphism to define the length of any subsegment of~$e$: this is just the length of the corresponding subsegment of $[0, \ell_e] $. A wider class of \emph{pseudometric graphs} is defined similarly but allowing the lengths of edges to be equal to zero. Only the edges of positive length carry a homeomorphism $ e \approx [0, \ell_e] $. All subsegments of edges of zero length have zero length. A \emph{pseudometric} (respectively, \emph{metric}) tree is a pseudometric (respectively, metric) graph which is path connected and has no cycles. The underlying set of a pseudometric tree~$L$ carries a \emph{path pseudometric} $d_L$: the distance $d_L(x,y) $ between any points $x,y \in L$ is the sum of the lengths of the edges and subedges forming an injective path from~$x$ to~$y$ in~$L$. The pseudometric $d_L$ is a metric if and only if~$L$ is a metric tree. Every pseudometric tree~$L$ yields a metric tree~$\overline L$ by contracting each edge of~$L$ of zero length to a point while keeping the homeomorphisms associated with the edges of positive length. The underlying metric space $({\overline L}, d_{\overline L})$ of~$\overline L$ is nothing but the metric quotient of $(L,d_L)$. \subsection{The leaf space}\label{Leaf} A \emph{leaf} of a pseudometric tree~$L$ is a vertex of~$L$ adjacent to a single edge of~$L$. The \emph{leaf space} $ {\partial L}$ is the set of all leaves of~$L$ together with the pseudometric~$d$ obtained by restricting $d_L$ to this set. The pseudometric space $(\partial L,d)$ depends only on the tree~$L$ and the lengths of its edges; it does not depend on the choice of homeomorphisms $ e \approx [0, \ell_e] $. The following lemma estimates (and in some cases computes) the function $\underline d:\partial L \to \mathbb{R}_+$. \begin{lemma}\label{cloleththree--} Let $(\partial L,d)$ be the leaf space of a pseudometric tree~$L$ such that $\operatorname{card} (\partial L) \geq 3$. Then for all $x\in \partial L$, we have ${\underline d}(x) \geq l_{e(x)} $ where $e(x) $ is the edge of~$L$ adjacent to~$x$. If the second vertex of $e(x)$ is adjacent to at least two leaves of $L$ besides~$x$, then ${\underline d}(x) = l_{e(x)} $. \end{lemma} \begin{proof} Let $v$ be the vertex of the edge $e(x)$ distinct from~$x$. Pick any $y,z \in \partial L \setminus \{x\}$. It is clear that an injective path from $x$ to~$y$ passes through~$v$. Therefore $ d_L (x,y)=l_{e(x)} + d_L (v,y) $ and, similarly, $d_L (x,z)=l_{e(x)} + d_L (v,z) $. Then $$ d( x,y,z)= \frac{ d_L(x,y)+ d_L(x,z)-d_L(y,z)}{2} $$ $$=l_{e(x)} +\frac{ d_L(v,y)+ d_L(v,z)-d_L(y,z)}{2} \geq l_{e(x)} .$$ Since this holds for all $y,z $ as above, ${\underline d}(x) \geq l_{e(x)} $. Suppose now that~$v$ is adjacent to distinct leaves $y,z \in \partial L \setminus \{x\}$. Then the edges $e(x)$, $e(y)$ form an injective path from $x$ to~$y$ and so $d_L(x,y)= l_{e(x)} +l_{e(y)}$. Similarly, $d_L(x,z)= l_{e(x)} +l_{e(x)}$ and $d_L(y,z)= l_{e(y)} +l_{e(z)}$. Then $ d( x,y,z)= l_{e(x)}$. Therefore ${\underline d}(x) = l_{e(x)} $. \end{proof} \subsection{Example} Let~$L$ be a metric tree whose every vertex adjacent to a leaf is adjacent to at least three leaves. Lemma~\ref{cloleththree--} implies that $t(\partial L)$ is the set of all vertices of~$L$ adjacent to a leaf with the metric $d_L$ restricted to this set. \subsection{Example} \label{A construction of rooted forests} Consider an infinite sequence of sets and surjective maps \begin{equation}\label{line++} X_0 \stackrel{p_0}{\longrightarrow} X_1 \stackrel{p_1}{\longrightarrow} X_2 \stackrel{p_2}{\longrightarrow} \cdots . \end{equation} Fix any functions $\{\delta_k :X_k\to (0, \infty)\}_{k\geq 0} $. Consider the metric graph~$L$ with the set of vertices $\amalg_{k\geq 0} X_k$ and with each $v\in X_k$ connected to $p_k(v)\in X_{k+1}$ by an edge of length $ {\delta_k} (v)$. Assume that $\operatorname{card} (p_{k}^{-1} (a)) \geq 3$ for all $k\geq 0$ and $a\in X_{k+1 }$ and that for any distinct $x,y \in X$ there is an integer $m=m(x,y)\geq 1$ such that $$(p_m \cdots p_1 p_0)(x)= (p_m \cdots p_1 p_0)(y) .$$ Then $L$ is a metric tree. For all $k\geq 0$, restricting the tree metric $d_L$ to $X_k\subset L$, we obtain a metric space $ (X_k, d_k)$ where $d_k$ is computed by $$d_k(x,y)= \sum_{i=k}^{m-1} \big ( \delta_i((p_{i-1} \cdots p_k)(x)) +\delta_i((p_{i-1} \cdots p_k)(y)) \big)$$ for any distinct $x,y\in X_k$, where $m>k$ is the smallest integer such that $$(p_{m-1} \cdots p_k)(x) = (p_{m-1} \cdots p_k)(y).$$ Lemma~\ref{cloleththree--} implies that $ \underline {d_k}=\delta_k$ and $(X_{k+1}, d_{k+1})$ is obtained from $(X_k, d_k)$ by trimming: $ (X_{k+1}, d_{k+1})=t(X_k, d_k)$. Then the sequence \eqref{line++} is the trimming sequence of $\partial L=(X_0,d_0)$ and $X_\infty$ is a singleton (i.e., $\operatorname{card} (X_\infty)=1$). \subsection{Example} \label{A ++ construction of rooted forests} We describe sets and maps satisfying the conditions of Example~\ref{A construction of rooted forests}. Let~$A$ be a set with $ \geq 3$ elements and let $a\in A$. For $k\geq 0$, let $X_k$ be the set of all infinite sequences $a_0,a_1,...$ of elements of $A$ such that $a_n=a$ for all sufficiently big~$n$. The map $p_k\colon X_k \to X_{k+1}$ drops the first element of a sequence. Given functions $\{\delta_k :X_k\to (0, \infty)\}_{k\geq 0} $, Example~\ref{A construction of rooted forests} provides, for each $k\geq 0$, a metric $d_k$ in $X_k$ such that $ (X_{k+1}, d_{k+1})=t(X_k, d_k)$ for all~$k$. \section{The trimming cylinder}\label{trimmcylinder} We introduce the trimming cylinder of a metric space. \subsection{The cylinder~$C$} Consider a metric space $(X,d)=(X_0, d_0)$ and its trimming sequence \eqref{line} where $ t^k(X)=(X_k, d_k)$ for all $k\geq 1$. We define a graph $C=C(X)$: take the disjoint union $\amalg_{k\geq 0} X_k$ as the set of vertices and connect each $v\in X_k$ to $p_k(v)\in X_{k+1}$ by an edge $e_v$ of length $ {\underline {d_k} } (v)\geq 0$. If $ {\underline {d_k} } (v)> 0$, then we take for $e_v$ a copy of the segment $[0, {\underline {d_k} } (v) ]\subset \mathbb{R}_+$. Clearly, $C$ is a pseudometric graph and $X =X_0$ is the set of its leaves. We call~$C$ the \emph{trimming cylinder} of~$X$. It is clear that all (path connected) components of $C$ are trees. Two points $x,y\in X \subset C$ lie in the same component of~$C$ if and only if $x \cong y$. We can therefore identify the set $\pi_0(C)$ of components of~$C$ with $X_\infty=X/\cong$. Each component of~$C$, being a pseudometric tree, carries the path pseudometric. Also, the set $X_k \subset C$ carries the metric $d_k$ for all $k\geq 0$. The following theorem extends all these pseudometrics and metrics to a pseudometric in~$C$. \begin{theor}\label{thmain} There is a unique pseudometric $\rho$ in~$C $ which restricts to the metric~$d $ in $X\subset C$, restricts to the path pseudometric in every component of~$C$, and is minimal in the class of such pseudometrics in~$C$. For all $k\geq 1$, we have $\rho\vert_{X_k}=d_k$. \end{theor} The minimality of $\rho$ means that for any pseudometric $\rho'$ in $C$ which restricts to~$d$ in~$X$ and to the path pseudometrics in the components of~$C$, we have $\rho(a,b) \leq \rho'(a,b)$ for all $a,b \in C$. The uniqueness of such~$\rho$ is obvious; we only need to prove the existence and the equality $\rho\vert_{X_k}=d_k$. We do it in Section~\ref{proof1}. In the proof of Theorem~\ref{thmain} we will use a partial order in~$C$ defined as follows. We say that a path in~$C$ \emph{goes down} if whenever it enters an edge connecting a point of $X_k$ to a point of $X_{k+1}$, it goes in this edge in the direction from $ X_k$ to $ X_{k+1}$. We say that a point $b\in C$ lies \emph{below} a point $a\in C $ if $a \neq b$ and there is a path in~$C$ going down from~$a$ to~$b$. For example, for $x\in X$ and integers $k < l$, the point $x_{(l)}\in X_l $ lies below $x_{(k)}\in X_k $. \subsection{The metric space $\overline C$}\label{overlC} Consider now the metric quotient $\overline C=(\overline C, \overline \rho)$ of the trimming cylinder $(C=C(X), \rho)$. Since~$X$ is a metric space, the embedding $X =X_0 \subset C$ induces a metric embedding $X\hookrightarrow \overline C$. We view~$X$ as a subspace of~$\overline C$ via this embedding. We say that a vertex $x_{(k)}\in X_k$ of~$C$ is \emph{special} if for all $l\geq k$, the edge of~$C$ connecting $x_{(l)}$ to $x_{(l+1)}$ has zero length, i.e., ${\underline {d_l}} (x_{(l)})=0$. A component~$L$ of~$C$ is \emph{special} if it has at least one special vertex. Then all special vertices of~$L$ and all edges between them form a tree which collapses into a single vertex in $\overline L\subset \overline C$. We call the latter vertex the \emph{root} of~$\overline L$. We can now describe~$\overline C$. If $\operatorname{card} (X_\infty)=1$, then $(C, \rho)$ is a pseudometric tree with the path pseudometric and $(\overline C, \overline \rho)$ is the associated metric tree with the path metric. If $\operatorname{card} (X_\infty)\geq 2$, then $(\overline C, \overline \rho)$ is obtained from $(C, \rho)$ in two steps. First, each component~$L$ of~$C$ is replaced with its metric quotient - the metric tree~$\overline L$ (which has a distinguished root if~$L$ is special). Second, for all $u,v\in X_\infty =\pi_0(C)$ such that $d_\infty (u,v)=0$ and both corresponding components $L_u, L_v$ of~$C$ are special, we identify (glue) the roots of $\overline{L_u}$ and $ \overline {L_v}$. This gives a metric graph whose components are trees. We endow this graph with the unique metric such that the projection from~$C$ to this graph is distance preserving. The resulting metric space is $(\overline C, \overline \rho)$ as is clear from the definition of~$\rho$ in Section~\ref{proof2}. \subsection{Example} The trimming cylinders arising in Examples~\ref{A construction of rooted forests} and \ref{A ++ construction of rooted forests} have no special vertices. If we modify the assumptions there to $\delta_k(X_k) \subset (0, \infty)$ for $k=0,1,..., N$ with some $N\geq 0$ and $\delta_k=0$ for all $k>N$, then the trimming sequence of the metric space $(X_0, d_0)$ consists of the metric spaces $\{(X_k, d_k)\}_{k=0}^{N}$, the set $X_{N+1} $ with zero pseudometric, and singletons corresponding to all $k>N+1$. All the vertices of the associated trimming cylinder belonging to $X_{N+1} $ are special. \section{The series~$\sigma$ and proof of Theorem~\ref{thmain}}\label{proof1} In this section, as above, $X=(X,d)$ is a metric space. \subsection{The series~$\sigma$}\label{The series} For each $x\in X$, consider the infinite series \begin{equation}\label{almotropdfdfdfds---} \sigma (x)= \sum_{k=0}^{\infty} \, \underline{d_k} (x_{(k)} )=\underline{d} (x ) + \sum_{k=1}^{\infty} \, \underline{d_k} (x_{(k)} ) . \end{equation} Here $(x_{(k)} \in X_k)_{k\geq 0}$ is the trimming sequence of~$x$ and $\underline{d_k}: X_k \to \mathbb{R}_+$ is the function induced by the metric $d_k$ in $X_k$. For any $n \geq 1$, set \begin{equation}\label{almotropdfdfdfds---part} \sigma^n (x)= \sum_{k=0}^{n-1} \, \underline{d_k} (x_{(k)} ) = \underline{d} (x ) + \sum_{k=1}^{n-1} \, \underline{d_k} (x_{(k)} ). \end{equation} This gives a well-defined function $\sigma^n:X\to \mathbb{R}_+$. In particular, $\sigma^1= \underline{d}$. \begin{lemma}\label{sigsig} (i) For any distinct $x,y\in X$ such that $x\cong y$, we have \begin{equation}\label{almotrops41aaas} d(x,y) = \sigma^{m}(x ) +\sigma^{m }(y ) \end{equation} where $m=m(x,y) \geq 1 $ is the meeting index of~$x$ and~$y$; (ii) For any $x,y\in X$ such that $x\ncong y$, both series $\sigma(x)$ and $\sigma (y)$ converge and \begin{equation}\label{almotrops41} d(x,y) = d_{\infty}(x_{(\infty)},y_{(\infty)}) +\sigma(x )+ \sigma(y ) .\end{equation} \end{lemma} \begin{proof} To prove (i), note that for all $s <m$, we have $x_{(s)} \neq y_{(s)}$ and therefore \begin{equation}\label{almotree} d_{s}(x_{(s)},y_{(s)})- d_{s+1}(x_{(s+1)},y_{(s+1)})= \underline{d_s} (x_{(s)} ) +\underline{d_s} (y_{(s)} ) . \end{equation} Summing up over $s=0, 1, ..., m-1$ and using that $d_{m}(x_{(m)},y_{(m)})=0$, we get $$ d(x,y) = d(x_{(0)},y_{(0)}) =\sigma^{m}(x ) +\sigma^{m}(y ).$$ To prove (ii), note that $x_{(s)} \neq y_{(s)}$ for all $s \geq 0$. Thus, we have \eqref{almotree} for all $s \geq 0$. Summing up over $s=0,1,..., n-1$, we obtain for all $n\geq 0$, \begin{equation}\label{almotrops} d(x,y)=d_{0}(x_{(0)},y_{(0)}) = d_{n}(x_{(n)},y_{(n)}) +\sigma^n (x)+ \sigma^n(y). \end{equation} Therefore all partial sums of the series $\sigma(x)$ and $\sigma(y)$ are bounded above by $ d(x,y) $. Hence, both series converge. Taking the limit $n\to \infty$ in \eqref{almotrops}, we obtain \eqref{almotrops41}. \end{proof} \begin{lemma}\label{sigsig+} If $\operatorname{card}(X_\infty)\geq 2 $, then the series $\sigma(x)$ converges for all $x\in X$. \end{lemma} This directly follows from Lemma~\ref{sigsig}.(ii) as we can pick $y\in X $ so that $x\ncong y$. \subsection{Proof of Theorem~\ref{thmain}}\label{proof2} Suppose first that $\operatorname{card}(X_\infty)=1$. Then~$C$ is (path) connected and~$\rho$ is the path pseudometric in~$C$. We verify that $\rho (x_{(k)}, y_{(k)})=d_k (x_{(k)}, y_{(k)})$ for all $x,y\in X$ and all $k\geq 0$. The connectedness of~$C$ implies that $x \cong y$. Let $m=m(x,y)$ be the meeting index of~$x$ and~$y$. If $k\geq m$, then $x_{(k)}= y_{(k)}$ and $\rho (x_{(k)}, y_{(k)})=d_k (x_{(k)}, y_{(k)})=0$. If $k <m$, then an injective path from $x_{(k)}$ to $y_{(k)}$ in~$C$ is formed by the edges $$ e_{x_{(k)} }, e_{x_{(k+1)} },..., e_{x_{(m-1)} }, e_{y_{(m-1)}}, ..., e_{y_{(k+1)}}, e_{y_{(k)}}.$$ Therefore \begin{equation}\label{almotr--} \rho(x_{(k)},y_{(k)}) =\sum_{s=k}^{m-1} (\underline{d_s} (x_{(s)} )+ \underline{d_s} (y_{(s)} )) . \end{equation} By the definition of the metric $d_{s+1}$, for all $s < m$, we have \eqref{almotree}. Summing up these equalities over $s=k, ..., m-1$, we obtain that $$ d_{k}(x_{(k)},y_{(k)})=d_{k}(x_{(k)},y_{(k)})- d_{m}(x_{(m)},y_{(m)})= \sum_{s=k}^{m-1} (\underline{d_s} (x_{(s)} )+ \underline{d_s} (y_{(s)} )) . $$ Comparing with \eqref{almotr--}, we obtain that $ \rho(x_{(k)},y_{(k)}) = d_{k}(x_{(k)},y_{(k)})$. Suppose now that $\operatorname{card}(X_\infty)\geq 2$. By Lemma~\ref{sigsig+}, the series \eqref{almotropdfdfdfds---} converges for all $x\in X$ and yields a function $\sigma: X\to \mathbb{R}_+$. This function extends to~$C $ as follows: for any $a\in C$ lying in the edge $e=e_{x_{(k)}}$ connecting the vertices $x_{(k)}$ and $x_{(k+1)}$ with $x\in X, k\geq 0$, set $$ \sigma(a)= d_e(a, x_{(k+1)})+\sum_{s=k+1}^{\infty} \, \underline{d_s} (x_{(s)} ) $$ where $d_e$ is the pseudometric in~$e$ induced by the fixed homeomorphism $e\approx [0, {\underline {d_k} } (x_{(k)})]$ if ${\underline {d_k} } (x_{(k)})>0$ and $d_e=0$ if ${\underline {d_k} } (x_{(k)})=0$. The infinite series in the expression for $\sigma(a)$ is majorated by $\sigma (x)$ and therefore converges. In particular, $\sigma(x_{(k)})= \sum_{s\geq k} \, \underline{d_s} (x_{(s)} ) $ for all $x\in X$, $k \geq 0$. It is clear that the function $\sigma \colon C\to \mathbb{R}_+$ is continuous. The next lemma implies that~$\sigma$ is monotonous in the sense that if $b\in C$ lies below $a\in C$, then $\sigma(b)\leq \sigma(a)$. \begin{lemma}\label{sigpointssig} If $a,b\in C$ lie in the same component~$L$ of~$C$, then \begin{equation}\label{ineqs} \vert \sigma (a) -\sigma (b) \vert \leq d_L(a,b) \leq \sigma (a) +\sigma (b)\end{equation} where $d_L$ is the path pseudometric in~$L$. Moreover, if~$b $ lies below~$a $, then \begin{equation}\label{ineqs+} \sigma (a) =d_L(a,b)+\sigma (b). \end{equation} \end{lemma} \begin{proof} Formula \eqref{ineqs+} follows from the definitions. For $a=b$ the formula \eqref{ineqs} is obvious. If one of the points $a,b$ lies below the other one, then \eqref{ineqs} follows from \eqref{ineqs+}. In all other cases, the injective path from~$a$ to~$b$ is V-shaped, i.e., goes down from~$a$ to a certain vertex~$v$ of~$L$ and then goes up to~$b$ (a path goes up if the inverse path goes down). Then $$\sigma(a)= d_L(a, v)+\sigma(v),\,\,\, \sigma(b)= d_L(b, v)+\sigma(v) , $$ and $d_L(a,b)=d_L(a, v)+ d_L(b, v)$. This easily implies \eqref{ineqs}. \end{proof} To complete the proof of the theorem, consider the map $q: C\to \pi_0(C)= X_\infty$ carrying each point to its path connected component. For any $a,b \in C$, set \begin{equation} \rho (a,b)=\left\{ \begin{array}{ll} d_L(a,b) \quad {\text {if}} \,\, a,b \,\, {\text {lie in the same component}} \, \, L \,\, {\rm {of}}\,\, C , \\ d_\infty( q(a), q(b) )+\sigma(a)+ \sigma(b) \quad {\text {if}} \,\, q(a) \neq q(b). \end{array} \right. \end{equation} We claim that the map $\rho: C \times C \to \mathbb{R}_+$ is a pseudometric. That $\rho(a,a)=0$ and $\rho(a,b)=\rho(b,a)$ for all $a,b\in C$ is clear. For points of~$C$ lying in the same component~$L$, the triangle inequality follows from the one for $d_L$. For points of~$C$ lying in three different components, the triangle inequality follows from the one for $d_\infty$. If $a,b \in L $ for a component $L\subset C $ and $c\in C \setminus L$, then by Lemma~\ref{sigpointssig}, $$\vert \rho(a,c)-\rho(b,c) \vert = \vert \sigma (a) -\sigma (b) \vert \leq d_L(a,b) =\rho(a,b) $$ and $$ \rho(a,b)=d_L(a,b) \leq \sigma (a) +\sigma (b) \leq \rho(a,c)+\rho(b,c).$$ We check that $\rho\vert_{X_k}=d_k$ for all~$k$. Pick any $x,y \in X$. If $x\cong y$, then the arguments given in the case of connected $C$ apply and show that $ \rho(x_{(k)},y_{(k)}) =d_{k}(x_{(k)},y_{(k)}) $. If $x\ncong y$, then similar arguments show that for all $n\geq k$, $$d_{k}(x_{(k)},y_{(k)})= d_{n}(x_{(n)},y_{(n)})+ \sum_{s=k}^{n-1} (\underline{d_s} (x_{(s)} )+ \underline{d_s} (y_{(s)} )). $$ Taking the limit $n \to \infty$, we get $$d_{k}(x_{(k)},y_{(k)}) = d_\infty( x_{(\infty)} , y_{(\infty)} )+\sigma(x_{(k)})+ \sigma(y_{(k)}) =\rho(x_{(k)},y_{(k)}). $$ Finally, we prove that $\rho(a,b) \leq \rho'(a,b)$ for any $a,b \in C$ and any pseudometric~$\rho'$ in~$C$ which restricts to~$d$ in~$X$ and to the path pseudometrics in the components of~$C$. It suffices to treat the case where $a,b$ lie in different components, say, $L, M$ of~$C$. Pick a path in~$L$ starting in~$a $, going up, and ending in some $x\in X$. Then $\sigma(x) =d_L(x,a) +\sigma(a) $. Similarly, pick a path in~$M$ starting in~$b $, going up, and ending in some $y\in X$. Then $\sigma(y) =d_M(y,b) +\sigma(b)$. We have $q(x)=q(a) \neq q(y)=q(b)$ and therefore $$d(x,y)=\rho(x,y)= d_\infty( q(x), q(y) )+\sigma(x)+ \sigma(y)$$ $$= d_\infty( q(a), q(b) )+d_L(x,a) +\sigma(a)+ d_M(y,b) +\sigma(b)$$ $$= \rho(a,b)+d_L(x,a) + d_M(y,b). $$ At the same time, the assumptions on $\rho'$ imply that $$d(x,y)=\rho'(x,y) \leq \rho'(x,a) + \rho'(a,b) +\rho'(y,b)= \rho'(a,b) +d_L(x,a) + d_M(y,b).$$ We conclude that $$\rho(a,b)= d(x,y)- d_L(x,a) - d_M(y,b) \leq \rho'(a,b) .$$ \subsection{Remark} If $\operatorname{card}(X_\infty)=1$, then the series $\sigma(x)$ may converge or not. For instance, if in Examples~\ref{A construction of rooted forests} and~\ref{A ++ construction of rooted forests} we set $ \delta_k(x)=1$ for all $x\in X,k\geq 0$, then $\sigma^n(x)=n$ for all~$n$ and $\sigma(x)$ does not converge. Setting $\delta_k (x)=2^{-k}$ for all $x,k$, we obtain examples where $\sigma (x) $ converges for all $x$. In general, if $\sigma(x)$ converges for some $x\in X$, then it converges for all $x\in X$. Indeed, the assumption $\operatorname{card}(X_\infty)=1$ ensures that any $x,y \in X$ project to the same element of $X_k$ for all sufficiently big~$k$ and therefore the series $\sigma(x), \sigma(y)$ differ only in a finite number of terms. Also, if (the only component of) $C$ is special, then all terms of the series $\sigma(x)$ starting from a certain place are equal to zero and $\sigma(x)$ converges. \section{The tight span versus the trimming cylinder} We recall the tight span following \cite{Dr}, \cite{DMT} and relate it to the trimming cylinder. We also discuss the tight spans of pseudometric spaces. \subsection{Tight span of a metric space}\label{Basics} The \emph{tight span} of a metric space $(X,d)$ is the metric space $(T(X), d_T)$ consisting of all functions $f \colon X\to \mathbb{R}_+$ such that \begin{equation}\label{titi} f(x)=\sup_{y\in X}(d(x,y)-f(y)) \,\,\, {\text {for all}} \, \,\, x\in X.\end{equation} This identity may be restated by saying that $(\ast)$ $ f(x)+ f(y) \geq d(x,y)$ for all $x,y \in X$ and $(\ast \ast)$ for any $x\in X$ and any real number $\varepsilon >0$, there is $ y \in X $ such that $ f(x)+ f(y) \leq d(x,y) +\varepsilon$. The metric $d_T$ in $T(X)$ is defined by \begin{equation}\label{metric} d_T(f,g)=\sup_{x\in X}\vert f(x)-g(x)\vert \,\, {\text {for any}} \,\,f,g\in T(X) \end{equation} (here the set $\{\vert f(x)-g(x)\vert\}_{x\in X}$ is bounded above and has a well-defined supremum). The map carrying any $x\in X$ to the function $X\to \mathbb{R}_+, y\mapsto d(x,y)$ is a metric embedding $X\hookrightarrow T(X)$. Each $f \in T(X)$ is minimal in the set of functions $f':X\to \mathbb{R}_+$ satisfying $( \ast)$: if $f\geq f' $ (in the sense that $f(x)\geq f'(x)$ for all $x\in X$), then $f=f'$. Indeed, $$f(x) \geq f'(x) \geq \sup_{y\in X}(d(x,y)-f'(y)) \geq \sup_{y\in X}(d(x,y)-f(y))=f(x)$$ for all $x\in X$, and so $f(x)=f'(x)$. If $X$ is a singleton, then $T(X)=\{0\}$. If $\operatorname{card} (X)\geq 2$, then \eqref{titi} may be reformulated as follows (see \cite{Dr}, Sect.\ 1.4): for every $x\in X$, \begin{equation}\label{titi+} f(x)=\sup_{y\in X\setminus \{x\}}(d(x,y)-f(y)).\end{equation} This identity may be restated by saying that $(\ast)'$ $ f(x)+ f(y) \geq d(x,y)$ for all distinct $x,y \in X$ and $(\ast \ast)'$ for any $x\in X$ and any real number $\varepsilon >0$, there is $ y \in X\setminus \{x\}$ such that $ f(x)+ f(y) \leq d(x,y) +\varepsilon$. We now relate the tight span $T(X)$ to the trimming cylinder $ C (X)$. \begin{theor}\label{cylli} For any metric space $X=(X,d)$, there is a canonical distance preserving map $ C(X)\to T(X)$ extending the standard embedding $X\hookrightarrow T(X)$. \end{theor} \begin{proof} For $a\in C=C(X) $, define a function $f=f_{a}:X \to \mathbb{R}_+$ by $f(x)= \rho(x,a) $ for all $x\in X \subset C$ where $\rho=\rho_X$ is the pseudometric in~$C$. We claim that $f\in T(X)$. Note that for any $x,y\in X$, the triangle inequality for~$\rho$ implies that $f(x)+f(y) \geq \rho(x,y)=d(x,y)$. Thus, $f$ satisfies Condition $( \ast)$ above. Instead of Condition $( \ast\ast)$, we check a stronger claim: for any $x\in X$, there is $y\in X$ such that $f(x)+f(y) = d(x,y)$. In other words, we find $y\in X$ such that~$a$ lies between~$x$ and~$y$ in~$C$ (see Section~\ref{Preliminaries} for \lq\lq betweenness"). Assume for concreteness that $a$ lies in an edge of~$C$ connecting vertices $v\in X_k$ and $p_k(v) \in X_{k+1}$ for some $k\geq 0$ (possibly, $a=v$). We separate several cases. (i) Let $x,v$ lie in different components of~$C$. Pick $y\in X$ such that $y_{(k)}=v$. The definition of~$\rho$ shows that $$f(x)+ f(y) =\rho(x,a)+\rho(y,a)=\rho(x,a)+ \sigma(y)-\sigma(a)=\rho(x,y)=d(x,y) .$$ (ii) Let $x,v$ lie in the same component of~$C$ and $x_{(k)} \neq v$. Pick $y\in X$ such that $y_{(k)}=v$. Then $x\cong y$. Consider the injective path in~$C$ going from $y$ down to $y_{(k)}=v$, then further down to $y_{(m)}=x_{(m)}$ where $m=m(x,y) >k$ is the meeting index of~$x, y$, and then up to $x$. This path from~$y$ to~$x$ has length $\rho(x,y)=d(x,y)$. Since the point~$a$ lies on this path, its length $d(x,y) $ is equal to $\rho(x,a)+\rho(y,a)=f(x)+ f(y)$. (iii) Let $x_{(k)}=v$ and $\operatorname{card}(X_k) \geq 2$. Pick $y\in X$ such that $x\ncong y$. The same argument as in (i) shows that $ f(x)+ f(y) =d(x,y) $. (iv) Let $\operatorname{card}(X_k)=1$, i.e., $X_k=\{v\}$. Then the edge containing~$a$ (i.e., the edge from $v$ to $p_k(v)$) has zero length, and $f=\rho(-, a)=\rho(-,v)$. Thus, it is enough to treat the case $a=v$. If $\operatorname{card}(X_{k-1})\geq 2$, then there is $y\in X$ such that $x_{(k-1)} \neq y_{(k-1)}$. The injective path from $y$ to~$x$ in~$C$ must pass through the only element~$v$ of $X_k$ and so $ f(x)+ f(y) =d(x,y) $. If $\operatorname{card}(X_{k-1})=1$, then $f= \rho(-,u)$ where~$u$ is the only element of $X_{k-1}$. Proceeding by induction, we eventually find some $l<k$ such that $\operatorname{card}(X_l) \geq 2$ and the argument above works or deduce that $\operatorname{card}(X)=1$ in which case the theorem is obvious. We next verify that $d_T(f_a, f_{b})= \rho(a,b)$ for all $a,b\in C$. The case $a=b$ is obvious as both sides are equal to zero. Assume that $a\neq b$ and recall that $$d_T(f_a, f_{b})=\sup_{x\in X}\vert f_{a}(x)-f_{b}(x)\vert=\sup_{x\in X}\vert \rho(x,a)-\rho(x,b)\vert.$$ The triangle inequality for~$\rho$ yields $d_T(f_a, f_{b}) \leq \rho(a,b)$. To prove that this is an equality, we find a point $x\in X\subset C$ such that either $b$ lies between~$a$ and~$x$ (with respect to the pseudometric~$\rho$ in~$C$) or $a$ lies between~$b$ and~$x$. In both cases $ \vert \rho(x,a)-\rho(x,b) \vert=\rho(a,b) $. Note that going up from $a,b$ in~$C$ we eventually hit certain elements, respectively, $x,y \in X\subset C$. If $a,b$ belong to different components of~$C$, then $a$ lies between~$b$ and~$x$ as follows from the definition of~$\rho$. Assume that $a,b$ belong to the same component~$L$ of~$C$. If~$b$ lies below~$a$, then~$a$ lies between~$b$ and~$x$ as follows from the definition of the path pseudometric in~$L$. In all other cases, an injective path from~$y$ to~$a$ in $L$ necessarily passes by $b$ and therefore~$b$ lies between~$a$ and~$y$. \end{proof} \begin{corol}\label{cyllicorol} For any metric space~$X$, there is a canonical metric embedding $ {\overline {C (X)}} \hookrightarrow T(X)$ extending the standard embedding $X\hookrightarrow T(X)$. \end{corol} The embedding $\overline C= {\overline {C (X)}} \hookrightarrow T(X)$ is induced by the distance preserving map from Theorem~\ref{cylli}. We will identify ${\overline {C }}$ with its image in $ T(X)$ under this embedding, i.e., view ${\overline {C }} $ as a metric subspace of $ T(X)$. \subsection{The tight span of a pseudometric space} The definition of the tight span via \eqref{titi} extends word for word to pseudometric spaces. This however does not give new metric spaces because, by the next lemma, a pseudometric space and its metric quotient have the same tight span. \begin{lemma}\label{psedotight11} Let $X=(X,d)$ be a pseudometric space and let $q:X \to \overline X = X/{\sim_d }$ be the projection from~$X$ to its metric quotient. Then the formula ${h}\mapsto {h} q$, where~${h}$ runs over $ T(\overline X)$, defines an isometry $T(\overline X) \to T(X)$. \end{lemma} \begin{proof} Let $\overline d$ be the metric in $\overline X$ induced by~$d$. We pick any function ${h} \colon \overline X \to \mathbb{R}_+$ in $ T(\overline X)$ and verify Conditions $(\ast)$, $(\ast\ast)$ for ${h}q\colon X \to \mathbb{R}_+$. For any $x,y \in X$, we have ${h}q(x)+ {h}q(y) \geq d(x,y)$ because if $q(x)=q(y)$, then $d(x,y)=0$, and if $q(x)\neq q(y)$, then ${h}q(x)+ {h}q(y) \geq \overline d (q(x), q(y))=d(x,y)$. To verify $(\ast\ast)$, pick any $x\in X$ and $\varepsilon>0$. Since $ {h} \in T(\overline X)$ and $q$ is onto, there is $y\in X$ such that $${h}q(x)+ {h}q(y) \leq \overline d (q(x), q(y))+\varepsilon =d(x,y)+\varepsilon .$$ Thus, ${h}q \in T(X)$. That the map $T(\overline X) \to T(X), {h}\mapsto {h} q$ is injective and metric preserving is clear from the definitions. To prove surjectivity, we show that each function $f\in T(X)$ takes equal values on $\sim_d$-equivalent points of~$X$. We have $f(y)-d(x,y) \leq f(x)$ for all $x, y \in X$ because (cf.\ \cite{DMT}, Section 2) $$f(y) -d(x,y)= \sup_{z\in X}(d(y,z)-f(z))- d(x,y)$$ $$=\sup_{z\in X}(d(y,z)-f(z) - d(x,y)) \leq \sup_{z\in X}(d(y,x)+d(x,z)-f(z) - d(x,y))$$ $$=\sup_{z\in X}(d(x,z)-f(z))= f(x).$$ If $d(x,y)=0$, then we get $f(y) \leq f(x)$. Exchanging $x,y$, we get $f(x) \leq f(y)$. Thus, if $d(x,y)=0$, then $f(x)=f(y)$. As a consequence, $f=hq$ for a function ${h} \colon \overline X \to \mathbb{R}_+$. Conditions $(\ast)$, $(\ast\ast)$ for~$f $ imply the same conditions for~$h $. \end{proof} \section{Trimming versus the tight span} We discuss further relations between trimming and the tight span. In this section, $X=(X,d)$ is a metric space. \subsection{The canonical embedding} Recall the function $\underline d: X \to \mathbb{R}_+$ induced by~$d$. \begin{theor}\label{tight11} There is a canonical metric embedding $T(t(X)) \hookrightarrow T(X)$ whose image consists of all $f\in T(X)$ such that $f \geq {\underline d}$. \end{theor} \begin{proof} Let $t(X)=(X_1, d_1)$ as in Section~\ref{Trimming}. Assume first that $X_1$ is a singleton. Then $T(X_1)=\{0\}$ and the embedding $T(X_1) \hookrightarrow T(X)$ carries $0$ to $\underline d$. We need to prove that ${\underline d} \in T(X)$. If $\operatorname{card} (X) =1$, then ${\underline d} =0 \in T(X)$. If $\operatorname{card} (X) \geq 2$, then our assumption on $X_1$ implies that $d(x,y)={\underline d}(x) +{\underline d}(y)$ for all distinct $x,y \in X$. Applying \eqref{titi+} to $f={\underline d}$, we deduce that $ {\underline d} \in T(X)$. By the minimality of any $f\in T(X)$, if $f \geq \underline d$, then $f=\underline d$. This gives the second claim of the theorem. Assume now that $\operatorname{card}(X_1) \geq 2$, and let $p\colon X\to X_1$ be the trimming projection. Given $ g\in T(X_1)$, we define a function $\hat g\colon X\to \mathbb{R}_+$ by $$\hat g (x)= gp(x)+ {\underline d}(x) =g(x_{(1)}) + {\underline d}(x)$$ for all $x\in X$. We claim that $\hat g \in T(X)$. To see it, we check Conditions $(\ast )'$ and $(\ast \ast)'$ of Section~\ref{Basics} for $f=\hat g$. To check $(\ast )'$, we separate two cases: $p(x)=p(y)$ and $p(x)\neq p(y)$. If $p(x)=p(y)$, then $(\ast )'$ holds because $$\hat g(x)+ \hat g(y) \geq {\underline d}(x) + {\underline d}(y) =d(x,y).$$ If $p(x)\neq p(y)$, then $(\ast )'$ holds because $$\hat g(x)+ \hat g(y) = gp(x)+ gp(y)+{\underline d}(x) + {\underline d}(y) $$ $$\geq d_1(p(x), p(y)) +{\underline d}(x) + {\underline d}(y)= d(x,y)$$ where the inequality follows from the inclusion $g\in T(X_1) $ and the final equality holds by the definition of $d_1$. To verify $(\ast \ast )'$, pick any $x\in X$ and $\varepsilon >0$. Using $(\ast \ast )'$ for $g\in T(X_1)$ and using the surjectivity of $p \colon X \to X_1$, we obtain an element $ y \in X$ such that $p(x)\neq p(y)$ and $$ d_1(p(x),p(y)) +\varepsilon \geq gp(x) +gp(y).$$ Then $ x \neq y$ and $(\ast \ast )'$ for $\hat g$ follows: $$d(x,y)+\varepsilon = d_1(p(x) ,p(y)) + {\underline d}(x) + {\underline d}( y) +\varepsilon $$ $$\geq gp(x) +gp(y)+ {\underline d}(x) + {\underline d}( y) =\hat g(x)+ \hat g(y). $$ The map $T(X_1)\to T(X), g\mapsto \hat g$ is a metric embedding: for any $g ,h \in T(X_1)$, $$d_T(\hat g , \hat h)= \sup_{x\in X}\vert \hat g (x)-\hat h(x)\vert = \sup_{x\in X}\vert g p(x)- hp(x)\vert$$ $$=\sup_{z\in X_1}\vert g (z)- h(z)\vert =(d_1)_T( g , h) .$$ To show that any $f\in T(X)$ satisfying $f \geq \underline d$ lies in the image of our embedding, set $f_-=f- \underline d\colon X \to \mathbb{R}_+$ and recall the pseudometric $d^-$ in~$X$ defined in Section~\ref{Trimming}. Since $\operatorname{card}(X) \geq \operatorname{card} (X_1)\geq 2$, we have by \eqref{titi+} that for any $x\in X$, $$f_-(x)= f(x) - {\underline d}(x)=\sup_{y\in X\setminus \{x\}}(d(x,y)- {\underline d}(x) - {\underline d}(y)+ {\underline d}(y)-f(y))$$ $$= \sup_{y\in X \setminus \{x\}}(d^{-} (x,y) - f_-(y) )= \sup_{y\in X }(d^{-} (x,y) - f_-(y) ) . $$ The last equality holds because its left-hand side is non-negative (being equal to $f_-(x)$) while $d^{-} (x,x) - f_-(x)= - f_-(x)\leq 0$. Thus, $f_- \in T(X, d^{-})$. By Lemma~\ref{psedotight11}, $f_-=gp$ for some $g\in T(X_1)$. Then $f=f_-+ \underline d=\hat g$. \end{proof} \subsection{The trimming filtration}\label{The trimming filtration} From now on, we view $T(t(X))=T(X_1)$ as a metric subspace of $T(X)$ via the embedding from Theorem~\ref{tight11}. Iterating the inclusion $T(X) \supset T(X_1)$, we obtain a filtration \begin{equation}\label{filt} T(X) \supset T(X_1) \supset T(X_2) \supset T(X_3) \supset \cdots \end{equation} where $t^n(X)= (X_n, d_n)$ for all $n \geq 1$. In \eqref{filt}, we identify each $T(X_n)$ with its image under the metric embedding $T(X_n) \hookrightarrow T(X)$ obtained as the composition of the embeddings $T(X_n) \hookrightarrow T(X_{n-1}) \hookrightarrow \cdots \hookrightarrow T(X)$. This composition carries any $f\in T(X_n)$ to the function $\hat f\colon X\to \mathbb{R}_+$ defined by $\hat f( x)= f( x_{(n)})+ \sigma^{n} (x ) $ for $x\in X$. Clearly, $ \hat f \geq \sigma^{n} $. An induction on~$n$ deduces from Theorem~\ref{tight11} that $$T(X_n)=\{g \in T(X)\, \vert \, g \geq \sigma^{n}\} \subset T(X).$$ \begin{theor}\label{tight11coroll} All terms of the filtration \eqref{filt} are closed subsets of $T(X)$. \end{theor} \begin{proof} It suffices to show that the set $T(X_1)\subset T(X)$ is closed in $T(X)$ and to apply induction. We prove that the complementary set $U=T(X) \setminus T(X_1)$ is open in $T(X)$. By Theorem~\ref{tight11}, for any $f \in U$, there is $a\in X$ such that $0 \leq f(a) <{\underline d}(a)$. We claim that the open ball $B\subset T(X)$ with center~$f$ and radius $r= {\underline d}(a)- f(a) >0 $ is contained in~$U$. Indeed, if $g\in B$, then $$r>d_T(f,g)=\sup_{x\in X}\vert f(x)-g(x)\vert\geq g(a)-f(a).$$ Therefore $g(a)< f(a)+r ={\underline d}(a)$. Thus, $g \ngeq \underline d$ and so $g\in U$. We conclude that~$U$ is open and $T(X_1) $ is closed. \end{proof} \subsection{The metric space $\tau$} Consider the set $$\tau=\tau(X)=\cap_{n\geq 1} \, T(X_n) \subset T(X) $$ and endow it with the metric obtained by restricting the one in $T(X)$. \begin{theor}\label{Xinfinity} If either $\operatorname{card}(X_\infty) \geq 2$ or $\operatorname{card}(X_\infty) =1$ and the series $\sigma(x)$ converges for all $x\in X$, then there is an isometry $ \tau \approx T(X_\infty)$. If $\operatorname{card}(X_\infty) =1$ and the series $\sigma(x)$ does not converge for some $x\in X$, then $\tau=\emptyset$. \end{theor} \begin{proof} Recall that $X_\infty =X/\cong$ where $\cong $ is the equivalence relation in~$X$ defined in Section~\ref{Trimming}. We start with the case $\operatorname{card}(X_\infty) \geq 2$. By Lemma~\ref{sigsig+}, the series $\sigma(x)$ converges for all $x\in X$ and yields a function $\sigma \colon X \to \mathbb{R}_+$. For any function $f: X_\infty \to \mathbb{R}_+$, we define a function $\tilde f : X\to \mathbb{R}_+$ by $\tilde f (x) = f(x_{(\infty)}) +\sigma(x)$. We claim that if $f\in T(X_\infty)$, then $\tilde f \in T(X)$. Note that $\operatorname{card}(X)\geq \operatorname{card} (X_\infty) \geq 2$. Therefore to verify the inclusion $\tilde f \in T(X)$, it suffices to verify for $\tilde f$ Conditions $(\ast)'$ and $(\ast \ast)'$ from Section~\ref{Basics}. To verify $(\ast)'$, pick any distinct $x,y \in X$. If $x\cong y$, then Formula~\eqref{almotrops41aaas} implies that \begin{equation}\label{sdf} d(x,y) \leq \sigma(x) +\sigma(y) \leq \tilde f(x) +\tilde f (y). \end{equation} If $x\ncong y$, then $x_{(\infty)} \neq y_{(\infty)}$. Formula~\eqref{almotrops41} and Condition $(\ast)'$ on~$f$ give $$d(x,y) \leq f(x_{(\infty)}) + f(y_{(\infty)}) +\sigma(x) +\sigma(y)= \tilde f(x) + \tilde f(y).$$ To verify $(\ast\ast)'$, pick any $x \in X$ and $\varepsilon >0$. Since $f: X_\infty \to \mathbb{R}_+$ satisfies $(\ast\ast)'$ and the projection $X\to X_\infty$ is onto, there is $y\in X$ such that $x_{(\infty)} \neq y_{(\infty)}$ and $$ f(x_{(\infty)})+ f(y_{(\infty)}) \leq d_\infty (x_{(\infty)}, y_{(\infty)}) +\varepsilon.$$ Then $x\ncong y$ and by \eqref{almotrops41}, $$\tilde f(x) + \tilde f(y) \leq d_\infty (x_{(\infty)}, y_{(\infty)}) +\varepsilon +\sigma(x) +\sigma(y)=d(x,y) +\varepsilon.$$ Thus, $\tilde f \in T(X)$. The map $T(X_\infty)\to T(X), f\mapsto \tilde f$ is a metric embedding: $$d_T(\tilde f_1, \tilde f_2)= \sup_{x\in X}\vert \tilde f_1(x)-\tilde f_2(x)\vert = \sup_{x\in X}\vert f_1(x_{(\infty)})- f_2(x_{(\infty)})\vert$$ $$=\sup_{z\in X_\infty}\vert f_1(z)- f_2(z)\vert =(d_\infty)_T( f_1, f_2) $$ for any $f_1,f_2 \in T(X_\infty)$. Also, for all $n\geq 1$, the obvious inequalities $\tilde f \geq \sigma \geq \sigma^n$ imply that $\tilde f \in T(X_n)\subset T(X)$. So, $\tilde f \in \tau$. It remains to show that for each $g \in \tau$ there is $f\in T(X_\infty)$ such that $g=\tilde f$. By Theorem~\ref{tight11}, the function $g-\sigma^1=g-\underline d $ is the composition of the projection $X\to t(X)$ and a certain function $g_1\in T(X_1)$. Proceeding by induction, we deduce that for each $n\geq 1$, the function $g-\sigma^n $ is the composition of the projection $X\to X_n$ and a function $g_n\in T(X_n)$. Thus, $g \geq \sigma^n$ for all $n \geq 1$ and therefore $g\geq \sigma$. We check now that the function $g-\sigma \geq 0$ is the composition of the projection $X\to X_\infty$ and a function $X_\infty \to \mathbb{R}_+ $. If two points $x,y \in X$ project to the same point of $X_\infty$, then there is an integer $n\geq 0$ such that $x_{(n)}=y_{(n)}$. Then $$(g - \sigma^n)(x) = g_n(x_{(n)})=g_n(y_{(n)})=(g - \sigma^n)(y).$$ Also, $x_{(s)}=y_{(s)}$ for all $s\geq n$ and so $$(\sigma-\sigma^n) (x)= \sum_{s\geq n} \underline{d_s} (x_{(s)} )= \sum_{s\geq n} \underline{d_s} (y_{(s)} )=(\sigma-\sigma^n) (y). $$ Therefore, $(g- \sigma)(x) = (g- \sigma)(y)$. Thus, the function $g-\sigma \geq 0$ is the composition of the projection $X\to X_\infty$ and a function $f: X_\infty \to \mathbb{R}_+$. So, $g=\tilde f$. We claim that $f \in T(X_\infty)$. To see it, we verify that $f$ satisfies Conditions $(\ast)$ and $(\ast \ast)$ from Section~\ref{Basics}. To verify $(\ast)$, pick any $x,y \in X$. If $x_{(\infty)} \neq y_{(\infty)} $, then $$f(x_{(\infty)} )+f(y_{(\infty)} )= g(x) - \sigma (x) + g(y)-\sigma(y) $$ $$\geq d(x,y) - \sigma (x) -\sigma(y)= d_{\infty}(x_{(\infty)},y_{(\infty)})$$ where the inequality follows from the assumption $g\in T(X)$ and the last equality holds by \eqref{almotrops41}. If $x_{(\infty)} = y_{(\infty)} $, then $$f(x_{(\infty)} )+f(y_{(\infty)} ) \geq 0= d_{\infty}(x_{(\infty)},y_{(\infty)}).$$ To verify $(\ast\ast)$, pick any $x \in X$ and $\varepsilon >0$. Since $g\in T(X)$, there is $y\in X$ such that $ g(x)+ g(y) \leq d(x,y) +\varepsilon$. Then $$f(x_{(\infty)} )+f(y_{(\infty)} )= g(x) - \sigma (x) + g(y)-\sigma(y) \leq d(x,y) +\varepsilon - \sigma (x) -\sigma(y).$$ If $x_{(\infty)} \neq y_{(\infty)} $, then the right-hand side is equal to $d_{\infty}(x_{(\infty)},y_{(\infty)}) +\varepsilon$ and we are done. If $x_{(\infty)} = y_{(\infty)} $, then the right-hand side is smaller than or equal to $\varepsilon= d_{\infty}(x_{(\infty)},y_{(\infty)}) +\varepsilon$ as follows from \eqref{sdf}. Assume that $\operatorname{card}(X_\infty) =1$, i.e., that $X_\infty$ is a singleton. If $\tau \neq \emptyset$, then any $f\in \tau=\cap_{n\geq 1}\, T^n(X) $ majorates all the functions $\{ \sigma^{n} \}_n$ on~$X$. Therefore the series $\sigma(x)$ converges for all $x\in X$. This implies the last claim of the theorem. Suppose now that the series $\sigma(x)$ converges for all $x\in X$. It defines then a function $\sigma:X \to \mathbb{R}_+$. Below we prove that $\sigma \in T(X)$. This will imply the claim of the theorem. Indeed, since $\sigma\geq \sigma^n$ for all $n\geq 1$, we have then $\sigma \in \tau $. Any $ f\in \tau $ satisfies $f\geq \sigma^n$ for all ${n\geq 1}$ and so $f \geq \sigma$. By the minimality of $f\in T(X)$ (see Section \ref{Basics}), the inequality $f \geq \sigma$ implies that $f=\sigma$. Thus, $ \tau=\{\sigma\}$ is isometric to $T(X_\infty)=\{0\}$. To prove the inclusion $\sigma \in T(X)$, we verify Conditions $(\ast)$ and $(\ast \ast)$ from Section~\ref{Basics}. The distance $d(x,y)$ between any points $x,y \in X$ may be computed from the equality $d(x,y)=\rho(x,y)$, where~$\rho$ is the pseudometric in the trimming cylinder of~$X$, and the expression~\eqref{almotr--} for $\rho(x,y)$ where $k=0$ and $m=m(x,y)$. This gives $$d(x,y)=\sigma^{m}(x)+\sigma^{m}(y) \leq \sigma(x) +\sigma(y)$$ which is Condition~$(\ast)$ for~$\sigma$. To check $(\ast \ast)$ for~$\sigma$, pick any $x\in X$ and $\varepsilon >0$. The assumption $\operatorname{card}(X_\infty) =1$ ensures that for every $y\in X $, there is an integer $k \geq 0$ such that $x_{(k)}=y_{(k)}$. Let $k_y$ be the smallest such integer. If the set of integers $\{k_y \geq 0\}_{y\in X}$ is finite, then either $\operatorname{card} (X)=1$ or there is an integer $K\geq 1$ such that $\operatorname{card} (X_{K-1})\geq 2$ and $\operatorname{card} (X_K)=1$. If $\operatorname{card} (X)=1$, then $\sigma=0\in T(X)=\{0\}$. If $\operatorname{card} (X_{K-1})\geq 2$ and $\operatorname{card} (X_K)=1$, then $\sigma=\sigma^{K}$ and there is $y\in X\setminus \{x\}$, such that $y_{(K-1)} \neq x_{(K-1)}$ and $y_{(K)} = x_{(K)}$. Then $$\sigma(x)+ \sigma(y)=\sigma^{K} (x) +\sigma^{K} (y)=\rho(x,y)=d(x,y)\leq d(x,y) +\varepsilon.$$ If the set $\{k_y\}_{y\in X}$ is infinite, then we can find $y\in X\setminus \{x\}$ with $k_y$ so big that $\sigma(x) - \sigma^{k_y}(x)<\varepsilon/2$. The equality $y_{(k_y)} = x_{(k_y)}$ implies that $$\sigma(y) - \sigma^{k_y}(y)= \sigma(x) - \sigma^{k_y}(x) <\varepsilon/2 .$$ Then $$ \sigma (x)+ \sigma (y) < \sigma^{k_y}(y)+ \sigma^{k_y}(x) +\varepsilon =\rho(x,y) +\varepsilon= d(x,y) +\varepsilon.$$ Here the equality in the middle follows from the definition of the path pseudometric $\rho$ and the conditions $y_{(k_y-1)} \neq x_{(k_y-1)}$, $ y_{(k_y)} = x_{(k_y)}$. Thus, Condition~$(\ast \ast)$ also holds for~$\sigma$. So, $\sigma \in T(X)$. \end{proof} \section{Main theorem} \subsection{Main theorem} We state our main result on the tight span $T(X)$ of a metric space $X=(X,d)$. Recall the subspaces $\overline C$ and $\tau$ of $T(X)$. Recall the special components of the trimming cylinder $C=(C, \rho)$ and the roots of their projections to~$\overline C$. The set of these roots is denoted ${\overline C}_\bullet$. \begin{theor}\label{cyllibb+} For any metric space $X=(X,d)$, $$T(X)= \tau \cup {\overline C } \quad {\text and} \quad \tau \cap {\overline C }= {\overline C }_\bullet.$$ \end{theor} \begin{proof} We first consider the case where $\operatorname{card}(X_\infty) \geq 2$. Consider the distance preserving map $C\to T(X), a \mapsto f_a$ from Theorem~\ref{cylli}. To prove the equality $T(X)= \tau \cup {\overline C }$, it suffices to show that each function $f:X\to \mathbb{R}_+$ belonging to $ T(X) \setminus \tau$ is equal to $f_a$ for some $a\in C $. We prove a stronger claim: $f=f_a$ for some $a\in C$ such that $\sigma(a)>0$ where $\sigma \colon C\to \mathbb{R}_+$ is the function defined in the proof of Theorem~\ref{thmain}. Suppose first that $f \notin T(X_1)$. By Theorem~\ref{tight11}, there is $y\in X$ such that $0 \leq f(y) < {\underline d}(y)$. The edge of~$C$ connecting $y=y_{(0)}$ to $y_{(1)}$ has length ${\underline d}(y)$ and contains a (unique) point~$a$ such that $ \rho(y,a)= f(y) $. Then $$\sigma(a)\geq \rho(a, y_{(1)})= {\underline d}(y) -f(y) >0$$ and $f_{a}(y)=\rho(y,a)=f(y) $. For any $x\in X\setminus \{y\}$, Condition ($\ast$) in Section~\ref{Basics} and the fact that~$a$ lies between~$x$ and~$y$ in~$C$ imply that $$f(x) \geq d(y,x) - f(y) =\rho(y,x) - \rho(y,a)= \rho(x,a)=f_{a}(x).$$ Thus, $f\geq f_{a}$. Since $f, f_{a } \in T(X)$, we conclude that $f=f_{a }$. Suppose now that $ f\in T(X_n) \setminus T(X_{n+1})$ with $n\geq 1$. By Section~\ref{The trimming filtration}, the embedding $T(X_n) \hookrightarrow T(X)$ carries any function $ g: X_n \to \mathbb{R}_+$ in $ T(X_n)$ to the function ${\hat g}:X \to \mathbb{R}_+$ defined by ${\hat g}( x) = g(x_{(n)}) +\sigma^{n}(x)$ for all $x\in X$. Since $f\notin T(X_{n+1})=T(t(X_n)) \subset T(X_n)$, Theorem~\ref{tight11} implies that there is $y\in X_n$ such that $0\leq f(y) < {\underline {d_n}}(y)$. The edge of~$C$ connecting $y\in X_n$ to the projection of~$y$ to $X_{n+1}$ has length ${\underline {d_n}}(y)$ and contains a (unique) point~$a$ such that $ \rho(y,a)= f(y) $. As above, $\sigma(a)\geq {\underline {d_n}}(y) -f(y) >0$. For any $x\in X$ such that $x_{(n)}=y$, the point~$y$ lies between~$x$ and~$a$ in~$C$, and therefore $${\hat f}(x)= f(y) +\sigma^{n}(x)=\rho(y,a)+\rho(x,y)=\rho(x,a)= f_a(x). $$ For any $x\in X$ such that $x_{(n)}\neq y$, the point $a$ lies between $x$ and~$y$ in~$C$. The inclusion $f\in T(X_n)$ and Condition ($\ast$) of Section~\ref{Basics} yield $${\hat f}(x)= f(x_{(n)}) +\sigma^{n}(x)\geq d_n(x_{(n)}, y) -f(y) +\sigma^{n}(x)=$$ $$=\rho(x_{(n)}, y) -\rho(y,a) +\rho(x,x_{(n)}) \geq \rho(x,y)-\rho(y,a)=\rho(x,a)= f_a(x) .$$ So, ${\hat f}\geq f_{a}$ and, since ${\hat f}, f_{a } \in T(X)$, we have ${\hat f}=f_{a }$. The arguments above prove the inclusion $T(X) \setminus \tau \subset {\overline C } \setminus \overline C_\bullet $. As a consequence, $T(X)= \tau \cup {\overline C } $ and $ \overline C_\bullet \subset \tau$. It remains only to show that $\tau \cap (\overline C\setminus \overline C_\bullet )=\emptyset $, i.e., that $f_a \notin \tau $ for any $a\in C$ such that $\sigma(a)>0$. Assume for concreteness that $a=x_{(k)}$ or~$a$ lies inside the edge of~$C$ connecting the vertices $x_{(k)}$, $x_{(k+1)}$ where $x\in X, k\geq 0$. Let $L\subset C$ be the component of~$C$ containing~$x$ and~$a$. The condition $\sigma(a)>0$ implies that there is $n \geq k$ such that $\underline {d_{n}} (x_{(n)})>0$. The definition of the path pseudometric $d_L$ in~$L$ implies that $d_L (a,x_{(n+1)}) \geq \underline {d_{n}} (x_{(n)}) >0$. It is also clear that~$a$ lies between the points~$x$ and $x_{(n+1)}$ of~$L$. Then $$\sigma^{n+1}(x) = d_L (x, x_{(n+1)})=d_L(x,a) + d_L(a,x_{(n+1)}) > d_L(x,a)=\rho(x,a)=f_a(x).$$ We conclude that the function $f_a$ does not majorate the function $\sigma^{n+1}:X \to \mathbb{R}_+$. By Section~\ref{The trimming filtration}, $f_a\notin T(X_{n+1} )\subset T(X)$. Therefore, $f_a \notin \tau$. Suppose now that $\operatorname{card}(X_\infty) = 1$ so that~$C$ is (path) connected. If the series $ \sigma(x)$ converges for all $x\in X$ (and defines a function $\sigma:X\to \mathbb{R}_+$), then the theorem is proved exactly as in the case $\operatorname{card}(X_\infty) \geq 2$. If the series $ \sigma(x)$ does not converge for some $x\in X$, then~$C$ is non-special and ${\overline C }_\bullet =\emptyset$. The equality $T(X)=\tau \cup \overline C$ is proved then as in the case $\operatorname{card}(X_\infty) \geq 2$ suppressing all references to ${\overline C }_\bullet, \sigma$. Also, $\tau= \emptyset$ by Theorem~\ref{Xinfinity} and so $ \tau \cap {\overline C }=\emptyset = {\overline C }_\bullet$. \end{proof} \subsection{Remarks} 1. When~$C$ is (path) connected, i.e., $\operatorname{card}(X_\infty) = 1$, we can explicitly compute the sets~$\tau$ and ${\overline C }_\bullet$. We consider three cases: (i) (the only component of) $C$ is special, or (ii) $C$ is not special but the series $ \sigma(x)$ converges for all $x\in X$ and defines a function $\sigma:X\to \mathbb{R}_+$, or (iii) the series $ \sigma(x)$ does not converge for some $x\in X$. In the cases (i) and (ii), the proof of Theorem~\ref{Xinfinity} shows that $\tau=\{\sigma\}$. In the case (iii), the same theorem gives $\tau =\emptyset$. The set $ \overline C_\bullet$ is computed from the definitions using the connectedness of~$C$. Namely, $ \overline C_\bullet =\{\sigma\} $ in the case (i) and $\overline C_\bullet= \emptyset$ in the cases (ii) and (iii). 2. For $x\in X $, let $L_x$ be the connected component of the trimming cylinder $C=C(X)$ containing $x\in X=X_0\subset C$. If the pseudometric tree $L_x$ is special, then the root of its metric quotient $\overline {L_x} $ belongs to the set ${\overline C }_\bullet \subset \tau $. Comparing the canonical embedding $\overline C\hookrightarrow T(X)$ with the isometry $T(X_\infty)=\tau $, one can check that the root in question is the point $x_{(\infty)} \in X_\infty \subset T(X_\infty)=\tau $. \end{document}	arXiv
Journal Home About Issues in Progress Current Issue All Issues Vol. 7, •https://doi.org/10.1364/OPTICA.7.000028 Waveguide-integrated three-dimensional quasi-phase-matching structures Jörg Imbrock, Lukas Wesemann, Sebastian Kroesen, Mousa Ayoub, and Cornelia Denz Jörg Imbrock,* Lukas Wesemann, Sebastian Kroesen, Mousa Ayoub, and Cornelia Denz Institute of Applied Physics and Center for Nonlinear Science, University of Münster, Corrensstr. 2, 48149 Münster, Germany Corresponding author: [email protected] Lukas Wesemann https://orcid.org/0000-0001-9142-1342 Mousa Ayoub https://orcid.org/0000-0002-2499-106X J Imbrock L Wesemann S Kroesen M Ayoub Jörg Imbrock, Lukas Wesemann, Sebastian Kroesen, Mousa Ayoub, and Cornelia Denz, "Waveguide-integrated three-dimensional quasi-phase-matching structures," Optica 7, 28-34 (2020) Ultrabroadband nonlinear optics in nanophotonic periodically poled lithium niobate waveguides Marc Jankowski, et al. Optica 7(1) 40-46 (2020) Chip-based self-referencing using integrated lithium niobate waveguides Yoshitomo Okawachi, et al. Optica 7(6) 702-707 (2020) Broadband quasi-phase-matching in dispersion-engineered all-optically poled silicon nitride... Edgars Nitiss, et al. Photon. Res. 8(9) 1475-1483 (2020) Beam shaping Nonlinear photonic crystals Parametric processes Phase matching Waveguide gratings Original Manuscript: October 1, 2019 Revised Manuscript: November 21, 2019 Manuscript Accepted: November 22, 2019 Published: January 7, 2020 PRINCIPLE OF LIGHT-INDUCED QUASI-PHASE MATCHING Suppl. Mat. (1) Nonlinear photonic structures with a modulated second-order nonlinearity are used widely for quasi-phase-matched parametric processes. Creating three-dimensional (3D) nonlinear photonic structures is promising but still challenging, since standard poling methods are limited to two-dimensional structures. Light-induced quasi-phase matching (QPM) can overcome this issue by a depletion of the second-order nonlinearity with focused femtosecond laser pulses. We report, to the best of our knowledge, the first integration of a 3D QPM structure in the core of a lithium niobate waveguide applying light-induced fabrication. Depressed-cladding waveguides and embedded QPM structures are fabricated by femtosecond laser lithography. The 3D capability is exploited by splitting the QPM gratings in the waveguide core into two or four parts, respectively. These monolithic nonlinear waveguides feature parallel multi-wavelength frequency conversion. Finally, we demonstrate a concept for second-harmonic beam shaping taking advantage of a helically twisted nonlinear structure. Our results open new avenues for creating highly efficient advanced QPM devices. Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Quasi-phase matching (QPM) is used in nonlinear optics for high-efficiency frequency conversion such as difference-frequency generation, sum-frequency generation, and second-harmonic generation (SHG), respectively [1,2]. Applications include single-frequency sources, broadband parametric processes, cascaded second-order interactions, and quantum optical devices [3–6]. Lithium niobate ($ {{\rm LiNbO}_3} $) is one of the most important crystals for photonic applications due to its high nonlinear optical and electro-optical coefficients. QPM can be realized in $ {{\rm LiNbO}_3} $ by periodic poling, i.e., inverting the direction of spontaneous polarization by applying an external electric field by patterned electrodes [7]. The modulation of the second-order nonlinearity between $ + {\chi ^{(2)}} $ and $ - {\chi ^{(2)}} $ allows for the realization of a variety of periodic, quasi-periodic, and also random nonlinear photonic structures [8–11]. However, electric field poling is limited to structure sizes of some micrometers and to 2D structures. 3D nonlinear photonic crystals could enable many new schemes of manipulation and control of nonlinear optical interactions such as simultaneous QPM of different nonlinear processes, volume nonlinear holography, and nonlinear beam shaping. To overcome the limitations of 2D nonlinear photonic structures, focused infrared femtosecond laser pulses have been applied to modulate the $ {\chi ^{(2)}} $ nonlinearity. Known principles are all-optical local domain inversion [12,13], local depletion of the nonlinearity [14], and pyroelectric field-assisted local domain inversion [15]. While the latter one can produce large 1D and 2D nonlinear photonic structures with high resolution, the two former ones are applicable for 3D structures. Recently, 3D nonlinear photonic crystals have been experimentally demonstrated using focused femtosecond laser pulses [16–18]. However, these structures are small in size and therefore low in efficiency. Higher efficiencies can be achieved by increasing the length of the nonlinear photonic crystal [19] or by embedding QPM structures into waveguides [20]. While fs laser lithography has been used to write waveguides into periodically poled $ {{\rm LiNbO}_3} $ [21–26], Kroesen et al. have fabricated efficient single-period quasi-phase-matched waveguides by structuring the $ {\chi ^{(2)}} $ nonlinearity in 1D using light-induced QPM (LiQPM) [27]. In this paper, we report the integration of 2D and 3D nonlinear structures in the core of a lithium niobate waveguide for efficient advanced QPM schemes. Depressed-cladding waveguides and embedded LiQPM structures are fabricated by femtosecond laser lithography (Fig. 1). First, we optimize the writing parameters for single-mode guiding and high conversion efficiency. Then, we demonstrate broadband SHG in a chirped grating and multi-wavelength SHG in two sequential gratings. Furthermore, we extend LiQPM to three dimensions by splitting the QPM gratings in the waveguide core into two and four parts, respectively. Finally, we propose a concept for the nonlinear generation of optical vortex beams by a helically twisted nonlinear susceptibility. Helically periodically poled crystals have been suggested for the generation of SH vortices and optical vortex converters [28,29]. However, such 3D structures cannot be produced by traditional electric field poling. Up to now, the experimental demonstration of nonlinear vortex generation from fundamental waves that do not carry angular optical momentum (OAM) was limited to 2D transverse geometries [30–33]. We show that this limit can be lifted by LiQPM. Fig. 1. Schematic design of a LiQPM waveguide that consists of a multiscan $ {\chi ^{(2)}} $ grating and a circular type-II waveguide fabricated in a single inscription sequence. Download Full Size \| PPT Slide \| PDF Fig. 2. Calculated SH build-up inside the nonlinear material assuming phase matching (PM) and QPM by PPLN and LiQPM structures where the second-order nonlinearity is periodically damped to a certain degree instead of domain inversion. The dashed lines are the analytical solutions of the second-harmonic power in the undepleted pump regime of the form $ {P_{2\omega }} \propto d_{{\rm eff}}^2{z^2} $, with the respective nonlinear coefficient $ {d_{{\rm eff}}} $. 2. PRINCIPLE OF LIGHT-INDUCED QUASI-PHASE MATCHING Under collinear phase matching, the change in the amplitudes $ {A_\omega } $ and $ {A_{2\omega }} $ of a fundamental wave and SH propagating along the $ y $ axis of the crystal (Fig. 1), respectively, are described by the following two coupled-mode equations [22]: (1)$$\frac{d}{{dy}}{A_\omega } = - i{\kappa ^}d({\textbf{r}})A_\omega ^*{A_{2\omega }}\exp ( - i\Delta ky) - {\alpha _\omega }{A_\omega },$$ (2)$$\frac{d}{{dy}}{A_{2\omega }} = - i\kappa d({\textbf{r}})A_\omega ^2\exp (i\Delta ky) - {\alpha _{2\omega }}{A_{2\omega }},$$ (3)$${\kappa ^2} = \frac{{2{\omega ^2}}}{{{ \epsilon _0}{c^3}}}\frac{1}{{n_\omega ^2{n_{2\omega }}{S_{{\rm eff}}}}},$$ with the phase mismatch (4)$$\Delta k = {k_{2\omega }} - 2{k_\omega },$$ and $ {S_{{\rm eff}}} $ being the cross section of the overlap of the fundamental and SH mode. The absorption coefficients $ {\alpha _\omega } $ and $ {\alpha _{2\omega }} $ of the fundamental and SH, respectively, account for the losses in the waveguide as well as in the QPM gratings. The amplitudes $ {A_\omega } $ and $ {A_{2\omega }} $ of the fundamental and SH are normalized such that $ {\vert {A_i} \vert ^2} = {P_i} $, with $ {P_i} $ representing the respective power. The nonlinear coefficient $ d({\textbf{r}}) $ can be in general modulated in any direction using LiQPM. For 1D QPM, the nonlinear coefficient is modulated only along the $ y $ axis $ d(y) = {d_{{\rm max}}}g(y) $. The normalized rectangular function $ g(y) $ incorporates the period $ \Lambda = 2{l_{\rm c}} = 2\pi / \Delta k $ to compensate for the phase mismatch, with the coherence length $ {l_{c}} $. The largest nonlinear coefficient in $ {{\rm LiNbO}_3} $ is $ {d_{{\max}}} = {d_{33}} $ for an ee-e process. A Fourier transform of $ d(y) $ yields an effective nonlinear coefficient: (5)$${d_{{\rm eff}}} = \frac{{{d_{{\rm max}}}}}{\pi }(1 - v),$$ with the visibility $ v = {d_{{\min}}}/{d_{{\max}}} $. Thus, a periodically poled structure with $ {d_{{\min}}} = - {d_{{\max}}} $ has the largest effective nonlinearity $ {d_{{\rm eff}}} = 2/\pi {d_{{\max}}} $, as can be seen in Fig. 2. However, a damping of the nonlinearity by LiQPM is already sufficient for effective QPM. We can realize a 3D modulation of the nonlinearity if we, for instance, split the waveguide core into two or four sections while each section is individually modulated, as shown in Fig. 3. We use this approach for SHG of multiple fundamental frequencies. We can also express the nonlinear coefficient in polar coordinates $ d({\textbf{r}},\theta ) $. If we modulate the rectangular function $ g({\textbf{r}},\theta ) $ in addition transversely with the phase $ 2\pi /\Lambda z + l\theta $, we will generate a SH vortex of charge $ l $ for first-order QPM [28]. Fig. 3. Three schemes to modulate the $ {\chi ^{(2)}} $ nonlinearity inside the waveguide core for QPM. One period (A), two periods (A, B), and four periods (A, B, C, D). For SHG of 1064 nm, the core diameter is typically 12 µm, and the period of the QPM grating is approximately 6.6 µm (cfg. Fig. 1). 3. METHODS A. Writing Waveguides and 3D Nonlinear Structures with fs Laser Pulses A schematic of the layout is depicted in Fig. 1. The proposed design consists of a circular waveguide and an embedded multiscan LiQPM grating of damped nonlinearity for the SHG process. We use undoped congruent $ {{\rm LiNbO}_3} $ samples cut from a $z$-cut wafer with the dimensions $ 12 \times 10 \times 0.5\,\,{{\rm mm}^3} $. The $z$ and $y$ surfaces are polished to optical quality. The waveguides and nonlinear structures are directly written into the crystals using a fs laser lithography setup [34]. The system consists of a regenerative Ti:sapphire amplifier operating at a central wavelength of 800 nm with a repetition rate of 1 kHz (Coherent Legend). The pulse duration is 110 fs, and the maximum pulse energy is 1 mJ. The beam is attenuated by neutral density filters followed by a motorized half-wave plate in combination with a polarizing beam splitter. The laser output is triggered synchronous to the position of the stage, which is crucial for the inscription of transverse grating structures. The beam is subsequently focused into the lithium niobate sample using a microscope objective with a numerical aperture (NA) of 0.8. All structures are inscribed with a writing beam linearly polarized along the $y$ axis and parallel to direction of the waveguides. The sample is placed on a computer-controlled Aerotech motion stage and thereby translated in three dimensions with nanometer precision relative to the focus of the laser beam. The multiscan LiQPM gratings are inscribed with 80 µm/s translation velocity. The transverse feature size along the $x$ and $y$ axes of the QPM grating is approximately 300 nm, and the feature size along the optical $z$ axis is 700 nm. We can achieve with this setup in periods down to 700 nm [34] enabling SHG into the ultraviolet spectral region. The pulse energies to write the waveguides range between 120 nJ and 200 nJ, while the core is modulated with lower pulse energies between 60 nJ and 90 nJ. B. Measurement of the Linear and Nonlinear Properties A setup with two continuous-wave lasers at 1059 nm (Sacher Lasertechnik DFB laser diode) and 532 nm (Coherent Compass 315M) is used in order to linearly characterize the fabricated waveguides regarding their insertion losses and mode profiles. The output of both lasers is coupled into single-mode fibers and guided to the lithium niobate sample. The near-field output of the waveguides is afterwards imaged onto a camera by a ${20 \times}$ microscope objective with an NA of 0.4, which allows screening the mode profile at the exit facet. We use a laser-scanning SHG microscope to image the fs laser-induced structures in 3D [35,36]. The fundamental beam of a Ti-Sapphire laser (800 nm, 80 MHz repetition rate, about 60 fs pulse duration, and up to 3.5 nJ pulse energy) is coupled to a commercial laser-scanning microscope (Nikon eclipse, Ti-U) and tightly focused by the microscope objective with an NA of 0.8 to a near-diffraction-limited spot in the sample. The position of the focus is raster-scanned in the $xy$ plane by a piezoelectric table (P-545, PI nano). The intensity of the SH signal is collected by a condenser lens (NA 0.9) and measured by a photomultiplier as a function of the focus position. Quasi-phase-matched SHG is performed with a $Q$-switched Nd:YAG laser with a wavelength of 1064 nm, pulse duration of 4.1 ns, and repetition rate of 100 Hz. Fabricated devices are mounted on a controllable heating element with a resolution and temperature stability of 0.1°C. The fundamental wave is extraordinarily polarized and coupled into the waveguides using a ${4} \times $ microscope objective with ${\rm NA}={0.1}$. The back-facet is imaged onto a camera by a ${20} \times $ microscope objective. Typical pulse energies are on the order of a few µJ, and the maximum fundamental peak power used is 400 W. A. Influence of the Writing Pulse Energy Efficient LiQPM waveguide devices should exhibit single-mode guiding and low losses at the fundamental frequency and SH, respectively, as well as a high nonlinearity. Therefore, in a first step, the insertion losses and mode profiles have been measured in dependence on the writing energy between 120 nJ and 200 nJ and as a function of the core diameter between 10 µm and 25 µm. A good trade-off between low losses and single-mode guiding is achieved with a core diameter of 12 µm at an energy of 150 nJ. These waveguides exhibit reasonable low insertion losses of 8.6 dB at 532 nm and only weakly multimode profiles as well as very symmetric profiles at 1059 nm accompanied by low insertion losses of 5.5 dB. Larger-core diameters support multimode guiding, while a smaller-core diameter leads to increased insertion losses. Fig. 4. Laser-scanning SHG microscope image of waveguide and LiQPM grating with a period of $ \Lambda = 6.6\,\, \unicode{x00B5}{\rm m} $. The first part of the grating is not surrounded by a waveguide in this example to illustrate that the grating is located in the middle of the waveguide. Fig. 5. Power of the SH and insertion losses at both wavelengths measured in dependence on the writing pulse energy. A total number of 17 individual devices with a 5 mm long LiQPM grating are fabricated using a 10.2 mm long sample. The LiQPM gratings are adjusted precisely in the middle of the waveguide core, as can be seen in Fig. 4. The performed analysis of SH power and insertion losses covers the full range from no visible detectable modification in the LiQPM grating to strong laser-induced damage. All temperature tuning curves are evaluated under the same conditions, and the characteristic build-up of the normalized SH power as a function of the writing pulse energy is indicated by the gray shaded area in Fig. 5. The presented data follow an almost linear slope that rapidly drops down when a certain energy threshold value around 80 nJ is exceeded. This behavior is associated with a significant increase in insertion losses at both wavelengths. Particularly, the power attenuation of the SH wave is subject to a steep change, and extremely high values of more than $- 30 \,\, {\rm dB}$ are obtained, although the grating length is 5 mm only. Considering the associated propagation losses, the $ {\chi ^{(2)}} $ modulation saturates for high-pulse energies. The optimum energy regime between 75 nJ and 80 nJ that is employed to obtain a localized damping of the $ {\chi ^{(2)}} $ nonlinearity is associated with severe material damage and high optical attenuation, as can be seen in Fig. 5 and related optical microscope images. This energy regime can be clearly identified with a starting type-II filamentation process. This is also reflected in the fact that LiQPM gratings do not show any signature of reversibility or reduction in efficiency upon thermal treatment. B. Chirped and Single-Period QPM Grating Chirped gratings can be used for the compression and shaping of ultrashort laser pulses [37–39]. Figure 6 shows the experimental temperature tuning and mode images of a chirped LiQPM device in comparison to a single-period device fabricated with equal parameters. Both gratings are 7 mm long and integrated into a 9.7 mm long waveguide. The single-period grating has a period of 6.6 µm, while the chirped grating has a period linearly increasing from $ {\Lambda _{{\rm start}}} = 6.586\,\, \unicode{x00B5}{\rm m} $ to $ {\Lambda _{{\rm end}}} = 6.614\,\, \unicode{x00B5}{\rm m} $. The chirp leads to an almost ideal top-hat profile and a significantly increased temperature acceptance width of approximately 11.3°C. This corresponds to an equivalent absolute increase in the spectral bandwidth from 200 pm to 0.94 nm. The integrals of the unchirped and chirped tuning curves are 77 W°C and 152 W°C, respectively. This is not in contradiction to Plancherel's theorem because the fundamental power of 291 W already exceeds the undepleted pump regime for the single-period grating [27], while the efficiency of the chirped grating is still linearly increasing with increasing pump power. It should be noted that the phase-matching temperature shifts by approximately $ - {30^\circ {\rm C}}$ with respect to the dispersion of congruent $ {{\rm LiNbO}_3} $ [40]. This shift corresponds to a change in the bulk material's dispersion of $ \vert {n_{2\omega }} - {n_\omega } \vert = 5.5 \times {10^{ - 4}} $ due to the refractive index profile of the waveguide [27]. However, the additional refractive index change in the periodic $ {\chi ^{(2)}} $ grating is small and does not influence the phases of the fundamental and SH waves much. Fig. 6. Experimental temperature tuning of a chirped LiQPM device for broadband SHG at 291 W fundamental power in comparison to a single-period device fabricated with equal parameters. Right: images of the fundamental and chirped second-harmonic mode at the end of the waveguide. C. Split-Core Structures for Multiple QPM LiQPM waveguide devices can be customized to allow for cascaded frequency conversion processes and parallel multi-wavelength SHG. The proposed LiQPM fabrication method based on direct laser writing allows monolithic inscription of such devices, as shown in Fig. 7. As indicated in the associated schematic, two separated LiQPM gratings with periods of 6.6 µm and 6.525 µm are inscribed into a 9.8 mm long waveguide. Hence, the temperature tuning curve shows two distinct maxima at 124.7°C and 167.9°C with an average SH power of 6.7 W at 291 W fundamental power. The two gray lines are numerical calculations by integrating Eqs. (1) and (2) taking into account different absorption coefficients of the gratings $ \alpha _\omega ^{{\rm grating}} = 6.7\,\,{\rm dB}/{\rm cm} $ and $ \alpha _{2\omega }^{{\rm grating}} = 17.2\,\,{\rm dB}/{\rm cm} $, and of the waveguides $ \alpha _\omega ^{{\rm waveguide}} = 4.6\,\,{\rm dB}/{\rm cm} $ and $ \alpha _{2\omega }^{{\rm waveguide}} = 7.7\,\,{\rm dB}/{\rm cm} $ for the fundamental and SH waves, respectively. As the fundamental wave propagates first through the grating with the shorter period and higher phase-matching temperature, this efficiency is larger than that of the second grating. This is in accordance with our numerical calculations. The temperature acceptance width of 6.8°C is in good agreement with the performed numerical calculation and the comparably short grating length of 3.5 mm. Since such a device is typically designed for simultaneous multi-wavelength operation rather than SHG of a single fundamental wave at different temperatures, the notation multi-wavelength LiQPM is used here. Assuming an operation temperature of 125°C, the presented LiQPM waveguide device allows for simultaneous SHG of 1064 nm and 1060.5 nm radiation. The employed periods and associated phase-matching wavelengths are selected here in a narrow region according to the 1064 nm laser source used for the nonlinear probe experiments. In general, fabrication of such a device can be on-the-fly adapted to any design wavelength. Furthermore, an arbitrary succession of LiQPM gratings is possible. Fig. 7. Sequential multi-wavelength SHG. The sequential scheme is composed of two successive LiQPM gratings inscribed into a single waveguide. A novel approach that allows for parallel instead of sequential multi-wavelength SHG is shown in Fig. 8. In this configuration, the LiQPM grating inside the waveguide core is split into two segments, each of which has its own period and associated design wavelength as indicated in the schematic (cf. Fig. 3). The obtained SH power of the split-core LiQPM device is similar to the former sequential configuration (Fig. 7), and an average efficiency of 2.4% is achieved at 291 W fundamental power. The temperature acceptance width of 3.25°C is in good agreement with the 7 mm long grating fabricated with 84 nJ, as indicated by the numerical calculation. To allow direct comparison, the numerical traces of the sequential gratings (Fig. 7) are shown again by the gray dashed lines. In this configuration, the reduced overlap between a LiQPM segment and the fundamental wave, which lowers the SHG efficiency, is almost exactly compensated for by the increased grating length. This is also reflected in the numerical calculation where doubling of the effective area from $ 75\,\, \unicode{x00B5}{\rm m}^2 $ to $ 150\,\, \unicode{x00B5}{\rm m}^2 $ is assumed. Fig. 8. Parallel multi-wavelength SHG. Parallel waveguide SHG is realized in a novel split-core approach as illustrated in the schematic. As a next step towards complex nonlinear devices, the number of LiQPM segments is increased, which allows for parallel multi-wavelength SHG of four individual wavelengths, as shown in Fig. 9. Fabrication parameters such as grating length and pulse energy are adopted from the previous configuration to allow direct comparison. All four distinct SHG maxima are observed at the predefined phase-matching temperatures, which clearly demonstrates the functionality of the fabricated 3D LiQPM device. According to the periods of 6.625, 6.6, 6.55, and 6.525 µm, simultaneously quasi-phase-matched SHG of 1065.3, 1064, 1061.6, and 1060.5 nm radiation is possible by this unique approach. An average SH power of 1.33 W is achieved, which corresponds to an efficiency of 0.46% at 291 W fundamental power. Similar to the former dual-segment device, the effective area is increased to $ 300\,\, \unicode{x00B5}{\rm m}^2 $ for each segment to account for the spatial overlap of the particular LiQPM grating and the fundamental wave. For the numerical integration of Eqs. (1) and (2), we used the absorption coefficients of the gratings and waveguides, as they have been measured separately (see Section 3.B). The final free parameter, which determines the effective nonlinearity in each grating, is the modulation depth $ v $. For each channel, the modulation depth $ v $ is fitted to match the measured power of the SH. The modulation depths $ v $ range between 0.866 and 0.902, and good agreement of the measured temperature acceptance width as well as SH power is obtained. The peak at 130°C is larger than predicted by theory, i.e., it is larger than the side lobe of the 6.6 µm grating. One reason for this might be that some part of grating B exhibits a period that is increased by about 20 nm. Fig. 9. Split-core LiQPM device for multi-wavelength SHG. A three-dimensional LiQPM grating composed of four segments is inscribed into the waveguide core to enable simultaneous frequency conversion of four individual design wavelengths. The significance of the presented four-segment LiQPM device is twofold. On one hand, the spatial overlap of the fundamental wave and a particular grating segment is intrinsically reduced, which in turn limits the maximum efficiency. This, however, is counterbalanced by the fact that dense packaging of multi-wavelength frequency conversion elements becomes possible. On the other hand, the presented four-segment LiQPM structure identifies the first experimental realization of a 3D QPM device integrated in a waveguide. Whereas the former dual-segment LiQPM structure could, in principle, also be fabricated using a specifically poled crystal, any kind of non-uniform $ {\chi ^{(2)}} $ modification along the crystallographic $c$ axis, as is the case for the A-D and B-C segments, is not possible by classic fabrication methods but becomes available using LiQPM. The 3D capability of LiQPM can be extended to more sophisticated applications such as nonlinear beam shaping. D. Helically Twisted Structure for Nonlinear Beam Shaping Finally, we propose a concept for the nonlinear generation of optical vortex beams by a helically twisted nonlinear susceptibility. As illustrated in Fig. 10(a), the $ {\chi ^{(2)}} $ modulation circuits around the optical axis in longitudinal direction with a period that fulfills the QPM condition for SHG. As a first experimental proof of principle, we want to show the fabrication of a helical nonlinear structure and its properties. Therefore, larger structures with a transverse diameter of 35 µm to 50 µm and a length of 6 mm have been induced first without waveguides. These structures can be better analyzed microscopically. The resolution of the setup allows us to inscribe helical structures with a diameter of 12 µm into waveguides. However, the waveguide has to support guiding of the shaped SH mode. To accomplish this, the waveguide parameters have to be adapted accordingly. Magnesium doped $x$-cut lithium niobate substrates are employed as a host material to account for threshold characteristics and focus splitting that occur for large-scale structures using $z$-cut samples [41,42]. According to the dopant concentration and associated change in dispersion, an increased QPM period of 6.7 µm is selected [43]. Similar to the 3D schematic [Fig. 10(a)], the 3D SHG microscope image reveals the twisted nature of the 3D pattern [Fig. 10(b)]. A projection of the realized structure is shown in Fig. 11(b). The bended periods are clearly visible. The detected SH signal as a function of the temperature is shown in Fig. 11(c) where a fundamental pulse energy of 2 µJ is employed. A phase-matching temperature of approximately 180°C is obtained for the volume structured material and extraordinary polarization. In this demonstration, the fabricated device provides limited homogeneity along the vertical direction, which is best seen in the microscope image of the front face of the helical structure [Fig. 11(a)]. Thus, the non-uniform conversion strength made it impossible to record the characteristic vortex pattern in this first experimental approach. One reason for the inhomogeneities is that the pulse energy drifts slightly over the long fabrication time of 36 h. The fabrication time will be reduced by two orders of magnitude if a pulse laser with a repetition rate of 100 kHz would be used. The second reason is the large refractive index of lithium niobate that leads to spherical aberrations. Spherical aberrations have to be compensated for in future experiments, for instance, by using a spatial light modulator [44]. Fig. 10. SHG with helical twisted structure. (a) Illustration of a helical LiQPM grating that transfers a fundamental Gaussian beam into a vortex second-harmonic beam. (b) 3D view of SHG microscope images of the helical LiQPM grating fabricated in magnesium doped $x$-cut lithium niobate (see Visualization 1). Fig. 11. Microscope images and SH power. (a) Microscope image of the front face of a large-scale helical QPM structure. (b) Microscope image from the top where the bended periods of the helical structure are visible. (c) Temperature tuning of SH power. We have demonstrated monolithic fabrication of waveguides with integrated 3D nonlinear photonic structures by LiQPM. The integration of 3D QPM structures into waveguides increases the conversion efficiency compared to nonlinear photonic crystals. Depressed-cladding waveguides with almost single-mode guiding and reasonable loss are fabricated by carefully choosing pulse energy and writing velocity. These parameters were optimized as well for the LiQPM structures to get the highest possible SHG power. We have fabricated chirped structures for broadband SHG as well as cascaded single-period structures for multi-wavelength SHG. As LiQPM allows for the fabrication of periods down to 2 µm, structures for SHG of 800 nm fs-laser pulses can be designed. Parallel multi-wavelength frequency conversion becomes possible by splitting the waveguide core into two or four parts, allowing for compact designs. Especially the quad-core structure cannot be realized by electric field poling. 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Mater. (1) Appl. Opt. (1) Appl. Phys. Lett. (9) IEEE J. Quantum. Electron. (1) IEEE J. Sel. Top. Quantum Electron. (1) IEEE Photon. J. (1) J. Opt. Soc. Am. B (1) J. Phys. D (1) Laser Photon. Rev. (4) Nat. Commun. (2) Nat. Photonics (2) Opt. Express (8) Opt. Lett. (8) Phys. Rev. (1) Phys. Rev. Lett. (3) Supplementary Material (1) Visualization 1 3D view of SHG microscope images of the helical LiQPM grating fabricated in magnesium doped x-cut lithium niobate. View in Article \| Download Full Size \| PPT Slide \| PDF Equations on this page are rendered with MathJax. Learn more. (1) d d y A ω = − i κ ∗ d ( r ) A ω ∗ A 2 ω exp ⁡ ( − i Δ k y ) − α ω A ω , (2) d d y A 2 ω = − i κ d ( r ) A ω 2 exp ⁡ ( i Δ k y ) − α 2 ω A 2 ω , (3) κ 2 = 2 ω 2 ϵ 0 c 3 1 n ω 2 n 2 ω S e f f , (4) Δ k = k 2 ω − 2 k ω , (5) d e f f = d m a x π ( 1 − v ) , Prem Kumar, Editor-in-Chief	CommonCrawl
Food Science of Animal Resources (한국축산식품학회지) Korean Society for Food Science of Animal Resources (한국축산식품학회) Agriculture, Fishery and Food > Agricultural, fisheries, livestock goods safety Agriculture, Fishery and Food > Food Science Korean J. Food Sci. An. is publishing original research articles and reviews on basic and applied aspects of the use of animal resources focusing on the livestock products including the meat, eggs, and dairy products. http://www.kosfaj.org/submission/Login.html KSCI KCI SCIE Effects of Mechanically Deboned Chicken Meat (MDCM) and Collagen on the Quality Characteristics of Semi-dried Chicken Jerky Song, Dong-Heon;Choi, Ji-Hun;Choi, Yun-Sang;Kim, Hyun-Wook;Hwang, Ko-Eun;Kim, Yong-Jae;Ham, Youn-Kyung;Kim, Cheon-Jei 727 https://doi.org/10.5851/kosfa.2014.34.6.727 PDF KSCI KPUBS JATS XML This study was conducted to determine the effects of using mechanically deboned chicken meat (MDCM) and collagen on quality characteristics of semi-dried chicken jerky. In experiment I, semi-dried chicken jerky was prepared with the replacement of chicken breast with MDCM (0, 10, 20, and 30%). The pH value of the jerky formulated with only chicken breast was 5.94, while the replacement of chicken breast with MDCM significantly increased the pH (p<0.05). The protein content and shear force of the jerkies decreased with increasing amounts of MDCM, whereas the fat, ash content and processing yield showed the opposite tendency (p<0.05). Replacement with up to 10% MDCM had no adverse effects on the sensory characteristics of the semi-dried chicken jerky. In experiment II, four levels of pork collagen (0, 1, 2, and 3%) were added to the semi-dried chicken jerky formulated with 90% chicken breast and 10% MDCM. The addition of collagen increased the moisture content, but decreased the ash content of the jerkies produced (p<0.05). The processing yield of the jerkies increased with increasing added amounts of collagen (p<0.05). It was found that the jerkies formulated with 0-2% collagen had significantly higher overall acceptance score than those prepared with 3% collagen (p<0.05). In conclusion, MDCM and collagen could be useful ingredients to reduce the production cost and improve the processing yield of semidried chicken jerky. The optimal levels of MDCM and collagen which could be added without adverse effects on the sensory characteristics were up to 10% and 2%, respectively. Probabilistic Models to Predict the Growth Initiation Time for Pseudomonas spp. in Processed Meats Formulated with NaCl and NaNO2 Jo, Hyunji;Park, Beomyoung;Oh, Mihwa;Gwak, Eunji;Lee, Heeyoung;Lee, Soomin;Yoon, Yohan 736 This study developed probabilistic models to determine the initiation time of growth of Pseudomonas spp. in combinations with $NaNO_2$ and NaCl concentrations during storage at different temperatures. The combination of 8 NaCl concentrations (0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, and 1.75%) and 9 $NaNO_2$ concentrations (0, 15, 30, 45, 60, 75, 90, 105, and 120 ppm) were prepared in a nutrient broth. The medium was placed in the wells of 96-well microtiter plates, followed by inoculation of a five-strain mixture of Pseudomonas in each well. All microtiter plates were incubated at 4, 7, 10, 12, and $15^{\circ}C$ for 528, 504, 504, 360 and 144 h, respectively. Growth (growth initiation; GI) or no growth was then determined by turbidity every 24 h. These growth response data were analyzed by a logistic regression to produce growth/no growth interface of Pseudomonas spp. and to calculate GI time. NaCl and $NaNO_2$ were significantly effective (p<0.05) on inhibiting Pseudomonas spp. growth when stored at $4-12^{\circ}C$. The developed model showed that at lower NaCl concentration, higher $NaNO_2$ level was required to inhibit Pseudomonas growth at $4-12^{\circ}C$. However, at $15^{\circ}C$, there was no significant effect of NaCl and $NaNO_2$. The model overestimated GI times by $58.2{\pm}17.5$ to $79.4{\pm}11%$. These results indicate that the probabilistic models developed in this study should be useful in calculating the GI times of Pseudomonas spp. in combination with NaCl and $NaNO_2$ concentrations, considering the over-prediction percentage. Evaluation of the Activities of Antioxidant Enzyme and Lysosomal Enzymes of the Longissimus dorsi Muscle from Hanwoo (Korean Cattle) in Various Freezing Conditions Kang, Sun Moon;Kang, Geunho;Seong, Pil-Nam;Park, Beomyoung;Kim, Donghun;Cho, Soohyun 742 This study was conducted to evaluate the activities of antioxidant enzyme (glutathione peroxidase (GSH-Px)) and lysosomal enzymes (alpha-glucopyranosidase (AGP) and beta-N-acetyl-glucosaminidase (BNAG)) of the longissimus dorsi (LD) muscle from Hanwoo (Korean cattle) in three freezing conditions. Following freezing at -20, -60, and $-196^{\circ}C$ (liquid nitrogen), LD samples (48 h post-slaughter) were treated as follows: 1) freezing for 14 d, 2) 1 to 4 freeze-thaw cycles (2 d of freezing in each cycle), and 3) refrigeration ($4^{\circ}C$) for 7 d after 7 d of freezing. The control was the fresh (non-frozen) LD. Freezing treatment at all temperatures significantly (p<0.05) increased the activities of GSH-Px, AGP, and BNAG. The $-196^{\circ}C$ freezing had similar effects to the $-20^{\circ}C$ and $-60^{\circ}C$ freezing. Higher (p<0.05) enzymes activities were sustained in frozen LD even after 4 freeze-thaw cycles and even for 7 d of refrigeration after freezing. These findings suggest that freezing has remarkable effects on the activities of antioxidant enzyme and lysosomal enzymes of Hanwoo beef in any condition. Rapid Determination of L-carnitine in Infant and Toddler Formulas by Liquid Chromatography Tandem Mass Spectrometry Ahn, Jang-Hyuk;Kwak, Byung-Man;Park, Jung-Min;Kim, Na-Kyeoung;Kim, Jin-Man 749 A rapid and simple analytical method for L-carnitine was developed for infant and toddler formulas by liquid chromatography tandem mass spectrometry (LC-MS/MS). A 0.3 g of infant formula and toddler formula sample was mixed in a 50 mL conical tube with 9 mL water and 1 mL 0.1 M hydrochloric acid (HCl) to chemical extraction. Then, chloroform was used for removing a lipid fraction. After centrifuged, L-carnitine was separated and quantified using LC-MS/MS with electrospray ionization (ESI) mode. The precursor ion for L-carnitine was m/z 162, and product ions were m/z 103 (quantitative) and m/z 85 (qualitative), respectively. The results for spiked recovery test were in the range of 93.18-95.64% and the result for certified reference material (SRM 1849a) was within the range of the certificated values. This method could be implemented in many laboratories that require time and labor saving. Studies on Physical and Sensory Properties of Premium Vanilla Ice Cream Distributed in Korean Market Choi, Mi-Jung;Shin, Kwang-Soon 757 The object of this study was to investigate the difference in physical and sensory properties of various premium ice creams. The physical properties of the various ice creams were compared by manufacturing brand. The water contents of the samples differed, with BR having the highest value at 60.5%, followed by NT and CS at 57.8% and 56.9%, respectively. The higher the water content, the lower Brix and milk fat contents in all samples. The density of the samples showed almost similar values in all samples (p>0.05). The viscosity of each ice cream had no effect on the water content in any of the brands. Before melting of the ice cream, the total color difference was dependent on the lightness, especially in the vanilla ice cream, owing to the reflection of light on the surface of the ice crystals. The CS product melted the fastest. In the sensory test, CS obtained a significantly higher sweetness intensity score but a lower score for color intensity, probably due to the smaller difference in total color, by which consumers might consider the color of CS as less intense. From this study, the cold chain system for ice cream distribution might be important to decide the physical properties although the concentration of milk fat is key factor in premium ice cream. Comparison of the Microsatellite and Single Nucleotide Polymorphism Methods for Discriminating among Hanwoo (Korean Native Cattle), Imported, and Crossbred Beef in Korea Heo, Eun-Jeong;Ko, Eun-Kyung;Seo, Kun-Ho;Chon, Jung-Whan;Kim, Young-Jo;Park, Hyun-Jung;Wee, Sung-Hwan;Moon, Jin-San 763 The identity of 45 Hanwo and 47 imported beef (non-Hanwoo) samples from USA and Australia were verified using the microsatellite (MS) marker and single nucleotide polymorphism (SNP) methods. Samples were collected from 19 supermarkets located in the city of Seoul and Gyeonggi province, South Korea, from 2009 to 2011. As a result, we obtained a 100% concordance rate between the MS and SNP methods for identifying Hanwoo and non-Hanwoo beef. The MS method presented a 95% higher individual discriminating value for Hanwoo (97.8%) than for non-Hanwoo (61.7%) beef. For further comparison of the MS and SNP methods, blood samples were collected and tested from 54 Hanwoo ${\times}$ Holstein crossbred cattle (first, second, and third generations). By using the SNP and MS methods, we correctly identified all of the first-generation crossbred cattle as non-Hanwoo; in addition, among the second and third generation crossbreds, the ratio identified as Hanwoo was 20% and 10%, respectively. The MS method used in our study provides more information, but requires sophisticated techniques during each experimental process. By contrast, the SNP method is simple and has a lower error rate. Our results suggest that the MS and SNP methods are useful for discriminating Hanwoo from non-Hanwoo breeds. Preparation and Characterization of Antioxidant Peptides from Fermented Goat Placenta Hou, Yinchen;Zhou, Jiejing;Liu, Wangwang;Cheng, Yongxia;Wu, Li;Yang, Gongming 769 The goat placenta was fermented by Bacillus subtilis and the optimal fermentation parameters of strongest antioxidant capacity of peptides were obtained using response surface methodology (RSM). The effects of fermentation time, initial pH value and glucose content on the 1,1-diphenyl-2-picrylhydrazyl (DPPH) radical scavenging capacity of the goat peptides were well fitted to a quadric equation with high determination coefficients. According to the data analysis of design expert, the strongest DPPH radical scavenging capacity value was obtained with the following conditions: content of glucose was 2.23%, initial pH value was 7.00 and fermentation time was 32.15 h. The DPPH radical scavenging capacity commonly referring antioxidant activity showed a concentration dependency and increased with increasing peptide concentration. The effects of temperature and pH were assessed to determine the stability of antioxidant peptides prepared from goat placenta. Antioxidant peptides showed good stabilities when temperature was lower than $70^{\circ}C$. However, the antioxidant peptides lost antioxidant activities rapidly under alkaline and excessive acid condition. Ultrafiltration technique was performed to separate fermentation broth with different Mw (molecular weight). It was found that peptides in the range of < 3 KDa mainly accounted for the antioxidant activities. Effect of Novel Quick Freezing Techniques Combined with Different Thawing Processes on Beef Quality Jo, Yeon-Ji;Jang, Min-Young;Jung, You-Kyoung;Kim, Jae-Hyeong;Sim, Jun-Bo;Chun, Ji-Yeon;Yoo, Seon-Mi;Han, Gui-Jung;Min, Sang-Gi 777 This study investigated the effect of various freezing and thawing techniques on the quality of beef. Meat samples were frozen using natural convection freezing (NF), individual quick freezing (IQF), or cryogenic freezing (CF) techniques, followed by natural convection thawing (NCT) or running water thawing (RT). The meat was frozen until the core temperature reached $-12^{\circ}C$ and then stored at $-24^{\circ}C$, followed by thawing until the temperature reached $5^{\circ}C$. Quality parameters, such as the pH, water binding properties, CIE color, shear force, and microstructure of the beef were elucidated. Although the freezing and thawing combinations did not cause remarkable changes in the quality parameters, rapid freezing, in the order of CF, IQF, and NF, was found to minimize the quality deterioration. In the case of thawing methods, NCT was better than RT and the meat quality was influence on the thawing temperature rather than the thawing rate. Although the microstructure of the frozen beef exhibited an excessive loss of integrity after the freezing and thawing, it did not cause any remarkable change in the beef quality. Taken together, these results demonstrate that CF and NCT form the best combination for beef processing; however, IQF and NCT may have practical applications in the frozen food industry. Effects of Ground, Concentrated, and Powdered Beef on the Quality of Noodle Products Jeon, Ki Hong;Hwang, Yoon Seon;Kim, Young Boong;Kim, Eun Mi;Park, Jong Dae;Choi, Jin Young 784 The aim of this study was to ascertain the effects of beef on the quality characteristics, such as color, texture profile, water absorption ratio, volume, turbidity, and sensory evaluation, of noodle products. Various types of beef were added to the flour at a mixture ratio of ground beef (BG) 10, 15, 20, 25%; concentrated beef (BC) 9, 11, 13, 15%; and powdered beef (BP) 1, 3, 5, 7%. Each treatment was analyzed and compared with a 100% flour noodle as a control. With increasing BG, BC, and BP ratios, the L and b values for color decreased, while the a value increased, from that in the control. The hardness of the noodles treated with BG increased with increasing mixture ratios, but hardness decreased in the BC and BP treatments with increasing mixture ratios (p<0.05). The noodles with the largest beef composition in the mixtures of each treatment exhibited the highest turbidity scores, which was believed to be because the solid contents would be transferred to the soup during heating. In the sensory evaluation of cooked noodles, the BG 10%, BC 9%, and BP 1% treatments exhibited the best color. In an overall preference test, 20% of BG and 3% of BP could be added to the noodles. The best palatability was exhibited by the BG 10%, BC 13%, and BP 3% treatments. Distribution Channel and Microbial Characteristics of Pig By-products in Korea Kang, Geunho;Seong, Pil-Nam;Moon, Sungsil;Cho, Soohyun;Ham, Hyoung-Joo;Park, Kyoungmi;Kang, Sun-Moon;Park, Beom-Young 792 The distribution channel of meat by-products from the pig farm to the final consumer can include a meat processor, wholesale market, wholesaler, retailer, and butcher shop. Bacterial contamination at any of these steps remains to be a serious public health concern. The aim of this study was to evaluate the distribution channel and microbial characteristics of pig by-products in Korea. Upon evaluation of pig by-products in cold storage, we found that the small and large intestine were significantly (p<0.05) higher in pH value compared to the heart and liver. The total plate counts were not significantly different among offals until cold storage for 7 d. The coliform count after 1 d of cold storage was significantly (p<0.05) higher in small and large intestine than in the other organs. The coliform count of heart, liver, and stomach showed a higher coliform count than small and large intestine until 7 d of cold storage. As determined by 16S rRNA sequencing, contamination of major pig by-products with Escherichia coli, Shigella spp., and other bacterial species occurred. Therefore, our results suggest that a more careful washing process is needed to maintain quality and hygiene and to ensure the safety of pig by-products, especially for small and large intestine. Meat Species Identification using Loop-mediated Isothermal Amplification Assay Targeting Species-specific Mitochondrial DNA Cho, Ae-Ri;Dong, Hee-Jin;Cho, Seongbeom 799 Meat source fraud and adulteration scandals have led to consumer demands for accurate meat identification methods. Nucleotide amplification assays have been proposed as an alternative method to protein-based assays for meat identification. In this study, we designed Loop-mediated isothermal amplification (LAMP) assays targeting species-specific mitochondrial DNA to identify and discriminate eight meat species; cattle, pig, horse, goat, sheep, chicken, duck, and turkey. The LAMP primer sets were designed and the target genes were discriminated according to their unique annealing temperature generated by annealing curve analysis. Their unique annealing temperatures were found to be $85.56{\pm}0.07^{\circ}C$ for cattle, $84.96{\pm}0.08^{\circ}C$ for pig, and $85.99{\pm}0.05^{\circ}C$ for horse in the BSE-LAMP set (Bos taurus, Sus scrofa domesticus and Equus caballus); $84.91{\pm}0.11^{\circ}C$ for goat and $83.90{\pm}0.11^{\circ}C$ for sheep in the CO-LAMP set (Capra hircus and Ovis aries); and $86.31{\pm}0.23^{\circ}C$ for chicken, $88.66{\pm}0.12^{\circ}C$ for duck, and $84.49{\pm}0.08^{\circ}C$ for turkey in the GAM-LAMP set (Gallus gallus, Anas platyrhynchos and Meleagris gallopavo). No cross-reactivity was observed in each set. The limits of detection (LODs) of the LAMP assays in raw and cooked meat were determined from $10pg/{\mu}L$ to $100fg/{\mu}L$ levels, and LODs in raw and cooked meat admixtures were determined from 0.01% to 0.0001% levels. The assays were performed within 30 min and showed greater sensitivity than that of the PCR assays. These novel LAMP assays provide a simple, rapid, accurate, and sensitive technology for discrimination of eight meat species. Effect of NaCl, Gum Arabic and Microbial Transglutaminase on the Gel and Emulsion Characteristics of Porcine Myofibrillar Proteins Davaatseren, Munkhtugs;Hong, Geun-Pyo 808 This study investigated the effect of gum arabic (GA) combined with microbial transglutaminase (TG) on the functional properties of porcine myofibrillar protein (MP). As an indicator of functional property, heat-set gel and emulsion characteristics of MP treated with GA and/or TG were explored under varying NaCl concentrations (0.1-0.6 M). The GA improved thermal gelling ability of MP during thermal processing and after cooling, and concomitantly added TG assisted the formation of viscoelastic MP gel formation. Meanwhile, the addition of GA decreased cooking yield of MP gel at 0.6 M NaCl concentration, and the yield was further decreased by TG addition, mainly attributed by enhancement of protein-protein interactions. Emulsion characteristics indicated that GA had emulsifying ability and the addition of GA increased the emulsification activity index (EAI) of MP-stabilized emulsion. However, GA showed a negative effect on emulsion stability, particularly great drop in the emulsion stability index (ESI) was found in GA treatment at 0.6 M NaCl. Consequently, the results indicated that GA had a potential advantage to form a viscoelastic MP gel. For the practical aspect, the application of GA in meat processing had to be limited to the purposes of texture enhancer such as restructured products, but not low-salt products and emulsion-type meat products. Changes in the Microbiological Characteristics of Korean Native Cattle (Hanwoo) Beef Exposed to Ultraviolet (UV) Irradiation Prior to Refrigeration Kim, Hyun-Jung;Lee, Yong-Jae;Eun, Jong-Bang 815 The effects of ultraviolet (UV) radiation were investigated with regards to the microbial growth inhibitory effect on the shelf life of Korean native cattle (Hanwoo) beef prior to refrigerated storage. The Hanwoo samples were exposed to UV radiation ($4.5mW/cm^2$) for 0, 5, 10, 15, and 20 min. The UV-irradiated beef that was exposed for 20 min showed significantly reduced mesophilic and psychrotrophic bacterial populations to the extent of approximately 3 log cycles, as compared to that of non-irradiated beef. About 2.5 Log CFU/g of mesophilic bacteria were different compared with UV-irradiated and non-irradiated meat. UV irradiation showed the most significant growth inhibition effects on mesophilic and psychrotrophic bacteria. Coliform and Gram-negative bacteria were also reduced by 1 log cycle. The population of L. monocytogenes, S. Typhimurium, and E. coli O157:H7 decreased significantly to 53.33, 39.68, and 45.76% after 10 min of UV irradiation. They decreased significantly to 84.64, 80.76, and 84.12%, respectively, after 20 min of UV irradiation. The results show that UV irradiation time and the inhibitory effect were proportional. These results verified that UV radiation prior to refrigeration can effectively reduce the number of pathogenic bacteria on the surface of meat and improve the meat's microbial safety. Skeletal Muscle Troponin I (TnI) in Animal Fat Tissues to Be Used as Biomarker for the Identification of Fat Adulteration Park, Bong-Sup;Oh, Young-Kyoung;Kim, Min-Jin;Shim, Won-Bo 822 In this study, the existence of skeletal muscle troponin I (smTnI), well-known as a muscle protein in fat tissues, and the utilization of smTnI as a biomarker for the identification of fat adulteration were investigated. A commercial antibody (ab97427) specific to all of animals smTnI was used in this study. Fat and meat samples (cooked and non-cooked) of pork and beef, and chicken considered as representative meats were well minced and extracted by heating and non-heating methods, and the extracts from fat and meat tissues were probed by the antibody used in both enzyme-linked immunosorbent assay (ELISA) and immunoblot. The antibody exhibited a strong reaction to all meat and fat extracts in ELISA test. On the other hand, the results of immunoblot analsis revealed a 23 kDa high intensity band corresponding to the molecular weight of smTnI (23786 Da). These results demonstrate that the existence of smTnI in all animal fat tissues. Since there are monoclonal antibodies specific to each species smTnI, smTnI in fat tissues could be used as a biomarker to identify or determine animal species adulterated in meat products. Therefore, an analytical method to identify fraudulent fat adulteration can be developed with an antibody specific to each species smTnI. Lysate of Probiotic Lactobacillus plantarum K8 Modulate the Mucosal Inflammatory System in Dextran Sulfate Sodium-induced Colitic Rats Ahn, Young-Sook;Park, Min Young;Shin, Jae-Ho;Kim, Ji Yeon;Kwon, Oran 829 Inflammatory bowel disease (IBD) is caused by dysregulation of colon mucosal immunity and mucosal epithelial barrier function. Recent studies have reported that lipoteichoic acid (LTA) from Lactobacillus plantarum K8 reduces excessive production of pro-inflammatory cytokine. In this study, we investigated the preventive effects of lysate of Lb. plantarum K8 in dextran sulfate sodium (DSS)-induced colitis. Male Sprague-Dawley rats were orally pretreated with lysate of Lb. plantarum K8 (low dose or high dose) or live Lb. plantarum K8 prior to the induction of colitis using 4% DSS. Disease progression was monitored by assessment of disease activity index (DAI). Histological changes of colonic tissues were evaluated by hematoxylin and eosin (HE) staining. Tumor necrosis factor-alpha (TNF-${\alpha}$), interleukin-6 (IL-6) levels were measured using enzyme-linked immunosorbent assay (ELISA). The colon mRNA expressions of TNF-${\alpha}$, IL-6, and toll like receptor-2 (TLR-2) were examined by quantitative real-time-transcription polymerase chain reaction (qPCR). Lysate of Lb. plantarum K8 suppressed colon shortening, edema, mucosal damage, and the loss of DSS-induced crypts. The groups that received lysate of Lb. plantarum K8 exhibited significantly decreased levels of the pro-inflammatory cytokines TNF-${\alpha}$ and IL-6 in the colon. Interestingly, colonic expression of toll like receptor-2 mRNA in the high-dose lysate of Lb. plantarum K8 group increased significantly. Our study demonstrates the protective effects of oral lysate of Lb. plantarum K8 administration on DSS-induced colitis via the modulation of pro-inflammatory mediators of the mucosal immune system. Application of Response Surface Methodology (RSM) for Optimization of Anti-Obesity Effect in Fermented Milk by Lactobacillus plantarum Q180 Park, Sun-Young;Cho, Seong-A;Lim, Sang-Dong 836 Obesity, a condition in which an abnormally large amount of fat is stored in adipose tissue, causing an increase in body weight, has become a major public health concern worldwide. The purpose of this study was to optimize the process for fermented milk for the production of a functional product with an anti-obesity effect by using Lactobacillus plantarum Q180 isolated from human feces. We used a 3-factor, 3-level central composite design (CCD) combined with the response surface methodology (RSM). Concentration of skim milk powder (%, $X_1$), incubation temperature ($^{\circ}C$, $X_2$), and incubation time (h, $X_3$) were used as the independent factors, whereas pH (pH, $Y_1$), anti-lipase activity (%, $Y_2$) and anti-adipogenetic activity (%, $Y_3$) were used as the dependent factors. The optimal conditions of fermented milk for the highest anti-lipase and anti-adipogenetic activity with pH 4.4 were the 9.5% of skim milk powder, $37^{\circ}C$ of incubation temperature, 28 h of incubation time. In the fermentation condition, the predicted values of pH, anti-lipase activity and anti-adipogenetic activity were 4.47, 55.55, and 20.48%, respectively. However, the actual values of pH, anti-lipase activity and anti-adipogenetic activity were 4.50, 52.86, and 19.25%, respectively. These results demonstrate that 9.5% of skim milk powder and incubation at $37^{\circ}C$ for 28 h were the optimum conditions for producing functional fermented milk with an anti-obesity effect. Anti-oxidation and Anti-wrinkling Effects of Jeju Horse Leg Bone Hydrolysates Kim, Dongwook;Kim, Hee-Jin;Chae, Hyun-Seok;Park, Nam-Gun;Kim, Young-Boong;Jang, Aera 844 This study focused on the anti-oxidative and collagenase- and elastase inhibition effects of low molecular weight peptides (LMP) from commercial Jeju horse leg bone hydrolysates (JHLB) on pancreatin, via enzymatic hydrolysis. Cell viability of dermal fibroblasts exposed to UVB radiation upon treatment with LMP from JHLB was evaluated. Determination of the antioxidant activity of various concentrations of LMP from JHLB were carried out by assessing 1,1-diphenyl-2-picrylhydrazyl (DPPH) and 2,2-azino-bis-3-ethybenzothiazoline-6-sulphonic acid (ABTS) radical scavenging activity, ferric reducing antioxidant power (FRAP), and oxygen radical absorbance capacity (ORAC). The DPPH radical scavenging activity of LMP from JHLB (20 mg/mL) was 92.21% and ABTS radical scavenging activity (15 mg/mL) was 99.50%. FRAP activity (30 mg/mL) was $364.72{\mu}M/TE$ and ORAC activity (1 mg/mL) was $101.85{\mu}M/TE$. The anti-wrinkle potential was assessed by evaluating the elastase- and collagenase inhibition potential of these LMP. We found that 200 mg/mL of LMP from JHLB inhibited elastase activity by 41.32%, and 100 mg/mL of LMP from JHLB inhibited collagenase activity by 91.32%. The cell viability of untreated HS68 human dermal fibroblasts was 45% when exposed to a UVB radiation dose of $100mJ/cm^2$. After 24 h of incubation with $500{\mu}g/mL$ LMP from JHLB, the cell viability increased to 60%. These results indicate that LMP from JHLB has potential utility as an anti-oxidant and anti-wrinkle agent in the food and cosmetic industry. Additional in vivo tests should be carried out to further characterize these potential benefits. Novel Convenient Method to Determine Wettability and Dispersibility of Dairy Powders Lee, Jeae;Chai, Changhoon;Park, Dong June;Lim, Kwangsei;Imm, Jee-Young 852 This study was carried out to develop a simple, convenient, and reproducible testing device to determine wettability and dispersibility of dairy powders. The testing device consists of a sieve ($150{\mu}m$) attached to a sample chamber, sensors mounted on a supporting body and a main control unit containing a display panel. The sensors detect the difference in electrical resistance between air and water. A timer is automatically triggered by the sensor when the bottom of sample-loaded chamber contacts water in the petri dish. Wettability and dispersibility of commercial skim milk powders (SMPs) produced at different heating strengths (low-, medium-, and high-heat SMP) are compared using the new testing device. Wettability of the SMPs were correlated with particle size and are found to increase in the order of medium-, low-, and high-heat SMP regardless of the amount of sample tested. Dispersibility of SMPs showed the same trend and high heat-SMP which has the smallest particle size resulted in the lowest dispersibility. Unlike existing methods, the new testing device can determine both wettability and dispersibility of powders and successfully detected differences among the samples.	CommonCrawl
One-relator group In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group theory by providing many explicit examples of finitely presented groups. Formal definition A one-relator group is a group G that admits a group presentation of the form $G=\langle X\mid r=1\,\rangle $ (1) where X is a set (in general possibly infinite), and where $r\in F(X)$ is a freely and cyclically reduced word. If Y is the set of all letters $x\in X$ that appear in r and $X'=X\setminus Y$ then $G=\langle Y\mid r=1\,\rangle \ast F(X').$ For that reason X in (1) is usually assumed to be finite where one-relator groups are discussed, in which case (1) can be rewritten more explicitly as $G=\langle x_{1},\dots ,x_{n}\mid r=1\,\rangle ,$ (2) where $X=\{x_{1},\dots ,x_{n}\}$ for some integer $n\geq 1.$ Freiheitssatz Main article: Freiheitssatz Let G be a one-relator group given by presentation (1) above. Recall that r is a freely and cyclically reduced word in F(X). Let $y\in X$ be a letter such that $y$ or $y^{-1}$ appears in r. Let $X_{1}\subseteq X\setminus \{y\}$. The subgroup $H=\langle X_{1}\rangle \leq G$ is called a Magnus subgroup of G. A famous 1930 theorem of Wilhelm Magnus,[1] known as Freiheitssatz, states that in this situation H is freely generated by $X_{1}$, that is, $H=F(X_{1})$. See also[2][3] for other proofs. Properties of one-relator groups Here we assume that a one-relator group G is given by presentation (2) with a finite generating set $X=\{x_{1},\dots ,x_{n}\}$ and a nontrivial freely and cyclically reduced defining relation $1\neq r\in F(X)$. • A one-relator group G is torsion-free if and only if $r\in F(x_{1},\ldots ,x_{n})$ is not a proper power. • Every one-relator group G is virtually torsion-free, that is, admits a torsion-free subgroup of finite index.[4] • A one-relator presentation is diagrammatically aspherical.[5] • If $r\in F(x_{1},\ldots ,x_{n})$ is not a proper power then the presentation complex P for presentation (2) is a finite Eilenberg–MacLane complex $K(G,1)$.[6] • If $r\in F(x_{1},\ldots ,x_{n})$ is not a proper power then a one-relator group G has cohomological dimension $\leq 2$. • A one-relator group G is free if and only if $r\in F(x_{1},\ldots ,x_{n})$ is a primitive element; in this case G is free of rank n − 1.[7] • Suppose the element $r\in F(x_{1},\ldots ,x_{n})$ is of minimal length under the action of $\operatorname {Aut} (F_{n})$, and suppose that for every $i=1,\dots ,n$ either $x_{i}$ or $x_{i}^{-1}$ occurs in r. Then the group G is freely indecomposable.[8] • If $r\in F(x_{1},\ldots ,x_{n})$ is not a proper power then a one-relator group G is locally indicable, that is, every nontrivial finitely generated subgroup of G admits a group homomorphism onto $\mathbb {Z} $.[9] • Every one-relator group G has algorithmically decidable word problem.[10] • If G is a one-relator group and $H\leq G$ is a Magnus subgroup then the subgroup membership problem for H in G is decidable.[10] • It is unknown if one-relator groups have solvable conjugacy problem. • It is unknown if the isomorphism problem is decidable for the class of one-relator groups. • A one-relator group G given by presentation (2) has rank n (that is, it cannot be generated by fewer than n elements) unless $r\in F(x_{1},\ldots ,x_{n})$ is a primitive element.[11] • Let G be a one-relator group given by presentation (2). If $n\geq 3$ then the center of G is trivial, $Z(G)=\{1\}$. If $n=2$ and G is non-abelian with non-trivial center, then the center of G is infinite cyclic.[12] • Let $r,s\in F(X)$ where $X=\{x_{1},\dots ,x_{n}\}$. Let $N_{1}=\langle \langle r\rangle \rangle _{F(X)}$ and $N_{2}=\langle \langle s\rangle \rangle _{F(X)}$ be the normal closures of r and s in F(X) accordingly. Then $N_{1}=N_{2}$ if and only if $r$ is conjugate to $s$ or $s^{-1}$ in F(X).[13][14] • There exists a finitely generated one-relator group that is not Hopfian and therefore not residually finite, for example the Baumslag–Solitar group $B(2,3)=\langle a,b\mid b^{-1}a^{2}b=a^{3}\rangle $.[15] • Let G be a one-relator group given by presentation (2). Then G satisfies the following version of the Tits alternative. If G is torsion-free then every subgroup of G either contains a free group of rank 2 or is solvable. If G has nontrivial torsion, then every subgroup of G either contains a free group of rank 2, or is cyclic, or is infinite dihedral.[16] • Let G be a one-relator group given by presentation (2). Then the normal subgroup $N=\langle \langle r\rangle \rangle _{F(X)}\leq F(X)$ admits a free basis of the form $\{u_{i}^{-1}ru_{i}\mid i\in I\}$ for some family of elements $\{u_{i}\in F(X)\mid i\in I\}$.[17] One-relator groups with torsion Suppose a one-relator group G given by presentation (2) where $r=s^{m}$ where $m\geq 2$ and where $1\neq s\in F(X)$ is not a proper power (and thus s is also freely and cyclically reduced). Then the following hold: • The element s has order m in G, and every element of finite order in G is conjugate to a power of s.[18] • Every finite subgroup of G is conjugate to a subgroup of $\langle s\rangle $ in G. Moreover, the subgroup of G generated by all torsion elements is a free product of a family of conjugates of $\langle s\rangle $ in G.[4] • G admits a torsion-free normal subgroup of finite index.[4] • Newman's "spelling theorem"[19][20] Let $1\neq w\in F(X)$ be a freely reduced word such that $w=1$ in G. Then w contains a subword v such that v is also a subword of $r$ or $r^{-1}$ of length $\|v\|=1+(m-1)\|s\|$. Since $m\geq 2$ that means that $\|v\|>\|r\|/2$ and presentation (2) of G is a Dehn presentation. • G has virtual cohomological dimension $\leq 2$.[21] • G is a word-hyperbolic group.[22] • G has decidable conjugacy problem.[19] • G is coherent, that is every finitely generated subgroup of G is finitely presentable.[23] • The isomorphism problem is decidable for finitely generated one-relator groups with torsion, by virtue of their hyperbolicity.[24] • G is residually finite.[25] Magnus–Moldavansky method Starting with the work of Magnus in the 1930s, most general results about one-relator groups are proved by induction on the length \|r\| of the defining relator r. The presentation below follows Section 6 of Chapter II of Lyndon and Schupp[26] and Section 4.4 of Magnus, Karrass and Solitar[27] for Magnus' original approach and Section 5 of Chapter IV of Lyndon and Schupp[28] for the Moldavansky's HNN-extension version of that approach.[29] Let G be a one-relator group given by presentation (1) with a finite generating set X. Assume also that every generator from X actually occurs in r. One can usually assume that $\#X\geq 2$ (since otherwise G is cyclic and whatever statement is being proved about G is usually obvious). The main case to consider when some generator, say t, from X occurs in r with exponent sum 0 on t. Say $X=\{t,a,b,\dots ,z\}$ in this case. For every generator $x\in X\setminus \{t\}$ one denotes $x_{i}=t^{-i}xt^{i}$ where $i\in \mathbb {Z} $. Then r can be rewritten as a word $r_{0}$ in these new generators $X_{\infty }=\{(a_{i})_{i},(b_{i})_{i},\dots ,(z_{i})_{i}\}$ with $\|r_{0}\|<\|r\|$. For example, if $r=t^{-2}btat^{3}b^{-2}a^{2}t^{-1}at^{-1}$ then $r_{0}=b_{2}a_{1}b_{-2}^{-2}a_{-2}^{2}a_{-1}$. Let $X_{0}$ be the alphabet consisting of the portion of $X_{\infty }$ given by all $x_{i}$ with $m(x)\leq i\leq M(x)$ where $m(x),M(x)$ are the minimum and the maximum subscripts with which $x_{i}^{\pm 1}$ occurs in $r_{0}$. Magnus observed that the subgroup $L=\langle X_{0}\rangle \leq G$ is itself a one-relator group with the one-relator presentation $L=\langle X_{0}\mid r_{0}=1\rangle $. Note that since $\|r_{0}\|<\|r\|$, one can usually apply the inductive hypothesis to $L$ when proving a particular statement about G. Moreover, if $X_{i}=t^{-i}X_{0}t^{i}$ for $i\in \mathbb {Z} $ then $L_{i}=\langle X_{i}\rangle =\langle X_{i}\|r_{i}=1\rangle $ is also a one-relator group, where $r_{i}$ is obtained from $r_{0}$ by shifting all subscripts by $i$. Then the normal closure $N=\langle \langle X_{0}\rangle \rangle _{G}$ of $X_{0}$ in G is $N=\left\langle \bigcup _{i\in \mathbb {Z} }L_{i}\right\rangle .$ Magnus' original approach exploited the fact that N is actually an iterated amalgamated product of the groups $L_{i}$, amalgamated along suitably chosen Magnus free subgroups. His proof of Freiheitssatz and of the solution of the word problem for one-relator groups was based on this approach. Later Moldavansky simplified the framework and noted that in this case G itself is an HNN-extension of L with associated subgroups being Magnus free subgroups of L. If for every generator from $X_{0}$ its minimum and maximum subscripts in $r_{0}$ are equal then $G=L\ast \langle t\rangle $ and the inductive step is usually easy to handle in this case. Suppose then that some generator from $X_{0}$ occurs in $r_{0}$ with at least two distinct subscripts. We put $Y_{-}$ to be the set of all generators from $X_{0}$ with non-maximal subscripts and we put $Y_{+}$ to be the set of all generators from $X_{0}$ with non-maximal subscripts. (Hence every generator from $Y_{-}$ and from $Y_{-}$ occurs in $r_{0}$ with a non-unique subscript.) Then $H_{-}=\langle Y_{-}\rangle $ and $H_{+}=\langle Y_{+}\rangle $ are free Magnus subgroups of L and $t^{-1}H_{-}t=H_{+}$. Moldavansky observed that in this situation $G=\langle L,t\mid t^{-1}H_{-}t=H_{+}\rangle $ is an HNN-extension of L. This fact often allows proving something about G using the inductive hypothesis about the one-relator group L via the use of normal form methods and structural algebraic properties for the HNN-extension G. The general case, both in Magnus' original setting and in Moldavansky's simplification of it, requires treating the situation where no generator from X occurs with exponent sum 0 in r. Suppose that distinct letters $x,y\in X$ occur in r with nonzero exponents $\alpha ,\beta $ accordingly. Consider a homomorphism $f:F(X)\to F(X)$ given by $f(x)=xy^{-\beta },f(y)=y^{\alpha }$ and fixing the other generators from X. Then for $r'=f(r)\in F(X)$ the exponent sum on y is equal to 0. The map f induces a group homomorphism $\phi :G\to G'=\langle X\mid r'=1\rangle $ that turns out to be an embedding. The one-relator group G' can then be treated using Moldavansky's approach. When $G'$ splits as an HNN-extension of a one-relator group L, the defining relator $r_{0}$ of L still turns out to be shorter than r, allowing for inductive arguments to proceed. Magnus' original approach used a similar version of an embedding trick for dealing with this case. Two-generator one-relator groups It turns out that many two-generator one-relator groups split as semidirect products $G=F_{m}\rtimes \mathbb {Z} $. This fact was observed by Ken Brown when analyzing the BNS-invariant of one-relator groups using the Magnus-Moldavansky method. Namely, let G be a one-relator group given by presentation (2) with $n=2$ and let $\phi :G\to \mathbb {Z} $ be an epimorphism. One can then change a free basis of $F(X)$ to a basis $t,a$ such that $\phi (t)=1,\phi (a)=0$ and rewrite the presentation of G in this generators as $G=\langle a,t\mid r=1\rangle $ where $1\neq r=r(a,t)\in F(a,t)$ is a freely and cyclically reduced word. Since $\phi (r)=0,\phi (t)=1$, the exponent sum on t in r is equal to 0. Again putting $a_{i}=t^{-i}at^{i}$, we can rewrite r as a word $r_{0}$ in $(a_{i})_{i\in \mathbb {Z} }.$ Let $m,M$ be the minimum and the maximum subscripts of the generators occurring in $r_{0}$. Brown showed[30] that $\ker(\phi )$ is finitely generated if and only if $m<M$ and both $a_{m}$ and $a_{M}$ occur exactly once in $r_{0}$, and moreover, in that case the group $\ker(\phi )$ is free. Therefore if $\phi :G\to \mathbb {Z} $ is an epimorphism with a finitely generated kernel, then G splits as $G=F_{m}\rtimes \mathbb {Z} $ where $F_{m}=\ker(\phi )$ is a finite rank free group. Later Dunfield and Thurston proved[31] that if a one-relator two-generator group $G=\langle x_{1},x_{2}\mid r=1\rangle $ is chosen "at random" (that is, a cyclically reduced word r of length n in $F(x_{1},x_{2})$ is chosen uniformly at random) then the probability $p_{n}$ that a homomorphism from G onto $\mathbb {Z} $ with a finitely generated kernel exists satisfies $0.0006<p_{n}<0.975$ for all sufficiently large n. Moreover, their experimental data indicates that the limiting value for $p_{n}$ is close to $0.94$. Examples of one-relator groups • Free abelian group $\mathbb {Z} \times \mathbb {Z} =\langle a,b\mid a^{-1}b^{-1}ab=1\rangle $ • Baumslag–Solitar group $B(m,n)=\langle a,b\mid b^{-1}a^{m}b=a^{n}\rangle $ where $m,n\neq 0$. • Torus knot group $G=\langle a,b\mid a^{p}=b^{q}\rangle $ where $p,q\geq 1$ are coprime integers. • Baumslag–Gersten group $G=\langle a,t\mid a^{a^{t}}=a^{2}\rangle =\langle a,t\mid (t^{-1}a^{-1}t)a(t^{-1}at)=a^{2}\rangle $ • Oriented surface group $G=\langle a_{1},b_{1},\dots ,a_{n},b_{n}\mid [a_{1},b_{1}]\dots [a_{n},b_{n}]=1\rangle $ where $[a,b]=a^{-1}b^{-1}ab$ and where $n\geq 1$. • Non-oriented surface group $G=\langle a_{1},\dots ,a_{n}\mid a_{1}^{2}\cdots a_{n}^{2}=1\rangle $, where $n\geq 1$. Generalizations and open problems • If A and B are two groups, and $r\in A\ast B$ is an element in their free product, one can consider a one-relator product $G=A\ast B/\langle \langle r\rangle \rangle =\langle A,B\mid r=1\rangle $. • The so-called Kervaire conjecture, also known as Kervaire–Laudenbach conjecture, asks if it is true that if A is a nontrivial group and $B=\langle t\rangle $ is infinite cyclic then for every $r\in A\ast B$ the one-relator product $G=\langle A,t\mid r=1\rangle $ is nontrivial.[32] • Klyachko proved the Kervaire conjecture for the case where A is torsion-free.[33] • A conjecture attributed to Gersten[22] says that a finitely generated one-relator group is word-hyperbolic if and only if it contains no Baumslag–Solitar subgroups. • If G is a finitely generated one-relator group (with or without torsion), $H\leq G$ is a torsion-free subgroup of finite index and $\phi :H\to \mathbb {Z} $ is an epimorphism then $\ker(\phi )$ has cohomological dimension 1 and therefore, by a result of Stallings, is locally free.[34] Baumslag, with co-authors, showed that in many cases, by a suitable choice of H and $\phi $ one can prove that that $\ker(\phi )$ is actually free (of infinite rank).[35][36] These results led to a conjecture[22] that every finitely generated one-relator group with torsion is virtually free-by-cyclic. See also • 3-manifolds • Geometric topology • Small cancellation theory Sources • Wilhelm Magnus, Abraham Karrass, Donald Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations, Reprint of the 1976 second edition, Dover Publications, Inc., Mineola, NY, 2004. ISBN 0-486-43830-9. MR2109550 • Lyndon, Roger C.; Schupp, Paul E. (2001). Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin. ISBN 3-540-41158-5. MR 1812024. References 1. Magnus, Wilhelm (1930). "Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz)". Journal für die reine und angewandte Mathematik. 1930 (163): 141–165. doi:10.1515/crll.1930.163.141. MR 1581238. S2CID 117245586. 2. Lyndon, Roger C. (1972). "On the Freiheitssatz". Journal of the London Mathematical Society. Second Series. 5: 95–101. doi:10.1112/jlms/s2-5.1.95. hdl:2027.42/135658. MR 0294465. 3. Weinbaum, C. M. (1972). "On relators and diagrams for groups with one defining relation". Illinois Journal of Mathematics. 16 (2): 308–322. doi:10.1215/ijm/1256052287. MR 0297849. 4. Fischer, J.; Karrass, A.; Solitar, D. (1972). "On one-relator groups having elements of finite order". Proceedings of the American Mathematical Society. 33 (2): 297–301. doi:10.2307/2038048. JSTOR 2038048. MR 0311780. 5. Lyndon & Schupp, Ch. III, Section 11, Proposition 11.1, p. 161 6. Dyer, Eldon; Vasquez, A. T. (1973). "Some small aspherical spaces". Journal of the Australian Mathematical Society. 16 (3): 332–352. doi:10.1017/S1446788700015147. MR 0341476. 7. Magnus, Karrass and Solitar, Theorem N3, p. 167 8. Shenitzer, Abe (1955). "Decomposition of a group with a single defining relation into a free product". Proceedings of the American Mathematical Society. 6 (2): 273–279. doi:10.2307/2032354. JSTOR 2032354. MR 0069174. 9. Howie, James (1980). "On locally indicable groups". 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London Math. Soc. Lecture Note Ser. Vol. 455. Cambridge University Press. pp. 119–157. ISBN 978-1-108-72874-4. MR 3931411. 23. Louder, Larsen; Wilton, Henry (2020). "One-relator groups with torsion are coherent". Mathematical Research Letters. 27 (5): 1499–1512. arXiv:1805.11976. doi:10.4310/MRL.2020.v27.n5.a9. MR 4216595. S2CID 119141737. 24. Dahmani, Francois; Guirardel, Vincent (2011). "The isomorphism problem for all hyperbolic groups". Geometric and Functional Analysis. 21 (2): 223–300. doi:10.1007/s00039-011-0120-0. MR 2795509. 25. Wise, Daniel T. (2009). "Research announcement: the structure of groups with a quasiconvex hierarchy". Electronic Research Announcements in Mathematical Sciences. 16: 44–55. doi:10.3934/era.2009.16.44. MR 2558631. 26. Lyndon& Schupp, Chapter II, Section 6, pp. 111-113 27. Magnus, Karrass, and Solitar, Section 4.4 28. Lyndon& Schupp, Chapter IV, Section 5, pp. 198-205 29. Moldavanskii, D.I. (1967). 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\begin{document} \newcounter{counter} \title{Approaching metric domains} \author{Gon\c{c}alo Gutierres} \address{CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal} \email{[email protected]} \author{Dirk Hofmann} \address{CIDMA, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal} \email{[email protected]} \thanks{Partial financial assistance by Centro de Matem\'{a}tica da Universidade de Coimbra/FCT, Centro de Investiga\c{c}\~ao e Desenvolvimento em Matem\'atica e Aplica\c{c}\~oes da Universidade de Aveiro/FCT and the project MONDRIAN (under the contract PTDC/EIA-CCO/108302/2008) is gratefully acknowledged.} \subjclass[2010]{54A05, 54A20, 54E35, 54B30, 18B35} \keywords{Continuous lattice, metric space, approach space, injective space, cocomplete space} \begin{abstract} In analogy to the situation for continuous lattices which were introduced by Dana Scott as precisely the injective T$_0$ spaces via the (nowadays called) Scott topology, we study those metric spaces which correspond to injective T$_0$ approach spaces and characterise them as precisely the continuous lattices equipped with an unitary and associative $[0,\infty]$-action. This result is achieved by a thorough analysis of the notion of cocompleteness for approach spaces. \end{abstract} \maketitle \section{Introduction} Domain theory is generally concerned with the study of \emph{ordered sets} admitting certain (typically up-directed) suprema and a notion of approximation, here the latter amounts to saying that each element is a (up-directed) supremum of suitably defined ``finite'' elements. From a different perspective, domains can be viewed as very particular \emph{topological spaces}; in fact, in his pioneering paper \citep{Sco_ContLat} Scott introduced the notion of continuous lattice precisely as injective topological T$_0$ space. Yet another point of view was added in \citep{Day_Filter,Wyl85} where continuous lattices are shown to be precisely the \emph{algebras} for the filter monad. Furthermore, suitable submonads of the filter monad have other types of domains as algebras (for instance, continuous Scott domains \citep{Book_ContLat} or Nachbin's ordered compact Hausdorff spaces \citep{Nach_TopOrd}), and, as for continuous lattices, these domains can be equally seen as objects of topology and of order theory. This interplay between topology and algebra is very nicely explained in \citep{EF_SemDom} where, employing a particular property of monads of filters, the authors obtain ``new proofs and [\ldots] new characterizations of semantic domains and topological spaces by injectivity''. Since \citeauthor{Law_MetLogClo}'s ground-breaking paper \citep{Law_MetLogClo} it is known that an individual metric spaces $X$ can be viewed as a category with objects the points of $X$, and the distance \[ d(x,y)\in[0,\infty] \] plays the role of the ``hom-set'' of $x$ and $y$. More modest, one can think of a metric $d:X\times X\to[0,\infty]$ as an order relation on $X$ with truth-values in $[0,\infty]$ rather than in the Boolean algebra $\catfont{2}=\{\mathsf{false},\mathsf{true}\}$. In fact, writing $0$ instead of $\mathsf{true}$, $\geqslant$ instead of $\Rightarrow$ and additon $+$ instead of and $\&$, the reflexivity and transitivity laws of an ordered set become \begin{align} 0\ge d(x,x) &&\text{and} && d(x,y)+d(y,z)\ge d(x,z) &&(x,y,x\in X), \end{align} and in this paper we follow Lawvere's point of view and assume no further properties of $d$. As pointed out in \citep{Law_MetLogClo}, ``this connection is more fruitful than a mere analogy, because it provides a sequence of mathematical theorems, so that enriched category theory can suggest new directions of research in metric space theory and conversely''. A striking example of commonality between category (resp.\ order) theory and metric theory was already given in \citep{Law_MetLogClo} where it is shown that Cauchy sequences correspond to adjoint (bi)modules and convergence of Cauchy sequences corresponds to representabilty of these modules. Eventually, this amounts to saying that a metric space is Cauchy complete if and only if it admits ``suprema'' of certain ``down-sets'' ($=$ morphisms of type $X^\mathrm{op}\to[0,\infty]$), here ``suprema'' has to be taken in the sense of weighted colimit of enriched category theory \citep{EK_CloCat,Kel_EnrCat}. Other types of ``down-sets'' $X^\mathrm{op}\to[0,\infty]$ specify other properties of metric spaces: forward Cauchy sequences (or nets) (see \citep{BBR_GenMet}) can be represented by so called flat modules (see \citep{Vic_LocComplI}) and their limit points as ``suprema'' of these ``down-sets'', and the formal ball model of a metric space relates to its cocompletion with respect to yet another type of ``down-sets'' (see \citep{Rut98a,KW_FormBall}). The particular concern of this paper is to contribute to the development of metric domain theory. Due to the many facets of domains, this can be pursued by either \begin{enumerate} \item formulation order-theoretic concepts in the logic of $[0,\infty]$, \item considering injective ``$[0,\infty]$-enriched topological spaces'', or \item studying the algebras of ``metric filter monads''. \end{enumerate} Inspired by \citep{Law_MetLogClo}, there is a rich literature employing the first point of view, including \citeauthor{Wag_PhD}'s Ph.D.\ thesis \citep{Wag_PhD}, the work of the Amsterdam research group at CWI \citep{BBR_GenMet,Rut98a}, the work of \citeauthor{FSW_QDT} on continuity spaces \citep{Kop88,Flagg_ComplCont,Flagg_QuantCont,FSW_QDT,FK_ContSp}, and the work of \citeauthor{Was_DomGirQant} with various coauthors on approximation and the formal ball model \citep{Was_DomGirQant,HW_AppVCat} and \citep{KW_FormBall}. However, in this paper we take a different approach and concentrate on the second and third aspect above. Our aim is to connect the theory of metric spaces with the theory of ``$[0,\infty]$-enriched topological spaces'' in a similar fashion as domain theory is supported by topology, where by ``$[0,\infty]$-enriched topological spaces'' we understand \citeauthor{Low_ApBook}'s approach spaces \citep{Low_ApBook}. (In a nutshell, an approach space is to a topological space what a metric space is to an ordered set: it can be defined in terms of ultrafilter convergence where one associates to an ultrafilter $\mathfrak{x}$ and a point $x$ a value of convergence $a(\mathfrak{x},x)\in[0,\infty]$ rather then just saying that $\mathfrak{x}$ converges to $x$ or not.) This idea was already pursued in \citep{Hof_Cocompl} and \citep{Hof_DualityDistSp} were among others it is shown that \begin{itemize} \item injective T$_0$ approach spaces correspond bijectively to a class of metric spaces, henceforth thought of as ``continuous metric spaces'', \item these ``continuous metric spaces'' are precisely the algebras for a certain monad on $\catfont{Set}$, henceforth thought of as the ``metric filter monad'', \item the category of injective approach spaces and approach maps is Cartesian closed. \end{itemize} Here we continue this path and \begin{itemize} \item recall the theory of metric and approach spaces as generalised orders (resp.\ categories), \item characterise metric compact Hausdorff spaces as the (suitably defined) stably compact approach spaces, and \item show that injective T$_0$ approach spaces (aka ``continuous metric spaces'') can be equivalently described as continuous lattices equipped with an unitary and associative action of the continuous lattice $[0,\infty]$. This result is achieved by a thorough analysis of the notion of cocompleteness for approach spaces. \end{itemize} \subsubsection{Warning} The underlying order of a topological space $X=(X,\mathcal{O})$ we define as \[ x\le y\quad\text{whenever}\quad \doo{x}\to y \] which is equivalent to $\mathcal{O}(y)\subseteq\mathcal{O}(x)$, hence it is the \emph{dual of the specialisation order}. As a consequence, the underlying order of an injective T$_0$ topological space is the dual of a continuous lattice; and our results are stated in terms of these op-continuous lattices. We hope this does not create confusion. \section{Metric spaces} \subsection{Preliminaries}\label{Subsect:Prelim} According to the Introduction, in this paper we consider \emph{metric spaces} in a more general sense: a metric $d:X\times X\to[0,\infty]$ on set $X$ is only required to satisfy \begin{align} 0\geqslant d(x,x) &&\text{and}&& d(x,y)+d(y,z)\geqslant d(x,z). \end{align} For convenience we often also assume $d$ to be \emph{separated} meaning that $d(x,y)=0=d(y,x)$ implies $x=y$ for all $x,y\in X$. With this nomenclature, ``classical'' metric spaces appear now as separated, symmetric ($d(x,y)=d(y,x)$) and finitary ($d(x,y)<\infty$) metric spaces. A map $f:X\to X'$ between metric spaces $X=(X,d)$ and $X'=(X',d')$ is a \emph{metric map} whenever $d(x,y)\geqslant d'(f(x),f(y))$ for all $x,y\in X$. The category of metric spaces and metric maps we denote as $\catfont{Met}$. To every metric space $X=(X,d)$ one associates its \emph{dual space} $X^\mathrm{op}=(X,d^\circ)$ where $d^\circ(x,y)=d(y,x)$, for all $x,y\in X$. Certainly, the metric $d$ on $X$ is symmetric if and only if $X=X^\mathrm{op}$. Every metric map $f$ between metric spaces $X$ and $Y$ is also a metric map of type $X^\mathrm{op}\to Y^\mathrm{op}$, hence taking duals is actually a functor $(-)^\mathrm{op}:\catfont{Met}\to\catfont{Met}$ which sends $f:X\to Y$ to $f^\mathrm{op}:X^\mathrm{op}\to Y^\mathrm{op}$. There is a canonical forgetful functor $(-)_p:\catfont{Met}\to\catfont{Ord}$: for a metric space $(X,d)$, put \[ x\le y \text{ whenever } 0\geqslant d(x,y), \] and every metric map preserves this order. Also note that $(-)_p:\catfont{Met}\to\catfont{Ord}$ has a left adjoint $\catfont{Ord}\to\catfont{Met}$ which interprets an order relation $\le$ on $X$ as the metric \[ d(x,y)= \begin{cases} 0 & \text{if }x\le y,\\ \infty & \text{else.} \end{cases} \] In particular, if $X$ is a discrete ordered set meaning that the order relation is just the equality relation on $X$, then one obtains the discrete metric on $X$ where $d(x,x)=0$ and $d(x,y)=\infty$ for $x\neq y$. The induced order of a metric space extends point-wise to metric maps making $\catfont{Met}$ an \emph{ordered category}, which enables us to talk about \emph{adjunctions}. Here metric maps $f:(X,d)\to (X',d')$ and $g:(X',d')\to(X,d)$ form an adjunction, written as $f\dashv g$, if $1_X\le g\cdot f$ and $f\cdot g\le 1_{X'}$. Equivalently, $f\dashv g$ if and only if, for all $x\in X$ and $x'\in X'$, \[ d'(f(x),x')=d(x,g(x')). \] The formula above explains the costume to call $f$ left adjoint and $g$ right adjoint. We also recall that adjoint maps determine each other meaning that $f\dashv g$ and $f\dashv g'$ imply $g\simeq g'$, and $f\dashv g$ and $f'\dashv g$ imply $f\simeq f'$. The category $\catfont{Met}$ is complete and, for instance, the product $X\times Y$ of metric spaces $X=(X,a)$ and $(Y,b)$ is given by the Cartesian product of the sets $X$ and $Y$ equipped with the $\max$-metric \[ d((x,y),(x',y'))=\max(a(x,x'),b(y,y')). \] More interestingly to us is the plus-metric \[ d'((x,y),(x',y'))=a(x,x')+b(y,y') \] on the set $X\times Y$, we write $a\oplus b$ for this metric and denote the resulting metric space as $X\oplus Y$. Note that the underlying order of $X\oplus Y$ is just the product order of $X_p$ and $Y_p$ in $\catfont{Ord}$. Furthermore, for metric maps $f:X\to Y$ and $g:X'\to Y'$, the product of $f$ and $g$ gives a metric map $f\oplus g:X\oplus X'\to Y\oplus Y'$, and we can view $\oplus$ as a functor $\oplus:\catfont{Met}\times\catfont{Met}\to\catfont{Met}$. This operation is better behaved then the product $\times$ in the sense that, for every metric space $X$, the functor $X\oplus-:\catfont{Met}\to\catfont{Met}$ has a right adjoint $(-)^X:\catfont{Met}\to\catfont{Met}$ sending a metric space $Y=(Y,b)$ to \begin{align} Y^X&=\{h:X\to Y\mid h\text{ in $\catfont{Met}$}\} &&\text{with distance}\qquad [h,k]=\sup_{x\in X}b(h(x),k(x)), \end{align} and a metric map $f:Y_1\to Y_2$ to \[ f^X:Y_1^X\to Y_2^X,\,h\mapsto f\cdot h. \] In particular, if $X$ is a discrete space, then $Y^X$ is just the $X$-fold power of $Y$. In the sequel we will pay particular attention to the metric space $[0,\infty]$, with metric $\mu$ defined by \[\mu(u,v)=v\ominus u:=\max\{v-u,0\},\] for all $u,v\in[0,\infty]$. Then the underlying order on $[0,\infty]$ is the ``greater or equal relation'' $\geqslant$. The ``turning around'' of the natural order of $[0,\infty]$ might look unmotivated at first sight but has its roots in the translation of ``$\mathsf{false}\le\mathsf{true}$'' in $\catfont{2}$ to ``$\infty\geqslant 0$'' in $[0,\infty]$. We also note that $u+-:[0,\infty]\to[0,\infty]$ is left adjoint to $\mu(u,-):[0,\infty]\to[0,\infty]$ with respect to $\geqslant$ in $[0,\infty]$. However, in the sequel we will usually refer to the natural order $\leqslant$ on $[0,\infty]$ with the effect that some formulas are dual to what one might expect. For instance, the underlying monotone map of a metric map of type $X^\mathrm{op}\to[0,\infty]$ is of type $X_p\to[0,\infty]$; and when we talk about a supremum ``$\bigvee$'' in the underlying order of a generic metric space, it specialises to taking infimum ``$\inf$'' with respect to the usual order $\leqslant$ on $[0,\infty]$. For every set $I$, the maps \begin{align} \inf:[0,\infty]^I&\to[0,\infty] &&\text{and}& \sup:[0,\infty]^I &\to[0,\infty]\\ \varphi &\mapsto \inf_{i\in I}\varphi(i) &&& \varphi &\mapsto \sup_{i\in I}\varphi(i) \intertext{are metric maps, and so are} +:[0,\infty]\oplus [0,\infty]&\to[0,\infty] &&\text{and}& \mu:[0,\infty]^\mathrm{op}\oplus [0,\infty]&\to[0,\infty].\\ (u,v) &\mapsto u+v &&& (u,v) &\mapsto v\ominus u \end{align} More general, for a metric space $X=(X,d)$, the metric $d$ is a metric map $d:X^\mathrm{op}\oplus X\to [0,\infty]$. Its mate is the \emph{Yoneda embedding} \[ \yoneda_X:=\mate{d}:X\to[0,\infty]^{X^\mathrm{op}},\,x\mapsto d(-,x), \] which satisfies indeed $d(x,y)=[\yoneda_X(x),\yoneda_X(y)]$ for all $x,y\in X$ thanks to the Yoneda lemma which states that \[ [\yoneda_X(x),\psi]=\psi(x), \] for all $x\in X$ and $\psi\in[0,\infty]^{X^\mathrm{op}}$. \subsection{Cocomplete metric spaces}\label{SubSect:CocomplMetSp} In this subsection we have a look at metric spaces ``through the eyes of category (resp.\ order) theory'' and study the existence of suprema of ``down-sets'' in a metric space. This is a particular case of the notion of weighted colimit of enriched categories (see \citep{EK_CloCat,Kel_EnrCat,KS_Colim}, for instance), and in this and the next subsection we spell out the meaning for metric spaces of general notions and results of enriched category theory. For a metric space $X=(X,d)$ and a ``down-set'' $\psi:X^\mathrm{op}\to[0,\infty]$ in $\catfont{Met}$, an element $x_0\in X$ is a \emph{supremum} of $\psi$ whenever, for all $x\in X$, \begin{equation}\label{Eq:SupremumMet} d(x_0,x)=\sup_{y\in X}(d(y,x)\ominus\psi(y)). \end{equation} Suprema are unique up to equivalence $\simeq$, we write $x_0\simeq\Sup_X(\psi)$ and will frequently say \emph{the} supremum. Furthermore, a metric map $f:(X,d)\to (X',d')$ preserves the supremum of $\psi\in[0,\infty]^{X^\mathrm{op}}$ whenever \[ d'(f(\Sup_X(\psi)),x')=\sup_{x\in X}(d'(f(x),x')\ominus\psi(x)) \] for all $x'\in X'$. As for ordered sets: \begin{lemma} Left adjoint metric maps preserve all suprema. \end{lemma} A metric space $X=(X,d)$ is called \emph{cocomplete} if every ``down-set'' $\psi:X^\mathrm{op}\to[0,\infty]$ has a supremum. This is the case precisely if, for all $\psi\in[0,\infty]^{X^\mathrm{op}}$ and all $x\in X$, \begin{equation} d(\Sup_X(\psi),x)=\sup_{y\in X}(d(y,x)\ominus\psi(y))=[\psi,\yoneda_X(x)]; \end{equation} hence $X$ is cocomplete if and only if the Yoneda embedding $\yoneda_X:X\to[0,\infty]^{X^\mathrm{op}}$ has a left adjoint $\Sup_X:[0,\infty]^{X^\mathrm{op}}\to X$ in $\catfont{Met}$. More generally, one has (see \citep{Hof_Cocompl}, for instance) \begin{proposition} For a metric space $X$, the following conditions are equivalent. \begin{eqcond} \item $X$ is injective (with respect to isometries). \item $\yoneda_X:X\to[0,\infty]^{X^\mathrm{op}}$ has a left inverse. \item $\yoneda_X$ has a left adjoint. \item $X$ is cocomplete. \end{eqcond} \end{proposition} Here a metric map $i:(A,d)\to(B,d')$ is called \emph{isometry} if one has $d(x,y)= d'(i(x),i(y))$ for all $x,y\in A$, and $X$ is injective if, for all isometries $i:A\to B$ and all $f:A\to X$ in $\catfont{Met}$, there exists a metric map $g:B\to X$ with $g\cdot i\simeq f$. Dually, an infimum of an ``up-set'' $\varphi:X\to[0,\infty]$ in $X=(X,d)$ is an element $x_0\in X$ such that, for all $x\in X$, \[ d(x,x_0)=\sup_{y\in X}(d(x,y)\ominus\varphi(y)). \] A metric space $X$ is \emph{complete} if every ``up-set'' has an infimum. By definition, an infimum of $\varphi:X\to[0,\infty]$ in $X$ is a supremum of $\varphi:(X^\mathrm{op})^\mathrm{op}\to[0,\infty]$ in $X^\mathrm{op}$, and everything said above can be repeated now in its dual form. In particular, with $\yonedaOP_X:X\to\left([0,\infty]^X\right)^\mathrm{op},\, x\mapsto d(x,-)$ denoting the \emph{contravariant Yoneda embedding} (which is the dual of $\yoneda_{X^\mathrm{op}}:X^\mathrm{op}\to[0,\infty]^X$): \begin{proposition} For a metric space $X$, the following conditions are equivalent. \begin{eqcond} \item $X$ is injective (with respect to isometries). \item $\yonedaOP_X:X\to\left([0,\infty]^X\right)^\mathrm{op}$ has a left inverse. \item $\yonedaOP_X$ has a left adjoint. \item $X$ is complete. \end{eqcond} \end{proposition} \begin{corollary} A metric space is complete if and only if it is cocomplete. \end{corollary} This latter fact can be also seen in a different way. To every ``down-set'' $\psi:X^\mathrm{op}\to[0,\infty]$ one assigns its ``up-set of upper bounds'' \[ \psi^+:X\to[0,\infty],\,x\mapsto\sup_{y\in X}(d(y,x)\ominus\psi(y)), \] and to every ``up-set'' $\varphi:X\to[0,\infty]$ its ``down-set of lower bounds'' \[ \varphi^-:X^\mathrm{op}\to[0,\infty],\,x\mapsto\sup_{y\in X}(d(x,y)\ominus\varphi(y)). \] This way one defines an adjunction $(-)^+\dashv (-)^-$ \[ \xymatrix{\left([0,\infty]^X\right)^\mathrm{op}\ar@/^1.5ex/[rr]^{(-)^-}\ar@{}[rr]\|\top && [0,\infty]^{X^\mathrm{op}}\ar@/^1.5ex/[ll]^{(-)^+}\\ & X\ar[ul]^{\yonedaOP_X}\ar[ur]_{\yoneda_X}} \] in $\catfont{Met}$ where both maps commute with the Yoneda embeddings. Therefore a left inverse of $\yoneda_X$ produces a left inverse of $\yonedaOP_X$, and vice versa. The adjunction $(-)^+\dashv(-)^-$ is also know as the \emph{Isbell conjugation adjunction}. For every (co)complete metric space $X=(X,d)$, its underlying ordered set $X_p$ is (co)complete as well. This follows for instance from the fact that $(-)_p:\catfont{Met}\to\catfont{Ord}$ preserves injective objects. Another argument goes as follows. For every (down-set) $A\subseteq X$, one defines a metric map \[ \psi_A:X^\mathrm{op}\to[0,\infty],\,x\mapsto\inf_{a\in A}d(x,a), \] and a supremum $x_0$ of $\psi_A$ must satisfy, for all $x\in X$, \[ d(x_0,x)=\sup_{y\in X}(d(y,x)\ominus\psi_A(y)) =\sup_{a\in A}\sup_{y\in X}(d(y,x)\ominus d(y,x)) =\sup_{a\in A}d(a,x). \] Therefore $x_0$ is not only a supremum of $A$ in the ordered set $X_p$, it is also preserved by every monotone map $d(-,x):X_p\to[0,\infty]$. \begin{lemma} Let $X=(X,d)$ be a metric space and let $x_0\in X$ and $A\subseteq X$. Then $x_0$ is the supremum of $\psi_A$ if and only if $x_0$ is the (order theoretic) supremum of $A$ and, for every $x\in X$, the monotone map $d(-,x):X_p\to[0,\infty]$ preserves this supremum. \end{lemma} \subsection{Tensored metric spaces}\label{Subsect:TensMet} We are now interested in those metric spaces $X=(X,d)$ which admit suprema of ``down-sets'' of the form $\psi=d(-,x)+u$ where $x\in X$ and $u\in[0,\infty]$. In the sequel we write $x+u$ instead of $\Sup_X(\psi)$. According to \eqref{Eq:SupremumMet}, the element $x+u\in X$ is characterised up to equivalence by \[ d(x+u,y)=d(x,y)\ominus u, \] for all $y\in X$. A metric map $f:(X,d)\to (X',d')$ preserves the supremum of $\psi=d(-,x)+u$ if and only if $f(x+u)\simeq f(x)+u$. Dually, an infimum of an ``up-set'' of the form $\varphi=d(x,-)+u$ we denote as $x\ominus u$, it is characterised up to equivalence by \[ d(y,x\ominus u)=d(y,x)\ominus u. \] One calls a metric space \emph{tensored} if it admits all suprema $x+u$, and \emph{cotensored} if $X$ admits all infima $x\ominus u$. \begin{example} The metric space $[0,\infty]$ is tensored and cotensored where $x+u$ is given by addition and $x\ominus u=\max\{x-u,0\}$. \end{example} Note that $X=(X,d)$ is tensored if and only if every $d(x,-):X\to[0,\infty]$ has a left adjoint $x+(-):[0,\infty]\to X$ in $\catfont{Met}$, and $X$ is cotensored if and only if every $d(-,x):X^\mathrm{op}\to[0,\infty]$ has a left adjoint $x\ominus(-):[0,\infty]\to X^\mathrm{op}$ in $\catfont{Met}$. Furthermore, if $X$ is tensored and cotensored, then $(-)+u:X\to X$ is left adjoint to $(-)\ominus u:X\to X$ in $\catfont{Met}$, for every $u\in[0,\infty]$. \begin{theorem} Let $X=(X,d)$ be a metric space. Then the following assertions are equivalent. \begin{eqcond} \item $X$ is cocomplete. \item $X$ has all order-theoretic suprema and is tensored and cotensored. \item $X$ has all (order theoretic) suprema, is tensored and, for every $u\in[0,\infty]$, the monotone map $(-)+u:X_p\to X_p$ has a right adjoint in $\catfont{Ord}$. \item $X$ has all (order theoretic) suprema, is tensored and, for every $u\in[0,\infty]$, the monotone map $(-)+u:X_p\to X_p$ preserves suprema. \item $X$ has all (order theoretic) suprema, is tensored and, for every $x\in X$, the monotone map $d(-,x):X_p\to[0,\infty]$ has a right adjoint in $\catfont{Ord}$. \item $X$ is has all (order theoretic) suprema, is tensored and, for every $x\in X$, the monotone map $d(-,x):X_p\to[0,\infty]$ preserves suprema. \end{eqcond} Under these conditions, the supremum of a ``down-set'' $\psi:X^\mathrm{op}\to[0,\infty]$ can be calculated as \begin{equation}\label{Eq:colimitSupMais} \Sup\psi=\inf_{x\in X}(x+\psi(x)). \end{equation} A metric map $f:X\to Y$ between cocomplete metric spaces preserves all colimits if and only if it preserves tensors and suprema. \end{theorem} \begin{proof} By the preceding discussion, the implications (i)$\Rightarrow$(ii) and (ii)$\Rightarrow$(iii) are obvious, and so are (iii)$\Leftrightarrow$(iv) and (v)$\Leftrightarrow$(vi). To see (iii)$\Rightarrow$(v), just note that a right adjoint $(-)\ominus u:X_p\to X_p$ of $(-)+u$ produces a right adjoint $x\ominus(-):[0,\infty]\to X_p$ of $d(-,x)$. Finally, (vi)$\Rightarrow$(i) can be shown by verifying that \eqref{Eq:colimitSupMais} calculates indeed a supremum of $\psi$. \end{proof} Every metric space $X=(X,d)$ induces metric maps \begin{align} X\oplus[0,\infty]\xrightarrow{\,\formalball_X\,}[0,\infty]^{X^\mathrm{op}} && \text{and} && X^I\xrightarrow{\,\fammod_{X,I}\,}[0,\infty]^{X^\mathrm{op}}\qquad\text{(where $I$ is any set)}. \end{align} Here $\formalball_X:X\oplus[0,\infty]\to[0,\infty]^{X^\mathrm{op}},(x,u)\mapsto d(-,x)+u$ is the mate of the composite \[ X^\mathrm{op}\oplus X\oplus [0,\infty]\xrightarrow{\,d\oplus 1\,} [0,\infty]\oplus [0,\infty]\xrightarrow{\,+\,}[0,\infty], \] and $\fammod_{X,I}:X^I\to[0,\infty]^{X^\mathrm{op}}$ is the mate of the composite \[ X^\mathrm{op}\oplus X^I\to[0,\infty]^I\xrightarrow{\,\inf\,}[0,\infty], \] where the first component is the mate of the composite \[ X^\mathrm{op}\oplus X^I\oplus I\xrightarrow{\,1\oplus\ev\,} X^\mathrm{op}\oplus X \xrightarrow{\,d\,}[0,\infty]. \] Spelled out, for $\varphi\in X^I$ and $x\in X$, $\fammod_{X,I}(\varphi)(x)=\inf_{i\in I}d(x,\varphi(i))$, and a supremum of $\fammod_{X,I}(\varphi)\in [0,\infty]^{X^\mathrm{op}}$ is also a (order-theoretic) supremum of the family $(\varphi(i))_{i\in I}$ in $X$. If $X$ is cocomplete, by composing with $\Sup_X$ one obtains metric maps \begin{align} X\oplus[0,\infty]\xrightarrow{\,+\,}X && && \text{and} && X^I\xrightarrow{\,\bigvee\,}X\qquad\text{(where $I$ is any set)}. \end{align} Finally, a categorical standart argument (see \citep[Lemma 4.10]{Joh_StoneSp} shows that with $Y$ also $Y^X$ is injective, hence, $Y^X$ is cocomplete. Furthermore, tensors and suprema in $Y^X$ can be calculated pointwise: \begin{align} h+u=(-+u)\cdot h &&\text{and}&& \left(\bigvee_{i\in I}h_i\right)=\bigvee\cdot\langle h_i\rangle_{i\in I}, \end{align} for $u\in[0,\infty]$, $h\in Y^X$ and $h_i\in Y^X$ ($i\in I$). Here $\langle h_i\rangle_{i\in I}:X\to Y^I$ denotes the map induced by the family $(h_i)_{i\in I}$. \subsection{$[0,\infty]$-actions on ordered sets} When $X=(X,d)$ is a tensored metric space, we might not have $\Sup_X$ defined on the whole space $[0,\infty]^{X^\mathrm{op}}$, but it still is defined on its subspace of all metric maps $\psi:X^\mathrm{op}\to[0,\infty]$ of the form $\psi=d(-,x)+u$. Hence, one still has a metric map \[ X\oplus[0,\infty]\to X,\,(x,u)\mapsto x+u, \] and one easily verifies the following properties. \begin{enumerate} \item\label{cond:action1} For all $x\in X$, $x+0\simeq x$. \item\label{cond:action2} For all $x\in X$ and all $u,v\in[0,\infty]$, $(x+u)+v\simeq x+(u+v)$. \item\label{cond:action3} $+:X_p\times[0,\infty]\to X_p$ is monotone in the first and anti-monotone in the second variable. \item\label{cond:action4} For all $x\in X$, $x+\infty$ is a bottom element of $X_p$. \item\label{cond:action5} For all $x\in X$ and $(u_i)_{i\in I}$ in $[0,\infty]$, $\displaystyle{x+\inf_{i\in I}u_i\simeq\bigvee_{i\in I}(x+u_i)}$. \end{enumerate} Of course, \eqref{cond:action4} is a special case of \eqref{cond:action5}. If $X$ is separated, then the first three conditions just tell us that $X_p$ is an algebra for the monad induced by the monoid $([0,\infty],\geqslant,+,0)$ on $\catfont{Ord}_\mathrm{sep}$. Hence, $X\mapsto X_p$ defines a forgetful functor \[ \catfont{Met}_{\mathrm{sep},+}\to\catfont{Ord}^{[0,\infty]}_\mathrm{sep}, \] where $\catfont{Met}_{\mathrm{sep},+}$ denotes the category of tensored and separated metric spaces and tensor preserving metric maps, and $\catfont{Ord}^{[0,\infty]}_\mathrm{sep}$ the category of separated ordered sets with an unitary (i.e.\ satisfying \eqref{cond:action1}) and associative (i.e.\ satisfying \eqref{cond:action2}) action of $([0,\infty],\geqslant,+,0)$ ($[0,\infty]$-algebras for short) and monotone maps which preserve this action. Conversely, let now $X$ be a $[0,\infty]$-algebra with action $+:X\times[0,\infty]\to X$. We define \begin{equation}\label{Eq:metric_from_action} d(x,y)=\inf\{u\in[0,\infty]\mid x+u\le y\}. \end{equation} Certainly, $x\le y$ implies $0\geqslant d(x,y)$, in particular one has $0\geqslant d(x,x)$. Since, for $x,y,z\in X$, \begin{align} d(x,y)+d(y,z) &=\inf\{u\in[0,\infty]\mid x+u\le y\}+\inf\{v\in[0,\infty]\mid y+v\le z\}\\ &=\inf\{u+v\mid u,v\in[0,\infty],\, x+u\le y,\, y+v\le z\}\\ &\geqslant\inf\{w\in[0,\infty]\mid x+w\le z=d(x,z), \end{align} we have seen that $(X,d)$ is a metric space. If the $[0,\infty]$-algebra $X$ comes from a tensored separated metric space, then we get the original metric back. If $X$ satisfies \eqref{cond:action4}, then the infimum in \eqref{Eq:metric_from_action} is non-empty, and therefore \begin{align} d(x+u,y) &= \inf\{v\in[0,\infty]\mid x+u+v\le y\}\\ &= \inf\{w\ominus u\mid w\in[0,\infty],\, x+w\le y\}\\ &= \inf\{w\mid w\in[0,\infty],\, x+w\le y\}\ominus u = d(x,y)\ominus u, \end{align} hence $(X,d)$ is tensored where $+$ is given by the algebra operation. Finally, if $X$ satisfies \eqref{cond:action5}, then the infimum in \eqref{Eq:metric_from_action} is actually a minimum, and therefore $0\geqslant d(x,y)$ implies $x\le y$. All told: \begin{theorem}\label{Thm:TensMetVsAction} The category $\catfont{Met}_{\mathrm{sep},+}$ is equivalent to the full subcategory of $\catfont{Ord}^{[0,\infty]}_\mathrm{sep}$ defined by those $[0,\infty]$-algebras satisfying (5). Under this correspondence, $(X,d)$ is a cocomplete separated metric space if and only if the $[0,\infty]$-algebra $X$ has all suprema and $(-)+u:X\to X$ preserves suprema, for all $u\in[0,\infty]$. \end{theorem} \begin{remark} The second part of the theorem above is essentially in \citep{PT89}, which actually states that cocomplete separated metric spaces correspond precisely to sup-lattices equipped with an unitary and associative action $+:X\times[0,\infty]\to X$ which is a bimorphism, meaning that it preserves suprema in each variable (where the order on $[0,\infty]$ is $\geqslant$) but not necessarily in both. Thanks to Freyd's Adjoint Functor Theorem (see \citep[Section V.6]{MacLane_WorkMath}), the category $\catfont{Sup}$ of sup-lattices and suprema preserving maps admits a tensor product $X\otimes Y$ which is characterised by \[ \Bimorph(X\times Y,Z)\simeq\catfont{Sup}(X\otimes Y,Z), \] naturally in $Z\in\catfont{Sup}$, for all sup-lattices $X,Y$. Hence, a cocomplete separated metric space can be identified with a sup-lattice $X$ equipped with an unitary and associative action $+:X\otimes[0,\infty]\to X$ in $\catfont{Sup}$. \end{remark} \begin{proposition}\label{Prop:ContrTensMet} Let $X$ and $Y$ be tensored metric spaces and $f:X\to Y$ be a map. Then $f:X\to Y$ is a metric map if and only if $f:X_p\to Y_p$ is monotone and, for all $x\in X$ and $u\in[0,\infty]$, $f(x)+u\le f(x+u)$. \end{proposition} \begin{proof} Every metric map is also monotone with respect to the underlying orders and satisfies $f(x)+u\le f(x+u)$, for all $x\in X$ and $u\in[0,\infty]$. To see the reverse implication, recall that the metric $d$ on $X$ satisfies \[ d(x,y)=\inf\{u\in[0,\infty]\mid x+u\le y\}, \] and for the metric $d'$ on $Y$ one has \[ d(f(x),f(y))=\inf\{v\in[0,\infty]\mid f(x)+v\le f(y)\}. \] If $x+u\le y$, then $f(x)+u\le f(x+u)\le f(y)$, and the assertion follows. \end{proof} \section{Metric compact Hausdorff spaces and approach spaces} \subsection{Continuous lattices}\label{Subsect:ContLat} Continuous lattices were introduced by D.\ Scott \citep{Sco_ContLat} as precisely those orders appearing as the \emph{specialisation order} of an injective topological T$_0$-space. Here, for an arbitrary topological space $X$ with topology $\mathcal{O}$, the specialisation order $\le$ on $X$ is defined as \[ x\le y\quad\text{whenever}\quad\mathcal{O}(x)\subseteq\mathcal{O}(y), \] for all $x,y\in X$. This relation is always reflexive and transitive, and it is anti-symmetric if and only if $X$ is T$_0$. If $X$ is an injective T$_0$-space, then the ordered set $(X,\le)$ is actually complete and, for all $x\in X$, \begin{equation} x=\bigvee\{y\in X\mid y\ll x\} \end{equation} (where $y\ll x$ whenever $x\le\bigvee D\,\Rightarrow\,y\in D$ for every up-directed down-set $D\subseteq X$); in general, a complete separated ordered set with this property is called \emph{continuous lattice}. In this particular case the specialisation order contains all information about the topology of $X$: $A\subseteq X$ is open if and only if $A$ is unreachable by up-directed suprema in the sense that \begin{equation}\label{eq:ScottTopology} \bigvee D\in A\,\Rightarrow\, D\cap A\neq\varnothing \end{equation} for every up-directed down-set $D\subseteq A$. Quite generally, \eqref{eq:ScottTopology} defines a topology on $X$ for any ordered set $X$, and a monotone map $f:X\to Y$ is continuous with respect to these topologies if and only if $f$ preserves all existing up-directed suprema. Furthermore, the specialisation order of this topology gives the original order back, and one obtains an injective topological T$_0$-space if and only if $X$ is a continuous lattice. In the sequel we will consider topological spaces mostly via ultrafilter convergence, and therefore define the \emph{underlying order} $\le$ of a topological space as the ``point shadow'' of this convergence: \begin{align} x\le y &\quad\text{whenever}\quad\doo{x}\to y, &&(\doo{x}=\{A\subseteq X\mid x\in A\}) \end{align} which is dual to the specialisation order. Consequently, the underlying order of an injective topological T$_0$ space is an \emph{op-continuous lattice} meaning that $(X,\le)$ is complete and, for any $x\in X$, \[ x=\bigwedge\{y\in X\mid y\succ x\}, \] where $y\succ x$ whenever $x\le\bigwedge D$ implies $y\in D$ for every down-directed up-set $D\subseteq X$. We also note that, with respect to the underlying order, the convergence relation of an injective T$_0$ space is given by \begin{align} \mathfrak{x}\to x &\iff \left(\bigwedge_{A\in\mathfrak{x}}\bigvee_{y\in A}y\right)\le x, \end{align} for all ultrafilters $\mathfrak{x}$ on $X$ and all $x\in X$. \subsection{Ordered compact Hausdorff spaces}\label{Subsect:OrdCompHaus} The class of domains with arguably the most direct generalisation to metric spaces is that of stably compact spaces, or equivalently, \emph{ordered compact Hausdorff spaces}. The latter were introduced by \citep{Nach_TopOrd} as triples $(X,\le,\mathcal{O})$ where $(X,\le)$ is an ordered set (we do not assume anti-symmetry here) and $\mathcal{O}$ is a compact Hausdorff topology on $X$ so that $\{(x,y)\mid x\le y\}$ is closed in $X\times X$. A morphism of ordered compact Hausdorff spaces is a map $f:X\to Y$ which is both monotone and continuous. The resulting category of ordered compact Hausdorff space and morphisms we denote as $\catfont{OrdCompHaus}$. If $(X,\le,\mathcal{O})$ is an ordered compact Hausdorff spaces, then the dual order $\le^\circ$ on $X$ together with the topology $\mathcal{O}$ defines an ordered compact Hausdorff spaces $(X,\le^\circ,\mathcal{O})$, and one obtains a functor $(-)^\mathrm{op}:\catfont{OrdCompHaus}\to\catfont{OrdCompHaus}$ which commutes with the canonical forgetful functor $\catfont{OrdCompHaus}\to\catfont{Set}$. Analogously to the fact that compact Hausdorff spaces and continuous maps form an algebraic category over $\catfont{Set}$ via ultrafilter convergence $UX\to X$ \citep{Man_TriplCompAlg}, it is shown in \citep{Fla97a} that the full subcategory $\catfont{OrdCompHaus}_\mathrm{sep}$ of $\catfont{OrdCompHaus}$ defined by those spaces with anti-symmetric order is the category of Eilenberg-Moore algebras for the prime filter monad of up-sets on $\catfont{Ord}_\mathrm{sep}$. The situation does not change much when we drop anti-symmetry, in \citep{Tho_OrderedTopStr} it is shown that ordered compact Hausdorff spaces are precisely the Eilenberg-Moore algebras for the ultrafilter monad $\monadfont{U}=(U,e,m)$ suitably defined on $\catfont{Ord}$. Here the functor $U:\catfont{Ord}\to\catfont{Ord}$ sends an ordered set $X=(X,\le)$ to the set $UX$ of all ultrafilters on the set $X$ equipped with the order relation \[ \mathfrak{x}\le\mathfrak{y}\quad\text{ whenever }\quad \forall A\in\mathfrak{x},B\in\mathfrak{y}\,\exists x\in A,y\in B\,.\,x\le y; \qquad (\mathfrak{x},\mathfrak{y}\in UX) \] and the maps \begin{align} e_X:X &\to UX & m_X:UUX &\to UX\\ x &\mapsto \doo{x}:=\{A\subseteq X\mid x\in A\} & \mathfrak{X} &\mapsto\{A\subseteq X\mid A^\#\in\mathfrak{X}\} \end{align} (where $A^\#:=\{\mathfrak{x}\in UX\mid A\in\mathfrak{x}\}$) are monotone with respect to this order relation. Then, for $\alpha:UX\to X$ denoting the convergence of the compact Hausdorff topology $\mathcal{O}$, $(X,\le,\mathcal{O})$ is an ordered compact Hausdorff space if and only if $\alpha:U(X,\le)\to(X,\le)$ is monotone. \subsection{Metric compact Hausdorff spaces} The presentation in \citep{Tho_OrderedTopStr} is even more general and gives also an extension of the ultrafilter monad $\monadfont{U}$ to $\catfont{Met}$. For a metric space $X=(X,d)$ and ultrafilters $\mathfrak{x},\mathfrak{y}\in UX$, one defines a distance \[ Ud(\mathfrak{x},\mathfrak{y})=\sup_{A\in\mathfrak{x},B\in\mathfrak{y}}\inf_{x\in A,y\in B}d(x,y) \] and turns this way $UX$ into a metric space. Then $e_X:X\to UX$ and $m_X:UUX\to UX$ are metric maps and $Uf:UX\to UY$ is a metric map if $f:X\to Y$ is so. Not surprisingly, we call an Eilenberg--Moore algebra for this monad \emph{metric compact Hausdorff space}. Such a space can be described as a triple $(X,d,\alpha)$ where $(X,d)$ is a metric space and $\alpha$ is (the convergence relation of) a compact Hausdorff topology on $X$ so that $\alpha:U(X,d)\to(X,d)$ is a metric map. We denote the category of metric compact Hausdorff spaces and morphisms (i.e.\ maps which are both metric maps and continuous) as $\catfont{MetCompHaus}$. The operation ``taking the dual metric space'' lifts to an endofunctor $(-)^\mathrm{op}:\catfont{MetCompHaus}\to\catfont{MetCompHaus}$ where $X^\mathrm{op}:=(X,d^\circ,\alpha)$, for every metric compact Hausdorff space $X=(X,d,\alpha)$. \begin{example}\label{Ex:PasMCHsp} The metric space $[0,\infty]$ with metric $\mu(u,v)=v\ominus u$ becomes a metric compact Hausdorff space with the Euclidean compact Hausdorff topology whose convergence is given by \[ \xi(\mathfrak{v})=\sup_{A\in\mathfrak{v}}\inf_{v\in A}v, \] for $\mathfrak{v}\in U[0,\infty]$. Consequently, $[0,\infty]^\mathrm{op}$ denotes the metric compact Hausdorff space $([0,\infty],\mu^\circ,\xi)$ with the same compact Hausdorff topology on $[0,\infty]$ and with the metric $\mu^\circ(u,v)=u\ominus v$. \end{example} \begin{lemma}\label{Lem:UXisTensored} If $(X,d)$ is a tensored metric space, then $(UX,Ud)$ is tensored too. \end{lemma} \begin{proof} For $u\in[0,\infty]$ and $\mathfrak{x}\in UX$, put $\mathfrak{x}+u=U(t_u)(\mathfrak{x})$ where $t_u:X\to X$ sends $x\in X$ to (a choice of) $x+u$. Then \begin{align} Ud(\mathfrak{x}+u,\mathfrak{y})&=\sup_{A\in\mathfrak{x},B\in\mathfrak{y}}\inf_{x\in A,y\in B}d(x+u,y)\\ &=\left(\sup_{A\in\mathfrak{x},B\in\mathfrak{y}}\inf_{x\in A,y\in B}d(x,y)\right)\ominus u\\ &=Ud(\mathfrak{x},\mathfrak{y})\ominus u, \end{align} for all $\mathfrak{y}\in UX$. Here we use the fact that $-\ominus u:[0,\infty]\to[0,\infty]$ preserves all suprema and non-empty infima. \end{proof} Clearly, if $f:X\to Y$ is a tensor preserving map between tensored metric spaces, then $Uf(\mathfrak{x}+u)\simeq Uf(\mathfrak{x})+u$, hence $U:\catfont{Met}\to\catfont{Met}$ restricts to an endofunctor on the category $\catfont{Met}_+$ of tensored metric spaces and tensor preserving maps. \subsection{Stably compact topological spaces}\label{Subsect:StablyCompSp} As we have already indicated at the beginning of Subsection \ref{Subsect:OrdCompHaus}, (anti-symmetric) ordered compact Hausdorff spaces can be equivalently seen as special topological spaces. In fact, both structures of an ordered compact Hausdorff space $X=(X,\le,\mathcal{O})$ can be combined to form a topology on $X$ whose opens are precisely those elements of $\mathcal{O}$ which are down-sets in $(X,\le)$, and this procedure defines indeed a functor $K:\catfont{OrdCompHaus}\to\catfont{Top}$. An ultrafilter $\mathfrak{x}\in UX$ converges to a point $x\in X$ with respect to this new topology if and only if $\alpha(\mathfrak{x})\le x$, where $\alpha:UX\to X$ denotes the convergence of $(X,\mathcal{O})$. Hence, $\le$ is just the underlying order of $\mathcal{O}$ and $\alpha(\mathfrak{x})$ is a smallest convergence point of $\mathfrak{x}\in UX$ with respect to this order. From that it follows at once that we can recover both $\le$ and $\alpha$ from $\mathcal{O}$. To be rigorous, this is true when $(X,\le)$ is anti-symmetric, in the general case $\alpha$ is determined only up to equivalence. In any case, we define the dual of a topological space $Y$ of the form $Y=K(X,\le,\alpha)$ as $Y^\mathrm{op}=K(X,\le^\circ,\alpha)$, and note that equivalent maps $\alpha$ lead to the same space $Y^\mathrm{op}$. A T$_0$ space $X=(X,\mathcal{O})$ comes from a anti-symmetric ordered compact Hausdorff space precisely if $X$ is \emph{stably compact}, that is, $X$ is sober, locally compact and stable. The latter property can be defined in different manners, we use here the one given in \citep{Sim82a}: $X$ is stable if, for open subsets $U_1,\ldots,U_n$ and $V_1,\ldots,V_n$ ($n\in\field{N}$) of $X$ with $U_i\ll V_i$ for each $1\le i\le n$, also $\bigcap_i U_i\ll\bigcap_i V_i$. As usual, it is enough to require stability under empty and binary intersections, and stability under empty intersection translates to compactness of $X$. Also note that a T$_0$ space is locally compact if and only it is exponentiable in $\catfont{Top}$. It is also shown in \citep[Lemma 3.7]{Sim82a} that, for $X$ exponentiable, $X$ is stable if and only if, for every ultrafilter $\mathfrak{x}\in UX$, the set of all limit points of $\mathfrak{x}$ is irreducible\footnote{Actually, Lemma 3.7 of \citep{Sim82a} states only one implication, but the other is obvious and even true without assuming exponentiability.}. For a nice introduction to these kinds of spaces we refer to \citep{Jung04}. If we start with a metric compact Hausdorff space $X=(X,d,\alpha)$ instead, the construction above produces, for every $\mathfrak{x}\in UX$ and $x\in X$, the \emph{value of convergence} \begin{equation}\label{Eq:ValConvMetCompHaus} a(\mathfrak{x},x)=d(\alpha(\mathfrak{x}),x)\in[0,\infty], \end{equation} which brings us into the realm of \subsection{Approach spaces}\label{Subsect:App} We will here give a quick overview of \emph{approach spaces} which were introduced in \citep{Low_Ap} and are extensively described in \citep{Low_ApBook}. An approach space is typically defined as a pair $(X,\delta)$ consisting of a set $X$ and an \emph{approach distance} $\delta$ on $X$, that is, a function $\delta:X\times \catfont{2}^X\to[0,\infty]$ satisfying \begin{enumerate} \item $\delta(x,\{x\})=0$, \item $\delta(x,\varnothing)=\infty$, \item $\delta(x,A\cup B)=\min\{\delta(x,A),\delta(x,B)\}$, \item $\delta(x,A)\leqslant\delta(x,A^{(\varepsilon)})+\varepsilon$, where $A^{(\varepsilon)}=\{x\in X\mid \delta(x,A)\leqslant\varepsilon\}$, \end{enumerate} for all $A,B\subseteq X$, $x\in X$ and $\varepsilon\in[0,\infty]$. For $\delta:X\times \catfont{2}^X\to[0,\infty]$ and $\delta':Y\times\catfont{2}^Y\to[0,\infty]$, a map $f:X\to Y$ is called \emph{approach map} $f:(X,\delta)\to(Y,\delta')$ if $\delta(x,A)\geqslant\delta'(f(x),f(A))$, for every $A\subseteq X$ and $x\in X$. Approach spaces and approach maps are the objects and morphisms of the category $\catfont{App}$. The canonical forgetful functor \[ \catfont{App}\to\catfont{Set} \] is topological, hence $\catfont{App}$ is complete and cocomplete and $\catfont{App}\to\catfont{Set}$ preserves both limits and colimits. Furthermore, the functor $\catfont{App}\to\catfont{Set}$ factors through $\catfont{Top}\to\catfont{Set}$ where $(-)_p:\catfont{App}\to\catfont{Top}$ sends an approach space $(X,\delta)$ to the topological space with the same underlying set $X$ and with \[ x\in\overline{A} \text{ whenever } \delta(x,A)=0. \] This functor has a left adjoint $\catfont{Top}\to\catfont{App}$ which one obtains by interpreting the closure operator of a topological space $X$ as \[ \delta(x,A)= \begin{cases} 0 & \text{if $x\in\overline{A}$,}\\ \infty & \text{else.} \end{cases} \] In fact, the image of this functor can be described as precisely those approach spaces where $\delta(x,A)\in\{0,\infty\}$, for all $x\in X$ and $A\subseteq X$. Being left adjoint, $\catfont{Top}\to\catfont{App}$ preserves all colimits, and it is not hard to see that this functor preserves also all limits (and hence has a left adjoint). As in the case of topological spaces, approach spaces can be described in terms of many other concepts such as ``closed sets'' or convergence. For instance, every approach distance $\delta:X\times\catfont{2}^X\to[0,\infty]$ defines a map \[ a:UX\times X\to[0,\infty],\,a(\mathfrak{x},x)=\sup_{A\in\mathfrak{x}}\delta(x,A), \] and vice versa, every $a:UX\times X\to[0,\infty]$ defines a function \[ \delta:X\times\catfont{2}^X\to[0,\infty],\,\delta(x,A)=\inf_{A\in\mathfrak{x}}a(\mathfrak{x},x). \] Furthermore, a mapping $f:X\to Y$ between approach spaces $X=(X,a)$ and $Y=(Y,b)$ is an approach map if and only if $a(\mathfrak{x},x)\geqslant b(Uf(\mathfrak{x}),f(x))$, for all $\mathfrak{x}\in UX$ and $x\in X$. Therefore one might take as well convergence as primitive notion, and axioms characterising those functions $a:UX\times X\to[0,\infty]$ coming from a approach distance can be already found in \citep{Low_Ap}. In this paper we will make use the characterisation (given in \citep{CH_TopFeat}) as precisely the functions $a:UX\times X\to[0,\infty]$ satisfying \begin{align}\label{Eq:AxiomsApp} 0\geqslant a(\doo{x},x) &&\text{and}&& Ua(\mathfrak{X},\mathfrak{x})+a(\mathfrak{x},x)\geqslant a(m_X(\mathfrak{X}),x), \end{align} where $\mathfrak{X}\in UUX$, $\mathfrak{x}\in UX$, $x\in X$ and \[ Ua(\mathfrak{X},\mathfrak{x})=\sup_{\mathcal{A}\in\mathfrak{X},A\in\mathfrak{x}}\inf_{\mathfrak{a}\in\mathcal{A},x\in A}a(\mathfrak{a},x). \] In the language of convergence, the underlying topological space $X_p$ of an approach space $X=(X,a)$ is defined by $\mathfrak{x}\to x\iff a(\mathfrak{x},x)=0$, and a topological space $X$ can be interpreted as an approach space by putting $a(\mathfrak{x},x)=0$ whenever $\mathfrak{x}\to x$ and $a(\mathfrak{x},x)=\infty$ otherwise. We can restrict $a:UX\times X\to[0,\infty]$ to principal ultrafilters and obtain a metric \[ a_0:X\times X\to[0,\infty],\,(x,y)\mapsto a(\doo{x},y) \] on $X$. Certainly, an approach map is also a metric map, therefore this construction defines a functor \[ (-)_0:\catfont{App}\to\catfont{Met}. \] which, combined with $(-)_p:\catfont{Met}\to\catfont{Ord}$, yields a functor \[ \catfont{App}\to\catfont{Ord} \] where $x\le y$ whenever $0\geqslant a(\doo{x},y)$. This order relation extends point-wise to approach maps, and we can consider $\catfont{App}$ as an ordered category. As before, this additional structure allows us to speak about adjunction in $\catfont{App}$: for approach maps $f:(X,a)\to (X',a')$ and $g:(X',a')\to(X,a)$, $f\dashv g$ if $1_X\le g\cdot f$ and $f\cdot g\le 1_{X'}$; equivalently, $f\dashv g$ if and only if, for all $\mathfrak{x}\in UX$ and $x'\in X'$, \[ a'(Uf(\mathfrak{x}),x')=a(\mathfrak{x},g(x')). \] One calls an approach space $X=(X,a)$ \emph{separated}, or T$_0$, if the underlying topology of $X$ is T$_0$, or, equivalently, if the underlying metric of $X$ is separated. Note that this is the case precisely if, for all $x,y\in X$, $a(\doo{x},y)=0=a(\doo{y},x)$ implies $x=y$. Similarly to the situation for metric spaces, besides the categorical product there is a further approach structure on the set \[ X\times Y \] for approach spaces $X=(X,a)$ and $Y=(Y,b)$, namely \[ c(\mathfrak{w},(x,y))=a(\mathfrak{x},x)+b(\mathfrak{y},y) \] where $\mathfrak{w}\in U(X\times Y)$, $(x,y)\in X\times Y$ and $\mathfrak{x}=U\pi_1(\mathfrak{w})$ and $\mathfrak{y}=U\pi_2(\mathfrak{w})$. The resulting approach space $(X\times Y,c)$ we denote as $X\oplus Y$, in fact, one obtains a functor $\oplus:\catfont{App}\times\catfont{App}\to\catfont{App}$. We also note that $1\oplus X\simeq X\simeq X\oplus 1$, for every approach space $X$. Unfortunately, the above described monoidal structure on $\catfont{App}$ is not closed, the functor $X\oplus-:\catfont{App}\to\catfont{App}$ does not have in general a right adjoint (see \citep{Hof_TopTh}). If it does, we say that the approach space $X=(X,a)$ is \emph{$+$-exponentiable} and denote this right adjoint as $(-)^X:\catfont{App}\to\catfont{App}$. Then, for any approach space $Y=(Y,b)$, the space $Y^X$ can be chosen as the set of all approach maps of type $X\to Y$, equipped with the convergence \begin{equation}\label{Eq:FunSpStr} \fspstr{\mathfrak{p}}{h}=\sup\{b(U\!\ev(\mathfrak{w}),h(x))\ominus a(\mathfrak{x},x)\mid x\in X,\mathfrak{w}\in U(Y^X\oplus X)\text{ with }\mathfrak{w}\mapsto\mathfrak{p}, (\mathfrak{w}\mapsto\mathfrak{x})\}, \end{equation} for all $\mathfrak{p}\in U(Y^X)$ and $h\in Y^X$. If $\mathfrak{p}=\doo{k}$ for some $k\in Y^X$, then \[ \fspstr{\doo{k}}{h}=\sup_{x\in X}b_0(k(x),h(x)), \] which tells us that $(Y^X)_0$ is a subspace of $Y^{X_0}$. If $X=(X,a)$ happens to be topological, i.e.\ $a$ only takes values in $\{0,\infty\}$, then \eqref{Eq:FunSpStr} simplifies to \[ \fspstr{\mathfrak{p}}{h}=\sup\{b(U\!\ev(\mathfrak{w}),h(x))\mid x\in X,\mathfrak{w}\in U(Y^X\oplus X)\text{ with }\mathfrak{w}\mapsto\mathfrak{p}, (\mathfrak{w}\mapsto\mathfrak{x}),a(\mathfrak{x},x)=0\}. \] Furthermore, a topological approach space is $+$-exponentiable if and only if it is exponentiable in $\catfont{Top}$, that is, core-compact. This follows for instance from the characterisation of exponentiable topological spaces given in \citep{Pis_ExpTop}, together with the characterisation of $+$-exponentiable approach spaces \citep{Hof_TopTh} as precisely the ones where the convergence structure $a:UX\times X\to[0,\infty]$ satisfies \[ a(m_X(\mathfrak{X}),x)=\inf_{\mathfrak{x}\in X}(Ua(\mathfrak{X},\mathfrak{x})+a(\mathfrak{x},x)), \] for all $\mathfrak{X}\in UUX$ and $x\in X$. Note that the left hand side is always smaller or equal to the right hand side. Via the embedding $\catfont{Top}\to\catfont{App}$ described earlier in this subsection, which is left adjoint to $(-)_p:\catfont{App}\to\catfont{Top}$, we can interpret every topological space $X$ as an approach space, also denoted as $X$, where the convergence structure takes only values in $\{0,\infty\}$. Then, for any approach space $Y$, $X\oplus Y=X\times Y$, which in particular tells us that the diagram \[ \xymatrix{\catfont{App}\ar[r]^{X\oplus-} & \catfont{App}\\ \catfont{Top}\ar[u]\ar[r]_{X\times-} &\catfont{Top}\ar[u]} \] commutes. Therefore, if $X$ is core-compact, then also the diagram of the corresponding right adjoints commutes, hence \begin{lemma}\label{Lem:PowerVsUnderlyingTop} For every core-compact topological space $X$ and every approach space $Y$, $(Y^X)_p=(Y_p)^X$. \end{lemma} \begin{remark} To be rigorous, the argument presented above only allows us to conclude $(Y^X)_p\simeq(Y_p)^X$. However, since we can choose the right adjoints $(-)^X$ and $(-)_p$ exactly as described earlier, one has indeed equality. \end{remark} The lack of good function spaces can be overcome by moving into a larger category where these constructions can be carried out. In the particular case of approach spaces, a good environment for doing so is the category $\catfont{PsApp}$ of \emph{pseudo-approach spaces} and approach maps \citep{LL_TopHulls}. Here a pseudo-approach space is pair $X=(X,a)$ consisting of a set $X$ and a convergence structure $a:UX\times X\to[0,\infty]$ which only needs to satisfy the first inequality of \eqref{Eq:AxiomsApp}: $0\geqslant a(\doo{x},x)$, for all $x\in X$. If $X=(X,a)$ and $Y=(Y,b)$ are pseudo-approach spaces, then one defines $X\oplus Y$ exactly as for approach spaces, and the formula \eqref{Eq:FunSpStr} defines a pseudo-approach structure on the set $Y^X$ of all approach maps from $X$ to $Y$, without any further assumptions on $X$ or $Y$. In fact, this construction leads now to an adjunction $X\oplus-\dashv(-)^X:\catfont{PsApp}\to\catfont{PsApp}$, for every pseudo-approach space $X=(X,a)$. \subsection{Stably compact approach spaces} Returning to metric compact Hausdorff spaces, one easily verifies that \eqref{Eq:ValConvMetCompHaus} defines an approach structure on $X$ (see \citep{Tho_OrderedTopStr}, for instance). Since a homomorphism between metric compact Hausdorff spaces becomes an approach map with respect to the corresponding approach structures, one obtains a functor \[ K:\catfont{MetCompHaus}\to\catfont{App}. \] The underlying metric of $KX$ is just the metric $d$ of the metric compact Hausdorff space $X=(X,d,\alpha)$, and $x=\alpha(\mathfrak{x})$ is a \emph{generic convergence point} of $\mathfrak{x}$ in $KX$ in the sense that \[ a(\mathfrak{x},y)=d(x,y), \] for all $y\in X$. The point $x$ is unique up to equivalence since, if one has $x'\in X$ with the same property, then \[ d(x,x')=a(\mathfrak{x},x')=d(x',x')=0 \] and, similarly, $d(x',x)=0$. In analogy to the topological case, we introduce the \emph{dual} $Y^\mathrm{op}$ of an approach space $Y=K(X,d,\alpha)$ as $Y^\mathrm{op}=K(X,d^\circ,\alpha)$, and we call an T$_0$ approach space \emph{stably compact} if it is of the form $KX$, for some metric compact Hausdorff space $X$. \begin{lemma}\label{Lem:MapsMetCompSp} Let $(X,d,\alpha)$, $(Y,d',\beta)$ be metric compact Hausdorff spaces with corresponding approach spaces $(X,a)$ and $(Y,b)$, and let $f:X\to Y$ be a map. Then $f$ is an approach map $f:(X,a)\to(Y,b)$ if and only if $f:(X,d)\to(Y,d')$ is a metric map and $\beta\cdot Uf(\mathfrak{x})\le f\cdot\alpha(\mathfrak{x})$, for all $\mathfrak{x}\in UX$. \end{lemma} \begin{proof} Assume first that $f:(X,d)\to(Y,d')$ is in $\catfont{Met}$ and that $\beta\cdot Uf(\mathfrak{x})\le f\cdot\alpha(\mathfrak{x})$, for all $\mathfrak{x}\in UX$. Then \[ a(\mathfrak{x},x)=d(\alpha(\mathfrak{x}),x)\geqslant d'(f\cdot\alpha(\mathfrak{x}),f(x)) \geqslant d'(\beta\cdot Uf(\mathfrak{x}),f(x))=b(Uf(\mathfrak{x}),f(x)). \] Suppose now that $f:(X,a)\to(Y,b)$ is in $\catfont{App}$ and let $\mathfrak{x}\in UX$. Then \[ 0\geqslant d(\alpha(\mathfrak{x}),\alpha(\mathfrak{x}))=a(\mathfrak{x},\alpha(\mathfrak{x}))\geqslant b(Uf(\mathfrak{x}),f\cdot \alpha(\mathfrak{x}))= d'(\beta\cdot Uf(\mathfrak{x}),f\cdot\alpha(\mathfrak{x})). \] Clearly, $f:(X,a)\to(Y,b)$ in $\catfont{App}$ implies $f:(X,d)\to(Y,d')$ in $\catfont{Met}$, and the assertion follows. \end{proof} It is an important fact that $K$ has a left adjoint \[ M:\catfont{App}\to\catfont{MetCompHaus} \] which can be described as follows (see \citep{Hof_DualityDistSp}). For an approach space $X=(X,a)$, $MX$ is the metric compact Hausdorff space with underlying set $UX$ equipped with the compact Hausdorff convergence $m_X:UUX\to UX$ and the metric \begin{equation}\label{Eq:MetricOnUX} d:UX\times UX\to[0,\infty],\ (\mathfrak{x},\mathfrak{y})\mapsto\inf\{\varepsilon\in[0,\infty]\mid \forall A\in\mathfrak{x}\,.\,A^{(\varepsilon)}\in\mathfrak{y}\}, \end{equation} and $Mf:=Uf:UX\to UY$ is a homomorphism between metric compact Hausdorff spaces provided that $f:X\to Y$ is an approach map between approach spaces. The unit and the counit of this adjunction are given by \begin{align} e_X:(X,a)\to(UX,d(m_X(-),-)) &&\text{and}&& \alpha:(UX,d,m_X)\to(X,d,\alpha) \end{align} respectively, for $(X,a)$ in $\catfont{App}$ and $(X,d,\alpha)$ in $\catfont{MetCompHaus}$. \begin{remark} All what was said here about metric compact Hausdorff spaces and approach space can be repeated, \emph{mutatis mutandis}, for ordered compact Hausdorff spaces and topological spaces. For instance, the funcor $K:\catfont{OrdCompHaus}\to\catfont{Top}$ (see Subsection \ref{Subsect:StablyCompSp}) has a left adjoint $M:\catfont{Top}\to\catfont{OrdCompHaus}$ which sends a topological space $X$ to $(UX,\le,m_X)$, where \[ \mathfrak{x}\le\mathfrak{y}\quad\text{whenever}\quad\forall A\in\mathfrak{x}\,.\,\overline{A}\in\mathfrak{y}, \] for all $\mathfrak{x},\mathfrak{y}\in UX$. Furthermore, Lemma \ref{Lem:MapsMetCompSp} reads now as follows: Let $(X,\le,\alpha)$, $(Y,\le,\beta)$ be ordered compact Hausdorff spaces with corresponding topological spaces $(X,a)$ and $(Y,b)$, and let $f:X\to Y$ be a map. Then $f$ is a continuous map $f:(X,a)\to(Y,b)$ in $\catfont{Top}$ if and only if $f:(X,d)\to(Y,d')$ is in $\catfont{Ord}$ and $\beta\cdot Uf(\mathfrak{x})\le f\cdot\alpha(\mathfrak{x})$, for all $\mathfrak{x}\in UX$. \end{remark} \begin{example} The ordered set $\catfont{2}=\{0,1\}$ with the discrete (compact Hausdorff) topology becomes an ordered compact Hausdorff space which induces the Sierpi\'nski space $\catfont{2}$ where $\{1\}$ is closed and $\{0\}$ is open. Then the maps \begin{enumerate} \item $\bigwedge:\catfont{2}^I\to\catfont{2}$ \item $v\Rightarrow-:\catfont{2}\to\catfont{2}$, \item $v\wedge-:\catfont{2}\to\catfont{2}$ \setcounter{counter}{\value{enumi}} \end{enumerate} are continuous, for every set $I$ and $v\in\catfont{2}$. Furthermore (see \citep{Nach_CompUnions,Esc_Synth}), \begin{enumerate} \setcounter{enumi}{\value{counter}} \item $\bigvee:\catfont{2}^I\to\catfont{2}$ is continuous if and only if $I$ is a compact topological space. \end{enumerate} Here the function space $\catfont{2}^I$ is possibly calculated in the category $\catfont{PsTop}$ of pseudotopological spaces (see \citep{HLCS_ImpTop}). In particular, if $I$ is a compact Hausdorff space, then $I$ is exponentiable in $\catfont{Top}$ and $\bigvee:\catfont{2}^I\to\catfont{2}$ belongs to $\catfont{Top}$. \end{example} \begin{example} The metric space $[0,\infty]$ with distance $\mu(x,y)=y\ominus x$ equipped with the Euclidean compact Hausdorff topology where $\mathfrak{x}$ converges to $\xi(\mathfrak{x}):=\sup_{A\in\mathfrak{x}}\inf A$ is a metric compact Hausdorff space (see Example \ref{Ex:PasMCHsp}) which gives the ``Sierpi\'nski approach space'' $[0,\infty]$ with approach convergence structure $\lambda(\mathfrak{x},x)=x\ominus\xi(\mathfrak{x})$. Then, with the help of subsection \ref{Subsect:Prelim}, one sees that \begin{enumerate} \item $\sup:[0,\infty]^I\to[0,\infty]$, \item $-\ominus v:[0,\infty]\to[0,\infty]$, \item $-+v:[0,\infty]\to[0,\infty]$ \setcounter{counter}{\value{enumi}} \end{enumerate} are approach maps, for every set $I$ and $v\in[0,\infty]$. If $I$ carries the structure $a:UI\times I\to[0,\infty]$ of an approach space, one defines the \emph{degree of compactness} \citep{Low_ApBook} of $I$ as \[ \comp(I)=\sup_{\mathfrak{x}\in UI}\inf_{x\in X}a(\mathfrak{x},x). \] Then (see \citep{Hof_TopTh}), \begin{enumerate} \setcounter{enumi}{\value{counter}} \item $\inf:[0,\infty]^I\to[0,\infty]$ is an approach map if and only if $\comp(I)=0$. \end{enumerate} As above, the function space $[0,\infty]^I$ is possibly calculated in $\catfont{PsApp}$, in fact, $(-)^I:\catfont{PsApp}\to\catfont{PsApp}$ is the right adjoint of $I\oplus-:\catfont{PsApp}\to\catfont{PsApp}$. \end{example} As any adjunction, $M\dashv K$ induces a monad on $\catfont{App}$ (respectively on $\catfont{Top}$). Here, for any approach space $X$, the space $KM(X)$ is the set $UX$ of all ultrafilters on the set $X$ equipped with an apporach structure, and the unit and the multiplication are essentially the ones of the ultrafilter monad. Therefore we denote this monad also as $\monadfont{U}=(U,e,m)$. In particular, one obtains a functor $U:=KM:\catfont{App}\to\catfont{App}$ (respectively $U:=KM:\catfont{Top}\to\catfont{Top}$). Surprisingly or not, the categories of algebras are equivalent to the Eilenberg--Moore categories on $\catfont{Ord}$ and $\catfont{Met}$: \begin{align} \catfont{Ord}^\monadfont{U}\simeq\catfont{Top}^\monadfont{U} &&\text{and}&& \catfont{Met}^\monadfont{U}\simeq\catfont{App}^\monadfont{U}. \end{align} More in detail (see \citep{Hof_DualityDistSp}), for any metric compact Hausdorff space $(X,d,\alpha)$ with corresponding approach space $(X,a)$, $\alpha:U(X,a)\to(X,a)$ is an approach contraction; and for an approach space $(X,a)$ with Eilenberg--Moore algebra structure $\alpha:U(X,a)\to(X,a)$, $(X,d,\alpha)$ is a metric compact Hausdorff space where $d$ is the underlying metric of $a$ and, moreover, $a$ is the approach structure induced by $d$ and $\alpha$. It is worthwhile to note that the monad $\monadfont{U}$ on $\catfont{Top}$ as well as on $\catfont{App}$ satisfies a pleasant technical property: it is of Kock-Z\"oberlein type \citep{Koc_MonAd,Zob_Doct}. In what follows we will not explore this further and refer instead for the definition and other information to \citep{EF_SemDom}. We just remark here that one important consequence of this property is that an Eilenberg--Moore algebra structure $\alpha:UX\to X$ on an $\{$approach, topological$\}$ space $X$ is necessarily left adjoint to $e_X:X\to UX$. If $X$ is T$_0$, then one even has that an approach map $\alpha:UX\to X$ is an Eilenberg--Moore algebra structure on $X$ if and only if $\alpha\cdot e_X=1_X$. Hence, a T$_0$ approach space $X=(X,a)$ is an $\monadfont{U}$-algebra if and only if \begin{enumerate} \item\label{C3} every ultrafilter $\mathfrak{x}\in UX$ has a generic convergence point $\alpha(\mathfrak{x})$ meaning that $a(\mathfrak{x},x)=a_0(\alpha(\mathfrak{x}),x)$, for all $x\in X$, and \item\label{C4} the map $\alpha:UX\to X$ is an approach map. \end{enumerate} We observed already in \citep{Hof_DualityDistSp} that the latter condition can be substituted by \begin{enumerate} \renewcommand{\ref{C3}b}{\ref{C4}'} \item $X$ is $+$-exponentiable. \end{enumerate} For the reader familiar with the notion of sober approach space \citep{BLO_AFrm} we remark that the former condition can be splitted into the following two conditions: \begin{enumerate} \renewcommand{\ref{C3}b}{\ref{C3}a} \item for every ultrafilter $\mathfrak{x}\in UX$, $a(\mathfrak{x},-)$ is an approach prime element, and \renewcommand{\ref{C3}b}{\ref{C3}b} \item $X$ is sober. \end{enumerate} Certainly, the two conditions above imply \eqref{C3}. For the reverse implication, just note that every approach prime element $\varphi:X\to[0,\infty]$ is the limit function of some ultrafilter $\mathfrak{x}\in UX$ (see \cite[Proposition 5.7]{BLO_AFrm}). Hence, every stably compact approach space is sober. We call an $+$-exponentiable approach space $X$ \emph{stable} if $X$ satisfies the condition (\ref{C3}a) above (compare with Subsection \ref{Subsect:StablyCompSp}), and with this nomenclature one has \begin{proposition} An T$_0$ approach space $X$ is stably compact if and only if $X$ is sober, $+$-exponentiable and stable. \end{proposition} \section{Injective approach spaces} \subsection{Yoneda embeddings}\label{SubSect:YonedaInAPP} Let $X=(X,a)$ be an approach space with convergence $a:UX\times X\to[0,\infty]$. Then $a$ is actually an approach map $a:(UX)^\mathrm{op}\oplus X\to[0,\infty]$, and we refer to its $+$-exponential mate $\yoneda_X:=\mate{a}:X\to[0,\infty]^{(UX)^\mathrm{op}}$ as the \emph{(covariant) Yoneda embedding} of $X$ (see \citep{CH_Compl} and \citep{Hof_DualityDistSp}). We denote the approach space $[0,\infty]^{(UX)^\mathrm{op}}$ as $PX$, and its approach convergence structure as $\fspstr{-}{-}$. One has $a(\mathfrak{x},x)=\fspstr{U\!\yoneda_X(\mathfrak{x})}{\yoneda_X(x)}$ for all $\mathfrak{x}\in UX$ and $x\in X$ (hence $\yoneda_X$ is indeed an embedding when $X$ is $T_0$) thanks to the \emph{Yoneda Lemma} which states here that, for all $\mathfrak{x}\in UX$ and $\psi\in PX$, \[ \fspstr{U\!\yoneda_X(\mathfrak{x})}{\psi}=\psi(\mathfrak{x}). \] The metric $d:UX\times UX\to[0,\infty]$ (see \eqref{Eq:MetricOnUX}) is actually an approach map $d:(UX)^\mathrm{op}\oplus UX\to[0,\infty]$, whose mate can be seen as a \emph{``second'' (covariant) Yoneda embedding} $\yonedaT_X:UX\to PX$, and the ``second'' Yoneda Lemma reads as (see \citep{Hof_DualityDistSp}) \[ \fspstr{U\!\yonedaT_X(\mathfrak{X})}{\psi}=\psi(m_X(\mathfrak{X})), \] for all $\mathfrak{X}\in UUX$ and $\psi\in PX$. \begin{remark}\label{rem:FilterOfOpens} Similarly, the convergence relation $\to:(UX)^\mathrm{op}\times X\to\catfont{2}$ of a topological space $X$ is continuous, and by taking its exponential transpose we obtain the Yoneda embedding $\yoneda_X:X\to\catfont{2}^{(UX)^\mathrm{op}}$. A continuous map $\psi:X^\mathrm{op}\to\catfont{2}$ can be identified with a closed subset $\mathcal{A}\subseteq UX$. In \citep{HT_LCls} it is shown that $\mathcal{A}$ corresponds to a filter on the lattice of opens of $X$, moreover, the space $\catfont{2}^{(UX)^\mathrm{op}}$ is homeomorphic to the space $F_0X$ of all such filters, where the topology on $F_0X$ has \begin{align} \{\mathfrak{f}\in F_0 X\mid A\in\mathfrak{f}\}&& \text{($A\subseteq X$ open)} \end{align} as basic open sets (see \citep{Esc_InjSp}). Under this identification, the Yoneda embedding $\yoneda_X:X\to\catfont{2}^{(UX)^\mathrm{op}}$ corresponds to the map $X\to F_0X$ sending every $x\in X$ to its neighbourhood filter, and $\yonedaT_X:UX\to F_0 X$ restricts an ultrafilter $\mathfrak{x}\in UX$ to its open elements. \end{remark} Since an approach space $X$ is in general not $+$-exponentiable, the set $[0,\infty]^X$ of all approach maps of type $X\to[0,\infty]$ does not admit a canonical approach structure. However, it still becomes a metric space when equipped with the sup-metric, that is, the metric space $[0,\infty]^X$ is a subspace of the $+$-exponential $[0,\infty]^{X_0}$ in $\catfont{Met}$ of underlying metric space $X_0$ of $X$. Recall from Subsection \ref{SubSect:CocomplMetSp} that the contravariant Yoneda embedding $\yonedaOP_{X_0}:X_0\to\left([0,\infty]^{X_0}\right)^\mathrm{op}$ of the metric space $X_0$ sends an element $x\in X_0$ to the metric map $X_0\to[0,\infty],\,x'\mapsto a_0(x,x')=a(\doo{x},x')$. But the map $\yonedaOP_{X_0}(x)$ can be also seen as an approach map of type $X\to[0,\infty]$, hence this construction defines also a metric map $\yonedaOP_X:X_0\to\left([0,\infty]^X\right)^\mathrm{op}$, for every approach space $X$. The inclusion map $[0,\infty]^X\hookrightarrow[0,\infty]^{X_0}$ has a left adjoint $[0,\infty]^{X_0}\to[0,\infty]^X$ which sends a metric map $\varphi:X_0\to[0,\infty]$ to the approach map $X\to[0,\infty]$ which sends $x$ to $\displaystyle{\inf_{\mathfrak{x}\in UX}a(\mathfrak{x},x)+\xi(U\varphi(\mathfrak{x}))}$ (where $\displaystyle{\xi(\mathfrak{u})=\sup_{A\in\mathfrak{u}}\inf_{u\in A}u}$). In particular, if $\varphi=a(e_X(x),-)$, then $\xi(U\varphi(\mathfrak{x}))=Ua(e_{UX}\cdot e_X(x),\mathfrak{x})$ and therefore $\displaystyle{\inf_{\mathfrak{x}\in UX}a(\mathfrak{x},x)+\xi(U\varphi(\mathfrak{x}))}=a(e_X(x),-)$. Hence, both the left and the right adjoint commute with the contravariant Yoneda embeddings. \[ \xymatrix{\left([0,\infty]^{X_0}\right)^\mathrm{op}\ar@/^1.5ex/[rr] && \left([0,\infty]^X\right)^\mathrm{op}\ar@/^1.5ex/[ll]\\ & X_0\ar[ul]^{\yonedaOP_{X_0}}\ar[ur]_{\yonedaOP_X}} \] Finally, one also have the \emph{Isbell conjugation adjunction} in this context: \[ \xymatrix{\left([0,\infty]^X\right)^\mathrm{op}\ar@/^1.5ex/[rr]^{(-)^-}\ar@{}[rr]\|\top && \left([0,\infty]^{X^\mathrm{op}}\right)_0\ar@/^1.5ex/[ll]^{(-)^+}\\ & X_0\ar[ul]^{\yonedaOP_X}\ar[ur]_{\yoneda_X}} \] where \begin{align} \varphi^-(\mathfrak{x})&=\sup_{x\in X}(a(\mathfrak{x},x)\ominus\varphi(x)) &\text{and}&& \psi^+(x)&=\sup_{\mathfrak{x}\in UX}(a(\mathfrak{x},x)\ominus\psi(\mathfrak{x})). \end{align} \begin{remark} In our considerations above we were very sparse on details and proofs. This is because (in our opinion) this material is best presented in the language of \emph{modules} (also called distributors or profunctors), but we decided not to include this concept here and refer for details to \citep{CH_Compl} and \citep{Hof_Cocompl} (for the particular context of this paper) and to \citep{BenDistWork} and \citep{Law_MetLogClo}) for the general concept. We note that $\psi:(UX)^\mathrm{op}\to[0,\infty]$ is the same thing as a module $\psi:X\kmodto 1$ from $X$ to $1$ and $\varphi:X\to[0,\infty]$ is the same thing as a module $\varphi:1\kmodto X$. Then $\psi^+$ is the \emph{extension} of $\psi$ along the identity module on $X$ (see \citep[1.3 and Remark 1.5]{Hof_Cocompl}), and $\varphi^-$ is the \emph{lifting} of $\varphi$ along the identity module on $X$ (see \citep[Lemma 5.11]{HW_AppVCat}); and this process defines quite generally an adjunction. \end{remark} \subsection{Cocomplete approach spaces} In this and the next subsection we study the notion of \emph{cocompleteness} for approach spaces, as initiated in \citep{CH_Compl,Hof_Cocompl,CH_CocomplII}. By analogy with ordered sets and metric spaces, we think of an approach map $\psi:(UX)^\mathrm{op}\to[0,\infty]$ as a ``down-set'' of $X$. A point $x_0\in X$ is a \emph{supremum} of $\psi$ if \[ a(\doo{x_0},x)=\sup_{\mathfrak{x}\in UX}a(\mathfrak{x},x)\ominus\psi(\mathfrak{x}), \] for all $x\in X$. As before, suprema are unique up to equivalence, and therefore we will often talk about the supremum. An approach map $f:(X,a)\to(Y,b)$ preserves the supremum of $\psi$ if \[ b(\doo{f(x_0)},y)=\sup_{\mathfrak{y}\in UY}b(Uf(\mathfrak{x}),y)\ominus\psi(\mathfrak{x}). \] Not surprisingly (see \citep{Hof_Cocompl}), \begin{lemma} Left adjoint approach maps $f:X\to Y$ between approach spaces preserve all suprema which exist in $X$. \end{lemma} We call an approach space $X$ \emph{cocomplete} if every ``down-set'' $\psi:(UX)^\mathrm{op}\to[0,\infty]$ has a supremum in $X$. If this is the case, then ``taking suprema'' defines a map $\Sup_X:PX\to X$, indeed, one has \begin{proposition} An approach space $X$ is cocomplete if and only if $\yoneda_X:X_0\to(PX)_0$ has a left adjoint $\Sup_X:(PX)_0\to X_0$ in $\catfont{Met}$. \end{proposition} \begin{remark} We deviate here from the notation used in previous work where a space $X$ was called cocomplete whenever $\yoneda_X:X\to PX$ has a left adjoint in $\catfont{App}$. Approach spaces satisfying this (stronger) condition will be called absolutely cocomplete (see Subsection \ref{SubSect:OpContLatAct} below) here. \end{remark} With the help of Subsection \ref{SubSect:YonedaInAPP}, one sees immediately that $\Sup_X:(PX)_0\to X_0$ produces a left inverse of $\yonedaOP_{X_0}:X_0\to\left([0,\infty]^{X_0}\right)^\mathrm{op}$ in $\catfont{Met}$, hence the underlying metric space $X_0$ is complete. Certainly, a left inverse of $ X_0\to\left([0,\infty]^{X_0}\right)^\mathrm{op}$ in $\catfont{Met}$ gives a left inverse of $\yoneda_X:X_0\to(PX)_0$ in $\catfont{Met}$, however, such a left inverse does not need to be a left adjoint (see Example \ref{ex:CocomplNotUCocompl}). In the following subsection we will see what is missing. \subsection{Special types of colimits} Similarly to what was done for metric spaces, we will be interested in approach spaces which admit certain types of suprema. Our first example are \emph{tensored} approach spaces which are defined exactly as their metric counterparts. Explicitly, to every point $x$ of an approach space $X=(X,a)$ and every $u\in[0,\infty]$ one associates a ``down-set'' $\psi:(UX)^\mathrm{op}\to[0,\infty],\,\mathfrak{x}\mapsto a(\mathfrak{x},x)+u$, and a supremum of $\psi$ (which, we recall, is unique up to equivalence) we denote as $x+u$. Then $X$ is called tensored if every such $\psi$ has a supremum in $X$. By definition, $x+u\in X$ is characterised by the equation \[ a(e_X(x+u),y)=\sup_{\mathfrak{x}\in UX}(a(\mathfrak{x},y)\ominus(a(\mathfrak{x},x)+u)), \] for all $y\in X$. This supremum is actually obtained for $\mathfrak{x}=\doo{x}$, so that the right hand side above reduces to $a(\doo{x},y)\ominus u$. Therefore: \begin{proposition} An approach space $X$ is tensored if and only if its underlying metric space $X_0$ is tensored. \end{proposition} We call an approach space $X=(X,a)$ \emph{$\monadfont{U}$-cocomplete} if every ``down-set'' $\psi:(UX)^\mathrm{op}\to[0,\infty]$ of the form $\psi=\yonedaT_X(\mathfrak{x})$ with $\mathfrak{x}\in UX$ has a supremum in $X$. Such a supremum, denoted as $\alpha(\mathfrak{x})$, is characterised by \[ a(\doo{\alpha(\mathfrak{x})},x)=\sup_{\mathfrak{y}\in UX}(a(\mathfrak{y},x)\ominus d(\mathfrak{y},\mathfrak{x})), \] for all $x\in X$. Here the supremum is obtained for $\mathfrak{y}=\mathfrak{x}$, hence the equality above translates to \[ a_0(\alpha(\mathfrak{x}),x)=a(\mathfrak{x},x), \] where $a_0$ denotes the underlying metric of $a$. Since $a(\mathfrak{x},x)=d(\mathfrak{x},\doo{x})$ where $d$ is the metric on $(UX)_0$, we conclude that \begin{proposition} An approach space $X$ is $\monadfont{U}$-cocomplete if and only if $e_X:X_0\to(UX)_0$ has a left adjoint $\alpha:(UX)_0\to X_0$ in $\catfont{Met}$. \end{proposition} Note that every metric compact Hausdorff space is $\monadfont{U}$-cocomplete. We are now in position to characterise cocomplete approach spaces. \begin{theorem} Let $X$ be an approach space. Then the following assertions are equivalent. \begin{eqcond} \item\label{C2} $X$ is cocomplete. \item The metric space $X_0$ is complete and and the approach space $X$ is $\monadfont{U}$-cocomplete. \item\label{C1} The metric space $X_0$ is complete and $e_X:X_0\to(UX)_0$ has a left adjoint $\alpha:(UX)_0\to X_0$ in $\catfont{Met}$. \end{eqcond} Furthermore, in this situation the supremum of a ``down-set'' $\psi:(UX)^\mathrm{op}\to[0,\infty]$ is given by \begin{equation}\label{Eq:FormWeightSupAp} \bigvee_{\mathfrak{x}\in UX}\alpha(\mathfrak{x})+\psi(\mathfrak{x}). \end{equation} \end{theorem} \begin{proof} To see the implication \eqref{C1}$\Rightarrow$\eqref{C2}, we only need to show that the formula \eqref{Eq:FormWeightSupAp} gives indeed a supremum of $\psi$. In fact, \[ a_0(\bigvee_{\mathfrak{x}\in UX}\alpha(\mathfrak{x})+\psi(\mathfrak{x}),x) =\sup_{\mathfrak{x}\in UX}a_0(\alpha(\mathfrak{x})+\psi(\mathfrak{x}),x) =\sup_{\mathfrak{x}\in UX}(a_0(\alpha(\mathfrak{x}),x)\ominus\psi(\mathfrak{x})) =\sup_{\mathfrak{x}\in UX}(a(\mathfrak{x},x)\ominus\psi(\mathfrak{x})), \] for all $x\in X$. \end{proof} \begin{example}\label{ex:CocomplNotUCocompl} Every metric compact Hausdorff space whose underlying metric is cocomplete gives rise to a cocomplete approach space. In particular, both $[0,\infty]$ and $[0,\infty]^\mathrm{op}$ are cocomplete (see Example \ref{Ex:PasMCHsp}). To each metric $d$ on a set $X$ one associates the approach convergence structure \[ a_d(\mathfrak{x},y)=\sup_{A\in\mathfrak{x}}\inf_{y\in A}d(x,y), \] and this construction defines a left adjoint to the forgetful functor $(-)_0:\catfont{App}\to\catfont{Met}$. Furthermore, note that $a_d(\doo{x},y)=d(x,y)$. In particular, for the metric space $[0,\infty]=([0,\infty],\mu)$ one obtains \[ a_\mu(\mathfrak{x},y)=\sup_{A\in\mathfrak{x}}\inf_{x\in A}(y\ominus x), \] and the approach space $([0,\infty],a_\mu)$ is not $\monadfont{U}$-cocomplete. To see this, consider any ultrafilter $\mathfrak{x}\in U[0,\infty]$ which contains all sets \[ \{x\in[0,\infty]\mid u\le x<\infty\}, \] $u\in[0,\infty]$ and $u<\infty$. Then $a_\mu(\mathfrak{x},\infty)=\infty$ and $a_\mu(\mathfrak{x},y)=0$ for all $y\in[0,\infty]$ with $y<\infty$, hence $a_\mu$ cannot be of the form $\mu(\alpha(-),-)$ for a map $\alpha:U[0,\infty]\to[0,\infty]$. However, for the metric space $[0,\infty]^\mathrm{op}=([0,\infty],\mu^\circ)$, the approach convergence structure $a_{\mu^\circ}$ is actually the structure induced by the metric compact Hausdorff space $([0,\infty],\mu^\circ,\xi)$ and therefore $([0,\infty],a_{\mu^\circ})$ is cocomplete. \end{example} $\monadfont{U}$-cocomplete approach spaces are closely related to metric compact Hausdorff spaces respectively stably compact approach space, in both cases the approach structure $a$ on $X$ can be decomposed into a metric $a_0$ and a map $\alpha:UX\to U$, and one recovers $a$ as $a(\mathfrak{x},x)=a_0(\alpha(\mathfrak{x}),x)$. In fact, every metric compact Hausdorff space is $\monadfont{U}$-cocomplete, but the reverse implication is in general false since, for instance, the map $\alpha:UX\to X$ does not need to be an Eilenberg--Moore algebra structure on $X$ (i.e.\ a compact Hausdorff topology). Fortunately, this property of $\alpha$ was not needed in the proof of Lemma \ref{Lem:MapsMetCompSp}, and we conclude \begin{lemma} Let $(X,a)$ and $(Y,b)$ be $\monadfont{U}$-cocomplete approach spaces and $f:X\to Y$ be a map. Then $f:(X,a)\to(Y,b)$ is an approach map if and only if $f:(X,a_0)\to(Y,b_0)$ is a metric map and, for all $\mathfrak{x}\in UX$, $\beta\cdot Uf(\mathfrak{x})\le f\cdot\alpha(\mathfrak{x})$. \end{lemma} \begin{remark} Once again, everything told here has its topological counterpart. For instance, we call a topological space $X$ $\monadfont{U}$-cocomplete whenever the monotone map $e_X:X_0\to(UX)_0$ has a left adjoint $\alpha:(UX)_0\to X_0$ in $\catfont{Ord}$. Then, with $\le$ denoting the underlying order of $X$, an ultrafilter $\mathfrak{x}\in UX$ converges to $x\in X$ if and only if $\alpha(\mathfrak{x})\le x$. Moreover, one also has an analog version of the lemma above. \end{remark} Recall from Subsection \ref{Subsect:App} that $(-)_p:\catfont{App}\to\catfont{Top}$ denotes the canonical forgetful functor from $\catfont{App}$ to $\catfont{Top}$, where $\mathfrak{x}\to x$ in $X_p$ if and only if $0=a(\mathfrak{x},x)$ in the approach space $X=(X,a)$. If $X=(X,a)$ is also $\monadfont{U}$-cocomplete with left adjoint $\alpha:(UX)_0\to X_0$, then, for any $\mathfrak{x}\in UX$ and $x\in X$, \[ \alpha(\mathfrak{x})\le x\iff 0=a_0(\alpha(\mathfrak{x}),x)\iff 0=a(\mathfrak{x},x)\iff\mathfrak{x}\to x. \] Here $\le$ denotes the underlying order of the underlying topology of $X$, which is the same as the underlying order of the underlying metric of $X$. Hence, $\alpha$ provides also a left adjoint to $e_X:X_{p0}\to U(X_p)_0$, and therefore the topological space $X_p$ is $\monadfont{U}$-cocomplete as well. An important consequence of this fact is \begin{proposition}\label{Prop:APcontrVsMetContr+Cont} Let $X=(X,a)$ and $Y=(Y,b)$ be $\monadfont{U}$-cocomplete approach spaces and $f:X\to Y$ be a map. Then $f:(X,a)\to(Y,b)$ is an approach map if and only if $f:(X,a_0)\to(Y,b_0)$ is a metric map and $f:X_p\to Y_p$ is continuous. \end{proposition} Finally, we also observe that $\monadfont{U}$-cocomplete approach spaces are stable under standard constructions: both $X\oplus Y$ and $X\times Y$ are $\monadfont{U}$-cocomplete, provided that $X=(X,a)$ and $Y=(Y,b)$ are so. \subsection{Op-continuous lattices with an $[0,\infty]$-action} \label{SubSect:OpContLatAct} We call an approach space $X$ \emph{absolutely cocomplete} if the Yoneda embedding $\yoneda_X:X\to PX$ has a left adjoint in $\catfont{App}$. This is to say, $X$ is cocomplete and the metric left adjoint $\Sup_X$ of $\yoneda_X$ is actually an approach map $\Sup_X:PX\to X$. It is shown in \citep{Hof_Cocompl} that \begin{itemize} \item the absolutely cocomplete approach spaces are precisely the injective ones, and that \item the category \[\catfont{InjApp}_\mathrm{sup}\] of absolutely cocomplete approach T$_0$ spaces and supremum preserving ($=$ left adjoint) approach maps is monadic over $\catfont{App}$, $\catfont{Met}$ and $\catfont{Set}$. The construction $X\mapsto PX$ is the object part of the left adjoint $P:\catfont{App}\to\catfont{InjApp}_\mathrm{sup}$ of the inclusion functor $\catfont{InjApp}_\mathrm{sup}\to\catfont{App}$, and the maps $\yoneda_X:X\to PX$ define the unit $\yoneda$ of the induced monad $\monadfont{P}=(P,\yoneda,\yonmult)$ on $\catfont{App}$. Composing this monad with the adjunction $(-)_d\dashv(-):\catfont{App}\leftrightarrows\catfont{Set}$ gives the corresponding monad on $\catfont{Set}$. \end{itemize} This resembles very much well-known properties of injective topological T$_0$ spaces, which are known to be the algebras for the filter monad on $\catfont{Top}$, $\catfont{Ord}$ and $\catfont{Set}$, hence, by Remark \ref{rem:FilterOfOpens}, are precisely the (accordingly defined) absolutely cocomplete topological T$_0$ spaces. Furthermore, all information about the topology of an injective T$_0$ space is contain in its underlying order, and the ordered sets occurring this way are the \emph{op-continuous lattices}, i.e. the duals of continuous lattices\footnote{Recall that our underlying order is dual to the specialisation order.}, as shown in \cite{Sco_ContLat} (see Subsection \ref{Subsect:ContLat}). In the sequel we will write $\catfont{ContLat_}$ to denote the category of op-continuous lattices and maps preserving all suprema and down-directed infima. Note that $\catfont{ContLat_}$ is equivalent to the category of absolutely cocomplete topological T$_0$ spaces and left adjoints in $\catfont{Top}$, and of course also to the category $\catfont{ContLat}$ of continuous lattices and maps preserving up-directed suprema and all infima. These analogies make us confident that absolutely cocomplete approach T$_0$ spaces provide an interesting metric counterpart to (op-)continuous lattices. In fact, in \citep{Hof_DualityDistSp} it is shown that the approach structure of such a space is determined by its underlying metric, hence we are talking essentially about metric spaces here. Moreover, every absolutely cocomplete approach space is exponentiable in $\catfont{App}$ and the \emph{full} subcategory of $\catfont{App}$ defined by these spaces is Cartesian closed. Theorem \ref{thm:CharAbsCocompII} below exposes now a tight connection with op-continuous lattices: the absolutely cocomplete approach T$_0$ spaces are precisely the op-continuous lattices equipped with an unitary and associative action of $[0,\infty]$ in the monoidal category $\catfont{ContLat_}$. Every approach space $X=(X,a)$ induces approach maps \begin{align} X\oplus[0,\infty]\xrightarrow{\,\formalball_X\,}PX, && UX\xrightarrow{\,\yonedaT_X\,}PX, && X^I\xrightarrow{\,\fammod_{X,I}\,}PX\qquad\text{(where $I$ is compact Hausdorff)}. \end{align} Exactly as in Subsection \ref{Subsect:TensMet}, $\formalball_X:X\oplus[0,\infty]\to PX$ is the mate of the composite \[ (UX)^\mathrm{op}\oplus X\oplus [0,\infty]\xrightarrow{\,a\oplus 1\,} [0,\infty]\oplus [0,\infty]\xrightarrow{\,+\,}[0,\infty], \] and $\fammod_{X,I}:X^I\to PX$ is the mate of the composite \[ (UX)^\mathrm{op}\oplus X^I\to[0,\infty]^I\xrightarrow{\,\inf\,}[0,\infty], \] where the first component is the mate of the composite \[ (UX)^\mathrm{op}\oplus X^I\oplus I\xrightarrow{\,1\oplus\ev\,} (UX)^\mathrm{op}\oplus X \xrightarrow{\,a\,}[0,\infty]. \] Explicitely, for $\varphi\in X^I$ and $\mathfrak{x}\in UX$, $\fammod_{X,I}(\varphi)(\mathfrak{x})=\inf_{i\in I}a(\mathfrak{x},\varphi(i))$. A supremum of the ``down-set'' $\fammod_{X,I}(\varphi)\in PX$ is necessarily a supremum of the family $(\varphi(i))_{i\in I}$ in the underlying order of $X$. If $X$ is cocomplete, one can compose the maps above with $\Sup_X$ and obtains metric maps \begin{align}\label{Eq:ThreeMaps} X_0\oplus[0,\infty]\xrightarrow{\,+\,}X_0, && (UX)_0\xrightarrow{\,\alpha\,} X_0, && (X^I)_0\xrightarrow{\,\bigvee\,}X_0\qquad\text{($I$ compact Hausdorff)}, \end{align} which are even morphisms in $\catfont{App}$ provided that $X$ is absolutely cocomplete. In fact, one has \begin{proposition}\label{Prop:CharAbsCocompI} Let $X$ be an approach space. Then $X$ is absolutely cocomplete if and only if $X$ is cocomplete and the three maps \eqref{Eq:ThreeMaps} are approach maps. \end{proposition} \begin{proof} The obtain the reverse implication, we have to show that the mapping \[ \Sup_X:PX\to X,\,\psi\mapsto\bigvee_{\mathfrak{x}\in UX}(\alpha(\mathfrak{x})+\psi(\mathfrak{x})) \] is an approach map. We write $X_d$ for the discrete approach space with underlying set $X$, then $U(X_d)$ is just a compact Hausdorff space, namely the \v{C}ech-Stone compactification of the set $X$. By assumption, $\bigvee:X^{U(X_d)}\to X$ is an approach map, therefore it is enough to show that \[ U(X_d)\oplus PX\to X,\,(\mathfrak{x},\psi)\mapsto\alpha(\mathfrak{x})+\psi(\mathfrak{x}) \] belonges to $\catfont{App}$. Since the diagonal $\Delta:U(X_d)\to U(X_d)\oplus U(X_d)$ as well as the identity maps $U(X_d)\to UX$ and $U(X_d)\to (UX)^\mathrm{op}$ are in $\catfont{App}$, we can express the map above as the composite \[ U(X_d)\oplus PX\xrightarrow{\,\Delta\oplus 1\,} UX\oplus (UX)^\mathrm{op}\oplus PX\xrightarrow{\,\alpha\oplus\ev\,} X\oplus[0,\infty]\xrightarrow{\,+\,} X \] of approach maps. \end{proof} \begin{example} The approach space $[0,\infty]$ is injective and hence absolutely cocomplete, but $[0,\infty]^\mathrm{op}$ is not injective. To see this, either observe that the map \[ f:\{0,\infty\}\to[0,\infty]^\mathrm{op},\, 0\mapsto\infty,\infty\mapsto 0 \] cannot be extended along the subspace inclusion $\{0,\infty\}\hookrightarrow[0,\infty]$, or that the mapping $(u,v)\mapsto u\ominus v$ (which is the tensor of the metric space $[0,\infty]^\mathrm{op}$) is not an approach map of type $[0,\infty]^\mathrm{op}\oplus[0,\infty]\to[0,\infty]^\mathrm{op}$. Therefore $[0,\infty]^\mathrm{op}$ is not absolutely cocomplete, however, recall from Example \ref{ex:CocomplNotUCocompl} that $[0,\infty]^\mathrm{op}$ is cocomplete. \end{example} \begin{remark} Similarly, a topological space $X$ is absolutely cocomplete if and only if $X$ is cocomplete and the latter two maps of \eqref{Eq:ThreeMaps} (accordingly defined) are continuous. \end{remark} \begin{lemma}\label{Lem:XpowerIisUcocomplete} Let $X$ be an approach space and $I$ be a compact Hausdorff space. If $X$ is cocomplete and $X_p$ is absolutely cocomplete, then $X^I$ is $\monadfont{U}$-cocomplete. \end{lemma} \begin{proof} We write $a:UX\times X\to[0,\infty]$ for the convergence structure of the approach space $X$, and $b:UI\times I\to\catfont{2}$ for the convergence structure of the compact Hausdorff space $I$. In the both cases there are maps $\alpha:UX\to X$ and $\beta:UI\to I$ respectively so that $a(\mathfrak{x},x)=a_0(\alpha(\mathfrak{x}),x)$ and $b(\mathfrak{u},i)=\mathsf{true}$ if and only if $\beta(\mathfrak{u})=i$, for all $\mathfrak{x}\in UX$, $x\in X$, $\mathfrak{u}\in UI$ and $i\in I$. For every $\mathfrak{p}\in U(X^I)$ and $h\in X^I$, \begin{align} \fspstr{\mathfrak{p}}{h} &=\sup\{a_0(\alpha(U\ev(\mathfrak{w})),h(\beta(\mathfrak{u}))\mid \mathfrak{w}\in U(X^I\times I),\mathfrak{w}\mapsto\mathfrak{p},(\mathfrak{w}\mapsto\mathfrak{u})\}\\ &=\sup_{i\in I}\sup_{\substack{\mathfrak{w}\in U(X^I\times I)\\ U\pi_1(\mathfrak{w})=\mathfrak{p}\\ \beta\cdot U\pi_2(\mathfrak{w})=i}}a_0(\alpha(U\ev(\mathfrak{w})),h(i))\\ &=\sup_{i\in I}a_0(\bigvee_{\substack{\mathfrak{w}\in U(X^I\times I)\\ U\pi_1(\mathfrak{w})=\mathfrak{p}\\ \beta\cdot U\pi_2(\mathfrak{w})=i}}\alpha(U\ev(\mathfrak{w})),h(i))\\ &=\sup_{i\in I}a_0(\gamma(\mathfrak{p})(i),h(i)), \end{align} where we define \[ \gamma(\mathfrak{p})(i)=\bigvee_{\substack{\mathfrak{w}\in U(X^I\times I)\\ U\pi_1(\mathfrak{w})=\mathfrak{p}\\ \beta\cdot U\pi_2(\mathfrak{w})=i}}\alpha(U\ev(\mathfrak{w})). \] In order to conclude that $\gamma$ is a map of type $U(X^I)\to X^I$, we have to show that $\gamma(\mathfrak{p})$ is a continuous map $\gamma(\mathfrak{p}):I\to X_p$, for every $\mathfrak{p}\in U(X^I)$. To this end, we note first that the supremum above can be rewritten as \[ \gamma(\mathfrak{p})(i)=\bigvee_{\substack{\mathfrak{w}\in U(X^I\times I)\\ U\pi_1(\mathfrak{w})=\mathfrak{p}}}\alpha(U\ev(\mathfrak{w}))\,\&\, b(U\pi_2(\mathfrak{w}),i). \] We put $Y=\{\mathfrak{w}\in U(X^I\times I)\mid U\pi_1(\mathfrak{w})=\mathfrak{p}\}$ and consider $Y$ as a subspace of $U((X^I\times I)_d)$, that is, the \v{C}ech-Stone compactification of the set $X^I\times I$. Note that $Y$ is compact, and one has continuous maps \begin{align} Y\xrightarrow{\,U\!\ev\,}U(X_d), && Y\xrightarrow{\,U\pi_2\,}U(I_d), && U(X_d)\xrightarrow{\,\alpha\,}X_p, && U(I_d)\times I\xrightarrow{\,b\,}\catfont{2}. \end{align} Therefore we can express the map $\gamma(\mathfrak{p})$ as the composite \[ I\longrightarrow X_p^I\xrightarrow{\,\bigvee\,}X_p \] of continuous maps, where the first component is the mate of the composite \[ Y\times I\xrightarrow{\,\Delta\times 1\,} Y\times Y\times I\xrightarrow{\,U\!\ev\times U\pi_2\times 1\,} U(X_d)\times U(I_d)\times I\xrightarrow{\,\alpha\times b\,} X_p\times\catfont{2}\xrightarrow{\,\&\,}X_p \] of continuous maps. \end{proof} \begin{proposition}\label{Prop:Cocompl+OPcont} Let $X=(X,d)$ be a cocomplete metric space whose underlying ordered set $X_p$ is a op-continuous lattice. Then $(X,d,\alpha)$ is a metric compact Hausdorff space where $\alpha:UX\to X$ is defined by $\displaystyle{\mathfrak{x}\mapsto\bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in X}x}$. \end{proposition} \begin{proof} Since $X_p$ is op-continuous, $X_p$ is even an ordered compact Hausdorff space with convergence $\alpha$. We have to show that $\alpha:U(X,d)\to(X,d)$ is a metric map. Recall from Lemma \ref{Lem:UXisTensored} that with $(X,d)$ also $U(X,d)$ is tensored, hence we can apply Proposition \ref{Prop:ContrTensMet}. Firstly, for $\mathfrak{x}\in UX$ and $u\in[0,\infty]$, \[ \alpha(\mathfrak{x})+u=\left(\bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in X}x\right)+u \le\bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in X}(x+u)=\alpha(\mathfrak{x}+u) \] since $-+u:X\to X$ preserves suprema. Secondly, let $\mathfrak{x},\mathfrak{y}\in UX$ and assume \[ 0=Ud(\mathfrak{x},\mathfrak{y})=\sup_{A\in\mathfrak{x},B\in\mathfrak{y}}\inf_{x\in A,y\in B}d(x,y) =\sup_{B\in\mathfrak{y}}\inf_{A\in\mathfrak{x}}\sup_{x\in A}\inf_{y\in B}d(x,y). \] For the last equality see \citep[Lemma 6.2]{SEAL_LaxAlg}, for instance. We wish to show that $\alpha(\mathfrak{x})\le\alpha(\mathfrak{y})$, that is, \[ \bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in A}x\le \bigwedge_{B\in\mathfrak{y}}\bigvee_{y\in B}y, \] which is equivalent to $\displaystyle{\bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in A}x\le\bigvee_{y\in B}y}$, for all $B\in\mathfrak{y}$. Let $B\in\mathfrak{y}$ and $\varepsilon>0$. By hypothesis, there exist some $A\in\mathfrak{x}$ with $\sup_{x\in A}\inf_{y\in B}d(x,y)<\varepsilon$, hence, for all $x\in A$, there exist some $y\in B$ with $d(x,y)<\varepsilon$ and therefore $x+\varepsilon\le y$. Consequently, for all $\varepsilon>0$, \[ \left(\bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in A}x\right)+\varepsilon\le \bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in A}(x+\varepsilon)\le \bigvee_{y\in B}y; \] and therefore also $\displaystyle{\bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in A}x\le\bigvee_{y\in B}y}$. \end{proof} \begin{theorem} Let $X$ be a T$_0$ approach space. Then the following assertions are equivalent. \begin{eqcond} \item $X$ is absolutely cocomplete. \item $X$ is cocomplete, $X_p$ is absolutely cocomplete and $+:X\oplus[0,\infty]\to X$ is an approach map. \item $X$ is cocomplete, $X_p$ is absolutely cocomplete and $+:X_p\times[0,\infty]_p\to X_p$ is continuous. \item $X$ is $\monadfont{U}$-cocomplete, $X_0$ is cocomplete, $X_p$ is absolutely cocomplete, and, for all $x\in X$ and $u\in[0,\infty ]$, the map $-+u:X\to X$ preserves down-directed infima and the map $x+-:[0,\infty]\to X$ sends up-directed suprema to down-directed infima. \end{eqcond} \end{theorem} \begin{proof} Clearly, (i)$\Rightarrow$(ii)$\Rightarrow$(iii)$\Rightarrow$(iv). Assume now (iv). According to Proposition \ref{Prop:CharAbsCocompI}, we have to show that the three maps \eqref{Eq:ThreeMaps} are approach maps. We write $a:UX\times X\to[0,\infty]$ for the convergence structure of the approach space $X$, by hypothesis, $a(\mathfrak{x},x)=d(\alpha(\mathfrak{x}),x)$ where $d$ is the underlying metric and $\alpha(\mathfrak{x})=\displaystyle{\bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in X}x}$. By Lemma \ref{Prop:Cocompl+OPcont}, $(X,d,\alpha)$ is a metric compact Hausdorff space and therefore $\alpha:UX\to X$ is an approach map. Since the metric space $X_0$ is cocomplete, $+:X_0\oplus[0,\infty]\to X_0$ and $\bigvee:X_0^I\to X_0$ ($I$ any set) are metric maps. If $I$ is a compact Hausdorff space, $(X^I)_0$ is a subspace of $X_0^{I_d}$, therefore also $\bigvee:(X^I)_0\to X_0$ is a metric map. Furthermore, since $X_p$ is absolutely cocomplete, $\bigvee:(X^I)_p=(X_p)^I\to X_p$ is continuous (see Lemma \ref{Lem:PowerVsUnderlyingTop}). Since $X^I$ is $\monadfont{U}$-cocomplete by Lemma \ref{Lem:XpowerIisUcocomplete}, $\bigvee:X^I\to X$ is actually an approach map by Proposition \ref{Prop:APcontrVsMetContr+Cont}. Similarly, $+:(X\oplus[0,\infty])_0\to X_0$ is a metric map since $(X\oplus[0,\infty])_0=X_0\oplus[0,\infty]$. Our hypothesis states that $+:(X\oplus[0,\infty])_p=X_p\times[0,\infty]_p\to X_p$ is continuous in each variable, and \citep[Proposition 2.6]{Sco_ContLat} tells us that it is indeed continuous. By applying Proposition \ref{Prop:APcontrVsMetContr+Cont} again we conclude that $+:X\oplus[0,\infty]\to X$ is an approach map. \end{proof} Note that the approach structure of an absolutely cocomplete T$_0$ approach space can be recovered from its underlying metric since the convergence $\alpha:UX\to X$ is defined by the underlying lattice structure. In fact, the Theorem above shows that an absolutely cocomplete T$_0$ approach space is essentially the same thing as a separated cocomplete metric space $X=(X,d)$ whose underlying ordered set is an op-continuous lattice (see Proposition \ref{Prop:Cocompl+OPcont}) and where the action $+:X\times[0,\infty]\to X$ preserves down-directed infima (in both variables). In the final part of this paper we combine this with Theorem \ref{Thm:TensMetVsAction} where separated cocomplete metric spaces are described as sup-lattices $X$ equipped with an unitary and associative action $+:X\times[0,\infty]\to X$ on the set $X$ which preserves suprema in each variable, or, equivalently, $+:X\otimes[0,\infty]\to X$ is in $\catfont{Sup}$. For $X,Y,Z$ in $\catfont{ContLat_}$, a map $h:X\times Y\to Z$ is a bimorphism if it is a morphism of $\catfont{ContLat_}$ in each variable. \begin{proposition} The category $\catfont{ContLat_}$ admits a tensor product which represents bimorphisms. That is, for all $X,Y$ in $\catfont{ContLat_}$, the functor \[ \Bimorph(X\times Y,-):\catfont{ContLat_}\to\catfont{Set} \] is representable by some object $X\otimes Y$ in $\catfont{ContLat_}$. \end{proposition} \begin{proof} One easily verifies that $\Bimorph(X\times Y,-)$ preserves limits. We check the solution set condition of Freyd's Adjoint Functor Theorem (in the form of \citep[Section V.3, Theorem 3]{MacLane_WorkMath}). Take $\mathcal{S}$ as any representing set of $\{L\in \catfont{ContLat_}\mid \|L\|\le\|F(X\times Y)\|\}$, where $F(X\times Y)$ denotes the set of all filters on the set $X\times Y$. Let $Z$ be an op-continuous lattice and $\varphi:X\times Y\to Z$ be a bimorphism. We have to find some $L\in\mathcal{S}$, a bimorphism $\varphi':X\times Y\to L$ and a morphism $m:L\to Z$ in $\catfont{ContLat_}$ with $m\cdot \varphi'=\varphi$. Since the map $e:X\times Y\to F(X\times Y)$ sending $(x,y)$ to its principal filter gives actually the reflection of $X\times Y$ to $\catfont{ContLat_}$, there exists some $f:F(X\times Y)\to Z$ in $\catfont{ContLat_}$ with $f\cdot e=\varphi$. \[ \xymatrix{ & F(X\times Y)\ar@{-->>}[dr]^q\ar[dd]^f\\ && L\ar@{>-->}[dl]^m\\ X\times Y\ar[uur]^e\ar[r]_\varphi\ar@{..>}[urr]^{\varphi'=q\cdot e\quad} & Z} \] Let $f=m\cdot q$ a (regular epi,mono)-factorisation of $f$ in $\catfont{ContLat_}$. Then $\varphi':=q\cdot e$ is a bimorphism as it is the corestriction of $\varphi$ to $L$, $m:L\to Z$ lies in $\catfont{ContLat_}$ and $L$ can be chosen in $\mathcal{S}$. \end{proof} By unicity of the representing object, $1\otimes X\simeq X\simeq X\otimes 1$ and $(X\otimes Y)\otimes Z\simeq X\otimes(Y\otimes Z)$. Furthermore, with the order $\geqslant$, $[0,\infty]$ is actually a monoid in $\catfont{ContLat_}$ since $+:[0,\infty]\times[0,\infty]\to[0,\infty]$ is a bimorphism and therefore it is a morphism $+:[0,\infty]\otimes[0,\infty]\to[0,\infty]$ in $\catfont{ContLat_}$, and so is $1\to [0,\infty],\,\star\mapsto 0$. We write \[ \catfont{ContLat_}^{[0,\infty]} \] for the category whose objects are op-continuous lattices $X$ equipped with a unitary and associative action $+:X\otimes[0,\infty]\to X$ in $\catfont{ContLat_}$, and whose morphisms are those $\catfont{ContLat_}$-morphisms $f:X\to Y$ which satisfy $f(x+u)=f(x)+u$, for all $x\in X$ and $u\in[0,\infty]$. Summing up, \begin{theorem}\label{thm:CharAbsCocompII} $\catfont{InjApp}_\mathrm{sup}$ is equivalent to $\catfont{ContLat_}^{[0,\infty]}$. \end{theorem} Here an absolutely cocomplete T$_0$ approach space $X=(X,a)$ is sent to its underlying ordered set where $x\le y\iff a(\doo{x},y)=0$ ($x,y\in X$) equipped with the tensor product of $X$, and an op-continuous lattice $X$ with action $+$ is sent to the approach space induced by the metric compact Hausdorff space $(X,d,\alpha)$ where $d(x,y)=\inf\{u\in[0,\infty]\mid x+u\le y\}$ and $\displaystyle{\alpha(\mathfrak{x})=\bigwedge_{A\in\mathfrak{x}}\bigvee_{x\in A}x}$, for all $x,y\in X$ and $\mathfrak{x}\in UX$. \begin{remark} By the theorem above, the diagram \[ \xymatrix{\catfont{InjApp}_\mathrm{sup}\simeq\catfont{ContLat_}^{[0,\infty]}\ar[dd]_\dashv\ar[rr]^-{\perp} & & \catfont{ContLat_*}\ar[ddll]^\rightthreetimes\ar@{.>}@/_4ex/[ll]_{-\otimes[0,\infty]}\\ \\ \catfont{Set}\ar@{.>}@/^4ex/[uu]^P\ar@{.>}@/_4ex/[uurr]_F} \] of right adjoints commutes, and therefore the diagram of the (dotted) left adjoints does so too. Here $FX$ is the set of all filters on the set $X$, ordered by $\supseteq$, and $PX=[0,\infty]^{UX}$ where $UX$ is equipped with the Zariski topology. In other words, $PX\simeq FX\otimes[0,\infty]$, for every set $X$. \end{remark} \def$'${$'$} \end{document}	arXiv
Berlin workshops on Babylonian mathematics The Berlin workshops were a series of six workshops that took place between 1983 and 1994 and focused on mathematical conceptualization and notation in a number of early writing systems. Although the names of the workshops varied slightly over time, most included the phase "conceptual development of Babylonian mathematics" and were supported by the Archaische Texte aus Uruk Project at Freie Universität Berlin[1] and the Max Planck Institute for the History of Science. The first meeting was held at the Altorientalisches Seminar und Seminar für Vorderasiatische Altertumskunde on August 5, 1983. Subsequent meetings were held in 1984, 1985, 1988 and 1994.[2] List of workshops The workshops played a significant role in advancing the decipherment of Proto-cuneiform and Proto-Elamite numerals as well as the comparative study of early mathematical notation.[3] • Workshop on Mathematical Concepts in Babylonian Mathematics Date: August 1-5, 1983. Place: Seminar für Vorderasiatische Altertumskunde und altorientalische Philologie der Freien Universität. • Second Workshop on Concept Development in Babylonian Mathematics Date: June 18-22, 1984. Place: Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin. • Third Workshop on Concept Development in Babylonian Mathematics Date: December 9-13, 1985. Place: Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin. • Fourth Workshop on Concept Development in Babylonian Mathematics Date: May 5-9, 1988. Place: Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin. • Fifth Workshop on Mathematical Concepts in Babylonian Mathematics Date: January 21-23, 1994. Place: Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin. • Standardisierung der elektronischen Transliteration von Keilschrifttexten Date: September 7-9, 1994. Place: Max Planck Institute for the History of Science[4] References 1. "Archaische Text aus Uruk". 2. Høyrup, Jens (2001). Changing Views on Ancient Near Eastern Mathematics. Reimer-Verlag. p. vii-viii. ISBN 3-496-02653-7. 3. Høyrup, Jens (1991). "Changing Trends in the Historiography of Mesopotamian Mathematics—An Insider's View". Roskilde University Preprints. 3: 30-31. 4. Høyrup, Jens (2001). Changing Views on Ancient Near Eastern Mathematics. Reimer-Verlag. p. xv-xvi. ISBN 3-496-02653-7.	Wikipedia
Low and high hierarchies In the computational complexity theory, the low hierarchy and high hierarchy of complexity levels were introduced in 1983 by Uwe Schöning to describe the internal structure of the complexity class NP.[1] The low hierarchy starts from complexity class P and grows "upwards", while the high hierarchy starts from class NP and grows "downwards".[2] Later these hierarchies were extended to sets outside NP. The framework of high/low hierarchies makes sense only under the assumption that P is not NP. On the other hand, if the low hierarchy consists of at least two levels, then P is not NP.[3] It is not known whether these hierarchies cover all NP. References 1. Uwe Schöning (1983). "A Low and a High Hierarchy within NP". J. Comput. Syst. Sci. 27 (1): 14–28. doi:10.1016/0022-0000(83)90027-2. 2. "Complexity Theory and Cryptology", by Jörg Rothe p. 232 3. Lane A. Hemaspaandra, "Lowness: a yardstick for NP-P", ACM SIGACT News, 1993, vol. 24, no.2, pp. 11-14. doi:10.1145/156063.156064	Wikipedia
\begin{document} \begin{center} {\Large \bf Odd length for even hyperoctahedral groups and\\ signed generating functions \footnote{2010 Mathematics Subject Classification: Primary 05A15; Secondary 05E15, 20F55.}} Francesco Brenti \\ Dipartimento di Matematica \\ Universit\`{a} di Roma ``Tor Vergata''\\ Via della Ricerca Scientifica, 1 \\ 00133 Roma, Italy \\ {\em [email protected] } \\ Angela Carnevale \footnote{Partially supported by German-Israeli Foundation for Scientific Research and Development, grant no. 1246.} \\ Fakultat f\"ur Mathematik \\ Universit\"at Bielefeld\\ D-33501 Bielefeld, Germany \\ {\em [email protected] } \\ \end{center} \begin{abstract} We define a new statistic on the even hyperoctahedral groups which is a natural analogue of the odd length statistic recently defined and studied on Coxeter groups of types $A$ and $B$. We compute the signed (by length) generating function of this statistic over the whole group and over its maximal and some other quotients and show that it always factors nicely. We also present some conjectures. \end{abstract} \section{Introduction} The signed (by length) enumeration of the symmetric group, and other finite Coxeter groups by various statistics is an active area of research (see, e.g., \cite{AGR, Bia, BC, Cas, DF, L, Man, Mon, Rei, Siv, Wa}). For example, the signed enumeration of classical Weyl groups by major index was carried out by Gessel-Simion in \cite{Wa} (type $A$), by Adin-Gessel-Roichman in \cite{AGR} (type $B$) and by Biagioli in \cite{Bia} (type $D$), that by descent by Desarmenian-Foata in \cite{DF} (type $A$) and by Reiner in \cite{Rei} (types $B$ and $D$), while that by excedance by Mantaci in \cite{Man} and independently by Sivasubramanian in \cite{Siv} (type $A$) and by Mongelli in \cite{Mon} (other types). In \cite{KV}, \cite{VS} and \cite{VS2} two statistics were introduced on the symmetric and hyperoctahedral groups, in connection with the enumeration of partial flags in a quadratic space and the study of local factors of representation zeta functions of certain groups, respectively (see \cite{KV} and \cite{VS2}, for details). These statistics combine combinatorial and parity conditions and have been called the ``odd length'' of the respective groups. In \cite{KV} and \cite{VS2} it was conjectured that the signed (by length) generating functions of these statistics over all the quotients of the corresponding groups always factor in a very nice way, and this was proved in \cite{BC} (see also \cite{CarT}) for types $A$ and $B$ and independently, and in a different way, in \cite{L} for type $B$. In this paper we define a natural analogue of these statistics for the even hyperoctahedral group and study the corresponding signed generating functions. More precisely, we show that certain general properties that these signed generating functions have in types $A$ and $B$ (namely ``shifting'' and ``compressing'') continue to hold in type $D$. We then show that these generating functions factor nicely for the whole group (i.e., for the trivial quotient) and for the maximal quotients. As a consequence of our results we show that the signed generating function over the whole even hyperoctahedral group is the square of the one for the symmetric group. The organization of the paper is as follows. In the next section we recall some definitions, notation, and results that are used in the sequel. In \S 3 we define a new statistic on the even hyperoctahedral group which is a natural analogue of the odd length statistics that have already been defined in types $A$ and $B$ in \cite{KV} and \cite{VS2}, and study some general properties of the corresponding signed generating functions. These include a complementation property, the identification of subsets of the quotients over which the corresponding signed generating function always vanishes, and operations on a quotient that leave the corresponding signed generating function unchanged. In \S 4 we show that the signed generating function over the whole even hyperoctahedral group factors nicely. As a consequence of this result we obtain that this signed generating function is the square of the corresponding one in type $A$. In \S 5 we compute the signed generating functions of the maximal, and some other, quotients and show that these also always factor nicely. Finally, in \S 6, we present some conjectures naturally arising from the present work, and the evidence that we have in their favor. \section{Preliminaries} In this section we recall some notation, definitions, and results that are used in the sequel. \noindent We let $\mathbb P:=\{1,\,2,\ldots\}$ be the set of positive integers and $\mathbb N:= \mathbb{P} \cup \{0\}$. For all $m,\,n \in \mathbb{Z}$, $m\leq n$ we let $[m,n]:= \{m,\,m+1,\ldots,\,n\}$ and $[n]:=[1,\,n]$. Given a set $I$ we denote by $\|I\|$ its cardinality. For a real number $x$ we denote by $\left\lfloor x \right\rfloor$ the greatest integer less than or equal to $x$ and by $\left\lceil x \right\rceil$ the smallest integer greater than or equal to $x$. Given $J \subseteq [0,n-1]$ there are unique integers $a_1 < \cdots < a_s$ and $b_1 < \cdots < b_s$ such that $J = [a_1,b_1] \cup \cdots \cup [a_s,b_s]$ and $a_{i+1} - b_{i} >1$ for $i=1, \ldots , s-1$. We call the intervals $[a_1,b_1], \ldots , [a_s,b_s]$ the {\em connected components} of $J$. \noindent For $n_1,\ldots,\,n_k \in \mathbb N$ and $n:=\sum_{i=1}^k n_i $, we let $\footnotesize \left[ \begin{array}{c} n \\ n_1,\ldots,n_k \end{array} \right]_{q}$ \normalsize denote the {\em $q$-multinomial coefficient} \[\left[ \begin{array}{c} n \\ n_1,\ldots,n_k \end{array} \right]_{q}:= \frac{[n]_{q}!}{[n_1]_q !\cdot \ldots\cdot [n_k]_q !},\] where \[[n]_q := \frac{1-q^n}{1-q},\qquad\qquad [n]_q ! := \prod_{i=1}^{n} [i]_q \qquad\qquad\mbox{and}\qquad [0]_q!:= 1. \] The symmetric group $S_n$ is the group of permutations of the set $[n]$. For $\sigma \in S_n$ we use both the one-line notation $\sigma=[\sigma(1),\,\ldots,\,\sigma(n)]$ and the disjoint cycle notation. We let $s_1,\ldots,\,s_{n-1}$ denote the standard generators of $S_n$, $s_i=(i,\,i+1)$. The hyperoctahedral group $B_n$ is the group of signed permutations, or permutations $\sigma$ of the set $[-n,n]$ such that $\sigma(j)=-\sigma(-j)$. For a signed permutation $\sigma$ we use the window notation $\sigma = [\sigma(1), \ldots ,\sigma(n)]$ and the disjoint cycle notation. The standard generating set of $B_n$ is $S=\{s_0,\,s_1,\,\ldots,\,s_{n-1}\}$, where $s_0=[-1,\,2,\,3,\ldots,\,n]$ and $s_1,\ldots,\,s_{n-1}$ are as above. We follow $\cite{BB}$ for notation and terminology about Coxeter groups. In particular, for $(W,S)$ a Coxeter system we let $\ell$ be the Coxeter length and for $I\subseteq S$ we define the quotients: \[ W^{I} := \{w\in W \;:\; D(w)\subseteq S\setminus I\}, \] and \[ ^{I}W := \{w\in W \;:\; D_L(w)\subseteq S\setminus I\}, \] where $D(w)=\{s\in S \;:\; \ell(ws)<\ell(w)\}$, and $D_L(w)=\{s\in S \;:\; \ell(sw)<\ell(w)\}$. The parabolic subgroup $W_I$ is the subgroup generated by $I$. The following result is well known (see, e.g., \cite[Proposition 2.4.4]{BB}). \begin{pro} Let $(W,S)$ be a Coxeter system, $J \subseteq S$, and $w \in W$. Then there exist unique elements $w^J \in W^J$ and $w_J \in W_J$ (resp., $^Jw \in ^JW$ and $_Jw \in W_J$) such that $w= w^J w_J$ (resp., $_Jw ^Jw$). Furthermore $\ell(w)= \ell(w^J)+\ell(w_J)$ (resp., $\ell(_Jw)+\ell(^Jw)$). \end{pro} It is well known that $S_n$ and $B_n$, with respect to the above generating sets, are Coxeter systems and that the following results hold (see, e.g., \cite[Propositions 1.5.2, 1.5.3, and \S 8.1]{BB}). \begin{pro} Let $\sigma \in S_n$. Then $ \ell_A(\sigma)=\| \{ (i,j) \in [n]^2 : i<j , \sigma(i) > \sigma(j) \} \| $ and $ D(\sigma) = \{ s_i : \sigma(i) > \sigma(i+1) \}. $ \end{pro} For $\sigma \in B_n$ let \begin{align}\noindent \ \inv(\sigma):=& \|\{(i,j)\in [n]\times[n] \; : \; i<j,\,\sigma(i)>\sigma(j)\}\|, \\ \negg(\sigma):=& \|\{ i\in [n]\; : \; \sigma(i)<0\}\|, \\ \nsp(\sigma):=& \|\{(i,j)\in [n]\times[n] \; : \; i<j, \, \sigma(i)+\sigma(j)<0\}\|.\end{align} \begin{pro} Let $\sigma \in B_n$. Then \[ \ell_B(\sigma)= \frac{1}{2} \| \{ (i,j) \in [-n,n]^2 : i<j , \sigma(i) > \sigma(j) \} \|=\ \inv(\sigma)+\negg(\sigma)+\nsp(\sigma) \] and $ D(\sigma) = \{ s_i : i \in [0,n-1] , \sigma(i) > \sigma(i+1) \}. $ \end{pro} The group $D_n$ of even-signed permutations is the subgroup of $B_n$ of elements with an even number of negative entries in the window notation: \[D_n=\{\sigma \in B_n\,:\, \negg(\sigma)\equiv 0 \pmod 2\}.\] This is a Coxeter group of type $D_{n}$, with set of generators $S=\{s_0 ^D,s^D _1,\ldots ,s^D _{n-1}\}$, where $s_0 ^D:=[-2,-1,3,\ldots n]$ and $ s^D_{i}:=s_i$ for $i\in [n-1]$. Moreover, the following holds (see, e.g., \cite[Propositions 8.2.1 and 8.2.3]{BB}). \begin{pro} \label{combD} Let $\sigma \in D_n$. Then $ \ell_D(\sigma)=\inv(\sigma)+\nsp(\sigma) $ and $ D(\sigma) = \{ s^D_i : i \in [0,n-1] , \sigma(i) > \sigma(i+1) \}, $ where $\sigma(0):=\sigma(-2)$. \end{pro} Thus, for a subset of the generators $I\subseteq S$, that we identify with the corresponding subset $I \subseteq [0,n-1]$, we have the following description of the quotient $$D_n^I=\{\sigma \in D_n \,:\, \sigma(i)<\sigma(i+1) \mbox{ for all } i \in I\}$$ where $\sigma(0):=-\sigma(2)$. The following statistic was first defined in \cite{KV}. Our definition is not the original one, but is equivalent to it (see \cite[Definition 5.1 and Lemma 5.2]{KV}) and is the one that is best suited for our purposes. \begin{defn} Let $n\in {\mathbb P}$. The statistic $L_{A}:S_n \rightarrow \mathbb N$ is defined as follows. For $\sigma \in S_n$ \[ L_{A}(\sigma):= \|\{(i,j) \in [n]^2 \; : \; i<j,\,\sigma(i)>\sigma(j),\,i\not\equiv j \pmod{2}\}\|. \] \end{defn} The following statistic was introduced in \cite{VS} and \cite{VS2}, and is a natural analogue of the statistic $L_A$ introduced above, for Coxeter groups of type $B$. \begin{defn} \label{defLB} Let $n\in {\mathbb P}$. The statistic $L_{B}:B_n \rightarrow \mathbb N$ is defined as follows. For $\sigma \in B_n$ \[ L_{B}(\sigma):= \frac{1}{2} \|\{(i,j)\in [-n,\,n]^2 \, : \, i<j,\,\sigma(i)>\sigma(j),\,i\not\equiv j \pmod{2}\}\|.\] \end{defn} \noindent For example, if $n=4$ and $\tau=[-2,4,3,-1]$ then $L_B (\tau)= \frac{1}{2}\|\{(-4,-3),\,(-4,1),\,(-3,-2),$ $\,(-1,0),\,(-1,4),\,(0,1),\,(2,3),\,(3,4)\}\|=4$. We call these statistics $L_A$ and $L_B$ the {\em odd length} of the symmetric and hyperoctahedral groups, respectively. Note that if $\sigma \in S_n \subset B_n $ then $ L_B (\sigma)=L_A(\sigma)$. The odd length of an element $\sigma \in B_n$ also has a description in terms of statistics of the window notation of $\sigma$. Given $\sigma \in B_n$ we let \begin{align}\noindent \ \oinv(\sigma):=& \|\{(i,j)\in [n]^2 \; : \; i<j,\,\sigma(i)>\sigma(j),\,i\not\equiv j \pmod{2}\}\|, \\ \oneg(\sigma):=& \|\{ i\in [n]\; : \; \sigma(i)<0,\,i\not\equiv 0 \pmod{2}\}\|, \\ \onsp(\sigma):=& \|\{(i,j)\in [n]^2 \; : \; \sigma(i)+\sigma(j)<0,\,i\not\equiv j \pmod{2}\}\|.\end{align} The following result appears in \cite[Proposition 5.1]{BC}. \begin{pro}\label{LB} Let $\sigma \in B_n$. Then $ L_B(\sigma)= \oinv(\sigma)+ \oneg(\sigma)+ \onsp(\sigma).$ \end{pro} The signed generating function of the odd length factors very nicely both on quotients of $S_n$ and of $B_n$. The following result was conjectured in \cite[Conjecture C]{KV} and proved in \cite{BC}. \begin{thm} \label{Aquot} Let $n \in {\mathbb P}$, $I \subseteq [n-1]$, and $I_{1}, \ldots , I_{s}$ be the connected components of $I$. Then \begin{align} \sum_{\sigma \in S_{n}^{I}} (-1)^{\ell_A (\sigma )} x^{L_A(\sigma )} &=\mc \prod_{k=2m+2}^n \left(1+(-1)^{k-1}x^{\left\lfloor\frac{k}{2}\right\rfloor}\right) \end{align} where $m := \sum_{k=1}^{s} \left\lfloor \frac{\|I_{k}\|+1}{2} \right\rfloor $. \end{thm} In particular, for the whole group we have the following. \begin{cor}\label{wgpA} Let $n \in {\mathbb P},\, n\geq 2$. Then \[ \sum_{\sigma \in S_n} (-1)^{\ell_A(\sigma)}x^{L_A(\sigma)} = \prod_{i=2}^{n} \left(1+(-1)^{i-1}x^{\left\lfloor\frac{i}{2}\right\rfloor}\right). \] \end{cor} For $J\subseteq [0,n-1]$ we define $J_0\subseteq J$ to be the connected component of $J$ which contains $0$, if $0\in J$, or $J_{0} := \emptyset$ otherwise. Let $J_1,\ldots,J_s$ be the remaining ordered connected components. The following result was conjectured in \cite[Conjecture 1.6]{VS2} and proved in \cite{BC} and independently in \cite{L}. \begin{thm} \label{Bquot} Let $n\in \mathbb P$, $J \subseteq [0,n-1]$, and $J_0,\ldots, J_s$ be the connected components of $J$ indexed as just described. Then \[ \sum_{\sigma \in {\qb}} (-1)^{\ell_B(\sigma)} x^{L_B(\sigma)}=\frac{\prod\limits_{j=a+1}^{n}(1-x^j)}{\prod\limits_{i=1}^{m}(1-x^{2i})} \mcb \] where $m:=\sum_{i=1}^s \left\lfloor \frac{\|J_{i}\|+1}{2}\right\rfloor$ and $a:=\min\{ [0, \,n-1] \setminus J\}$. \end{thm} \section{Definition and general properties} In this section we define a new statistic, on the even hyperoctahedral group $D_n$, which is a natural analogue of the odd length statistics that have already been defined and studied in types $A$ and $B$, and study some of its basic properties. Given the descriptions of $L_A$ and $L_B$ in terms of odd inversions, odd negatives and odd negative sum pairs, and the relation between the Coxeter lengths of the Weyl groups of types $B$ and $D$ (see, e.g., \cite[Propositions 8.1.1 and 8.2.1]{BB}), the following definition is natural. \begin{defn} Let $\sigma \in D_n$. We let \[L_D(\sigma):=L_B(\sigma)- \oneg(\sigma)= \oinv(\sigma)+ \onsp(\sigma). \] \end{defn} For example let $n=5$, $\sigma=[2,-1,5,-4,3]$. Then $L_D(\sigma)=5$. We call $L_D$ the {\em odd length} of type $D$. Note that the statistic $L_D$ is well defined also on $S_n$ (where it coincides with $L_A$) and on $B_n$. In fact, the signed distribution of $L_D$ over any quotient of $D_{n}$ and over its ``complement'' in $B_n$, is exactly the same, as we now show. For $I \subseteq [0,n-1]$ let $(B_{n} \setminus D_{n})^{I}:= \{ \sigma \in B_n \setminus D_n : \; \sigma (i) < \sigma (i+1) \mbox{ for all } i \in I \}$ where $\sigma (0) := - \sigma (2)$. Note that $(B_n \setminus D_n)^{I}=B_{n}^{I} \setminus D_n^I$ if $I \subseteq [n-1]$. \begin{lem}\label{compl} Let $n\in \mathbb P$ and $I \subseteq [0,n-1]$. Then \[\sum_{\sigma \in D_n ^{I}}{y^{\ell_D(\sigma)}x^{L_D(\sigma)}}=\sum_{\sigma \in (B_n \setminus D_n)^{I}}{y^{\ell_D(\sigma)}x^{L_D(\sigma)}}. \] In particular, $\sum_{\sigma \in D_n ^{I}}{(-1)^{\ell_D(\sigma)}x^{L_D(\sigma)}}=\sum_{\sigma \in (B_n \setminus D_n)^{I}}{(-1)^{\ell_D(\sigma)}x^{L_D(\sigma)}}$. \end{lem} \begin{proof} Left multiplication by $s_0$ (that is, changing the sign of $1$ in the window notation) is a bijection between $D_n^{I}$ and $(B_n\setminus D_n)^{I}$. Moreover, (odd) inversions and (odd) negative sum pairs are preserved by this operation so $L_D(s_0 \sigma )=L_D(\sigma)$, and $ \ell_D(s_0 \sigma )=\ell_D(\sigma)$, for all $ \sigma \in D_{n}$ and the result follows. \end{proof} In what follows, since we are mainly concerned with distributions in type $D$, we omit the subscript and write just $\ell$ and $L$ for the length and odd length, respectively, on $D_n$. We now show that the generating function of $(-1)^{\ell(\cdot)}x^{L(\cdot)}$ over any quotient of $D_{n}$ such that $s_{0}^{D} \in D_{n}^{I}$ can be reduced to elements for which the maximum (or the minimum) is in certain positions. More precisely, we prove that, for a given quotient, our generating function is zero over all elements for which the maximum (or minimum) is sufficiently far from $I$. For a subset $I\subseteq [0,n-1]$ we let $\delta_0 (I)=1$ if $0 \in I$ and $\delta_0 (I)=0$ otherwise. \begin{lem}\label{zerod} Let $n\in \mathbb P$, $n\geq 3$, $I\subseteq [0,n-1]$. Let $a \in [2+\delta_0 (I),n-1]$ be such that $[a-2,a+1]\cap I=\emptyset$. Then \[\sum_{\substack{\{\sigma \in D_n^I :\\\sigma(a)=n\}}}(-1)^{\ell(\sigma)}x^{L(\sigma)} =\sum_{\substack{\{\sigma \in D_n^I :\\\sigma(a)=-n\}}}(-1)^{\ell(\sigma)}x^{L(\sigma)}=0.\] \end{lem} \begin{proof} In our hypotheses, if $\sigma \in D_n ^I$ then also $\sigma^{a}:= \sigma (-a-1,-a+1) (a-1,a+1)$ is in the same quotient. Clearly $(\sigma^a)^a=\sigma$ and $\|\ell(\sigma)-\ell(\sigma^a)\|=1$, while, since $\sigma(a)=n$, $L(\sigma^a)=L(\sigma)$. Therefore we have that \begin{align}\sum_{\substack{\{\sigma \in {D}^{I}_{n}: \\ \sigma(a)=n \} }} (-1)^{\ell (\sigma )}x^{L(\sigma )}&=& \sum_{\substack{\{\sigma \in {D}^{I}_n: \sigma(a)=n, \\ \sigma (a-1) < \sigma (a+1) \} }}\left( (-1)^{\ell (\sigma )}x^{L(\sigma )} + (-1)^{\ell (\sigma^a )}x^{L(\sigma^a )}\right) =0.\end{align} The proof of the second equality is exactly analogous and is therefore omitted. \end{proof} Although we do not know of any definition of our (or of any other) odd length statistics in Coxeter theoretic language, it is natural to expect that the only automorphism of the Dynkin diagram of $D_n$ preserves the corresponding signed generating function. This is indeed the case, as we now show. \begin{pro}\label{zerouno} Let $n \in \mathbb P$, $n\geq 2$, and $I \subseteq [2,n-1]$. Then \[ \sum_{\sigma \in D_n ^{I \cup \{0\}}}{y^{\ell(\sigma)}x^{L(\sigma)}}= \sum_{\sigma \in D_n ^{I \cup \{1\}}}{y^{\ell(\sigma)}x^{L(\sigma)}}. \] In particular, $\sum_{\sigma \in D_n ^{I \cup \{0\}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in D_n ^{I \cup \{1\}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}$. \end{pro} \begin{proof} Right multiplication by $s_{0}$ (i.e., changing the sign of the leftmost element in the window notation) is a bijection between $D_{n} ^{I \cup \{ 0 \}}$ and $(B_{n} \setminus D_{n} )^{I \cup \{ 1 \}}$. Furthermore, if $\sigma \in D_{n}$, then \begin{eqnarray} \oinv (\sigma s_{0}) & = & \oinv (\sigma ) - \|\{ i \in [2,n]: i \equiv 0 \pmod{2}, \; \sigma (1)>\sigma (i) \} \| \\ & & + \|\{ i \in [2,n]: i \equiv 0 \pmod{2}, \; -\sigma (1) > \sigma (i) \} \| ,\\ \onsp (\sigma s_{0}) & = & \onsp (\sigma ) - \|\{ i \in [2,n]: i \equiv 0 \pmod{2}, \; \sigma (1) + \sigma (i) <0 \} \| \\ & & + \| \{ i \in [2,n]: i \equiv 0 \pmod{2}, \; -\sigma (1)+ \sigma (i) <0 \} \| ,\\ \inv (\sigma s_{0} )& =& \inv (\sigma ) -\|\{ i \in [2,n]: \; \sigma (1) > \sigma (i) \} \| + \|\{ i \in [2,n]: \; - \sigma (1)> \sigma (i) \} \|,\\ \mbox{and} \quad \qquad& & \\ \nsp (\sigma s_{0} ) &=& \nsp (\sigma ) -\|\{ i \in [2,n]: \; \sigma (1) + \sigma (i)<0 \} \| + \|\{ i \in [2,n]: \; - \sigma (1)+ \sigma (i) <0 \} \|. \end{eqnarray} Therefore $L(\sigma s_{0})=L(\sigma )$ and $\ell (\sigma s_{0})=\ell (\sigma )$. Hence \[ \sum_{\sigma \in (B_{n} \setminus D_{n}) ^{I \cup \{ 1\}}}y^{\ell (\sigma )} \; x^{L(\sigma )} = \sum _{\sigma \in D_{n}^{I \cup \{ 0 \}}}y^{\ell (\sigma s_{0})} \; x^{L(\sigma s_{0})} = \sum _{\sigma \in D_{n}^{I \cup \{ 0 \}}}y^{\ell (\sigma )} \; x^{L(\sigma )} , \] and the result follows from Lemma \ref{compl}. \end{proof} We conclude by showing that when $I$ does not contain $0$, each connected component can be shifted to the left or to the right, as long as it remains a connected component, without changing the generating function over the corresponding quotient. The proof is identical to that of \cite[Proposition 3.3]{BC}, and is therefore omitted. \begin{pro}\label{shd} Let $I\subseteq [n-1]$, $i\in \mathbb P$, $k\in \mathbb N$ be such that $[i,\,i+2k]$ is a connected component of $I$ and $i+2k+2 \notin I$. Then \[ \sum_{\sigma \in D_n ^I}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in D_n ^{I\cup \tilde{I}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in D_n ^{\tilde{I}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} \] where $\tilde{I}:=(I\setminus\{i\})\cup\{i+2k+1\}$. \end{pro} Shifting is also allowed when $I$ contains $0$, but only for connected components which are sufficiently far from it, as stated in the next result. \begin{pro} Let $I\subseteq [0,n-1]$, $i\in \mathbb P$, $i>2$ and $k\in \mathbb N$ such that $[i,\,i+2k]$ is a connected component of $I$ and $i+2k+2 \notin I$. Then \[ \sum_{\sigma \in D_n ^I}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in D_n ^{I\cup \tilde{I}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in D_n ^{\tilde{I}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} \] where $\tilde{I}:=(I\setminus\{i\})\cup\{i+2k+1\}$. \end{pro} \begin{proof} The proof is analogous to that of \cite[Proposition 3.3]{BC} noting that , since $i>2$, $\sigma \in D_n^{I}$ if and only if $\sigma(i+2j,i+2k+2)(-i-2j,-i-2k-2)\in D_n^I$. \end{proof} \section{Trivial quotient} In this section, using the results in the previous one, we compute the generating function of $(-1)^{\ell(\cdot)}x^{L(\cdot)}$ over the whole even hyperoctahedral group $D_n$. In particular, we obtain that this generating function is the square of the corresponding one in type $A$ (i.e., for the symmetric group). \begin{thm}\label{sq} Let $n\in \mathbb{P}$, $n\geq 2$. Then \[ \sum_{\sigma \in D_n}{(-1)^{\ell (\sigma)}x^{L(\sigma)}}=\prod_{j=2}^{n}(1+(-1)^{j-1}x^{\left\lfloor\frac{j}{2}\right\rfloor})^2 . \] \end{thm} \begin{proof} We proceed by induction on $n$. By Lemma \ref{zerod}, the sum over all elements for which $n$ or $-n$ appears in positions different from $1$ and $n$ is zero. So the generating function over $D_n$ reduces to \begin{align} \sum_{\sigma \in D_n} {(-1)^{\ell(\sigma)}x^{L(\sigma)}}&= \sum_{\substack{ \{\sigma \in D_n: \\ \sigma(1)=n\}}} {(-1)^{\ell(\sigma)}x^{L(\sigma)}} +\sum_{\substack{ \{\sigma \in D_n: \\ \sigma(n)=n\} } }{(-1)^{\ell(\sigma)}x^{L(\sigma)}} \\ &+ \sum_{\substack{ \{\sigma \in D_n: \\ \sigma(1)=-n\} }}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\substack{\{\sigma \in D_n: \\ \sigma(n)=-n\} }}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} \\ &= \sum_{\sigma \in D_{n-1}}{(-1)^{\ell(\tilde{\sigma})}x^{L(\tilde{\sigma})}} + \sum_{\sigma \in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} \\ &+ \!\!\!\!\sum_{\sigma \in B_{n-1}\setminus D_{n-1}}\!\!{(-1)^{\ell(\hat{\sigma})}x^{L(\hat{\sigma})}}+\!\!\!\!\sum_{\sigma \in B_{n-1}\setminus D_{n-1}}\!\!{(-1)^{\ell(\check{\sigma})}x^{L(\check{\sigma})}} , \end{align} \normalsize where $\tilde\sigma:=[n,\sigma(1),\ldots,\sigma(n-1)]$, $\hat\sigma:=[-n,\sigma(1),\ldots,\sigma(n-1)],$ and $\check\sigma:=[\sigma(1),\ldots,\sigma(n-1),-n]$. But, by our definition and Proposition 8.2.1 of \cite{BB}, we have that \begin{align} L(\tilde\sigma)& =L(\sigma)+m, \qquad \ell(\tilde\sigma)=\ell(\sigma)+n-1\\ \label{ha} L(\hat\sigma)& =L(\sigma)+m, \qquad \ell(\hat\sigma)=\ell(\sigma)+n-1\\ \label{che} L(\check\sigma)& =L(\sigma)+2m,\:\: \quad \ell(\check\sigma)=\ell(\sigma)+2(n-1), \end{align} where $m := \left\lfloor \frac{n}{2} \right\rfloor $. Therefore \[\sum_{\sigma \in D_{n-1}}{(-1)^{\ell(\tilde{\sigma})}x^{L(\tilde{\sigma})}}=(-1)^{n-1}x^m\sum_{\sigma \in D_{n-1}}{(-1)^{\ell({\sigma})}x^{L({\sigma})}}\] and, similarly, \begin{eqnarray} \sum_{\sigma \in B_{n-1}\setminus D_{n-1}}{(-1)^{\ell(\hat{\sigma})}x^{L(\hat{\sigma})}}&=&(-1)^{n-1}x^m\sum_{\sigma \in B_{n-1}\setminus D_{n-1}}{(-1)^{\ell({\sigma})}x^{L({\sigma})}}\\ \sum_{\sigma \in B_{n-1}\setminus D_{n-1}}{(-1)^{\ell(\check{\sigma})}x^{L(\check{\sigma})}}&=&x^{2m}\sum_{\sigma \in B_{n-1}\setminus D_{n-1}}{(-1)^{\ell({\sigma})}x^{L({\sigma})}}. \end{eqnarray} So by Lemma \ref{compl} and our induction hypothesis we obtain that \begin{eqnarray} \sum_{\sigma \in D_n} {(-1)^{\ell(\sigma)}x^{L(\sigma)}}&=&(1+(-1)^{n-1}x^m) \sum_{\sigma \in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} +\\ &+&((-1)^{n-1}x^m +x^{2m})\!\!\!\!\!\sum_{\sigma \in B_{n-1}\setminus D_{n-1}}{\!\!\!\!\!\!(-1)^{\ell(\sigma)}x^{L(\sigma)}}\\ &=&(1+2(-1)^{n-1}x^{m} +x^{2m})\sum_{\sigma\in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}\\ &=&\left(1+(-1)^{n-1}x^{\left\lfloor \frac{n}{2} \right\rfloor }\right)^2\sum_{\sigma \in D_{n-1}} (-1)^{\ell(\sigma)}x^{L(\sigma)},\\ \end{eqnarray} and the result follows by induction. \end{proof} As an immediate consequence of Theorem \ref{sq} and of Corollary \ref{wgpA} we obtain the following result. \begin{cor} \label{DA} Let $n \in \mathbb{P}$, $n \geq 2$. Then \[ \sum _{\sigma \in D_{n}} (-1)^{\ell (\sigma )} \, x^{L}(\sigma ) = \left( \sum _{\sigma \in S_{n}} (-1)^{\ell _{A}(\sigma )} \, x^{L_{A}(\sigma )} \right)^{2}. \Box \] \end{cor} It would be interesting to have a direct proof of Corollary \ref{DA}. \section{Maximal and other quotients} In this section we compute, using the results in \S 3, the signed generating function of the odd length over the maximal, and some other, quotients of $D_{n}$. In particular, we obtain that these generating functions always factor nicely. \begin{thm}\label{maxquod} Let $n\in \mathbb P$, $n\geq 3$ and $i\in [0,n-1]$. Then \[ \sum_{\sigma \in D_n^{\{i\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=(1-x^2)\prod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2. \] \end{thm} \begin{proof} By Propositions \ref{zerouno} (with $I = \emptyset$) and \ref{shd}, we may assume $i=1$. We proceed by induction on $n \geq 3$. By Lemma \ref{zerod} we have that the sum over $\sigma\in D^{\{1\}}_n$ such that $n$ or $-n$ appear in the window in any position but $1,2,3$, or $n$ is zero. Furthermore, if $\sigma \in D_n^{\{1\}}$ then $\sigma^{-1}(n)\neq 1$ and $\sigma^{-1}(-n)\neq 2$. Moreover, the map $\sigma \mapsto \sigma (1,3) (-1,-3)$ is a bijection of $\{ \sigma \in D_{n}^{\{ 1\} }: \; \sigma (2) =n \}$ in itself. But $L (\sigma (1,3)(-1,-3))=L(\sigma )$ and $\ell (\sigma ) \not \equiv \ell (\sigma (1,3)(-1,-3)) \pmod{2} $ for all $\sigma \in D_{n}^{\{ 1\} }$ such that $\sigma (2)=n$ so the sum is zero also over this kind of elements. Thus we have that: \begin{align} &\sum_{\sigma \in D_n^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\substack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\substack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(n)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} +\\ &+ \sum_{\substack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(1)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\substack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\substack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(n)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} \end{align} \begin{align}&=\sum_{\substack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} + \sum_{\sigma \in B_{n-1}\setminus D_{n-1}}{ (-1)^{\ell(\hat\sigma)}x^{L(\hat\sigma)}}+\\ &+\sum_{\substack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+\sum_{\sigma \in (B_{n-1}\setminus D_{n-1})^{\{ 1\}}}{ (-1)^{\ell(\check\sigma)}x^{L(\check\sigma)}} \end{align} where $\hat\sigma:=[-n,\sigma(1),\ldots,\sigma(n-1)]$ and $\check\sigma:=[\sigma(1),\ldots,\sigma(n-1),-n]$. Now \[\sum_{\substack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\substack{ \{\sigma \in D_{n-1}: \\ \sigma(1)>\sigma(2)\}}}{ (-1)^{\ell(\bar\sigma)}x^{L(\bar\sigma)}}\] where $\bar\sigma:=[\sigma(2),\sigma(1),n,\sigma(3),\ldots,\sigma(n-1)]$. But $\ell(\bar\sigma)= \inv(\sigma)+n-4+\nsp(\sigma)=\ell(\sigma)+n-4$, and $L(\bar\sigma)= \oinv(\sigma)-1+\left\lceil\frac{n-3}{2}\right\rceil+ \onsp(\sigma)=L(\sigma)+\left\lceil\frac{n-5}{2}\right\rceil=L(\sigma)+m-2,$ where $m := \left\lceil \frac{n-1}{2} \right\rceil$, so \begin{align} &\sum_{\substack{\{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=(-1)^n x^{m-1}\sum_{\substack{ \{\sigma \in D_{n-1}: \\ \sigma(1)>\sigma(2)\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\\ &=(-1)^n x^{m-2}\left(\sum_{\sigma \in D_{n-1}}(-1)^{\ell(\sigma)}x^{L(\sigma)} -\sum_{\sigma \in D_{n-1}^{\{1\}}} (-1)^{\ell(\sigma)}x^{L(\sigma)}\right) . \end{align} Similarly, \[\sum_{\substack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\substack{ \{\sigma \in B_{n-1}\setminus D_{n-1}: \\ \sigma(1)>\sigma(2)\}}}{ (-1)^{\ell(\bar{\bar{\sigma}})}x^{L(\bar{\bar{\sigma}})}}\] where $\bar{\bar{\sigma}}:=[\sigma(2),\sigma(1),-n,\sigma(3),\ldots,\sigma(n-1)]$ and $\ell(\bar{\bar{\sigma}})= \inv(\sigma)+1+\nsp(\sigma)+n-1=\ell(\sigma)+n$, $L(\bar{\bar{\sigma}})= \oinv(\sigma)+ \onsp(\sigma)+1+\left\lceil\frac{n-3}{2}\right\rceil=L(\sigma)+\left\lceil\frac{n-1}{2}\right\rceil=L(\sigma)+m$. So, by Lemma \ref{compl}, \begin{align} &\sum_{\substack{ \{\sigma \in D_n^{\{1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=(-1)^n x^{m} \sum_{\substack{ \{\sigma \in B_{n-1}\setminus D_{n-1}: \\ \sigma(1)>\sigma(2)\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\\ &=(-1)^{n}x^{m}\left(\sum_{\sigma\in B_{n-1} \setminus D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}-\!\!\!\!\!\sum_{ (B_{n-1} \setminus D_{n-1})^{\{ 1\}}}{\!\!\! (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right)\\ &=(-1)^{n}x^{m}\left(\sum_{\sigma \in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}-\sum_{ \sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right). \end{align} Moreover, by \eqref{ha} and \eqref{che} we have \[\sum_{\sigma \in B_{n-1}\setminus D_{n-1}} {(-1)^{\ell(\hat\sigma)}x^{L(\hat\sigma)}}=(-1)^{n-1}x^m \sum_{\sigma \in B_{n-1}\setminus D_{n-1}} {(-1)^{\ell(\sigma)}x^{L(\sigma)}}\] and \[ \sum_{\sigma \in (B_{n-1} \setminus D_{n-1})^{\{ 1\}}} {(-1)^{\ell(\check\sigma)}x^{L(\check\sigma)}}=x^{2m} \sum_{\sigma \in (B_{n-1}\setminus D_{n-1})^{\{ 1 \}}} {(-1)^{\ell(\sigma)}x^{L(\sigma)}}. \] Thus we get, again by Lemma \ref{compl}, \small \begin{eqnarray} \sum_{\sigma \in D_n^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}&=& (-1)^n x^{m-2}\left(\sum_{\sigma \in D_{n-1}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}-\sum_{\sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right)\\ &+&\sum_{\sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+(-1)^{n-1}x^{m}\sum_{\sigma \in D_{n-1}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\\ &+&(-1)^{n}x^{m}\left(\sum_{\sigma \in D_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}-\sum_{\sigma \in D_{n-1}^{\{1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right) \\ &+&x^{2m}\sum_{\sigma \in D_{n-1}^{\{ 1 \}}}{ \!\!\!(-1)^{\ell(\sigma)}x^{L(\sigma)}}\\ &=&(-1)^n x^{m-2}\sum_{\sigma \in D_{n-1}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\\ &+&\left(1+(-1)^{n-1}x^{m-2}+(-1)^{n-1}x^{m}+x^{2m}\right)\sum_{\sigma\in D^{\{1\}}_{n-1}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} \end{eqnarray} \normalsize and the result follows by Theorem \ref{sq} and our induction hypothesis. \end{proof} We note the following consequence of Theorems \ref{sq} and \ref{maxquod}. \begin{cor} Let $n \in \mathbb{P}$, $n \geq 3$, and $i \in [0,n-1]$. Then \[ \sum _{\sigma \in D_{n}}(-1)^{\ell (\sigma )} \, x^{L(\sigma )}=(1-x^{2}) \, \sum _{\sigma \in D_{n}^{\{ i\}}}(-1)^{\ell (\sigma )} \, x^{L(\sigma )} . \] \end{cor} \begin{proof} This follows immediately from Theorems \ref{sq} and \ref{maxquod}. \end{proof} The results obtained up to now compute $\sum_{\sigma \in D_{n}^{I}} (-1)^{\ell (\sigma )} x^{L(\sigma )}$ when $\|I\| \leq 1$. A natural next step is to try to compute these generating functions if $\|I \setminus \{ 0 \}\| \leq 1$. We are able to do this for $I=\{ 0,1 \}$, and $I=\{ 0,2 \}$. The computation for $I=\{ 0,2 \}$ follows easily from results that we have already obtained. \begin{cor} \label{pro02} Let $n \in {\mathbb P}$, $n \geq 4$. Then \[\sum_{\sigma \in D_{n}^{\{ 0, \, 2 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} = (1-x^2) \prod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2. \] \end{cor} \begin{proof} By Proposition \ref{zerouno} and Proposition \ref{shd} we have \[\sum_{\sigma \in D_{n}^{\{ 0,2 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} =\sum_{\sigma \in D_{n}^{\{ 1,2 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} = \sum_{\sigma \in D_{n}^{\{ 1 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}}, \] and the result follows by Theorem \ref{maxquod}. \end{proof} We conclude this section by computing $\sum_{\sigma \in D_{n}^{I}} (-1)^{\ell (\sigma )} x^{L(\sigma )}$ when $I =\{ 0,1 \}$. \begin{thm} \label{pro01} Let $n \in {\mathbb P}$, $n \geq 3$. Then \[\sum_{\sigma \in D_{n}^{\{ 0, \, 1 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} = (1+x^2) \prod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2. \] \end{thm} \begin{proof} We proceed by induction on $n \geq 3$. By Lemma \ref{zerod} we have that the sum over $\sigma\in D^{\{0,\,1\}}_n$ such that $n$ or $-n$ appear in the window in any position but $1,2,3$, or $n$ is zero; moreover for $\sigma \in D_n^{\{0,\,1\}}$ we always have $\sigma^{-1}(\pm n)\neq 1$ and $\sigma^{-1}(-n)\neq 2$. Also, the map $\sigma \mapsto \sigma (1,3) (-1,-3)$ is a bijection of $\{ \sigma \in D_{n}^{\{ 0,\,1\} }: \, \sigma (2) =n \}$ in itself, so by the same argument as in the proof of Theorem \ref{maxquod} the sum over this kind of elements is zero. Thus we have that: \begin{align} \sum_{\sigma \in D_n^{\{0,\,1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}&= \sum_{\substack{\{\sigma \in D_n^{\{0,\,1\}}: \\ \| \sigma(3) \|=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}+ \sum_{\substack{ \{\sigma \in D_n^{\{0,\,1\}}: \\ \| \sigma(n) \|=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}. \end{align} By \eqref{che} and Lemma \ref{compl} we have that \begin{align} \sum_{\substack{ \{\sigma \in D_n^{\{0,\,1\}}: \\ \| \sigma(n) \|= n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} & = \left(1+x^{2\left\lfloor\frac{n}{2}\right\rfloor}\right)\sum_{\sigma \in D_{n-1}^{\{0,\,1\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}} \\ & = \left(1+x^{2\left\lfloor\frac{n}{2}\right\rfloor}\right) (1+x^2) \prod_{j=4}^{n-1} (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2 \end{align} by our induction hypothesis. Moreover, \[ \sum_{\substack{\{\sigma \in D_n^{\{0,\,1\}}: \\ \sigma(3)=n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\sigma \in D_{n-1}^{\{1\}}\setminus D_{n-1}^{\{0,\,1\}}}{ (-1)^{\ell(\bar\sigma)}x^{L(\bar\sigma)}} \] where $\bar\sigma:=[-\sigma(2),-\sigma(1),n,\sigma(3),\ldots,\sigma(n-1)]$. But \begin{eqnarray} \inv (\bar\sigma )& =& \inv ([n,\sigma (3), \ldots, \sigma (n-1)]) + \|\{ j \in [3,n-1]: \; - \sigma (1) > \sigma (j) \} \| \\ & & + \|\{ j \in [3,n-1]: \; - \sigma (2)> \sigma (j) \} \|,\\ \nsp (\bar\sigma ) &=& \nsp ([n,\sigma (3), \ldots, \sigma (n-1)]) + \|\{ j \in [3,n-1]: \; - \sigma (1) + \sigma (j) <0 \} \| \\ & & + \|\{ j \in [3,n-1]: \; - \sigma (2)+ \sigma (j) <0 \} \|, \\ \oinv (\bar\sigma ) & = & \oinv ([n,\sigma (3), \ldots, \sigma (n-1)] ) + \|\{ j \in [3,n-1]: j \equiv 0 \pmod{2}, \; \sigma (1)+\sigma (j) <0 \} \| \\ & & + \|\{ j \in [3,n-1]: j \equiv 1 \pmod{2}, \; \sigma (2) + \sigma (j) <0 \} \| ,\\ \mbox{and} \quad \qquad& & \\ \onsp (\bar\sigma ) & = & \onsp ([n,\sigma (3), \ldots, \sigma (n-1)] ) + \|\{ j \in [3,n-1]: j \equiv 0 \pmod{2}, \; \sigma (1) > \sigma (j) \} \| \\ & & + \| \{ j \in [3,n-1]: j \equiv 1 \pmod{2}, \; \sigma (2) > \sigma (j) \} \|. \end{eqnarray} Therefore, $\ell(\bar\sigma)=\ell(\sigma)+n-4$, and $L(\bar\sigma)= \oinv(\sigma)-1+\left\lceil\frac{n-3}{2}\right\rceil+ \onsp(\sigma)=L(\sigma)+m-2,$ where $m := \left\lfloor \frac{n}{2} \right\rfloor$. Similarly, \[ \sum_{\substack{ \{\sigma \in D_n^{\{0,\,1\}}: \\ \sigma(3)=-n\}}}{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}=\sum_{\substack{ \{\sigma \in B_{n-1}\setminus D_{n-1}: \\ \sigma(1)<\sigma(2)<-\sigma(1)\}}}{ (-1)^{\ell(\bar{\bar{\sigma}})}x^{L(\bar{\bar{\sigma}})}} \] where $\bar{\bar{\sigma}}:=[-\sigma(2),-\sigma(1),-n,\sigma(3),\ldots,\sigma(n-1)]$ and $\ell(\bar{\bar{\sigma}})=\ell(\sigma)+n$, $L(\bar{\bar{\sigma}})= \oinv(\sigma)+ \onsp(\sigma)+1+\left\lceil\frac{n-3}{2}\right\rceil=L(\sigma)+m$. But $\{ \sigma \in B_{n-1} \setminus D_{n-1} \, : \,\sigma (1) < \sigma (2) < - \sigma (1) \} = (B_{n-1} \setminus D_{n-1})^{ \{ 1 \} } \setminus (B_{n-1} \setminus D_{n-1})^{ \{ 0,1 \} }$, so by Lemma \ref{compl}, Theorem \ref{maxquod} and our induction hypothesis \begin{align} \hspace{-2.5em}\sum_{\substack{ \{\sigma \in D_n^{\{0,\,1\}}: \\ \| \sigma(3) \|= n\}}}\!\!\!{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}& = (-1)^n x^ {m-2} (1+x^2) \left(\sum_{\sigma \in D_{n-1}^{\{1\}}}{\! (-1)^{\ell(\sigma)}x^{L(\sigma)}}-\!\!\sum_{\sigma \in D_{n-1}^{\{0,\,1\}}}\!\!{ (-1)^{\ell(\sigma)}x^{L(\sigma)}}\right)\\ &=2(-1)^{n-1}x^m (1+x^2) \prod_{j=4}^{n-1} (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2 \end{align} Thus \[ \sum_{\sigma \in D_{n}^{\{ 0, \, 1 \}}}{(-1)^{\ell(\sigma)}x^{L(\sigma)}} = (1+(-1)^{n-1}x^m)^2 (1+x^2) \prod_{j=4}^{n-1} (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2 \] and the result follows. \end{proof} \section{Open problems} In this section we present some conjectures naturally arising from the present work and the evidence that we have in their favor. In this paper we have given closed product formulas for $\sum_{\sigma \in D_{n}^{I}} (-1)^{\ell (\sigma )} x^{L(\sigma )}$ when $\|I\| \leq 1$, $I=\{0,1\}$ and $I=\{0,2\}$. We feel that such formulas always exist. In particular, if $\|I \setminus \{ 0,1 \}\| \leq 1$, we feel that the following holds. For $n \in {\mathbb P}$ and $I \subseteq [0,n-1]$ let, for brevity, $D_{n}^{I}(x) := \sum_{\sigma \in D_{n}^{I}} (-1)^{\ell (\sigma )} x^{L (\sigma )}$. \begin{con} \label{con0i} Let $n \in {\mathbb P}$, $n \geq 5$, and $i \in [3,n-1]$. Then \[ D_{n}^{\{ 0, \, i \}}(x) = \prod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2. \] \end{con} \begin{con} \label{con01i} Let $n \in {\mathbb P}$, $n \geq 5$, and $i \in [3,n-1]$. Then \[ D_{n}^{\{ 0,1, \, i \}} (x) = \frac{1+x^2}{1-x^2} \; \prod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2. \] \end{con} We have verified these conjectures for $n \leq 8$. Note that, by Proposition \ref{shd}, it is enough to prove Conjectures \ref{con0i} and \ref{con01i} for $i=3$. Note that, by Theorem \ref{Aquot}, Conjectures \ref{con0i} and \ref{con01i} may be formulated in the following equivalent way. For $n \in {\mathbb P}$ and $J \subseteq [n-1]$ let $S_{n}^{J}(x) := \sum_{\sigma \in S_{n}^{J}} (-1)^{\ell_A (\sigma )} x^{L_{A}(\sigma )}$. \begin{con} \label{con0ib} Let $n \in {\mathbb P}$, $n \geq 5$, and $i \in [3,n-1]$. Then \[ D_{n}^{\{ 0, \, i \}} (x) = (S_n^{ \{ i \} } (x))^2. \] \end{con} \begin{con} \label{con01ib} Let $n \in {\mathbb P}$, $n \geq 5$, and $i \in [3,n-1]$. Then \[ D_{n}^{\{ 0,1,i \}} (x) = (1-x^4) \, (S_n^{ \{ 1, \, i \} } (x))^2. \] \end{con} We feel that the presence of the factor $\prod_{j=4}^n (1+(-1)^{j-1} x^{\left\lfloor\frac{j}{2} \right\rfloor})^2$ in Conjectures \ref{con0i} and \ref{con01i} is not a coincidence. More generally, we feel that the following holds. \begin{con} Let $n \in {\mathbb P}$, $n \geq 3$, and $J \subseteq [0,n-1]$. Let $J_0, J_1, \ldots , J_s$ be the connected components of $J$ indexed as described before Theorem \ref{Bquot}. Then there exists a polynomial $M_J (x) \in {\mathbb Z}[x]$ such that \[ D_{n}^{J} (x) = M_J (x) \; \prod_{j=2m+2}^{n} (1+(-1)^{j-1} x^{\left\lfloor \frac{j}{2} \right\rfloor })^2 , \] where $m:=\sum_{i=0}^s \left\lfloor \frac{\|J_{i}\|+1}{2}\right\rfloor$. Furthermore, $M_J(x)$ only depends on $( \|J_0\|, \|J_1\|, \ldots , \|J_s\| )$ and is a symmetric function of $ \|J_1\|, \ldots , \|J_s\| $. \end{con} This conjecture has been verified for $n\leq 8$. \end{document}	arXiv
Thickening Mars's Atmosphere if we were to send a mission to mars (unlimited budget WE WISH!) to explore different methods that would enable us to thicken the martian atmosphere, what different types of data should this mission capture and how could this help is in exploring potential methods to thicken the martian atmosphere. atmosphere mars Ahmed AbdullaAhmed Abdulla $\begingroup$ Despite this site being called Earth Science, some question about planetary science & other planets can be on top here. You may want to see what information is available on SE Space Exploration looking at questions that ask about Mars' atmosphere & terraforming, such Would Terraforming Mars Be Possible. $\endgroup$ – Fred Oct 22 '19 at 4:07 $\begingroup$ On the impossibility of releasing enough CO2 for a noticeable pressure change of the Marsian atmopshere, even if all reservoirs (including crustal) are being released (which we can't): nature.com/articles/s41550-018-0529-6. $\endgroup$ – ebv Jan 7 at 10:50 We are already knowledgeable about some aspects of the requirements needed to terraform Mars & the Moon. As @David_Garcia_Bodego mentions in his comments, gravity is a major factor in celestial bodies being able to retain an atmosphere. This is because the gravity of a celestial body affects the escape velocity of that body. The other important factor is the surface temperature of the celestial body. This is why Titan, a satellite of Saturn, has a thick atmosphere, despite having a similar escape velocity to the Moon (Luna), it is much colder and has a thick atmosphere. Another factor that assists Titan in maintaining a thick atmosphere is its distance from the Sun and the weaker effect of the solar wind at that distance. Also, Titan is protected by Saturn's magnetosphere. I recommend reading: What will happen to the air released from the Moon's Surface? Could the Moon keep an atmosphere? and the links contained within those questions. Yes, these questions discuss the Moon and not Mars, but the information provided in those questions is applicable to Mars. The graph below provides a visual of the gases a celestial body can retain in a atmosphere based on escape velocity and temperature. Factors that act in favor of Mars being able to have a thick atmosphere are: Mars is farther away from the Sun than Earth & Luna so it experiences a weaker solar wind. The surface temperature of Mars is cold, which reduces the energy of atmospheric molecules. Factors that act against Mars being able to have a thick atmosphere are: The gravity of Mars and subsequently the escape velocity of Mars is small. The lack of a magnetosphere to shield the atmosphere from solar wind. FredFred This is an Earth Science site, not a Mars Science site. However, there will be a manned expedition to Mars within the next 25 years provided a world war doesn't throw a spanner in the works. They won't be exploring methods of thickening the Martian atmosphere, partly because it is impossible to thicken it to any useful degree, by which I mean to a degree which would enable astronauts to wander about in the open without wearing pressure suits. Even if you could find a way to melt all the $\small\mathsf{CO_2}$ in the polar caps (mainly the south pole), there isn't enough to create an atmosphere as substantial as you would find at the top of Mount Everest. There might be enough to raise average temperatures by a few degrees, but what use would that be? The first astronauts will be doing the usual things. mainly geological exploration, particularly with a view to establishing whether there was ever any primitive, unicellular life on the planet. Another task will be to find sites which might be suitable for establishing a permanent base, one of the main requirements being a good water supply. They might also experiment with ways of using local materials for building purposes, in preparation for future expeditions building a permanent base. The fact that liquid water can't exist in the open might be a problem when it comes to making cement. The Romans made an excellent cement out of volcanic tuff, so in the right place there could be plenty of that. Measuring the effects of the Martian environment on their bodies will take up much of the astronaut's time. They will not be exploring methods of geo-engineering. Michael WalsbyMichael Walsby $\begingroup$ Main problem is not the gasses themselves, it is gravity! The gravity on Mars is 3.711m/sg2, so a third that the one we have on the Earth. That one is not enough to keep one breathable atmosphere. Even if we try to Earth-form it, the gasses on the artificial atmosphere will be lost due to the low gravity. $\endgroup$ – David García Bodego Oct 22 '19 at 6:40 $\begingroup$ Martian atmosphere is mostly CO2 ,so whatever you do to it, it won't be breathable however high the pressure. In small quantities the astronauts can refine it to make it breathable, but only in small quantities. $\endgroup$ – Michael Walsby Oct 22 '19 at 8:35 $\begingroup$ Can you edit your questions to add some supporting sources for your claims? $\endgroup$ – samcarter Oct 22 '19 at 9:11 $\begingroup$ That´s why I am saying "Even if we try to Earth-form it...". Gravity of Mars, Gravity of Earth, Mars Terraforming $\endgroup$ – David García Bodego Oct 22 '19 at 9:22 $\begingroup$ Terraforming Mars is for science fiction. This site is supposed to deal with the real world. I say again: the expedition to Mars will not be carrying out any activity connected to terraforming. $\endgroup$ – Michael Walsby Oct 22 '19 at 12:49 Not the answer you're looking for? Browse other questions tagged atmosphere mars or ask your own question. (How long) would Earth's atmosphere last without a global magnetic field? How is the equilibrium of 21% oxygen in Earth's atmosphere established? What causes clouds to appear blue? How long would it take for earth's core to cool down and solidify? Can the atmosphere affect the composition of igneous rocks? What are the most similar Earth analogues to Mars' seasonally recurring slope linea? Is the impact of carbon emision dependent on the location on the earth? What is serpentinization, in the context of disappearance of surface water on Mars? Methane Reservoir	CommonCrawl
Periodic continued fraction In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form $x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\quad \ddots \quad a_{k}+{\cfrac {1}{a_{k+1}+{\cfrac {\ddots }{\quad \ddots \quad a_{k+m-1}+{\cfrac {1}{a_{k+m}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{a_{k+2}+{\ddots }}}}}}}}}}}}}}}}}$ where the initial block of k + 1 partial denominators is followed by a block [ak+1, ak+2,...ak+m] of partial denominators that repeats ad infinitum. For example, ${\sqrt {2}}$ can be expanded to a periodic continued fraction, namely as [1,2,2,2,...]. The partial denominators {ai} can in general be any real or complex numbers. That general case is treated in the article convergence problem. The remainder of this article is devoted to the subject of simple continued fractions that are also periodic. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers. Purely periodic and periodic fractions Since all the partial numerators in a regular continued fraction are equal to unity we can adopt a shorthand notation in which the continued fraction shown above is written as ${\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},a_{k+1},a_{k+2},\dots ,a_{k+m},a_{k+1},a_{k+2},\dots ,a_{k+m},\dots ]\\&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\overline {a_{k+1},a_{k+2},\dots ,a_{k+m}}}]\end{aligned}}$ where, in the second line, a vinculum marks the repeating block.[1] Some textbooks use the notation ${\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\dot {a}}_{k+1},a_{k+2},\dots ,{\dot {a}}_{k+m}]\end{aligned}}$ where the repeating block is indicated by dots over its first and last terms.[2] If the initial non-repeating block is not present – that is, if k = -1, a0 = am and $x=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}],$ the regular continued fraction x is said to be purely periodic. For example, the regular continued fraction for the golden ratio φ – given by [1; 1, 1, 1, ...] – is purely periodic, while the regular continued fraction for the square root of two – [1; 2, 2, 2, ...] – is periodic, but not purely periodic. As unimodular matrices Such periodic fractions are in one-to-one correspondence with the real quadratic irrationals. The correspondence is explicitly provided by Minkowski's question-mark function. That article also reviews tools that make it easy to work with such continued fractions. Consider first the purely periodic part $x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}],$ This can, in fact, be written as $x={\frac {\alpha x+\beta }{\gamma x+\delta }}$ with the $\alpha ,\beta ,\gamma ,\delta $ being integers, and satisfying $\alpha \delta -\beta \gamma =1.$ Explicit values can be obtained by writing $S={\begin{pmatrix}1&0\\1&1\end{pmatrix}}$ which is termed a "shift", so that $S^{n}={\begin{pmatrix}1&0\\n&1\end{pmatrix}}$ and similarly a reflection, given by $T\mapsto {\begin{pmatrix}-1&1\\0&1\end{pmatrix}}$ so that $T^{2}=I$. Both of these matrices are unimodular, arbitrary products remain unimodular. Then, given $x$ as above, the corresponding matrix is of the form [3] $S^{a_{1}}TS^{a_{2}}T\cdots TS^{a_{m}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}$ and one has $x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}]={\frac {\alpha x+\beta }{\gamma x+\delta }}$ as the explicit form. As all of the matrix entries are integers, this matrix belongs to the modular group $SL(2,\mathbb {Z} ).$ Relation to quadratic irrationals A quadratic irrational number is an irrational real root of the quadratic equation $ax^{2}+bx+c=0$ where the coefficients a, b, and c are integers, and the discriminant, b2 − 4ac, is greater than zero. By the quadratic formula every quadratic irrational can be written in the form $\zeta ={\frac {P+{\sqrt {D}}}{Q}}$ where P, D, and Q are integers, D > 0 is not a perfect square (but not necessarily square-free), and Q divides the quantity P2 − D (for example (6+√8)/4). Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example (3+√2)/2) as explained for quadratic irrationals. By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy. Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic.[4] Given a quadratic irrational x one can construct m different quadratic equations, each with the same discriminant, that relate the successive complete quotients of the regular continued fraction expansion of x to one another. Since there are only finitely many of these equations (the coefficients are bounded), the complete quotients (and also the partial denominators) in the regular continued fraction that represents x must eventually repeat. Reduced surds The quadratic surd $\zeta ={\frac {P+{\sqrt {D}}}{Q}}$ is said to be reduced if $\zeta >1$ and its conjugate $\eta ={\frac {P-{\sqrt {D}}}{Q}}$ satisfies the inequalities $-1<\eta <0$. For instance, the golden ratio $\phi =(1+{\sqrt {5}})/2=1.618033...$ is a reduced surd because it is greater than one and its conjugate $(1-{\sqrt {5}})/2=-0.618033...$ is greater than −1 and less than zero. On the other hand, the square root of two ${\sqrt {2}}=(0+{\sqrt {8}})/2$ is greater than one but is not a reduced surd because its conjugate $-{\sqrt {2}}=(0-{\sqrt {8}})/2$ is less than −1. Galois proved that the regular continued fraction which represents a quadratic surd ζ is purely periodic if and only if ζ is a reduced surd. In fact, Galois showed more than this. He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have ${\begin{aligned}\zeta &=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}]\\[3pt]{\frac {-1}{\eta }}&=[{\overline {a_{m-1};a_{m-2},a_{m-3},\dots ,a_{0}}}]\,\end{aligned}}$ where ζ is any reduced quadratic surd, and η is its conjugate. From these two theorems of Galois a result already known to Lagrange can be deduced. If r > 1 is a rational number that is not a perfect square, then ${\sqrt {r}}=[a_{0};{\overline {a_{1},a_{2},\dots ,a_{2},a_{1},2a_{0}}}].$ In particular, if n is any non-square positive integer, the regular continued fraction expansion of √n contains a repeating block of length m, in which the first m − 1 partial denominators form a palindromic string. Length of the repeating block By analyzing the sequence of combinations ${\frac {P_{n}+{\sqrt {D}}}{Q_{n}}}$ that can possibly arise when ζ = (P + √D)/Q is expanded as a regular continued fraction, Lagrange showed that the largest partial denominator ai in the expansion is less than 2√D, and that the length of the repeating block is less than 2D. More recently, sharper arguments [5][6][7] based on the divisor function have shown that L(D), the length of the repeating block for a quadratic surd of discriminant D, is given by $L(D)={\mathcal {O}}({\sqrt {D}}\ln {D})$ where the big O means "on the order of", or "asymptotically proportional to" (see big O notation). Canonical form and repetend The following iterative algorithm [8] can be used to obtain the continued fraction expansion in canonical form (S is any natural number that is not a perfect square): $m_{0}=0\,\!$ $d_{0}=1\,\!$ $a_{0}=\left\lfloor {\sqrt {S}}\right\rfloor \,\!$ $m_{n+1}=d_{n}a_{n}-m_{n}\,\!$ $d_{n+1}={\frac {S-m_{n+1}^{2}}{d_{n}}}\,\!$ $a_{n+1}=\left\lfloor {\frac {{\sqrt {S}}+m_{n+1}}{d_{n+1}}}\right\rfloor =\left\lfloor {\frac {a_{0}+m_{n+1}}{d_{n+1}}}\right\rfloor \!.$ Notice that mn, dn, and an are always integers. The algorithm terminates when this triplet is the same as one encountered before. The algorithm can also terminate on ai when ai = 2 a0,[9] which is easier to implement. The expansion will repeat from then on. The sequence [a0; a1, a2, a3, ...] is the continued fraction expansion: ${\sqrt {S}}=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+\,\ddots }}}}}}$ Example To obtain √114 as a continued fraction, begin with m0 = 0; d0 = 1; and a0 = 10 (102 = 100 and 112 = 121 > 114 so 10 chosen). ${\begin{aligned}{\sqrt {114}}&={\frac {{\sqrt {114}}+0}{1}}=10+{\frac {{\sqrt {114}}-10}{1}}=10+{\frac {({\sqrt {114}}-10)({\sqrt {114}}+10)}{{\sqrt {114}}+10}}\\&=10+{\frac {114-100}{{\sqrt {114}}+10}}=10+{\frac {1}{\frac {{\sqrt {114}}+10}{14}}}.\end{aligned}}$ $m_{1}=d_{0}\cdot a_{0}-m_{0}=1\cdot 10-0=10\,.$ $d_{1}={\frac {S-m_{1}^{2}}{d_{0}}}={\frac {114-10^{2}}{1}}=14\,.$ $a_{1}=\left\lfloor {\frac {a_{0}+m_{1}}{d_{1}}}\right\rfloor =\left\lfloor {\frac {10+10}{14}}\right\rfloor =\left\lfloor {\frac {20}{14}}\right\rfloor =1\,.$ So, m1 = 10; d1 = 14; and a1 = 1. ${\frac {{\sqrt {114}}+10}{14}}=1+{\frac {{\sqrt {114}}-4}{14}}=1+{\frac {114-16}{14({\sqrt {114}}+4)}}=1+{\frac {1}{\frac {{\sqrt {114}}+4}{7}}}.$ Next, m2 = 4; d2 = 7; and a2 = 2. ${\frac {{\sqrt {114}}+4}{7}}=2+{\frac {{\sqrt {114}}-10}{7}}=2+{\frac {14}{7({\sqrt {114}}+10)}}=2+{\frac {1}{\frac {{\sqrt {114}}+10}{2}}}.$ ${\frac {{\sqrt {114}}+10}{2}}=10+{\frac {{\sqrt {114}}-10}{2}}=10+{\frac {14}{2({\sqrt {114}}+10)}}=10+{\frac {1}{\frac {{\sqrt {114}}+10}{7}}}.$ ${\frac {{\sqrt {114}}+10}{7}}=2+{\frac {{\sqrt {114}}-4}{7}}=2+{\frac {98}{7({\sqrt {114}}+4)}}=2+{\frac {1}{\frac {{\sqrt {114}}+4}{14}}}.$ ${\frac {{\sqrt {114}}+4}{14}}=1+{\frac {{\sqrt {114}}-10}{14}}=1+{\frac {14}{14({\sqrt {114}}+10)}}=1+{\frac {1}{\frac {{\sqrt {114}}+10}{1}}}.$ ${\frac {{\sqrt {114}}+10}{1}}=20+{\frac {{\sqrt {114}}-10}{1}}=20+{\frac {14}{{\sqrt {114}}+10}}=20+{\frac {1}{\frac {{\sqrt {114}}+10}{14}}}.$ Now, loop back to the second equation above. Consequently, the simple continued fraction for the square root of 114 is ${\sqrt {114}}=[10;{\overline {1,2,10,2,1,20}}].\,$ (sequence A010179 in the OEIS) √114 is approximately 10.67707 82520. After one expansion of the repetend, the continued fraction yields the rational fraction ${\frac {21194}{1985}}$ whose decimal value is approx. 10.67707 80856, a relative error of 0.0000016% or 1.6 parts in 100,000,000. Generalized continued fraction A more rapid method is to evaluate its generalized continued fraction. From the formula derived there: ${\sqrt {z}}={\sqrt {x^{2}+y}}=x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {y}{2x+\ddots }}}}}}=x+{\cfrac {2x\cdot y}{2(2z-y)-y-{\cfrac {y^{2}}{2(2z-y)-{\cfrac {y^{2}}{2(2z-y)-\ddots }}}}}}$ and the fact that 114 is 2/3 of the way between 102=100 and 112=121 results in ${\sqrt {114}}={\cfrac {\sqrt {1026}}{3}}={\cfrac {\sqrt {32^{2}+2}}{3}}={\cfrac {32}{3}}+{\cfrac {2/3}{64+{\cfrac {2}{64+{\cfrac {2}{64+{\cfrac {2}{64+\ddots }}}}}}}}={\cfrac {32}{3}}+{\cfrac {2}{192+{\cfrac {18}{192+{\cfrac {18}{192+\ddots }}}}}},$ which is simply the aforementioned [10;1,2, 10,2,1, 20,1,2] evaluated at every third term. Combining pairs of fractions produces ${\sqrt {114}}={\cfrac {\sqrt {32^{2}+2}}{3}}={\cfrac {32}{3}}+{\cfrac {64/3}{2050-1-{\cfrac {1}{2050-{\cfrac {1}{2050-\ddots }}}}}}={\cfrac {32}{3}}+{\cfrac {64}{6150-3-{\cfrac {9}{6150-{\cfrac {9}{6150-\ddots }}}}}},$ which is now $[10;1,2,{\overline {10,2,1,20,1,2}}]$ evaluated at the third term and every six terms thereafter. See also • Continued fraction – Number represented as a0+1/(a1+1/...) • Generalized continued fraction – generalization of continued fractions in which the partial numerators and partial denominators can assume arbitrary complex valuesPages displaying wikidata descriptions as a fallback • Hermite's problem • Continued fraction method of computing square roots – Algorithms for calculating square roots • Restricted partial quotients • Continued fraction factorization – an integer factorization algorithmPages displaying wikidata descriptions as a fallback Notes 1. Pettofrezzo & Byrkit 1970, p. 158. 2. Long 1972, p. 187. 3. Khinchin 1964. 4. Davenport 1982, p. 104. 5. Hickerson 1973. 6. Cohn 1977. 7. Podsypanin 1982. 8. Beceanu 2003. 9. Gliga 2006. References • Beceanu, Marius (5 February 2003). "Period of the Continued Fraction of sqrt(n)" (PDF). Theorem 2.3. Archived from the original (PDF) on 21 December 2015. Retrieved 3 May 2022. • Cohn, J. H. E. (1977). "The length of the period of the simple continued fraction expansion of d1/2". Pacific J. Math. 71: 21–32. doi:10.2140/pjm.1977.71.21. • Davenport, H. (December 1982). The Higher Arithmetic. Cambridge University Press. ISBN 0-521-28678-6. • Gliga, Alexandra Ioana (17 March 2006). On continued fractions of the square root of prime numbers (PDF). Corollary 3.3. Retrieved 3 May 2022. • Hickerson, Dean R. (1973). "Length of period of simple continued fraction expansion of vd". Pacific J. Math. 46: 429–432. doi:10.2140/pjm.1973.46.429. • Khinchin, A. Ya. (1964) [Originally published in Russian, 1935]. Continued Fractions. University of Chicago Press. ISBN 0-486-69630-8. (This is now available as a reprint from Dover Publications.) • Long, Calvin T. (1972). Elementary Introduction to Number Theory (3 Sub ed.). Waveland Pr Inc. LCCN 77-171950. • Pettofrezzo, Anthony Joseph; Byrkit, Donald R. (1970). Elements of Number Theory (11 ed.). Englewood Cliffs: Prentice Hall. ISBN 9780132683005. LCCN 77-81766. • Podsypanin, E.V. (1982). "Length of the period of a quadratic irrational". Journal of Soviet Mathematics. 18 (6): 919–923. doi:10.1007/BF01763963. S2CID 119567810. • Rockett, Andrew M.; Szüsz, Peter (1992). CONTINUED FRACTIONS. World Scientific Publishing Company. ISBN 9789810210526.	Wikipedia
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. In an AC circuit which voltage lags the current in a capacitor: Source voltage or capacitor voltage? In a circuit with just a resistor and a capacitor I'm trying to figure out which voltage is being referred to that is lagging the current(which is the same current throughout the entire SERIES AC circuit). The voltage which leads or lags is the same voltage referred to by $$I(t)=C\frac{dV(t)}{dt}$$ which comes from the derivative of $$V(t) = \frac{Q(t)}{C} = \frac{1}{C}\int_{t_0}^t I(\tau) \mathrm{d}\tau + V(t_0)$$ Therefore the voltage which lags is the voltage drop across the capacitor because the charges are added above to the plates of the capacitor so the voltage refers to it. In an AC circuit, the voltage source is forced to alternate with a cosine wave and the phase difference between the source current driven by $$V_0\cos(\omega t)\tag{1}$$ and the voltage which I'm asking about comes from: $$I = C \frac{dV}{dt} = -\omega {C}{V_\text{0}}\sin(\omega t)\tag{2}$$ which is the same as $$I = {I_\text{0}}{\cos({\omega t} + {90^\circ})}$$ The voltage used in this formula was the source voltage not the voltage drop across the capacitor which defines Capacitance. The voltage across the capacitor is not instantaneous and in fact exponentially decays up to the applied voltage as shown by a constant DC voltage circuit where : $$V_0 = v_\text{resistor}(t) + v_\text{capacitor}(t) = i(t)R + \frac{1}{C}\int_{t_0}^t i(\tau) \mathrm{d}\tau$$ Taking the derivative: $$RC\frac{\mathrm{d}i(t)}{\mathrm{d}t} + i(t) = 0$$ Solving the first order: $$I(t) = \frac{V_0}{R} \cdot e^{\frac{-t}{\tau_0}}$$ Assuming initially the resitor is $V_0$ the voltage of capacitor: $$V(t) = V_0 \left( 1 - e^{\frac{-t}{\tau_0}}\right)$$ Thus I'm confused about where the $90^\circ$ voltage lag comes from. If it's because of the derivative of the source voltage why is formula 2 even applicable to the source voltage. Second question: What is the formula for the voltage reached by the capacitor in an ac circuit. It appears as if it is the source max voltage but I don't believe/understand that. Here is an identically solved using a sin source Voltage: $$I_C+I_{max}\sin(\omega t +90^\circ)$$ In the above derivation, the source voltage is again mixed with the formula for the voltage stored across a capacitor or I'm to believe the maximum source voltage is somehow reached on the exponential decay to the voltage on a capacitor during a cycle. Could someone either explain why the source voltage is used as if it was the capacitor voltage or the identically reversed inductor or refer me to a source that explains it? (1)https://en.wikipedia.org/wiki/Capacitor (2)http://www.electronics-tutorials.ws/accircuits/ac-capacitance.html electric-circuits user5389726598465 user5389726598465user5389726598465 $\begingroup$ Note that, for AC analysis, sinusoidal steady state is assumed, i.e, all transients have decayed away. You seem to be mixing AC analysis and transient analysis and this is most likely causing some confusion. $\endgroup$ – Alfred Centauri The defining equation for a capacitor is $Q=CV_{\rm C}$ and when that equation is differentiated with respect to time one gets $\dfrac{dQ}{dt} = I = C\dfrac{dV_{\rm C}}{dt}$ So the current is proportional to the rate of change of voltage across the capacitor Applying a sinusoidal voltage to a capacitor results it the following current and voltage graphs. Notice that the current is determined by the gradient of the voltage against time graph. being a maximum at time $a$ and zero at times $b$and $d$. Whatever the current is doing the voltage does a quarter of a period (equivalent to $90^\circ$) later. So the current is a maximum at time $a$ and the voltage is a maximum at a later time $b$. We say that current leads the voltage across a capacitor by $90^\circ$. In the graph $V_{\rm C}(t)=V_{\rm max} \sin \omega t$ and so the current is $I(t) =\omega CV_{\rm max} \cos\omega t$ with a peak current $I_{\rm max}=\omega CV_{\rm max}$. When you add a series resistor to the circuit the current is the same in all parts of the circuit. The voltage across the capacitor still lags the current by $90^\circ$ and the voltage across the resistor will be in phase with the current. The (applied) voltage across both components will lag the current through the circuit at some value between $0^\circ$ and $90^\circ$ depending on the values of the capacitance of the capacitor, the resistance of the resistor and the frequency of the applied voltage. Here I have considered what are called steady state conditions and so there are no transients which would be characterised by an exponential function and a time constant. The difference for an inductor is that the defining equation is $V_{\rm L} = L \dfrac{dI}{dt}$ and the voltage across an inductor leads the current by $90^\circ$. Update as a result of a comment I think that what you are asking about is the transient behaviour which occurs when you first connect a capacitor to the voltage source. If by chance you make this connection to an uncharged capacitor when the voltage of the supply is zero then there is no transient and the circuit currents and voltages are as per the graph shown above. If on the other that is not so you will have a combination of the transient (the exponential function you have described) and the steady state. After about 10 time constants (10CR) the transients would have decayed away and all that is left is steady state Now with an "ideal" circuit with no resistance the time constant is zero and the circuit settles down to steady state behaviour "instantly". However with a finite resistance in the circuit then there will be a transient behaviour which you tend to to see because it decays away. I can show you this idea of a transient in action by using the "Circuit Sandbox" which is available in the edX Circuits and Electronics course, a course I thoroughly recommend even if it just to be able to used the circuit simulator. Here is the result of a simulation where there is dc voltage of 1 V across a capacitor and after one second a sinusoidal voltage of peak value 1 V and frequency 1 Hz is applied across a resistor and a capacitor connected in series. The graph is voltage across the capacitor in volts against time in seconds. You can see very clearly the transient behaviour (the exponential decay) and then the steady state behaviour. The 10RC s just a rule of thumb where $e^{-10} \approx 4.5 \times 10^{-5}$ and the decay has effective finished. Others use 5RC which corresponds to a decrease of $e^{-5} \approx 6.7 \times 10^{-3}$. Here is the supply voltage shown in red and the voltage across the capacitor shown in cyan. The supply voltage and voltage across the capacitor start at $+1 \, \rm V$ and then a $\pm 1 \, \rm V$ sinusoidal voltage is added after 1 second. It clearly shows the $90^\circ$ phase shift. FarcherFarcher $\begingroup$ This is very good but it explains everything that I already learned and identically skips the critical information that all the text books I consulted also skip. You say "So the current is proportional to the rate of change of voltage across the capacitor". The voltage of the capacitor is from the charge on the plates which has thus far accumulated over time which is different from the sinusoidal source voltage which is driven by some form of EM generator. When the applied voltage is say 1, the voltage caused by accumulated electrons on the capacitor could start out at 0. $\endgroup$ – user5389726598465 $\begingroup$ By the time the applied voltage equals zero(if frequency=4), enough electrons are on the capacitor to still be less than (1-e^{-1}) in a 1 ohm 1F circuit had the voltage been constant. The question is why is the sinusoidal voltage applied, plugged into the formula which represents the voltage caused by electrons on the plates of the capacitor $I=CV'$ They are two different values from two unrelated processes. $\endgroup$ $\begingroup$ I see, so the capacitor will have enough electrons on it that if I put a voltmeter across it (eventually) during one time cycle the voltage of the electrons on the plates WILL equal the max voltage applied by the source at exactly $1/4$ and $3/4$ of the cycle through? That means the electrons on the capacitor get on one plate and then on the other plate at the same rate as the voltage cycles, regardless of how great the voltage is that is applied. You said "After about 10 time constants (10CR)" do you know of a book that does these obscure calculations of that nature that I can read? $\endgroup$ $\begingroup$ Because the ac has a +1 V dc offset. $\endgroup$ – Farcher $\begingroup$ $\pm 1 \,\rm V$ and remember that it is the voltage across the capacitor which has been graphed. I can add the supply voltage graph if you wish? $\endgroup$ When we have a series RC circuit we know that: $$\text{V}_\text{in}\left(t\right)=\text{V}_\text{R}\left(t\right)+\text{V}_\text{C}\left(t\right)\tag1$$ We now can use: $$\text{V}_\text{R}\left(t\right)=\text{I}_\text{R}\left(t\right)\cdot\text{R}\tag2$$ $$\text{I}_\text{C}\left(t\right)=\text{V}_\text{C}'\left(t\right)\cdot\text{C}\tag3$$ So, we get: $$\text{V}_\text{in}'\left(t\right)=\text{I}_\text{R}'\left(t\right)\cdot\text{R}+\text{I}_\text{C}\left(t\right)\cdot\frac{1}{\text{C}}\tag4$$ And for the current we can write: $$\text{I}_\text{in}\left(t\right)=\text{I}_\text{R}\left(t\right)=\text{I}_\text{C}\left(t\right)\tag5$$ So, we end up with: $$\text{V}_\text{in}'\left(t\right)=\text{I}_\text{in}'\left(t\right)\cdot\text{R}+\text{I}_\text{in}\left(t\right)\cdot\frac{1}{\text{C}}\tag6$$ Now, when we have an input voltage that looks like: $$\text{V}_\text{in}\left(t\right)=\hat{\text{u}}\cdot\cos\left(\omega t+\varphi\right)\tag7$$ We find: $$\text{V}_\text{in}'\left(t\right)=\frac{\partial}{\partial t}\left(\hat{\text{u}}\cdot\cos\left(\omega t+\varphi\right)\right)=\text{I}_\text{in}'\left(t\right)\cdot\text{R}+\text{I}_\text{in}\left(t\right)\cdot\frac{1}{\text{C}}\tag8$$ You can use, for example, Laplace transform to solve $\left(8\right)$ in the general case. Assuming that $\varphi=0$, $\text{I}_\text{in}\left(0\right)=0$ and $\hat{\text{u}}=1$: $$\text{I}_\text{in}\left(t\right)=\frac{\frac{\text{W}}{\text{R}}\cdot\left(\text{W}\cdot\left(\cos\left(\omega t\right)-\exp\left(-t\cdot\frac{\omega}{\text{W}}\right)\right)-\sin\left(\omega t\right)\right)}{1+\text{W}^2}\tag9$$ Where $\text{W}=\text{RC}\omega$ J.W.L. Jan EerlandJ.W.L. Jan Eerland $\begingroup$ This is very interesting. I wonder why the current is different from $\frac{û\cdot \cos(\omega t+\phi)}{\sqrt{R^2+(\omega L-\frac{1}{\omega C})^2}}$ $\endgroup$ Thanks for contributing an answer to Physics Stack Exchange! Not the answer you're looking for? Browse other questions tagged electric-circuits or ask your own question. Using the loop rule while charging circuits Determining energy stored in capacitor and inductor in RLC circuit Series / parallel capacitor network: find two capacitances and source voltage given some measured data Cylindrical capacitor in an electric circuit In an RLC series circuit on resonance, how can the voltages over the capacitor and the inductor be larger than the source voltage? In RLC circuit, if source voltage $V(t)=V_p \sin(\omega t)$ then $V_p= \sqrt{V_{R,p}^2+(V_{L,p}-V_{C,p})^2}$? LCR circuit (AC source) potential difference across capacitor Transformers and basic inductor physics If no current is able to flow through a capacitor what will be the node voltage on the other end of the capacitor opposite the voltage source?	CommonCrawl
\begin{document} \def\mathbb{R}{\mathbb{R}} \def\text{Exp}{\text{Exp}} \def\mathcal{F}_\al{\mathcal{F}_\alpha} \title[] {Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball -- Preasymptotics, asymptotics, and tractability} \author[]{Jia Chen, Heping Wang} \address{School of Mathematical Sciences, Capital Normal University, Beijing 100048, China.} \email{ [email protected];\ \ \ [email protected].} \keywords{Approximation number; Sobolev spaces; Gevrey type spaces, preasymptotics and asymptotics; tractability} \begin{abstract} In this paper, we investigate optimal linear approximations ($n$-approximation numbers ) of the embeddings from the Sobolev spaces $H^r\ (r>0)$ for various equivalent norms and the Gevrey type spaces $G^{\alpha,\beta}\ (\alpha,\beta>0)$ on the sphere ${\Bbb S}^{d}$ and on the ball ${\Bbb B^d}$, where the approximation error is measured in the $L_2$-norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in $n$ and the dimension $d$. We emphasis that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension $d$ and $n$. As a consequence we obtain that for the absolute error criterion the approximation problems $I_d: H^{r}\to L_2$ are weakly tractable if and only if $r>1$, not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any $\alpha,\beta>0$, the approximation problems $I_d: G^{\alpha,\beta}\to L_2$ are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if $\alpha\ge 1$. \end{abstract} \maketitle \input amssym.def \section{Introduction} This paper is devoted to investigating the behavior of the approximation numbers of embeddings of Sobolev spaces and Gevrey type spaces on the sphere ${\Bbb S}^{d}$ and on the ball ${\Bbb B^d}$ into $L_2$. The approximation numbers of a bounded linear operator $T:X\rightarrow Y$ between two Banach spaces are defined as \begin{align}\label{1.1} a_n(T:X\rightarrow Y):&=\inf_{rank A< n}\sup_{\\|x\|X\\|\leq 1}\\|Tx-Ax\|Y\\|\notag\\ &=\inf_{rank A< n}\\|T-A:X\rightarrow Y\\|,\ \ n\in \Bbb N_+, \end{align} where $\Bbb N_+=\{1,2,3,\dots\},\ \Bbb N=\{0,1,2,3,\dots\}$. They describe the best approximation of $T$ by finite rank operators. If $X$ and $Y$ are Hilbert spaces and $T$ is compact, then $a_n(T )$ is the $n$th singular number of $T$. Also $a_n(T)$ is the $n$th minimal worst-case error with respect to arbitrary algorithms and general information in the Hilbert setting. On the torus $\Bbb T^d$, there are many results concerning asymptotics of the approximation numbers of smooth function spaces, see the monographs \cite{Te} by Temlyakov and the references therein. However, the obtained asymptotics often hide dependencies on the dimension $d$ in the constants, and can only be seen after ``waiting exponentially long ($n\ge 2^d)$'' if $d$ is large. In order to overcome this deficiency, K\"uhn and other authors obtained preasymptotics and asymptotics of the approximation numbers of the classical isotropic Sobolev spaces, Sobolev spaces of dominating mixed smoothness, periodic Gevrey spaces, and anisotropic Sobolev spaces (see \cite{KSU1, KSU2, KMU, CW}). Note that in these preasymptotics and asymptotics, the equivalence constants are independent of the dimension $d$ and $n$. On the sphere ${\Bbb S}^{d}$ and on the ball ${\Bbb B^d}$, the following two-sided estimates can be found in \cite{Ka} and \cite{WH} in a slightly more general setting: \begin{equation}\label{1.2}C_1(r,d)n^{-r/d}\le a_n(I_d: H^r({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))\le C_2(r,d)n^{-r/d},\ n\in\Bbb N_+, \end{equation} and \begin{equation}\label{1.3}C_3(r,d)n^{-r/d}\le a_n(I_d: H^r({\Bbb B^d})\to L_2({\Bbb B^d}))\le C_4(r,d)n^{-r/d}, \ n\in\Bbb N_+,\end{equation} where $I_d$ is the identity (embedding) operator, $H^r({\Bbb S}^{d}),\ H^r({\Bbb B^d})$ are the Sobolev spaces on the sphere ${\Bbb S}^{d}$ and on the ball ${\Bbb B^d}$, the constants $C_i(r,d), i=1,2,3,4$, only depending on the smoothness index $r$ and the dimension $d$, were not explicitly determined. In the paper we discuss preasymptotics and asymptotics of the approximation numbers of the embeddings $I_d$ of the Sobolev spaces $H^r\ (r>0)$ and the Gevrey type spaces $G^{\alpha,\beta}\ (\alpha,\beta>0)$ on the sphere ${\Bbb S}^{d}$ and on the ball ${\Bbb B^d}$ into $L_2$. We remark that the Gevrey type spaces have a long history and have been used in numerous problems related to partial differential equations. Our main focus in this paper is to clarify, for arbitrary but fixed $r>0$ and $\alpha,\beta>0$, the dependence of these approximation numbers $a_n(I_d)$ on $d$. In fact, it is necessary to fix the norms on the spaces $H^r$ on ${\Bbb S}^{d}$ and on ${\Bbb B^d}$ in advance, since the constants $C_i(r,d),\ i=1,2,3,4 $ in \eqref{1.2} and \eqref{1.3} depend on the size of the respective unit balls. Surprisingly, for a collection of quite natural norms of $H^r$ (see Sections 2.1 and 6.1), and for sufficiently large $n$, say $n\ge 2^d$, it turns out that the optimal constants decay polynomially in $d$, i.e., $$C_1(r,d)\asymp_r C_2(r,d)\asymp_r C_3(r,d)\asymp_r C_4(r,d)\asymp_r d^{-r},$$where $A\asymp B$ means that there exist two constants $c$ and $C$ which are called the equivalence constants such that $c A\le B\le C A$, and $\asymp_{r}$ indicates that the equivalence constants depend only on $r$. This means that on ${\Bbb S}^{d}$ and on ${\Bbb B^d}$, for $n\ge 2^d$, $$ a_n(I_d: H^r\to L_2)\asymp_r d^{-r} n^{-r/d},$$where the equivalence constants are independent of $d$ and $n$. We also show that on ${\Bbb S}^{d}$ and on ${\Bbb B^d}$, for $n\ge 2^d$, $$ \ln( a_n(I_d: G^{\alpha,\beta}\to L_2))\asymp_\alpha -\beta d^{\alpha} n^{\alpha/d},$$ where the equivalence constants depend only on $\alpha$, but not on $d$ and $n$. Specially, we prove that the limits \begin{equation}\lim_{n\to\infty}n^{r/d}a_n(I_d: H^{r}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=\Big(\frac2{d\,!}\Big)^{r/d}\end{equation} and \begin{equation}\lim_{n\to\infty}n^{r/d}a_n(I_d: H^{r}({\Bbb B^d})\to L_2({\Bbb B^d}))=\Big(\frac1{d\,!}\Big)^{r/d}\end{equation} exist, having the same value for various norms. We also prove that for $0<\alpha<1$, $\beta>0$, $ \gamma=\Big(\frac2{d\,!}\Big)^{-\alpha/d}$, $\tilde \gamma=\Big(\frac1{d\,!}\Big)^{-\alpha/d}$, \begin{equation}\lim_{n\to \infty }e^{\beta \gamma n^{\alpha/d}}a_n(I_d: G^{\alpha, \beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=1.\end{equation} and \begin{equation}\lim_{n\to \infty }e^{\beta \tilde\gamma n^{\alpha/d}}a_n(I_d: G^{\alpha, \beta}({\Bbb B^d})\to L_2({\Bbb B^d}))=1.\end{equation} For small $n, 1\le n\le 2^d$, we also determine explicitly how these approximation numbers $ a_n(I_d)$ behave preasymptotically. We emphasize that the preasymptotic behavior of $ a_n(I_d)$ is completely different from its asymptotic behavior. For example, we show that \begin{align} a_n(I_d: H^{r,}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))&\asymp_r a_n(I_d: H^{r,}({\Bbb B^d})\to L_2({\Bbb B^d}))\\ & \asymp_r \left\{\begin{matrix} 1,\ \ \ &n=1,\\ d^{-r/2}, & \ \ 2\le n\leq d,\\ d^{-r/2}\Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r/2},&\ \ d\le n\le 2^d, \end{matrix}\right. \end{align} and \begin{align} \ln \big(a_n(I_d: G^{\alpha,\beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))\big) &\asymp_\alpha \ln \big(a_n(I_d: G^{\alpha,\beta}({\Bbb B^d})\to L_2({\Bbb B^d}))\big) \\ &\asymp_\alpha -\beta \left\{\begin{matrix} 1, & \ \ 1\le n\leq d,\\ \Big(\frac{\log n}{\log(1+\frac{d}{\log n})}\Big)^{\alpha},&\ \ d\le n\le 2^d, \end{matrix}\right. \end{align} where the equivalence constants depend only on $r$ or $\alpha$, but not on $d$ and $n$. Here ``$$'' stands for a specific (but natural) norm in $H^r$ on ${\Bbb S}^{d}$ and on ${\Bbb B^d}$. Finally we consider tractability results for the approximation problems of the Sobolev embeddings and the Gevrey type embeddings on ${\Bbb S}^{d}$ and on ${\Bbb B^d}$. Based on the asymptotic and preasymptotic behavior of $ a_n(I_d : H^{r}\to L_2)$ and $a_n(I_d:G^{\alpha,\beta}\to L_2)$, we show that for the absolute error criterion the approximation problems $I_d: H^{r}\to L_2$ are weakly tractable if and only if $r>1$, not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any $\alpha,\beta>0$, the approximation problems $I_d: G^{\alpha,\beta}\to L_2$ are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if $\alpha\ge 1$ and the exponent of quasi-polynomial tractability is $$t^{\rm qpol}=\sup_{m\in \Bbb N}\frac m {1+\beta m^\alpha},\ \ \alpha\ge1.$$ The paper is organized as follows. In Section 2.1 we give definitions of the Sobolev spaces with various equivalent norms and the Gevrey type spaces on the sphere. Section 2.2 is devoted to some basics on the approximation numbers on the sphere. In Section 3, we study strong equivalence of the approximation numbers $ a_n(I_d : H^{r}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))$ and $a_n(I_d:G^{\alpha,\beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))$. Section 4 contains results concerning preasymptotics and asymptotics of the above approximation numbers. Section 5 transfers our approximation results into the tractability ones of the respective approximation problems. In the final Section 6, we obtain the corresponding results on ${\Bbb B^d}$ such as strong equivalence, preasymptotics and asymptotics of the approximation numbers $ a_n(I_d : H^{r}({\Bbb B^d})\to L_2({\Bbb B^d}))$ and $a_n(I_d:G^{\alpha,\beta}({\Bbb B^d})\to L_2({\Bbb B^d}))$, and tractability of the respective approximation problems. \section{Preliminaries on the sphere} \subsection{Sobolev spaces and Gevrey type spaces on the sphere} \ Let ${\Bbb S}^{d}=\{x\in\mathbb{R}^{d+1}:\, \|x\|=1\}$ ($d\ge 2$) be the unit sphere of $\mathbb{R}^{d+1}$ (with $\|x\|$ denoting the Euclidean norm in $\mathbb{R}^{d+1}$) endowed with the rotationally invariant measure $\sigma$ normalized by $\int_{{\Bbb S}^{d}}d\sigma=1$. Denote by $L_2({\Bbb S}^{d})$ the collection of real measurable functions $f$ on ${\Bbb S}^{d}$ with finite norm $$\\|f\|L_2({\Bbb S}^{d})\\|=\Big(\int_{{\Bbb S}^{d}}\|f(x)\|^2\, d\sigma(x)\Big)^\frac{1}{2}<+ \infty.$$ We denote by $\mathcal{H}_\ell^d$ the space of all spherical harmonics of degree $l$ on ${\Bbb S}^{d}$. Denote by $\Pi^{d+1}_m(\Bbb S^d)$ the set of spherical polynomials on $\mathbb{S}^d$ of degree $\le m$, which is just the set of polynomials of degree $\le m$ on $\mathbb{R}^{d+1}$ restricted to $\mathbb{S}^d$. It is well known (see \cite[Corollaries 1.1.4 and 1.1.5]{DaiX}) that the dimension of $\mathcal{H}_\ell^d$ is \begin{equation}\label{2.1} Z(d,\ell):={\rm dim}\,\mathcal{H}_\ell^d=\left\{\begin{array}{cl} 1,\ \ \ & {\rm if}\ \ \ell=0,\\ \frac{(2\ell +d-1)\,(\ell +d-2)!}{(d-1)!\ l!},\ \ & {\rm if}\ \ \ell=1,2,\dots, \end{array}\right.\end{equation} and \begin{equation}\label{2.2}C(d,m):={\rm dim}\,\Pi_m^{d+1}(\Bbb S^d)=\frac{(2m+d)(m+d-1)!}{m!\,d!},\ m\in\Bbb N.\end{equation} Let $$\{Y_{\ell,k}\equiv Y_{\ell,k}^{d}\ \|\ k=1,\dots, Z(d,\ell)\}$$ be a fixed orthonormal basis for $\mathcal{H}_\ell^d$. Then $$\{Y_{\ell,k}\ \|\ k=1,\dots, Z(d,\ell),\ \ell=0,1,2,\dots\}$$ is an orthonormal basis for the Hilbert space $L_2({\Bbb S}^{d})$. Thus any $f\in L_2({\Bbb S}^{d})$ can be expressed by its Fourier (or Laplace) series$$f=\sum_{\ell=0}^{\infty} H_\ell(f)=\sum_{\ell=0}^{\infty}{\sum_{k=1}^{Z(d,\ell)} \langle f,Y_{\ell,k}\rangle Y_{\ell,k}},$$ where $H_\ell^d(f)=\sum\limits_{k=1}^{Z(d,\ell)} \langle f,Y_{\ell,k}\rangle Y_{\ell,k},\ \ell=0,1,\dots,$ denote the orthogonal projections of $f$ onto $\mathcal{H}_\ell^d$, and $$\langle f,Y_{\ell,k}\rangle =\int_{{\Bbb S}^{d}} f(x) Y_{\ell,k}(x) \, d\sigma(x)$$ are the Fourier coefficients of $f$. We have the following Parseval equality: $$\\|f\|L_2({\Bbb S}^{d})\\|=\Big(\sum_{\ell=0}^{\infty}\sum_{k=1}^{Z(d,\ell)} \|\langle f,Y_{\ell,k}\rangle \|^2\Big)^{1/2}.$$ Let $\Lambda=\{\lambda_k\}_{k=0}^\infty$ be a bounded sequence, and let $T^\Lambda$ be a multiplier operator on $L_2({\Bbb S}^{d})$ defined by $$T^\Lambda (f)=\sum_{\ell=0}^{\infty}\lambda_\ell H_\ell(f)=\sum_{\ell=0}^{\infty}\lambda_\ell {\sum_{k=1}^{Z(d,\ell)} \langle f,Y_{\ell,k}\rangle Y_{\ell,k}}.$$ Let $SO(d+1)$ be the special rotation group of order ${d+1}$, i.e., the set of all rotation on $\Bbb R^{d+1}$. For any $\rho \in SO(d+1)$ and $f\in L_2({\Bbb S}^{d})$, we define $\rho(f)(x)=f(\rho x)$. It is well known (see \cite[Proposition 2.2.9]{DaiX}) that a bounded linear operator $T$ on $L_2({\Bbb S}^{d})$ is a multiplier operator if and only if $T\rho=\rho T$ for any $\rho \in SO(d+1)$. \begin{defn}\label{d2.1}Let $\Lambda=\{\lambda_k\}_{k=0}^\infty$ be a non-increasing positive sequence with $\lim\limits_{k\to\infty}\lambda_k=0$. We define the multiplier space $H^\Lambda({\Bbb S}^{d})$ by \begin{align}H^\Lambda({\Bbb S}^{d})&:=\Big\{T^\Lambda f\, \big\|\, f\in L_2({\Bbb S}^{d}) \ {\rm and }\ \\|T^\Lambda f\,\big\|\,H^\Lambda({\Bbb S}^{d})\\|=\\|f\|L_2({\Bbb S}^{d})\\|<\infty\Big\}\\& := \Big\{f\in L_2({\Bbb S}^{d})\, \big\|\, \\|f\,\big\|\,H^\Lambda({\Bbb S}^{d})\\|:=\Big(\sum_{\ell=0}^{\infty}\frac1{\lambda_\ell^2}\sum_{k=1}^{Z(d,\ell)} \|\langle f,Y_{\ell,k}\rangle \|^2\Big)^{1/2}<\infty\Big\}. \end{align}\end{defn} Clearly, the multiplier space $H^\Lambda({\Bbb S}^{d})$ is a Hilbert space with inner product $$\langle f,g \rangle_{H^\Lambda({\Bbb S}^{d})}= \sum_{\ell=0}^{\infty}\frac1{\lambda_\ell^2}\sum_{k=1}^{Z(d,\ell)} \langle f,Y_{\ell,k}\rangle\, \langle g,Y_{\ell,k}\rangle.$$ We remark that Sobolev spaces and Gevrey type spaces on the sphere ${\Bbb S}^{d}$ are special multiplier spaces whose definitions are given as follows. \begin{defn}\label{d2.2}Let $r>0$ and $\square\in\{,+,\#,-\}$. The Sobolev space $H^{r,\square}({\Bbb S}^{d})$ is the collection of all $f\in L_2({\Bbb S}^{d})$ such that $$\\|f\,\big\|\,H^{r,\square}({\Bbb S}^{d})\\|=\Big(\sum_{\ell=0}^{\infty}(r_{\ell,d}^{\square})^{-2}\sum_{k=1}^{Z(d,\ell)} \|\langle f,Y_{\ell,k}\rangle \|^2\Big)^{1/2}<\infty,$$ where \begin{align} r_{\ell,d}^{}&=(1+(\ell(\ell+d-1))^r)^{-1/2},\ \ r_{\ell,d}^{+}=(1+\ell(\ell+d-1))^{-r/2},\\ r_{\ell,d}^{\#}&=(1+\ell)^{-r},\ \ \ \ \qquad \qquad\qquad r_{\ell,d}^{-}=(\ell+(d-1)/2)^{-r},\ \ \ l=0,1,\dots. \end{align} If we set $\Lambda^{\square}=\{r_{k,d}^\square\}_{k=0}^\infty$, then the Sobolev space $H^{r,\square}({\Bbb S}^{d})$ is just the multiplier space $H^{\Lambda^\square}({\Bbb S}^{d})$. \end{defn} \begin{rem}We note that the above four Sobolev norms are equivalent with the equivalence constants depending on $d$ and $r$. The natural Sobolev space on the sphere is $H^{r,}({\Bbb S}^{d})$. The Sobolev space $H^{r,-}({\Bbb S}^{d})$ is given in \cite[Definition 3.23]{AH}. \end{rem} \begin{rem}Let $\triangle$ be the Laplace-Beltrami operator on the sphere. It is well known that the spaces $\mathcal{H}_\ell^d, \ \ell=0,1,2, \dots$, are just the eigenspaces corresponding to the eigenvalues $-\ell(\ell +d-1)$ of the operator $\triangle$. Given $s>0$, we define the $s$-th (fractional) order Laplace-Beltrami operator $(-\triangle)^s$ on ${\Bbb S}^{d}$ in a distributional sense by $$H_0((-\triangle)^s(f))=0,\ \ \ H_\ell((-\triangle)^s(f))=(\ell(\ell +d-1))^s H_\ell(f), \ \ \ell=1,2,\dots\ ,$$ where $f$ is a distribution on ${\Bbb S}^{d}$. We can define $(I-\triangle)^s, \ (((d-1)/2)^2I-\triangle)^s $ analogously, where $I$ is the identity operator. For $r>0$ and $f\in H^{r}({\Bbb S}^{d})$, we have the following equalities. \begin{align} \\|f\,\big\|\,H^{r,}({\Bbb S}^{d})\\|&=\Big(\\|f\|L_2({\Bbb S}^{d})\\|^2+\\|(-\triangle)^{r/2}f\|L_2({\Bbb S}^{d})\\|^2\Big)^{1/2},\\ \\|f\,\big\|\,H^{r,+}({\Bbb S}^{d})\\|&=\Big(\\|(I-\triangle)^{r/2}f\|L_2({\Bbb S}^{d})\\|^2\Big)^{1/2},\\ \\|f\,\big\|\,H^{r,-}({\Bbb S}^{d})\\|&=\Big(\\|(((d-1)/2)^2I-\triangle)^{r/2}f\|L_2({\Bbb S}^{d})\\|^2\Big)^{1/2}.\end{align} \end{rem} \begin{defn}\label{d2.3}Let $0<\alpha,\beta<\infty$. The Gevrey type space $G^{\alpha,\beta}({\Bbb S}^{d})$ is the collection of all $f\in L_2({\Bbb S}^{d})$ such that $$\\|f\,\big\|\,G^{\alpha,\beta}({\Bbb S}^{d})\\|=\Big(\sum_{\ell=0}^{\infty}e^{2\beta \ell^\alpha}\sum_{k=1}^{Z(d,\ell)} \|\langle f,Y_{\ell,k}\rangle \|^2\Big)^{1/2}<\infty.$$ If we set $\Lambda_{\alpha,\beta}=\{e^{-\beta k^\alpha}\}_{k=0}^\infty$, then the Gevrey type space $G^{\alpha,\beta}({\Bbb S}^{d})$ is just the multiplier space $H^{\Lambda_{\alpha,\beta}}({\Bbb S}^{d})$. \end{defn} \begin{rem} M. Gevrey \cite{G} introduced in 1918 the classes of smooth functions on $\Bbb R^d$ that are nowadays called Gevrey classes. They have played an important role in study of partial differential equation. A standard reference on Gevrey spaces is Rodino¡¯s book \cite{R}. The Gevrey spaces on $d$-dimensional torus were introduced and investigated in \cite{KMU}. Our definition is a natural generalization from $\Bbb R^d$ to ${\Bbb S}^{d}$. When $d=1$, the unit ball of $G^{\alpha,\beta}({\Bbb S}^{d})$ recedes to the class $A_q^{\tau,b}$ with $q=2, \ b=\alpha$ and $\tau=\beta$ introduced in \cite[p. 73]{Te}. When $\alpha=1$, the action of the Gevrey type multiplier $T^{\Lambda_{1,\beta}}$ on $f\in L_2({\Bbb S}^{d})$ is just the Possion integral of $f$ on the sphere (see \cite[pp. 34-35]{DaiX}). \end{rem} \subsection{Approximation numbers} \ Let ${H}$ and ${G}$ be two Hilbert spaces and $S$ be a compact linear operator from $H$ to $G$. The fact concerning the approximation numbers $a_n(S: H\to G)$ is well-known, see e.g. K\"onig \cite[Section 1.b]{K}, Pinkus \cite[Theorem IV.2.2]{P}, and Novak and Wo\'niakowski \cite[Corollary 4.12]{NW1}. The following lemma is a simple form of the above fact. Let $H$ be a separable Hilbert space, $\{e_k\}_{k=1}^\infty $ an orthonormal basis in $H$, and ${\tau}=\{\tau_k\}_{k=1}^\infty$ a sequence of positive numbers with $\tau_1\ge\tau_2\ge \dots\ge \tau_k\ge\cdots$. Let $H^{ \tau}$ be a Hilbert space defined by $$H^\tau=\Big\{x\in H\ \|\ \\|x\|H\\|=\Big(\sum_{k=1}^\infty \frac{\|(x,e_k)\|^2}{\tau_k^2}\Big)^{1/2}<\infty\Big\}.$$ According to \cite[Corollary 2.6]{P} we have the following lemma. \begin{lem}\label{l2.1} Let $H, \tau $ and $H^{\tau }$ be defined as above. Then $$a_n(I_d: H^\tau\to H)= \tau_n, \ \ n\in\Bbb N_+.$$ \end{lem} In the sequel, we always suppose that $\Lambda=\{\lambda_{k,d}\}_{k=0}^\infty$ is a non-increasing positive sequence with $\lim\limits_{k\to\infty}\lambda_{k,d}=0$. It follows from Definition \ref{d2.1} that the mutiplier space $H^\Lambda({\Bbb S}^{d})$ is of form $H^\tau=(L_2({\Bbb S}^{d}))^\tau$ with \begin{align} \{\tau_k\}_{k=0}^\infty=\{\lambda_{0,d},\underbrace{\lambda_{1,d},\cdots,\lambda_{1,d}}_{Z(d,1)},\underbrace{\lambda_{2,d}, \cdots,\lambda_{2,d}}_{Z(d,2)},\cdots,\underbrace{\lambda_{k,d},\cdots,\lambda_{k,d}}_{Z(d,k)},\cdots\}, \end{align} where $Z(d,m)$ and $C(d,m)$ are given in \eqref{2.1} and \eqref{2.2}. According to Lemma \ref{l2.1} we obtain \begin{thm} \label{t2.1} For $C(d,k-1)<n\le C(d,k),\ k=0, 1,2,\dots,$ we have $$a_n(T^\Lambda: L_2({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))= a_n(I_d: H^\Lambda({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=\lambda_{k,d}, $$where we set $C(d,-1)=0$. Specially, let $r>0$, $\square\in\{,+,\#,-\}$, and $0<\alpha,\beta<\infty$. Then for $C(d,k-1)<n\le C(d,k),\ k=0,1,2,\dots,$ we have \begin{equation}\label{2.3} a_n(I_d: H^{r,\square}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=r_{k,d}^\square,\end{equation} and \begin{equation}\label{2.4} a_n(I_d: G^{\alpha,\beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=e^{-\beta k^\alpha},\end{equation}where the definitions of $H^{r,\square}({\Bbb S}^{d}),\ r_{k,d}^\square$, $\square\in\{,+,\#,-\}$ are given in Definition \ref{d2.2}, and the definition of $G^{\alpha,\beta}({\Bbb S}^{d})$ is in Definition \ref{d2.3}. \end{thm} \section{Strong equivalences of approximation numbers} This section is devoted to giving strong equivalence of the approximation numbers of the Sobolev embeddings and the Gevrey type embeddings on the sphere. \begin{thm}\label{t3.1}Suppose that $\lim\limits_{k\to\infty}\lambda_{k,d}k^s=1$ for some $s>0$. Then \begin{equation}\label{3.1}\lim_{n\to\infty}n^{s/d}a_n(I_d: H^\Lambda({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=\Big(\frac2{d\,!}\Big)^{s/d},\end{equation} Specially, for $r>0$ and $\square\in\{,+,\#,-\}$, we have \begin{equation}\label{3.2}\lim_{n\to\infty}n^{r/d}a_n(I_d: H^{r,\square}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=\Big(\frac2{d\,!}\Big)^{r/d}.\end{equation} \end{thm} \begin{proof}For $C(d,k-1)<n\le C(d,k),\ k=0,1,2,\dots,$ we have $$a_n(I_d: H^\Lambda({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=\lambda_{k,d}.$$ It follows that \begin{align}(C(d,k-1))^{s/d}\lambda_{k,d}< n^{s/d}a_n(I_d: H^\Lambda({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))\le (C(d,k))^{s/d}\lambda_{k,d}.\end{align} We recall from \eqref{2.2} that $$C(d,k)= \frac{(2k+d)(k+d-1)\,!}{k\,!\,d\,!}, $$which yields that $$\lim_{k\to\infty}\frac{C(d,k)}{k^d}=\lim_{k\to\infty}\frac{C(d,k-1)}{k^d}=\frac2{d\,!}.$$ Since \begin{align}&\lim_{k\to\infty}(C(d,k-1))^{s/d}\lambda_{k,d}=\lim_{k\to\infty}(C(d,k))^{s/d}\lambda_{k,d}\\ &\quad\ = \lim_{k\to\infty}\Big(\frac{C(d,k)}{k^d}\Big)^{s/d}k^s\lambda_{k,d}=\Big(\frac2{d\,!}\Big)^{s/d}, \end{align} we obtain \eqref{3.1}. Theorem \ref{t3.1} is proved. \end{proof} \begin{rem} One can rephrase \eqref{3.2} as strong equivalences $$a_n(I_d: H^{r,\square}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))\sim n^{-r/d}\Big(\frac2{d\,!}\Big)^{r/d} $$for $r>0$ and $\square\in\{,+,\#,-\}$. The novelty of Theorems \ref{t3.1} is that they give strong equivalences of $a_n(I_d: H^{r,\square}({\Bbb S}^{d})\to L_2({\Bbb S}^{d})) $ and provide asymptotically optimal constants, for arbitrary fixed $d$ and $r>0$. \end{rem} \begin{thm}\label{t3.3}Let $0<\alpha<1$, $\beta>0$, and $\gamma=\Big(\frac2{d\,!}\Big)^{-\alpha/d}$. Then we have \begin{equation}\label{3.3}\lim_{n\to \infty }e^{\beta \gamma n^{\alpha/d}}a_n(I_d: G^{\alpha, \beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=1.\end{equation}\end{thm} \begin{proof}It follows from \eqref{2.4} that for $C(d,k-1)<n\le C(d,k),\ k=0,1,2,\dots,$ \begin{equation} a_n\equiv a_n(I_d: G^{\alpha,\beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=e^{-\beta k^\alpha}.\end{equation}Therefore, we have \begin{equation}\label{3.4} e^{\beta \gamma (C(d,k-1))^{\alpha/d}}e^{-\beta k^\alpha}< e^{\beta \gamma n^{\alpha/d}}a_n\le e^{\beta \gamma (C(d,k))^{\alpha/d}} e^{-\beta k^\alpha}.\end{equation} Since for $0<\alpha<1$, $$\lim_{k\to\infty}(\gamma (C(d,k))^{\alpha/d} -k^\alpha)=\lim_{k\to\infty}k^\alpha\Big(\Big( \big(1+\frac d{2k}\big)\prod_{j=1}^{d-1}(1+\frac jk)\Big)^{\frac \alpha d}-1\Big)=0,$$ we get \begin{equation}\label{3.5}\lim_{n\to\infty}e^{\beta \gamma (C(d,k))^{\alpha/d}} e^{-\beta k^\alpha}=e^{\beta\lim\limits_{k\to\infty}(\gamma (C(d,k))^{\alpha/d} -k^\alpha)}=1.\end{equation} Similarly, we can show that $$\lim_{n\to\infty}e^{\beta \gamma (C(d,k-1))^{\alpha/d}} e^{-\beta k^\alpha}=1,$$which combining with \eqref{3.5} and \eqref{3.4}, yields \eqref{3.3}. Theorem \ref{t3.3} is proved. \end{proof} \begin{rem} One can rephrase \eqref{3.3} as a strong equivalence $$a_n(I_d: G^{\alpha, \beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))\sim e^{-\beta \gamma n^{\alpha/d}}$$ for $0<\alpha<1$ and $\beta>0$, where $\gamma=\Big(\frac2{d\,!}\Big)^{-\alpha/d}$. The novelty of Theorems \ref{t3.3} is that they give a strong equivalence of $ a_n(I_d: G^{\alpha, \beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))$ and provide asymptotically optimal constants, for arbitrary fixed $d$, $0<\alpha<1$, and $\beta>0$. \end{rem} \begin{rem}For $\alpha=1$, we have $$\lim_{n\to\infty}e^{\beta \gamma (C(d,k-1))^{\alpha/d}} e^{-\beta k^\alpha}=e^{ \frac{\beta(d-1)^2}{2d}}\neq e^{ \frac{\beta d}{2}}= \lim_{n\to\infty}e^{\beta \gamma (C(d,k))^{\alpha/d}} e^{-\beta k^\alpha}, $$ which means that the strong equivalence $$a_n(I_d: G^{\alpha, \beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))\sim e^{-\beta \gamma n^{\alpha/d}}$$ does not hold. However, we have the weak equivalence \begin{equation}\label{3.6}a_n(I_d: G^{\alpha, \beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))\asymp e^{-\beta (nd\,!/2)^{1/d}}, \end{equation} where the equivalence constants may depend on $d$, but not on $n$. For $\alpha>1$, there seems even no weak asymptotics of $a_n(I_d: G^{\alpha, \beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))$ as \eqref{3.6}. \end{rem} \section{Preasymptotics and asymptotics of the approximation numbers} This section is devoted to giving preasymptotics and asymptotics of the approximation numbers of the Sobolev embeddings and the Gevrey type embeddings on the sphere. \begin{lem}\label{l4.1} For $m\in \Bbb N $ and $d\in\Bbb N_+$ we have \begin{align}\label{4.1}\max\Big\{\big(1+\frac{m}{d}\big)^{d},\,\big(1+\frac{d}{m}\big)^{m}\Big\}\leq C(d,m)\le \min\Big\{e^d\big(1+\frac{m}{d}\big)^{d},\,e^m\big(1+\frac{d}{m}\big)^{m}\Big\}. \end{align} \end{lem} \begin{proof} We note that $$\binom{m+d}d\le C(d,m)=\binom{m+d}d \frac{2m+d}{m+d}\le 2\binom{m+d}d. $$ Using the inequality (see \cite[(3.6)]{KSU1}) \begin{align}\label{4.2} \binom{m+d}{d}\leq e^{d-1}{(1+\frac{m}{d})}^{d}, \end{align}we get the upper estimate of $C(d,m)$. Using the inequality (see \cite[(3.5)]{KSU1})$$\max\Big\{\big(1+\frac{m}{d}\big)^{d},\,\big(1+\frac{d}{m}\big)^{m}\Big\}\leq \binom{m+d}d, $$we get the lower estimate of $C(d,m)$. Lemma \ref{l4.1} is proved. \end{proof} \begin{thm}\label{t4.1} Let $r>0$. We have \begin{equation}\label{4.2} a_n(I_d: H^{r,}({\Bbb S}^{d})\to L_2({\Bbb S}^{d})) \asymp \left\{\begin{matrix} 1,\ \ \ &n=1,\\ d^{-r/2}, & \ \ 2\le n\leq d,\\ d^{-r/2}\Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r/2},&\ \ d\le n\le 2^d, \\ d^{-r}n^{-r/d},&\ \ n\ge 2^d, \end{matrix}\right. \end{equation} where the equivalence constants depend only on $r$, but not on $d$ and $n$. \end{thm} \begin{proof} We have for $n=1$, $$a_n(I_d: H^{r,}({\Bbb S}^{d})\to L_2({\Bbb S}^{d})) =1.$$ For $2\le n\le C(d,1)=d+2$, we have $$ a_n(I_d: H^{r,}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d}))=r^_{1,d}=(1+d^r)^{-1/2}\asymp d^{-r/2}.$$This means that $$a_n(I_d: H^{r,}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d}))\asymp d^{-r/2}\asymp \left\{\begin{matrix} d^{-r/2}, & 2\le n\leq d,\\ d^{-r/2}\Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r/2},& d\le n\le C(d,1). \end{matrix}\right.$$ For $C(d,m-1)<n\le C(d,m),\ 2\le m\le d$, we have \begin{equation}\label{4.3}a_n(I_d: H^{r,}({\Bbb S}^{d})\rightarrow L_2( {\Bbb S}^{d}))=(1+(m(m+d-1))^r)^{-1/2}\asymp m^{-r/2}d^{-r/2}.\end{equation} By \eqref{4.1} we get that $$n\le C(d,m)\le e^m(1+d/m)^m\le \Big(\frac{2ed}m\Big)^m.$$ It follows that $$ \log n\le m\log (2ed/m), $$ which implies \begin{equation}\label{4.4} m\ge \frac{\log n}{\log (2ed/m)}\end{equation} and $$\log \Big(\frac{2ed}{\log n}\Big)\ge \log\Big(\frac{2ed}{m\log (2ed/m)}\Big)=\log(\frac{2ed}m)-\log\Big(\log(\frac {2ed}m)\Big). $$ Using the inequality $x\ge 2\log x$ for $x\ge 2$, we obtain $$\log \Big(\frac{2ed}{\log n}\Big)\ge \frac12\log(\frac{2ed}m). $$ This combining with \eqref{4.4} yields \begin{equation}\label{4.5} m\ge \frac{\log n}{2\log (2ed/(\log n))}.\end{equation} On other hand, it follows from \eqref{4.1} that $$n>C(d,m-1)\ge \Big(1+\frac d{m-1}\Big)^{m-1}.$$ This yields $$m-1\le\frac{ \log n}{\log \big(1+\frac {d}{m-1}\big)}\le\log n.$$ It follows that $$m\le \frac{ \log n}{\log \big(\frac {2d}{\log n}\big)}+1,$$ which combining with \eqref{4.5}, leads to \begin{equation}\label{4.6} m\asymp \frac{ \log n}{1+\log \big(\frac {d}{\log n}\big)}.\end{equation}It follows from \eqref{4.3} that for $C(d,m-1)<n\le C(d,m),\ 2\le m\le d$, \begin{align} a_n(I_d:\ H^{r,}({\Bbb S}^{d})\rightarrow L_2( {\Bbb S}^{d}))\asymp m^{-r/2}d^{-r/2}\asymp d^{-r/2}\Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r/2}.\end{align} Note that $$ 2^d\le C(d,d)\le (2e)^d$$ and for $2^d\le n\le (2e)^d$, $$ \frac{\log n} {\log(1+\frac{d}{\log n})}\asymp d\ \ {\rm and}\ \ n^{-r/d}\asymp 1.$$ We obtain that for $C(d,1)<n\le C(d,d)$, \begin{align} a_n(I_d:\ H^{r,}({\Bbb S}^{d})\rightarrow L_2( {\Bbb S}^{d}))&\asymp d^{-r/2}\Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r/2}\\ &\asymp \left\{\begin{matrix} d^{-r/2}\Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r/2},&\ \ d\le n\le 2^d, \\ d^{-r}n^{-r/d},&\ \ 2^d\le n\le C(d,d). \end{matrix}\right..\end{align} For $C(d,m-1)<n\le C(d,m),\ m>d$, we have \begin{equation}\label{4.7}a_n(I_d:\, H^{r,}({\Bbb S}^{d})\rightarrow L_2( {\Bbb S}^{d}))=(1+(m(m+d-1))^r)^{-1/2}\asymp m^{-r}.\end{equation} It follows from \eqref{4.1} that \begin{align}\frac md\le 1+\frac{m-1}d \le (C(d,m-1))^{\frac 1d}\le n^{\frac 1d}\le (C(d,m))^{\frac 1d}\le e(1+\frac md)\le \frac{2e m}d. \end{align} Therefore, we get \begin{equation}\label{4.8}m\asymp dn^{1/d},\end{equation} which combining with \eqref{4.7} yields that for $n>C(d,d)$, $$a_n(I_d:\, H^{r,}({\Bbb S}^{d})\rightarrow L_2( {\Bbb S}^{d}))\asymp d^{-r}n^{-r/d}.$$The proof of Theorem \ref{t4.1} is complete. \end{proof} Using the same method as in the proof of Theorem \ref{t4.1}, we can obtain the following two theorems. \begin{thm}\label{t4.2} Let $r>0$. We have \begin{equation}\label{4.9} a_n(I_d: H^{r,+}({\Bbb S}^{d})\to L_2({\Bbb S}^{d})) \asymp \left\{\begin{matrix} 1,\ \ \ &n=1,\\ d^{-r/2}, & \ \ 2\le n\leq d,\\ d^{-r/2}\Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r/2},&\ \ d\le n\le 2^d, \\ d^{-r}n^{-r/d},&\ \ n\ge 2^d, \end{matrix}\right. \end{equation} \begin{equation}\label{4.10} a_n(I_d: H^{r,\#}({\Bbb S}^{d})\to L_2({\Bbb S}^{d})) \asymp \left\{\begin{matrix} 1, & \ \ 1\le n\leq d,\\ \Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r},&\ \ d\le n\le 2^d, \\ d^{-r}n^{-r/d},&\ \ n\ge 2^d, \end{matrix}\right. \end{equation} and \begin{equation}\label{4.11} a_n(I_d: H^{r,-}({\Bbb S}^{d})\to L_2({\Bbb S}^{d})) \asymp \left\{\begin{matrix} d^{-r},&\ \ 1\le n\le 2^d, \\ d^{-r}n^{-r/d},&\ \ n\ge 2^d, \end{matrix}\right. \end{equation} where all above equivalence constants depend only on $r$, but not on $d$ and $n$. \end{thm} \begin{thm}\label{t4.3} Let $\alpha,\beta>0$. We have \begin{equation}\label{4.12} \ln \big(a_n(I_d: G^{\alpha,\beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))\big) \asymp -\beta \left\{\begin{matrix} 1, & \ \ 1\le n\leq d,\\ \Big(\frac{\log n}{\log(1+\frac{d}{\log n})}\Big)^{\alpha},&\ \ d\le n\le 2^d, \\ d^{\alpha}n^{\alpha/d},&\ \ n\ge 2^d, \end{matrix}\right. \end{equation} where the equivalence constants depend only on $\alpha$, but not on $d$ and $n$. \end{thm} \section{Tractability analysis} Recently, there has been an increasing interest in $d$-variate computational problems with large or even huge $d$. Such problems are usually solved by algorithms that use finitely many information operations. In this paper, we limit ourselves to the worst case setting, information operation is defined as the evaluation of a continuous linear functional, and we deal with a Hilbert space setting (source and target space). The information complexity $n(\varepsilon, d)$ is defined as the minimal number of information operations which are needed to find an approximating solution to within an error threshold $\varepsilon$. A central issue is the study of how the information complexity depends on $\varepsilon^{-1}$ and $d$. Such problem is called the tractable problem. Nowadays tractability of multivariate problems is a very active research area (see \cite{NW1, NW2,NW3} and the references therein). Let ${H_d}$ and ${G_d}$ be two sequences of Hilbert spaces and for each $d\in \Bbb N_+$, $F_d$ be the unit ball of $H_d$. Assume a sequence of bounded linear operators (solution operators) $$S_d : H_d\rightarrow G_d$$ for all $d \in \Bbb N_+$. For $n\in \Bbb N_+$ and $f\in F_d$, $S_d f$ can be approximated by algorithms $$A_{n,d}(f)=\Phi _{n,d}(L_1(f),...,L_n(f)),$$ where $L_j,\ j=1,\dots,n$ are continuous linear functionals on $F_d$ which are called general information, and $\Phi _{n,d} : \Bbb R^n\rightarrow G_d$ is an arbitrary mapping. The worst case error $e(A_{n,d})$ of the algorithm $A_{n,d}$ is defined as $$e(A_{n,d})=\sup_{f\in F_d} \\|S_d(f)-A_{n,d}(f)\\|_{G_d}.$$ Furthermore, we define the $n$th minimal worst-case error as $$e(n,d )=\inf_{A_{n,d}}e(A_{n,d}),$$ where the infimum is taken over all algorithms using $n$ information operators $L_1,L_2,...,L_n$. For $n=0$, we use $A_{0,d}=0$. The error of $A_{0,d}$ is called the initial error and is given by $$e(0,d )=\sup_{f\in F_d}\\|S_d f\\|_{G_d}.$$ The $n$th minimal worst-case error $e(n,d)$ with respect to arbitrary algorithms and general information in the Hilbert setting is just the $n+1$-approximation number $a_{n+1}(S_d:H_d\to G_d)$ (see \cite[p. 118]{NW1}), i.e., $$e(n,d)=a_{n+1}(S_d:H_d\to G_d).$$ In this paper, we consider the embedding operators $S_d=I_d$ (formal identity operators). For $\varepsilon \in (0,1)$ and $d\in \Bbb N_+$, let $n(\varepsilon, d)$ be the information complexity which is defined as the minimal number of continuous linear functionals which are necessary to obtain an $\varepsilon -$approximation of $I_d$, i.e., $$n(\varepsilon ,d)=\min\{n\,\|\,e(n,d)\leq \varepsilon CRI_d \},$$ where $$CRI_d=\begin{cases} 1,&\ \ \text{for the absolute error criterion},\\ e(0,d), &\ \ \text{for the normalized error criterion}.\end{cases}$$ Next, we list the concepts of tractability below. We say that the approximation problem is $\bullet$ \textbf {weakly tractable}, if \begin{equation}\label{t1.2} \lim_{\varepsilon ^{-1}+d\rightarrow \infty }\frac{\ln n(\varepsilon ,d)}{\varepsilon ^{-1}+d}=0. \end{equation}Otherwise, the approximation problem is called intractable. $\bullet$ \textbf {uniformly weakly tractable}, if for all $s,t>0$ \begin{equation}\label{t1.3} \lim_{\varepsilon ^{-1}+d\rightarrow \infty }\frac{\ln n(\varepsilon ,d)}{(\varepsilon ^{-1})^{s }+d^{t }}=0. \end{equation} $\bullet$ \textbf {quasi-polynomially tractable}, if there exist two positive constants $C,\, t$ such that for all $d\in \Bbb N_+, \ \varepsilon \in(0,1)$, \begin{equation}\label{t1.4} n(\varepsilon ,d)\leq C\exp(t(1+\ln\varepsilon ^{-1})(1+\ln d)). \end{equation} The infimum of $t$ satisfying \eqref{t1.4} is called the exponent of quasi-polynomial tractability and is denoted by $t^{\rm qpol}$. $\bullet$ \textbf {polynomially tractable}, if there exist non-negative numbers $C, p$ and $q$ such that for all $d\in \Bbb N_+, \ \varepsilon \in(0,1)$, \begin{equation}\label{t1.5} n(\varepsilon ,d )\leq Cd^q(\varepsilon ^{-1})^p. \end{equation} $\bullet$ \textbf {strongly polynomially tractable}, if there exist non-negative numbers $C$ and $p$ such that for all $d\in \Bbb N_+,\ \varepsilon \in (0,1)$, \begin{equation}\label{t1.6} n(\varepsilon ,d )\leq C(\varepsilon ^{-1})^p. \end{equation} Of course, the latter tractability implies the former tractability. $\bullet$ The approximation problem suffers from \textbf {the curse of dimensionality}, if there exist positive numbers $C, \varepsilon _0, \gamma $ such that for all $0<\varepsilon\leq \varepsilon _{0}$ and infinitely many $d\in \Bbb N_+$, \begin{equation}\label{1.7} n(\varepsilon ,d)\geq C(1+\gamma )^{d}. \end{equation} Recently, Siedlecki and Weimar introduced the notion of $(s,t)$-weak tractability in \cite{SW}. If for some fixed $s,\ t>0$ it holds \begin{equation} \lim_{\varepsilon ^{-1}+d\rightarrow \infty }\frac{\ln n(\varepsilon ,d)}{(\varepsilon ^{-1})^{s}+d^{t}}=0, \end{equation} then the approximation problem is called $(s, t)$-weakly tractable. Clearly, $(1,1)$-weak tractability is just weak tractability, whereas the approximation problem is uniformly weakly tractable if it is $(s,t)$-weakly tractable for all positive $s$ and $t$ (see \cite{S}). Also, if the approximation problem suffers from the curse of dimensionality, then for any $s>0, \ 0<t\le1,$ it is not $(s,t)$-weakly tractable. We introduce the following lemma which is used in the proofs of main results. \begin{lem}\label{l5.1} Let $m,d\in\Bbb N_+$, $s>0,\ t>1$. Then we have $$\lim_{m+d\to\infty}\frac{m\ln(m+d)}{m^t+d^s}=\lim_{m+d\to\infty}\frac{d\ln(m+d)}{m^s+d^t}=0.$$ \end{lem} {\begin{proof}We set $\gamma=s/t$. Let $x=m+d^{\gamma}$. Then $m+d\to\infty$ if and only if $x\to\infty$. If $\gamma\ge 1$, then $m+d\le x$, and if $\gamma <1$, then $$m+d\le m^{1/\gamma}+d\le (m+d^\gamma)^{1/\gamma}=x^{1/\gamma}.$$It follows that $$m\ln(m+d)\le \max\{1,1/\gamma\} x\ln x.$$Using the inequality $$(a+b)^p\le 2^p(a^p+b^p),\ a,b\ge0, p>0,$$ we get $$m^t+d^s\ge 2^{-t}(m+d^{s/t})^t=2^{-t}x^t.$$ Hence, we have $$0\le \lim_{m+d\to\infty}\frac{m\ln(m+d)}{m^t+d^s}\le\lim_{x\to\infty}\frac{\max\{1,1/\gamma\} x\ln x}{2^{-t}x^t}=0,$$which means that $$ \lim_{m+d\to\infty}\frac{m\ln(m+d)}{m^t+d^s}=0. $$ Similarly, we can prove $$\lim\limits_{m+d\to\infty}\frac{d\ln(m+d)}{m^s+d^t}=0.$$ Lemma \ref{l5.1} is proved. \end{proof} We remark that if $e(0,d)=\\|I_d\\|=1$, then the normalized error criterion and the absolute error criterion coincide. We emphasize that $$e(0,d)=1$$ for the approximation problems $$I_d: H^{r,\square}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d}),\ \ r>0,\ \square\in \{,+,\#\},$$ and $$ I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d}),\ \ \alpha,\beta>0.$$ However, we have $$e(0,d)=\big(\frac{d-1}2\big)^{-r}$$ for the approximation problems $$I_d: H^{r,-}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d}),\ \ r>0.$$ \begin{thm}\label{t5.1} Let $r>0$ and $s,\,t>0$. Then (1) the approximation problems $$I_d: H^{r,\square}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d}), \ \ \square\in \{,+,\#\}$$ and for the absolute error criterion the approximation problem $$I_d: H^{r,-}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$$ are $(s, t)$-weakly tractable if and only if $r>1/s$ and $t>0$ or $s>0$ and $t>1$. Specially, for the absolute error criterion the approximation problems $I_d: H^{r,\square}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d}), \ \square\in \{,+,\#,-\}$ are weakly tractable if and only if $r>1$, not uniformly weakly tractable, and do not suffer from the curse of dimensionality. (2) for the normalized error criterion, the approximation problem $$I_d: H^{r,-}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$$ suffers from the curse of dimensionality. \end{thm} \begin{proof} (1) First we show that for the absolute error criterion the approximation problems $I_d: H^{r,\square}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ are not $(s, t)$-weakly tractable if $0<s\le 1/r$ and $0<t\le1$, where $r>0$,\ $\square\in \{,+,\#,-\}$. We note that $$e(n,d)=a_{n+1}(I_d: H^{r,\square}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})).$$It follows from Theorem \ref{t2.1} that for $C(d,m-1)<n+1\le C(d,m),\ m\ge1 $, we have \begin{equation}e(n,d)=r_{m,d}^{\square},\end{equation}where $r_{m,d}^{\square}$ are given in Definition \ref{d2.2}. This implies that $$e(C(d,d)-1,d)=r_{d,d}^\square.$$ Choose $$\varepsilon=\varepsilon_d=r_{d+1,d}^\square<r_{d,d}^\square.$$ Then $$n(\varepsilon_d,d)=\inf\{n\in\Bbb N\ \|\ e(n,d)\le \varepsilon_d\}\ge C(d,d)\ge 2^d.$$If $0<s\le 1/r$ and $0<t\le1$, then we have $$\lim_{1/\varepsilon_d+d\to\infty}\frac{\ln (n(\varepsilon_d,d))}{(\varepsilon_d)^{-s}+d^t}\ge \lim_{d\to\infty}\frac {d}{(r_{d+1,d}^\square)^{-1/r}+d}\neq0,$$ which implies that for the absolute error criterion the approximation problems $I_d: H^{r,\square}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ are not $(s, t)$-weakly tractable if $0<s\le 1/r$ and $0<t\le1$. Next we show that if $s>1/r$ and $t>0$ or $s>0$ and $t>1$, then for the absolute error criterion the approximation problems $I_d: H^{r,\square}({\Bbb S}^{d}) \rightarrow L_2({\Bbb S}^{d})$ are $(s, t)$-weakly tractable. Let $0<\varepsilon<1$ be given and select $m\in\Bbb N_+$ such that $$r_{m,d}^\square\le\varepsilon< r_{m-1,d}^\square.$$Since \begin{equation}e(n,d)=r_{m,d}^\square\end{equation} for $C(d,m-1)<n+1\le C(d,m)$, we get $$n(\varepsilon,d)\le n(r_{m,d}^\square,d)=\inf\{n\in\Bbb N\ \|\ e(n,d)\le r_{m,d}^\square\}=C(d,m-1)< C(d,m).$$ For $s>0$ and $t>1$, it follows from \eqref{4.1} and Lemma \ref{l5.1} that $$\lim_{\frac 1\varepsilon+d\to\infty}\frac{\ln n(\varepsilon,d)}{\varepsilon^{-s}+d^t}\le \lim_{m+d\to\infty}\frac {d(1+\ln(\frac{m+d}d))}{(r_{m-1,d}^\square)^{-s}+d^t}\le \lim_{m+d\to\infty}\frac {d(1+\ln({m+d}))}{(m-1)^{rs}+d^t}=0.$$ For $s>1/r$ and $t>0$, it follows from \eqref{4.1} and Lemma \ref{l5.1} that $$\lim_{\frac 1\varepsilon+d\to\infty}\frac{\ln n(\varepsilon,d)}{\varepsilon^{-s}+d^t}\le \lim_{m+d\to\infty}\frac {m(1+\ln(\frac{m+d}m))}{(r_{m-1,d}^\square)^{-s}+d^t}\le \lim_{m+d\to\infty}\frac {m(1+\ln({m+d}))}{(m-1)^{rs}+d^t}=0.$$ Hence, for the absolute error criterion the approximation problems $I_d: H^{r,\square}({\Bbb S}^{d}) \rightarrow L_2({\Bbb S}^{d})$ are $(s, t)$-weakly tractable if and only if $s>1/r$ and $t>0$ or $s>0$ and $t>1$. (2) It follows from Theorem \ref{t4.2} that for $0\le n\le 2^d$, $$e(n,d)=a_{n+1}(I_d: H^{r,-}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d}))\asymp d^{-r}.$$ This means that there is a positive constant $c$ depending only on $r$ such that $$e(2^d,d)\ge c\, e(0,d).$$ Choose $\varepsilon\in(0,c)$. Then for the normalized criterion we have $$n(\varepsilon,d)=\min\{n\,\|\,e(n,d)\leq \varepsilon\, e(0,d)\}\ge 2^d.$$ Hence, for the normalized error criterion the approximation problem $I_d: H^{r,-}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ suffers from the curse of dimensionality. Theorem \ref{t5.1} is proved. \end{proof} \begin{thm}\label{t5.2} Let $\alpha,\beta>0$. Then the approximation problem $$I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$$ (1) is uniformly weakly tractable. (2) is not polynomially tractable. (3) is quasi-polynomially tractable if and only if $\alpha\ge 1$ and the exponent of quasi-polynomial tractability is $$t^{\rm qpol}=\sup_{m\in \Bbb N}\frac m {1+\beta m^\alpha},\ \ \alpha\ge1.$$Specially, if $\alpha=1$, then $t^{\rm qpol}=\frac1\beta$. \end{thm} \begin{proof} (1) For $0<\varepsilon<1$, we choose $m\in\Bbb N$ such that $$e^{-\beta(m+1)^{\alpha}}\le \varepsilon<e^{-\beta m^{\alpha}}.$$Since for $C(d,m)<n+1\le C(d,m+1),\ m\in\Bbb N $, \begin{equation}e(n,d)=a_{n+1}(I_d:G^{\alpha,\beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=e^{-\beta(m+1)^{\alpha}},\end{equation}we get $$n(\varepsilon,d)\le n(e^{-\beta(m+1)^{\alpha}}, d)=C(d,m).$$ For any $s,t>0$, by \eqref{4.1} we have $$\frac{\ln(n(\varepsilon,d))}{\varepsilon^{-s}+d^t}\le \frac{\ln(C(d,m))}{e^{s\beta m^\alpha}+d^t}\le \frac{m(1+\ln(m+d))}{e^{s\beta m^\alpha}+d^t}.$$ We note that $$\lim_{m\to\infty}\frac{m^2}{e^{s\beta m^\alpha}}=\lim_{x\to+\infty}\frac{x^{2/\alpha}}{e^{s\beta x}}=0.$$ This implies that there exists a positive constant $c$ such that $$e^{s\beta m^\alpha}\ge c m^2. $$It follows from Lemma \ref{l5.1} that $$\lim_{\frac1\varepsilon+d\to\infty}\frac{\ln(n(\varepsilon,d))}{\varepsilon^{-s}+d^t}\le \lim_{m+d\to\infty}\frac {m\ln(m+d)}{cm^2+d^t}=0.$$ Hence, the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is uniformly weakly tractable for any $\alpha,\beta>0$. (2) For any $p,q>0$, we choose $$\varepsilon_d=e^{-\beta (m_d+1)^\alpha},\ \ m_d=[(\frac q{\beta p}\ln d)^{1/\alpha}]-1,$$where $[x]$ denotes the largest integer not exceeding $x\in\Bbb R$. Such selection is to make $$(\varepsilon_d)^{-p}\le d^q.$$ Since for $C(d,m_d)<n+1\le C(d,m_d+1),\ m\in\Bbb N $, \begin{equation}e(n,d)=a_{n+1}(I_d:G^{\alpha,\beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=\varepsilon_d,\end{equation}we get $$n(\varepsilon_d,d)=C(d,m_d)\ge (1+\frac d{m_d})^{m_d}\ge (\frac d{m_d})^{m_d}=e^{m_d(\ln d-\ln m_d)} .$$ Since $\lim\limits_{d\to\infty}m_d=+\infty$ and $\lim\limits_{d\to\infty}\frac{\ln m_d}{\ln d}=0$, we get for sufficiently large $d$, $$\ln d-\ln m_d\ge \frac12\ln d\ \ {\rm and}\ \ \frac {m_d}2\ge 2q+1.$$ It follows that $$\lim_{d\to\infty}\frac{n(\varepsilon_d,d)}{(\varepsilon_d)^{-p}d^q}\ge \lim_{d\to\infty}\frac{e^{(2q+1)\ln d}}{d^{2q}}=\lim_{d\to\infty}d= +\infty,$$which implies that \eqref{t1.5} does not hold. Hence, the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is not polynomially tractable for any $\alpha,\beta>0$. (3) Let $0<\alpha<1$ and $\beta>0$. We choose $\varepsilon_d$ and $m_d$ as above, i.e., $$\varepsilon_d=e^{-\beta (m_d+1)^\alpha},\ \ m_d=[(\frac q{\beta p}\ln d)^{1/\alpha}]-1.$$ Then for sufficiently large $d$, $$n(\varepsilon_d,d)=C(d,m_d)\ge e^{\frac12m_d\ln d}=d^{m_d/2}.$$ It follows that for any $t>0$, $$\lim_{d\to\infty}\frac{n(\varepsilon_d,d)}{e^{t(1+\ln d)(1+\ln \frac 1{\varepsilon_d})}}\ge \lim_{d\to\infty}\frac{d^{m_d/2}}{e^{4t 2^\alpha\beta (m_d)^\alpha\ln d}}=\lim_{d\to\infty}d^{\frac{m_d}2-42^\alpha t\beta (m_d)^\alpha}=+\infty, $$which means that \eqref{t1.4} is not valid. Hence, the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is not quasi-polynomially tractable if $0<\alpha<1$. Let $\alpha\ge 1$ and $\beta>0$. We set \begin{equation}\label{5.2}t_0=\sup_{m\in \Bbb N}\frac m {1+\beta m^\alpha}.\end{equation} For any $0<\varepsilon<1$, we choose $m\in\Bbb N$ such that $$e^{-\beta(m+1)^{\alpha}}\le \varepsilon<e^{-\beta m^{\alpha}}.$$Then we have $$n(\varepsilon,d)\le n(e^{-\beta(m+1)^{\alpha}}, d)=C(d,m).$$ We note from \eqref{5.2} that $$\sup_{d\in\Bbb N_+,\ \varepsilon\in(0,1)}\frac{n(\varepsilon,d)}{e^{t_0(1+\ln d)(1+\ln \frac 1{\varepsilon})}}\le \sup_{d\in\Bbb N_+,\ m\in \Bbb N} \frac{C(d,m)}{(ed)^{t_0(1+\beta m^\alpha)}}\le \sup_{d\in\Bbb N_+,\ m\in \Bbb N} \frac{C(d,m)}{(ed)^m}. $$ From \eqref{2.2} we get $$C(d,0)=1,\ C(d,1)=d+2,\ C(1,m)=2m+1,$$ which yields $$\sup_{d\in\Bbb N_+,\ m=0,1} \frac{C(d,m)}{(ed)^m}\le 2 \ \ {\rm and}\ \ \sup_{m\in \Bbb N,\ d=1} \frac{C(d,m)}{(ed)^m}\le 2.$$ If $m,d\ge 2$, then by \eqref{4.1} we get $$\frac{C(d,m)}{(ed)^m}\le \frac{e^m(1+d/m)^m}{(ed)^m}=(\frac1d+\frac1m)^m\le1.$$ This means that for any $d\in\Bbb N_+$ and any $\varepsilon\in(0,1)$, $$n(\varepsilon,d)\le 2e^{t_0(1+\ln d)(1+\ln \frac 1{\varepsilon})}.$$Hence, the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is quasi-polynomially tractable if $\alpha\ge 1$ and the exponent $t^{\rm qpol}$ of quasi-polynomial tractability satisfies $$t^{\rm qpol}\le t_0=\sup_{m\in \Bbb N}\frac m {1+\beta m^\alpha}.$$ Let $\alpha=1$. Then $t_0=1/\beta$. For $t<t_0=1/\beta$, we choose $\varepsilon_d$ and $m_d$ as above. Such $m_d$ satisfies $\lim\limits_{d\to\infty}m_d=+\infty$ and $\lim\limits_{d\to \infty}\frac{\ln m_d}{\ln d}=0$. Then $n(\varepsilon_d,d)=C(d,m_d)$. It follows that \begin{align}\lim_{d\to \infty}\frac{n(\varepsilon_d,d)}{e^{t(1+\ln d)(1+\ln \frac1{\varepsilon_d})}}&=\lim_{d\to \infty}\frac{C(d,m_d)}{e^{t(1+\ln d)(1+\beta (m_d+1))}}\\ &\ge \lim_{d\to \infty}\frac{(1+d/m_d)^{m_d}}{(ed)^{t+t\beta (m_d+1)}}. \end{align} Let $\gamma$ be such that $\gamma\in (t\beta,1)$. Then for sufficiently large $d$, we have $$\ln d-\ln m_d\ge \gamma \ln d,$$ which means that for sufficiently large $d$, $$(1+d/m_d)^{m_d}\ge e^{m_d(\ln d-\ln m_d)}\ge e^{\gamma m_d\ln d}=d^{\gamma m_d}.$$ Using the facts $$a^b=d^{b\frac{\ln a}{\ln d}},\ \ \gamma-t\beta>0, \ \ {\rm and}\ \ \lim_{d\to\infty}m_d=+\infty,$$we get \begin{align}\lim_{d\to \infty}\frac{n(\varepsilon_d,d)}{e^{t(1+\ln d)(1+\ln\frac1{\varepsilon_d})}}&\ge\lim_{d\to\infty} \frac{d^{\gamma m_d}}{(ed)^{t+t\beta (m_d+1)}}\\ &=\lim_{d\to\infty}d^{(\gamma-t\beta-\frac{t\beta}{\ln d})m_d-(t+t\beta)(1+\frac1{\ln d})}=+ \infty, \end{align}which means that \eqref{t1.4} is not true with $t<t_0=1/\beta$. Hence, we have for $\alpha=1$, $$t^{\rm qpol}=t_0=1/\beta.$$ Let $\alpha>1$. Since $\lim\limits_{m\to\infty}\frac m{1+\beta m^\alpha}=0$, there exists a positive integer $m_0$ depending only on $\alpha>1$ and $\beta>0$ such that $$t_0=\sup_{m\in \Bbb N}\frac m {1+\beta m^\alpha}=\frac {m_0} {1+\beta (m_0)^\alpha}.$$ We choose $\varepsilon_d\in (0,1)$ such that $$e^{-\beta(m_0+1)^{\alpha}}\le \varepsilon_d<e^{-\beta (m_0)^{\alpha}}\ \ {\rm and}\ \ \lim_{d\to\infty}\varepsilon_d=e^{-\beta (m_0)^{\alpha}}.$$ Since for $C(d,m)<n+1\le C(d,m+1),\ m\in\Bbb N $, \begin{equation}e(n,d)=a_{n+1}(I_d:G^{\alpha,\beta}({\Bbb S}^{d})\to L_2({\Bbb S}^{d}))=e^{-\beta(m+1)^{\alpha}},\end{equation}we get $$n(\varepsilon_d,d)=\inf\{n\in\Bbb N\ \|\ e(n,d)\le \varepsilon_d\}=C(d,m_0).$$ We have for any $t<t_0=\frac {m_0} {1+\beta (m_0)^\alpha}$, \begin{align}\lim_{d\to \infty}\frac{n(\varepsilon_d,d)}{e^{t(1+\ln d)(1+\ln\frac1{\varepsilon_d})}}=\lim_{d\to\infty} \frac{C(d,m_0)}{(ed)^{t(1+\ln \frac 1{\varepsilon_d})}}\ge \lim_{d\to\infty} \frac {d^{m_0-t(1+\ln\frac1{\varepsilon_d})\frac{\ln ed}{\ln d}}}{m_0\,!}. \end{align}Noting that $$\lim_{d\to\infty}\Big(m_0-t(1+\ln\frac1{\varepsilon_d})\frac{\ln ed}{\ln d}\Big)=(1+\beta (m_0)^\alpha)(t_0-t)>0, $$ we obtain $$\lim_{d\to \infty}\frac{n(\varepsilon_d,d)}{e^{t(1+\ln d)(1+\ln\frac1{\varepsilon_d})}}=+\infty,$$ which means that \eqref{t1.4} is not true with $t<t_0$. Hence, we have for $\alpha>1$, $$t^{\rm qpol}=t_0.$$ The proof of Theorem \ref{t5.2} is complete. \end{proof} \begin{rem} From \eqref{4.12} we know that $$e(n,d)\le C_1(d)e^{-C_2(d)n^{\alpha/d}},$$where $C_1(d),\,C_2(d)$ are two constants depending on $d$ but not on $n$. This means that the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ has exponential convergence (see \cite{PW}, \cite{IKPW}). So we can consider exponential convergence tractability. Exponential convergence tractability has been considered in many papers (see for example, \cite{PW}, \cite{IKPW}, \cite{PPW}). For $t,s>0$, we say the approximation problem is $(t,\ln^s)$-weakly tractable (see \cite{PPW}), if \begin{equation} \lim_{\varepsilon ^{-1}+d\rightarrow \infty }\frac{\ln n(\varepsilon ,d)}{(\ln\varepsilon ^{-1})^s+d^t}=0. \end{equation}Otherwise, the approximation problem is called $(t,\ln^{s})$-intractable. Specially if $s=t=1$, $(t,\ln^{s})$-weak tractability is just exponential convergence-weak tractability. Similarly we can define exponential convergence-uniformly weak tractability. Let $n^{G^{\alpha,\beta}}(\varepsilon,d)$ and $n^{H^{\alpha,\#}}(\varepsilon,d)$ be the information-complexities of the approximation problems $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ and $I_d: H^{\alpha,\#}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$. For $0<\varepsilon<1$, we choose $m\in\Bbb N$ such that $$e^{-\beta(m+1)^{\alpha}}\le \varepsilon<e^{-\beta m^{\alpha}}.$$Then $$n^{G^{\alpha,\beta}}(\varepsilon,d)=C(d,m).$$ It follows that $$\lim_{\frac 1\varepsilon+d\to\infty}\frac{\ln(n^{G^{\alpha,\beta}}(\varepsilon,d))}{(\ln \varepsilon^{-1})^s+d^t}= \lim_{m+d\to\infty}\frac{\ln(C(d,m))}{\beta^s m^{s\alpha}+d^t}.$$ Similarly, for $0<\varepsilon<1$, let $k\in\Bbb N$ be such that $$(k+2)^{-\alpha}\le \varepsilon< (k+1)^{-\alpha}. $$Then we have $$n^{H^{\alpha,\#}}(\varepsilon,d)=C(d,k),$$and $$\lim_{\frac 1\varepsilon+d\to\infty}\frac{\ln(n^{H^{\alpha,\#}}(\varepsilon,d))}{ \varepsilon^{-s}+d^t}= \lim_{k+d\to\infty}\frac{\ln(C(d,k))}{ (k+1)^{s\alpha}+d^t}.$$ Clearly, $$\lim_{\frac 1\varepsilon+d\to\infty}\frac{\ln(n^{G^{\alpha,\beta}}(\varepsilon,d))}{(\ln \varepsilon^{-1})^s+d^t}=0\ \ {\rm if \ and \ only\ if}\ \ \lim_{\frac 1\varepsilon+d\to\infty}\frac{\ln(n^{H^{\alpha,\#}}(\varepsilon,d))}{ \varepsilon^{-s}+d^t}=0,$$which means that the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is $(t,\ln^s)$-weakly tractable if and only if the approximation problem $I_d: H^{\alpha,\#}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is $(s,t)$-weakly tractable, and if and only if $\alpha>1/s$ and $t>0$ or $s>0$ and $t>1$. Hence, the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is $(t,\ln^s)$-weakly tractable if and only if $\alpha>1/s$ and $t>0$ or $s>0$ and $t>1$. Specially, it is exponential convergence-weakly tractable if and only if $\alpha>1$. Using the same reasoning, we can prove that the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is exponential convergence-uniformly weakly tractable if and only if the approximation problem $I_d: H^{\alpha,\#}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is uniformly weakly tractable. However, the approximation problem $I_d: H^{\alpha,\#}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is not uniformly weakly tractable. Hence, the approximation problem $I_d: G^{\alpha,\beta}({\Bbb S}^{d})\rightarrow L_2({\Bbb S}^{d})$ is not exponential convergence-uniformly weakly tractable for any $\alpha, \beta>0$. \end{rem} \section{Asymtotics, Preasymtotics, and tractability on the ball} \subsection{Sobolev spaces and Gevrey type spaces on the ball} \ Let ${\Bbb B^d}=\{x\in\mathbb{R}^d:\ \ \|x\|\le 1\}$ denote the unit ball in $\mathbb{R}^d$, where $x\cdot y$ is the usual inner product, and $\|x\|=(x\cdot x)^{1/2}$ is the usual Euclidean norm. For the weight $W_\mu(x)=(1-\|x\|^2)^{\mu-1/2}\ (\mu\ge 0)$, denote by $L_{2,\mu}({\Bbb B^d}) \equiv L_2({\Bbb B^d}, W_\mu(x)\,dx)$ the space of measurable functions defined on ${\Bbb B^d}$ with the finite norm $$ \\|f\|L_{2,\mu}({\Bbb B^d}) \\|:=\Big(\int_{{\Bbb B^d}}\|f(x)\|^2\,W_\mu(x)dx\Big)^{1/2}.$$ When $\mu=1/2$, $W_\mu(x)\equiv 1$, and $L_{2,1/2}({\Bbb B^d})$ recedes to $L_2({\Bbb B^d})$. We denote by $\Pi_n^d({\Bbb B^d})$ the space of all polynomials in $d$ variables of degree at most $n$ restricted to ${\Bbb B^d}$, and by $\mathcal{V}_{n,\mu}^d({\Bbb B^d})$ the space of all polynomials of degree $n$ which are orthogonal to polynomials of low degree in $L_{2,\mu}({\Bbb B^d})$. Note that \begin{equation}\label{6.1} N(n,d):={\rm dim}\,\mathcal{V}_{n,\mu}^d({\Bbb B^d})=\binom {n+d-1} n.\end{equation} and \begin{equation}\label{6.2} D(n,d):={\rm dim}\,\Pi_n^d({\Bbb B^d})=\binom {n+d} n.\end{equation} It is well known (see \cite[p. 38 or p. 229]{DX} or \cite[p. 268]{DaiX}) that the spaces $\mathcal{V}_{n,\mu}^d({\Bbb B^d})$ are just the eigenspaces corresponding to the eigenvalues $-n(n+2\mu+d-1)$ of the second-order differential operator $$D_{\mu,d}:=\triangle-(x\cdot\nabla)^2-(2\mu+d-1)\,x\cdot \nabla,$$ where the $\triangle$ and $\nabla$ are the Laplace operator and gradient operator respectively. More precisely, $$D_{\mu,d} P=-n(n+2\mu+d-1)P\ {\rm for} \ P\in \mathcal{V}_{n,\mu}^d({\Bbb B^d}).$$ Also, the spaces $\mathcal{V}_{n,\mu}^d({\Bbb B^d})$ are mutually orthogonal in $L_{2,\mu}({\Bbb B^d})$ and \begin{equation} L_{2,\mu}({\Bbb B^d})= \bigoplus_{n=0}^\infty \mathcal{V}_{n,\mu}^d({\Bbb B^d}), \quad \ \ \ \ \Pi_n^d({\Bbb B^d})= \bigoplus_{k=0}^n \mathcal{V}_{n,\mu}^{d}({\Bbb B^d}) . \end{equation} Let $$\{\phi_{nk}\equiv \phi_{nk}^{d,\mu}\ \|\ k=1,\dots, N(n,d)\}$$ be a fixed orthonormal basis for $\mathcal{V}_{n,\mu}^d({\Bbb B^d})$ in $L_{2,\mu}({\Bbb B^d})$. Then $$\{\phi_{nk}\ \|\ k=1,\dots, N(n,d),\ n=0,1,2,\dots\}$$ is an orthonormal basis for $L_{2,\mu}({\Bbb B^d})$. Evidently, any $ f\in L_{2,\mu}({\Bbb B^d})$ can be expressed by its Fourier series \begin{equation}f=\sum_{n=0}^\infty Proj_{n,\mu} f=\sum_{n=0}^\infty{\sum_{k=1}^{N(n,d)}} \langle \phi_{nk},f\rangle_\mu\phi_{nk}, \end{equation} where $Proj_{n,\mu}(f)={\sum\limits_{k=1}^{N(n,d)}} \langle \phi_{nk},f\rangle_\mu\phi_{nk}$ is the orthogonal projection of $f$ from $L_{2,\mu}({\Bbb B^d})$ onto $\mathcal{V}_{n,\mu}^d({\Bbb B^d})$, and $$ \langle \phi_{nk},f\rangle_\mu :=\int_{{\Bbb B^d}} f(x) \phi_{nk}W_\mu(x)\, dx$$ are the Fourier coefficients of $f$. We have the following Parseval equality: $$\\|f\,\|\,L_{2,\mu}({\Bbb B^d})\\|=\Big(\sum_{n=0}^\infty{\sum_{k=1}^{N(n,d)}} \|\langle \phi_{nk},f\rangle_\mu\|^2\Big)^{1/2}.$$ \begin{defn}\label{d6.1}Let $\Lambda=\{\lambda_k\}_{k=0}^\infty$ be a non-increasing positive sequence with $\lim\limits_{k\to\infty}\lambda_k=0$, and let $T^\Lambda$ be a multiplier operator on $L_{2,\mu}({\Bbb B^d})$ defined by $$T^\Lambda (f)=\sum_{n=0}^\infty{\sum_{k=1}^{N(n,d)}} \lambda_n\langle \phi_{nk},f\rangle_\mu\phi_{nk}.$$ We define the multiplier space $H_{\mu}^\Lambda({\Bbb B^d})$ by \begin{align}H_{\mu}^\Lambda({\Bbb B^d})&=\Big\{T^\Lambda f\, \big\|\, f\in L_{2,\mu}({\Bbb B^d}) \ {\rm and }\ \\|T^\Lambda f\,\big\|\,H_{\mu}^\Lambda({\Bbb B^d})\\|=\\|f\|L_{2,\mu}({\Bbb B^d})\\|<\infty\Big\}\\&:=\Big\{f\in L_{2,\mu}({\Bbb B^d})\, \big\|\, \\|f\,\big\|\,H_{\mu}^\Lambda({\Bbb B^d})\\|:= \Big(\sum_{n=0}^{\infty}\frac1{\lambda_n^2}\sum_{k=1}^{N(n,d)} \|\langle \phi_{nk},f\rangle_\mu\|^2\Big)^{1/2}<\infty\Big\}. \end{align}\end{defn} Similar to the case on the sphere, we can define the Sobolev spaces $H^{r,\square}_\mu({\Bbb B^d}),\ r>0,\ \square\in\{,+,\#,-\}$ and the Gevrey type spaces $G_\mu^{\alpha,\beta}({\Bbb B^d}),\ \alpha,\beta>0$ analogously. However, in this section we deal only with the most important and interesting case $\mu=1/2$, and the corresponding spaces $H^{r,}({\Bbb B^d})$ and $G^{\alpha,\beta}({\Bbb B^d})$. We remark that there is no difference for the cases $\mu=1/2$ and $\mu\neq 1/2$ concerning with results about strong equivalences, asymptotics and preasymptotics, and tractability. We also remark that the corresponding results on $H^{r,\square}_\mu({\Bbb B^d}),\ \square\in\{+,\#,-\}$ is similar to the ones on $H^{r,\square}({\Bbb S}^{d}),\ \square\in\{+,\#,-\}$. The proofs go through with hardly any change. \begin{defn}\label{d6.2}Let $r>0$. The Sobolev space $H^{r,}({\Bbb B^d}) \equiv H_{1/2}^{r,}({\Bbb B^d})$ is the collection of all $f\in L_2({\Bbb B^d})$ such that $$\\|f\,\big\|\,H^{r,}({\Bbb B^d})\\|:= \Big(\sum_{n=0}^{\infty}\frac1{1+(n(n+d))^r}\sum_{k=1}^{N(n,d)} \|\langle \phi_{nk},f\rangle_{1/2}\|^2\Big)^{1/2}<\infty,$$ If we set $\tilde \Lambda^=\{\tilde r_{k,d}^\}_{k=0}^\infty,\ \tilde r_{k,d}^=(1+(k(k+d))^r)^{-1/2}$, then the Sobolev space $H^{r,}({\Bbb B^d})$ is just the multiplier space $H_{1/2}^{\tilde\Lambda^}({\Bbb B^d})$. \end{defn} \begin{rem} Given $r>0$, we define the fractional power $(-D_{1/2,d})^{r/2}$ of the operator $-D_{1/2,d}$ on $f$ by $$(-D_{1/2,d})^{r/2} (f)= \sum_{k=1}^\infty (k(k+d))^{r/2} Proj_{k,1/2}(f) $$ in the sense of distribution. Then for $f\in H^{r,}({\Bbb B^d})$, we have $$\\|f\,\big\|\,H^{r,}({\Bbb B^d})\\|= \Big(\\|f\,\|\,L_{2}({\Bbb B^d})\\|^2+\\|(-D_{1/2,d})^{r/2} (f)\,\|\,L_{2}({\Bbb B^d})\\|^2\Big)^{1/2}.$$ \end{rem} \begin{defn}\label{d6.3}Let $\alpha,\beta>0$. The Gevrey type space $G^{\alpha,\beta}({\Bbb B^d})\equiv G_{1/2}^{\alpha,\beta}({\Bbb B^d}) $ is the collection of all $f\in L_2({\Bbb B^d})$ such that $$\\|f\,\big\|\,G^{\alpha,\beta}({\Bbb B^d})\\|:= \Big(\sum_{n=0}^{\infty}e^{2\beta n^\alpha}\sum_{k=1}^{N(n,d)} \|\langle \phi_{nk},f\rangle_{1/2}\|^2\Big)^{1/2}<\infty.$$ If we set $\Lambda_{\alpha,\beta}=\{e^{-\beta k^\alpha}\}_{k=0}^\infty$, then the Gevrey type space $G^{\alpha,\beta}({\Bbb B^d})$ is just the multiplier space $H_{1/2}^{\Lambda_{\alpha,\beta}}({\Bbb B^d})$. \end{defn} \subsection{Strong equivalences on the ball} \ Let $\Lambda=\{\lambda_k\}_{k=0}^\infty$ be a non-increasing positive sequence with $\lim\limits_{k\to\infty}\lambda_k=0$. We note from Definition \ref{d6.1} that the mutiplier space $H_{\mu}^\Lambda({\Bbb B^d})\ (\mu\ge 0)$ is also of form $H^\tau=(L_{2,\mu}({\Bbb B^d}))^\tau$ with \begin{align} \{\tau_k\}_{k=0}^\infty=\{\lambda_{0,d},\underbrace{\lambda_{1,d},\cdots,\lambda_{1,d}}_{N(1,d)},\underbrace{\lambda_{2,d}, \cdots,\lambda_{2,d}}_{N(2,d)},\cdots,\underbrace{\lambda_{k,d},\cdots,\lambda_{k,d}}_{N(k,d)},\cdots\}, \end{align} where $N(m,d)$ and $D(m,d)$ are given in \eqref{6.1} and \eqref{6.2}. According to Lemma \ref{l2.1} we obtain \begin{thm} \label{t6.1}For $D(k-1,d)<n\le D(k,d),\ k=0,1,2,\dots,$ we have $$a_n(T^\Lambda: L_{2,\mu}({\Bbb B^d})\to L_{2,\mu}({\Bbb B^d}))=a_n(I_d: H_{\mu}^\Lambda({\Bbb B^d})\to L_{2,\mu}({\Bbb B^d}))=\lambda_{k,d},\ \ \mu\ge 0.$$ where we set $D(-1,d)=0$. Specially, let $r>0$ and $\alpha,\beta>0$. Then for $D(k-1,d)<n\le D(k,d),\ k=0,1,2,\dots,$ we have \begin{equation}\label{6.3}a_n(I_d: H^{r,}({\Bbb B^d})\to L_2({\Bbb B^d}))=(1+(k(k+d))^r)^{-1/2},\end{equation}and \begin{equation}\label{6.4}a_n(I_d: G^{\alpha,\beta}({\Bbb B^d})\to L_2({\Bbb B^d}))=e^{-\beta k^\alpha}.\end{equation} \end{thm} \begin{thm}\label{t6.2}Suppose that $\lim\limits_{k\to\infty}\lambda_{k,d}k^s=1$ for some $s>0$. Then \begin{equation}\label{6.5} \lim_{n\to\infty}n^{s/d} a_n(I_d: H_{\mu}^\Lambda({\Bbb B^d})\to L_{2,\mu}({\Bbb B^d}))=\Big(\frac1{d\,!}\Big)^{s/d},\ \mu\ge 0.\end{equation}Specially, we have for $r>0$, \begin{equation}\label{6.6}\lim_{n\to\infty}n^{r/d}a_n(I_d: H^{r,}({\Bbb B^d})\to L_2({\Bbb B^d}))=\Big(\frac1{d\,!}\Big)^{r/d}\end{equation} \end{thm} \begin{proof} The proof is similar to the one of \eqref{3.1}. For $D(k-1,d)<n\le D(k,d),\ k=0, 1,2,\dots,$ we have $$a_n(I_d: H_{\mu}^\Lambda({\Bbb B^d})\to L_{2,\mu}({\Bbb B^d}))=\lambda_{k,d},\ \mu\ge 0,$$where $D(k,d)=\binom{k+d}k$. It follows that $$(D(k-1,d))^{s/d}\lambda_{k,d}\le n^{s/d} a_n(I_d: W_{2,\mu}^\Lambda({\Bbb B^d})\to L_{2,\mu}({\Bbb B^d}))\le (D(k,d))^{s/d}\lambda_{k,d}. $$ Using the argument of \eqref{3.1} and noting that $\lim\limits_{k\to\infty}D(k,d)k^{-d}=\frac 1{d\,!} $, we get \eqref{6.5}. Theorem \ref{t6.2} is proved. \end{proof} \begin{rem} One can rephrase \eqref{6.6} as a strong equivalences $$a_n(I_d: H^{r,}({\Bbb B^d})\to L_2({\Bbb B^d}))\sim n^{-r/d}\Big(\frac1{d\,!}\Big)^{r/d} $$for $r>0$. The novelty of Theorems \ref{t6.2} is that they give a strong equivalence of $a_n(I_d: H^{r,}({\Bbb B^d})\to L_2({\Bbb B^d})) $ and provide asymptotically optimal constants, for arbitrary fixed $d$ and $r>0$. \end{rem} \begin{thm}\label{t6.3}Let $0<\alpha<1$, $\beta>0$, and $\tilde \gamma=\Big(\frac1{d\,!}\Big)^{-\alpha/d}$. Then we have \begin{equation}\label{6.7}\lim_{n\to \infty }e^{\beta \tilde\gamma n^{\alpha/d}}a_n(I_d: G^{\alpha, \beta}({\Bbb B^d})\to L_2({\Bbb B^d}))=1.\end{equation}\end{thm} \begin{proof}It follows from \eqref{6.4} that for $D(k-1,d)<n\le D(k,d),\ k=0,1,2,\dots,$ \begin{equation} a_n\equiv a_n(I_d: G^{\alpha,\beta}({\Bbb B^d})\to L_2({\Bbb B^d}))=e^{-\beta k^\alpha}.\end{equation}Therefore, we have \begin{equation} e^{\beta \tilde\gamma (D(k-1,d))^{\alpha/d}}e^{-\beta k^\alpha}< e^{\beta \tilde\gamma n^{\alpha/d}} a_n\le e^{\beta \tilde\gamma (D(k,d))^{\alpha/d}} e^{-\beta k^\alpha}.\end{equation} Since for $0<\alpha<1$, $$\lim_{k\to\infty}(\tilde \gamma (D(k,d))^{\alpha/d} -k^\alpha)=\lim_{k\to\infty}k^\alpha\Big(\Big( \prod_{j=1}^{d}(1+\frac jk)\Big)^{\frac \alpha d}-1\Big)=0,$$ similar to the proof of \eqref{3.3}, we get \eqref{6.7}. Theorem \ref{t6.3} is proved. \end{proof} \begin{rem} One can rephrase \eqref{6.7} as a strong equivalence $$a_n(I_d: G^{\alpha, \beta}({\Bbb B^d})\to L_2({\Bbb B^d}))\sim e^{-\beta \tilde\gamma n^{\alpha/d}}$$ for $0<\alpha<1$ and $\beta>0$, where $\tilde\gamma=\Big(\frac1{d\,!}\Big)^{-\alpha/d}$. The novelty of Theorems \ref{t6.3} is that they give a strong equivalence of $ a_n(I_d: G^{\alpha, \beta}({\Bbb B^d})\to L_2({\Bbb B^d}))$ and provide asymptotically optimal constants, for arbitrary fixed $d$, $0<\alpha<1$, and $\beta>0$. \end{rem} \subsection{Preasymptotics and asymptotics on the ball} \ Let $r>0$ and $\alpha,\beta>0$. Then for $D(m-1,d)<n\le D(m,d),\ m=0,1,2,\dots,$ we have \begin{equation}a_n(I_d: H^{r,}({\Bbb B^d})\to L_2({\Bbb B^d}))=(1+(m(m+d))^r)^{-1/2},\end{equation}and \begin{equation}a_n(I_d: G^{\alpha,\beta}({\Bbb B^d})\to L_2({\Bbb B^d}))=e^{-\beta m^\alpha},\end{equation} where $D(-1,d)=0$ and $D(m,d)=\binom {d+m}d$. We note that \begin{align}\max\Big\{\big(1+\frac{m}{d}\big)^{d},\,\big(1+\frac{d}{m}\big)^{m}\Big\}\leq D(m,d) \le \min\Big\{e^d\big(1+\frac{m}{d}\big)^{d},\,e^m\big(1+\frac{d}{m}\big)^{m}\Big\}. \end{align} Using the same reasoning as in the proof of Theorem \ref{t4.1}, we obtain that for $D(m-1,d)<n\le D(m,d),\ 1\le m \le d,$ $$ m\asymp \frac{ \log n}{1+\log \big(\frac {d}{\log n}\big)},$$and for $D(m-1,d)<n\le D(m,d),\ m> d,$ $$m\asymp dn^{1/d}.$$By the above two equivalences we can obtain the following two theorems. \begin{thm}\label{t6.4} Let $r>0$. We have \begin{equation}\label{6.8} a_n(I_d: H^{r,}({\Bbb B^d})\to L_2({\Bbb B^d})) \asymp \left\{\begin{matrix} 1,\ \ \ &n=1,\\ d^{-r/2}, & \ \ 2\le n\leq d,\\ d^{-r/2}\Big(\frac{\log(1+\frac{d}{\log n})}{\log n}\Big)^{r/2},&\ \ d\le n\le 2^d, \\ d^{-r}n^{-r/d},&\ \ n\ge 2^d, \end{matrix}\right. \end{equation} where the equivalence constants depend only on $r$, but not on $d$ and $n$. \end{thm} \begin{thm}\label{t6.5} Let $\alpha,\beta>0$. We have \begin{equation}\label{6.9} \ln \big(a_n(I_d: G^{\alpha,\beta}({\Bbb B^d})\to L_2({\Bbb B^d}))\big) \asymp -\beta \left\{\begin{matrix} 1, & \ \ 1\le n\leq d,\\ \Big(\frac{\log n}{\log(1+\frac{d}{\log n})}\Big)^{\alpha},&\ \ d\le n\le 2^d, \\ d^{\alpha}n^{\alpha/d},&\ \ n\ge 2^d, \end{matrix}\right. \end{equation} where the equivalence constants depend only on $\alpha$, but not on $d$ and $n$. \end{thm} \subsection{Tractability on the ball} \ Using the same methods as in Theorems \ref{t5.1} and \ref{t5.2}, we obtain the two theorems. \begin{thm}\label{t6.6} Let $r>0$ and $s,\,t>0$. Then the approximation problem $$I_d: H^{r,}({\Bbb B^d})\rightarrow L_2({\Bbb B^d})$$ is $(s, t)$-weakly tractable if and only if $s>1/r$ and $t>0$ or $s>0$ and $t>1$. Specially, the approximation problem $I_d: H^{r,}({\Bbb B^d})\rightarrow L_2({\Bbb B^d})$ is weakly tractable if and only if $r>1$, not uniformly weakly tractable, and does not suffer from the curse of dimensionality. \end{thm} \begin{thm}\label{t6.7} Let $\alpha,\beta>0$. Then the approximation problem $$I_d: G^{\alpha,\beta}({\Bbb B^d})\rightarrow L_2({\Bbb B^d})$$ (1) is uniformly weakly tractable. (2) is not polynomially tractable. (3) is quasi-polynomially tractable if and only if $\alpha\ge 1$ and the exponent of quasi-polynomial tractability is $$t^{\rm qpol}=\sup_{m\in \Bbb N}\frac m {1+\beta m^\alpha},\ \ \alpha\ge1.$$Specially, if $\alpha=1$, then $t^{\rm qpol}=\frac1\beta$.\end{thm} \begin{rem} We can also consider exponential convergence tractability for the approximation problem $$I_d: G^{\alpha,\beta}({\Bbb B^d})\rightarrow L_2({\Bbb B^d})\ \ (\alpha,\beta>0).$$ We can prove that the approximation problem $I_d: G^{\alpha,\beta}({\Bbb B^d})\rightarrow L_2({\Bbb B^d})$ is $(t,\ln^s)$-weakly tractable if and only if $\alpha>1/s$ and $t>0$ or $s>0$ and $t>1$, and is not exponential convergence-uniformly weakly tractable for any $\alpha, \beta>0$. Specially, it is exponential convergence-weakly tractable if and only if $\alpha>1$. \end{rem} \section{Acknowledgments} The authors were supported by the National Natural Science Foundation of China (Project no. 11671271, 11271263), the Beijing Natural Science Foundation (1172004, 1132001). \end{document}	arXiv
My first dose on 1 March 2017, at the recommended 0.5ml/1.5mg was miserable, as I felt like I had the flu and had to nap for several hours before I felt well again, requiring 6h to return to normal; after waiting a month, I tried again, but after a week of daily dosing in May, I noticed no benefits; I tried increasing to 3x1.5mg but this immediately caused another afternoon crash/nap on 18 May. So I scrapped my cytisine. Oh well. Absorption of nicotine across biological membranes depends on pH. Nicotine is a weak base with a pKa of 8.0 (Fowler, 1954). In its ionized state, such as in acidic environments, nicotine does not rapidly cross membranes…About 80 to 90% of inhaled nicotine is absorbed during smoking as assessed using C14-nicotine (Armitage et al., 1975). The efficacy of absorption of nicotine from environmental smoke in nonsmoking women has been measured to be 60 to 80% (Iwase et al., 1991)…The various formulations of nicotine replacement therapy (NRT), such as nicotine gum, transdermal patch, nasal spray, inhaler, sublingual tablets, and lozenges, are buffered to alkaline pH to facilitate the absorption of nicotine through cell membranes. Absorption of nicotine from all NRTs is slower and the increase in nicotine blood levels more gradual than from smoking (Table 1). This slow increase in blood and especially brain levels results in low abuse liability of NRTs (Henningfield and Keenan, 1993; West et al., 2000). Only nasal spray provides a rapid delivery of nicotine that is closer to the rate of nicotine delivery achieved with smoking (Sutherland et al., 1992; Gourlay and Benowitz, 1997; Guthrie et al., 1999). The absolute dose of nicotine absorbed systemically from nicotine gum is much less than the nicotine content of the gum, in part, because considerable nicotine is swallowed with subsequent first-pass metabolism (Benowitz et al., 1987). Some nicotine is also retained in chewed gum. A portion of the nicotine dose is swallowed and subjected to first-pass metabolism when using other NRTs, inhaler, sublingual tablets, nasal spray, and lozenges (Johansson et al., 1991; Bergstrom et al., 1995; Lunell et al., 1996; Molander and Lunell, 2001; Choi et al., 2003). Bioavailability for these products with absorption mainly through the mucosa of the oral cavity and a considerable swallowed portion is about 50 to 80% (Table 1)…Nicotine is poorly absorbed from the stomach because it is protonated (ionized) in the acidic gastric fluid, but is well absorbed in the small intestine, which has a more alkaline pH and a large surface area. Following the administration of nicotine capsules or nicotine in solution, peak concentrations are reached in about 1 h (Benowitz et al., 1991; Zins et al., 1997; Dempsey et al., 2004). The oral bioavailability of nicotine is about 20 to 45% (Benowitz et al., 1991; Compton et al., 1997; Zins et al., 1997). Oral bioavailability is incomplete because of the hepatic first-pass metabolism. Also the bioavailability after colonic (enema) administration of nicotine (examined as a potential therapy for ulcerative colitis) is low, around 15 to 25%, presumably due to hepatic first-pass metabolism (Zins et al., 1997). Cotinine is much more polar than nicotine, is metabolized more slowly, and undergoes little, if any, first-pass metabolism after oral dosing (Benowitz et al., 1983b; De Schepper et al., 1987; Zevin et al., 1997). For instance, they point to the U.S. Army's use of stimulants for soldiers to stave off sleep and to stay sharp. But the Army cares little about the long-term health effects of soldiers, who come home scarred physically or mentally, if they come home at all. It's a risk-benefit decision for the Army, and in a life-or-death situation, stimulants help. And as before, around 9 AM I began to feel the peculiar feeling that I was mentally able and apathetic (in a sort of aboulia way); so I decided to try what helped last time, a short nap. But this time, though I took a full hour, I slept not a wink and my Zeo recorded only 2 transient episodes of light sleep! A back-handed sort of proof of alertness, I suppose. I didn't bother trying again. The rest of the day was mediocre, and I wound up spending much of it on chores and whatnot out of my control. Mentally, I felt better past 3 PM. Smart drugs, formally known as nootropics, are medications, supplements, and other substances that improve some aspect of mental function. In the broadest sense, smart drugs can include common stimulants such as caffeine, herbal supplements like ginseng, and prescription medications for conditions such as ADHD, Alzheimer's disease, and narcolepsy. These substances can enhance concentration, memory, and learning. A total of 330 randomly selected Saudi adolescents were included. Anthropometrics were recorded and fasting blood samples were analyzed for routine analysis of fasting glucose, lipid levels, calcium, albumin and phosphorous. Frequency of coffee and tea intake was noted. 25-hydroxyvitamin D levels were measured using enzyme-linked immunosorbent assays…Vitamin D levels were significantly highest among those consuming 9-12 cups of tea/week in all subjects (p-value 0.009) independent of age, gender, BMI, physical activity and sun exposure. Two additional studies assessed the effects of d-AMP on visual–motor sequence learning, a form of nondeclarative, procedural learning, and found no effect (Kumari et al., 1997; Makris, Rush, Frederich, Taylor, & Kelly, 2007). In a related experimental paradigm, Ward, Kelly, Foltin, and Fischman (1997) assessed the effect of d-AMP on the learning of motor sequences from immediate feedback and also failed to find an effect. Much better than I had expected. One of the best superhero movies so far, better than Thor or Watchmen (and especially better than the Iron Man movies). I especially appreciated how it didn't launch right into the usual hackneyed creation of the hero plot-line but made Captain America cool his heels performing & selling war bonds for 10 or 20 minutes. The ending left me a little nonplussed, although I sort of knew it was envisioned as a franchise and I would have to admit that showing Captain America wondering at Times Square is much better an ending than something as cliche as a close-up of his suddenly-opened eyes and then a fade out. (The movie continued the lamentable trend in superhero movies of having a strong female love interest… who only gets the hots for the hero after they get muscles or powers. It was particularly bad in CA because she knows him and his heart of gold beforehand! What is the point of a feminist character who is immediately forced to do that?)↩ It is often associated with Ritalin and Adderall because they are all CNS stimulants and are prescribed for the treatment of similar brain-related conditions. In the past, ADHD patients reported prolonged attention while studying upon Dexedrine consumption, which is why this smart pill is further studied for its concentration and motivation-boosting properties. Methylphenidate, commonly known as Ritalin, is a stimulant first synthesised in the 1940s. More accurately, it's a psychostimulant - often prescribed for ADHD - that is intended as a drug to help focus and concentration. It also reduces fatigue and (potentially) enhances cognition. Similar to Modafinil, Ritalin is believed to reduce dissipation of dopamine to help focus. Ritalin is a Class B drug in the UK, and possession without a prescription can result in a 5 year prison sentence. Please note: Side Effects Possible. See this article for more on Ritalin. "A system that will monitor their behavior and send signals out of their body and notify their doctor? You would think that, whether in psychiatry or general medicine, drugs for almost any other condition would be a better place to start than a drug for schizophrenia," says Paul Appelbaum, director of Columbia University's psychiatry department in an interview with the New York Times. He used to get his edge from Adderall, but after moving from New Jersey to San Francisco, he says, he couldn't find a doctor who would write him a prescription. Driven to the Internet, he discovered a world of cognition-enhancing drugs known as nootropics — some prescription, some over-the-counter, others available on a worldwide gray market of private sellers — said to improve memory, attention, creativity and motivation. Now, what is the expected value (EV) of simply taking iodine, without the additional work of the experiment? 4 cans of 0.15mg x 200 is $20 for 2.1 years' worth or ~$10 a year or a NPV cost of $205 (\frac{10}{\ln 1.05}) versus a 20% chance of $2000 or $400. So the expected value is greater than the NPV cost of taking it, so I should start taking iodine. The absence of a suitable home for this needed research on the current research funding landscape exemplifies a more general problem emerging now, as applications of neuroscience begin to reach out of the clinical setting and into classrooms, offices, courtrooms, nurseries, marketplaces, and battlefields (Farah, 2011). Most of the longstanding sources of public support for neuroscience research are dedicated to basic research or medical applications. As neuroscience is increasingly applied to solving problems outside the medical realm, it loses access to public funding. The result is products and systems reaching the public with less than adequate information about effectiveness and/or safety. Examples include cognitive enhancement with prescription stimulants, event-related potential and fMRI-based lie detection, neuroscience-based educational software, and anti-brain-aging computer programs. Research and development in nonmedical neuroscience are now primarily the responsibility of private corporations, which have an interest in promoting their products. Greater public support of nonmedical neuroscience research, including methods of cognitive enhancement, will encourage greater knowledge and transparency concerning the efficacy and safety of these products and will encourage the development of products based on social value rather than profit value. Medication can be ineffective if the drug payload is not delivered at its intended place and time. Since an oral medication travels through a broad pH spectrum, the pill encapsulation could dissolve at the wrong time. However, a smart pill with environmental sensors, a feedback algorithm and a drug release mechanism can give rise to smart drug delivery systems. This can ensure optimal drug delivery and prevent accidental overdose. Most people would describe school as a place where they go to learn, so learning is an especially relevant cognitive process for students to enhance. Even outside of school, however, learning plays a role in most activities, and the ability to enhance the retention of information would be of value in many different occupational and recreational contexts. Nootropics – sometimes called smart drugs – are compounds that enhance brain function. They're becoming a popular way to give your mind an extra boost. According to one Telegraph report, up to 25% of students at leading UK universities have taken the prescription smart drug modafinil [1], and California tech startup employees are trying everything from Adderall to LSD to push their brains into a higher gear [2].	CommonCrawl
\begin{document} \renewcommand\footnotemark{} \title{Variational formulas and cocycle solutions\\ for directed polymer and percolation models} \titlerunning{Variational formulas for polymers and percolation} \author{Nicos Georgiou\inst{1} \and Firas Rassoul-Agha\inst{2} \and Timo Sepp\"al\"ainen\inst{3} \thanks{F.\ Rassoul-Agha and N.\ Georgiou were partially supported by National Science Foundation grant DMS-0747758.} \thanks{F.\ Rassoul-Agha was partially supported by National Science Foundation grant DMS-1407574 and by Simons Foundation grant 306576.} \thanks{T.\ Sepp\"al\"ainen was partially supported by National Science Foundation grant DMS-1306777, by Simons Foundation grant 338287, and by the Wisconsin Alumni Research Foundation.} } \authorrunning{N.~Georgiou \and F.~Rassoul-Agha \and T.~Sepp\"al\"ainen} \institute{Mathematics, University of Sussex, Falmer Campus, Brighton BN1 9QH, UK.\\ \email{[email protected]} \and Mathematics, University of Utah, 155S 1400E, Salt Lake City, UT 84112, USA.\\ \email{[email protected]} \and Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Dr.,\\ Madison WI 53706-1388, USA.\\ \email{[email protected]} } \date{Received: June 20, 2015 / Accepted: January 19, 2016} \maketitle \begin{abstract} We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder. \end{abstract} \setcounter{tocdepth}{1} \tableofcontents \section{Introduction} Existence of limit shapes has been foundational for the study of growth models and percolation type processes. These limits are complicated, often coming from subadditive sequences. Beyond a handful of exactly solvable models, very little information is available about the limit shapes. This article develops and studies variational formulas for the limiting free energies of directed random paths in a random medium, both for positive temperature directed polymer models and for zero-temperature last-passage percolation models. Earlier papers \cite{Ras-Sep-14} and \cite{Ras-Sep-Yil-13} proved variational formulas for positive temperature directed polymers, without addressing solutions of these formulas. Article \cite{Ras-Sep-Yil-15-} gives simpler proofs of some of the results of \cite{Ras-Sep-Yil-13}. The present paper continues the project in two directions: (i) We extend the variational formulas from positive to zero temperature, that is, we derive variational formulas for the limiting time constants of directed last-passage percolation models. (ii) We develop an approach for finding minimizers for one type of variational formula in terms of cocycles, for both positive temperature and zero temperature models. Our paper, and the concurrent and independent work of Krishnan \cite{Kri-14,Kri-15-}, are the first to provide general formulas for the limits of first- and last-passage percolation models. The variational formulas we present come in two types. (a) One formula minimizes over gradient-like cocycle functions. In the positive temperature case this formula mimics the commonly known min-max formula of the Perron-Frobenius eigenvalue of a nonnegative matrix. In the case of a periodic environment this cocycle variational formula reduces to the min-max formula from linear algebra. The origins of this formula go back to the PhD thesis of Rosenbluth \cite{Ros-06}. He adapted homogenization work \cite{Kos-Rez-Var-06} to deduce a formula of this type for the quenched large deviation rate function for random walk in random environment. (b) The second formula maximizes over invariant measures on the space of environments and paths. The positive temperature version of this formula is of the familiar type that gives the dual of entropy as a function of the potential. In zero temperature the entropy disappears and only the expected potential is left, maximized over invariant measures that are absolutely continuous with respect to the background measure. In a periodic environment this zero-temperature formula reduces to the maximal average circuit weight formula of a max-plus eigenvalue. The next example illustrates the two types of variational formulas for the two-dimensional corner growth model. The notation and the details are made precise in the sequel. \begin{example}\label{ex:corner} Let $\Omega=\R^{\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ be the space of weight configurations $\omega=(\omega_x)_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ on the planar integer lattice $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$, and let $\bP$ be an i.i.d.\ product probability measure on $\Omega$. Assume $\bE(\abs{\omega_x}^{p})<\infty$ for some $p>2$. Let $h\in\R^2$ be an external field parameter. The point-to-line last-passage time is defined by \begin{equation} G^\infty_{0,(n)}(h) =\max_{x_{0,n}: \,x_0=0}\Bigl\{ \,\sum_{k=0}^{n-1}\omega_{x_k} + h\cdot x_n\Bigr\} \label{gpl1.0}\end{equation} where the maximum is over paths $x_{0,n}=(x_0,\dotsc, x_n)$ that begin at the origin $x_0=0$ and take directed nearest-neighbor steps $x_k-x_{k-1}\in\{e_1, e_2\}$. There is a law of large numbers \begin{equation}\label{p2lh-lim1.0} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)= \lim_{n\to\infty} n^{-1}G_{0,(n)}^\infty(h) \quad \text{ $\bP$-almost surely, simultaneously $\forall h\in\R^2$. } \end{equation} This defines a deterministic convex Lipschitz function $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty:\R^2\to\R$. (The subscript pl is for point-to-line and the superscript $\infty$ is for zero temperature.) The results to be described give the following two characterizations of the limit. Theorem \ref{th:K-var} gives the cocycle variational formula \begin{align} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)&=\inf_{F} \,\bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \max_{i=1,2} \; \bigl\{\omega_0+ h\cdot e_i+F(\omega, 0,e_i)\bigr\}.\label{eq:g:K-var1.0} \end{align} The infimum is over centered stationary cocycles $F$. These are mean-zero functions $F:\Omega\times(\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2)^2\to\R$ that satisfy additivity $F(\omega, x,y)+F(\omega, y,z)=F(\omega, x,z)$ and stationarity $F(T_z\omega, x,y)=F(\omega, z+x,z+y)$ (Definition \ref{def:cK}). The second formula is over measures and comes as a special case of Theorem \ref{th:g=Hstar}: \begin{align} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)&=\sup\bigl\{E^\mu[\omega_0+h\cdot z]:\mu\in\mathcal{M}_s(\Omega\times\{e_1,e_2\}),\,\mu\vert_{\Omega}\ll\bP,\, E^\mu[\omega_0^-]<\infty\bigr\}.\label{eq:g:H-var1.0} \end{align} The supremum is over probability measures $\mu$ on pairs $(\omega, z)\in\Omega\times\{e_1,e_2\}$ that are invariant in a natural way (described in Proposition \ref{lm:S}) and whose $\Omega$-marginal is absolutely continuous with respect to the environment distribution $\bP$. $E^\mu$ denotes expectation under $\mu$. As we will see, these formulas are valid quite generally in all dimensions, for general walks, ergodic environments, and more complicated potentials, provided certain moment assumptions are satisfied. $\triangle$ \end{example} In addition to deriving the formulas, we develop a solution approach for the cocycle formula in terms of stationary cocycles suitably adapted to the potential. Such cocycles can be obtained from limits of gradients of free energies and last-passage times. These limits are called {\it Busemann functions}. Their existence is in general a nontrivial problem. Along the way we show that, once Busemann functions exist as almost sure limits, their integrability follows from the $L^1$ shape theorem which a priori is a much cruder result. Over the last two decades Busemann functions have become an important tool in the study of the geometry of percolation and invariant distributions of related particle systems. Study of Busemann functions is also motivated by fluctuation questions. One approach to quantifying fluctuations of free energy and the paths goes through control of fluctuations of Busemann functions. In 1+1 dimension these models are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class and there are well-supported conjectures for universal fluctuation exponents and limit distributions. Some of these conjectures have been verified for a handful of exactly solvable models. (See surveys \cite{corw-rev, quas-icm,spoh-12,trac-wido-02}.) In dimensions 3+1 and higher, high temperature behavior of directed polymers has been proved to be diffusive \cite{come-yosh-aop-06}, but otherwise conjectures beyond 1+1 dimension are murky. To summarize, the purpose of this paper is to develop the variational formulas, illustrate them with examples, and set an agenda for future study with the Busemann solution. We show how the formulas work in weak disorder, in exactly solvable 1+1 dimensional models, and in periodic environments. Applications that go beyond these cases cannot be covered within the scope of this paper and will follow in future work. Minimizing cocycles for \eqref{eq:g:K-var1.0} have been constructed for the two-dimensional corner growth model with general i.i.d.\ weights in \cite{geor-rass-sepp-lppbuse}. In the sequel \cite{geor-rass-sepp-lppgeo} these cocycles are used to construct geodesics and to prove existence, uniqueness and coalescence properties of directional geodesics and to study the competition interface. In another direction of work on these formulas, article \cite{Ras-Sep-Yil-15-} proves the cocycle variational formula for the annealed free energy of a directed polymer and uses it to characterize the so-called weak disorder phase of the model. {\bf Overview of related literature.} Independently of the present work and with a different methodology, Krishnan \cite{Kri-14,Kri-15-} proves a variational formula for undirected first passage bond percolation with bounded ergodic weights. Taking an optimal control approach, he embeds the lattice problem into $\R^d$ and applies the recent stochastic homogenization results of Lions and Souganidis \cite{Lio-Sou-05} to derive a variational formula. The resulting formula is a first passage percolation version of our formula \eqref{eq:g:K-var}. The homogenization parallel of our work is \cite{Kos-Rez-Var-06, Kos-Var-08} rather than \cite{Arm-Sou-12, Lio-Sou-05}. The quantity homogenized corresponds in our world to the finite-volume free energy. We run through a selection of highlights from past study of limiting shapes and free energies. For directed polymers Vargas \cite{varg-07} proved the a.s.\ existence of the limiting free energy under moment assumptions similar to the ones we use. Earlier proofs with stronger assumptions appeared in \cite{carm-hu-02, come-shig-yosh-03}. In weak disorder the limiting polymer free energy is the same as the annealed one. In strong disorder no general formulas appeared in the literature before \cite{Ras-Sep-14, Ras-Sep-Yil-13}. Carmona and Hu \cite{carm-hu-02} gave some bounds in the Gaussian case. Lacoin \cite{laco-10} gave small-$\beta$ asymptotics in dimensions $d=1,2$. The earliest explicit free energy for an exactly solvable directed polymer model is the calculation in \cite{mori-oconn-07} for the semi-discrete polymer in a Brownian environment. Explicit limits for the exactly solvable log-gamma polymer appear in \cite{geor-rass-sepp-yilm-15, sepp-12-aop}. The study of Lyapunov exponents and large deviations for random walks in random environments is a related direction of literature. \cite{Var-03-cpam,Zer-98-aop} are two early papers in the multidimensional setting. A seminal paper in the study of directed last-passage percolation is Rost 1981 \cite{rost}. He deduced the limit shape of the corner growth model with exponential weights in conjunction with a hydrodynamic limit for TASEP (totally asymmetric simple exclusion process) with the step initial condition. However, the last passage representation of this model was discovered only later. The study of directed last-passage percolation bloomed in the 1990s, with the first shape results for exactly solvable cases in \cite{aldo-diac95, cohn-elki-prop-96, jock-prop-shor-98, sepp98mprf, sepp-96}. Early motivation for \cite{aldo-diac95} came from Hammersley 1972 \cite{hamm}. The breakthroughs of \cite{baik-deif-joha-99, joha} transformed the study of exactly solvable last-passage models and led to the first rigorous KPZ fluctuation results. The only universal shape result is the asymptotic result on the boundary of $\R_+^2$ for the corner growth model by Martin \cite{mart-04}. In undirected first passage percolation the fundamental shape theorem is due to Cox and Durrett \cite{cox-durr-81}. A classic in the field is the flat edge result of Durrett and Liggett \cite{durr-ligg-81}. Marchand \cite{Mar-02} sharpened this result and Auffinger and Damron \cite{auff-damr-13} built on it to prove differentiability of the shape at the edge of the percolation cone. Busemann functions came on the percolation scene in the work of Newman and coauthors \cite{How-New-01, Lic-New-96, New-95}. Busemann functions were shown to exist as almost sure limits of passage time gradients as a consequence of uniqueness and coalescence of infinite directional geodesics, under uniform curvature assumptions on the limit shape. These assumptions were relaxed through a weak convergence approach of Damron and Hanson \cite{Dam-Han-14}. Busemann functions have been used to study competition in percolation models and properties of particle systems and randomly driven equations. For a selection of the literature, see \cite{bakh-cato-khan-14, Cat-Pim-11, Cat-Pim-12, cato-pime-13, ferr-pime-05, ferr-mart-pime-09, hoff-05, hoff-08, pime-07}. {\bf Organization of the paper.} Section \ref{sec:free} defines the models and states the existence theorems for the limiting free energies whose description is the purpose of the paper. Section \ref{sec:corr} derives the cocycle variational formula for the point-to-level case and develops an approach for solving these formulas. Section \ref{sec:tilt} extends this to point-to-point free energy via a duality between tilt and velocity. Section \ref{sec:bus} demonstrates how minimizing cocycles arise from Busemann functions. Section \ref{sec:lg+exp} explains how the theory of the paper works in explicitly solvable 1+1 dimensional models, namely the log-gamma polymer and the corner growth model with exponential weights. Section \ref{sec:entr} develops variational formulas in terms of measures. In the positive temperature case these formulas involve relative entropy. Section \ref{sec:finite} illustrates the results of the paper for periodic environments where our variational formulas become elements of Perron-Frobenius theory. {\bf Notation and conventions.} We collect here some items for later reference. $\N=\{1,2,3,\dotsc\}$, $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+=\{0,1,2,\dotsc\}$, $\R_+=[0,\infty)$. $\abs{x}=(\sum_i \abs{x_i}^2)^{1/2}$ denotes Euclidean norm. The standard basis vectors of $\R^d$ are $e_1=(1,0,\dotsc,0), e_2=(0,1,0,\dotsc,0), \dotsc,e_d=(0,\dotsc,0,1)$. $\mathcal{M}_1(\mathcal X)$ denotes the space of Borel probability measures on a space $\mathcal X$ and $b\mathcal X$ the space of bounded Borel functions $f:\mathcal X\to\R$. $\bP$ is a probability measure on environments $\omega$, with expectation operation $\bE$. Expectation with respect to $\omega$ of a multivariate function $F(\omega, x, y)$ can be expressed as $\bE F(\omega,x,y)=\bE F(x,y)=\int F(\omega,x,y)\,\bP(d\omega)$. $\triangle$ marks the end of an example and a remark. \section{Free energy in positive and zero temperature}\label{sec:free} In this section we describe the setting and state the limit theorems for free energy and last-passage percolation. The positive temperature limits are quoted from past work and then extended to last-passage percolation via a zero-temperature limit. Fix the dimension $d\in\N$. Let $p:\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d\to[0,1]$ be a random walk probability kernel: $\sum_{z\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}p(z)=1$. Assume $p$ has finite support $\mathcal R=\{z\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d: p(z)>0\}$. $\mathcal R$ must contain at least one nonzero point, and $\mathcal R$ may contain $0$. A path $x_{0,n}=(x_k)_{k=0}^n$ in $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$ is {\sl admissible} if its steps satisfy $z_k\equiv x_k-x_{k-1}\in\mathcal R$. The probability of an admissible path from a fixed initial point $x_0$ is $p(x_{0,n})=p(z_{1,n})=\prod_{i=1}^n p(z_i)$. Let $\delta=\min_{z\in\mathcal R}p(z)>0$. $\mathcal R$ generates the additive subgroup $\mathcal G= \{\sum_{z\in\mathcal R}a_z z:a_z\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN\}$ of $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$. $\mathcal G$ is isomorphic to some $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^k$ (Prop.~P1 on p.~65 in \cite{spitzer}). $\mathcal U$ is the convex hull of $\mathcal R$ in $\R^d$, and $\ri\mathcal U$ the relative interior of $\mathcal U$. The common affine hull of $\mathcal R$ and $\mathcal U$ is denoted by $\aff\mathcal R=\aff\mathcal U$. An {\sl environment} $\omega$ is a sample point from a Polish probability space $(\Omega, \mathfrak{S}, \bP)$ where $\mathfrak{S}$ is the Borel $\sigma$-algebra of $\Omega$. $\Omega$ comes equipped with a group $\{T_x:{x\in\mathcal G}\}$ of measurable commuting bijections that satisfy $T_{x+y}=T_xT_y$ and $T_0$ is the identity. $\bP$ is a $\{T_x\}_{x\in\mathcal G}$-invariant probability measure on $(\Omega,\mathfrak{S})$. This is summarized by the statement that $(\Omega,\mathfrak{S},\bP,\{T_x\}_{x\in\mathcal G})$Ê is a measurable dynamical system. We assume $\bP$ {\sl ergodic}. As usual this means that $\bP(A)=0$ or $1$ for all events $A\in\mathfrak{S}$ that satisfy $T_z^{-1}A=A$ for all $z\in\mathcal R$. Occasionally we make stronger assumptions on $\bP$. $\bE$ denotes expectation under $\bP$. A {\sl potential} is a measurable function $V:\Omega\times\mathcal R^\ell\to\R$ for some $\ell\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+$, denoted by $V(\omega,z_{1,\ell})$ for an environment $\omega$ and a vector of admissible steps $z_{1,\ell}=(z_1,\dotsc, z_\ell)\in\mathcal R^\ell$. The case $\ell=0$ corresponds to a potential $V:\Omega\to\R$ that is a function of $\omega$ alone. The variational formulas from \cite{Ras-Sep-14} and \cite{Ras-Sep-Yil-13} that this article relies upon were proved under the following assumption on $V$. \begin{definition}[Class $\mathcal{L}$]\label{def:cL} A function $V:\Omega\times\mathcal R^\ell\to\R$ is in class $\mathcal{L}$ if for every $ \tilde z_{1,\ell}=(\tilde z_1, \dotsc, \tilde z_\ell)\in\mathcal R^\ell$ and for every nonzero $z\in\mathcal R$, $V(\,\cdot\,,\tilde z_{1,\ell})\in L^1(\bP)$ and \begin{equation} \label{cL-cond} \varlimsup_{\varepsilon\searrow0}\;\varlimsup_{n\to\infty} \;\max_{x\in\mathcal G:\abs{x}\le n}\;\frac1n \sum_{0\le k\le\varepsilon n} \abs{V(T_{x+kz}\omega, \tilde z_{1,\ell})}=0\quad\text{for $\bP$-a.e.\ $\omega$.}\end{equation} \end{definition} Membership $V\in\mathcal{L}$ depends on a combination of mixing of $\bP$ and moments of $V$. See Lemma A.4 of \cite{Ras-Sep-Yil-13} for a precise statement. Boundedness of $V$ is of course sufficient. \begin{remark}\label{rm:prod}({\it Canonical settings}) Often the natural choice for $\Omega$ is a product space $\Omega=\mathcal{S}^{\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}$ with a Polish space $\mathcal{S}$, product topology, and Borel $\sigma$-algebra $\mathfrak{S}$. A generic point of $\Omega$ is then denoted by $\omega=(\omega_x)_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}$. The mappings are shifts $(T_x\omega)_y=\omega_{x+y}$. For example, random weights assigned to the vertices of $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$ would be modeled by $\Omega=\R^{\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}$ and $V(\omega)=\omega_0$. In fact, it would be sufficient to take $\Omega=\R^{\mathcal G}$ since the coordinates outside $\mathcal G$ are not needed as long as paths begin at points in $\mathcal G$. To represent directed edge weights we can take $\Omega=\mathcal{S}^{\mathcal G}$ with $\mathcal{S}=\R^\mathcal R$ where an element $s\in\mathcal{S}$ represents the weights of the admissible edges out of the origin: $s=(\omega_{(0,z)}: z\in\mathcal R)$. Then $\omega_x=(\omega_{(x,x+z)}: z\in\mathcal R)$ is the vector of edge weights out of vertex $x$. Shifts act by $(T_u\omega)_{(x,y)} =\omega_{(x+u,y+u)}$ for $u\in\mathcal{G}$. The potential is $V(\omega,z)=\omega_{(0,z)}=$ the weight of the edge $(0,z)$. To have weights on undirected nearest-neighbor edges take $\Omega=\R^{{\mathcal E}}$ where ${\mathcal E}=\{ \{x,y\}\subset\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d: \abs{y-x}=1\}$ is the set of undirected nearest-neighbor edges on $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$. Now $\mathcal R=\{ \pm e_i: i=1,\dotsc,d\}$, $V(\omega,z)=\omega_{\{0,z\}}$ and $(T_u\omega)_{\{x,y\}} =\omega_{\{x+u,y+u\}}$ for $u\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$. $\bP$ is an {\sl i.i.d.}\ or {\sl product measure} if the coordinates $\{\omega_x\}_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}$ (or $\{\omega_x\}_{x\in\mathcal G}$ or $\{\omega_e\}_{e\in{\mathcal E}}$) are independent and identically distributed (i.i.d.) random variables under $\bP$. With an i.i.d.\ $\bP$ and {\sl local} $V$ (that is, $V$ depends on only finitely many coordinates of $\omega$), for $V\in\mathcal{L}$ it suffices to assume $V(\cdot\,, z_{1,\ell})\in L^p(\bP)$ for some $p>d$ and all $z_{1,\ell}\in\mathcal R^\ell$. $\triangle$ \end{remark} For inverse temperature parameter $0<\beta<\infty$ define the $n$-step {\sl quenched partition function} \begin{equation}\label{gpl8} Z^\beta_{0,(n)}=\sum_{x_{0,n+\ell-1}:\,x_0=0} p(x_{0,n+\ell-1}) \,e^{\beta\sum_{k=0}^{n-1}V(T_{x_k}\omega,\, z_{k+1, k+\ell})}.\end{equation} The sum is over admissible $(n+\ell-1)$-step paths $x_{0,n+\ell-1}$ that start at $x_0=0$. The second argument of $V$ is the $\ell$-vector $z_{k+1, k+\ell}=(z_{k+1}, z_{k+2},\dotsc, z_{k+\ell})$ of steps, and it is not present if $\ell=0$. The corresponding free energy is defined by \begin{equation}\label{gpl5} G^\beta_{0,(n)}= \beta^{-1} \log Z^\beta_{0,(n)}. \end{equation} In the $\beta\to\infty$ limit this turns into the $n$-step {\sl last-passage time} \begin{equation} G^\infty_{0,(n)}=\max_{x_{0,n+\ell-1}: \,x_0=0}\sum_{k=0}^{n-1}V(T_{x_k}\omega,\, z_{k+1, k+\ell}). \label{gpl1}\end{equation} As in the definitions above we shall consistently use the subscript $(n)$ with parentheses to indicate number of steps. In the most basic situation where $d=2$ and $\mathcal R=\{e_1,e_2\}$ the quantity $G^\infty_{0,(n)}$ is a {\sl point-to-line} last-passage value because admissible paths $x_{0,n}$ go from $0$ to the line $\{(i,j): i+j=n\}$. We shall call the general case \eqref{gpl5}--\eqref{gpl1} {\sl point-to-level}. The $n$-step {\sl quenched point-to-point partition function} is for $x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$ \begin{equation}\label{zpp1} Z^\beta_{0,(n),x}=\sum_{x_{0,n+\ell-1}:\,x_0=0,\,x_n=x}p(x_{0,n+\ell-1})\, e^{\beta\sum_{k=0}^{n-1}V(T_{x_k}\omega,\, z_{k+1, k+\ell})}\end{equation} with free energy \[ G^\beta_{0,(n),x}= \beta^{-1} \log Z^\beta_{0,(n),x}. \] Its zero-temperature limit is the $n$-step {\sl point-to-point last-passage time} \begin{equation}G^\infty_{0,(n),x}=\max_{x_{0,n+\ell-1}:\, x_0=0,\,x_n=x}\sum_{k=0}^{n-1}V(T_{x_k}\omega,\, z_{k+1,k+\ell}).\label{gpp1}\end{equation} \begin{remark}\label{path-rm} The formulas for limits presented in this paper are for the case where the length of the path is restricted, as in \eqref{zpp1} and \eqref{gpp1}, so that only those paths that reach $x$ from $0$ in exactly $n$ steps are considered. This is indicated by the subscript $(n)$. Extension to paths of unrestricted length from $0$ to $x$ or from $0$ to a hyperplane is left for future work. In the most-studied directed models this restriction can be dropped because each path between two given points has the same number of steps. Examples where this is the case are $\mathcal R=\{e_1,\dotsc,e_d\}$ and $\mathcal R=\{(z',1): z'\in\mathcal R'\}$ for a finite subset $\mathcal R'\subset\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^{d-1}$. $\triangle$ \end{remark} To take limits of point-to-point quantities we specify lattice points $\hat x_n(\xi)$ that approximate $n\xi$ for $\xi\in\mathcal U$. For each point $\xi\in\mathcal U$ fix weights $\alpha_z(\xi)\in[0,1]$ such that $\sum_{z\in\mathcal R}\alpha_z(\xi) =1$ and $\xi=\sum_{z\in\mathcal R}\alpha_z(\xi) z$. Then define a path \begin{align}\label{eq:def:xhat} \hat x_n(\xi)=\sum_{z\in\mathcal R}\bigl(\lfloor n\alpha_z(\xi)\rfloor +b_z^{(n)}(\xi)\bigr) z, \quad n\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+, \end{align} where $b_z^{(n)}(\xi)\in\{0,1\}$ are arbitrary but subject to these constraints: if $\alpha_z(\xi)=0$ then $b_z^{(n)}(\xi)=0$, and $\sum_{z\in\mathcal R} b_z^{(n)}(\xi) = n-\sum_{z\in\mathcal R}\fl{n\alpha_z(\xi)}$. In other words, $\hat x_n(\xi)$ is a lattice point that approximates $n\xi$ to within a constant independent of $n$, can be reached in $n$ $\mathcal R$-steps from the origin, and uses only those steps that appear in the pre-specified convex representation $\xi=\sum_z \alpha_z z$. When $\xi\in\mathcal U\cap\Q^d$ we require that $\alpha_z(\xi)$ be rational. This is possible by Lemma A.1 of \cite{Ras-Sep-Yil-13}. The next theorem defines the limits whose study is the purpose of the paper. We state it so that it covers simultaneously both the positive temperature ($0<\beta<\infty$) and the zero-temperature case (last-passage percolation, or $\beta=\infty$). The subscripts are pl for point-to-level and pp for point-to-point. \begin{theorem}\label{th:p2p} Let $V\in\mathcal{L}$ and assume $\bP$ ergodic. Let $\beta\in(0,\infty]$. {\rm (a)} The nonrandom limit \begin{align}\label{eq:g:p2l} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta=\lim_{n\to\infty}n^{-1}G^\beta_{0,(n)} \end{align} exists $\bP$-a.s.\ in $(-\infty,\infty]$. {\rm (b)} There exists an event $\Omega_0$ with $\bP(\Omega_0)=1$ such that the following holds for all $\omega\in\Omega_0$. For all $\xi\in\mathcal U$ and any choices made in the definition of $\hat x_n(\xi)$ in \eqref{eq:def:xhat}, the limit \begin{align}\label{eq:g:p2p} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)=\lim_{n\to\infty}n^{-1}G^\beta_{0,(n),\hat x_n(\xi)} \end{align} exists in $(-\infty,\infty]$. For a particular $\xi$ the limit is independent of the choice of convex representation $\xi=\sum_z\alpha_z(\xi) z$ and the numbers $b^{(n)}_z(\xi)$ that define $\hat x_n(\xi)$ in \eqref{eq:def:xhat}. We have the almost sure identity \begin{align}g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta =\sup_{\xi\in\Q^d\cap\mathcal U}g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi) =\sup_{\xi\in\mathcal U}g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi).\label{eq:sup p2p:lpp} \end{align} \end{theorem} \begin{proof} The case $0<\beta<\infty$ is covered by Theorem 2.2 of \cite{Ras-Sep-14}. (The kernel there is the uniform one $p(z)=\abs{\mathcal R}^{-1}$ but this makes no difference to the arguments. Alternatively, the kernel can be moved into the potential.) For any $0<\beta<\infty$, \begin{align} G^\infty_{0,(n)}+\beta^{-1}(n+\ell-1)\log\delta\;&\le\;\beta^{-1}\logZ^\beta_{0,(n)}\;\le\; G^\infty_{0,(n)}\\ \text{ and }\qquad G^\infty_{0,(n),x}+\beta^{-1}(n+\ell-1)\log\delta\;&\le\;\beta^{-1}\logZ_{0,(n),x}^\beta \;\le\;G^\infty_{0,(n),x}. \end{align} Divide by $n$, let first $n\to\infty$ and then $\beta\to\infty$. This gives the existence of the limits for the case $\beta=\infty$. We also get these bounds, uniformly in $\omega$ and $\xi\in\mathcal U$: \begin{equation} \label{eq:g-Lambda} \begin{aligned} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty+\beta^{-1}\log\delta\;&\le\; \Lapl^\beta\;\le\; g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty \\ \text{ and}\qquad g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\infty(\xi)+\beta^{-1}\log\delta\;&\le\; \Lapp^\beta(\xi)\;\le\; g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\infty(\xi). \end{aligned} \end{equation} These bounds extend \eqref{eq:sup p2p:lpp} from $0<\beta<\infty$ to $\beta=\infty$. \qed \end{proof} Since our hypotheses are fairly general, we need to address the randomness, finiteness, and regularity of the limits. For $0<\beta<\infty$ the remarks below repeat claims proved in \cite{Ras-Sep-14}. The properties extend to $\beta=\infty$ by way of bounds \eqref{eq:g-Lambda} as $\beta\to\infty$. \begin{remark}($\bP$ {\it ergodic}) If we only assume $\bP$ ergodic and place no further restrictions on admissible paths then we need to begin by assuming that $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta\in\R$. An obvious way to guarantee this would be to assume that $V$ is bounded above (in addition to what is assumed to have $V\in\mathcal{L}$). Under the assumption $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta\in\R$ the point-to-point limit $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)$ is a nonrandom, real-valued, concave and continuous function on the relative interior $\ri\mathcal U$. Boundary values $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)$ for $\xi\in\mathcal U\smallsetminus\ri\mathcal U$ can be random, but on the whole of $\mathcal U$, for $\bP$-a.e.~$\omega$, the (possibly random) function $\xi\mapsto g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi;\omega)$ is lower semicontinuous and bounded. The upper semicontinuous regularization of $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ and its unique continuous extension from $\ri\mathcal U$ to $\mathcal U$ are equal and nonrandom. $\triangle$ \end{remark} \begin{remark}\label{rmk:dir-iid}({\it Directed i.i.d.\ $L^{d+\varepsilon}$ case}) Assume the canonical setting from Remark \ref{rm:prod}: $\Omega$ is a product space, $\bP$ is i.i.d., $V$ is local, and $\bE[\abs{V(\omega,z_{1,\ell})}^p]<\infty$ for some $p>d$ and $\forall z_{1,\ell}\in\mathcal R^\ell$. Assume additionally that $0\not\in\mathcal U$. We call this the {\sl directed i.i.d.\ $L^{d+\varepsilon}$ case}. Then $V\in\mathcal{L}$, $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta\in\R$, and the point-to-point limit $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)$ is a nonrandom, real-valued, concave and continuous function on all of $\mathcal U$ (Theorem 3.2(a) of \cite{Ras-Sep-14}). $\triangle$ \end{remark} \section{Cocycle variational formula for the point-to-level case} \label{sec:corr} In Sections \ref{sec:corr}--\ref{sec:bus} we study potentials of the form \begin{equation}\label{V0h} V(\omega, z)=V_0(\omega, z)+h\cdot z, \qquad (\omega, z) \in \Omega\times\mathcal R \end{equation} for a measurable function $V_0:\Omega\times\mathcal R\to\R$ and a vector $h\in\R^d$. We think of $V_0$ as fixed and $h$ as a variable and hence amend our notation as follows. As before the steps of admissible paths are $z_k=x_k-x_{k-1}\in\mathcal R$. \begin{equation} G_{0,(n)}^\beta(h)= \beta^{-1}\log \sum_{x_{0,n}: \,x_0=0} p(x_{0,n})\,e^{\beta \sum_{k=0}^{n-1}V_0(T_{x_k}\omega, \,z_{k+1}) + \beta h\cdot x_n} \label{p2lh-beta}\end{equation} for $0<\beta<\infty$, \begin{equation} G_{0,(n)}^\infty(h)= \max_{x_{0,n}: \,x_0=0} \Bigl\{ \;\sum_{k=0}^{n-1}V_0(T_{x_k}\omega, z_{k+1}) + h\cdot x_n \Bigr\}, \label{p2lh}\end{equation} and \begin{equation}\label{p2lh-lim} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)= \lim_{n\to\infty} n^{-1}G_{0,(n)}^\beta(h) \quad \text{a.s. for all $0<\beta\le \infty$}. \end{equation} Limit \eqref{p2lh-lim} is a special case of \eqref{eq:g:p2l}. By \eqref{eq:g-Lambda}, if $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)$ is finite for one $\beta\in(0,\infty]$, it is finite for all $\beta\in(0,\infty]$. This can be guaranteed by assuming $V_0$ bounded above, or by the directed i.i.d.\ $L^{d+\varepsilon}$ assumption of Remark \ref{rmk:dir-iid}, or by some other case-specific assumption. If $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)$ is finite, it is clear from the expressions above that $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ is a real-valued convex Lipschitz function of $h\in\R^d$. We develop a variational formula for $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ for $\beta\in(0,\infty]$ in terms of gradient-like cocycles, and identify a condition that singles out extremal cocycles. For $0<\beta<\infty$ this variational formula appeared in \cite{Ras-Sep-Yil-13} and here we extend it to $\beta=\infty$. The solution proposal is new for all $\beta$. \begin{definition}[Cocycles] \label{def:cK} A measurable function $F:\Omega\times\mathcal G^2\to\R$ is a {\rm stationary cocycle} if it satisfies these two conditions for $\bP$-a.e.\ $\omega$ and all $x,y,z\in\mathcal G$: \begin{align} F(\omega, z+x,z+y)&=F(T_z\omega, x,y) \qquad \text{{\rm(}stationarity{\rm)}} \\ F(\omega,x,y)+F(\omega,y,z)&=F(\omega, x,z) \qquad \;\ \ \text{{\rm(}additivity{\rm)}.} \end{align} If $\bE\abs{F(x,y)} <\infty$ $\forall x,y\in\mathcal G$ then $F$ is an {\rm $L^1(\bP)$ cocycle}, and if also $\bE[F(x,y)]=0$ $\forall x,y\in\mathcal G$ then $F$ is {\rm centered}. $\mathcal{K}$ denotes the space of stationary $L^1(\bP)$ cocycles, and $\mathcal{K}_0$ denotes the subspace of centered stationary $L^1(\bP)$ cocycles. \end{definition} As illustrated above, $\omega$ can be dropped from the notation $F(\omega,x,y)$. The term cocyle is borrowed from differential forms terminology, see e.g.\ \cite{Ken-09}. One could also use the term {\sl conservative flow} or {\sl curl-free flow} following vector fields terminology. The space $\mathcal{K}_0$ is the $L^1(\bP)$ closure of gradients $F(\omega,x,y)=\varphi(T_y\omega)-\varphi(T_x\omega)$ \cite[Lemma C.3]{Ras-Sep-Yil-13}. For $B\in\mathcal{K}$ there exists a vector $h({B})\in\R^d$ such that \begin{equation} \bE[{B}(0,z)]=-h({B})\cdot z \qquad \text{ for all $z\in\mathcal R$. } \label{EB}\end{equation} Existence of $h(B)$ follows because $c(x)=\bE[{B}(0,x)]$ is an additive function on the group $\mathcal G\cong\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^k$. $h(B)$ is not unique unless $\mathcal R$ spans $\R^d$, but the inner products $h(B)\cdot x$ for $x\in\mathcal G$ are uniquely defined. Then \begin{equation} F(\omega, x,y)= h({B})\cdot (x-y)-{B}(\omega, x,y), \qquad x,y\in\mathcal G \label{FF}\end{equation} is a centered stationary $L^1(\bP)$ cocycle. \begin{theorem}\label{th:K-var} Let $V_0\in\mathcal{L}$ and assume $\bP$ ergodic. Then the limits in \eqref{p2lh-lim} have these variational representations: for $0<\beta<\infty$ \begin{equation} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)=\inf_{F\in\mathcal{K}_0} \,\bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \beta^{-1}\log \sum_{z\in\mathcal R} p(z)e^{\beta V_0(\omega,z)+\beta h\cdot z+\beta F(\omega, 0,z)} \label{eq:Lambda:K-var} \end{equation} and \begin{align} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)&=\inf_{F\in\mathcal{K}_0} \,\bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \max_{z\in\mathcal R} \{V_0(\omega,z)+ h\cdot z+F(\omega, 0,z)\}.\label{eq:g:K-var} \end{align} A minimizing $F\in\mathcal{K}_0$ exists for each $0<\beta\le\infty$ and $h\in\R^2$. \end{theorem} \begin{proof} Theorem 2.1 of \cite{Ras-Sep-Yil-15-} gives formula \eqref{eq:Lambda:K-var} for $0<\beta<\infty$. The kernel in that reference is the uniform one $p(z)=\abs\mathcal R^{-1}$ but changing the kernel makes no difference to the proof. To get the formula for $\beta=\infty$, note that for $\beta>0$ and $F\in\mathcal{K}_0$, \begin{align} &\beta^{-1}\log\sum_z p(z)e^{\beta V(\omega,z)+\beta F(\omega,0,z)} \le\max_z \{ V(\omega,z)+ F(\omega,0,z)\} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\le \beta^{-1}\log\sum_z p(z)e^{\beta V(\omega,z)+\beta F(\omega,0,z)}+\beta^{-1}\log\delta^{-1}.\end{align} Thus \begin{align} \Lapl^\beta \;\le\; \inf_{F\in\mathcal{K}_0} \bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \max_z \{V(\omega,z)+F(\omega,0,z)\} \le \; \Lapl^\beta+\beta^{-1}\log\delta^{-1}.\end{align} Formula \eqref{eq:g:K-var} follows from this and \eqref{eq:g-Lambda}, upon letting $\beta\to\infty$. Theorem 2.3 of \cite{Ras-Sep-Yil-15-} gives the existence of a minimizer for $0<\beta<\infty$, and the same proof works also for $\beta=\infty$. \qed \end{proof} Assuming $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)$ finite is not necessary for Theorem \ref{th:K-var}. By the assumption $V_0\in\mathcal{L}$, any $F\in\mathcal{K}_0$ that makes the right-hand side of \eqref{eq:g:K-var} finite satisfies the ergodic theorem (Theorem \ref{th:Atilla}) in the appendix. Then potential $V(\omega,z)$ can be replaced by $V(\omega,z)+F(\omega,0,z)$ without altering $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$, and consequently $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ is finite. Formulas \eqref{eq:Lambda:K-var} and \eqref{eq:g:K-var} can be viewed as infinite-dimensional versions of the min-max variational formula for the Perron-Frobenius eigenvalue of a nonnegative matrix. This connection is discussed in Section \ref{sec:finite}. The next definition and theorem offer a way to identify a minimizing $F$ for \eqref{eq:Lambda:K-var} and \eqref{eq:g:K-var}. Later we explain how Busemann functions provide minimizers that match this recipe. That this approach is feasible will be demonstrated by examples: weak disorder (Example \ref{ex:weak1}), the exactly solvable log-gamma polymer (Section \ref{sec:lg} below) and the corner growth model with exponential weights (Section \ref{sec:exp}). This strategy is carried out for the two-dimensional corner growth model with general weights (a non-solvable case) in article \cite{geor-rass-sepp-lppbuse}. \begin{definition}\label{def:bdry-model} Fix $\beta\in(0,\infty]$. A stationary $L^1$ cocycle $B$ {\rm is adapted to} potential $V_0$ if the following condition holds. If $0<\beta<\infty$ the requirement is \begin{equation} \sum_{z\in\mathcal R}p(z)\,e^{\beta V_0(\omega,z)-\beta B(\omega, 0,z)} =1 \quad\text{ for }\bP\text{-a.e.\ }\omega, \label{VBbeta}\end{equation} while if $\beta=\infty$ then the condition is \begin{equation} \max_{z\in\mathcal R} \{ V_0(\omega,z) -B(\omega, 0,z)\}=0 \quad\text{ for }\bP\text{-a.e.\ }\omega. \label{VB}\end{equation} \end{definition} \begin{theorem}\label{thm:minimizer} Fix $\beta\in(0,\infty]$, assume $\bP$ ergodic and $V_0\in\mathcal{L}$. Suppose we have a stationary $L^1$ cocycle $B$ that is adapted to $V_0$ in the sense of Definition \ref{def:bdry-model}. Define $h(B)$ and $F$ as in \eqref{EB}--\eqref{FF}. Then we have conclusions {\rm (i)--(ii)} below. {\rm (i)} $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h(B))=0$. $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ is finite for all $h\in\R^d$. {\rm (ii)} $F$ solves the variational formula. Precisely, assume $h\in\R^d$ satisfies \begin{equation} (h-h(B))\cdot(z-z')=0 \quad\text{ for all $z,z'\in\mathcal R$. } \label{h(B)-1}\end{equation} Under this assumption we have the two cases below. {\rm (ii-a)} Case $0<\beta<\infty$. $ F$ is a minimizer in \eqref{eq:Lambda:K-var} for potential $V(\omega,z)=V_0(\omega,z)+h\cdot z$. The essential supremum in \eqref{eq:Lambda:K-var} disappears and we have, for $\bP$-a.e.\ $\omega$ and any $z'\in\mathcal R$, \begin{equation} \label{eq:Kvar:minbeta} \begin{aligned} \Lapl^\beta(h)&\;=\;\beta^{-1}\log\sum_{z\in\mathcal R} p(z)\,e^{\betaV_0(\omega,z)+\beta h\cdot z+\beta F(\omega, 0,z)} \;=\;(h-h(B))\cdot z'. \end{aligned}\end{equation} {\rm (ii-b)} Case $\beta=\infty$. Then $F$ is a minimizer in \eqref{eq:g:K-var} for potential $V(\omega,z)=V_0(\omega,z)+h\cdot z$. The essential supremum in \eqref{eq:g:K-var} disappears and we have, for $\bP$-a.e.\ $\omega$ and any $z'\in\mathcal R$, \begin{equation} \label{eq:Kvar:min} \begin{aligned} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)&\;=\;\max_{z\in\mathcal R} \; \{V_0(\omega,z)+h\cdot z+F(\omega, 0,z)\} \;=\;(h-h(B))\cdot z'. \end{aligned}\end{equation} \end{theorem} Condition \eqref{h(B)-1} says that $h-h(B)$ is orthogonal to the affine hull of $\mathcal R$ in $\R^d$. If $0\in\mathcal U$ this affine hull is the linear span of $\mathcal R$ in $\R^d$. \begin{remark}({\it Correctors}) A mean-zero cocycle that minimizes in \eqref{eq:Lambda:K-var} or \eqref{eq:g:K-var} without the essential supremum (that is, satisfies the first equality of \eqref{eq:Kvar:minbeta} or \eqref{eq:Kvar:min}) could be called a {\it corrector} by analogy with the homogenization literature (see for example Section 7 in \cite{Kos-07} and top of page 468 in \cite{Arm-Sou-12}). These correctors have been useful in the study of infinite geodesics in the corner growth model \cite{geor-rass-sepp-lppgeo} and infinite directed polymers \cite{geor-rass-sepp-yilm-15}. $\triangle$ \end{remark} \begin{proofof}{of Theorem \ref{thm:minimizer}} {{\sl Case $0<\beta<\infty$.}} From assumption \eqref{VBbeta} and definition \eqref{FF} of $F$ \begin{align} \log\sum_{z\in\mathcal R} p(z)\,e^{\beta V_0(\omega,\,z)+\beta h(B)\cdot z+ \beta F(\omega, 0,z)}=0\quad \text{ for $\bP$-a.e.\ $\omega$}.\label{eq:auxbeta} \end{align} Iterating this gives (with $x_k=z_1+\dotsm+z_k$) \begin{align} \log\sum_{z_{1,n}\in\mathcal R^n}p(x_{0,n})\,e^{\beta \sum_{k=0}^{n-1} V_0(T_{x_k}\omega, \,z_{k+1})+\beta h(B)\cdot x_n+\beta F(\omega, 0, x_{n})}=0.\label{eq:stat-tiltbeta} \end{align} Assumption \eqref{VBbeta} gives the bound $ F(\omega, 0,z) \le V_0^(\omega)+C $ for $z\in\mathcal R$, with \begin{equation}\label{V} V_0^(\omega)=\max_{z\in\mathcal R} \abs{V_0(\omega,z)} \end{equation} that satisfies $V_0^\in\mathcal{L}$ and a constant $C$. By Theorem \ref{th:Atilla} in the appendix, $F(\omega, 0, x_{n})=o(n)$ uniformly in $z_{1,n}$, $\bP$-almost surely. It follows from \eqref{eq:stat-tiltbeta} that $\Lapl^\beta(h(B))=0$. Since the steps of the walks are bounded, finiteness of $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ for all $h$ follows from the definition \eqref{p2lh-beta}. Assume \eqref{h(B)-1} for $h$. Then \eqref{eq:auxbeta} gives \[ \beta^{-1} \log\sum_{z\in\mathcal R}p(z)\, e^{\beta V_0(\omega,\,z)+\beta h\cdot z+ \beta F(\omega, 0,z)}= (h-h(B))\cdot z' \] while from \eqref{p2lh-beta} and \eqref{p2lh-lim} \[ \Lapl^\beta(h)=\Lapl^\beta(h(B)) + (h-h(B))\cdot z' = (h-h(B))\cdot z'. \] We have verified \eqref{eq:Kvar:minbeta}. {\sl Case $\beta=\infty$.} From assumption \eqref{VB} and definition \eqref{FF} of $F$ \begin{align} \max_{z\in\mathcal R} \{V_0(\omega,z)+h(B)\cdot z+ F(\omega, 0,z)\}=0\quad \text{ for $\bP$-a.e.\ $\omega$}.\label{eq:aux} \end{align} Iterating this gives (with $x_k=z_1+\dotsm+z_k$) \begin{align} \max_{z_{1,n}\in\mathcal R^n}\Bigl\{\;\sum_{k=0}^{n-1} V_0(T_{x_k}\omega, z_{k+1})+h(B)\cdot x_n+F(\omega, 0, x_{n}) \Bigr\}=0.\label{eq:stationary-tilt} \end{align} By Theorem \ref{th:Atilla}, $F(\omega, 0, x_{n})=o(n)$ uniformly in $z_{1,n}$ $\bP$-a.s. It follows that $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}(h(B))=0$. Assume \eqref{h(B)-1} for $h$. Then \eqref{eq:aux} gives \[ \max_{z\in\mathcal R} \{V_0(\omega,z)+h\cdot z+ F(\omega, 0,z)\}= (h-h(B))\cdot z' \] while from \eqref{p2lh}--\eqref{p2lh-lim} \[ g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)=g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h(B)) + (h-h(B))\cdot z' = (h-h(B))\cdot z'.\tag{\qed}\] \end{proofof} \begin{remark} The results of this section extend to the more general potentials $V(T_{x_k}\omega,z_{k+1,k+\ell})$ discussed in Section \ref{sec:free}. For the definition of the cocycle see Definition 2.2 of \cite{Ras-Sep-Yil-13}. We do not pursue these generalizations to avoid becoming overly technical and because presently we do not have an interesting example of this more general potential. $\triangle$ \end{remark} The remainder of this section discusses an example that illustrates Theorem \ref{thm:minimizer}. \begin{example}\label{ex:weak1}({\it Directed polymer in weak disorder}) We consider the standard $k+1$ dimensional directed polymer in an i.i.d.\ random environment, or ``bulk disorder''. (For references see \cite{come-shig-yosh-04, come-yosh-aop-06, denholl-polymer}.) We show that the condition of weak disorder itself gives the corrector that solves the variational formula for the point-to-level free energy. The background walk is a simple random walk in $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^k$, and we use an additional $(k+1)$st coordinate to represent time. So $d=k+1$, $\Omega=\R^{\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}$, $\bP$ is i.i.d., $\mathcal R=\{(\pm e_i,1): 1\le i\le k\}$ and $p(z)=\abs{\mathcal R}^{-1}$ for $z\in\mathcal R$. The potential is simply the environment at the site: $V_0(\omega)=\omega_0$. Define the logarithmic moment generating functions \begin{equation} \lambda(\beta)=\log\bE(e^{\beta\omega_0}) \quad\text{for $\beta\in\R$} \label{lambdabeta}\end{equation} and \begin{equation} \kappa(h)=\log \sum_{z\in\mathcal R} p(z)\, e^{h\cdot z} \quad\text{for $h\in\R^{d}$.} \label{kappa1}\end{equation} Consider only $\beta$-values such that $\lambda(\beta)<\infty$. The normalized partition function \[ W_n = e^{-n(\lambda(\beta)+\kappa(\beta h))} \sum_{x_{0,n}} p(x_{0,n})\,e^{\beta \sum_{k=0}^{n-1}\omega_{x_k} + \beta h\cdot x_n} \] is a positive mean 1 martingale. The weak disorder assumption is this: \begin{equation}\label{weak:ui} \text{the martingale $W_n$ is uniformly integrable. } \end{equation} Given $h\in\R^d$, this can be guaranteed by taking $k\ge 3$ and small enough $\beta>0$ (see Lemma 5.3 in \cite{Ras-Sep-12-arxiv}). Then $W_n\to W_\infty$ a.s.\ and in $L^1(\bP)$, $W_\infty\ge 0$ and $\bE W_\infty=1$. The event $\{W_\infty>0\}$ is a tail event in the product space of environments, and hence by Kolmogorov's 0-1 law we must have $\bP(W_\infty>0)=1$. This gives us the limiting point-to-level free energy: \begin{equation} \label{weak:g} \begin{aligned} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h) &= \lim_{n\to\infty} n^{-1} \beta^{-1}\log \sum_{x_{0,n}} p(x_{0,n})\,e^{\beta \sum_{k=0}^{n-1} \omega_{x_k} + \beta h\cdot x_n}\\ &= \lim_{n\to\infty} n^{-1} \beta^{-1}\log W_n + \beta^{-1}(\lambda(\beta)+\kappa(\beta h) ) \\ &= \beta^{-1}(\lambda(\beta)+\kappa(\beta h) ) . \end{aligned} \end{equation} Decomposition according to the first step (Markov property) gives \[ W_n(\omega) = e^{-\lambda(\beta)-\kappa(\beta h)} \sum_{z\in\mathcal R} p(z) e^{\beta \omega_{0} + \beta h\cdot z} W_{n-1}(T_z\omega) \] and a passage to the limit \begin{equation}\label{weak:W} W_\infty(\omega) = e^{-\lambda(\beta)-\kappa(\beta h)} \sum_{z\in\mathcal R} p(z) e^{\beta \omega_{0} + \beta h\cdot z} W_{\infty}(T_z\omega) \quad \text{$\bP$-a.s.} \end{equation} Combining \eqref{weak:g} and \eqref{weak:W} gives \begin{equation}\label{weak:g3} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h) = \beta^{-1}\log \sum_{z\in\mathcal R} p(z)e^{\beta \omega_0+\beta h\cdot z+\beta F(\omega, 0,z)} \quad \text{$\bP$-a.s.} \end{equation} with the gradient \begin{equation}\label{weak:F} F(\omega, x,y)=\beta^{-1} \log W_{\infty}(T_y\omega) - \beta^{-1} \log W_{\infty}(T_x\omega). \end{equation} In order to check that $F$ is a centered cocycle it remains to verify that $F(\omega, 0, z)$ is integrable and mean-zero. Equation \eqref{weak:g3} gives an upper bound that shows $\bE[F(\omega,0,z)^+]<\infty$. We argue indirectly that also $\bE[F(\omega,0,z)^-]<\infty$. The first limit in probability below comes from stationarity. \begin{align} 0\; &\overset{\text{prob}}=\lim_{n\to\infty} \;\bigl[ n^{-1}\beta^{-1}\log W_{\infty}(T_{nz}\omega) - n^{-1} \beta^{-1}\log W_{\infty}(\omega)\bigr]\\ &= \lim_{n\to\infty} \, n^{-1}\sum_{k=0}^{n-1} F(T_{kz}\omega, 0, z) . \end{align} Since $\bE[F(\omega,0,z)^+]<\infty$, the assumption $\bE[F(\omega,0,z)^-]=\infty$ and the ergodic theorem would force the limit above to $-\infty$. Hence it must be that $F(\omega, 0, z)\in L^1(\bP)$. The limit above then gives $\bE[F(\omega,0,z)]=0$. To summarize, \eqref{weak:g3} shows that the centered cocycle $F$ satisfies \eqref{eq:Kvar:minbeta} for $V(\omega,z)=\omega_0+h\cdot z$ for this particular value $(\beta,h)$. $F$ is the corrector given in Theorem \ref{thm:minimizer}, from the cocycle $B$ that is adapted to $V_0$ given by \[ B(\omega, x,y)= g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h) e_d\cdot (y-x)- h\cdot (y-x) -F(\omega, x,y) \] with $ h(B)= h- g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h) e_d. $ A vector $\tilde h$ satisfies \eqref{h(B)-1} if and only if $\tilde h=h+\alpha e_d$ for some $\alpha\in\R$. The conclusion of the theorem, that $F$ is a corrector for potential $ V_0(\omega)= \omega_0$ and all such tilts $\tilde h$, is obvious because $e_d\cdot x_n=n$ for admissible paths. $\triangle$ \end{example} \section{Tilt-velocity duality} \label{sec:tilt} Section \ref{sec:corr} gave a variational description of the point-to-level limit in terms of stationary cocycles. Theorem \ref{th:K-var-pp} below extends this description to point-to-point limits via tilt-velocity duality. Tilt-velocity duality is the familiar idea from large deviation theory that pinning the path is dual to tilting the energy by an external field. In the positive temperature setting this is exactly the convex duality of the quenched large deviation principle for the endpoint of the path (see Remark 4.2 in \cite{Ras-Sep-14}). We continue to consider potentials of the form $V(\omega,z)=V_0(\omega,z)+h\cdot z$ in general dimension $d\in\N$, with $\beta\in(0,\infty]$ and $\bP$ ergodic. As above, the point-to-level limits $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ are defined by \eqref{p2lh-lim}. For the point-to-point limits $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)$ we use only the $V_0$-part of the potential. So for $\xi\in\mathcal U$ define \begin{equation} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)= \lim_{n\to\infty} n^{-1} \beta^{-1}\log \sum_{\substack{x_{0,n}:\, x_0=0,\\ \quad x_n=\hat x_n(\xi)}} p(x_{0,n})\,e^{\beta \sum_{k=0}^{n-1}V_0(T_{x_k}\omega, \,z_{k+1}) } \label{p2pV_0}\end{equation} for $0<\beta<\infty$ and \begin{equation} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\infty(\xi)= \lim_{n\to\infty} \max_{\substack{x_{0,n}:\, x_0=0,\\ \quad x_n=\hat x_n(\xi)}} \; n^{-1}\sum_{k=0}^{n-1}V_0(T_{x_k}\omega, z_{k+1}) . \label{p2pV_0LPP}\end{equation} In this context we call the vector $h\in\R^d$ a {\sl tilt} and elements $\xi\in\mathcal U$ {\sl directions} or {\sl velocities}. Let us assume $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)$ finite. Then for $\xi\in\ri\mathcal U$, the a.s.\ point-to-point limits \eqref{p2pV_0}--\eqref{p2pV_0LPP} define nonrandom, bounded, concave, continuous functions $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta:\ri\mathcal U\to\R$ for $\beta\in(0,\infty]$ (see Theorem 2.4 and 2.6 and Remark 2.5 of \cite{Ras-Sep-14}). The results of this section do not touch the relative boundary of $\mathcal U$. Consequently we do not need additional assumptions that guarantee regularity of $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ up to the boundary. One sufficient assumption would be the directed i.i.d.\ $L^{d+\varepsilon}$ of Remark \ref{rmk:dir-iid} (Theorem 3.2 of \cite{Ras-Sep-14}). \begin{remark} To illustrate what can go wrong on the boundary of $\mathcal U$, suppose $z\in\mathcal R$ is an extreme point of $\mathcal U$. Then the only path from $0$ to $nz$ is $x_k=kz$, and we get $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(z)=\beta^{-1}\log p(z) + \bE[V_0(\omega, z)\,\vert\,\mathcal{I}_z]$ where $\mathcal{I}_z$ is the $\sigma$-algebra of events invariant under the mapping $T_z$. This can be random even if $\bP$ is assumed ergodic under the full group $\{T_x\}$. In general $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ is lower semicontinuous on all of $\mathcal U$, for a.e.\ fixed $\omega$ (Theorem 2.6 of \cite{Ras-Sep-14}). $\triangle$ \end{remark} With definitions \eqref{p2lh-beta}--\eqref{p2lh-lim} and \eqref{p2pV_0}--\eqref{p2pV_0LPP}, equation \eqref{eq:sup p2p:lpp} becomes \begin{align}\label{eq:velocity-tilt} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)=\sup_{\xi\in\mathcal U}\bigl\{g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)+h\cdot\xi\bigr\}, \qquad h\in\R^d. \end{align} In order to invert this relationship between $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta$ and $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ we turn it into a convex (or rather, concave) duality. First extend $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ outside $\mathcal U$ via $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)=-\infty$ for $\xi\in\mathcal U^c$, and then replace $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ with its upper semicontinuous regularization $\bargpp^\beta(\xi)= g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)\vee \varlimsup_{\zeta\to\xi}g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\zeta) $. Now \eqref{eq:velocity-tilt} extends to \[ g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)=\sup_{\xi\in\R^d}\{\bargpp^\beta(\xi)+h\cdot\xi\}, \qquad h\in\R^d, \] which standard convex duality \cite{rock-ca} inverts to \[ \bargpp^\beta(\xi)=\inf_{h\in\R^d}\{g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)-h\cdot\xi\}, \qquad \xi\in\R^d. \] By the continuity of $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ on $\ri\mathcal U$, the last display gives \begin{align}\label{eq:tilt-velocity} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)=\inf_{h\in\R^d}\big\{g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)-h\cdot\xi\big\} \quad \text{ for $\xi\in\ri\mathcal U$.} \end{align} Equations \eqref{eq:velocity-tilt} and \eqref{eq:tilt-velocity} suggest the next definition, and then Lemma \ref{lm:zeta->h} answers part of the natural next question. \begin{definition}\label{def:h-zeta} At a fixed $\beta\in(0,\infty]$, we say that tilt $h\in\R^d$ and velocity $\xi\in\ri\mathcal U$ are {\rm dual} to each other if \begin{align} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)=g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)+h\cdot\xi.\label{h-zeta}\end{align} \end{definition} \begin{lemma}\label{lm:zeta->h} Fix $\beta\in(0,\infty]$. Assume $\bP$ ergodic, $V_0\in\mathcal{L}$ and $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)<\infty$. Then every $\xi\in\ri\mathcal U$ has a dual $h\in\R^d$. Furthermore, if $h$ is dual to $\xi\in\mathcal U$ and $h'$ is such that \begin{align} (h-h')\cdot(z-z')=0\text{ for all $z,z'\in\mathcal R$}\label{eq:h-unique} \end{align} then $h'$ is also dual to $\xi$. \end{lemma} \begin{proof} We start with the proof of the second claim. If \eqref{eq:h-unique} holds then directly from \eqref{p2lh-beta}--\eqref{p2lh-lim}, $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h')=g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)+(h'-h)\cdot z$ for all $z\in\mathcal R$. Hence, $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h')-h'\cdot\xi=g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)-h\cdot\xi$ and $h$ is dual to $\xi$ if and only if $h'$ is. The equality above also implies that any $h$ in \eqref{eq:tilt-velocity} can be replaced by any $h'$ satisfying \eqref{eq:h-unique}. Fix $z_0\in\mathcal R$. One way to satisfy \eqref{eq:h-unique} is to let $h'$ be the orthogonal projection of $h$ onto the linear span $\cV$ of $\mathcal R-z_0$. Consequently we can restrict the infimum in \eqref{eq:tilt-velocity} to $h\in\cV$. (This can be all of $\R^d$.) For any $z\in\mathcal R$, $h\in\R^d$, and $\beta\in(0,\infty]$, \[g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)\ge\bE[V_0(\omega,z)]+h\cdot z +\beta^{-1}\log p(z).\] To see this, for $z\ne 0$ consider the path $x_k=kz$ and use the ergodic theorem. For $z=0$ consider a path that finds $V_0(T_x\omega, 0)$ within $\varepsilon$ of $\mathop{\mathrm{ess\,sup}}V_0(\cdot\,,0)$ and stays there. Furthermore, \eqref{eq:sup p2p:lpp} gives $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)\le g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0).$ Consequently we can restrict the infimum in \eqref{eq:tilt-velocity} to $h\in\cV$ that satisfy \begin{align} {h}\cdot(z-\xi)\le g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)+1-\bE[V_0(\omega,z)] -\beta^{-1}\log p(z) \, \le\, c \end{align} for all $z\in\mathcal R$ and a constant $c$. Convex combinations over $z$ lead to ${h}\cdot(\eta-\xi)\le c$ for all $\eta\in\mathcal U$. By the definition of relative interior, $\xi\in\ri\mathcal U$ implies that for some $\varepsilon>0$, $\zeta\in\mathcal U$ for all $\zeta\in\aff\mathcal U$ such that $\abs{\xi-\zeta}\le\varepsilon$. Since $h\in\cV$, $\eta=\xi+\varepsilon\abs{h}^{-1} h$ lies in $\aff\mathcal U$ and then by choice of $\varepsilon$ also in $\mathcal U$. We conclude that $\varepsilon\abs{h}\le c$ and thereby that the infimum in \eqref{eq:tilt-velocity} can be restricted to a compact set. Continuity of $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta$ implies that the infimum is achieved and existence of an $h$ dual to $\xi$ has been established. \qed \end{proof} With these preliminaries we extend Theorem \ref{th:K-var} to the point-to-point case. Recall Definition \ref{def:cK} of the space $\mathcal{K}$ of stationary $L^1$ cocycles. \begin{theorem}\label{th:K-var-pp} Assume $V_0\in\mathcal{L}$, $\bP$ ergodic and $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)$ finite. Then we have these variational formulas for $\xi\in\ri\mathcal U$. \begin{equation} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)\;=\;\inf_{B\in\mathcal{K}} \,\bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \beta^{-1}\log \sum_{z\in\mathcal R} p(z)e^{\beta V_0(\omega,z)-\beta B(\omega, 0,z)-\beta h(B)\cdot \xi} \label{eq:K-var-pp} \end{equation} for $0<\beta<\infty$ and \begin{align} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\infty(\xi)&\;=\;\inf_{B\in\mathcal{K}}\, \bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \max_{z\in\mathcal R}\, \{V_0(\omega,z)- B(\omega, 0,z)- h(B)\cdot \xi\}. \label{eq:g:K-var-pp} \end{align} The infimum in \eqref{eq:K-var-pp}--\eqref{eq:g:K-var-pp} can be restricted to $B\in\mathcal{K}$ such that $h(B)$ is dual to $\xi$. For each $\xi\in\ri\mathcal U$ and $0<\beta\le\infty$, there exists a minimizing $B\in\mathcal{K}$ such that $h(B)$ is dual to $\xi$. \end{theorem} \begin{proof} We write the proof for $0<\beta<\infty$, the case $\beta=\infty$ being similar enough. The right-hand side of \eqref{eq:K-var-pp} equals \begin{align} &\inf_h \Bigl\{ \, \inf_{B: h(B)=h} \bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \beta^{-1}\log \sum_{z\in\mathcal R} p(z)e^{\beta V_0(\omega,z)-\beta B(\omega, 0,z)} - h\cdot \xi \Bigr\} \\ &=\inf_h \Bigl\{ \, \inf_{F\in\mathcal{K}_0} \bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \beta^{-1}\log \sum_{z\in\mathcal R} p(z)e^{\beta V_0(\omega,z)+\beta h\cdot z +\beta F(\omega, 0,z)} - h\cdot \xi \Bigr\} \\ &=\inf_h \{ \, g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h) - h\cdot \xi\} = g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi). \end{align} The middle equality is true because $B$ is a cocycle with $h(B)=h$ if and only if $F(\omega,0,z)=- B(\omega, 0,z)-h\cdot z$ is a centered cocycle. For the existence, use Lemma \ref{lm:zeta->h} to pick $h$ dual to $\xi$, and then Theorem \ref{th:K-var} to find a minimizing $F\in\mathcal{K}_0$ for $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$. Then $B(\omega,0,z)=-h\cdot z-F(\omega,0,z)$ is a minimizer for $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)$ and $h(B)=h$. \qed \end{proof} Combining Theorems \ref{thm:minimizer} and \ref{th:K-var-pp} with \eqref{h-zeta} gives: \begin{corollary} \label{cor:h-xi} Assume $V_0\in\mathcal{L}$, $\bP$ ergodic and $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)$ finite. Let $\beta\in(0,\infty]$ and $\xi\in\ri\mathcal U$. Suppose there exists $B\in\mathcal{K}$ adapted to $V_0$ {\rm(}Definition \ref{def:bdry-model}{\rm)} and such that $h(B)$ is dual to $\xi$. Then $B$ minimizes in \eqref{eq:K-var-pp} or \eqref{eq:g:K-var-pp} without the essential supremum over $\omega$ and \begin{align}\label{eq:p2p=B.xi} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)=-h(B)\cdot\xi . \end{align} \end{corollary} If $\nablag_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ exists at $\xi$, the duality of $h(B)$ and $\xi$ implies that \begin{align}\label{eq:grad g=-h} \nablag_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)=-h(B). \end{align} In some situations $\mathcal U$ has empty interior but $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta$ extends as a homogeneous function to an open neighborhood of $\mathcal U$, and \eqref{eq:grad g=-h} makes sense for the extended function. Such is the case for example when $\mathcal R=\{e_1,\dotsc,e_d\}$. In the 1+1 dimensional exactly solvable models discussed in Section \ref{sec:lg+exp} below, for each $\xi\in\ri\mathcal U$ there exists a cocycle $B=B^\xi$ that satisfies \eqref{eq:p2p=B.xi} and \eqref {eq:grad g=-h}. Modulo some regularity issues, this is the case also for the 1+1 dimensional corner growth model with general weights \cite{geor-rass-sepp-lppbuse}. \section{Cocycles from Busemann functions} \label{sec:bus} The solution approach advanced in this paper for the cocycle variational formulas relies on cocycles that are adapted to $V_0$ (Definition \ref{def:bdry-model}). This section describes how to obtain such cocycles from limits of gradients of free energy, called {\sl Busemann functions}, provided such limits exist. Busemann functions come in two variants, point-to-point and point-to-level. These are treated in the next two theorems. Proofs of the theorems are at the end of the section. We assume now that every admissible path between two given points $x$ and $y$ has the same number of steps. This prevents loops. The natural examples are $\mathcal R=\{e_1,e_2,\dotsc, e_d\}$ and $\mathcal R=\{ (z',1): z^\prime\in\mathcal R^\prime\}$ for some finite $\mathcal R^\prime\subset\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^{d-1}$. For $x,y\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$ such that $y$ can be reached from $x$ define the free energy \begin{equation}\label{Gxy5} G^\beta_{x,y} \;= \; \beta^{-1}\log\sum_{\substack{n\ge1\\x_{0,n}:\,x_0=x,\, x_n=y}}p(x_{0,n})\,e^{\beta\sum_{k=0}^{n-1}V_0(T_{x_k}\omega, \,z_{k+1})} \quad\text{for $0<\beta<\infty$}\end{equation} and the last-passage time \begin{equation}\label{Gxy6} G^\infty_{x,y} \;= \;\max_{\substack{n\ge1\\x_{0,n}: \,x_0=x,\, x_n=y}}\;\;\sum_{k=0}^{n-1}V_0(T_{x_k}\omega, z_{k+1}). \end{equation} The sum and the maximum are taken over all admissible paths from $x$ to $y$, and then there is a unique $n$, namely the number of steps from $x$ to $y$. Recall definition \eqref{eq:def:xhat} of the path $\hat x_n(\xi)$. A point-to-point Busemann function in direction $\xi\in\ri\mathcal U$ is defined by \begin{equation} B_{\text{\rm pp}}^\xi(x,y)=\lim_{n\to\infty}\bigl[ G^\beta_{x,\,\hat x_{n}(\xi)+z}-G^\beta_{y,\,\hat x_{n}(\xi)+z}\,\bigr] , \quad x,y\in\mathcal{G}, \ z\in\mathcal R\cup\{0\}, \label{bus3}\end{equation} provided that the limit exists $\bP$-almost surely and does not depend on $z$. The extra perturbation by $z$ on the right-hand side will be used to establish stationarity of the limit. $\beta$ is now fixed and we omit the dependence of $B_{\text{\rm pp}}^\xi$ on $\beta$ from the notation. To ensure that paths to $\hat x_n(\xi)$ from both $x$ and $y$ exist in \eqref{bus3}, in the definition \eqref{eq:def:xhat} of $\hat x_n(\xi)$ pick $\alpha_z(\xi)>0$ for all $z\in\mathcal R$. (For $\xi\in\ri\mathcal U$ this is possible by Theorem 6.4 in \cite{rock-ca}.) Then, any point $x\in\mathcal G$ can reach $\hat x_n(\xi)$ with steps in $\mathcal R$ for large enough $n$. \begin{theorem}\label{th:Bus=grad(a)} Let $\beta\in(0,\infty]$, $V_0\in\mathcal{L}$, $\bP$ ergodic and $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)$ finite. Assume that every admissible path between two given points $x$ and $y$ has the same number of steps. Fix $\xi\in\ri\mathcal U$ and choose $\alpha_z(\xi)>0$ for each $z\in\mathcal R$ in \eqref{eq:def:xhat}. Assume that for all $x,y\in\mathcal G$ and $\bP$-a.e.\ $\omega$, the limits \eqref{bus3} exist for $z\in\mathcal R\cup\{0\}$ and are independent of $z$. Then $B_{\text{\rm pp}}^\xi(x,y)$ is a stationary cocycle that is adapted to $V_0$ in the sense of Definition \ref{def:bdry-model}. Assume additionally \begin{align}\label{Gppliminf} \varlimsup_{n\to\infty} n^{-1}\bE[G^\beta_{0,\,\hat x_n(\xi)}]\leg_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi). \end{align} Then $B_{\text{\rm pp}}^\xi(x,y)\in L^1(\bP)$ $\forall x,y\in\mathcal G$, $h(B_{\text{\rm pp}}^\xi)$ is dual to $\xi$ {\rm(}Definition \ref{def:h-zeta}{\rm)}, and $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)=-h(B_{\text{\rm pp}}^\xi)\cdot\xi$. \end{theorem} The point of the theorem is that the Busemann function furnishes correctors for the variational formulas. Once the assumptions of Theorem \ref{th:Bus=grad(a)} are satisfied, (i) Theorem \ref{thm:minimizer} implies that $F(x,y)=h(B_{\text{\rm pp}}^\xi)\cdot (x-y)- B_{\text{\rm pp}}^\xi(x,y)$ is a corrector for $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ for any $h$ such that $h-h(B_{\text{\rm pp}}^\xi)\perp\aff\mathcal R$, and (ii) depending on $\beta$, $B_{\text{\rm pp}}^\xi$ minimizes either \eqref{eq:K-var-pp} or \eqref{eq:g:K-var-pp} without the $\bP$-essential supremum. In the point-to-level case the free energy and last-passage time for paths of length $n$ started at $x$ are defined by a shift $G^\beta_{x,(n)}(h)(\omega)=G^\beta_{0,(n)}(h)(T_x\omega)$. Point-to-level Busemann functions are defined by \begin{equation} B_{\text{\rm pl}}^h(0,z)=\lim_{n\to\infty}\bigl[G^\beta_{0,(n)}(h)-G^\beta_{z,(n-1)}(h)\bigr], \qquad z\in\mathcal R, \label{buse4}\end{equation} omitting again the $\beta$-dependence from the notation. \begin{theorem}\label{th:Bus=grad(b)} Let $\beta\in(0,\infty]$, $V_0\in\mathcal{L}$, $\bP$ ergodic and $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(0)$ finite. Assume that every admissible path between any two given points $x$ and $y$ has the same number of steps. Fix $h\in\R^d$. Assume the $\bP$-a.s.\ limits \eqref{buse4} exist for all $z\in\mathcal R$. Then we can extend $\{B_{\text{\rm pl}}^h(0,z)\}_{z\in\mathcal R}$ to a stationary cocycle $\{B_{\text{\rm pl}}^h(x,y)\}_{x,y\in\mathcal G}$, and cocycle $B_{\text{\rm pl}}^h(x,y)-h\cdot(y-x)$ is adapted to $V_0$ in the sense of Definition \ref{def:bdry-model}. Assume additionally \begin{align}\label{Gplliminf} \varliminf_{n\to\infty} n^{-1}\bE[G^\beta_{0,(n)}(h)]\leg_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h). \end{align} Then $B_{\text{\rm pl}}^h(x,y)\in L^1(\bP)$ for $x,y\in\mathcal G$. $F(\omega,x,y)=h(B_{\text{\rm pl}}^h)\cdot (x-y)- B_{\text{\rm pl}}^h(\omega,x,y)$ is a minimizer in \eqref{eq:Lambda:K-var} for $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ if $0<\beta<\infty$ and in \eqref{eq:g:K-var} if $\beta=\infty$. \end{theorem} Remark \ref{pl-cor} below indicates how the theorem could be upgraded to state that the minimizer $F$ is also a corrector, in other words satisfies \eqref{eq:Kvar:minbeta} or \eqref{eq:Kvar:min}. \begin{remark} Assumptions \eqref{Gppliminf} and \eqref{Gplliminf} need to be verified separately for the case at hand. In the directed i.i.d.\ $L^{d+\varepsilon}$ case of Remark \ref{rmk:dir-iid}, we can use lattice animal bounds: Lemma 3 from page 85 of \cite{Gand-Kest-94} gives $ \sup_n \bE\bigl[ \bigl( n^{-1} G^\beta_{0,\,\hat x_{n}(\xi)}\bigr)^2\,\bigr] < \infty$ and $\sup_n \bE\bigl[ \bigl( n^{-1} G^\beta_{0,(n)}(h)\bigr)^2\,\bigr] < \infty$, which imply $L^1$ convergence in \eqref{p2pV_0}--\eqref{p2pV_0LPP} and \eqref{p2lh-lim}, respectively. A completely general sufficient condition is to have $V_0$ bounded above. $\triangle$ \end{remark} \begin{remark} All of the assumptions and conclusions of Theorems \ref{th:Bus=grad(a)}--\ref{th:Bus=grad(b)} can be verified in the exactly solvable cases. In the explicitly solvable 1+1 dimensional cases the Busemann limits $B_{\text{\rm pp}}^\xi$ and $B_{\text{\rm pl}}^h$ are connected by the duality of $\xi$ and $h$, and lead to the same set of cocycles, as described in the next section. This also holds for the general 1+1 dimensional corner growth model under local regularity assumptions on the shape that ensure the existence of Busemann functions \cite{geor-rass-sepp-lppbuse}. We would expect this feature to be true very generally. $\triangle$ \end{remark} \begin{remark} According to \eqref{bus3}, $B_{\text{\rm pp}}^\xi$ is a microscopic gradient of free energy and passage times in direction $\xi$, and by \eqref{eq:grad g=-h} its average gives the macroscopic gradient. This form of \eqref{eq:grad g=-h} was anticipated in \cite{How-New-01} in the context of Euclidean first passage percolation (FPP), where $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}(x,y)=c\sqrt{x^2+y^2}$ for some $c>0$. (See the paragraph after the proof of Theorem 1.13 in \cite{How-New-01}.) A version of the formula also appears in Theorem 3.5 of \cite{Dam-Han-14} in the context of nearest-neighbor FPP. $\triangle$ \end{remark} \begin{example}\label{ex:weak1.01}({\it Directed polymer in weak disorder}) The directed polymer in weak disorder illustrates Theorem \ref{th:Bus=grad(b)}. We continue with the notation from Example \ref{ex:weak1} and take $\beta>0$ small enough. Then $\bP$-almost surely for $z\in\mathcal R$, \begin{align} &G^\beta_{0,(n)}(h)-G^\beta_{z,(n-1)}(h) \\ &\qquad= \beta^{-1}\log W_n - \beta^{-1}\log W_{n-1}\circ T_z + \beta^{-1}(\lambda(\beta)+\kappa(\beta h) ) \\ &\qquad\underset{n\to\infty}\longrightarrow \; \; \beta^{-1}\log W_\infty - \beta^{-1}\log W_{\infty}\circ T_z + \beta^{-1}(\lambda(\beta)+\kappa(\beta h) ) \\ &\qquad= -F(0,z)+ g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h), \end{align} with $F$ defined by \eqref{weak:F}. Thus the Busemann function is $B_{\text{\rm pl}}^h(0,z)=- F(0,z)+g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$. By Theorem \ref{th:Bus=grad(b)}, cocycle $B_{\text{\rm pl}}^h(0,z)-h\cdot z$ is adapted to $V_0$, as already observed in Example \ref{ex:weak1}. The Busemann function recovers the corrector $F$ identified in Example \ref{ex:weak1}. $\triangle$ \end{example} In the remainder of the section we prove Theorems \ref{th:Bus=grad(a)} and \ref{th:Bus=grad(b)} and then comment on getting a corrector in Theorem \ref{th:Bus=grad(b)}. \begin{proofof}{of Theorem \ref{th:Bus=grad(a)}} To check stationarity, for $z\in\mathcal R$ \begin{align} B_{\text{\rm pp}}^\xi(z+x,z+y)&=\lim_{n\to\infty}[ G^\beta_{z+x,\,z+\hat x_{n}(\xi)}-G^\beta_{z+y,\,z+\hat x_{n}(\xi)}]\\ &=\lim_{n\to\infty}[G^\beta_{x,\,\hat x_{n}(\xi)}-G^\beta_{y,\,\hat x_{n}(\xi)}]\circ T_z =B_{\text{\rm pp}}^\xi(x,y)\circ T_z.\end{align} Additivity is satisfied by telescoping sums. The condition of Definition \ref{def:bdry-model} is readily checked. For example, in the $\beta=\infty$ case, if $x$ is reachable from $0$ and from every $z\in\mathcal R$, $\max_{z\in\mathcal R}\{V_0(\omega,z)+G^\infty_{z,x}-G^\infty_{0,x}\}=0$ because some $z\in\mathcal R$ is the first step of a maximizing path from $0$ to $x$. Assume \eqref{Gppliminf}. Recall \eqref{V}. Fix $\ell\in\N$ large enough so that, for each $k\ge m\ge 1$, there exists an admissible path $\{y_i^{m,k}\}_{i=0}^{\ell}$ from $\hat x_{k-m}(\xi)$ to $\hat x_{k+\ell}(\xi)-\hat x_{m}(\xi)$. Then \begin{equation}\label{bus-77} G^\beta_{0,\, \hat x_{k+\ell}(\xi)-\hat x_{m}(\xi)}(\omega) \ \ge \ G^\beta_{0,\, \hat x_{k-m}(\xi)}(\omega) +\beta^{-1}\log p(y^{m,k}_{0,\ell}) - \sum_{i=0}^{\ell-1} V_0^(T_{y_i^{m,k}}\omega) . \end{equation} By \eqref{bus-77}, for $0<m<n$, \begin{align} &\frac1{(m+\ell)n} \sum_{k=m}^n \bE\bigl[ G^\beta_{0,\, \hat x_{k+\ell}(\xi)} - G^\beta_{\hat x_{m}(\xi),\, \hat x_{k+\ell}(\xi)} \, \bigr] \\ &\qquad = \frac1{(m+\ell)n} \sum_{k=m}^n \bE\bigl[ G^\beta_{0,\, \hat x_{k+\ell}(\xi)} - G^\beta_{0,\, \hat x_{k+\ell}(\xi)-\hat x_{m}(\xi)} \, \bigr] \\ &\qquad \le \frac1{(m+\ell)n} \sum_{k=m}^n \bE\bigl[ G^\beta_{0,\, \hat x_{k+\ell}(\xi)} - G^\beta_{0,\, \hat x_{k-m}(\xi)} \, \bigr] \; + \; \frac{\log p(y^{m,k}_{0,\ell})}{\beta (m+\ell)} \; + \; \frac{\ell\bE(V_0^)}{m+\ell}\\ &\qquad \le \frac1{(m+\ell)n} \sum_{k=n-m+1}^{n+\ell} \bE[ G^\beta_{0,\, \hat x_{k}(\xi)} ] \;-\; \frac1{(m+\ell)n} \sum_{k=0}^{m+\ell-1} \bE[ G^\beta_{0,\, \hat x_{k}(\xi)} ] \; + \; \frac{C}{m} \end{align} where the last $C$ depends on the fixed $\ell$. By \eqref{Gppliminf} we get the upper bound \begin{align} \label{upper007} \varliminf_{n\to\infty} \frac1{(m+\ell)n} \sum_{k=m}^n \bE\bigl[ G^\beta_{0,\, \hat x_{k+\ell}(\xi)} - G^\beta_{\hat x_{m}(\xi),\, \hat x_{k+\ell}(\xi)} \, \bigr] \le g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)+\frac{C}m. \end{align} On the other hand, by superadditivity, \[G^\beta_{0,\, \hat x_{k+\ell}(\xi)} - G^\beta_{\hat x_{m}(\xi),\, \hat x_{k+\ell}(\xi)}\geG^\beta_{0,\,\hat x_m(\xi)}\] and hence $\frac1{(m+\ell)n} \bigl[ \sum_{k=m}^n \bigl( G^\beta_{0,\, \hat x_{k+\ell}(\xi)} - G^\beta_{\hat x_{m}(\xi),\, \hat x_{k+\ell}(\xi)} \, \bigr)\bigr]^-$ is uniformly integrable as $n\to\infty$. Since by assumption \eqref{bus3} \[ \frac1{m+\ell} \, B_{\text{\rm pp}}^\xi(0,\hat x_m(\xi)) = \lim_{n\to\infty} \frac1{(m+\ell)n} \sum_{k=m}^n \bigl[ G^\beta_{0,\, \hat x_{k+\ell}(\xi)} - G^\beta_{\hat x_{m}(\xi),\, \hat x_{k+\ell}(\xi)} \, \bigr] \quad\text{$\bP$-a.s.} \] we can apply Lemma \ref{app:lm1} from the appendix to conclude that $B_{\text{\rm pp}}^\xi(0,\hat x_m(\xi))$ is integrable and satisfies \begin{align}\label{bus8} \frac1{m+\ell} \,\bE [B_{\text{\rm pp}}^\xi(0,\hat x_m(\xi))] \le g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi) \; + \; \frac{C}{m}. \end{align} Now we can show $B_{\text{\rm pp}}^\xi(0,z)\in L^1(\bP)$ $\forall z\in\mathcal R$. We have assumed that each step $z$ appears along the path $\hat x_m(\xi)$, so it suffices to observe that \begin{align} & B_{\text{\rm pp}}^\xi(0, \hat x_m(\xi)-\hat x_{m-1}(\xi)) \circ T_{\hat x_{m-1}(\xi)} \\ &\qquad\qquad\qquad\qquad = B_{\text{\rm pp}}^\xi(0, \hat x_m(\xi)) - B_{\text{\rm pp}}^\xi(0, \hat x_{m-1}(\xi)) \; \in \; L^1(\bP). \end{align} We have established that $B_{\text{\rm pp}}^\xi$ is a stationary $L^1(\bP)$ cocycle that is adapted to $V_0$ in the sense of Definition \ref{def:bdry-model}. By definition \eqref{EB}, the left-hand side of \eqref{bus8} equals \[ -(m+\ell)^{-1} h(B_{\text{\rm pp}}^\xi)\cdot \hat x_m(\xi) \; \to \; - h(B_{\text{\rm pp}}^\xi)\cdot \xi \qquad\text{as $m\to\infty$.} \] We have $- h(B_{\text{\rm pp}}^\xi)\cdot \xi\le g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi) $. Since $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h(B_{\text{\rm pp}}^\xi))=0$ by Theorem \ref{thm:minimizer}, variational formula \eqref{eq:tilt-velocity} gives the opposite inequality $- h(B_{\text{\rm pp}}^\xi)\cdot \xi\ge g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi) $. Duality of $h(B_{\text{\rm pp}}^\xi)$ and $ \xi$ has been established. \qed \end{proofof} \begin{proofof}{of Theorem \ref{th:Bus=grad(b)}} We check that limits \eqref{buse4} define a stationary cocycle $B_{\text{\rm pl}}^h(\omega,x,y)$. Fix $x,y\in\mathcal G$ such that there is a path $x_{0,\ell}$ with increments $z_i=x_i-x_{i-1}\in\mathcal R$ that goes from $x=x_0$ to $y=x_\ell$. By shifting the $n$-index, \begin{equation}\label{bus15} \begin{aligned} \sum_{i=0}^{\ell-1} B_{\text{\rm pl}}^h(T_{x_i}\omega, 0, z_{i+1}) &= \lim_{n\to\infty} \sum_{i=0}^{\ell-1} [G^\beta_{x_i,(n)}(h)-G^\beta_{x_{i+1},(n-1)}(h)] \\ &= \lim_{n\to\infty} \sum_{i=0}^{\ell-1} [G^\beta_{x_i,(n-i)}(h)-G^\beta_{x_{i+1},(n-i-1)}(h)]\\ &= \lim_{n\to\infty} [G^\beta_{x_0,(n)}(h)-G^\beta_{x_{\ell},(n-\ell)}(h)]\\ &= \lim_{n\to\infty} [G^\beta_{0,(n)}(h)-G^\beta_{y-x,(n-\ell)}(h)]\circ T_{x}. \end{aligned}\end{equation} By assumption each path from $x$ to $y$ has the same number $\ell$ of steps. Hence we can define $B_{\text{\rm pl}}^h(\omega,x,y)=\sum_{i=0}^{\ell-1} B_{\text{\rm pl}}^h(T_{x_i}\omega,0, z_{i+1})$ independently of the particular steps $z_i$ taken, and with the property $B_{\text{\rm pl}}^h(\omega,x,y)=B_{\text{\rm pl}}^h(T_x\omega,0,y-x)$. If $y$ is not accessible from $x$, pick a point $\bar x$ from which both $x$ and $y$ are accessible and set $B_{\text{\rm pl}}^h(\omega,x,y)=B_{\text{\rm pl}}^h(\omega,\bar x,y)-B_{\text{\rm pl}}^h(\omega,\bar x,x)$. This definition is independent of the point $\bar x$. Now we have a stationary cocyle $B_{\text{\rm pl}}^h$. A first step decomposition of $G^\beta_{0,(n)}(h)$ shows that cocycle \begin{equation}\label{bus13} \widetilde} \def\wh{\widehat} \def\wb{\overline B(0,z)=B_{\text{\rm pl}}^h(0,z)-h\cdot z \end{equation} satisfies Definition \ref{def:bdry-model}. Under assumption \eqref{Gplliminf} the integrability of $B_{\text{\rm pl}}^h(0,z)$ is proved exactly as in the proof of Theorem \ref{th:Bus=grad(a)}. First an upper bound: \begin{align}\label{L1-upper} \begin{split} &\varliminf_{n\to\infty}n^{-1}\sum_{k=1}^n \bE[G^\beta_{0,(k)}(h)-G^\beta_{z,(k-1)}(h)]\\ &\qquad=\varliminf_{n\to\infty}n^{-1}\sum_{k=1}^n \bE[G^\beta_{0,(k)}(h)-G^\beta_{0,(k-1)}(h)]\\ &\qquad=\varliminf_{n\to\infty}n^{-1}\bE[G^\beta_{0,(n)}(h)] \leg_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h). \end{split} \end{align} Then uniform integrability of $\bigl[n^{-1}\sum_{k=1}^n (G^\beta_{0,(k)}(h)-G^\beta_{z,(k-1)}(h))\bigr]^-$ from the lower bound \begin{equation}\label{pl-ui} G^\beta_{0,(n)}(h)-G^\beta_{z,(n-1)}(h)\ge V_0(\omega,z)+h\cdot z+\beta^{-1}\log p(z).\end{equation} By Lemma \ref{app:lm1}, $B_{\text{\rm pl}}^h(0,z)\in L^1(\bP)$ and \begin{equation}\label{bus17} -h(B_{\text{\rm pl}}^h)\cdot z = \bE[B_{\text{\rm pl}}^h(0,z)]\leg_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)\quad \text{for}\quad z\in\mathcal R. \end{equation} Define the centered stationary $L^1$ cocycle \begin{equation}\label{bus-F} F(\omega, x,y)= h(\widetilde} \def\wh{\widehat} \def\wb{\overline B)\cdot (x-y)- \widetilde} \def\wh{\widehat} \def\wb{\overline B (\omega, x,y)= h(B_{\text{\rm pl}}^h)\cdot (x-y)- B_{\text{\rm pl}}^h(\omega, x,y). \end{equation} By variational formula \eqref{eq:Lambda:K-var}, \eqref{bus13}, \eqref{bus17}, and \eqref{VBbeta} applied to $\widetilde} \def\wh{\widehat} \def\wb{\overline B$, \begin{equation}\label{bus19}\begin{aligned} &g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h) \le \bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \beta^{-1}\log \sum_{z\in\mathcal R} p(z)e^{\beta V_0(\omega,z)+\beta h\cdot z+\beta F(\omega, 0,z)}\\ &= \bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \beta^{-1}\log \sum_{z\in\mathcal R} p(z)e^{\beta V_0(\omega,z)-\beta h(B_{\text{\rm pl}}^h)\cdot z-\beta \widetilde} \def\wh{\widehat} \def\wb{\overline B(\omega, 0,z)}\\ &\le g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h) \; + \; \bP\text{-}\mathop{\mathrm{ess\,sup}}_\omega\; \beta^{-1}\log \sum_{z\in\mathcal R} p(z)e^{ \beta V_0(\omega,z) -\beta \widetilde} \def\wh{\widehat} \def\wb{\overline B(\omega, 0,z)} = g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h). \end{aligned}\end{equation} This shows that $F$ is a minimizer in \eqref{eq:Lambda:K-var}. A similar proof works for the case $\beta=\infty$. \qed \end{proofof} \begin{remark}\label{pl-cor}({\it Corrector in Theorem \ref{th:Bus=grad(b)}}) Continue with the assumptions of Theorem \ref{th:Bus=grad(b)}. We point out two sufficient conditions for concluding that $F$ of \eqref{bus-F} is not merely a minimizing cocycle for $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ as stated in Theorem \ref{th:Bus=grad(b)}, but also a corrector for $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$. By Theorem \ref{thm:minimizer}, $F$ is a corrector for $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h')$ for any $h'$ such that $h'-h(\widetilde} \def\wh{\widehat} \def\wb{\overline B)\perp\aff\mathcal R$. Since $h'-h(\widetilde} \def\wh{\widehat} \def\wb{\overline B)= h'-h(B_{\text{\rm pl}}^h)-h$, for $h'=h$ the condition is $h(B_{\text{\rm pl}}^h)\perp\aff\mathcal R$, or equivalently that $h(B_{\text{\rm pl}}^h)\cdot z$ is constant over $z\in\mathcal R$. \eqref{bus17} and \eqref{bus19} (and its analogue for $\beta=\infty$) imply that $-h(B_{\text{\rm pl}}^h)\cdot z=g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$ for at least one $z\in\mathcal R$. Hence the condition is \begin{align}\label{-h.z=g} -h(B_{\text{\rm pl}}^h)\cdot z=g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)\quad\text{for all }z\in\mathcal R. \end{align} Here are two ways to satisfy \eqref{-h.z=g}. {\rm(a)} By the first two equalities in \eqref{L1-upper}, \eqref{-h.z=g} would follow from convergence of expectations in \eqref{p2lh-lim} and Ces\`aro convergence of expectations in \eqref{buse4}: \begin{align} \bE[B_{\text{\rm pl}}^h(0,z)] &= \lim_{n\to\infty}\bE\Bigl[\,\frac1n\sum_{k=1}^n\bigl(G^\beta_{0,(k)}(h)-G^\beta_{z,(k-1)}(h)\bigr)\Bigr] \\ &= \lim_{n\to\infty}\bE\bigl[n^{-1}G^\beta_{0,(n)}(h)\bigr] = g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h). \end{align} {\rm(b)} Suppose $h$ is dual to some $\bar\xi\in\ri\mathcal U$. Then \eqref{-h.z=g} follows by this argument. First $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h(B_{\text{\rm pl}}^h)+h)=g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h(\widetilde} \def\wh{\widehat} \def\wb{\overline B))=0$ by Theorem \ref{thm:minimizer}. Then combining \eqref{eq:velocity-tilt} and \eqref{bus17} gives \begin{align}\label{auxauxaux} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}(\xi)+h\cdot\xi\le-h(B_{\text{\rm pl}}^h)\cdot\xi\leg_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h) \qquad \forall \xi\in\mathcal U. \end{align} From this $-h(B_{\text{\rm pl}}^h)\cdot\bar\xi=g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h)$. Since $\bar\xi\in\ri\mathcal U$ we can write $\bar\xi=\sum_{z\in\mathcal R} \alpha_z z $ where each $\alpha_z>0$, and now \eqref{bus17} forces \eqref{-h.z=g}. $\triangle$ \end{remark} \section{Exactly solvable models in 1+1 dimensions} \label{sec:lg+exp} We describe how the theory developed manifests itself in two well-known 1+1 dimensional exactly solvable models. The setting is the canonical one with $\Omega=\R^{\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$, $\mathcal R=\{e_1,e_2\}$, $\mathcal U=\{(s,1-s): 0\le s\le 1\}$, and i.i.d.\ weights $\{\omega_x\}_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ under $\bP$. The distributions of the weights are specified in the examples below. \subsection{Log-gamma polymer} \label{sec:lg} The log-gamma polymer \cite{sepp-12-aop} is an explicitly solvable 1+1 dimensional directed polymer model for which the approach of this paper can be carried out explicitly. Some details are in \cite{geor-rass-sepp-yilm-15}. We describe the results briefly. Fix $0<\rho<\infty$ and let $\omega_x$ be Gamma($\rho$)-distributed, i.e.\ $\bP\{{\omega_x}\le r\}=\Gamma(\rho)^{-1} \int_0^r t^{\rho-1}e^{-t}\,dt$ for $0\le r<\infty$. Inverse temperature is fixed at $\beta=1$. (Parameter $\rho$ can be viewed as temperature, see Remark 3.2 in \cite{geor-rass-sepp-yilm-15}.) The potential is $V_0(\omega)=-\log \omega_0+\log 2$. Let $\Psi_0=\Gamma'/\Gamma$ and $\Psi_1=\Psi_0'$ be the digamma and trigamma function. Utilizing the stationary version of the log-gamma polymer one can compute the point-to-point limit for $\xi=(s,1-s)$ as \begin{equation} \label{eq:lfed} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^1(\xi) =\inf_{\theta \in (0,\rho)}\{ -s\Psi_0(\theta) -(1-s)\Psi_0(\rho-\theta)\}= -s\Psi_0(\theta(\xi)) -(1-s)\Psi_0(\rho-\theta(\xi)) \end{equation} where $\theta=\theta(\xi)\in(0,\rho)$ is the unique solution of the equation \[ s\Psi_1(\theta)-(1-s)\Psi_1(\rho-\theta)=0.\] (See Theorem 2.4 in \cite{sepp-12-aop} or Theorem 2.1 in \cite{geor-sepp-13}.) From this we solve the tilt-velocity duality explicitly: tilt $h=(h_1, h_2)\in\R^2$ and velocity $\xi\in \ri\mathcal U$ are dual (Definition \ref{def:h-zeta}) if and only if \begin{equation} \label{lg-dual} h_1-h_2 =\Psi_0(\theta(\xi))-\Psi_0(\rho-\theta(\xi)). \end{equation} Then \begin{equation}\label{lg-gpl} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^1(h)=h_1-\Psi_0(\theta(\xi))=h_2-\Psi_0(\rho-\theta(\xi)). \end{equation} For all $\xi \in \ri\mathcal U$ and $h\in\R^2$, the point-to-point and point-to-line Busemann functions $B_{\text{\rm pp}}^\xi(\omega,0,z)$ and $B_{\text{\rm pl}}^{h}(\omega,0,z)$ exist as the a.s.\ limits defined by \eqref{bus3} and \eqref{buse4} (Theorems 4.1 and 6.1 in \cite{geor-rass-sepp-yilm-15}). They satisfy \begin{equation} \label{buse6} B_{\text{\rm pl}}^h(\omega, 0,z)= B_{\text{\rm pp}}^\xi(\omega,0,z)+h\cdot z \quad\text{for $z\in\mathcal R$} \end{equation} whenever $\xi$ and $h$ are dual (\cite{geor-rass-sepp-yilm-15}, Theorem 6.1). All the assumptions and conclusions of Theorems \ref{th:Bus=grad(a)}--\ref{th:Bus=grad(b)} and Remark \ref{pl-cor} are valid. The marginal distributions of the Busemann functions are given by \[ e^{-B_{\text{\rm pp}}^\xi(x,x+e_1)}\sim \textrm{Gamma}(\theta(\xi)) \quad \textrm{and} \quad e^{-B_{\text{\rm pp}}^\xi(x,x+e_2)}\sim \textrm{Gamma}(\rho-\theta(\xi)). \] Vector \[ h(B_{\text{\rm pp}}^\xi)=-\bigl(\,\bE[B_{\text{\rm pp}}^\xi(0,e_1)]\,, \bE[B_{\text{\rm pp}}^\xi(0,e_2)]\,\bigr)=\bigl(\Psi_0(\theta(\xi)), \Psi_0(\rho-\theta(\xi))\bigr) \] is dual to $\xi$ and $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^1(\xi)=-h(B_{\text{\rm pp}}^\xi)\cdot\xi$ gives the point-to-point free energy \eqref{eq:lfed}. From \eqref{buse6} we deduce $\bE[B_{\text{\rm pl}}^h(0,z)]=g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^1(h)$ for $z\in\{e_1,e_2\}$. \subsection{Corner growth model with exponential weights} \label{sec:exp} This is last-passage percolation on $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ with admissible steps $\{e_1, e_2\}$ and i.i.d.\ weights $\{ \omega_x\} _{x\in \bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2}$ with rate 1 exponential distribution. That is, $\bP\{\omega_x>s\}=e^{-s}$ for $0\le s<\infty$. The potential is $V_0(\omega)=\omega_0$ and then $G^\infty_{x,y}$ is as in \eqref{Gxy6}. This model can be viewed as the zero-temperature limit of the log-gamma polymer (Remark 4.3 in \cite{geor-rass-sepp-yilm-15}). Since the limit shape of the exponential corner growth model is known explicitly and has curvature, Busemann functions can be derived with the approach of Newman et al.\ by first proving coalescence of geodesics. This approach was carried out by Ferrari and Pimentel \cite{ferr-pime-05} (see also Sect.~8 of \cite{Cat-Pim-12}). An alternative approach that begins by constructing stationary cocycles from queueing fixed points is in \cite{geor-rass-sepp-lppbuse}. Velocity $\xi = (s,1-s)$ now selects a parameter $\alpha(\xi) =\frac{\sqrt{s}}{ \sqrt{s} + \sqrt{1-s}}\in (0,1)$ that characterizes the marginal distributions of the Busemann functions: \[ B_{\text{\rm pp}}^{\xi}(x,x+e_1) \sim \textrm{Exp}(\alpha(\xi)) \quad \textrm{ and } \quad B_{\text{\rm pp}}^{\xi}(x,x+e_2) \sim \textrm{Exp}(1-\alpha(\xi)). \] A tilt dual to $\xi \in \mathcal U$ is given by \[ h(\xi)=-\bigl(\,\bE[B_{\text{\rm pp}}^\xi(0,e_1)], \bE[B_{\text{\rm pp}}^\xi(0,e_2)]\,\bigr)=-\Bigl(\,\frac{1}{\alpha(\xi)}\,,\, \frac{1}{1- \alpha(\xi)} \,\Bigr). \] Substituting in \eqref{eq:p2p=B.xi} we obtain the well-known limit formula from Rost \cite{rost}: \[ g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^{\infty}(s,1-s) = 1 + 2\sqrt{s(1-s)}. \] \section{Variational formulas in terms of measures} \label{sec:entr} In this section we derive variational formulas for last-passage percolation in terms of probability measures on the spaces $\Omega_\ell=\Omega\times\mathcal R^\ell$ for $\ell\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+$. This section contains no new results for positive temperature models, but positive temperature results are recalled and rewritten for taking a zero-temperature limit. The formulas we obtain are zero-temperature limits of polymer variational formulas that involve entropy. A maximizing measure can be identified for polymers in weak (enough) disorder (Example \ref{ex:weak2} below). In the final Section \ref{sec:finite} we relate these measure variational formulas to Perron-Frobenius theory, the classical one for $0<\beta<\infty$ and max-plus theory for $\beta=\infty$. Return now to the setting of Section \ref{sec:free}, with general $\ell\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+$ and measurable potential $V:\Omega_\ell\to\R$. For $\beta\in(0,\infty]$ define the point-to-level and point-to-point limits $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta$ and $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)$ by Theorem \ref{th:p2p}. A generic element of $\Omega_\ell$ is denoted by $\eta=(\omega,z_{1,\ell})$ with $\omega\in\Omega$ and $z_{1,\ell}=(z_1,\dotsc, z_\ell)\in\mathcal R^\ell$. For $1\le j\le \ell$ let $Z_j(\omega, z_{1,\ell})=z_j$ denote the $j$th step variable on $\Omega_\ell$. On $\Omega_\ell$ introduce the mappings \begin{align}\label{eq:Sz} S_z(\omega,z_{1,\ell})=(T_{z_1}\omega,(z_{2,\ell-1},z)),\quad z\in\mathcal R. \end{align} When $\ell=0$, always take $\Omega_0=\Omega$, $\eta=\omega$ and $S_z=T_z$. In general, let $b\mathcal X$ denote the space of bounded measurable real-valued functions on the space $\mathcal X$. The probability measures that appear in the variational formula possess a natural invariance. This is described in the next proposition, proved at the end of the section. One manifestation of the invariance will be the following property of a probability measure $\mu\in{\mathcal M}_1(\Omega_\ell)$ for any $\ell\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+$: \begin{align} \label{eq:max>} E^\mu\big[ \max_{z\in\mathcal R} f\circ S_z \big] \ge E^\mu[ f ] \quad \text{ $\forall\, f\in b\Omega_\ell$.} \end{align} If $\ell\ge1$ and $\mu\in\mathcal{M}_1(\Omega_\ell)$, let $\mu_\ell(\cdot\,\vert\, \omega, z_{1,\ell-1})$ denote the conditional distribution of $Z_\ell$ under $\mu$, given $(\omega, z_{1,\ell-1})$. We associate to $\mu$ the following Markov transition kernel on the space $\Omega_\ell$: \begin{equation} q_z(\omega, z_{1,\ell}) \equiv q\bigl( (\omega, z_{1,\ell}), (T_{z_1}\omega, (z_{2,\ell},z))\bigr) =\mu_\ell(z\,\vert\,T_{z_1}\omega, z_{2,\ell}) . \label{q-ker1}\end{equation} The first notation is a convenient abbreviation. Under this kernel the state of the Markov chain on $\Omega_\ell$ jumps from $(\omega, z_{1,\ell})$ to $(T_{z_1}\omega, (z_{2,\ell},z))$ with probability $\mu_\ell(z\,\vert\,T_{z_1}\omega, z_{2,\ell})$ for $z\in\mathcal R$. Let $z_{k,\infty}=(z_i)_{k\le i<\infty}$ denote an infinite sequence of steps indexed by $\{k,k+1,k+2, \dotsc\}$. It is an element of $\mathcal R^{\{k,k+1,k+2, \dotsc\}}$ which we identify with $\mathcal R^\N$ in the obvious way. On the space $\Omega_\N=\Omega\times\mathcal R^\N$ define a shift mapping $S$ by $S(\omega,z_{1,\infty})=(T_{z_1}\omega,z_{2,\infty})$. Let $\mathcal{M}_s(\Omega_\N)$ denote the set of $S$-invariant probability measures on $\Omega_\N$. \begin{proposition} \label{lm:S} $ $ \hbox{} {\rm Case (a).} Let $\ell\in\N$ and $\mu\in\mathcal{M}_1(\Omega_\ell)$. Then properties {\rm(a.i)}--{\rm(a.iv)} below are equivalent. {\rm (a.i)} $\mu$ is invariant under kernel \eqref{q-ker1} defined in terms of $\mu$ itself. {\rm (a.ii)} $\mu$ is the $\Omega_{\ell}$-marginal of an $S$-invariant probability measure $\nu\in\mathcal{M}_s(\Omega_\N)$. {\rm (a.iii)} $\mu$ has property \eqref{eq:max>}. {\rm (a.iv)} $\mu$ satisfies this condition: \begin{align} E^\mu[ f(\omega, Z_{1,\ell-1}) ] = E^\mu[ f(T_{Z_1}\omega, Z_{2,\ell}) ] \quad \text{ $\forall\, f\in b\Omega_{\ell-1}$.} \label{Sell-1}\end{align} {\rm Case (b).} Let $\ell=0$ and $\mu\in\mathcal{M}_1(\Omega)$. Then properties {\rm(b.i)}--{\rm(b.iii)} below are equivalent. {\rm (b.i)} There exists a Markov kernel of the form $\{q_z(\omega)\equiv q(\omega,T_z\omega):z\in\mathcal R\}$ on $\Omega$ that fixes $\mu$. {\rm (b.ii)} $\mu$ is the $\Omega$-marginal of an $S$-invariant probability measure $\nu\in\mathcal{M}_s(\Omega_\N)$. {\rm (b.iii)} $\mu$ has property \eqref{eq:max>} with $S_z=T_z$. \end{proposition} For $\ell\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+$ let ${\mathcal M}_s(\Omega_\ell)$ denote the space of probability measures described in Proposition \ref{lm:S} above. To illustrate, if $\ell=0$ then ${\mathcal M}_s(\Omega)$ contains all $\{T_x\}$-invariant measures, and if also $0\in\mathcal R$ then ${\mathcal M}_s(\Omega)$ contains all probability measures on $\Omega$. We can now state the measure variational formulas for point-to-level and point-to-point last-passage percolation limits. For a probability measure $\mu$ on $\Omega_\ell$, $\mu_0$ denotes the $\Omega$-marginal: $\mu_0(A)=\mu(A\times\mathcal R^\ell)$. If $\ell=0$ then $\mu_0=\mu$. $V^-=-\min\{V,0\}$ is the negative part of the function $V$. \begin{theorem}\label{th:g=Hstar} Let $\bP$ be ergodic, $\ell\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+$, and assume $V\in\mathcal{L}$. Then \begin{align} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty&=\sup\Big\{E^\mu[V]:\mu\in\mathcal{M}_s(\Omega_\ell),\,\mu_0\ll\bP,\, E^\mu[V^-]<\infty\Big\}.\label{eq:g:H-var} \end{align} \end{theorem} The set in braces in \eqref{eq:g:H-var} is not empty because the measure $\mu(d\omega, z_{1,\ell})$ $=$ $\bP(d\omega)\alpha(z_1)\dotsm\alpha(z_\ell)$ is a member of $\mathcal{M}_s(\Omega_\ell)$ for any probability $\alpha$ on $\mathcal R$ and $V(\cdot\,, z_{1,\ell})\in L^1(\bP)$ by the assumption $V\in\mathcal{L}$. We state the point-to-point version only for the directed i.i.d.~$L^{d+\varepsilon}$ case defined in Remark \ref{rmk:dir-iid}. \begin{theorem}\label{th:ent2:lpp} Let $\Omega=\mathcal{S}^{\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}$ be a product of Polish spaces with shifts $\{T_x\}_{x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d}$ and an i.i.d.\ product measure $\bP$. Let $\ell\in\N$ and assume $0\notin\mathcal U$. Assume that $\forall z_{1,\ell}\in\mathcal R^\ell$, $V(\omega,z_{1,\ell}) $ is a local function of $\omega$ and a member of $L^p(\bP)$ for some $p>d$. Then for all $\xi\in\mathcal U$, \begin{align} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\infty(\xi)&=\sup\Big\{E^\mu[V]:\mu\in\mathcal{M}_s(\Omega_\ell),\,\mu_0\ll\bP,E^\mu[V^-]<\infty,E^\mu[Z_1]=\xi\Big\}.\label{eq:g:H-var:p2p} \end{align} \end{theorem} Note that even if $V$ is a function on $\Omega$ only, variational formula \eqref{eq:g:H-var:p2p} uses measures on $\Omega_\ell$ with $\ell\ge 1$ in order for the mean step condition $E^\mu[Z_1]=\xi$ to make sense. Remark \ref{rmk:tight} below explains why Theorem \ref{th:ent2:lpp} is stated only for the directed i.i.d.~$L^{d+\varepsilon}$ case. In the general setting of Theorem \ref{th:g=Hstar} the point-to-point formula \eqref{eq:g:H-var:p2p} is valid for compact $\Omega$ and $\xi\in\ri\mathcal U$. It can be derived by applying the argument given below to the results in \cite{Ras-Sep-12-arxiv}. To prepare for the proofs we discuss the positive temperature setting. In the end we take $\beta\to\infty$ to prove Theorems \ref{th:g=Hstar}--\ref{th:ent2:lpp}. Recall the random walk kernel $p$ from the beginning of Section \ref{sec:free} with ellipticity constant $\delta=\min_{z\in\mathcal R}p(z)>0$. It acts as a Markov transition kernel on $\Omega_\ell$ through \begin{align}\label{eq:def:p} &p(\eta,S_z\eta)=p(z) \, \text{ for }z\in\mathcal R\text{ and }\eta=(\omega,z_{1,\ell})\in\Omega_\ell. \end{align} This kernel defines a joint Markovian evolution $(T_{X_n}\omega, Z_{n+1,n+\ell})$ of the environment seen by the $p$-walk $X_n$ and the vector $Z_{n+1,n+\ell}=(Z_{n+1},\dotsc, Z_{n+\ell})$ of the next $\ell$ steps $Z_k=X_k-X_{k-1}$ of the walk. As before if $\ell=0$ then $S_z=T_z$ and the Markov chain is $T_{X_n}\omega$. We define an entropy $\bar H(\mu)$ for probability measures $\mu\in{\mathcal M}_1(\Omega_\ell)$, associated to this Markov chain and the background measure $\bP$. If $q(\eta,\cdot\,)$ is a Markov kernel on $\Omega_\ell$ such that $q(\eta,\cdot\,)\ll p(\eta,\cdot\,)$ $\mu$-a.s., then $q(\eta,\cdot\,)$ is supported on $\{S_z\eta\}_{z\in\mathcal R}$ and the familiar relative entropy is \[H(\mu\times q\,\vert\,\mu\times p) = \int_{\Omega_\ell} \sum_{z\in\mathcal R}q(\eta,S_z\eta)\,\log\frac{q(\eta,S_z\eta)}{p(\eta,S_z\eta)}\,\mu(d\eta).\] Set \begin{align}\label{eq:def:H} \bar H(\mu)= \begin{cases} \displaystyle\inf_{q:\,\mu q=\mu} H(\mu\times q\,\|\,\mu\times p) &\text{if }\mu_0\ll\bP\\ \infty&\text{otherwise,} \end{cases} \end{align} where the infimum is over Markov kernels $q$ on $\Omega_\ell$ that fix $\mu$, i.e.\ $\mu q(\cdot)\equiv\int q(\eta,\cdot)\mu(d\eta)=\mu(\cdot)$. The function $\bar H: \mathcal{M}_1(\Omega_\ell)\to[0,\infty] $ is convex \cite[Sect.~4]{Ras-Sep-11}. \begin{remark} When $\mu\in{\mathcal M}_s(\Omega_\ell)$ for some $\ell\ge 1$ and $\mu_0\ll\bP$, the minimizing kernel in \eqref{eq:def:H} is the one defined in \eqref{q-ker1}, and \begin{equation}\label{H79} \bar H(\mu)= H(\mu\,\vert\,\mu_{\ell-1}\otimes p) = \int_{\Omega_\ell} \,\mu(d\omega, dz_{1,\ell}) \log\frac{\mu_\ell(z_\ell\,\vert\,\omega,z_{1,\ell-1})}{p(z_\ell)} \end{equation} where $\mu_{\ell-1}$ is the distribution of $(\omega, Z_{1,\ell-1})$ under $\mu$ and $\mu_{\ell-1}\otimes p$ is the product measure on $\Omega_\ell$. Here is the argument. Let $q(\eta, S_z\eta)=q_z(\eta)$ be an arbitrary kernel that fixes $\mu$ and is supported on $\{S_z\eta\}_{z\in\mathcal R}$. The first equality below is the convex dual representation of relative entropy (see for example Theorem 5.4 in \cite{Ras-Sep-15-ldp}). In the second last equality use both $q$-invariance and \eqref{Sell-1}. \begin{align} &H(\mu\times q\,\vert\, \mu\times p)\\ &= \sup_{h\in b\Omega_\ell^2} \biggl\{ \sum_z \int\limits_{\Omega_\ell} h(\eta, S_z\eta)\,q_z(\eta) \,\mu(d\eta) - \log \sum_z p(z) \int\limits_{\Omega_\ell} e^{h(\eta, S_z\eta)} \,\mu(d\eta) \biggr\} \\ &\ge \sup_{f\in b\Omega_\ell} \biggl\{ \sum_z \int\limits_{\Omega_\ell} f( S_z\eta)\,q_z(\eta) \,\mu(d\eta) - \log \sum_z p(z) \int\limits_{\Omega_\ell} e^{f( T_{z_1}\omega, (z_{2,\ell}, z))} \,\mu(d\omega, dz_{1,\ell}) \biggr\} \\ &= \sup_{f\in b\Omega_\ell} \biggl\{ \; \int\limits_{\Omega_\ell} f \,d\mu - \log \sum_z p(z) \!\! \int\limits_{\Omega_{\ell-1}} \!\! e^{f( \omega, (z_{1,\ell-1}, z))} \,\mu_{\ell-1}(d\omega, d z_{1,\ell-1}) \biggr\} \\ &= \; H(\mu\,\vert\, \mu_{\ell-1}\otimes p).\tag{$\triangle$} \end{align} \end{remark} We state the measure variational formulas for point-to-level and point-to-point polymers in positive temperature. These are slightly altered versions of Theorem 2.3 of \cite{Ras-Sep-Yil-13} and Theorem 5.3 of \cite{Ras-Sep-14}. \begin{theorem}\label{th:var9} Let $\bP$ be ergodic, $\ell\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN_+$, $0<\beta<\infty$, and assume $V\in\mathcal{L}$. Then \begin{align} \Lapl^\beta&= \sup\Big\{ E^\mu[V]-\beta^{-1}\bar H(\mu):\mu\in\mathcal{M}_s(\Omega_\ell),\,\mu_0\ll\bP, \, E^\mu[ V^- ] < \infty\Big\} .\label{eq:var9.1} \end{align} \end{theorem} The quantity inside the braces cannot be $\infty-\infty$ for the following reason. By Proposition \ref{lm:S} every $\mu\in{\mathcal M}_s(\Omega_\ell)$ is fixed by some kernel $q$ supported on shifts. Thereby, if also $\mu_0\ll\bP$, the definition of entropy gives \begin{equation}\label{H-9} 0\le\bar H(\mu)\le \log\delta^{-1} . \end{equation} As above, we state the point-to-point version only for the directed i.i.d.~$L^{d+\varepsilon}$ case defined in Remark \ref{rmk:dir-iid}. See Remark \ref{rmk:tight} below for an explanation. \begin{theorem}\label{th:var10} Repeat the assumptions of Theorem \ref{th:ent2:lpp}. Then for $0<\beta<\infty$ and $\xi\in\mathcal U$, \begin{align} \begin{split} g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)=\sup\Big\{E^\mu[V]-\beta^{-1}\bar H(\mu):\,&\mu\in\mathcal{M}_s(\Omega_\ell),\,\mu_0\ll\bP,\\ &E^\mu[V^-]<\infty,E^\mu[Z_1]=\xi\Big\}. \end{split}\label{eq:var10.1} \end{align} \end{theorem} We illustrate formulas \eqref{eq:var9.1} and \eqref{eq:var10.1} in the case of weak disorder. \begin{example}\label{ex:weak2}({\it Directed polymer in weak disorder}) We identify first the measure $\mu$ that maximizes variational formula \eqref{eq:var9.1} for the directed polymer in weak disorder, with potential $V(\omega,z)=V_0(\omega)+h\cdot z=\omega_0+h\cdot z$ and small enough $0<\beta<\infty$. This measure will be invariant for the Markov transition implicitly contained in equation \eqref{weak:W}. We continue with the notation and assumptions from Example \ref{ex:weak1}. To define the measure we need a backward path and a martingale in the reverse time direction. The backward path $(x_k)_{k\le 0}$ satisfies $x_0=0$ and $z_k=x_k-x_{k-1}\in\mathcal R$, and the corresponding martingale is \[ W^-_n = e^{-n(\lambda(\beta)+\kappa(\beta h))} \sum_{x_{-n,0}} \abs{\mathcal R}^{-n}\,e^{\beta \sum_{k=-n}^{-1}\omega_{x_k} - \beta h\,\cdot\, x_{-n}}. \] $W^-_n$ is the same as $W_n$ composed with the reflection $\omega_x\mapsto \omega_{-x}$, and so \eqref{weak:ui} guarantees also $W^-_n\to W^-_\infty$ with the same properties. (Recall that in this example we took the uniform kernel $p(z)=\abs{\mathcal R}^{-1}$.) By \eqref{weak:W} \[ q^h_0(\omega, z)= p(z)\, e^{\beta\omega_0-\lambda(\beta)+\beta h\cdot z-\kappa(\beta h)} \frac{W_\infty(T_z\omega)}{W_\infty(\omega)} \] defines a stochastic kernel from $\Omega$ to $\mathcal R$. Define a Markov transition kernel on $\Omega\times\mathcal R$ by \begin{equation} q^h((\omega, z_{1}), (T_{z_1}\omega, z))= q^h_0(T_{z_1}\omega, z) . \label{qtheta}\end{equation} Define the probability measure $\mu^h$ on $\Omega\times\mathcal R$ as follows. For a bounded Borel function $\varphi$ \[ \sum_{z\in\mathcal R} \int_\Omega \varphi(\omega, z)\,\mu^h(d\omega, z) = \sum_{z\in\mathcal R} \int_\Omega W_\infty^-(\omega)\,W_\infty(\omega) \,q^h_0(\omega, z)\, \varphi(\omega, z) \,\bP(d\omega). \] Using the 1-step decomposition of $W_\infty^-$ (analogue of \eqref{weak:W}) one shows that $q^h$ fixes $\mu^h$. Let us strengthen assumption \eqref{weak:ui} to also include $\bE[W_\infty\log^+ W_\infty]<\infty$. This is true for small enough $\beta$. Then the entropy can be calculated: \begin{align} &H(\mu^h\times q^h\vert\mu^h\times p)\\ &\qquad=\beta E^{\mu^h}[V] -\lambda(\beta)-\kappa(\beta h) +\sum_z \int \mu^h_0(d\omega)q^h_0(\omega, z) \log \frac{W_\infty(T_z\omega)}{W_\infty(\omega)}\\ &\qquad=\beta E^{\mu^h}[V] -\lambda(\beta)-\kappa(\beta h) \end{align} because the last term of the middle member vanishes by the invariance. $E^{\mu^h}[V] $ is finite because, by independence and Fatou's lemma, \begin{align} E^{\mu^h}(\abs{\omega_0}) = \bE( \abs{\omega_0}W_\infty^-\,W_\infty)\le \varliminf_{n\to\infty} \bE( \abs{\omega_0}W_n) \end{align} while the last sequence is bounded, as can be seen by utilizing the 1-step decomposition \eqref{weak:W} and by taking $\beta$ in the interior of the region $\lambda(\beta)<\infty$. Consequently \begin{equation}\label{weak:H9} E^{\mu^h}[V] - \beta^{-1} H(\mu^h\times q^h\vert\mu^h\times p) = \beta^{-1}(\lambda(\beta)+\kappa(\beta h) ) = g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta(h). \end{equation} The pair $(\mu^h, q^h)$ is the unique one that satisfies \eqref{weak:H9}, by virtue of the strict convexity of entropy. The maximizer for the point-to-point formula \eqref{eq:var10.1} can also be found. Let $g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi)$ be as in \eqref{p2pV_0} with $V_0(\omega)=\omega_0$. Given $\xi\in\ri\mathcal U$, $h\in\R^d$ can be chosen so that $\nabla\kappa(\beta h)=\xi$. If $\beta$ is small enough, uniform integrability of the martingales $W_n$ can be ensured, and thereby $\mu^h$ and $q^h$ are again well-defined. The choice of $h$ implies that $E^{\mu^h}[Z_1]=\xi$, and we can turn \eqref{weak:H9} into \begin{align} E^{\mu^h}[V_0] - \beta^{-1} H(\mu^h\times q^h\vert\mu^h\times p) &= - h\cdot E^{\mu^h}[Z_1] + \beta^{-1}(\lambda(\beta)+\kappa(\beta h) ) \\ & = \beta^{-1}\lambda(\beta) - \beta^{-1}\kappa^(\xi)= g_{\text{\rm pp}}} \def\bargpp{\bar g_{\text{\rm pp}}} \def\Lapp{g_{\text{\rm pp}}^\beta(\xi). \end{align} The last equality can be seen for example from duality \eqref{eq:tilt-velocity}. Markov chain \eqref{qtheta} appeared in \cite{come-yosh-aop-06}. Under some restrictions on the environment and with $h=0$, \cite{more-10} showed that $\mu^0_0$ is the limit of the environment seen by the particle. $\triangle$ \end{example} We prove the theorems of this section, beginning with the positive temperature statements. \begin{proofof}{of Theorems \ref{th:var9} and \ref{th:var10}} Let $V: \Omega_\ell\to\R$ be a member of $\mathcal{L}$ (Definition \ref{def:cL}), $\bP$ ergodic and $0<\beta<\infty$. Theorem 2.3 of \cite{Ras-Sep-Yil-13} gives the variational formula \begin{align} \Lapl^\beta&= \sup\Big\{ E^\mu[\min(V,c)]-\beta^{-1}\bar H(\mu):\mu\in\mathcal{M}_1(\Omega_\ell),\ c>0\Big\} .\label{eq:Lambda:H-var} \end{align} Note that \cite{Ras-Sep-Yil-13} used the uniform kernel $p(z)=\abs{\mathcal R}^{-1}$ but this makes no difference to the proofs, and in any case the kernel can be included in the potential to extend the result to an arbitrary kernel supported on $\mathcal R$. We convert \eqref{eq:Lambda:H-var} into \eqref{eq:var9.1} in a few steps. The measure $\mu=\bP\otimes\alpha$ with $\alpha(z_{1,\ell})=p(z_{1,\ell})$ satisfies $\mu\in{\mathcal M}_s(\Omega_\ell)$, $\mu p=\mu$, and $\bar H(\mu)=0$. Since $V(\cdot\,,z_{1,\ell})\in L^1(\bP)$, this gives the finite lower bound $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\beta\ge E^{\bP\otimes\alpha}[V]$ for \eqref{eq:Lambda:H-var}. (If $\ell=0$ the $\alpha$-factor is not there.) Hence we can restrict the supremum in \eqref{eq:Lambda:H-var} to $\mu$ such that $E^\mu[ V^- ] + \bar H(\mu) < \infty$. Since $E^\mu[V]$ is well-defined in $(-\infty,\infty]$ for all such $\mu$, we can drop the truncation at $c$. Entropy has the following representation: for $\mu\in{\mathcal M}_1(\Omega_\ell)$, \begin{equation}\label{entr-5} \inf_{q: \mu q=\mu}H(\mu\times q\,\|\,\mu\times p) =-\inf_{f\in b\Omega_\ell} E^\mu\big[\log\sum_zp(z)e^{f\circ S_z-f}\big]. \end{equation} The infimum on the left is over Markov kernels $q$ on $\Omega_\ell$ that fix $\mu$. $S_z$ is the shift mapping defined in \eqref{eq:Sz}. For a proof of \eqref{entr-5} see Theorem 2.1 of \cite{Don-Var-76}, Lemma 2.19 of \cite{Sep-93-ptrf-1}, or Theorem 14.2 of \cite{Ras-Sep-15-ldp}. Recall the definition of $\bar H$ in \eqref{eq:def:H}. From the inequality \[\log\sum_zp(z)e^{f\circ S_z-f}\le \max_z \{f\circ S_z-f\}\le\log\sum_zp(z)e^{f\circ S_z-f}+\log\delta^{-1}\] follows, for $\mu_0\ll\bP$, \begin{align}\label{eq:interesting} \bar H(\mu)-\log\delta^{-1}\le -\inf_{f\in b\Omega_\ell} E^\mu\big[\max_z \{f\circ S_z-f\}\big]\le \bar H(\mu). \end{align} If there exists $f\in b\Omega_\ell$ such that $E^\mu[\max_z \{f\circ S_z-f\}]<0$ then replacing $f$ by $cf$ and taking $c\to\infty$ shows that the infimum over $f$ is actually $-\infty$. This makes $\bar H(\mu)=\infty$. Thus, relevant measures $\mu$ in \eqref{eq:Lambda:H-var} are ones that satisfy \eqref{eq:max>} and so we can insert the restriction $\mu\in\mathcal{M}_s(\Omega_\ell)$ into \eqref{eq:Lambda:H-var}. \eqref{eq:Lambda:H-var} has been converted into \eqref{eq:var9.1}. Assuming the directed i.i.d.~$L^{d+\varepsilon}$ setting described in Theorem \ref{th:ent2:lpp}, Theorem 5.3 of \cite{Ras-Sep-14} gives the point-to-point version: for $\xi\in\mathcal U$, \begin{equation} \label{eq:rhs-var-p2p} \Lapp^\beta(\xi)=\sup\bigl\{ E^\mu[\min(V,c)]-\beta^{-1}\bar H(\mu):\mu\in\mathcal{M}_1(\Omega_\ell),\ E^\mu[Z_1]=\xi,\ c>0\bigr\}. \end{equation} This is converted into \eqref{eq:var10.1} by the same reasoning used above. \qed \end{proofof} \begin{remark}\label{rmk:tight} We can state \eqref{eq:rhs-var-p2p} only for the directed i.i.d.~$L^{d+\varepsilon}$ setting for the following reason. The point-to-level formula \eqref{eq:Lambda:H-var} is proved directly in \cite{Ras-Sep-Yil-13}. By contrast, the point-to-point formula \eqref{eq:rhs-var-p2p} is derived in \cite{Ras-Sep-14} via a contraction applied to a quenched large deviation principle (LDP) for polymer measures. This LDP is proved in \cite{Ras-Sep-Yil-13}. In the general setting the upper bound of this LDP has been proved only for compact sets (weak LDP). However, in the directed i.i.d.\ case the LDP is a full LDP, and the contraction works without additional assumptions. Consequently in the directed i.i.d.~$L^{d+\varepsilon}$ setting \eqref{eq:rhs-var-p2p} is valid for Polish spaces $\Omega$, but in the general setting $\Omega$ would need to be compact. $\triangle$ \end{remark} \begin{proofof}{of Theorems \ref{th:g=Hstar} and \ref{th:ent2:lpp}} Take $\beta\to\infty$ in \eqref{eq:var9.1} and \eqref{eq:var10.1}, utilizing bounds \eqref{H-9} and \eqref{eq:g-Lambda}. \qed \end{proofof} \begin{proofof}{of Proposition \ref{lm:S}} Each $f$ below is a $b\Omega_\ell$ test function on the appropriate space $\Omega_\ell$. First we work with the case $\ell\ge1$. We argue the implications (a.i)$\Rightarrow$(a.ii)$\Rightarrow$(a.iii)$\Rightarrow$(a.iv)$\Rightarrow$(a.i). (a.i)$\Rightarrow$(a.ii): An $S$-invariant probability measure $\nu$ on $\Omega_\N=\Omega\times\mathcal R^{\mathbb{N}}$ that extends $\mu$ can be defined by writing, for any $m\ge \ell$, \begin{equation} \int f(\omega, z_{1,m})\,d\nu = \sum_{z_{1,m}} \int_\Omega f(\omega, z_{1,m})\, \prod_{i=\ell+1}^m q_{z_{i}}(T_{x_{i-\ell-1}}\omega, z_{i-\ell, i-1}) \, \mu(d\omega, z_{1,\ell}). \label{nu1}\end{equation} (a.ii)$\Rightarrow$(a.iii): From the $S$-invariance of $\nu$, \begin{align} &E^\mu\big[\max_z f(T_{Z_1}\omega,(Z_{2,\ell},z))\big]=E^\nu\big[\max_z f(T_{Z_1}\omega,(Z_{2,\ell},z))\big]\\ &\qquad =E^\nu\big[\max_z f(\omega,(Z_{1,\ell-1},z))\big] \ge E^\nu\big[ f(\omega,Z_{1,\ell})\big] =E^\mu[f]. \end{align} (a.iii)$\Rightarrow$(a.iv): If $f$ is only a function of $(\omega,z_{1,\ell-1})$, then $f(S_z(\omega,z_{1,\ell}))=f(T_{z_1}\omega,z_{2,\ell})$ does not depend on $z$. \eqref{eq:max>} then implies $E^\mu\big[f(T_{Z_1}\omega,Z_{2,\ell})]\ge E^\mu[ f ]$. Replacing $f$ by $-f$ makes this an equality and \eqref{Sell-1} follows. (a.iv)$\Rightarrow$(a.i): Use property (a.iv) in the second equality below to show that $\mu q=\mu$. \begin{align} &\int_{\Omega\times\mathcal R^{\ell}} \sum_z q_z(\omega, z_{1,\ell}) f(T_{z_1}\omega, (z_{2,\ell},z)) \, \mu(d\omega, dz_{1,\ell})\\ &= \sum_z \int_{\Omega\times\mathcal R^{\ell}} f(T_{z_1}\omega, (z_{2,\ell},z)) \, \mu_\ell(z\,\vert\, T_{z_1}\omega, z_{2,\ell})\, \mu(d\omega, dz_{1,\ell})\\ &= \sum_z \int_{\Omega\times\mathcal R^{\ell}} f(\omega, (z_{1,\ell-1},z)) \, \mu_\ell(z\,\vert\, \omega, z_{1,\ell-1})\, \mu(d\omega, dz_{1,\ell})\\ &= \int_{\Omega\times\mathcal R^{\ell}} f(\omega, z_{1,\ell}) \, \mu(d\omega, dz_{1,\ell}) . \end{align} We turn to the case $\ell=0$ and show (b.i)$\Rightarrow$(b.ii)$\Rightarrow$(b.iii)$\Rightarrow$(b.i). (b.i)$\Rightarrow$(b.ii): Now define $\nu$ on $\Omega\times\mathcal R^{\mathbb{N}}$ by \[E^\nu[f(\omega,Z_{1,m})]=\sum_{z_{1,m}}\int f(\omega,z_{1,m}) \prod_{i=1}^{m} q_{z_i}(T_{x_{i-1}}\omega)\,\mu(d\omega).\] (b.ii)$\Rightarrow$(b.iii): Analogously to (a.ii)$\Rightarrow$(a.iii) above, \begin{align} E^\mu\big[\max_z f(T_{z}\omega)\big]&=E^\nu\big[\max_z f(T_{z}\omega)\big] \ge E^\nu\big[ f(T_{Z_1}\omega)\big]=E^\nu[f(\omega)] =E^\mu[f]. \end{align} (b.iii)$\Rightarrow$(b.i): Observe that for $f\in b\Omega$ we have \[E^\mu\big[\max_z \{f\circ T_z-f\}\big]\le E^\mu\big[\log\sum_zp(z)e^{f\circ T_z-f}\big]+\log\delta^{-1}.\] By assumption \eqref{eq:max>} the left-hand side is nonnegative. Then by \eqref{entr-5} \begin{align} \inf\{H(\mu\times q\,\|\,\mu\times p):\mu q=\mu\} =-\inf_{f\in b\Omega} E^\mu\big[\log\sum_zp(z)e^{f\circ T_z-f}\big]\le\log\delta^{-1}. \end{align} Since the infimum is not $+\infty$ there must exist a Markov kernel $q$ that fixes $\mu$ and for which $H(\mu\times q\,\|\,\mu\times p)<\infty$. This implies that for $\mu$-a.e.\ $\omega$ the kernel is supported on $\{T_z\omega:z\in\mathcal R\}$. \qed \end{proofof} \section{Periodic environments} \label{sec:finite} The case of finite $\Omega$ provides explicit illustration of the theory developed in the paper. The point-to-level limits and solutions to the variational formulas come from Perron-Frobenius theory, the classical theory for $0<\beta<\infty$ and the max-plus theory for $\beta=\infty$. (See \cite{bacc-cohe-etal-book, berm-plem-book, heid-olds-woud, sene-book} for expositions.) We consider a potential $V(\omega,z)=V_0(\omega)+h\cdot z$ for $(\omega,z)\in\Omega\times\mathcal R$, $h\in\R^d$. Let $\Omega$ be a finite set of $m$ elements. As all along, $\{T_x\}_{x\in\mathcal G}$ is a group of commuting bijections on $\Omega$ that act irreducibly. That is, for each pair $(\omega,\omega')\in\Omega\times\Omega$ there exist $z_1,\dotsc, z_k\in\mathcal R$ such that $T_{z_1+\dotsm+ z_k}\omega=\omega'$. The ergodic probability measure is $\bP(\omega)=m^{-1}$. A basic example is a periodic environment indexed by $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$. Take a vector $a>0$ in $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$ (coordinatewise inequalities), define the rectangle $\Lambda=\{ x\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d: 0\le x<a \}$, fix a finite configuration $(\bar\omega_x)_{x\in\Lambda}$, and then extend $\bar\omega$ to all of $\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$ periodically: $\bar\omega_{x+k\circ a}=\bar\omega_x$ for $k\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^d$, where $k\circ a=(k_ia_i)_{1\le i\le d}$ is the coordinatewise product of two vectors. Irreducibility holds for example if $\mathcal R$ contains $\{e_1,\dotsc, e_d\}$. \subsection{Case $0<\beta<\infty$} We take $\beta=1$ and drop it from the notation. Define a nonnegative irreducible matrix indexed by $\Omega$ by \begin{equation}\label{pf-R} A_{\omega, \omega'}= \sum_{z\in\mathcal R}p(z)\, \mbox{\mymathbb{1}}\{ T_z\omega=\omega'\,\}e^{V_0(\omega)+h\cdot z} \quad \text{for $\omega,\omega'\in\Omega$.} \end{equation} Let $\rho$ be the Perron-Frobenius eigenvalue (spectral radius) of $A$. Then by standard asymptotics the limiting point-to-level free energy is \begin{equation}\begin{aligned} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}(h) &= \lim_{n\to\infty} n^{-1}\log \sum_{x_{0,n}: \,x_0=0} p(x_{0,n}) e^{\sum_{k=0}^{n-1} V_0(T_{x_k}\omega) + h\cdot x_n} \\ &=\lim_{n\to\infty} n^{-1}\log \sum_{\omega'\in\Omega} A^n_{\omega, \omega'} =\log\rho. \end{aligned}\label{pf:gpl}\end{equation} On a finite $\Omega$ every cocycle is a gradient (proof left to the reader). Hence we can replace the general cocycle $F$ with a gradient $F(\omega, 0, z)=f(T_z\omega)-f(\omega)$ and write the cocycle variational formula \eqref{eq:Lambda:K-var} as \begin{equation}\label{pf:var-h} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}(h)=\inf_{f\in\R^\Omega} \,\max_\omega\; \log \sum_{z\in\mathcal R} p(z)e^{V_0(\omega)+ h\cdot z+f(T_z\omega)-f(\omega)}. \end{equation} This is now exactly the same as the following textbook characterization of the Perron-Frobenius eigenvalue: \begin{equation}\label{pf:char1} \rho = \inf_{\varphi\in\R^\Omega:\,\varphi>0} \; \max_\omega \; \frac1{\varphi(\omega)} \sum_{\omega'} A_{\omega, \omega'} \varphi(\omega'). \end{equation} Let $\sigma} \def\rev{\tau$ and $\rev$ be the left and right (strictly positive) Perron-Frobenius eigenvectors of $A$ normalized so that $\sum_{\omega\in\Omega} \sigma} \def\rev{\tau(\omega)\rev(\omega)=1$. For each $\omega\in\Omega$ the left eigenvector equation is \begin{equation}\label{lev} \sum_{z\in\mathcal R}p(z)\, e^{V_0(T_{-z}\omega)+h\cdot z} \sigma} \def\rev{\tau(T_{-z}\omega)=\rho\sigma} \def\rev{\tau(\omega) \end{equation} and the right eigenvector equation is \begin{equation}\label{rev} e^{V_0(\omega)}\sum_{z\in\mathcal R} p(z) e^{h\cdot z} \rev(T_z\omega)=\rho\rev(\omega). \end{equation} The right eigenvector equation \eqref{rev} says that the gradient \begin{equation}\label{pf:F} F(\omega,x,y)=\log\rev(T_y\omega)-\log\rev(T_x\omega)\end{equation} minimizes in \eqref{pf:var-h} without the maximum over $\omega$ (the right-hand side of \eqref{pf:var-h} is constant in $\omega$). In other words, $F$ is a corrector for $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}(h)$. Compare this to \eqref{eq:Kvar:minbeta}. Define a probability measure on $\Omega$ by $\mu_0(\omega)=\sigma} \def\rev{\tau(\omega)\rev(\omega)$. The left eigenvector equation \eqref{lev} says that $\mu_0$ is invariant under the stochastic kernel \begin{equation}\label{pf-q} q_0(\omega,\omega')= \sum_{z\in\mathcal R}p(z)\, \mbox{\mymathbb{1}}\{ T_z\omega=\omega'\}e^{V_0(\omega)+h\cdot z+ F(\omega,0,z)-g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}(h)}, \quad \omega, \omega'\in\Omega. \end{equation} Using this one can check that the measure \[ \mu(\omega, z_1) = p(z) {\mu_0(\omega)} e^{V_0(\omega)+h\cdot z_1+ F(\omega,0,z_1)-g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}(h)} \] is a member of ${\mathcal M}_s(\Omega\times\mathcal R)$ and invariant under the kernel \[q( (\omega, z_1), (T_{z_1}\omega, z)) = p(z) e^{V_0(T_{z_1}\omega)+h\cdot z+ F(T_{z_1}\omega,0,z)-g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}(h)}. \] Another computation checks that \[ E^\mu[V_0(\omega)+h\cdot Z_1]- H(\mu\times q\,\vert \,\mu\times p) = g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}(h). \] Hence $\mu$ is a maximizer in the entropy variational formula \eqref{eq:Lambda:H-var}. Assume additionally that matrix $A$ is aperiodic on $\Omega$. Then $A$ is primitive, that is, $A^n$ is strictly positive for large enough $n$. Perron-Frobenius asymptotics (for example, Theorem 1.2 in \cite{sene-book}) give the Busemann function $B_{\text{\rm pl}}^h$ of \eqref{buse4}. \begin{align} & B_{\text{\rm pl}}^h(\omega,0,z) =\lim_{n\to\infty} \biggl\{ \log \sum_{x_{0,n}: \, x_0=0} p(x_{0,n}) e^{\sum_{k=0}^{n-1}V_0(T_{x_k}\omega)+h\cdot x_n} \\ &\qquad\qquad \qquad\qquad \;-\; \log \sum_{x_{0,n-1}: \,x_0=z} p(x_{0,n-1}) e^{\sum_{k=0}^{n-2}V_0(T_{x_k}\omega)+h\cdot (x_{n-1}-z)} \biggr\} \\ & =\lim_{n\to\infty} \biggl\{ \log \sum_{\omega'\in\Omega} A^n_{\omega, \omega'} - \log \sum_{\omega'\in\Omega} A^{n-1}_{T_z\omega, \omega'}\biggr\}\\ &=\lim_{n\to\infty} \biggl\{ \log\rho + \log \Bigl( \sum_{\omega'\in\Omega} \rev(\omega)\sigma} \def\rev{\tau(\omega') + o(1)\Bigr) - \log \Bigl( \sum_{\omega'\in\Omega} \rev(T_z\omega)\sigma} \def\rev{\tau(\omega') + o(1)\Bigr)\biggr\} \\ &= \log\rho + \log \rev(\omega) - \log \rev(T_z\omega) . \end{align} If we assume that all admissible paths between two given points have the same number of steps, then $ B_{\text{\rm pl}}^h(\omega,0,z)$ extends to a stationary $L^1$ cocycle, as showed in Theorem \ref{th:Bus=grad(b)}. Then this situation fits the development of Sections \ref{sec:corr}--\ref{sec:bus}. Equation \eqref{rev} shows that cocycle \begin{equation}\label{pf:B5} \widetilde} \def\wh{\widehat} \def\wb{\overline B(\omega,0,z)=B_{\text{\rm pl}}^h(\omega,0,z)-h\cdot z\end{equation} is adapted to $V_0$, illustrating Theorem \ref{th:Bus=grad(b)}. Definition \eqref{EB} applied to the explicit formulas above gives \[ h(\widetilde} \def\wh{\widehat} \def\wb{\overline B)\cdot z=-\bE[\widetilde} \def\wh{\widehat} \def\wb{\overline B(\omega,0,z)]= -\log \rho+ h\cdot z \qquad \text{ for each $z\in\mathcal R$. } \] Consequently $h(B_{\text{\rm pl}}^h)\perp\aff\mathcal R$. By Theorem \ref{thm:minimizer} the cocycle \begin{align}\label{pf:F5} \begin{split} \widetilde} \def\wh{\widehat} \def\wb{\overline F(\omega, 0,z)&=-h(\widetilde} \def\wh{\widehat} \def\wb{\overline B)\cdot z - \widetilde} \def\wh{\widehat} \def\wb{\overline B(\omega,0,z)=\log \rho-B_{\text{\rm pl}}^h(\omega, 0,z)\\ &=\log\rev(T_z\omega)-\log\rev(\omega), \end{split} \end{align} that appeared in \eqref{pf:F}, is the minimizer in \eqref{pf:var-h} for any tilt $ h'$ such that $(h'-h(\widetilde} \def\wh{\widehat} \def\wb{\overline B))\cdot z= (h'-h)\cdot z +\log \rho $ is constant over $z\in\mathcal R$. Connection \eqref{pf:gpl} between the limiting free energy and the Perron-Frobenius eigenvalue is standard fare in textbook treatments of the large deviation theory of finite Markov chains \cite{demb-zeit, stro-ldp-84, Ras-Sep-15-ldp}. \subsection{Point-to-level last-passage case} The {\sl max-plus algebra} is the semiring $\R_{\text{max}}$ $=$ $\R\cup\{-\infty\}$ under the operations $ x\oplus y=x\vee y $ and $x\otimes y=x+y$. Define an irreducible $\R_{\text{max}}$-valued matrix by \begin{equation}\label{pf-R2} A(\omega, \omega')= \begin{cases} \displaystyle V_0(\omega)+\max_{z:T_z\omega=\omega'} h\cdot z, &\omega'\in\{T_z\omega: z\in\mathcal R\} \\ -\infty, &\omega'\notin\{T_z\omega: z\in\mathcal R\} .\end{cases} \end{equation} As an irreducible matrix $A$ has a unique finite max-plus eigenvalue $\lambda$ together with a (not necessarily unique even up to an additive constant) finite eigenvector $\sigma$ that satisfy \begin{equation}\label{pf:max+4} \max_{\omega'\in\Omega}\, [ A(\omega,\omega')+\sigma(\omega')] = \lambda + \sigma(\omega), \qquad \omega\in\Omega. \end{equation} Inductively \begin{equation}\label{pf:max+5} \max_{\omega=\omega_0,\,\omega_1,\dotsc,\,\omega_n} \Bigl\{ \; \sum_{k=0}^{n-1} A(\omega_k,\omega_{k+1}) +\sigma(\omega_n) \Bigr\} =n \lambda + \sigma(\omega), \qquad \omega\in\Omega. \end{equation} The last-passage value from \eqref{p2lh} can be expressed as \begin{align} \label{pf:max+5.5} G_{0,(n)}^\infty(h)&= \max_{x_{0,n}} \sum_{k=0}^{n-1}\bigl( V_0(T_{x_k}\omega) + h\cdot (x_{k+1}-x_k) \bigr) \\ \nonumber &=\max_{\omega=\omega_0,\,\omega_1,\dotsc,\,\omega_n} \sum_{k=0}^{n-1} A(\omega_k,\omega_{k+1}) . \end{align} Dividing through \eqref{pf:max+5} by $n$ gives the limit \begin{align} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)= \lim_{n\to\infty} n^{-1}G_{0,(n)}^\infty(h) = \lambda. \end{align} The eigenvalue equation \eqref{pf:max+4} now rewrites as \begin{equation}\label{pf:max+6} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)= \max_{z\in\mathcal R} \{V_0(\omega)+ h\cdot z+ \sigma(T_z\omega)-\sigma(\omega) \}. \end{equation} This is the cocycle variational formula \eqref{eq:g:K-var} (without the supremum over $\omega$) and shows that a corrector is given by the gradient \begin{equation}\label{pf:F13} F(\omega,0,z)= \sigma(T_z\omega)-\sigma(\omega). \end{equation} Compare \eqref{pf:max+6} to \eqref{eq:Kvar:min}. The measure variational formula \eqref{eq:var9.1} links with an alternative characterization of the max-plus eigenvalue as the maximal average weight of an elementary circuit. To describe this, consider the directed graph $(\Omega, {\mathcal E})$ with vertex set $\Omega$ and edges ${\mathcal E}=\{(\omega,T_z\omega):\omega\in\Omega, z\in\mathcal R\}$. This allows multiple edges from $\omega$ to $\omega'$ and loops from $\omega$ to itself. Loops happen in particular if $0\in\mathcal R$. Identify edge $(\omega,T_z\omega)$ with the pair $(\omega, z)$. An {\sl elementary circuit} of length $N$ is a sequence of edges $(\omega_0, z_1), (\omega_1, z_2), \dotsc, (\omega_{N-1}, z_N)$ such that $\omega_i=T_{z_i}\omega_{i-1}$ with $\omega_N=\omega_0$, but $\omega_i\ne \omega_j$ for $0\le i<j<N$. Given any fixed $\omega$, all elementary circuits can be represented as admissible paths $x_0, x_1$, $\dotsc$, $x_N$ in $\mathcal G$ by choosing $x_0$ so that $\omega_0=T_{x_0}\omega$ and $x_i=x_{i-1}+z_i$ for $1\le i\le N$. Conversely, an admissible path $x_0, x_1, \dotsc, x_N$ in $\mathcal G$ represents an elementary circuit if $T_{x_0}\omega, T_{x_1}\omega, \dotsc, T_{x_{N-1}}\omega$ are distinct, but $T_{x_0}\omega=T_{x_N}\omega$. Let $\mathcal{C}} \def\cD{\mathcal{D}$ denote the set of elementary circuits. The average weight formula for the eigenvalue is (Thm.~2.9 in \cite{heid-olds-woud}) \begin{equation}\label{pf:aa} \lambda= \max_{N\in\N, \, x_{0,N}\in\mathcal{C}} \def\cD{\mathcal{D}} N^{-1}\sum_{k=0}^{N-1} \bigl( V_0(T_{x_k}\omega)+h\cdot z_{k+1}\bigr). \end{equation} The right-hand side is independent of $\omega$ because switching $\omega$ amounts to translating the circuit, by the assumption of irreducible action by $\{T_z\}_{z\in\mathcal R}$. It is elementary to verify from definitions that $g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^\infty(h)$ equals the right-hand side of \eqref{pf:aa}. (The sum on the right-hand side of \eqref{pf:max+5.5} decomposes into circuits and a bounded part, while an asymptotically optimal path finds a maximizing circuit and repeats it forever.) If we take \eqref{pf:aa} as the definition of $\lambda$, then the identity \begin{equation}\label{pf:max+9} \lambda = \max\Bigl\{\, \sum_{(\omega,z) \in\Omega\times\mathcal R}\mu(\omega,z)(V_0(\omega) +h\cdot z) : \mu\in\mathcal{M}_s(\Omega\times\mathcal R)\Bigr\} \end{equation} follows from the fact that the extreme points of the convex set $\mathcal{M}_s(\Omega\times\mathcal R)$ are exactly those uniform probability measures whose support is a single elementary circuit. We omit the proof. Equation \eqref{pf:max+9} is the measure variational formula \eqref{eq:var9.1} which has now been (re)derived in the finite setting from max-plus theory. As in the finite temperature case, existence of point-to-level Busemann functions follows from asymptotics of matrices. The {\sl critical graph} of the max-plus matrix $A$ is the subgraph of $(\Omega, {\mathcal E})$ consisting of those nodes and edges that belong to elementary circuits that maximize in \eqref{pf:aa}. Matrix $A$ is {\sl primitive} if it is irreducible and if its critical graph has a unique strongly connected component with cyclicity $1$ (that is, a unique irreducible and aperiodic component in Markov chain terminology). This implies that the eigenvector is unique up to an additive constant and these asymptotics hold as $n\to\infty$: \begin{equation} \begin{split} G_{0,(n)}^\infty(h)- G_{z,(n-1)}^\infty(h) &= (A^{\otimes n}\otimes\mathbf0)(\omega) - (A^{\otimes (n-1)}\otimes\mathbf0)(T_z\omega) \\ &\phantom{xxxxx}\longrightarrow \lambda + \sigma(\omega) - \sigma(T_z\omega)\equiv B_{\text{\rm pl}}^h(\omega,0,z). \label{eq:per:p2l:bus} \end{split} \end{equation} (From \cite{heid-olds-woud} apply Thm.~3.9 with cyclicity 1 and section 4.3.) Above $\mathbf0 = (0,\ldots,0)^T$ and operations $\otimes$ are in the max-plus sense. Equation \eqref{pf:max+6} shows that cocycle $\widetilde} \def\wh{\widehat} \def\wb{\overline B(\omega,0,z)=B_{\text{\rm pl}}^h(\omega,0,z)-h\cdot z$ is adapted to $V_0$, as an example of Theorem \ref{th:Bus=grad(b)} for $\beta=\infty$. The next simple example illustrates the max-plus case. All the previous results of this paper identify correctors that solve the variational formulas of Theorem \ref{th:K-var} so that the essential supremum over $\omega$ can be dropped. This example shows that there can be additional minimizing cocycles $F$ for which the function of $\omega$ on the right in \eqref{eq:g:K-var} is not constant in $\omega$. \begin{figure} \caption{Environment configuration $\omega^{(1)}$ indexed by $\bZ} \label{fig:tile} \end{figure} \begin{example} \label{ex:stripes1} Take $d=2$ and a two-point environment space $\Omega=\{\omega^{(1)}, \omega^{(2)}=T_{e_1}\omega^{(1)}\}$ where $\omega^{(1)}_{i,j}=\tfrac12(1+(-1)^i)$ for $(i,j)\in\bZ} \def\Q{\bQ} \def\R{\bR}\def\N{\bN^2$ is a vertically striped configuration of zeroes and ones, with a one at the origin (Figure \ref{fig:tile}). Admissible steps are $\mathcal R=\{e_1, e_2\}$ and $T_{e_2}$ acts as an identity. The ergodic measure is $\bP=\tfrac12(\delta_{\omega^{(1)}}+ \delta_{\omega^{(2)}})$ and the potential $V_0(\omega)=\omega_0$ with tilts $h=(h_1, h_2)\in\R^2$. Matrix $A(\omega^{(i)}, \omega^{(j)})$ of \eqref{pf-R2} is \[ A= \begin{bmatrix} 1+h_2 &1+ h_1\\ h_1 & h_2\\ \end{bmatrix} \] and the directed graph $(\Omega, {\mathcal E})$ is in Figure \ref{pf:gr-fig}. \begin{figure} \caption{Graph $(\Omega, {\mathcal E})$ for Example \ref{ex:stripes1}.} \label{pf:gr-fig} \end{figure} Since $A$ is irreducible its unique max-plus eigenvalue is the maximum average value of elementary circuits and this gives the point-to-line last-passage limit: \begin{equation} \label{eq:lambda:R} g_{\text{\rm pl}}} \def\Lapl{g_{\text{\rm pl}}^{\infty}(h)= \lambda = \max\{ \tfrac12 +h_1, 1+h_2 \}. \end{equation} There are two cases to consider, and in both cases there is a unique eigenvector (up to an additive constant) $\sigma=(\sigma(\omega^{(1)}), \sigma(\omega^{(2)}))$: \begin{enumerate} \item[(i)] $ \tfrac12+ h_1 \le 1+ h_2= \lambda$, $\sigma=(1, h_1-h_2)$, the critical graph has cyclicity 1. \item[(ii)] $1+h_2 < \tfrac12+ h_1 = \lambda$, $\sigma=(1, \tfrac12)$, the critical graph has cyclicity 2. \end{enumerate} {\sl Case {\rm(}i{\rm)}.} One can verify by hand that variational formula \eqref{eq:g:K-var} is minimized by the cocycles \begin{equation}\label{se:F9} F(\omega^{(1)}, 0, e_1)=a=-F(\omega^{(2)}, 0, e_1), \quad F(\omega^{(1)}, 0, e_2)=F(\omega^{(2)}, 0, e_2)=0 \end{equation} for $a\in[h_1-h_2-1, h_2-h_1]$. Let $\widetilde} \def\wh{\widehat} \def\wb{\overline F$ denote the cocycle for $a=h_1-h_2-1$ which is the one consistent with \eqref{pf:F13} for the eigenvector $\sigma$. Among the minimizing cocycles only $\widetilde} \def\wh{\widehat} \def\wb{\overline F$ satisfies \eqref{eq:g:K-var} without $\max_\omega$, that is, in the form \eqref{eq:Kvar:min}. And indeed this corrector comes from Theorem \ref{thm:minimizer}(ii-b). $\widetilde} \def\wh{\widehat} \def\wb{\overline F$ is given by equation \eqref{FF} with a cocycle $\widetilde} \def\wh{\widehat} \def\wb{\overline B$ that is adapted to $V_0$ (as defined in \eqref{VB}) if and only if $1+h_2\ge \tfrac12+h_1$. In case (i) matrix $A$ is primitive and limit \eqref{eq:per:p2l:bus} gives an explicit Busemann function $ B_{\text{\rm pl}}^h(\omega,0,z)$. From this Busemann function \eqref{pf:B5} gives cocycle $\widetilde} \def\wh{\widehat} \def\wb{\overline B$. {\sl Case {\rm(}ii{\rm)}.} In this case there is a unique minimizing corrector $\check F$ which is \eqref{se:F9} with $a=-1/2$, the one that satisfies \eqref{pf:F13} for the eigenvector $\sigma$. $\check F$ comes via equation \eqref{FF} from a cocycle that is adapted to $V_0$ if and only if $ \tfrac12+h_1\ge 1+h_2$. So the variational formula \eqref{eq:g:K-var} is again satisfied without $\max_\omega$. However, this time $\check F$ cannot come from Busemann functions because some Busemann functions do not exist. Maximizing $n$-step paths use only $e_1$-steps and consequently \[ G_{0,(n)}^\infty(h)- G_{e_2,(n-1)}^\infty(h) = h_1 + \mbox{\mymathbb{1}}\{\text{$n$ is odd}\} \] does not converge as $n\to\infty$. Note that $\check F$ is a minimizing cocycle in both cases (i) and (ii), but only in case (ii) it satisfies \eqref{eq:g:K-var} without $\max_\omega$. $\triangle$ \end{example} \appendix \section{Auxiliary lemmas}\label{sec:cK} Centered cocycles satisfy a uniform ergodic theorem. The following is a special case of Theorem 9.3 of \cite{geor-rass-sepp-yilm-15}. Note that a one-sided bound suffices for a hypothesis. Recall Definition \ref{def:cL} for class $\mathcal{L}$ and Definition \ref{def:cK} for the space $\mathcal{K}_0$ of centered cocycles. \begin{theorem}\label{th:Atilla} Assume $\bP$ is ergodic under the transformations $\{T_z:z\in\mathcal R\}$. Let $F\in\mathcal{K}_0$. Assume there exists $V\in\mathcal{L}$ such that $\max_{z\in\mathcal R} F(\omega,0,z)\le V(\omega)$ for $\bP$-a.e.\ $\omega$. Then for $\bP$-a.e.\ $\omega$ \[\lim_{n\to\infty}\;\max_{\substack{x=z_1+\dotsm+z_n\\z_{1,n}\in\mathcal R^n}} \;\frac{\abs{F(\omega,0,x)}}n=0.\] \end{theorem} \begin{lemma}\label{app:lm1} Let $X_n\in L^1$, $X_n\to X$ a.s., $\displaystyle\varliminf_{n\to\infty} EX_n\le c<\infty$, and $X_n^-$ uniformly integrable. Then $ X\in L^1$ and $EX\le c$. \end{lemma} \begin{proof} Since $X_n^-\to X^-$ a.s. and $X_n^-$ is uniformly integrable, $X_n^-\to X^-$ in $L^1$ and in particular $X^-\in L^1$. By Fatou's lemma and by the assumption, \begin{align} E(X^+)&=E(\lim_{n\to\infty} X_n^+) \le \varliminf_{n\to\infty} E(X_n^+) = \varliminf_{n\to\infty} E(X_n + X_n^-) \le c + E(X^-) < \infty \end{align} from which we conclude that $X\in L^1$ and then $EX\le c$. \qed \end{proof} \footnotesize \end{document}	arXiv
SOTIP is a versatile method for microenvironment modeling with spatial omics data Deep learning and alignment of spatially resolved single-cell transcriptomes with Tangram Tommaso Biancalani, Gabriele Scalia, … Aviv Regev Iterative single-cell multi-omic integration using online learning Chao Gao, Jialin Liu, … Joshua D. Welch Overcoming false-positive gene-category enrichment in the analysis of spatially resolved transcriptomic brain atlas data Ben D. Fulcher, Aurina Arnatkeviciute & Alex Fornito A multiresolution framework to characterize single-cell state landscapes Shahin Mohammadi, Jose Davila-Velderrain & Manolis Kellis Comprehensive in situ mapping of human cortical transcriptomic cell types Christoffer Mattsson Langseth, Daniel Gyllborg, … Mats Nilsson neuromaps: structural and functional interpretation of brain maps Ross D. Markello, Justine Y. Hansen, … Bratislav Misic Transcriptome-scale spatial gene expression in the human dorsolateral prefrontal cortex Kristen R. Maynard, Leonardo Collado-Torres, … Andrew E. Jaffe Probabilistic cell typing enables fine mapping of closely related cell types in situ Xiaoyan Qian, Kenneth D. Harris, … Mats Nilsson Multiscale 3D phenotyping of human cerebral organoids Alexandre Albanese, Justin M. Swaney, … Kwanghun Chung Zhiyuan Yuan ORCID: orcid.org/0000-0002-9367-42361,2,3 na1, Yisi Li3 na1, Minglei Shi4, Fan Yang2, Juntao Gao3, Jianhua Yao ORCID: orcid.org/0000-0001-9157-95962 & Michael Q. Zhang ORCID: orcid.org/0000-0002-7022-61153,4,5 Cancer microenvironment Cellular signalling networks The rapidly developing spatial omics generated datasets with diverse scales and modalities. However, most existing methods focus on modeling dynamics of single cells while ignore microenvironments (MEs). Here we present SOTIP (Spatial Omics mulTIPle-task analysis), a versatile method incorporating MEs and their interrelationships into a unified graph. Based on this graph, spatial heterogeneity quantification, spatial domain identification, differential microenvironment analysis, and other downstream tasks can be performed. We validate each module's accuracy, robustness, scalability and interpretability on various spatial omics datasets. In two independent mouse cerebral cortex spatial transcriptomics datasets, we reveal a gradient spatial heterogeneity pattern strongly correlated with the cortical depth. In human triple-negative breast cancer spatial proteomics datasets, we identify molecular polarizations and MEs associated with different patient survivals. Overall, by modeling biologically explainable MEs, SOTIP outperforms state-of-art methods and provides some perspectives for spatial omics data exploration and interpretation. Tissue and cell state are jointly modulated by the intracellular gene regulatory network and extracellular ecosystem (i.e., the microenvironment). Recent studies highlighted the importance of such microenvironment in tissue homeostasis1,2,3,4, disease occurrence5,6,7, and tumor progression8,9,10. For instance, in healthy vertebrate liver, the zonation patterns of hepatocytes perform mutual effect on the microenvironment within liver lobules4. While during liver fibrosis, there occurs metabolic alteration around the liver-fibrotic microenvironment7. In cancer, researchers also reported that different degrees of immune cell infiltration was associated with patient survival11,12,13. Recent years, various spatial omics technologies have been developed, including spatial transcriptomics (e.g., in situ sequencing (ISS)14, single-molecule fluorescence in situ hybridization (smFISH)15, multiplexed error-robust fluorescence in situ hybridization (MERFISH)16, sequential fluorescence in situ hybridization (seqFISH)17, spatial transcriptomics (ST)18, slide-seq19,20, High-definition spatial transcriptomics (HDST)21, stereo-seq22), spatial proteomics (e.g., iterative indirect immunofluorescence imaging (4i)23, co-detection by indexing (CODEX)24, multiplexed ion beam imaging by time-of-flight (MIBI-TOF)12,25, imaging mass cytometry (IMC)13,26), as well as spatial metabolomics (e.g., airflow-assisted desorption electrospray ionization mass spectrometry imaging (AFADESI-MSI)27, SpaceM28, spatial single nuclear metabolomics (SEAM)7). These advances provide fertile biomolecular profiles of each observation unit (viz., spots, pixel, or single cells) as well as the corresponding spatial coordination, thus facilitating deep and systematic investigation of tissue microenvironment. Although above technologies have achieved unprecedented profiling coverage and/or spatial resolution, only a few studies focused on the quantitative description of microenvironment (ME)29,30,31. Existing studies typically represent an ME as a vector of cell type frequency within cellular neighborhood, and use Euclidean distance among ME representations to construct a graph (e.g., KNN graph32), which is further exploited to perform downstream analysis such as clustering (termed classical cell neighborhood representation, CCNR, Supplementary Table 1)29,30,31. While easily implemented, this practice fails to consider the mutual relationship of cells within the ME, which may undermine the reliability of the subsequently constructed ME graph. This defect would easily hinder the downstream analytical performance especially in cases containing continuous cell states, such as liver2 and cortex33. The complexity of the cell composition within the spatial context is reported to be associated with disease progression34 and tissue development16,33. Classical methods quantify the heterogeneity of an ME by counting the number of unique cell clusters within it (termed NUCC, Supplementary Table 1)16,33. Such practice assumes equal importance of different clusters, potentially leading to the inaccuracy of heterogeneity quantification. For example, they may fail to distinguish the heterogeneity difference between two MEs (one is composed of tumor cells and immune cells, and another is composed of two subsets of immune cells). Partitioning tissues into spatial domains with the integration of both gene expression and spatial information is an important task in tissue biology. Existing methods can be divided into three categories based on their computational principles: Zhu et al.35 and BayesSpace36 (Supplementary Table 1) applied Hidden-Markov random field (HMRF) to model the spatial dependency of hidden variables associated with spots. SpaGCN37 (Supplementary Table 1) utilized existing structures in the field of graph convolutional network (GCN) to aggregate gene expression features according to the spatial graph. StLearn38 (Supplementary Table 1) normalized the spatial information, tissue histology, and gene expression features to perform clustering. While these methods and other very recent preprints39,40,41 achieved good performance in their cases, they may lack proper interpretation for the parameters (e.g., weight trade-off between spatial prior and conditional distributions for HMRF-based methods) or features (e.g., features learned from deep learning-based methods) of biological entities. In addition, they may not be flexible enough to be compatible with different spatial omics data types. Here, we present SOTIP (Fig. 1), a scalable method to jointly perform three main spatial omics tasks, namely, spatial heterogeneity (SHN) quantification (Fig. 1e), spatial domain (SDM) identification (Fig. 1f), differential microenvironment (DME) analysis (Fig. 1g), and other downstream spatial omics tasks (Fig. 1h) within a unified and biologically interpretable framework. With SOTIP, we propose to use MECN (molecular-expression-aware cellular neighborhood) as an agent to abstract and mathematically represent the microenvironment (Fig. 1b). SOTIP's algorithm modules perform these tasks based on an MECN graph (MEG, Fig. 1d), in which each node represents an MECN and the edge weight between two nodes was the function of cell states discrepancy (Fig. 1c) and cell state composition (Fig. 1b) within MEs. In this manner, gene expressions, single cells, microenvironments, and tissue regions can be linked in a biologically meaningful way. Then SHN is calculated by measuring the gene expression variation within each node, SDM was identified by hierarchical merging nodes guided by the connectivity derived from edges, and DME analysis was performed by assessing nodes' relative densities between conditions on the MEG. We demonstrate all SOTIP modules with various types of spatial omics data42,43,44,45 (Fig. 1i, Supplementary Table 3), including FISH-based46,47 and sequencing-based48 spatial transcriptomics, mass-spectrometry-based and fluorescence-based spatial proteomics12,49, and secondary ion mass spectrometry (SIMS) based spatial metabolomics7. We summarize the datasets in Supplementary Table 8 and the demonstration logic in Supplementary Figs. 13–15. Fig. 1: Schematic overview. a–h Overview of SOTIP. SOTIP takes 2D/3D spatial omics data (spatially resolved gene expression, protein or metabolic profiles) as input (a). Next, the cell (spot) type clustering is obtained solely by the expression matrix. Then the histogram of cell clusters is obtained combing the spatial information and clustering result (b). The manifold distance is approximated by combing clustering result and the expression matrix (c). The ME graph is built using optimal transport between histograms characterizing microenvironments (MEs) (d). Finally, multiple tasks can be performed based on the MECN graph, including spatial heterogeneity quantification (e), spatial domain identification (f), differential microenvironment analysis (g), and other downstream tasks (h). i The supported spatial data. Model rational To smooth the reading, we describe SOTIP's rational in this section (see "Methods" for more technical description). In the following, we use "cells" to denote the measurement units in spatial omics data for ease of explanation, which can be freely replaced to"spots" in technologies like 10X Visium or ST. For any data with N cells and P features (e.g., gene expression vector with length P), traditional (non-spatial) clustering methods32 directly construct a graph for the N cells, by computing distances on the P feature vectors. Then Leiden50 algorithm is performed on this graph to get the clustering result. Such traditional clustering methods only take into account the gene expression profile of each cell, without considering the neighborhood information of each cell. On the contrary, SOTIP considers both gene expression profiles of cells, and the neighborhood information of cells. The main idea of SOTIP is to represent the neighborhood of each cell (neighborhood indexed by cell-i is called MECN-i) as a histogram of cell types (following figure shows example of 3 MECNs characterized by histograms), then SOTIP uses the earth mover's distance (EMD, a distance metrics for histograms) based on an optimal transport principle to construct a graph (termed MEG) between all MECNs, and finally a clustering procedure is performed on MEG to get the clustering result (i.e., spatial domain identification result). Here, it is worth re-emphasizing that MEG is the graph of all MECNs, each node of MEG is a MECN, and edge between nodes is the distance between two MECNs, which is the EMD distance between the two histograms characterizing the two MECNs. The key design of SOTIP is construction of the MECN graph (MEG) using EMD. In this way, when measuring the distance between two MECNs (e.g., MECN-i and MECN-j, indexed by cell-i and cell-j), SOTIP considers how to design an optimal transportation plan on the gene expression manifold to move cells (characterized by cell type) within MECN-i to match cells within MECN-j, so that the histogram characterizing MECN-i after the transportation is exactly the same as the histogram characterizing MECN-j. Another key design is that when designing the optimal transportation plan, the cells must be moved along the gene expression manifold. The optimal transportation cost is used as the distance between MECN-i and MECN-j. One can imagine that the transportation cost is determined by two factors: (1) the number of cells to be moved, and (2) the cost of moving cells between two positions along the gene expression manifold. The first factor has already been encoded in the histograms associated with MECNs, and the second factor (named ground distance in optimal transport theorem) is the geodesic distance between cell types along the gene expression manifold. SOTIP framework demonstration with in silico spatial transcriptomics data To demonstrate SOTIP's utility and to conduct adequate comparisons with other methods, we generated three sets of in silico spatial transcriptomics data by simulating scRNA-seq data, followed by arranging cell spatial coordinates (see "Methods" and Supplementary Fig. 1). It is worth noting that since each MECN is generated by searching the spatially nearest neighbors of each cell, in the following sections, every MECN (consisted with multiple cells within the neighborhood of a center cell) is associated with its center cell. In the first simulation, we aimed to compare SOTIP with other methods in SHN quantification. Other methods include NUCC16,33 and IGD (see Methods "NUCC and IGD"). To achieve this goal, we simulated three clusters of single cells with inter-cluster discrepancy (Fig. 2a left). Then we randomly positioned these three cell clusters as a sequential tissue band on a two dimensional plane (Fig. 2a right). With this simulation, we expected higher spatial heterogeneity around tissue boundaries than inner tissue, which corresponded to higher SHN values between C2 (middle band in Fig. 2a right) and C3 (right band in Fig. 2a right) than between C1 (left band in Fig. 2a right) and C2. We next separately performed SOTIP, IGD and NUCC16,33 (Supplementary Table 1) to quantify the spatial heterogeneity across the tissue sample. The result showed that all three methods highlighted two tissue boundaries (Fig. 2b), while only SOTIP successfully quantified the difference between two boundaries as expected (Fig. 2b right). We also tested how cluster size influence the analysis, and the result also showed higher SHN peaks around highlighted SHNs at C1-C2 boundary, and lower SHN peaks at C2-C3 boundary, both peaks are significantly higher than background (Supplementary Fig. 12). Fig. 2: Simulation study. a PCA plot (left) and spatial distribution (right) of simulation data 1. In both plots, each point represents a cell. Both plots share the same color-coding scheme. b Comparison of NUCC (left), IGD (middle) and SOTIP (right) of SHN quantification performance on simulation data 1. Each panel consists of the two parts, the top part shows the SHN value of each MECN as a function of horizontal coordinate, and the bottom part shows the same set of cells with spatial coordinates as in Fig. 2a right, colored according to SHN value of associated MECN. c PCA plot (left), spatial distribution (middle) and tissue regions (right) of simulation data 2. In all plots, each point represents a cell. PCA plot (left) and spatial distribution (middle) share the same color-coding scheme, different with tissue region (right). d MECN graphs generated by CCNR (top row) and SOTIP (bottom row) are embedded with different algorithms, i.e. diffusion map (left column), UMAP (middle column) and PHATE (right column). In each embedding plot, each point represents a MECN, colored according to the clustering label of its center cell in Fig. 2c right. e PCA plot and two version of spatial arrangement of simulation data 3. Each point represents a cell. All the plots share the same color-coding scheme. The red arrow points to C1-C2 boundary in sample 1, and blue arrow points to C3-C1 boundary in sample 2. f Sample-specific microenvironments (left panel for sample 1, right panel for sample 2) are highlighted by MEG constructed with SOTIP. Each panel consists of two parts. Take the left panel as an example, the bottom part shows the same set of cells as in Fig. 2e middle, colored according to the relative likelihood of observing each microenvironment in sample 1, and the top part shows the value of relative likelihood of each MECN as a function of horizontal coordinate. The right panel shares similar configuration. Same as (e), the red arrow points to C1-C2 boundary in sample 1, and blue arrow points to C3-C1 boundary in sample 2. Since SOTIP relies on the MECN graph (MEG) to represent MECN relationships, the quality of MEG determines the performance of downstream tasks. In the second simulation, we aimed to evaluate the quality of the MECN graph constructed by SOTIP. To do this, we generated five single cell clusters along a continuous manifold (Fig. 2c left), and regularly mixed these five clusters in five sequential bands on a two-dimensional plane (Fig. 2c right). In this data, each cell type occupies a major component in a band, and other cell types occupy minor components in its band. The arrangement of major cell types along these tissue bands (R1 to R5) is in accordance with the order of cell types along the gene expression manifold (C1 to C5) (Fig. 2c middle). With this simulation, we expected that R1 to R5 microenvironments should display a continuous pattern when embedded into low-dimensional space. We separately performed SOTIP and CCNR29,30,31 (Supplementary Table 1) on this simulation data, and embedded their generated graphs with three popular embedding algorithms (diffusion maps51, UMAP52 and PHATE53, Fig. 2d). The results showed that both methods can separate these 5 MECNs in the embedded space (Fig. 2d), while only SOTIP well preserved the continuous order of R1 to R5 with different embedding methods (Fig. 2d bottom row). In the third simulation, we want to ask whether SOTIP can perform differential microenvironment (DME) analysis to identify specific microenvironments between samples, even if these samples shared exactly the same cellular composition. Since there is currently no other method for this task, it was not possible to compare different methods on this type of data. We firstly generated three discrete single-cell clusters (Fig. 2e left), then positioned them in two different orders, thus in silico generating two different tissue samples with exactly the same cell composition but different spatial organizations (Fig. 2e middle and right). With this simulation, differential abundance analysis based on non-spatial scRNA-seq54,55,56 failed to detect any difference between the two samples, but SOTIP successfully highlighted (Fig. 2f) the major differential MECNs (C1-C2 boundary in sample 1, and C1-C3 boundary in sample 2), by estimating the relative likelihood of observing each microenvironment in the two tissue samples55 (see "MELD analysis" in "Methods"). We also tested the case where C1 and C2 are more similar than C3 in gene expression (Supplementary Fig. 11), and the analysis result also highlighted unique microenvironments of each sample. All these simulation results supported that SOTIP accurately described the microenvironments and corresponding properties, and detected finer biological differences, complementing with the existing methods. SOTIP quantifies accurate SHNs in accordance with subcellular and tissue structures In eukaryotic cells, the nuclear envelope compartmentalizes the DNA in nucleus and the protein synthesis machine in the cytoplasm, thus separating transcription from translation57. The spatial separation of two fundamentally different compositions makes the nuclear envelope to be most heterogeneous region within a eukaryotic cell58 (Fig. 3a). Iterative indirect immunofluorescence imaging (4i)23 is a robust protocol that can achieve multiplexed protein staining at nanometer resolution, thus providing opportunities to characterize the covariance among molecular, spatial, and morphological properties of subcellular structures. Squidpy59 provided a subset of original data, consisting of 270,876 pixels from 13 HeLa cells (~20,836 pixels per cell) with 40 protein assayed23. The data also provided pixel-level nucleus/cytoplasm annotations. Within each cell, since the nuclear envelope is expected to display the highest heterogeneity, if we take the scaled heterogeneity as the probability of being the nuclear envelope, the obtained area under curve (AUC) given ground truth should be high for a good heterogeneity estimation method (see "Methods"). Fig. 3: SOTIP accurately quantifies spatial heterogeneity. a Schematic diagram of a eukaryotic cell. b Summary of the SHN quantification performance on all 13 samples. The AUC is used to estimate the performance. Two-sided Wilcoxon rank-sum test. The detailed definition of box plot elements is described in Methods. The boxplots show n = 13 biologically independent cells. c Visualization of the ground truth (1st column) and SHN values computed by NUCC (2nd column), IGD (3rd column), and SOTIP (4th column) in cell 118. White arrows point out the same position of nuclear envelope. Orange arrow points out a computed potential intracellular membrane structure. d Similar with c but for cell 119. e Overlay of SOTIP computed SHN values with PCNA protein expression in cell 118. White arrow points out the same position with Fig. 3c. f Overlay of clipped SOTIP computed SHN values with p-RPS6 expression in cell 118. Orange arrow points to the same position as in Fig. 3c 4th column. g Histological image of the zebrafish data. Tumor region is annotated with black dashed line. Scale bar: 2 mm. h 10x Visium spots are paired with Fig. 3g, colored according to SOTIP computed SHN values. i The stability (y-axis) of SHN computed using different methods, with different clustering resolutions (x-axis). The stability is evaluated by computing the correlations between SHN values under resolution 1 and SHN values under different resolutions. j Schematic diagram of the cerebral cortex. k Region annotation of the osmFISH mouse cortex dataset. Each point is a cell colored by its layer. l 5 FOVs of seqFISH+ cortex data. The layer depth (Layer 2–6) increased with the FOV label (0–4). m Distribution of SHN values (y-axis) computed using SOTIP in each cortical layer of osmFISH dataset (x-axis). Each violin represents a distribution of SHN values (y-axis) of MECNs in one layer (x-axis), and each point in one violin represents a MECN's SHN value. The violin-plot is colored by the layer label of the MECN. The x-axis is arranged in ascending order of the cortical depth. n Similar with (n) but with seqFISH+ dataset. The gradient pattern of SHN along the cortical layer depth. Violin-plots were colored according to (m). Source data are provided as a Source Data file. The experiment results in (g, h) were similar with three independent repeats. For all 13 cells, we compared SOTIP with NUCC16,33 and IGD (Supplementary Table 1), the quantitative results (Fig. 3b) showed that SOTIP (median AUC 0.85) significantly outperformed NUCC (median AUC is 0.68, p = 7.34 × 10−6, two-sided Wilcoxon rank-sum test, n = 13 cells), and IGD (median AUC is 0.71, p = 1.16 × 10−5, two-sided Wilcoxon rank-sum test, n = 13 cells). To dive into details of above methods, we unbiasedly selected two cells as examples to qualitatively compare the computationally delineated nuclear envelopes with the ground truth (Fig. 3c, d). In cell 118, both IGD (AUC = 0.66, Fig. 3c third column) and SOTIP (AUC = 0.77, Fig. 3c fourth column) delineated clear curves with closed pattern, while NUCC (AUC = 0.64, Fig. 3c second column) tended to output blurred curves with irregular and unsmooth pattern. When further comparing IGD and SOTIP, we found that SOTIP exhibited the highest SHN value around the nuclear envelope (white arrows in Fig. 3c fourth column) with strong specificity, but IGD failed to differentiate the relative values of SHN around different regions (white and orange arrows in Fig. 3c third column). Similar results have also been found in cell 119 (Fig. 3d). To further confirm these results, we inspected the relationship between the spatial localization of SOTIP-computed SHN values and protein enrichment. We implemented this comparison by encoding SHN values into red channel, and targeted protein intensities into green channel, then plotting the encoded colors for each cell. Again we used cell 118 as an example, and found that SOTIP-identified SHN values aligned well with proliferating cell nuclear antigen (PCNA), a known nuclear marker60 (Fig. 3e). Although SOTIP successfully highlighted the region of the highest heterogeneity within a cell, we additionally asked whether SOTIP also delineated other cellular structures (e.g., orange arrow in Fig. 3c fourth column) with relatively lower heterogeneity. To answer this and to check whether SOTIP could reflect different levels of heterogeneity, we first clipped the highest SHN values, then encoded targeted protein intensities and the clipped SHN values in colors as before (Fig. 3f). It turned out that SOTIP-identified structure aligned well with ribosomal protein S6 (RPS6) (Fig. 3f), which is reported as a marker of endoplasmic reticulum (ER)60. To test whether SOTIP can quantify SHN values on tissue samples, we used a spatial transcriptomics dataset containing sections from BRAFV600E melanoma zebrafish model61. We firstly applied NUCC, IGD and SOTIP on the sample containing a clear tumor-muscle interface (Fig. 3g). The results showed that SOTIP-computed SHN values delineated the tumor-muscle interface, which was expected to be the most heterogeneous region61, in accordance with histological annotation (Fig. 3g, h white arrow) with high specificity. On the contrary, both NUCC-computed and IGD-computed SHN values exhibited indistinguishable pattern within tissue bulk and the tumor-muscle interface (Supplementary Fig. 2e, f). Since SOTIP involved a clustering step (i.e., Leiden50) before MEG construction (see "Methods"), we wanted to ask whether it could output stable results for different clustering resolutions. To evaluate the robustness of SOTIP regarding to the clustering resolution, we firstly quantified the SHN values of the zebrafish sample (Fig. 3g, h) by SOTIP with different Leiden clustering resolutions ranging from 1 to 10. Then we computed the Pearson correlation coefficient (Pearson's r) between SHN values on resolution 1 with SHN values on different resolutions. Finally we plotted the computed correlation as the function of Leiden resolution (Fig. 3i left). With this procedure, we compared the robustness of SOTIP, NUCC, and IGD, and showed that SOTIP have a substantially stronger robustness than the existing methods, and could output stable SHN values even with extreme over-clustering (Fig. 3i left). The other two samples of the zebrafish61 study also supported this claim (Fig. 3i middle and right). SOTIP reveals gradient spatial heterogeneity along cerebral cortical layer axis The laminar structure (Fig. 3j) of mammalian cerebral cortex is reported to exhibit layer-specific properties on gene expression, morphology and connectivity in single cell level62,63,64. One study33 of the recent BRAIN Initiative Cell Census Network (BICCN) program64 proposed an observation that the composition complexity of cell neighborhoods increased towards deeper layers. They counted the number of different cell clusters within cell neighborhood to define the spatial heterogeneity (like NUCC does). As our aforementioned analysis, the reliability of their statement could be compromised on different clustering resolutions (Fig. 3i). To confirm and refine their conclusion, we applied SOTIP, which was proved robust and accurate in SHN quantification, on other mouse cortex datasets of independent experiment protocols. To this end, we firstly adopted a single-cell spatial transcriptomics dataset (cyclic-ouroboros single-molecule fluorescence in situ hybridization, osmFISH46) which covered the main layers of mouse somatosensory cortex with layer annotation46. This dataset contains 5328 single cells across cell-wise annotated cortical layers (Fig. 3k) ranging from pia layer 1 (n = 159), layer 2–3 medial (n = 549), layer 2–3 lateral (n = 254), layer 3–4 (n = 131), layer 4 (n = 1002), layer 5 (n = 295), layer 6 (n = 1015), and other regions. We applied SOTIP on this dataset to quantify each ME's SHN value and plotted the distribution of SHN values as violin-plots to display the trends along the cortical depth axis (Fig. 3m). Specifically, each violin-plot is a distribution of SOTIP-computed SHN values within a specific cortical layer (x-axis), and the x-axis conforms to the order of the cortex from shallow (viz. pia layer 1) to deep (viz. Layer 6) layer. The result visually showed a gradient pattern of SHN values from shallow layers to deep (Fig. 3m). To further quantify the ordinal agreement between the SHN values and layer depth, we used both ME-wise and layer-wise Spearman's rank correlation coefficients (Spearman's ⍴) between them (see "Methods"). The results showed a strong correlation between the SHN value and the cortical depth, in both manner of ME-wise (Spearman's ⍴ = 0.674, p < 1 × 10−15) and layer-wise (Spearman's ⍴ = 0.847, p = 0.016). To strengthen our conclusion, we additionally applied SOTIP on another independent single-cell spatial transcriptomics dataset (evolution of sequential fluorescence in situ hybridization, seqFISH+17) of mouse cortex covering different layers17. This dataset contains 913 single cells from five spatially continuous field of views (FOVs) ranging from layer 2 to layer 6 (Fig. 3l). We conducted the same SOTIP analysis as with osmFISH dataset, the result also confirmed a continuous gradient pattern of SHN values along the cortical axis (Fig. 3n), and a strong and positive correlation between the SHN value and the layer depth (ME-wise Spearman's ⍴ = 0.719, p < 1 × 10−15 and layer-wise Spearman's ⍴ = 1, p < 1 × 10−15). For osmFISH and seqFISH+, the performance of SOTIP were not disrupted by neither different over-clustering levels, nor different neighborhood sizes, being more robust than other methods, e.g., NUCC, IGD (Supplementary Fig. 2a–d). SOTIP stratifies known layers in brain tissues To test the capability of SOTIP in the task of spatial domain identification, we took the SpatialLIBD dataset, which provided 10x Visium (Fig. 4a for illustration) spatial transcriptomics data on 12 tissue sections of human dorsolateral prefrontal cortex (DLPFC)48. For each tissue section, the original research48 also provided spot-level annotations of cortical layers as well as white matter regions, which can be used as ground truth to numerally evaluate performance. Since other methods36,37,38 have already demonstrated the improved performance of integrating spatial information by comparing with non-spatial clustering algorithms, e.g., k-means, Louvain50, and SC365, on this dataset, we compared SOTIP only with the spatial methods, including stLearn38, BayesSpace36, and SpaGCN37, Giotto66, SEDR39, STAGATE40, and SpatialPCA41 (Supplementary Table 1). Following the comparison methodology of previous studies36,37,48, we input to the algorithms the number of clusters, and used adjusted rand index (ARI) to quantify the similarity between the partition results and manual annotations. Fig. 4: SOTIP stratifies known layers in brain tissues. a–d Human dorsolateral prefrontal cortex dataset with 10x Visium. a Spatial arrangement illustration for typical 10x Visium dataset. b Ground truth annotation for slice 151673. c Spatial domain detected by SOTIP. The labels are not consecutive because SOTIP follows a hierarchical merging scheme. d Boxplots summarize the detection accuracy (y-axis) of four methods (x-axis) in all 12 samples. The definition of each box can be found in Methods. e–h mouse somatosensory cortex dataset with osmFISH. e Spatial arrangement illustration for typical osmFISH dataset. f Ground truth of regional annotation for the osmFISH sample. g, h Spatial domains identified by SpaGCN (g) and SOTIP (h). Source data are provided as a Source Data file. We took one of the representative tissue sample, sample 151673 (n = 1639 spots, shown in Fig. 4b) for an example. Visually, SOTIP identified SDM exhibited a layered pattern consistent with the ground truth (Fig. 4c). Since it's difficult to reproduce the original ARI (also reported by other group39), especially for deep learning based methods37,38,39, we directly used the ARI and clustering results as claimed in the original paper. When we applied these algorithms to all 12 tissue sections, we found that SOTIP outperformed all other methods (Fig. 4d, n = 12) in terms of mean and median. Specifically, SpaGCN (mean/median: 0.43/0.45) and BayesSpace (mean/median: 0.43/0.42) are two best peer-reviewed methods, have higher ARI than older methods, e.g., stLearn (mean/median: 0.36/0.36), and Giotto (mean/median: 0.36/0.37), while lower than STAGATE (mean/median: 0.49/0.51) and SpatialPCA (mean/median: 0.52/0.55), the two best preprinted methods. SOTIP (mean/median: 0.58/0.57) performed best than all of them. For the compatibility, BayesSpace was restricted to the lattice-shaped spatial data like 10x Visium (Fig. 4a), while SOTIP and SpaGCN does not rely on the spatial arrangement of spots/cells. To demonstrate whether SOTIP surpass other methods on non-lattice-shaped spatial transcriptomics data (Fig. 4e), we examined the two algorithms on a single-cell spatial transcriptomics dataset (osmFISH46) of the mouse brain somatosensory cortex. We evaluated the performance with ARI taking the known cell-wise region annotations as the ground truth (Fig. 4f). The result showed that SOTIP (ARI = 0.566, Fig. 4h) substantially outperformed SpaGCN (ARI = 0.257, Fig. 4g). In addition to all the above advantages, SOTIP is suitable for spatial proteomics, which is hardly manageable for any existing methods. We will further demonstrate SOTIP utility and performance with spatial proteomics technologies in the next section. SOTIP characterizes spatial and molecular tumor-immune organization After demonstrating the superior performance of SOTIP on various spatial transcriptomics datasets, we next ask whether SOTIP is also compatible with spatial proteomics datasets. Since there are currently few methods that was explicitly applied to perform spatial domain identification on spatial proteomics data, we only compared SpaGCN with SOTIP. We found two spatial proteomics datasets, of which the first was produced by MIBI-TOF technique on 41 triple negative breast cancer (TNBC) patients12, and the second was produced by scMEP technique on two colorectal carcinoma (CRC) patients and two healthy controls49. With these datasets, we attempted to identify the tumor-immune interplay (Fig. 5a for illustration), which has been widely regarded as a prognostic indicator in tumor progression67,68,69. Fig. 5: SOTIP applications on two spatial proteomics datasets. a–c Schematic diagram to show how SOTIP works spatial proteomics datasets. Illustration of tumor-immune interplay (a), Voronoi diagram used to visualize spatial domains (b), and molecular polarization to tissue boundary (c). a Tumor and immune cells have complex interplay around the tumor-immune boundary. b Voronoi diagram (as implemented in https://stackoverflow.com/questions/20515554/colorize-voronoi-diagram/20678647#20678647%203.18.2019) is a visualization method for spatial omics data29. c Molecular polarization is a phenomenon that some molecules exhibit gradient changes perpendicular to tissue boundaries. d Ground truth of tumor-immune boundary from the original paper of CRC samples. Top panel: point 16. Bottom panel: point 23. Edited from original paper. Scale bar: 100 μm. e Voronoi diagram of SpaGCN-detected (left column), and SOTIP-detected (right column) spatial domains for point 16 (top row) and point 23 (bottom row). f Comparison of SpaGCN-detected and SOTIP-detected spatial domains to find polarized proteins to/away tumor region, in CRC point 16 (top row) and CRC point 23 (bottom row). Left column: The power plots show the proportion of true positives (y axis) detected by different methods at a range of FDRs (x axis) for point 16 (top row) and point 23 (bottom row). Right column: The overlay of three representative polarized proteins found by SOTIP-detected SDM in point 16 (top row) and point 23 (bottom row). Scale bar: 100 μm. g Ground truth of tumor-immune boundary from the original paper of TNBC samples. Top panel: patient 4. Bottom panel: patient 9. Edited from original paper. Scale bar: 100 μm. h Voronoi diagram of SpaGCN-detected (left column), and SOTIP-detected (right column) spatial domains for patient 4 (top row) and patient 9 (bottom row). i Similar with (f) but on TNBC patient 4 (top row) and patient 9 (bottom row). In d, e, g, h, all white arrows point to tumor regions. Scale bar: 100um. Source data are provided as a Source Data file. The experiment results in (d, f, g, i) were similar with three independent repeats. As the original papers provided tumor/immune partition annotations for 2 samples in CRC (Fig. 5d), and two samples in TNBC (Fig. 5g), we used these four samples to evaluate the performance of different methods on stratifying tumor and immune regions. For better visualization and qualitative comparison29, we plotted the detected spatial domains using Voronoi diagrams (Fig. 5b for illustration). The results (Fig. 5e, h) showed that in all datasets, SOTIP achieved better results in terms of visual coherence compared with SpaGCN, and almost perfectly matched with the originally reported region annotations (Fig. 5d, g). Tumor and immune cells are reported to perform mutual modulation by frequent cross-talk within the tumor-immune ecosystem70. Their intercellular communications and intracellular regulation networks generate patterned phenotypic cell identities and molecular distribution71. Molecular polarization (a form of spatial molecular distribution pattern that certain molecule tends to differentially enriched distal/proximal to a tissue landmark) was recently reported to appear nearby pathological tissue boundaries7,12,61 (Fig. 5c for illustration). To show that SOTIP-detected tumor/immune regions better fit for downstream analysis, we next inspected the relationship between molecular polarization and tissue architecture. To be clear, in the following statements, when we say that a protein is polarized to/away a region, we mean that the protein tends to distributed proximally/distally to that region. In the CRC datasets, we first classified immune cells as distal and proximal to tumor-immune boundary (see "Methods", Supplementary Fig. 4a, b) using SOTIP and SpaGCN results, respectively. Next, Wilcoxon rank sum test was adopted to estimate the protein enrichment/depletion with respect to the boundary, and polarized proteins with statistical significance were detected. We measured the detection power on the basis of false-discovery rate (FDR) in both datasets to conduct a fair comparison across methods. The results showed that SOTIP was more powerful than SpaGCN across a range of FDR cutoffs (Fig. 5f left). We also performed similar analysis on TNBC datasets (see "Methods", Supplementary Fig. 4c, d), to identify molecular polarization to/away tumor (Fig. 5i left) and immune regions (Supplementary Fig. 4e–g), confirming the superior detection power of SOTIP. Last, we showed representative (see Methods) polarized proteins detected by SOTIP in CRC (Fig. 5f right), and in TNBC (Fig. 5i right, Supplementary Fig. 4e–g) samples. Apart from consistency with the original paper (e.g., CD98, CD11c polarization in CRC immune region49, and CD45RO, CD11c polarization in TNBC immune region12, and Keratin6, Keratin17, HLA-DR polarization in TNBC tumor region12), we also detected some unexpected polarizations. For example, in TNBC patient 9, we found that CD45 was significantly polarized to the tumor region (p = 6.88 × 10−44, two-sided Wilcoxon rank sum test, Fig. 5i bottom right), which was also uncovered in patient 4 (p = 1.30 × 10−9, two-sided Wilcoxon rank sum test). CD45 plays an significant role in T cell receptor signaling pathway72, while CD45RO, an isoform of CD45, is of vital importance for the production of TNF-α and IFN-γ73. TNF-α and IFN-γ are powerful in monitoring tumor proliferation and trigger cell death74. TNF-α involves in complex immune response to induce tumor cell apoptosis and destroy the blood vessels which supports the tumor growth by transporting essential nutrients75. IFN-γ, a member of the type II interferon group, possibly contributes to the tumor cell apoptosis via JAK‐STAT pathway in a microenvironment-dependent manner76,77. The enrichment of CD45 and CD45RO around the tumor-immune boundaries (Fig. 5i), which was also described in the original paper, indicated that they may function during the battle of immune tissue against the tumor tissue via multiple apoptosis-associated immune signaling pathways. In both TNBC patient 4 and 9, CD138 has significantly polarized away the tumor-immune boundary (p = 5.12 × 10−27 in patient 4 and p = 7.17 × 10−52 in patient 9, two-sided Wilcoxon rank sum test, Fig. 5i top right and bottom right). CD138, also known as heparan sulfate (HS) proteoglycan Syndecan-1, is a key regulator responsible for inflammatory cytokines modulation78,79. Cytokines have been found to play anti-tumor effects by binding to the receptors to mediate cell-cell communications, supported by both animal studies and clinical researches80. Specifically, CD138 could upgrade the survival of mature plasma cells, which is critical for long-term humoral immune81. This function is in accordance with our observation that CD138 expression is highly enriched in the core of immune cell cluster (Fig. 5i). It indicates that activated B cells may play an active anti-cancer role in a persistent way with recruitment of other immune cells. SOTIP recovers known differential microenvironments in cirrhotic liver The last task of SOTIP is differential microenvironment analysis (DMA), which aims to identify microenvironments between conditions (e.g. disease status, drug treatment, and experimental perturbation) with high specificity. This task is in analogy with the differential abundance analysis task54,55,56 in single cell analysis, but unavailable in spatial omics analysis. To demonstrate the utility of SOTIP on this task, and as a sanity check, we firstly applied DMA in a spatial metabolomics dataset including two samples from healthy and fibrotic liver (Fig. 6a for illustration). They provided paired H&E staining, which could be taken as a positive control for our detection result (Fig. 6b). The healthy liver sample (Fig. 6b left) was a section from a standard liver lobule, which is a repeating hexagonal-shaped units of liver. The fibrotic sample (Fig. 6b right) was a liver section containing both regions of fibrotic niche (red arrow) and hepatic tissue bulk. The original paper provided cell type annotations (Fig. 6c). In this case, we expected that SOTIP could spot the fibrotic-specific microenvironments, e.g., the fibrotic region and tissue boundary. Fig. 6: SOTIP recovers differential microenvironments between liver samples. a Illustration of healthy and fibrotic livers. b Histological images of the healthy (left) and fibrotic (right) liver samples. The red arrows indicate fibrotic niche. The green dashed line indicates the fibrotic boundary. Scale bar: 100 μm. c Spatial single cell map in healthy (left) and fibrotic (right) liver sample paired with Fig. 6b. The red arrows and green dashed line are consistent with those in Fig. 6b. d MEG abstraction. Each node is a cluster of densely connected MECNs. The width of edges can be considered as similarity between nodes (see "Methods"). Each node (i.e., MECN cluster) is assigned an unique color. The size of each node is proportional to the number of MECNs belonging to that MECN cluster. e MECN clusters of the fibrotic liver sample showed in Voronoi diagram. The colors of MECN clusters are consistent with those in Fig. 6d. The green dashed line is consistent with that in Fig. 6b. f Entropy-SHN (EH) plot (see "Methods"). Each node is a MECN cluster with the same color scheme as Fig. 6d, e. The y-axis is the entropy (HC) of the MECN cluster C. The x-axis is the average SHN values of the MECN cluster. MECN cluster 0, 3, 9, 11, 12, 14, 15 with low entropy and high SHN values lie at the right bottom corner of EH plot. g Left: The same MECN clusters as Fig. 6e, but only MECN cluster 0, 9, 11, 12, 14 in highlight. The red arrows and green dashed line are consistent with those in Fig. 6b. Right: Cell composition of MECN cluster 0, 9, 11, 12, 14, with the same color scheme as Fig. 6c. h Top: The same MECN cluster map as Fig. 6e, but only MECN cluster 3 in highlight. The red arrows and green dashed line are consistent with those in Fig. 6b. Bottom: Cell composition of MECN cluster 3, with the same color scheme as Fig. 6c. i Similar with Fig. 6h but using MECN cluster 15. Source data are provided as a Source Data file. The experiment results in (b, c, d) were similar with three independent repeats. To investigate on that, we firstly built an MEG of the joint MECNs from both samples, resulting in total 1608 nodes and 237,604 edges (Supplementary Fig. 5). The complete high-resolution graph was too complicated to read and interpreted, so we next detected communities in this graph and analyzed the abstracted topological structure (see "Methods"). The resulted MEG abstraction (a simplified version of MEG, Fig. 6d) consisted of 16 nodes, each of which is an MECN cluster (a group of similar MECNs). The edge width can be considered as similarity between nodes. To get a global view of the spatial organization of various microenvironments within the tissue field, we plotted the clustered MECN groups on physical space with Voronoi diagram (Fig. 6e), so that MECNs from the same cluster share the same color according to the MEG abstraction (Fig. 6d). From the MECN cluster map (Fig. 6e), we observed that certain microenvironments showed consistency with the histological images (Fig. 6b right). To select the MECN clusters with high sample-specificity and cell type complexity, we plotted each MECN cluster on an EH plot (abbreviation for "entropy-of-ME-cluster (EMC)" versus "spatial heterogeneity (SHN)" plot, see "Methods"). On the EH plot, MECN clusters with higher sample-specificity corresponded to lower EMC (y-axis), and MECN clusters with higher compositional complexity corresponded to higher SHN value (x-axis), so that MECN clusters at the bottom right corner should be of special interest (Fig. 6f). Based on this criterion, we selected 7 MECN clusters, and all of them were specifically occurred in the fibrotic liver sample. We first looked at MECNs with highest cell composition complexity, i.e., the MECN cluster 15 (Fig. 6f, i). The hepatocytes account for more than half of the cell counts, followed by all other major cell types, fibroblast, Kupffer cell, immune cell, and endothelial cell. This high level of SHN has been expected, since the metabolomic profile of hepatocytes is reported to exhibit substantial difference with other non-hepatic parenchymal cells (NPCs)2,3,7. Given that MECN cluster 0, 9, 11, 12, 14 were densely connected in the MEG abstraction (Fig. 6d), we jointly plotted them in a single MECN cluster map (Fig. 6g left). The result indicated that these MECNs displayed fibrotic-niche phenotype in cell type composition (Fig. 6g right) and also exhibited high consistency with the histological information (right arrows in Fig. 6b, g right) in spatial localization. We also investigate MECN cluster 3 (Fig. 6h), whose cell type composition (Fig. 6h bottom) showed a balanced mixture of the fibrotic niche and the hepatic tissue bulk. Its spatial distribution confirmed this composition by approximately delineating the hepatic-fibrotic tissue boundary (green dashed line in Fig. 6b right and Fig. 6h). In summary, these analyses demonstrate the capability of SOTIP to identify samples associated microenvironments with high specificity, and interrogate the relationship between the phenotypic and spatial features of them. SOTIP identifies highly specific MECNs associated with prognosis in subtypes of TNBC Triple-negative breast cancer (TNBC), defined by lack of therapeutic targets (estrogen receptor, progesterone receptor, and Her2), is considered to be a form of breast cancer with more aggression and poorer prognosis82. Keren et al.12 applied multiplexed ion beam imaging by time-of-flight (MIBI-TOF) to generate a 36-plex spatial single cell proteomics dataset from 41 TNBC patients. In that paper, the researchers classified TNBC patients into three archetypical subtypes (viz., cold, mixed, and compartmentalized), by quantifying the mixture degree of tumor and immune cells12. The dataset hasn't been fully mined since the original paper mainly concentrated their analysis on protein co-expression and basic spatial statistics12, we here focused on the microenvironment to draw a spatial organization of the tumor-immune landscape and to inspect interesting microenvironment with strong specificity. Since the "cold" samples, lack of immune infiltration, can be easily differentiated from other two subtypes. To demonstrate SOTIP's capability, we choose the more challenging case to identify differential MECNs between mixed (immune cells mixed with tumor cells)12 and compartmentalized (immune cells spatially separated from tumor cells)12 samples. The challenge to distinguish the mixed from the compartmentalized came from the fact that these two subtypes of TNBC samples typically contained similar cell type compositions12, such that they could not be easily identified by current single-cell based analysis54,56, or protein co-expression analysis67. To this end, we chose the most representative samples, suggested by the original research12, from the two subtypes of TNBC, i.e., patient 4 for compartmentalized and patient 12 for mixed (Fig. 7a, b). As with previous section, we firstly built an MEG of the joint MECNs from both samples (Supplementary Fig. 6a), then generated an MEG abstraction (Fig. 7c), consisting of 29 MECN clusters as nodes and the corresponding edges as similarities between MECN clusters. We further plotted the MECN cluster map for the two samples (Fig. 7d for compartmentalized and Fig. 7e for mixed), and selected the MECN clusters with high SHN and low EMC on EH plot (Fig. 7f). Interestingly, we identified several pronounced MECN clusters, of which MECN cluster 9 and MECN cluster 24 were strictly enriched in specific samples with disappeared EMC. Fig. 7: SOTIP identifies differential microenvironments between subtypes of TNBC. a Illustration of compartmentalized and mixed subtypes of TNBC. b Spatial single cell map in compartmentalized (left) and mixed (right) TNBC samples. The black dashed lines in Fig. 7b, d, g, h, j annotate tumor regions according to the original paper. c MEG abstraction. Each node is a cluster of densely connected MECNs. The width of edges can be considered as similarity between nodes (see Methods). The size of each node is proportional to the number of MECNs. d, e MECN clusters of the compartmentalized (d) and mixed (e) sample shown in Voronoi diagram. f Entropy-SHN (EH) plot (see "Methods"). Each node is a MECN cluster with the same color scheme as Fig. 7d, e. The y-axis is the entropy (HC) of the MECN cluster C. The x-axis is the average SHN values of the MECN cluster. g Top: The same MECN cluster as Fig. 7d, but only MECN cluster 9 in highlight. The black arrows point to the tumor region as in Fig. 5g top. Bottom: Cell composition of MECN cluster 9, with the same color scheme as Fig. 7d, e. h The same single cell cluster map with Fig. 7b, but only highlights Tumor and Keratin+ tumor clusters. i Top: The same MECN cluster map with Fig. 7d, but only highlights MECN cluster 24. Bottom: Cell composition of MECN cluster 24, with the same color scheme with Fig. 7d, e. j The same single cell cluster map with Fig. 7b, but only highlights CD8 T, Macrophage and Keratin+ tumor clusters. All three clusters occupy a substantial abundance in both samples. k Occurrence score (y-axis) of MECN 9 comparison between Mixed and Compartmentalized patients (x-axis). Two-sided Wilcoxon rank-sum test, no significance. Boxplots are defined in "Methods". N = 34 independent patients. l Occurrence score (y-axis) of MECN 24 comparison between Mixed and Compartmentalized patients (x-axis). One-sided Wilcoxon rank-sum test. Boxplots are defined in "Methods". N = 34 independent patients. m Survival analysis with Kaplan–Meier curves shows survival as a function of time (days) for patients between two groups (high MKT vs low MKT). Source data are provided as a Source Data file. MECN cluster 9 is specifically restricted in the compartmentalized sample. To inspect the cellular complexity of MECN cluster 9, we found that its SHN value was fairly low, since it only consisted of two subtypes of cells, tumor cell (Tumor cluster) and its Keratin (pan-Keratin, Keratin6, Keratin17, Supplementary Fig. 6b) positive counterpart (Keratin+ tumor cluster). Since there lacks cells of Tumor cluster in mixed sample (Supplementary Fig. 6c, Fig. 7g, h), it's not surprising to detect this MECN differential between samples. More interestingly, MECN cluster 24 is specifically restricted in the mixed sample (Fig. 7i), and is composed of a high percentage of keratin+ tumor cells, as well as a mixture of CD8 T cells and macrophages (Fig. 7i bottom). We noticed that all the three cell types occupied a substantial proportion within both mixed and compartmentalized samples (Supplementary Fig. 6c), so we reasoned that these three cell types may distribute exclusively in the compartmentalized sample but closely in the mixed sample, which was confirmed by the spatial relationships of them (Fig. 7j). We also looked at other differential MECNs (with non-zero EMCs, but still differentiated between samples by occurrence (with EMC cutoff of 0.5)), and found that they were mostly derivative from MECN cluster 9, and 24 (Supplementary Fig. 6d–g). Specifically, for the MECN clusters mainly occurred in the compartmentalized sample (Supplementary Fig. 6d, e), they were dominated by the same two cell types (i.e. Tumor and Keratin+ tumor) as MECN cluster 9. Correspondingly, for those MECN clusters mainly occurring in the mixed samples (Supplementary Fig. 6f, g), they were dominated by the same three cell types (i.e., CD8 T, macrophage, and Keratin+ tumor) as MECN cluster 24, except for a rather minor proportion of CD4 T cells. This indicated that MECN cluster 9 (Tumor and Keratin+ tumor) and/or 24 (CD8 T, macrophage, and Keratin+ tumor) may be the main driver for the differentiation of mixed and compartmentalized forms of TNBC. To examine this statement, we evaluated the occurrence of MECN cluster 9 and 24 in all compartmentalized (n = 15) and mixed (n = 19) patients, and compared the occurrence score of these MECNs between two patient groups (see "Methods"). The results showed that the occurrence score of MECN cluster 9 (Tumor and Keratin+ tumor) displayed more consistency in mixed samples than in compartmentalized samples, with lower variance (Fig. 7k), but there is no significant difference of the occurrence of MECN cluster 9 between the mixed and compartmentalized groups (p = 0.986, two-sided Wilcoxon rank-sum test, Fig. 7k). On the contrary, MECN cluster 24 (macrophage, and Keratin+ tumor, CD8 T) significantly showed more occurrence in mixed samples than in compartmentalized samples (p = 9.20 × 10−6, one-sided Wilcoxon rank-sum test, Fig. 7l). This indicated that MECN cluster 24 (macrophage, and Keratin+ tumor, CD8 T) might be used as a clinical indicator, so we termed MECN cluster 24 as MKT. To investigate the relevance between the occurrence of MKT and prognosis, we partitioned the patients according to the occurrence score of MKT. Survival analysis showed that patients with lower occurrence of MKT was associated with better survival outcomes (Fig. 7m), regardless of the specific threshold used to separate two groups of patients (Supplementary Fig. 7). Note that the MKT is a kind of microenvironment consisting of triple-cell-type-interactions, it could not be identified by existing spatial interaction analysis, which focused on double-cell-type interactions. In this study, we presented SOTIP, a unified framework to perform multiple important tasks with various spatial omics technologies. The core of SOTIP is the construction of MECN graph (MEG), of which each node represents an MECN and each edge encodes the relationship between MEs. Based on MEG, spatial heterogeneity (SHN) quantification, spatial domain (SDM) identification, and differential microenvironment (DME) analysis can be performed. Note that the second task is essentially a clustering task, we named it with "SDM identification" to keep consensus with other related methods (for example SpaGCN37 and STAGATE83), so that readers may not be confused when reading and comparing them. We conducted simulation experiments to demonstrate the utility of SOTIP, and compared the performance with different methods. We also performed exhaustive comparisons between SOTIP and state-of-art methods on different spatially resolved datasets. In the task of SHN quantification, we benchmarked SOTIP against NUCC and IGD by delineating the nuclear envelope in HeLa cell line (spatial proteomics, 4i23) and recovering the gradient heterogeneity pattern in mouse brain cortex (spatial transcriptomics, osmFISH46). In the task of SDM identification, we firstly showed SOTIP's superior performance compared with other spatial clustering algorithms on the commonly used SpatialLIBD dataset of human brain cortex48 (spatial transcriptomics, 10x Visium). On another spatial transcriptomics dataset of mouse brain cortex46, SOTIP also outperformed others in terms of ARI. We validated the compatibility of SOTIP on other spatial omics data by collecting samples from CRC (spatial proteomics, scMEP49) and TNBC (spatial proteomics, MIBI12). Although SpaGCN is also extendable to spatial proteomics dataset, SOTIP showed better performance when comparing with the ground truth in multiple cases (Fig. 4). Based on SOTIP detected tumor-immune boundary, we also detected proteins with spatial polarization. In DME analysis, we firstly compared a fibrotic liver sample with healthy liver sample7, then identified MECNs specifically enriched in the fibrotic sample. The spatial localization and cellular composition of the detected MECN is consistent with the histology image. We finally applied SOTIP to compare the MECN differences between compartmentalized and mixed subtypes of triple negative breast cancer (TNBC), consistent with previously reported tumor-immune interactions and revealing possible theory to explain the driven factor for the differentiation of these two subtypes. Apart from the aforementioned performance advantages and discoveries, there are other several points that make SOTIP a distinct method from others. As for interpretability, deep learning-based algorithms are typically considered as black boxes and hard to interpret the result and parameters. On the contrary, each component of SOTIP can be easily mapped back to a biological entity, which is beneficial for model diagnosis and biological interpretation. As for clustering procedure, current methods36,37,38 need multiple runs to search for the optimal result when the true number of clusters is not accessible, while SOTIP maintains the intermediate result for every level during the one-time hierarchical merging process. As for scalability, unlike methods whose applications may be limited to certain range of modalities, SOTIP can be freely generalized to different spatial techniques. The core of SOTIP, MEG, was designed to only require distance/similarity matrix, without full accessibility of the coordination of the two spaces (i.e., physical space and molecular expression space). Several variants of SOTIP can be quickly implemented with different configurations of the two spaces. For example, SOTIP can be used to characterize microenvironment of other spatial coordination, for example 3D (given 3D spatially resolved techniques), and virtually inferred spatial affinity84,85. For another example, SOTIP can use the physical space of the high resolution histological images and the molecule expression space of the high throughput scRNA-seq, to integrate their benefits. With the most advanced multi-modal biotechnologies86,87, which can obtain multi-omics measurements within single cells, the physical space of SOTIP can be even replaced with protein expression, chromatin accessibility or metabolites profiles. One limitation is that SOTIP defines MECNs with predefined shape and size, which might lead to false negative discoveries. The high complexity and diversity of microenvironments in different tissue states and disease progressions make it intricate to fully understand the mechanistic properties, for example, the shapes and sizes of MEs, while SOTIP's rigid manner of MECN definition reduces the searching space of MEs, since those MEs falling out of the beforehand definition (e.g., MEs with non-spherical shapes) could be overlooked. This problem could especially stand out when the real applications need investigations of finer resolutions, for example when comparing the MECN differences between an early stage of disease (with fairly subtle alterations) and the healthy control. To alleviate this problem, two lines of methods can be considered. One is to formalize the geometric and connectomic properties as priors to be incorporated into MECN modeling. This approach might be hindered by the divergence of MEs. Another one is to expend the searching space by considering as many shapes and sizes and combinations of MEs. This approach might be troubled by computational efficiency, and supervisions need to be properly used to guide the searching process. We believe that methods belonging to this niche is essential for future research. MEG construction SOTIP takes spatial omics data (gene expression/protein/metabolomic profiles of spots/cells/pixels) as input. For ease of explanation, we will use spatial single cell transcriptomics data (formed as gene expression matrix and 2D spatial coordinates for each cell) to illustrate the method, which can be extended to other spatial omics modalities without loss of generality. The core of SOTIP is the definition of the MECN graph (MEG). With the MEG, the relationships between every pair of MECNs are encoded by considering both the occurrence, and the gene expression dissimilarities of constituted single cell identities. In this way, gene expressions, single cells, microenvironments, and tissue regions could be linked in a biological meaningful way. To describe MEG, its nodes and edges should be defined. Specifically, each node is defined as an MECN, which is a bag of cells encoded by cell clusters within the spatial neighborhood. Then each MECN is represented by counting the frequency of each cell cluster (generated by user-defined clustering algorithm, e.g., Leiden) within the MECN, resulting a histogram. In this way, for a spatial omics dataset of a tissue sample, the number of MECNs is the same as the number of cells, and every MECN are represented by a vector with the length of number of cell clusters. To define the edge between two nodes in MEG, SOTIP computes earth mover's distance (EMD)88 between them. Considering the cell discrepancy within high-dimensional gene expression space, we derive a connectivity guided minimum graph distance (CGMGD) as the ground distance which approximates the distance in the transcriptome state space. We elaborate CGMGD 's better performance compared with other choices in "Connectivity guided minimum graph distance (CGMGD)" of the "Methods" section. With simulation data, we also demonstrate the better quality of MEG constructed based on CGMGD than based on an isotropic ground distance (IGD), which assumes equal distances among cell groups (Fig. 2c). After computing the pairwise distance, SOTIP estimates the connectivity between nodes using an efficient neighbor search similar with UMAP32,89, and the MEG is then constructed. We next formally define MEG and associated terms. Different with general application of earth mover's distance, in the same biological sample, SOTIP represents every MECNs using single cells with the same configuration of clusters. In a spatial omics sample with n cells and t cell clusters, we can formulize the definition of MECN as a histogram: $M{{ECN}}_{p}=\{({T}_{i},\,{w}_{{T}_{i}}^{p})\},{T}_{i}\in \left[1,\,t\right],\,p\in [1,\,n]$. Ti enumerates all possible cell cluster labels in the tissue sample, and ${w}_{{T}_{i}}^{p}$ is the total-normalized count of cluster Ti in MECNp. The distance between two MECNs, e.g. MECNp and MECNq is defined as (1): $${dist}\left(M{{ECN}}_{p},\,M{{ECN}}_{q}\right)=\frac{\mathop{\sum }\nolimits_{i=1}^{t}\mathop{\sum }\nolimits_{j=1}^{t}{f}_{{ij}}{CGMGD}(i,\,j)}{\mathop{\sum }\nolimits_{i=1}^{t}\mathop{\sum }\nolimits_{j=1}^{t}{f}_{{ij}}}$$ where CGMGD(i, j) is precomputed between all pairs of cell clusters according to Methods section "Connectivity guided minimum graph distance (CGMGD)". And $\mathop{\sum }\nolimits_{i=1}^{t}\mathop{\sum }\nolimits_{j=1}^{t}{f}_{{ij}}{CGMGD}(i,\,j)$ is minimized subject to (2–4). $${f}_{{ij}}\ge 0,\,i,\,\,j\in [1,\,t],$$ $$\mathop{\sum }\limits_{i=1}^{t}{f}_{{ij}}\le {w}_{j}^{M{{ECN}}_{q}},\,i,\,j\in \left[1,\,t\right],\,q\in [1,\,n],$$ $$\mathop{\sum }\limits_{j=1}^{t}{f}_{{ij}}\le {w}_{i}^{M{{ECN}}_{p}},\,i,\,j\in \left[1,\,t\right],\,p\in \left[1,\,n\right].$$ Spatial heterogeneity (SHN) quantification We define the spatial heterogeneity (SHN) as a numerical property of a node in MEG, to assess the total gene expression variation within the represented MECN. For a MECN, suppose it consists of k single cells, that is (5). $${CONSIST}({MECN})=({c}_{1}\ldots {c}_{i}\ldots {c}_{k}),\,i\in [1,\,k]$$ ci is vectorized by a high-dimensional gene expression feature. Each single cell ci can be encoded with a categorical variable Ti, representing the belonging cell group, which is precomputed by clustering with the gene expression feature vector (6). $${GROUP}\left({c}_{i}\right)={T}_{i},\,{i\in \left[1,\,k\right],T}_{i}\in [1,\,t]$$ Since we have also precomputed the pairwise CGMGD among cells, and the CGMGD between two single cells is a function of their belonging cell groups. We denote the pairwise gene expression variation (PGEV) as (7). $${PGEV}\left({c}_{i},\,{c}_{j}\right)={CGMGD}\left({GROUP}\left({c}_{i}\right),\,{GROUP}\left({c}_{j}\right)\right),\,i,\,j\in [1,\,k]$$ The SHN of a MECN is computed as the total gene expression variation by summing over i and j (8). $${SHN}({MECN})=\mathop{\sum}\limits_{{c}_{i},\,{c}_{j}\in {CONSIST}({MECN})}{PGEV}\left({c}_{i},\,{c}_{j}\right)$$ When estimating SHN with an isotropic ground distance (IGD), the pairwise gene expression variation is simply replaced with (9). $${PGE}{V}_{{IGD}}\left({c}_{i},\,{c}_{j}\right)={IDENTITY}\left({GROUP}\left({c}_{i}\right),{GROUP}\left({c}_{j}\right)\right),\,i,\,j\in [1,\,k]$$ Where IDENTITY is an indicator function defined as (10). $${IDENTITY}\left(p,\,q\right)={{{{{{\bf{1}}}}}}}_{p!=q}(p,\,q)$$ Spatial domain (SDM) identification The relationship among MECNs encoded by MEG considers both the cellular composition by representing each MECN as a histogram, and the discrepancy among cells described by gene expression profiles. To connect the local microenvironment with the tissue domain at a relatively broader scale, SOTIP performs the task of spatial domain identification based on the pre-constructed MEG. Leiden community detection50 is firstly performed on the MEG to partition MECNs into clusters, to this end the clustered result already reflect meaningful tissue organization. To reach the pre-defined number of clusters, SOTIP next performs cluster merging with a hierarchical scheme based on the connectivity between MECN clusters. Specifically, similar to modularity90,91, SOTIP measures the degree of connectivity of two MECN clusters by considering the ratio between the number of inter-edges and the expected number of inter-edges under random assignment. SOTIP then iterates between two sub-steps: assess the pairwise connectivity between MECN clusters, and update MECN cluster assignment by merging MECN clusters with largest connectivity, until the number clusters reached the predefined. Since the process is guided by the finely designed MEG, in each step the algorithm merges two most similar sub regions of tissue. After the hierarchical merging, users can get a multi-level tissue partitioning hierarchy, from which tissue regions with different resolutions can be investigated. This is particularly an advantage over other SDM identification algorithms, e.g., stLearn38, bayesSpace36, and SpaGCN37, especially in real cases that the ground truth of number of tissue regions is not known in advance. Differential microenvironment (DME) analysis The aim of DME analysis is to identify specifically enriched microenvironments between samples (or conditions, disease status, drug treatment, etc.). This task is also based on the MEG construction, but from joining MECNs of two samples. After that, Leiden50 is used for community detection based on MEG, resulting a cluster assignment for MEs. Then an abstract version of MEG is constructed with partition-based graph abstraction (PAGA)91. Each node of the resulted MEG abstraction is an MECN cluster, and the connectivity between nodes is computed as the ratio between the number of inter-edges and the expected number of inter-edges under random assignment, similar to modularity90. To identify interesting MECN clusters with specific sample associated enrichment and more complex cellular composition, an EH plot (abbreviation for "entropy-of-MECN-cluster (EMC)" versus "spatial heterogeneity (SHN)" plot) is drawn for each DME analysis application. EH plot is a scatter plot, in which each point is an MECN cluster, the x-axis is defined as average spatial heterogeneity (SHN) across MECNs of the MECN cluster, and the y-axis is defined as the entropy of the MECN cluster (EMC) between the compared two samples. Suppose we want to compare the MECN differences of two samples: sample 0 and sample 1, the EMC (EMCC) of an MECN cluster C (C is an MECN cluster, viz., a set of MECNs) is defined on the Bernoulli distribution (parameterized by p) modeling the probability of observing an MECN from either samples. That is: $${{EMC}}_{C}\left(p\right)=-{p}_{C}{lo}{g}_{2}{p}_{C}-\left(1-{p}_{C}\right){{{\log }}}_{2}\left(1-{p}_{C}\right),{where\; C\; is\; a\; MECN\; set}$$ The parameter of the distribution is estimated as the proportion of normalized count of MECNs (NCME) from sample 0, that is (11). $${p}_{C}=\frac{{NCME}(C,\,{sample}0)}{{NCME}\left(C,\,{sample}0\right)+{NCME}(C,\,{sample}1)},$$ where NCME of sample k is defined as (12). $${NCME}\left(C,\,{sample\; k}\right)=\frac{\left\|\left\{{MECN}\|{MECN}\in {C\; and\; SAMPLE}\left({MECN}\right)={sample\; k}\right\}\right\|}{\left\|\left\{{MECN}\|{SAMPLE}\left({MECN}\right)={sample\; k}\right\}\right\|},$$ where SAMPLE(MECN) is defined as which sample the MECN comes from. Interesting MECNs (strong specificity and high SHN) lies in the bottom right corner of the EH plot. Above analysis can be extended to multiple samples/conditions with multinomial distribution. To summarize, the input of SOTIP-DME is two tissue samples (spatial omics data), and the output is the EH plot. Using the EH plot, each point is a MECN cluster, one can visually assess those MECN clusters with high sample-specificity and high spatial heterogeneity. Connectivity guided minimum graph distance (CGMGD) Computing the earth mover's distance between microenvironments needs to define the ground distance between cell clusters. Directly computing Euclidean distance in the high-dimensional gene expression space would be cursed by the dimensionality. Early single cell analysis92 attempted to compute distance in the embedded space, e.g. t-SNE93, or UMAP89, to perform downstream analysis such as clustering, but was proved problematic since the principle of these algorithms made it poorly preserving global topological data structure94,95,96. Diffusion pseudo-time (DPT)97 computes the geodesic distance between each data point and a given root point in a diffusion map embedded space, which seems consistent with our objective. However, since we need to compute the ground distance between cell clusters, we need to run DPT (e.g., SCANPY implementation32) for multiple times by setting the root point to the center cell of every cell cluster. So the computational efficiency is a problem, not to mention the choice of the cluster centers. More recently, PAGA91 is reported to preserve the topological data structure of the high-dimensional scRNA-seq data, and has been widely used as an official module of SCANPY32. It is also not suitable for us since it assesses the connectivity between clusters, instead of the distance we need. PHATE53 is reported to preserve better distance than t-SNE and UMAP, there are also researches applying Euclidean distance within PHATE embedding to assess the distance between single cells, we found it produced errors with our simulated data. To more accurately approximate the distance in the transcriptome state space, and also accounting for computational efficiency, we propose connectivity guided minimum graph distance (CGMGD). Specifically, since we only need the cluster level distance instead of single cell level, CGMGD firstly assesses the connectivity between cell clusters with PAGA to produce a binary connectivity matrix (BCM) justifying whether there are edges (1) or not (0) between two cell clusters. In the second step, CGMGD embeds the single cells into UMAP space, and calculates the pairwise distance of cluster centers within the embedding space to produce a UMAP distance matrix (UDM). In the third step, a connectivity-guided cluster graph (CGG) is constructed, of which each node is defined as a cell cluster. The edge of the graph is defined with the adjacent matrix computed by element-wise multiplication between BCM and UDM. Finally, the CGMGD is computed by searching pairwise minimum distance of the CGG. With this procedure, CGMGD approximates the distance in the transcriptome state space by integrating the global topological preservation of PAGA, and the local data structure preservation of UMAP. We used simulation data 4 (see "Methods") to demonstrate the advantage of CGMGD over six manifold learning algorithms, PCA, UMAP89, DPT97, ForceAtlas98, PHATE53, and PAGA91 (Supplementary Fig. 8). Correlation analysis between SHN and cortical depth To quantify the ordinal agreement between the SHN values and the cortical layer, we used both ME-wise and layer-wise Spearman's rank correlation coefficients (Spearman's ⍴) between them. For ME-wise manner, the SHN was computed for each MECN, and the layer depth of each MECN was assigned by setting a number for each layer, Pia Layer 1 (1), Layer 2–3 medial (2.5), Layer 2–3 lateral (2.5), Layer 3–4 (3.5), Layer 4 (4), Layer 5 (5), Layer 6 (6). The Spearman's ⍴ was computed between the SHNs and layer depths of MEs. For layer-wise manner, the SHN was computed as median SHN of MECNs for each layer, and the layer depth was assigned as with ME-wise manner. The Spearman's ⍴ was computed between the SHNs and layer depths of layers. Validation of identified DMEs in all TNBC patients To validate that our identified differential MEs, i.e., MECN cluster 9 (Tumor and Keratin+ tumor) and 24 (CD8 T, macrophage, and Keratin+ tumor) consistently differentiated between compartmentalized and mixed samples from other patients, we performed MECN occurrence analysis as follows. For ease of explanation, we used MECN cluster 9 as an example. For each patient sample, we firstly counted the number of target MECNs (MECNs which contained cells from both Tumor and Keratin+ tumor clusters) as kobs. In order to account for difference in number of cells, the expected number of target MECNs (kexp) under background distribution was computed by random permutation of cell cluster labels, followed by averaging across their number of target MEs. Formally, ${\bar{k}}_{{\exp }}=\frac{1}{N}\times \mathop{\sum }\nolimits_{i=1}^{N}{\bar{k}}_{i}$, where N is the number of permutations, and ${\bar{k}}_{i}$ is the count of target MECNs in each permutation. The occurrence score of MECN cluster 9 is computed as ${k}_{{obs}}/{\bar{k}}_{{\exp }}$. Following this procedure, we computed the occurrence score for every patient, and compare the occurrence scores between compartmentalized (N = 15) and mixed (N = 19) patients. The significance is estimated using one-sided Wilcoxon rank-sum test. Simulation datasets There are four simulation datasets used in this study. Simulation 1~3 were firstly generated with Splatter99, a R package for scRNA-seq data simulation, then arrange the spatial coordination for the generated cells. Simulation 4 is generated by Splatter as single-cell data, without spatial coordination. The full information for these datasets is summarized in Supplementary Table 2. In the analysis of simulation 3 (Fig. 2e, f), the relative likelihood of observing each microenvironment in specific sample is performed by MELD55, with the precomputed MECN distance matrix as input. MELD analysis MELD55 is an algorithm for estimating the relative likelihood of observing each cell state between different conditions (e.g., different disease state, before/after drug treatment or other experimental perturbations). In original MELD publication, the input of MELD is (1) the single cell graph on gene expression space, and (2) the condition label for each single cell. The output of MELD is the relative likelihood of observing each cell in each condition. In our application (e.g., Fig. 2f), we instead input the microenvironment graph, and the condition label for each microenvironment. In this way, by combining SOTIP's mathematical definition of microenvironment and MELD, we can estimate the relative likelihood of observing each microenvironment between different conditions. PAGA analysis PAGA91 is an algorithm for trajectory inference through a topology-preserving map of single cells. In the original PAGA publication, the input of PAGA is (1) the single-cell graph on gene expression space, (2) a clustering assignment for the single cell data. The output of PAGA is graph, in which each node is a cluster, and the edge between two nodes is the strength of connectivity (the connectivity could also be interpreted as similarity) between two clusters. In our application (e.g., Supplementary Fig. 3), we instead input the microenvironment graph, and the clustering label. So that we can get the topological map of the targeted tissue. This study involved multiple datasets from seven different spatial omics technologies. All datasets are publicly available. Human HeLa cell line 4i dataset: https://squidpy.readthedocs.io/en/stable/23. Mouse brain cortex osmFISH dataset: http://linnarssonlab.org/osmFISH46. Mouse brain cortex seqFISH+ dataset: https://github.com/CaiGroup/seqFISH-PLUS17. Zebrafish melanoma Visium dataset: GSE15970961. Human brain cortex Visium dataset: http://research.libd.or g/spatialLIBD/48. Human colorectal carcinoma scMEP dataset: https://zenodo.org/record/395161349. Human triple negative breast cancer MIBI dataset: https://mibi-share.ionpath.com/12. TOF-SIMS liver dataset: https://github.com/yuanzhiyuan/SEAM/tree/master/SEAM/data/raw_tar7. The full information of public datasets used in this study can be found in Supplementary Table 3. NUCC and IGD NUCC16,33 and IGD are both control methods for spatial heterogeneity (SHN) quantification. NUCC is a simple and widely used method. The main idea is to quantify the spatial heterogeneity around a cell by counting the number of unique cell clusters within the neighborhood. Its rationale is straightforward, if a cellular neighborhood contains many unique cell types, then the SHN of that neighborhood is high. The "cell clusters" could be defined by either single-cell clustering algorithm, or by cell type annotation. The"cellular neighborhood" could be defined by either k-nearest-neighbor like SOTIP-SHN, or by a user-defined radius. IGD is a variant of SOTIP-SHN, and is also proposed as a control method for SHN quantification by this paper. We propose IGD to prove the advantage of using CGMGD in SOTIP-SHN. The only difference between IGD and SOTIP-SHN is that, SOTIP-SHN used the sum of CGMGD distance matrix within the cellular neighborhood, while IGD replaced the CGMGD distance matrix with IDENTITY distance matrix (please refer to "Spatial heterogeneity (SHN) quantification" section). More explanations are in "Further explanations on SHN" in Supplementary Notes. Method comparison To evaluate the performance in spatial heterogeneity (SHN) quantification, NUCC and IGD was compared with SOTIP. As with SOTIP, NUCC, IGD both need to perform clustering and define spatial neighborhood. For a fair comparison, NUCC, IGD, and SOTIP utilized exactly the same parameters (MECN size and cluster resolution). We also compared the relative performance under different combinations of these parameters, the parameters were also consistent in every comparison. With 4i dataset (Fig. 3a–f), since the original data (Supplementary Table 3) provided ground truth of nucleus and cytoplasm, we defined their boundary as width of 2 pixels. We used area under curve (AUC) to evaluate and compare the performance. With osmFISH dataset (Fig. 3k), the original data (Supplementary Table 3) provided cell-wise annotations of different cortical layers. We used Spearman's rank correlation coefficient to evaluate the ordinal agreement between the layer order and the computed SHN. To digitalize the layer order, we set a number for each layer, Pia Layer 1 (1), Layer 2–3 medial (2.5), Layer 2–3 lateral (2.5), Layer 3–4 (3.5), Layer 4 (4), Layer 5 (5), Layer 6 (6). To test the impact of over-clustering on SHN quantification (Supplementary Fig. 2), on osmFISH data and seqFISH data, we set the clustering resolution to moderate resolution, over-clustering (3 × moderate), and extreme over-clustering (5 × moderate). The moderate resolution was determined by resolution researching given true number of clusters (provided by original paper17,46). To evaluate the performance in spatial domain (SDM) identification, for both brain cortex datasets (Fig. 4), we set the true number of clusters to run compared algorithms, and used adjusted Rand index (ARI) to compare the similarity between clustering result and ground truth. Note that for human sample, since original ground truth labels contain "N/A" (Fig. 4b), we filtered out these spots before the comparison. The ARI of the human sample was directly adopted from the ARI as they reported in the original paper36,37. We also applied the same gene filtering and data normalization procedures as with code provided by SpaGCN. For the mouse cortex osmFISH dataset, the original ground truth labels contain "excluded" (Fig. 4e), we filtered out these single cells before the comparison, and the parameter of SpaGCN was set as default. As for the spatial proteomics datasets (Fig. 5), the original paper provided ground truth for tumor-immune boundaries, based on which we compared SOTIP and SpaGCN qualitatively. For better visualization29, we plotted the results by Voronoi diagrams, adopted from https://stackoverflow.com/questions/20515554/colorize-voronoi-diagram/20678647#20678647%203.18.2019. Parameter settings There are three parameters for user to choose according to their applications. These parameters are set with a consistent manner except when evaluating the robustness to different combinations of parameters. Across the manuscript, the resolution of Leiden clustering is set by searching according to the true number of cell clusters if available, otherwise 2. The parameter of MECN size (k-NN parameter) is set to 10. The number of spatial domains is set according to the ground truth. More discussions of parameter settings can be found in Supplementary Notes. Molecular polarization with spatial proteomics data In the CRC datasets, following the consistent manner with Hartmann et al.49, we first classified immune cells within a 51 pixels (20 μm) radius to tumor region as proximal, and other immune cells as distal (Supplementary Fig. 4a, b). In the TNBC datasets, and also consistent with Keren et al.12, the definition of proximity of both "tumor cells to immune region" and "immune cells to tumor region" is defined with radius of 100 pixels (39 μm) (Supplementary Fig. 4c, d). The criterion for selection of representative polarized proteins (Fig. 4f, l, Supplementary Fig. 4g) is by ascendingly sorting proteins by FDR, and the representative proteins are overlay and plotted by the interactive plotting webserver Ionpath (https://www.ionpath.com/). Steps of Fig. 5f–i We identified the polarized proteins following the standard procedures of previous studies12,49, with the following steps: Define spatial domains using either SpaGCN or SOTIP's SDM module. Classify immune cells into two groups. Group1: Proximal (those immune cells whose smallest distance from any tumor cells are smaller than a radius), and Group2: Distal (those immune cells whose smallest distance from any tumor cells are larger than a radius). For each protein, we compared the protein abundance between Group1 and Group2, and assessed the significance using two-sample rank-sum test, which is further corrected by Benjamini–Hochberg (BH) procedure for multiple hypotheses testing. We set different FDR cutoffs (the x-axis in Fig. 5f, i) to filter those statistical significant polarized proteins and obtained the identification power by comparing with the true positives. The true positives are based on the polarized protein list identified in original papers12,49. All boxplots in the manuscript share the same settings: the lower and upper hinges show the first and third quartiles (the 25th and 75th percentiles); the center lines correspond to the median; the upper whisker extends from the upper hinge to the largest value, which should be <1.5× the interquartile range (or distance between the first and third quartiles) and the lower whisker extends from the lower hinge to the smallest value, which is at most the 1.5× interquartile range. Data beyond the end of the whiskers are 'outlying' points and are plotted individually. Quantitative and statistical analysis All statistical tests used in this study are described in detail in the corresponding figure legends. Spearman's rank correlation coefficients, Pearson correlation coefficient, and Wilcoxon rank-sum test are performed using scipy100. To account for multiple hypotheses testing, we applied the Benjamini–Hochberg (BH) procedure to report the associated FDR101, which is performed using pingouin102. ARI and AUC are performed using scikit-learn103. Survival analysis was performed using scikit-survival104. Numpy, Pandas, and Scikit-learn were used to perform scientific computing. Matplotlib, seaborn, palettable were used to generate figures. SCANPY and Squidpy were used to analyze spatial data. Networkx and shapely was used to deal with graph representation. Pyemd was used to compute EMD distance. Statistics and reproducibility For Figs. 3g, h, 5d, f, g, I, 6b, c, d, and Supplementary Fig. 4e, g, the experiment results were similar with at least three independent repeats. All raw data are freely available at following links: Human HeLa cell line (4i): [https://squidpy.readthedocs.io/en/stable/] Mouse Brain cortex (osmFISH): [http://linnarssonlab.org/osmFISH] Mouse Brain cortex (seqFISH+): [https://github.com/CaiGroup/seqFISH-PLUS] Zebrafish melanoma (10X Visium): GSE159709 Human Brain cortex (10X Visium): [http://research.libd.org/spatialLIBD] Human colorectal carcinoma (scMEP): [https://zenodo.org/record/3951613] Human triple-negative breast cancer (MIBI): [https://mibishare.ionpath.com/] Liver (SIMS): [https://github.com/yuanzhiyuan/SEAM/] Mouse brain (EASI-FISH): [https://janelia.figshare.com/articles/dataset/EASI-FISH_enabled_spatial_analysis_of_molecular_cell_types_in_the_lateral_hypothalamus/13749154] Human breast cancer (3D IMC): [https://doi.org/10.5281/zenodo.4752030] The processed data in this manuscript can be downloaded at figshare [https://doi.org/10.6084/m9.figshare.18516128]. Source data are provided with this paper. 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Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011). Pölsterl, S. scikit-survival: A Library for Time-to-Event Analysis Built on Top of scikit-learn. J. Mach. Learn. Res. 21, 1–6 (2020). This research was supported by the National Key Research and Development Program of China (2018YFA0801402), National Natural Science Foundation of China (81890994) to M.L.S., National Natural Science Foundation of China (81890991), Beijing Municipal Natural Science Foundation (Z200021), CAS Interdisciplinary Innovation Team (JCTD-2020-04), and the State Key Research Development Program of China (2021YFE0201100) to J.T.G. M.Q.Z. acknowledges the support by the Cecil H. and Ida Green Distinguished Chair. Z.Y.Y. acknowledges the support by Shanghai Municipal Science and Technology Major Project (No.2018SHZDZX01), ZJ Lab, Shanghai Center for Brain Science and Brain-Inspired Technology, and 111 Project (No.B18015). These authors contributed equally: Zhiyuan Yuan, Yisi Li. Institute of Science and Technology for Brain-Inspired Intelligence; MOE Key Laboratory of Computational Neuroscience and Brain-Inspired Intelligence; MOE Frontiers Center for Brain Science, Fudan University, Shanghai, 200433, China Zhiyuan Yuan Tencent AI Lab, Shenzhen, China Zhiyuan Yuan, Fan Yang & Jianhua Yao MOE Key Laboratory of Bioinformatics; Bioinformatics Division and Center for Synthetic & Systems Biology, BNRist; Department of Automation, Tsinghua University, Beijing, 100084, China Zhiyuan Yuan, Yisi Li, Juntao Gao & Michael Q. Zhang MOE Key Laboratory of Bioinformatics; Bioinformatics Division and Center for Synthetic & Systems Biology, School of Medicine, Tsinghua University, Beijing, 100084, China Minglei Shi & Michael Q. Zhang Department of Biological Sciences, Center for Systems Biology, The University of Texas, Richardson, TX, 75080-3021, USA Michael Q. Zhang Yisi Li Minglei Shi Fan Yang Juntao Gao Jianhua Yao M.Q.Z. and Y.L. conceived and designed the project. Z.Y. and Y.L. developed and implemented the algorithms under the guidance of M.Q.Z. and J.Y. Z.Y. and Y.L. collected and processed public datasets. Z.Y. and Y.L. conducted the data analysis and methods comparisons. Y.L. and M.S. did the biological interpretation. M.S. and J.G. gave suggestions on the applications of the method. Z.Y. and Y.L. completed the figures and manuscript with the guidance of J.Y., M.S., and M.Q.Z. F.Y. helped with the figure generation. All authors approved the manuscript. Correspondence to Zhiyuan Yuan, Jianhua Yao or Michael Q. Zhang. F.Y. and J.Y. are employees of Tencent. The remaining authors declare no conflicts of interest. Nature Communications thanks Guiyan Ni and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. This article has been peer reviewed as part of Springer Nature's Guided Open Access initiative. Supplementary Figures Tables and Notes Transparent Peer Review File Editorial Assessment Report source data for figure 3~7 Yuan, Z., Li, Y., Shi, M. et al. SOTIP is a versatile method for microenvironment modeling with spatial omics data. Nat Commun 13, 7330 (2022). https://doi.org/10.1038/s41467-022-34867-5 Sign up for the Nature Briefing: Cancer newsletter — what matters in cancer research, free to your inbox weekly. Get what matters in cancer research, free to your inbox weekly. Sign up for Nature Briefing: Cancer	CommonCrawl
The areas of three squares are 16, 49 and 169. What is the average (mean) of their side lengths? Since the areas of the three squares are 16, 49 and 169, then their side lengths are $\sqrt{16}=4$, $\sqrt{49}=7$ and $\sqrt{169}=13$, respectively. Thus, the average of their side lengths is $$\frac{4+7+13}{3}=\boxed{8}.$$	Math Dataset
The expression $x^2 + 18x - 63$ can be written in the form $(x - a)(x + b)$, where $a$ and $b$ are both nonnegative real numbers. What is the value of $b$? Factoring, we find that $x^2 + 18x - 63 = (x - 3)(x + 21).$ Therefore, $b = \boxed{21}.$	Math Dataset
\begin{document} \title{f Asymptotic spreading of interacting species with multiple fronts II: Exponentially decaying initial data} \begin{abstract} This is part two of our study on the spreading properties of the Lotka-Volterra competition-diffusion systems with a stable coexistence state. We focus on the case when the initial data are exponential decaying. By establishing a comparison principle for Hamilton-Jacobi equations, we are able to apply the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. As a result, the exact formulas of spreading speeds and their dependence on initial data are derived. Our results indicate that sometimes the spreading speed of the slower species is nonlocally determined. Connections of our results with the traveling profile due to Tang and Fife, as well as the more recent spreading result of Girardin and Lam, will be discussed. \end{abstract} {\small {\makeatletter \def\@@underline#1{#1} \def\@@overline#1{#1} \tableofcontents \makeatother} } \section{Introduction} For monotone dynamical systems, the pioneering work of Weinberger et al. \cite{Weinberger_1982,Weinberger_2002b} (see also \cite{Lui_1989}) relates the spreading speed of the population to the minimal speed of (monostable) traveling wave solutions. Their result can be applied to the diffusive Lotka-Volterra competition system. Suitably non-dimensionalized, the system is given by \begin{equation}\label{eq:1-1} \left\{ \begin{array}{ll} \partial_t u-\partial_{xx}u=u(1-u-av),& \text{ in }(0,\infty)\times \mathbb{R},\\ \partial _t v-d\partial_{xx}v=r v(1-bu-v),& \text{ in }(0,\infty)\times \mathbb{R},\\ u(0,x)=u_0(x), & \text { on } \mathbb{R},\\ v(0,x)=v_0(x), & \text { on } \mathbb{R}, \end{array} \right . \end{equation} with $a,b\in(0,1)$. It is clear that \eqref{eq:1-1} admits a trivial equilibrium $(0, 0)$, two semi-trivial equilibria $(1,0)$ and $(0,1)$, and further a linearly stable equilibrium $$(k_1,k_2)=\left(\frac{1-a}{1-ab},\frac{1-b}{1-ab}\right).$$ \begin{theorem}[Lewis et al.{\cite{Lewis_2002}}]\label{thm:LLW} Let $(u,v)$ be the solution of \eqref{eq:1-1} with initial data $$u(0,x)=\rho_1(x), \,\,\,\, v(0,x)=1-\rho_2(x), $$ where $0\leq \rho_i<1$ $(i=1,2)$ are compactly supported functions in $\mathbb{R}$. Then there exists some $c_{\rm LLW} \in [2\sqrt{1-a}, 2]$ such that \begin{align} \left\{ \begin{array}{ll} \lim\limits_{t\rightarrow \infty} \sup\limits_{\|x\|<c t} (\|u(t,x)-k_1\|+\|v(t,x)-k_2\|)=0 &\text{ for each } c<c_{\rm LLW},\\ \lim\limits_{t\rightarrow \infty} \sup\limits_{\|x\|>c t} (\|u(t,x)\|+\|v(t,x)-1\|)=0 &\text{ for each }~ c>c_{\rm LLW}. \end{array} \right. \end{align} In this case, we say that $u$ spreads at speed $c_{\rm LLW}$. \end{theorem} \begin{remark}\label{rmk:LLW} If the initial data $(u,v)(0,x)$ is a compact perturbation of $(1,0)$, then there exists $\tilde{c}_{\rm LLW} \in [2\sqrt{dr(1-b)}, 2\sqrt{dr}]$ such that the species $v$ spreads at speed $\tilde{c}_{\rm LLW}$. \end{remark} It is shown in \cite{Li_2005, Liang_2007} that the spreading speed $c_{\rm LLW}$ (resp. $\tilde{c}_{\rm LLW}$) is identical to the minimum wave speed of traveling wave solution connecting the pair of equilibria $(k_1,k_2)$ and $(0,1)$ (resp. $(1,0)$). It is crucial for the theory that the pair of equilibria forms an ordered pair of equilibria (regarding the comparability of steady states in the theory of monotone semi-flows, see \cite{Smith_1995}). For the weak competitive diffusive system \eqref{eq:1-1}, Tang and Fife \cite{Tang_1980} proved an additional class of traveling wave solutions connecting the positive equilibrium $(k_1,k_2)$ with the trivial equilibrium $(0,0)$. In this case, the equilibria $(0,0)$ and $(k_1,k_2)$ are un-ordered, and hence the existence of traveling wave, due to Tang and Fife \cite{Tang_1980}, does not directly follow from the monotone dynamical systems framework due to Weinberger et al. \cite{Weinberger_2002,Weinberger_2002b} (see also \cite{Fang_2014,Liang_2007}). A natural question is whether the speed traveling wave solutions due to Tang and Fife, which connect $(k_1,k_2)$ to $(0,0)$, determine the spreading speed of the populations in the Cauchy problem \eqref{eq:1-1}, provided the initial data $(u_0,v_0)$ has the same asymptotics at $x=\pm \infty$ as the traveling wave solution? What happens for more general exponentially decaying initial data? Does the two species spread with different speeds? In this paper, we continue our investigation in \cite{LLL2019} on the spreading properties of solutions of the Cauchy problem \eqref{eq:1-1}. We are interested in determining the spreading speeds of each of the populations $u$ and $v$, for the class of initial data $(u_0,v_0)$ satisfying $(u_0,v_0)(-\infty) = (1,0)$, $(u_0,v_0)(\infty) = (0,0)$ and such that $u_0 \to 0$ exponentially at $\infty$ with rate $\lambda_u>0$; $v_0 \to 0$ decays exponentially at $ \infty$ (resp. $-\infty$) with rate $\lambda^{+}_v>0$ (resp. $\lambda^-_v>0$). We introduce the Hamilton-Jacobi approach to study the spreading of two-interacting species into an open habitat, and resolve a conjecture by Shigesada \cite[Ch. 7]{Shigesada_1997}. Inspired by the pioneering work of Freidlin \cite{Freidlin_1985} and of Evans and Souganidis \cite{Evans_1989} on the Fisher-KPP equation, we shall derive, via the thin-front limit, a couple of Hamilton-Jacobi equations for which solutions have to be understood in the viscosity sense. In our previous work \cite{LLL2019}, we considered the Cauchy problem \eqref{eq:1-1} endowed with compactly supported initial data, and used the dynamics programming approach to show the uniqueness of the limiting Hamilton-Jacobi equations, and to evaluate the solution by determining the path that minimizes certain action functional. In contrast to our previous paper, we will tackle the Cauchy problem with exponentially decaying initial data using entirely PDE arguments. For this purpose, we establish a general comparison principle for discontinuous viscosity solutions associated with piecewise Lipschitz Hamiltonians, the latter arising naturally in the spreading of multiple species. The proof of the comparison result is based on combining the ideas due to Ishii \cite{Ishii_1997} and Tourin \cite{Tourin_1992}. With this comparison principle at our disposal, we are able to obtain large-deviation type estimates of the solutions $(u,v)$ to the Cauchy problem \eqref{eq:1-1} by explicit construction of simple piecewise linear super- and sub-solutions. \begin{comment} Recall from the classical literature \cite{Aronson_1975, Fisher_1937, Kolmogorov_1937} that the scalar Fisher-KPP equation \begin{equation} \left\{ \begin{array}{ll} \partial _t u_{KPP}-d\partial_{xx}u_{KPP}=ru_{KPP}(1-u_{KPP})& in~\mathbb{R}^+\times \mathbb{R},\\ u_{KPP}(0,x)=\tilde{u}_0(x) & for ~all~ x\in \mathbb{R},\\ \end{array} \right . \end{equation} with $d,r>0$. If $\tilde{u}_0\in \mathscr{C}(\mathbb{R})$ with nonempty compact support has the following spreading property : there exists a unique $c_{KPP}>0$ satisfying $$\lim\limits_{t\rightarrow \infty} \sup_{\|x\|<c t} (\|u_{KPP}(t,x)-1\|)=0 ~~ for~ all~ c<c_{KPP},$$ $$\lim\limits_{t\rightarrow \infty} \sup_{\|x\|>c t} (\|u_{KPP}(t,x)\|)=0 ~~for~ all~ c>c_{KPP}.$$ \end{comment} \subsection{Known results of a single population} We first recall some classical asymptotic spreading results concerning the single Fisher-KPP equation: \begin{equation}\label{eq:single} \left\{ \begin{array}{ll} \partial _t \phi-\tilde{d}\partial_{xx}\phi=\tilde{r}\phi(1 - \phi),& \text{ in }(0,\infty)\times \mathbb{R},\\ \phi(0,x)=\phi_0(x), & \text{ on } \mathbb{R},\\ \end{array} \right . \end{equation} where $\tilde{d},\tilde{r}$ are positive constants. If the initial data is a Heaviside function, supported on $(-\infty,0]$, it is shown \cite{Aronson_1975,Fisher_1937,Kolmogorov_1937} that the population, whose density is given by $\phi(t,x)$ has the spreading speed $c^* = 2\sqrt{\tilde{d} \tilde{r}}$, i.e., \begin{equation} \left\{ \begin{array}{ll} \lim\limits_{t\rightarrow \infty} \sup\limits_{x<c t} \|\phi(t,x)-1\|=0 &\text{ for all } c<c^,\\ \lim\limits_{t\rightarrow \infty} \sup\limits_{x>c t} \|\phi(t,x)\|=0 &\text{ for all } c>c^.\\ \end{array} \right . \end{equation} In addition, the spreading speed $c^$ coincides with the minimal speed of the traveling wave solutions to \eqref{eq:single} in this case. If we broaden the scope of initial data $\phi_0$ to include the class of exponentially decaying data, then the asymptotic behavior of the solution to \eqref{eq:single} is sensitive to the rate of decay of $\phi_0$ at $x=\pm\infty$ (see e.g. \cite[pp.42]{Frank_1975}), which is the {leading edge} of the front. This is related to the fact that $0$ is a saddle for \eqref{eq:single}, see \cite{Kametaka_1976,Mckean_1975,Booty_1993,Ebert_2000,Saarloos_2003}. Precisely, denoting $\lambda^=\sqrt{{\tilde{r}}/{\tilde{d}}}$. It is proved \cite{Kametaka_1976, Mckean_1975} that: \begin{itemize} \item [\rm{(i)}] When the initial data $\phi_0(x)$ decays faster than $\exp\{-\lambda^x\}$ at $x=\infty$, then the spreading speed $c^=2\sqrt{\tilde{d}\tilde{r}}$; \item [\rm{(ii)}] When the initial data $\phi_0(x)$ is the form of $\exp\{-(\lambda+o(1)) x\}$ at $x=\infty$ with $\lambda<\lambda^$, then the population has the spreading speed $c(\lambda) = \tilde{d}\lambda + \frac{\tilde{r}}{\lambda}$ which is strictly greater than $2\sqrt{\tilde{d}\tilde{r}}$. \end{itemize} For recent developments in asymptotic spreading of a single population in heterogeneous environments, we refer to \cite{Berestycki_2012,Berestycki_2018, Fang_2016} for the one-dimensional case, and to \cite{Berestycki_2008,BNpreprint,Nolen_2008,Weinberger_2002} for higher-dimensional case. \subsection{Known results of multiple populations} For close to three decades, researchers have been trying to extend these results to reaction-diffusion systems describing two or more interacting populations. Motivated by the northward spreading of several tree species into the newly de-glaciated North American continent at the end of the last ice age, Shigesada et al. \cite[Ch. 7]{Shigesada_1997} formulated the question of spreading of two or more competing species into an open habitat, i.e., one that is unoccupied by either species. In case of two competing species, it is conjectured that for large time, the solution behaves like stacked traveling fronts, i.e., it exhibits two transition layers moving at two different speeds $c_1 > c_2$, connecting three homogeneous equilibrium states $(0,0)$, $E_1$ and $E_2$. Here $E_1$ is the semi-trivial equilibrium where the faster species is present, and $E_2$ is either the other semi-trivial equilibrium or the coexistence equilibrium (if the latter exists). While it is not difficult to see that the spreading speed $c_1$ of the faster species can be predicted by the underlying single equation (since the slower species is essentially absent at the leading edge of the front), the determination of the second speed remained open over a decade. Lin and Li \cite{Lin_2012} first worked on the spreading properties of \eqref{eq:1-1} in the weak competition case $0<a,b<1$ with compactly supported initial condition $(u_0,v_0)$ and obtained estimates for the spreading speed $c_2$ of the slower species. For the strong competition case $a,b>1$, Carr\`{e}re \cite{Carrere_2018} determined both of the spreading speeds, where $c_2$ is determined by the unique speed of traveling wave solutions connecting the semi-trivial steady state $(1,0)$ and $(0,1)$. The predator-prey system was considered by Ducrot et al. \cite{Ducrot_preprint}. For cooperative systems with equal diffusion coefficients, the existence of stacked fronts for cooperative systems was also studied by \cite{Iida_2011}. In these cases, the spreading speeds of each individual species can be determined locally and is not influenced by the presence of other invasion fronts. However, the second speed $c_2$ can in general be influenced by the first front with speed $c_1$, as demonstrated by the work of Holzer and Scheel \cite{Holzer_2014} which applies in particular to \eqref{eq:1-1} for the case $a=0$ and $b>0$. They showed that the second speed $c_2$ can be determined by the linear instability of the zero solution of a single equation with space-time inhomogeneous coefficient. For coupled systems, the case $0 < a < 1 < b$ was treated in a recently appeared paper of Girardin and the third author \cite{Girardin_2018}. By deriving an explicit formula for $c_2$, it is observed that $c_2$ can sometimes be strictly greater than the minimal speed of traveling wave connecting $E_1$ and $E_2$, and that it depends on the first speed $c_1$ in a non-increasing manner. The proof in \cite{Girardin_2018} is based on a delicate construction of (piecewise smooth) super- and sub-solutions for the parabolic system. In our previous paper \cite{LLL2019}, we showed that in the weak competition case $0< a,b<1$ the formula for $c_2$ is exactly the same as the one in \cite{Girardin_2018} but with a novel strategy of proof based on obtaining large deviation estimates via analyzing the Hamilton-Jacobi equations obtained in the thin-front limit. We also mention that coupled parabolic systems were also treated in \cite{Evans_1989b,Freidlin_1991} based on the large deviations approach, but in these papers all components spread with a single spreading speed. \subsection{Main results} In this paper, we study the spreading of two competing species into an open habitat with exponentially decaying (in space) initial data, with attention to how the spreading speeds are influenced by the exponential rates of decay at infinity. For a function $g: \mathbb{R} \to \mathbb{R}$ and $\lambda \in \mathbb{R}$, we say that $g(x) \sim e^{-\lambda x}$ at $\infty$ if $$ 0 < \liminf_{x \to \infty} e^{\lambda x}g(x) \leq \limsup_{x \to \infty} e^{\lambda x}g(x) <\infty. $$ Definition for $g(x) \sim e^{\lambda x}$ at $-\infty$ is similar. We now state our hypothesis for the initial data $(u_0,v_0)$. $$ \rm{(H_\lambda)}\begin{cases} \text{The initial value } (u_0, v_0)\in C(\mathbb{R};[0,1])^2 \text{ is strictly positive on }\mathbb{R},\\ \text{ and there exist positive constants } \theta_0,\lambda_u, \lambda^+_v, \lambda^-_v \text{ such that } \\ u_0(x) \geq \theta_0\quad \text{ in }(-\infty,0], \quad u_0(x) \sim e^{-\lambda_u x} \,\, \text{ at }\,\,\infty,\\ v_0(x) \sim e^{\lambda_v^- x} \,\,\text{ at }\,\, -\infty,\quad \text{ and }\quad v_0(x) \sim e^{-\lambda_v^+ x} \,\, \text{ at }\,\, \infty. \end{cases}. $$ We denote \begin{align}\label{eq:sigma} \begin{cases}\displaystyle \sigma_1=d(\lambda_v^+\wedge \sqrt{\frac{r}{d}})+\frac{r}{\lambda_v^+\wedge\sqrt{\frac{r}{d}}},\quad \sigma_2=(\lambda_u\wedge 1)+\frac{1}{\lambda_u\wedge 1},\\ \displaystyle \sigma_3=d(\lambda_v^-\wedge \sqrt{\frac{r(1-b)}{d}})+\frac{r(1-b)}{\lambda_v^-\wedge\sqrt{\frac{r(1-b)}{d}}}, \end{cases} \end{align} where $a\wedge b = \min\{a,b\}$ for $a,b \in \mathbb{R}$. Here the quantity $\sigma_1$ (resp. $\sigma_2$) denotes the spreading speed of $v$ (resp. $u$) in the absence of the competitor \cite{Kametaka_1976,Mckean_1975}. Without loss of generality, we assume $\sigma_1\geq \sigma_2 $ throughout this paper. This amounts to fixing the choice of $v$ to be the faster spreading species. \begin{comment} \begin{definition}\label{cLLW} Speeds $c_{\rm LLW}>0$ and $\tilde{c}_{\rm LLW}>0$ are the minimal speeds of traveling wave solution of \eqref{eq:1-1} such that the traveling wave profile $(\varphi(x),\psi(x))$ satisfies $$ (\varphi(-\infty),\psi(-\infty))=(k_1,k_2),\quad (\varphi(\infty),\psi(\infty))=(0,1) $$ and $$ (\varphi(-\infty),\psi(-\infty))=(k_1,k_2),\quad (\varphi(\infty),\psi(\infty))=(1,0), $$ respectively. It was proved by \cite{Li_2005} that the minimal wave speed coincides with the speed of propagation under given initial value. And denote $\lambda_{\rm{LLW}}=\frac{c_{\rm LLW}-\sqrt{c_{\rm LLW}^2-4(1-a)}}{2}$ and $\tilde{\lambda}_{LLW}=\frac{\tilde{c}_{\rm LLW}-\sqrt{\tilde{c}_{\rm LLW}^2-4dr(1-b)}}{2d}$. \end{definition} \end{comment} Our main result is stated as follows. \begin{theorem}\label{thm:1-2} Assume $\sigma_1 > \sigma_2$. Let $(u,v)$ be the solution of \eqref{eq:1-1} such that the initial data satisfies $\mathrm{(H_\lambda)}$. Then there exist $c_1, c_2, c_3\in \mathbb{R}$ such that $c_3 < 0 < c_2 < c_1$, and for each small $\eta>0$, the following spreading results hold: \begin{equation} \begin{cases} \lim\limits_{t\rightarrow \infty} \sup\limits_{ x>(c_{1}+\eta) t} (\|u(t,x)\|+\|v(t,x)\|)=0, \\%&\,\text{ for } \epsilon>0, \\ \lim\limits_{t\rightarrow \infty} \sup\limits_{(c_2+\eta) t< x<(c_{1}-\eta) t} (\|u(t,x)\|+\|v(t,x)-1\|)=0, \\% &\,\text{ for }\epsilon\in \left(0,\frac{c_{1}-c_2}{2}\right), \\ \lim\limits_{t\rightarrow \infty} \sup\limits_{(c_{3}+\eta)t< x<(c_2-\eta) t} (\|u(t,x)-k_1\|+\|v(t,x)-k_2\|)=0 , \\%&\,\text{ for }\epsilon\in \left(0,\frac{c_2-c_3}{2}\right), \\ \lim\limits_{t\rightarrow \infty} \sup\limits_{x<(c_{3}-\eta)t} (\|u(t,x)-1\|+\|v(t,x)\|)=0. \end{cases} \label{eq:spreadingly} \end{equation} Precisely, the spreading speeds $c_3 < 0 < c_2 < c_1$ can be determined as follows: \begin{equation}\label{eq:c123} c_1 = \sigma_1, \quad c_2 = \max\{c_{\rm LLW}, \hat{c}_{\rm nlp}\},\quad c_3 = - \max\{ \tilde{c}_{\rm LLW}, \sigma_3\}, \end{equation} where $c_{\rm LLW}$ $($resp. $\tilde{c}_{\rm LLW})$ is given in Theorem \ref{thm:LLW} $($resp. Remark \ref{rmk:LLW}$)$, and \begin{equation}\label{eq:hcacc} \hat{c}_{\rm nlp} = \begin{cases} \frac{\sigma_1}{2} - \sqrt{a} + \frac{1-a}{\frac{\sigma_1}{2} - \sqrt{a}}, & \text{ if }\sigma_1 < 2\lambda_u\,\, \text{ and }\,\, \sigma_1 \leq 2 (\sqrt a + \sqrt{1-a}),\\ \tilde\lambda_{\rm nlp} + \frac{1-a}{\tilde\lambda_{\rm nlp}}, & \text{ if }\sigma_1 \geq 2\lambda_u \,\,\text{ and }\,\, \tilde\lambda_{\rm nlp} \leq \sqrt{1-a},\\ 2\sqrt{1-a}, &\text{ otherwise,} \end{cases} \end{equation} with the quantity $\tilde\lambda_{\rm nlp}$ being given by \begin{equation}\label{eq:lambdaacc} \tilde\lambda_{\rm nlp} = \frac{1}{2}\left[\sigma_1 - \sqrt{(\sigma_1-2\lambda_u)^2 + 4a}\right]. \end{equation} \end{theorem} To visualize the spreading result \eqref{eq:spreadingly} visually, we consider the scaling $$(\hat u,\hat v)(t,x)=\lim_{\epsilon\to0}(u,v)\left(\frac{t}{\epsilon},\frac{x}{\epsilon}\right) \quad \text{ for } (t,x)\in (0,\infty)\times \mathbb{R}, $$ whose asymptotic behaviors can be given in Figure \ref{figure5}. \begin{figure} \caption{The asymptotic behaviors of $(\hat u,\hat v)$.} \label{figure5} \end{figure} Note that while the spreading speed $c_1$ of the faster species $v$ is entirely determined by $\lambda_v^+$ (the exponential decay of $v_0$ at $x \approx \infty$), and is unaffected by the slower species $u$, the corresponding speed $c_2$ of species $u$ depends upon $\sigma_1$ and $\lambda_u$ (the exponential decay of $u_0$ at $x \approx \infty$). In particular, when $\lambda_v^+\geq\sqrt{\frac{r}{d}}$ and $\lambda_u> \frac{\sigma_1}{2}$, i.e., $v_0(x)$ and $u_0(x)$ decay fast enough, the speeds $c_1$ and $c_2$ are the same as that of the case of compactly supported initial data (see \cite[Theorem 1.2]{LLL2019}). \begin{remark} We point out that the speed $c_2$ in Theorem \ref{thm:1-2} is non-increasing in both $\sigma_1$ and $\lambda_u$, which follows from the following observations: {\rm(i)} $\tilde\lambda_{\rm nlp}$ given by \eqref{eq:lambdaacc} is non-decreasing in both $\sigma_1$ and $\lambda_u$; {\rm(ii)} $s + \frac{1-a}{s}$ is non-increasing in $(0,\sqrt{1-a}]$. This fact makes intuitive sense: {\rm{(i)}} a higher $\sigma_1$ means the region dominated by species $v$, which is roughly $\{(t,x): c_2t < x < \sigma_1 t\}$, is larger and thus rendering it more difficult for species $u$ to invade; {\rm{(ii)}} a higher $\lambda_u$ means there are less population at the front to pull the invasion wave, which also makes it difficult for species $u$ to invade. \end{remark} Fix $\sigma_1,\, \lambda_u >0$ and $0 <a <1$, such that $\sigma_1 >\sigma_2$ holds. We shall see that the quantity $\hat{c}_{\rm nlp}$ in \eqref{eq:hcacc} can be equivalently defined by $$ \{(t,x): \overline w_2 (t,x) =0\} = \{ (t,x): t>0 \, \text{ and }\, x \leq \hat{c}_{\rm nlp} t\}, $$ where $\overline w_2(t,x)$ is the unique viscosity solution of the Hamilton-Jacobi equation \begin{equation}\label{eq:hj_j11} \left\{\begin{array}{ll} \min\{\partial_t w + \|\partial_x w\|^2 + 1 - a \chi_{\{x < \sigma_1 t\}},w\} = 0, & \text{ in } (0,\infty)\times \mathbb{R},\\ w(0,x) = \lambda_u \max\{x, 0\}, &\text{ on }\mathbb{R}. \end{array} \right. \end{equation} Here $\chi_{S}$ is the indicator function of the set $S\in (0,\infty)\times \mathbb{R}$. A further point of interest is the involvement of $(0,0)$ and $(k_1,k_2)$ in co-invasion process of \eqref{eq:1-1}, which happens only in the weak competition case $0<a,b<1$. In this case, the equilibrium states $(0,0)$ and $(k_1,k_2)$ are un-ordered, and hence the existence of traveling wave, due to Tang and Fife \cite{Tang_1980}, cannot be established by monotone dynamical systems framework due to Weinberger et al. \cite{Weinberger_2002b} (see also \cite{Fang_2014,Liang_2007}). We will see that the invasion front $(k_1,k_2)$ into $(0,0)$ is indeed realized in \eqref{eq:1-1} for initial data with certain values of exponential decay rates $\lambda_u, \lambda^{+}_v$ at infinity, namely, when $\sigma_1=\sigma_2$. \begin{theorem}\label{thm:1-2b} Assume $\sigma_1 = \sigma_2$. Let $(u,v)$ be the solution of \eqref{eq:1-1} such that the initial data satisfies $\mathrm{(H_\lambda)}$. Then for each small $\eta>0$, it holds that \begin{equation} \begin{cases} \lim\limits_{t\rightarrow \infty} \sup\limits_{ x>(\sigma_{1}+\eta) t} (\|u(t,x)\|+\|v(t,x)\|)=0, \\%&\,\text{ for } \epsilon>0, \\ \lim\limits_{t\rightarrow \infty} \sup\limits_{(c_{3}+\eta)t< x<(\sigma_1-\eta) t} (\|u(t,x)-k_1\|+\|v(t,x)-k_2\|)=0 , \\%&\,\text{ for }\epsilon\in \left(0,\frac{c_2-c_3}{2}\right), \\ \lim\limits_{t\rightarrow \infty} \sup\limits_{x<(c_{3}-\eta)t} (\|u(t,x)-1\|+\|v(t,x)\|)=0, \end{cases} \label{eq:spreadingly'} \end{equation} where $c_3 = - \max\{ \tilde{c}_{\rm LLW}, \sigma_3\}$ and that $\tilde{c}_{\rm LLW}$ is given in Remark \ref{rmk:LLW}. \end{theorem} For initial data with general exponential decay rates, Theorem \ref{thm:1-2} demonstrates that there are two separate monostable fronts where each of the two species invades with distinct speeds. Moreover, if the parameters of \eqref{eq:1-1} changes in such a way that $\|\sigma_1-\sigma_2\| \to 0$, the distance of the two fronts tends to zero. Therefore, the invasion front of $(k_1,k_2)$ transitioning directly into $(0,0)$, due to Tang and Fife, is in fact the special case when these two monostable fronts coincide (Theorem \ref{thm:1-2b}). \begin{remark} As in \cite{Evans_1989,LLL2019}, our approach can be applied to the spreading problem of competing species in higher dimensions under minor modifications. However, we choose to focus here on the one-dimensional case to keep our exposition simple, and close to the original formulation of the conjecture in \cite[Chapter 7]{Shigesada_1997}. \end{remark} \subsection{Outline of main ideas} To determine $c_1,\,c_2,\,c_3$, we introduce large deviation approach and construct appropriate viscosity super- and sub-solutions for certain Hamilton-Jacobi equations, and then apply the comparison principle (Theorem \ref{eq:D1}) to obtain the desired estimations. We outline the main steps leading to the determination of the nonlocally pulled spreading speed $c_2$, as stated Theorem \ref{thm:1-2}, and remark that $c_1,c_3$ can be obtained by a similar even simpler argument as $c_2$. \begin{enumerate} \item To estimate $c_2$ from below, we consider the transformation $w_2^\epsilon(t,x) = -\epsilon \log u\left( \frac{t}{\epsilon}, \frac{x}{\epsilon}\right)$ and show that the half-relaxed limits $$ w_{2,}(t,x) = \hspace{-0.3cm} \liminf_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.3cm}w_2^\epsilon(t',x') \quad \text{ and } \quad w_2^(t,x) = \hspace{-0.3cm} \limsup_{\scriptsize\begin{array}{c} \epsilon \to 0 \\ (t',x') \to (t,x) \end{array}} \hspace{-0.3cm} w_2^\epsilon(t',x') $$ exist, upon establishing uniform bounds in $C_{\mathrm{loc}}$ (see Lemma \ref{lem:3-1'}). By constructing viscosity super-solution $\overline{w}_2$, which satisfies $$ \{(t,x):\overline w_2(t,x) =0\} = \{ (t,x): t>0 \, \text{ and }\, x \leq \hat{c}_{\rm nlp} t\}, $$ and using the comparison principle (Theorem \ref{thm:D}), we can show that $w_2^\leq \overline{w}_2$, and thus $w_2^\epsilon \to 0$ locally uniformly in $\{(t,x): \, x < \hat{c}_{\rm nlp} t\}.$ One can then apply the arguments in \cite[Section 4]{Evans_1989} to show that $$ \liminf_{\epsilon \to 0} u\left( \frac{t}{\epsilon}, \frac{x}{\epsilon}\right) >0 \quad \text{ in } \left\{ (t,x): t>0 \, \text{ and }\, x < \hat{c}_{\rm nlp} t\right\}. $$ This implies that ${c}_2 \geq \hat{c}_{\rm nlp}$ (see Lemma \ref{lem:underlinec2}). \item To estimate $c_{2}$ from above, we construct viscosity sub-solution $\underline{w}_2$ and apply Theorem \ref{thm:D} to estimate $w_{2}$ from below, see Proposition \ref{prop:overlinec2}. This enables us to obtain a large deviation estimate of $u$. Namely, for each small $\delta>0$, let $\hat{c}_\delta = \sigma_1-\delta$, we have $$ u(t, \hat{c}_\delta t) \leq \exp \left( -[ \hat{\mu}_\delta + o(1)]t \right) \quad \text{ for } t \gg 1, $$ where $\hat{\mu}_\delta = \overline w_2(1, \hat{c}_\delta)= \overline w_2(1, \sigma_1 - \delta)$. Now, recalling that $(u,v)$ is a solution to \eqref{eq:1-1} restricted to the domain $\{(t,x): 0\leq x \leq \hat{c}_\delta t\}$, with boundary condition satisfying $$ \lim_{t\to\infty} (u,v)(t,0) = (k_1,k_2) \quad \text{ and }\quad \lim_{t\to\infty} (u,v)(t, \hat{c}_\delta t) = (0,1), $$ we may apply Lemma \ref{lem:appen1} in Appendix to show that $\hat{c}_\delta$ and $\hat{\mu}_\delta$ completely controls the spreading speed $c_2$ of $u$ from above.\end{enumerate} The rest of the paper is organized as follows: In Section \ref{S2}, we give upper estimates $c_i$ for $i=1,\,2,\,3$ and $c_2\geq c_{\rm LLW}$. In Section \ref{S3}, we give lower estimates of $c_1,\,c_2$. The approximate asymptotic expressions of $u$ and $v$ are established in Section \ref{S4}, where we also determine $c_2,\, c_3$. In Section \ref{S4b}, we discuss the relation of our results with the invasion mode due to Tang and Fife \cite{Tang_1980}. In Section \ref{S5}, we discuss the relation of our result with that of \cite{Girardin_2018} due to Girardin and the last author. In Section \ref{S6}, we prove an extension which is associated to the spreading speeds of the three-species competition systems. We conclude the article with the Appendix. Therein we give the comparison principle of Hamilton-Jacobi equation with piecewise Lipschitz continuous Hamiltonian and two other useful lemmas. This paper concerns the Cauchy problem of a system of reaction-diffusion equations modeling two competing species. For the spreading of two species into an open habitat, we refer to \cite{Li_2018} for an integro-difference competition model, and to\cite{Du_2018} for a competition model with free-boundaries. See also \cite{Guo_2015,Wu_2015,Wang_2017,Wang_2018,Liu_2019} for other related results in free-boundary problems. We also note that in those works the spreading speeds are always locally determined and thus do not interact. \section{Estimating the maximal and minimal speeds}\label{S2} The concepts of maximal and minimal spreading speeds are introduced in \cite[Definition 1.2]{Hamel2012} for a single species; see also \cite{Garnier_2012,LLL2019}. In our setting, we define \begin{equation}\label{eq:speeds} \begin{cases} \overline{c}_1=\inf{\{c>0~\|~\limsup \limits_{t\rightarrow \infty}\sup\limits_{x>ct} v(t,x)=0\}},\\ \underline{c}_1=\sup{\{c>0~~\|\liminf \limits_{t\rightarrow \infty}\inf\limits_{ct-1<x<ct} v(t,x)>0\}}, \\ \overline{c}_2=\inf\{c>0~\|~\limsup\limits_{t\rightarrow \infty}\sup\limits_{ x> ct}u(t,x)=0\}, \\ \underline{c}_2=\sup{\{c>0~~\|\liminf \limits_{t\rightarrow \infty}\inf\limits_{ct-1<x<ct} u(t,x)>0\}},\\ \overline{c}_3=\inf\{c<0~\|~\liminf\limits_{t\rightarrow \infty}\inf\limits_{ ct<x< ct+1}v(t,x) >0\},\\ \underline{c}_3=\sup\{c<0~\|~\limsup\limits_{t\rightarrow \infty}\sup\limits_{ x< ct}v(t,x)=0\}, \end{cases} \end{equation} where $\overline{c}_1$ and $\underline{c}_1$ (resp. $\overline{c}_2$ and $\underline{c}_2$) are the maximal and minimal rightward spreading speeds of species $v$ (resp. species $u$), whereas $-\underline{c}_3$ and $-\overline{c}_3$ are the maximal and minimal leftward spreading speeds of $v$, respectively. In this section, for initial data satisfying ($\mathrm{H_\lambda})$, we will give some estimates of the maximal and minimal spreading speeds. The main result of this section can be precisely stated as follows. \begin{proposition}\label{prop:1} Let $(u,v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then the spreading speeds defined in \eqref{eq:speeds} satisfy \begin{itemize} \item[\rm{(i)}] $\overline{c}_i\leq \sigma_i$ for $i=1,\,2$ and $\overline{c}_3\leq -\sigma_3$; \item[\rm{(ii)}] $\underline{c}_2\geq c_{\rm LLW}$, and $\overline{c}_3\leq -\tilde{c}_{\rm LLW}$, \end{itemize} where $\sigma_1, \sigma_2, \sigma_3$ are defined in \eqref{eq:sigma} and $c_{\rm LLW},\, \tilde{c}_{\rm LLW}$ are given respectively in Theorem \ref{thm:LLW} and Remark \ref{rmk:LLW}. Furthermore, we have \begin{equation}\label{eq:spreadingly2.1} \lim\limits_{t\to \infty} (\|u(t,0)-k_1\| + \|v(t,0)-k_2\|) =0. \end{equation} \end{proposition} \begin{proof} We will complete the proof in the following order: \rm{(1)} $\overline{c}_2 \leq \sigma_2$, \rm{(2)} $\overline{c}_1\leq \sigma_1$, \rm{(3)} $\overline c_3\leq -\sigma_3$, \rm{(4)} $\overline c_3\leq -\tilde c_{\rm LLW}$, \rm{(5)} $\underline c_2 \geq c_{\rm LLW}$, \rm{(6)} \eqref{eq:spreadingly2.1} holds. \\ \noindent{\bf Step 1.} We show assertions \rm{(1)}, \rm{(2)} and \rm{(3)}. Observe that for some $M>0$ the function $$ \overline{u}(t,x):= \min\{1, M \exp(-\min\{\lambda_u,1\}(x - \sigma_2 t))\} $$ is a weak super-solution to the single KPP-type equation \begin{equation} \partial_t \overline{u} - \partial_{xx} \overline{u} = \overline{u} (1-\overline{u} ) \,\quad \text{in }(0,\infty) \times \mathbb{R}, \end{equation} of which $u(t,x)$ is clearly a sub-solution. By choosing the constant $M>0$ so large that $u_0(x)\leq \overline{u}(0,x)$ in $\mathbb{R}$, it follows by comparison that \begin{equation}\label{eq:ukpp00} u(t,x) \leq \overline{u}(t,x)= \min\left\{1, M \exp(-\min\{\lambda_u,1\}\left(x -\sigma_2 t\right) \right\} \end{equation} for $(t,x) \in [0,\infty) \times \mathbb{R}$. In particular, \begin{equation}\label{eq:ukpp} \lim_{t \to \infty} \sup_{x > (\sigma_2+\eta)t} \|u(t,x)\| =0\,\quad \text{ for each }\eta >0. \end{equation} This proves $\overline{c}_2 \leq \sigma_2$, i.e., assertion \rm{(1)} holds. Similarly, we deduce assertion \rm{(2)} by comparison with $$ \bar{v}(t,x):= \min\{1, M \exp(-\min\{\lambda^+_v, \sqrt{r/d}\}(x-\sigma_1t))\} $$ which is the solution of \begin{equation} \left\{ \begin{array}{ll} \partial_t \overline{v} - d\partial_{xx} \overline{v} = r\overline{v}(1-\overline{v}), &\text{in }(0,\infty) \times \mathbb{R},\\ \overline{v}(0,x)=\min(1,M e^{-\min\{\lambda_v^+,\sqrt{\frac{r}{d}}\}x}), & x\in\mathbb{R}. \end{array} \right. \end{equation} To prove assertion \rm{(3)}, let $\tilde v(t,x)=v(t,-x)$, we turn to consider another single KPP-type equation \begin{equation} \left\{ \begin{array}{ll} \partial_t \underline{v} - d\partial_{xx} \underline{v} = r\underline{v}(1-b-\underline{v}), &\text{in }(0,\infty) \times \mathbb{R},\\ \underline{v}(0,x)=v_0(-x), & x\in\mathbb{R}. \end{array} \right. \end{equation} Again the scalar comparison principle implies $v(t,-x)=\tilde v(t,x)\geq \underline{v}$. By the results in \cite{Kametaka_1976} or \cite{Mckean_1975}, we have \begin{equation}\label{estim_3} \liminf\limits_{t\to\infty} \inf\limits_{ (-\sigma_3+\eta)t<x\leq 0} v\geq \liminf\limits_{t\to\infty} \inf\limits_{ \|x\|<(\sigma_3-\eta)t} \tilde v\geq \frac{1-b}{2}, \end{equation} which means $\overline{c}_3\leq -\sigma_3$. \noindent{\bf Step 2.} We show assertions \rm{(4)} and \rm{(5)}. Given any non-trivial, compactly supported function $\tilde {v}_0$ such that $0 \leq \tilde{v}_0 \leq v_0$. Then $$ (u_0(x),v_0(x)) \preceq (1, \tilde{v}_0(x)) \,\quad \text{ in }\mathbb{R}. $$ Let $(\tilde u_{\rm LLW}, \tilde v_{\rm LLW})$ be the solution to \eqref{eq:1-1} with initial value $(1, \tilde{v}_{0}(x))$. Then Theorem \ref{thm:LLW} and Remark \ref{rmk:LLW} guarantee the existence of $\tilde{c}_{\rm LLW} \ \geq 2\sqrt{dr(1-b)}$, such that $$ \liminf_{t \to \infty} \inf_{\|x\| < \|c\|t } \tilde v_{\rm LLW}(t,x) >0 \quad \text{ for each } c \in (-\tilde{c}_{\rm LLW}, 0). $$ By the comparison principle for \eqref{eq:1-1}, we have $(u,v) \preceq (\tilde u_{\rm LLW},\tilde v_{\rm LLW})$ for all $(t,x) \in (0,\infty) \times \mathbb{R}$, which yields, for each $c \in (- \tilde{c}_{\rm LLW}, 0)$, $$ \liminf_{t \to \infty} \inf_{ct < x < ct + 1} v(t,x) \geq \liminf_{t \to \infty} \inf_{ct < x < ct + 1} \tilde v_{\rm LLW}(t,x) >0. $$ This proves $\overline{c}_3 \leq - \tilde{c}_{\rm LLW}$ and thus assertion \rm{(4)} holds. Similarly, we can get show assertion (5), i.e., $\underline{c}_2 \geq c_{\rm LLW}$. By comparing $(u,v)$ with the solution $(u_{\rm LLW}, v_{\rm LLW})$ of \eqref{eq:1-1} with initial condition $(\tilde{u}_0, 1)$, for some compactly supported $\tilde{u}_0$ satisfying $0 \leq \tilde{u}_0 \leq u_0$, and then using Theorem \ref{thm:LLW}. In this way, we get \begin{align}\label{eq:positiveu'} \liminf_{t \to \infty} \inf_{\|x\| < c t } u \geq \liminf_{t \to \infty} \inf_{\|x\| < c t } u_{\rm LLW} >0 \quad \text{ for each } c \in (0, c_{\rm LLW}). \end{align} \noindent{\bf Step 3.} We show assertion \rm{(6)}. In view of \eqref{estim_3} and \eqref{eq:positiveu'}, one can deduce \eqref{eq:spreadingly2.1} from items \rm{(a)} and \rm{(c)} of Lemma \ref{lem:entire1}. \end{proof} \section{Estimating $\displaystyle \overline c_1$ and $\displaystyle \overline c_2$ from below}\label{S3} We assume $\sigma_1>\sigma_2$ throughout this section. In this section, we estimate $\underline c_1$ and $\underline c_2$ from below via the large deviation approach and applying Theorem \ref{thm:D}. To this end, we introduce a small parameter $\epsilon$ via the following scaling \begin{equation}\label{scaling} u^\epsilon(t,x)=u\left(\frac{t}{\epsilon},\frac{x}{\epsilon}\right)\quad \mathrm{and}\quad v^\epsilon(t,x)=v\left(\frac{t}{\epsilon},\frac{x}{\epsilon}\right). \end{equation} Under the new scaling, we rewrite the equation of $u^\epsilon$ and $v^\epsilon$ in \eqref{eq:1-1} as \begin{align}\label{eq:1-1'} \left \{ \begin{array}{ll} \partial_tu^\epsilon=\epsilon\partial_{xx} u^\epsilon+\frac{u^{\epsilon}}{\epsilon}(1-u^\epsilon-av^\epsilon), & \text{ in } (0,\infty)\times\mathbb{R},\\ \partial_tv^\epsilon=\epsilon d\partial_{xx} v^\epsilon+r\frac{v^{\epsilon}}{\epsilon}(1-bu^\epsilon-v^\epsilon), & \text{ in } (0,\infty)\times\mathbb{R},\\ u^\epsilon(0,x)=u_0(\frac{x}{\epsilon}),& \text{ on } \mathbb{R},\\ v^\epsilon(0,x)=v_0(\frac{x}{\epsilon}), & \text{ on } \mathbb{R}. \end{array} \right. \end{align} To obtain the asymptotic behaviors of $v^\epsilon$ and $u^\epsilon$ as $\epsilon\rightarrow 0$, the idea is to consider the WKB ansatz $w_1^\epsilon$ and $w_2^\epsilon$, which are given respectively by \begin{equation}\label{eq:w} w_1^\epsilon(t,x)=-\epsilon\log{v^\epsilon(t,x)}, \quad w_2^\epsilon(t,x)=-\epsilon\log{u^\epsilon(t,x)}, \end{equation} and satisfy, respectively, the equations \begin{align}\label{eq:epsilonw1} \left \{ \begin{array}{ll} \partial_t w^\epsilon-\epsilon d\partial_{xx} w^\epsilon+d\| \partial_xw^\epsilon\|^2+r(1-bu^\epsilon-v^\epsilon)=0, & \text{ in } (0,\infty)\times\mathbb{R},\\ w^\epsilon(0,x)=-\epsilon\log{v^\epsilon(0,x)}, & \text{ on } ~\mathbb{R},\\ \end{array} \right. \end{align} and \begin{align}\label{eq:epsilonw2} \left \{ \begin{array}{ll} \partial_tw^\epsilon-\epsilon\partial_{xx} w^\epsilon+\| \partial_xw^\epsilon\|^2+1-u^\epsilon-av^\epsilon=0, & \text{ in } (0,\infty)\times\mathbb{R},\\ w^\epsilon(0,x)=-\epsilon\log{u^\epsilon(0,x)}, & \text{ on } \mathbb{R}.\\ \end{array} \right. \end{align} \begin{lemma}\label{lem:underlinec} Let $G$ be an open set in $(0,\infty)\times \mathbb{R}$ and $K, K'$ be compact sets such that $K \subset {\rm Int}\,K' \subset K' \subset G.$ \begin{itemize} \item[\rm{(a)}] If $w_2^\epsilon \to 0$ uniformly in $K'$ as $\epsilon \to 0$, then \begin{equation}\label{eq:underlineu} \liminf_{\epsilon\to 0} \inf_{K} u^\epsilon \geq 1-a\limsup_{\epsilon\to 0} \sup_{K'} v^\epsilon; \end{equation} \item[\rm{(b)}] If $w_1^\epsilon \to 0$ uniformly in $K' $ as $\epsilon \to 0$, then \begin{equation}\label{eq:underlinev} \liminf_{\epsilon\to 0} \inf_{K} v^\epsilon \geq 1-b\limsup_{\epsilon\to 0} \sup_{K'} u^\epsilon. \end{equation} \end{itemize} \end{lemma} \begin{proof} We first prove (a) by adapting the arguments from \cite[Section 4]{Evans_1989}. Let $K, K'$ and $G$ be given as above. Fix an arbitrary $(t_0,x_0) \in K$ and define the test function $$ \rho(t,x) = \|x-x_0\|^2 + (t-t_0)^2. $$ Since (i) $(t_0,x_0) \in K \subset {\rm Int}\, K'$ and (ii) $w^\epsilon_2 \to 0$ uniformly in $K'$, the function $w^\epsilon_2 - \rho$ attains global maximum over $K'$ at $(t_\epsilon,x_\epsilon)\in {\rm Int}\,K'$ such that $(t_\epsilon,x_\epsilon) \to (t_0,x_0)$ as $\epsilon \to 0$. Furthermore, $\partial_t \rho(t_\epsilon,x_\epsilon), \partial_x \rho(t_\epsilon,x_\epsilon) \to 0$, so that at the point $(t_\epsilon,x_\epsilon)$, $$ o(1) = \partial_t \rho - \epsilon \partial_{xx} \rho + \|\partial_x\rho\|^2 \leq \partial_t w_2^\epsilon - \epsilon \partial_{xx} w_2^\epsilon + \|\partial_x w_2^\epsilon\|^2 \leq u^\epsilon - 1 + a\limsup_{\epsilon\to 0} \sup_{K'} v^\epsilon. $$ This yields $$ u^\epsilon(t_\epsilon, x_\epsilon) \geq 1-a\limsup_{\epsilon\to 0} \sup_{K'} v^\epsilon + o(1). $$ Since $w^\epsilon_2 - \rho$ attains maximum over $K'$ at $(t_\epsilon,x_\epsilon)$, we have in particular $$ w_2^\epsilon(t_\epsilon,x_\epsilon) \geq (w_2^\epsilon - \rho)(t_\epsilon,x_\epsilon) \geq(w_2^\epsilon - \rho)(t_0,x_0) = w_2^\epsilon(t_0,x_0). $$ Recalling $u^\epsilon(t_0,x_0) = e^{-\epsilon w^\epsilon_2(t_0,x_0)}$ and $u^\epsilon(t_\epsilon,x_\epsilon) = e^{-\epsilon w^\epsilon_2(t_\epsilon,x_\epsilon)}$, we therefore have $$ u^\epsilon(t_0,x_0) \geq u^\epsilon(t_\epsilon,x_\epsilon) \geq 1-a\limsup_{\epsilon\to 0} \sup_{K'} v^\epsilon + o(1). $$ Since this argument is uniform for $(t_0,x_0) \in K$ (depends only on $K, K'$ and $G$), we deduce assertion \rm{(a)}. The proof for {\rm(b)} is analogous. \end{proof} Next, we will pass to the (upper and lower) limits using the half-relaxed limit method, which is due to Barles and Perthame \cite{BP1987}. Define $$w_1^(t,x)=\limsup\limits_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.3cm} w_1^\epsilon (t',x'),$$ $$w_2^(t,x)=\limsup\limits_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.3cm} w_2^\epsilon (t',x')\quad \mathrm{and} \quad w_{2,}(t,x)=\liminf\limits_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.3cm} w_2^\epsilon (t',x').$$ That the above are well defined is due to the following lemma: \begin{lemma}\label{lem:3-1'} Let $w_1^\epsilon$ and $w_2^\epsilon$ be the solutions to \eqref{eq:epsilonw1} and \eqref{eq:epsilonw2}, respectively. Then there exits some $Q>0$, independent of $\epsilon$ small, such that \begin{subequations}\label{eq:spreadinglyprop} \begin{equation}\label{estamition1} \max\{\lambda_v^+x_++\lambda_v^-x_-- Q(t+\epsilon),0\} \leq {w}_1^{\epsilon}(t,x) \leq\lambda_v^+x_++\lambda_v^-x_-+Q(t+\epsilon), \end{equation} \begin{equation}\label{estamition2} \max\{\lambda_u x_+- Q(t+\epsilon),0\} \leq {w}_2^{\epsilon}(t,x) \leq \lambda_u x_++ Q(t+\epsilon), \end{equation} \begin{equation}\label{estamition3} 0 \leq {w}_1^{\epsilon}(t,x) \leq Q(\lambda_v^+x_++\lambda_v^-x_-+\epsilon), \end{equation} \begin{equation}\label{estamition4} 0 \leq {w}_2^{\epsilon}(t,x) \leq Q(\lambda_ux_++\epsilon), \end{equation} \end{subequations} for $ (t,x) \in [0,\infty)\times \mathbb{R}$, where $x_+=\max\{x,0\}$ and $x_-=\max\{-x,0\}$. \end{lemma} \begin{proof} We only prove \eqref{estamition1} and the estimations \eqref{estamition2}-\eqref{estamition4} follow from a quite similar argument. Since $v^\epsilon\leq 1$, we have $w_1^\epsilon\geq 0$ by definition. By ${(\rm{H_\lambda})}$, there exist positive constants $C_1$ and $C_2$ such that $$ C_2e^{-(\lambda_v^+x_++\lambda_v^- x_-)}\leq v(0,x)\leq C_1e^{-(\lambda_v^+x_++\lambda_v^- x_-)} \quad \text{ for } x \in \mathbb{R}. $$ By definition \eqref{eq:w}, we have \begin{equation}\label{eq:w10} \lambda_v^+x_++\lambda_v^- x_--\epsilon\log C_1\leq w_1^\epsilon(0,x)\leq\lambda_v^+x_++\lambda_v^- x_--\epsilon\log C_2 . \end{equation} Define $$\overline{z}^\epsilon_1=\lambda_v^+ x+Q(t+\epsilon).$$ We shall choose large $Q$ independent of $\epsilon$ such that \begin{equation}\label{estimate} w_1^\epsilon(t,x)\leq \overline{z}^\epsilon_1 \,\quad\text{ in } [0,\infty)\times [0,\infty). \end{equation} To this end, observe that $\overline{z}^\epsilon_1$ is a (classical) super-solution of \eqref{eq:epsilonw1} in $(0,\infty)\times (0,\infty)$ provided $Q\geq r$. By \eqref{eq:spreadingly2.1} in Proposition \ref{prop:1}, we find $-\log v(t,0)$ is uniformly bounded in $[0,\infty)$ (since $v(0,x)>0$ in $\mathbb{R}$), so that we may choose \begin{equation}\label{Q} Q=\max\left\{\sup_{t\in[0,\infty)}[-\log v(t,0)], \,\|\log C_2\|,\, r\right\}, \end{equation} such that $$w^\epsilon_1(t,0)\leq \overline{z}^\epsilon_1(t,0)\,\quad \text{ for all }t\geq 0, \quad \,w^\epsilon_1(0,x)\leq \overline{z}^\epsilon_1(0,x)\quad \text{ for all }x\geq 0,$$ where the last inequality is due to \eqref{eq:w10}. By comparison, \eqref{estimate} thus holds. By a similar argument, we can verify $$\overline{z}^\epsilon_2=-\lambda_v^- x+Q(t+\epsilon)$$ is a super-solution of \eqref{eq:epsilonw1} in $(0,\infty)\times (-\infty,0)$, so that \begin{equation}\label{estimate1} w_1^\epsilon(t,x)\leq \overline{z}^\epsilon_2(t,x) \,\quad\text{ in } [0,\infty)\times (-\infty, 0], \end{equation} where $Q$ is defined by \eqref{Q}. Combining with \eqref{estimate} and \eqref{estimate1} gives the desired upper bound of $ w_1^\epsilon$. To obtain the lower bound of $ w_1^\epsilon$, we may define functions $$\underline{z}^\epsilon_1=\lambda_v^+ x-Q(t+\epsilon)\,\quad \text{ and }\,\quad\underline{z}^\epsilon_2=-\lambda_v^+ x-Q(t+\epsilon).$$ By the same arguments as before, we can check \begin{equation} w_1^\epsilon(t,x)\geq \underline{z}^\epsilon_1 \,\, \text{ in } [0,\infty)\times \mathbb{R}\,\,\text{ and } \,w_1^\epsilon(t,x)\geq \underline{z}^\epsilon_2 \,\, \text{ in } [0,\infty)\times \mathbb{R}, \end{equation} by choosing $Q=\max\left\{ \,\|\log C_1\|,\, d(\lambda_v^+)^2+d(\lambda_v^-)^2 +r \right\}$. This completes the proof of \eqref{estamition1}. \end{proof} \begin{remark}\label{rmk:w1w20} According to Lemma \ref{lem:3-1'}, by letting $t=0$ and then $\epsilon \to 0$ in \eqref{estamition1} and \eqref{estamition2}, we deduce that \begin{equation} w^_1(0,x)= \begin{cases} \lambda^+_v x, & \text{for }x\in [0,\infty),\\ \lambda^-_v x, &\text{for }x\in (-\infty,0], \end{cases} \end{equation} and \begin{equation} w^_2(0,x)=w_{2,}(0,x)= \begin{cases} \lambda_u x, & \text{for }x\in [0,\infty),\\ 0, &\text{for }x\in (-\infty,0]. \end{cases} \end{equation} Similarly, by setting $x=0$ and then $\epsilon \to 0$ in \eqref{estamition3} and \eqref{estamition4}, we have $$w_1^(t,0)=w_2^(t,0)=w_{2,}(t,0)=0\,\,\text{ for }\,t\geq 0.$$ \end{remark} \subsection{Estimating $\underline{c}_1$ from below} By Proposition \ref{prop:1}, $\overline c_2 \leq \sigma_2$, so we deduce \begin{equation}\label{eq:u_ep} 0\leq \hspace{-.3cm}\limsup_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.5cm} u^\epsilon(t',x')\leq \chi_{\{x \leq \sigma_2 t\}}. \end{equation} \begin{lemma}\label{lem:sub-solutionw1} Let $(u,v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then \begin{itemize} \item[{\rm (a)}] $w_1^$ is a viscosity sub-solution of \begin{align}\label{eq:sub-solutionw1} \begin{cases} \min\{\partial_t w+d\|\partial_xw\|^2+r(1-b\chi_{\{x\leq \sigma_2 t\}}) ,w\}=0,&\text{ in } (0,\infty)\times(0,\infty),\\ w(0,x)=\lambda_v^+x, & \text{ on }[0,\infty),\\ w(t,0)= 0, &\text{ on }[0,\infty); \end{cases} \end{align} \item[{\rm (b)}] $w_1^$ is a viscosity sub-solution of \begin{align}\label{eq:sub-solutionw1'} \begin{cases} \min\{\partial_t w+d\|\partial_xw\|^2+r(1-b\chi_{\{x\leq \sigma_2 t\}}) ,w\}=0,&\text{ in } (0,\infty)\times(-\infty,0),\\ w(0,x)=-\lambda_v^-x, & \text{ on }(-\infty,0],\\ w(t,0)= 0, &\text{ on }[0,\infty), \end{cases} \end{align} \end{itemize} where $\sigma_2$ is defined by \eqref{eq:sigma} and $\lambda_v^-,\,\lambda_v^+ \in (0,\infty)$ are given in $\mathrm{(H_\lambda)}$. \end{lemma} \begin{proof} First, observe that $w^_1$ is upper semicontinuous (usc) by construction. By Remark \ref{rmk:w1w20}, the initial and boundary conditions of \eqref{eq:sub-solutionw1} and \eqref{eq:sub-solutionw1'} are satisfied. It remains to show that $w^_1$ is a viscosity sub-solution of $\min\{\partial_t w+d\|\partial_x w\|^2+r(1-b\chi_{\{x\leq \sigma_2 t\}}) ,w\}=0$ in the domain $(0,\infty)\times \mathbb{R}$. According to definition of viscosity sub-solution of Hamilton-Jacobi equation, (see Appendix \ref{SD}), let $ \varphi \in C^\infty((0,\infty)\times\mathbb{R})$ and let $(t_0,x_0)$ be a strict local maximum point of $w_1^-\varphi$ such that $w_1^(t_0,x_0)>0$. By passing to a sequence $\epsilon = \epsilon_k$ if necessary, $ w_1^\epsilon-\varphi$ has a local maximum point at $(t_\epsilon,x_\epsilon)$ such that $w_1^\epsilon(t_\epsilon,x_\epsilon) \to w_1^(t_0,x_0)$ and $(t_\epsilon,x_\epsilon)\to (t_0,x_0)$ uniformly as $\epsilon\to 0$. At the point $(t_\epsilon, x_\epsilon)$, we have \begin{align} \epsilon d\partial_{xx}\varphi \geq \epsilon d\partial_{xx} w_1^\epsilon &= \partial_t w_1^\epsilon+d\|\partial_x w_1^\epsilon\|^2+r(1-bu^\epsilon -e^{-\frac{w_1^\epsilon}{\epsilon}})\\ &= \partial_t \varphi+d\|\partial_x \varphi\|^2+r(1-bu^\epsilon -e^{-\frac{w_1^\epsilon}{\epsilon}}). \end{align} By the fact that $e^{-w_1^\epsilon(t_\epsilon,x_\epsilon)/\epsilon}\to 0$ (as $w_1^\epsilon(t_\epsilon,x_\epsilon) \to w_1^(t_0,x_0)>0$), we may pass to the limit $\epsilon = \epsilon_k \to 0$ so that $$ 0 \geq \partial_t \varphi(t_0,x_0)+d\|\partial_x \varphi(t_0,x_0)\|^2+r(1-b\chi_{\{(t,x):\,x \leq \sigma_2 t\} (t_0,x_0)}-0). $$ Hence $w_1^$ is a viscosity sub-solution of \eqref{eq:sub-solutionw1} and \eqref{eq:sub-solutionw1'}. \end{proof} \begin{lemma}\label{lem:underlinec1} Let $(u,v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then $$\underline{c}_1\geq \sigma_1,$$ where $\sigma_1$ is defined by \eqref{eq:sigma}. \end{lemma} \begin{proof} Define the function $\overline{w}_1: [0,\infty) \times [0,\infty) \to [0,\infty)$ by \begin{equation} \begin{split} \overline w_1(t,x) = \left\{ \begin{array}{ll} \lambda_v^+(x-(d\lambda_v^++\frac{r}{\lambda_v^+})t), & \text{for}~ \frac{x}{t}>d\lambda_v^++\frac{r}{\lambda_v^+},\\ 0, & \text{for}~0\leq \frac{x}{t}\leq d\lambda_v^++\frac{r}{\lambda_v^+},\\ \end{array} \right. \end{split} \end{equation} when $ \lambda_v^+ \leq \sqrt{\frac{r}{d}}$, and by \begin{equation} \begin{split} \overline w_1(t,x) = \left\{ \begin{array}{ll} \lambda_v^+(x-(d\lambda_v^++\frac{r}{\lambda_v^+})t), & \text{for}~ \frac{x}{t}> 2d\lambda_v^+,\\ \frac{t}{4d}(\frac{x^2}{t^2}-4dr), &\text{for}~ 2\sqrt{dr}< \frac{x}{t}\leq 2d\lambda_v^+,\\ 0, & \text{for}~0\leq \frac{x}{t}\leq 2\sqrt{dr},\\ \end{array} \right. \end{split} \end{equation} when $ \lambda_v^+ > \sqrt{\frac{r}{d}}$. By construction, $\overline{w}_1$ is continuous in $[0,\infty)\times [0,\infty)$. Next, we claim that the continuous $\overline{w}_1$ is a viscosity super-solution of \eqref{eq:sub-solutionw1}. We will check the latter case of $ \lambda_v^+ > \sqrt{\frac{r}{d}}$ as the former case can be verified analogously. Under the condition $ \lambda_v^+ > \sqrt{\frac{r}{d}}$, we have $\sigma_1=2\sqrt{dr}$. According to definition of viscosity super-solution of Hamilton-Jacobi equation (see Appendix \ref{SD}), let $\varphi \in C^\infty((0,\infty)\times\mathbb{R})$ and let $(t_0,x_0)$ be a strict local minimum point of $\overline w_1-\varphi$. If $x_0/t_0 \neq 2\sqrt{dr}$, then $\overline{w}_1$ is a classical solution of \eqref{eq:sub-solutionw1}. If $x_0/t_0 = 2\sqrt{dr}$, then $\overline{w}_1(t_0,x_0) = 0$ by definition. Moreover, $$ -\varphi(t, 2\sqrt{dr} t) = (\overline{w}_1 -\varphi)(t, 2\sqrt{dr} t) \geq (\overline{w}_1 -\varphi)(t_0,x_0) = - \varphi(t_0,x_0) \quad \text{ for }t \approx t_0,$$ and we must have $\partial_t \varphi(t_0,x_0)+2\sqrt{dr} \partial_x \varphi(t_0,x_0)) = 0$, and hence \begin{align} &\quad \partial_t \varphi(t_0,x_0)+ d\|\partial_x \varphi(t_0,x_0)\|^2 + r(1-b \chi_{\{(t,x):\,x \leq \sigma_2 t\}} (t_0,x_0))\\ &= -2\sqrt{dr} \partial_x \varphi(t_0,x_0)+ d\|\partial_x \varphi(t_0,x_0)\|^2 + r\\ &= \left(\sqrt{d}\partial_x \varphi(t_0,x_0) - \sqrt{r}\right)^2 \geq 0, \end{align} where the first equality follows from the fact that $x_0/t_0 = 2\sqrt{dr}=\sigma_1 > \sigma_2$. By Remark \ref{rmk:w1w20} and the expression of $\overline{w}_1$, we have $$ \overline{w}_1(t,x) = \lambda^+_v x=w_1^(t,x) \,\,\text{ on } \partial [(0,\infty) \times (0,\infty)]. $$ And recalling Lemma \ref{lem:sub-solutionw1}(a), $\overline{w}_1$ and $w_1^$ is a pair of viscosity super and sub-solutions of \eqref{eq:sub-solutionw1}. Then, we may apply Theorem \ref{thm:D} to get $$0 \leq w^_1 \leq \overline{w}_1 \quad \quad \text{ in } [0,\infty)\times[0,\infty),$$ which implies that $$\{(t,x):w_1^(t,x)=0\}\supset\{(t,x):\overline w_1(t,x)=0\}= \{(t,x):0\leq x\leq \sigma_1t\}.$$ Letting $\epsilon \to 0$, we arrive at $$w_1^\epsilon(t,x) = -\epsilon \log{v^\epsilon(t,x)} \to 0 \text{ locally uniformly on } \{(t,x):0\leq x < \sigma_1t\}.$$ Hence for each small $\eta>0$, by choosing the compact sets $K=\{(1,x): \eta \leq x \leq \sigma_1 - \eta \}$ and $K'=\{(1,x): \frac{\eta}{2} \leq x \leq \sigma_1 - \frac{\eta}{2} \}$, we may apply Lemma \ref{lem:underlinec}\rm{(b)} to deduce that \begin{equation}\label{eq:asymptoticvc1'} \liminf_{t\to \infty}\inf_{\eta t< x < (\sigma_1 - \eta) t} v(t,x)=\liminf_{\epsilon\to 0}\inf_{K} v^\epsilon (t,x)\geq \frac{1-b}{2}>0. \end{equation} This implies $\underline{c}_1\geq \sigma_1$. \end{proof} \begin{corollary}\label{cor:underlinec1} Let $\sigma_1 > \sigma_2$ and let $(u,v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then for each small $\eta>0$, \begin{subequations} \begin{align} &\lim\limits_{t\to\infty}\sup\limits_{x>(\sigma_1+\eta)t} (\|u\|+\|v\|)=0,\label{eq:spreadc1_a} \\ &\lim\limits_{t\to\infty}\sup\limits_{(\overline{c}_2+\eta)t<x<(\sigma_1-\eta)t} (\|u\|+\|v-1\|)=0, \label{eq:spreadc1_b} \end{align} \end{subequations} where $\sigma_1$ is defined by \eqref{eq:sigma}. \end{corollary} \begin{proof} By definition, $\underline{c}_1 \leq \overline{c}_1$. It follows from Proposition \ref{prop:1} and Lemma \ref{lem:underlinec1} that $\sigma_1 \leq \underline{c}_1\leq \overline{c}_1 \leq \sigma_1$. Hence, $\underline{c}_1=\overline{c}_1=\sigma_1$. By Proposition \ref{prop:1}(i), $\overline{c}_2 \leq \sigma_2 < \sigma_1$, so that \eqref{eq:spreadc1_a} holds. In view of \eqref{eq:asymptoticvc1'} and definition of $\overline{c}_2$, we have, for each small $\eta>0$, $$ \liminf_{t\to \infty}\inf_{\eta t< x < (\sigma_1 - \eta) t} v(t,x)>0,\quad \text{ and }\quad \lim_{t\to\infty} \sup_{x > (\overline{c}_2 + \eta)t} u = 0. $$ We may then apply Lemma \ref{lem:entire1}{\rm{(d)}} to deduce \eqref{eq:spreadc1_b}. \end{proof} \begin{comment} For each small $\eta >0$, choose $$ K'=\{(1,x):~\sigma_2\leq x\leq \sigma_1-\eta\}\subset \{(t,x):0\leq x\leq \sigma_1t\}. $$ By \eqref{eq:asymptoticvc1} and $v^\epsilon(t,x)=v(\frac{t}{\epsilon},\frac{x}{\epsilon})$, we can deduce $\text{ for } 0<\epsilon\ll1$, $\forall c\in [\overline c_2,\sigma_1-\eta]$, $$v(\frac{1}{\epsilon},\frac{c}{\epsilon})\geq 1-b-o(1) \quad \text{ for each } \eta\in(0,\frac{\sigma_1-\overline c_2}{2}),$$ which implies $$\liminf\limits_{t\to\infty}\inf\limits_{\overline c_2 t\leq x<(\sigma_1-\eta)t} v(t,x)\geq 1-b \quad \text{ for each } \eta\in(0,\frac{\sigma_1-\overline c_2}{2}).$$ \end{comment} \subsection{Estimating $\underline c_2$ from below} By Corollary \ref{cor:underlinec1}, we have \begin{equation}\label{eq:v_ep} \chi_{\{\sigma_2 t<x<\sigma_1t\}}\leq\hspace{-.5cm}\liminf_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.5cm} v^\epsilon(t',x')\leq\hspace{-.5cm}\limsup_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.5cm} v^\epsilon(t',x')\leq \chi_{\{x \leq \sigma_1 t\}}. \end{equation} \begin{lemma}\label{lem:sub-solutionw2} Let $( u, v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then, $w_2^$ is a viscosity sub-solution of \begin{align}\label{eq:sub-solutionw2} \left \{ \begin{array}{ll} \min\{\partial_t w+\|\partial_xw\|^2+1-a\chi_{\{x\leq \sigma_1 t\}} ,w\}=0,&\text{ in } (0,\infty)\times\mathbb{R},\\ w(0,x)=\lambda_u\max\{x, 0\}, & \text{ on }\mathbb{R}, \end{array} \right. \end{align} where $\sigma_1$ is defined by \eqref{eq:sigma} and $\lambda_u >0$ is given in $\mathrm{(H_\lambda)}$. \end{lemma} \begin{proof} The proof is analogous to Lemma \ref{lem:sub-solutionw1} and we omit the details. \end{proof} \begin{lemma}\label{lem:underlinec2} Let $(u,v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then $$\underline c_2\geq \hat{c}_{\rm nlp},$$ where $\hat{c}_{\rm nlp}$ is defined in Theorem \ref{thm:1-2}. \end{lemma} \begin{proof} According to definition of $\hat{c}_{\rm nlp}$ in Theorem \ref{thm:1-2}, we consider the three cases separately: (a) $\sigma_1 < 2\lambda_u$ and $\sigma_1 < 2(\sqrt a + \sqrt{1-a})$; (b) $\sigma_1 \geq 2\lambda_u$ and $\tilde\lambda_{\rm nlp}\leq \sqrt{1-a}$; (c) otherwise. First, we claim that we are done in case (c). Since in that case $\hat{c}_{\rm nlp} = 2\sqrt{1-a}$, and, according to Proposition \ref{prop:1}(ii), $\underline{c}_2 \geq c_{\rm LLW}$ where $c_{\rm LLW} \geq 2\sqrt{1-a}$ by Theorem \ref{thm:LLW}. Thus $$ \underline{c}_2 \geq c_{\rm LLW} \geq 2\sqrt{1-a} = \hat{c}_{\rm nlp}. $$ It remains to consider cases (a) and (b). We start by defining $$\bar{c}_{\rm nlp} = \frac{\sigma_1}{2} - \sqrt{a} + \frac{1-a}{\frac{\sigma_1}{2} - \sqrt{a}}$$ and \begin{equation}\label{eq:tlambdanlp} \tilde{c}_{\rm nlp} = \tilde\lambda_{\rm nlp} + \frac{1-a}{\tilde\lambda_{\rm nlp}} \quad \text{ where } \quad \tilde\lambda_{\rm nlp}= \frac{1}{2}\left(\sigma_1 -\sqrt{(\sigma_1-2\lambda_u)^2 + 4a}\right). \end{equation} Suppose case (a) holds, then $\hat{c}_{\rm nlp}=\bar{c}_{\rm nlp}$. Define $\overline{w}_2$ by \begin{equation} \begin{split} \overline w_2(t,x) = \left\{ \begin{array}{ll} \lambda_u(x-(\lambda_u+\frac{1}{\lambda_u})t), & \text{for}~\frac{x}{t}\geq 2\lambda_u,\\ \frac{t}{4}(\frac{x^2}{t^2}-4), &\text{for}~ \sigma_1\leq \frac{x}{t}<2\lambda_u,\\ (\frac{\sigma_1}{2}-\sqrt{a})(x-\bar{c}_{\rm nlp}t), & \text{for}~\bar{c}_{\rm nlp}<\frac{x}{t}<\sigma_1,\\ 0, & \text{for}~\frac{x}{t}\leq \bar{c}_{\rm nlp}. \end{array} \right. \end{split} \end{equation} By construction, $\overline{w}_2$ is continuous in $[0,\infty)\times \mathbb{R}$. We claim that continuous $\overline{w}_2$ is a viscosity super-solution of \eqref{eq:sub-solutionw2}. (Actually, it is the unique viscosity solution of \eqref{eq:sub-solutionw2}, but we do not need this fact.) Indeed, $\overline{w}_2$ is a classical solution for \eqref{eq:sub-solutionw2} whenever $\frac{x}{t} \not \in \{\sigma_1, \bar{c}_{\rm nlp}\}$. Now, it remains to consider the case when $\overline{w}_2 - \varphi$ attains a strict local minimum at $(t_0,x_0)$ for $\forall \varphi\in C^\infty(0,\infty)\times\mathbb{R})$, when $\frac{x_0}{t_0} = \sigma_1$ or $\bar{c}_{\rm nlp}$. In case $\frac{x_0}{t_0} = \sigma_1$, $(\overline{w}_2-\varphi)(t,\sigma_1 t) \geq (\overline{w}_2-\varphi)(t_0, x_0)$ for all $t\approx t_0$, so that $\partial_t \varphi(t_0,x_0) + \sigma_1\partial_x\varphi(t_0,x_0) = \frac{\sigma^2_1}{4} -1$. Hence, at $(t_0,x_0)$, (note that $(-a\chi_{\{x \leq \sigma_1 t\}})^ = - a\chi_{\{x < \sigma_1 t\}}$) \begin{align} \partial_t\varphi +\|\partial_x \varphi\|^2 + 1 - a\chi_{\{x < \sigma_1 t\}} &= \frac{\sigma_1^2}{4} - 1 - \sigma_1 \partial_x \varphi + \|\partial_x \varphi\|^2 + 1\\ &= \left( \partial_x \varphi - \frac{\sigma_1}{2}\right)^2 \geq 0. \end{align} On the other hand, if $\frac{x_0}{t_0} = \bar{c}_{\rm nlp}$, then $\nabla \varphi(t_0,x_0) \cdot (1,\bar{c}_{\rm nlp}) = 0$, and $$ 0 \leq \nabla \varphi(t_0,x_0) \cdot (-\bar{c}_{\rm nlp}t,1) \leq \nabla[({\frac{\sigma_1}{2}} - \sqrt{a})(x - \bar{c}_{\rm nlp}t)] \cdot(-\bar{c}_{\rm nlp},1), $$ which means $\partial_t \varphi(t_0,x_0) = -\bar{c}_{\rm nlp}\partial_x \varphi$ and $0 \leq \partial_x \varphi(t_0,x_0) \leq \frac{\sigma_1}{2} - \sqrt{a}$, whence \begin{align} \partial_t\varphi +\|\partial_x \varphi\|^2 + 1 - a\chi_{\{x < \sigma_1 t\}} &= -\bar{c}_{\rm nlp} \partial_x \varphi + \|\partial_x \varphi\|^2 + 1-a\\ &= \left( \partial_x \varphi - \frac{1-a}{\frac{\sigma_1}{2}-\sqrt{a}}\right)\left( \partial_x \varphi - {\frac{\sigma_1}{2}+\sqrt{a}} \right) \geq 0 \end{align} at $(t_0,x_0)$. The last inequality holds because $\partial_x \varphi \leq \frac{\sigma_1}{2} - \sqrt{a} \leq \sqrt{1-a} \leq \frac{1-a}{\frac{\sigma_1}{2}-\sqrt{a}}$. Hence, $\overline{w}_2$ is a viscosity super-solution of \eqref{eq:sub-solutionw2}. By Remark \ref{rmk:w1w20} and the express of $\overline{w}_2$, we have $$ \overline{w}_2(0,x) = \lambda_u \max\{x,0\}=w_2^(0,x) \,\quad \text{ for } x\in \mathbb{R}. $$ And recalling that $w_2^$ is a viscosity sub-solution of \eqref{eq:sub-solutionw2}, we may deduce by Theorem \ref{thm:D} that \begin{equation}\label{eq:ineqw2u} 0 \leq w_2^\leq \overline w_2 \quad\quad \text{ in } [0,\infty)\times\mathbb{R}. \end{equation} Now, $$\{(t,x):w_2^(t,x)=0\}\supset\{(t,x):\overline w_2(t,x)=0\}= \{(t,x): x\leq\hat c_{\rm nlp}t\}. $$ Hence, $$ w_2^\epsilon(t,x) = -\epsilon \log{u^\epsilon(t,x)} \to 0 \quad\text{ locally uniformly on } \{(t,x):~x< \hat c_{\rm nlp}t\}.$$ Hence for each small $\eta>0$, by choosing the compact sets $K=\{(1,x): \eta\leq x \leq \hat c_{\rm nlp} - \eta \}$ and $K'=\{(1,x): \frac{\eta}{2} \leq x \leq \hat c_{\rm nlp} - \frac{\eta}{2} \}$, we may apply Lemma \ref{lem:underlinec}\rm{(a)} to get $$\liminf_{t\to \infty}\inf_{\eta t\leq x\leq ( \hat c_{\rm nlp} - \eta)t} u(t,x)=\liminf_{\epsilon\to 0}\inf_{K} u^\epsilon (t,x)\geq \frac{1-a}{2}>0,$$ which implies $\underline c_2\geq \hat{c}_{\rm nlp}$. Finally, for case (b), then we have $\hat{c}_{\rm nlp}=\tilde{c}_{\rm nlp}$. We define \begin{equation} \begin{split} \overline w_2 (t,x) = \left\{ \begin{array}{ll} \lambda_u(x-(\lambda_u+\frac{1}{\lambda_u})t), & \text{for}~\frac{x}{t}\geq \sigma_1,\\ \tilde \lambda_{\rm nlp} (x-\tilde {c}_{\rm nlp}t), & \text{for}~\tilde{c}_{\rm nlp}< \frac{x}{t}<\sigma_1,\\ 0, & \text{for}~\frac{x}{t}\leq \tilde{c}_{\rm nlp}. \end{array} \right. \end{split} \end{equation} Then one can verify that $\overline{w}_2$ is likewise a viscosity super-solution of \eqref{eq:sub-solutionw2}, so that one can repeat the arguments for case (a) to show, again, that $\underline c_2\geq \hat{c}_{\rm nlp}$. \end{proof} \section{Estimating $\displaystyle \overline c_2$ from above and $\displaystyle \underline c_3$ from below}\label{S4} We assume $\sigma_1>\sigma_2$ throughout this section. It remains to show $$\overline c_2\leq \max\{c_{\rm LLW}, \hat{c}_{\rm nlp}\} \quad \text{ and } \quad \underline c_3\geq -\max\{\tilde c_{\rm LLW}, \sigma_3\}.$$ \subsection{Estimating $\overline c_2$ from above} For $\delta \geq 0$, we will construct an exponent $\hat\mu_\delta$ depending continuously on $\delta$ such that $$ u(t, (\sigma_1-\delta) t)\leq \exp\left( - (\hat \mu_\delta+ o(1))t\right) \quad \text{ for }t \gg 1, $$ so that we may apply Lemma \ref{lem:appen1}\rm{(a)} to estimate $\overline{c}_2$ from above. \begin{lemma}\label{lem:supsolutionw2} Let $( u, v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then $w_{2,}$ is a viscosity super-solution of \begin{align}\label{eq:supsolutionw2} \left\{ \begin{array}{ll} \min\{\partial_tw+\|\partial_xw\|^2+1-a\chi_{\{\sigma_2 t<x<\sigma_1t\}},w\}=0,&\text{ in } (0,\infty)\times\mathbb{R},\\ w(0,x)=\lambda_u\max\{x, 0\}, & \text{ on } \mathbb{R}, \end{array} \right. \end{align} where $\sigma_1$ and $\sigma_2$ are defined in \eqref{eq:sigma}. \end{lemma} \begin{proof} It follows from standard arguments as in Lemma \ref{lem:sub-solutionw1}. \end{proof} \begin{proposition}\label{prop:overlinec2} Let $(u,v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then $$\overline c_2\leq \max\{c_{\rm LLW}, \hat{c}_{\rm nlp}\},$$ \end{proposition} where $c_{\rm LLW}$ and $\hat{c}_{\rm nlp}$ are defined respectively in Theorem \ref{thm:LLW} and \ref{thm:1-2}. \begin{proof} \noindent{\bf Step 1.} Define $\underline{w}_2: [0,\infty) \times \mathbb{R}$ by \begin{equation}\label{eq:underw2} \begin{split} \underline{w}_2(t,x) = \left\{ \begin{array}{ll} \lambda_u(x-(\lambda_u+\frac{1}{\lambda_u})t), & \text{for}~\frac{x}{t}\geq 2\lambda_u,\\ \frac{t}{4}(\frac{x^2}{t^2}-4), &\text{for}~ 2\leq \frac{x}{t}<2\lambda_u,\\ 0, & \text{for}~{\frac{x}{t}<2}, \end{array} \right. \end{split} \end{equation} in case $\lambda_u >1$, and by \begin{equation} \underline{w}_2(t,x) = \lambda_u \max\left\{x-(\lambda_u+\frac{1}{\lambda_u})t, 0\right\}, \end{equation} in case $\lambda _u\leq 1$. Then it is straightforward to verify that $\underline{w}_2$ is a viscosity sub-solution of \eqref{eq:supsolutionw2}. Since, $w_{2,}(0,x) = \lambda_u \max\{x,0\} = \underline{w}_2(0,x)$ in $\mathbb{R}$ (by Remark \ref{rmk:w1w20}), we may apply Theorem \ref{thm:D} to deduce \begin{equation}\label{eq:equivaw2} w_{2,}(t, x)\geq \underline w_2(t,x) \quad \text{ for } [0,\infty)\times\mathbb{R}. \end{equation} \noindent{\bf Step 2.} To show that, for each $\hat{c}\geq 0$, \begin{equation}\label{eq:inequ} u(t,\hat{c}t)\leq \exp\{-(\underline w_2(1,\hat{c})+o(1))t\} \quad \text{ for } t\gg1. \end{equation} And that $\underline w_2(1,\sigma_1)$ is given by \begin{equation}\label{eq:muc1} \underline w_2(1,\sigma_1)=\left\{ \begin{array}{ll} (\frac{\sigma_1}{2}-\sqrt{a})(\sigma_1-\bar c_{\rm nlp}), &\text{ for } \sigma_1< 2\lambda_u,\\ \tilde\lambda_{\rm nlp}(\sigma_1-\tilde c_{\rm nlp}), &\text{ for } \sigma_1\geq 2\lambda_u,\\ \end{array} \right. \end{equation} and $\bar{c}_{\rm nlp}, \,\tilde{c}_{\rm nlp},\,\tilde\lambda_{\rm nlp}$ are all given in Lemma \ref{lem:underlinec2}. By definition of $w_{2,}$ and $w_2^\epsilon(t,x)=-\epsilon \log{u^\epsilon(t,x)}$, for each small $\epsilon>0$, by applying Step 1, we have \begin{equation} -\epsilon\log{u\left(\frac{1}{\epsilon},\frac{\hat c}{\epsilon}\right)}\geq w_{2,}(1,\hat c)+o(1) \geq \underline w_2(1,\hat c)+o(1) \end{equation} $$\Longleftrightarrow \,\, u\left(\frac{1}{\epsilon},\frac{\hat c}{\epsilon}\right) \leq \exp\left(-\frac{\underline w_2(1,\hat c) + o(1)}{\epsilon}\right), $$ which implies \eqref{eq:inequ}. By the formula of $\underline w_2$, we can show \begin{itemize} \item[\rm{(i)}] For $\sigma_1<2\lambda_u$, we substitute $(t,x) = (1,\sigma_1)$ in \eqref{eq:underw2} to obtain \begin{equation}\label{eq:4.3b}\underline w_2(1,\sigma_1)=\frac{1}{4}(\sigma_1^2-4)=(\frac{\sigma_1}{2}-\sqrt{a})(\sigma_1-\bar c_{\rm nlp}), \end{equation} where $\bar c_{\rm nlp}=\frac{\sigma_1}{2}-\sqrt{a}+\frac{1-a}{\frac{\sigma_1}{2}-\sqrt{a}}$; \item[\rm{(ii)}] For $\sigma_1\geq 2\lambda_u$, we substitute $(t,x) = (1,\sigma_1)$ in \eqref{eq:underw2} to obtain \begin{equation}\label{eq:4.3ccc} \underline w_2(1,\sigma_1)=\lambda_u\left(\sigma_1-(\lambda_u+\frac{1}{\lambda_u})\right). \end{equation} Recalling the definition of $\tilde{\lambda}_{\rm nlp}$ in \eqref{eq:tlambdanlp}, we have $$ \tilde{\lambda}_{\rm nlp} - \lambda_u = \frac{1}{2} \left[(\sigma_1 - 2\lambda_u) - \sqrt{\sigma_1 - 2\lambda_u) ^2 + 4a }\right], $$ so that \begin{equation}\label{eq:4.3cc} (\tilde{\lambda}_{\rm nlp} - \lambda_u)^2 - (\sigma_1 - 2\lambda_u)(\tilde{\lambda}_{\rm nlp} - \lambda_u) - a=0. \end{equation} Hence, \eqref{eq:4.3ccc} becomes \begin{equation}\label{eq:4.3c} \underline w_2(1,\sigma_1)=\lambda_u\left(\sigma_1-(\lambda_u+\frac{1}{\lambda_u})\right)=\tilde\lambda_{\rm nlp}(\sigma_1-\tilde c_{\rm nlp}), \end{equation} where $\tilde c_{\rm nlp}, \tilde\lambda_{\rm nlp}$ are as in \eqref{eq:tlambdanlp}. \end{itemize} This implies \eqref{eq:muc1} holds, which completes Step 2. \noindent{\bf Step 3.} To show $\overline c_2\leq \max\{c_{\rm LLW}, \hat{c}_{\rm nlp}\}$. It follows from Proposition \ref{prop:1} and Corollary \ref{cor:underlinec1} that for $\hat c\in (\sigma_2,\sigma_1)$, $$\lim\limits_{t\to\infty}(u,v)(t,0)=(k_1,k_2)\,\,\text{ and }\,\,\lim\limits_{t\to\infty}(u,v)(t, \hat c t)=(0,1).$$ By Step 2 and observation $ \lambda_{\rm LLW}{c}_{\rm LLW}= \lambda_{\rm LLW}^2+1-a$, then we apply Lemma \ref{lem:appen1}\rm{(a)} in Appendix to conclude that for $\hat c\in (\sigma_2,\sigma_1)$, \begin{equation}\label{estim} \overline c_2\leq c_{\hat{c},\underline w_2(1,\hat{c})} =\left\{ \begin{array}{ll} c_{\rm LLW},& \text{ if } \underline w_2(1,\hat{c})\geq -\lambda_{\rm LLW}^2+\lambda_{\rm LLW} \hat{c}-(1-a),\\ \hat{c}-\frac{2{\underline w_2(1,\hat{c})}}{\hat{c}-\sqrt{\hat{c}^2-4({\underline w_2(1,\hat{c})}+1-a)}},& \text{ if } \underline w_2(1,\hat{c})< -\lambda_{\rm LLW}^2+\lambda_{\rm LLW} \hat{c}-(1-a).\\ \end{array} \right . \end{equation} Letting $\hat{c}\nearrow \sigma_1$, \eqref{estim} can be expressed as (denote $\hat\mu=\underline w_2(1,\sigma_1)$) \begin{equation}\label{estimc2} \overline c_2\leq c_{\sigma_1,{\hat\mu}} =\left\{ \begin{array}{ll} c_{\rm LLW},& \text{ if } {\hat\mu}\geq -\lambda_{\rm LLW}^2+\lambda_{\rm LLW}\sigma_1-(1-a),\\ \sigma_1-\frac{2{\hat\mu}}{\sigma_1-\sqrt{\sigma_1^2-4({\hat\mu}+1-a)}},& \text{ if } {\hat\mu}< -\lambda_{\rm LLW}^2+\lambda_{\rm LLW}\sigma_1-(1-a).\\ \end{array} \right . \end{equation} It remains to verify $c_{\sigma_1,{\hat\mu}}=\max\{c_{\rm LLW},\hat c_{\rm nlp}\}$, where $\hat c_{\rm nlp}=\hat\lambda_{\rm nlp}+\frac{1-a}{\hat\lambda_{\rm nlp}}$ and \begin{equation}\label{eq:lcacc} \hat{\lambda}_{\rm nlp} = \begin{cases} \frac{\sigma_1}{2}-\sqrt{a},& \text{ if }\sigma_1 < 2\lambda_u\,\, \text{ and }\,\, \sigma_1 \leq 2 (\sqrt a + \sqrt{1-a}),\\ \tilde\lambda_{\rm nlp}, & \text{ if }\sigma_1 \geq 2\lambda_u \,\,\text{ and }\,\, \tilde\lambda_{\rm nlp} \leq \sqrt{1-a},\\ \sqrt{1-a}, &\text{ otherwise,} \end{cases} \end{equation} and $\tilde\lambda_{\rm nlp}$ is given in Lemma \ref{lem:underlinec2}. Note that \begin{equation}\label{eq:lambdamuhat} \hat{\lambda}_{\rm nlp} = \min\{\lambda_{\hat\mu}, \sqrt{1-a}\}, \quad \text{ where }\quad \lambda_{\hat\mu}:= \left\{ \begin{array}{ll} \frac{\sigma_1}{2}-\sqrt{a}, &\text{ for } \sigma_1< 2\lambda_u,\\ \tilde\lambda_{\rm nlp}, &\text{ for } \sigma_1\geq 2\lambda_u.\\ \end{array} \right. \end{equation} By \eqref{eq:4.3b} and \eqref{eq:4.3c}, $\hat\mu = \underline{w}_2(1,\sigma_1)$ can be expressed as \begin{equation}\label{eq:muc1'} {\hat\mu}=G(\lambda_{\hat\mu}), \quad \text {where }\quad G(\lambda): = -\lambda^2+\sigma_1\lambda-(1-a) \end{equation} and $\lambda_{\hat\mu}$ is as defined in \eqref{eq:lambdamuhat}. Note that $G(\lambda)$ is strictly increasing on $[0,\frac{\sigma_1}{2}]$. We note for later purposes that \eqref{eq:muc1'} is a quadratic equation in $\lambda_{\hat\mu}$, so that \begin{equation}\label{eq:lambdac1} \lambda_{\hat\mu} = \frac{\sigma_1 - \sqrt{\sigma_1^2 - 4(\hat\mu + 1 - a)}}{2}. \end{equation} Since $\lambda_{\rm LLW} \in (0, \sqrt{1-a}]$, we divide our discussion into two cases: (i) $\lambda_{\hat\mu} < \lambda_{\rm LLW}$; (ii) $\lambda_{\rm LLW} \leq \lambda_{\hat\mu}$. \begin{itemize} \item[(i)] Case $\lambda_{\hat\mu} < \lambda_{\rm LLW}$. (Recall that $\lambda_{\rm LLW} \leq \sqrt{1-a}$.) By \eqref{eq:lambdamuhat}, $\hat\lambda_{\rm nlp} = \lambda_{\hat\mu}< \lambda_{\rm LLW}$, whence it follows from the observation \begin{equation}\label{eq:oob} \hat{c}_{\rm nlp} = \hat\lambda_{\rm nlp} + \frac{1-a}{\hat\lambda_{\rm nlp}}\,\, \text{ and }\,\, c_{\rm LLW} = \lambda_{\rm LLW} + \frac{1-a}{\lambda_{\rm LLW}}, \end{equation} and the monotonicity of $s + \frac{1-a}{s}$ in $(0,\sqrt{1-a}]$ that $\hat{c}_{\rm nlp} \geq c_{\rm LLW}$. It remains to show that $c_{\sigma_1,\hat\mu} = \hat{c}_{\rm nlp}$. Now, by monotonicity of $G$, we have $${\hat\mu}=G(\lambda_{\hat\mu}) <G(\lambda_{\rm{LLW}})=-\lambda_{ \rm LLW}^2+\lambda_{\rm LLW}\sigma_1-(1-a).$$ By \eqref{estimc2}, we have $c_{\sigma_1,{\hat\mu}}= \sigma_1-\frac{2{\hat\mu}}{\sigma_1-\sqrt{ \sigma_1^2-4({\hat\mu}+1-a)}}$. Hence, $$c_{\sigma_1,{\hat\mu}} =\sigma_1- \frac{{\hat\mu}}{\lambda_{\hat\mu}}=\lambda_{\hat\mu}+\frac{1-a}{\lambda_{\hat\mu}}=\hat{c}_{\rm nlp},$$ where the first and second equalities follow from \eqref{eq:lambdac1} and \eqref{eq:muc1'}, respectively. \item[(ii)] Case $\lambda_{\rm LLW} \leq \lambda_{\hat\mu}$. By \eqref{eq:lambdamuhat}, $$ \hat\lambda_{\rm nlp} = \min\{\lambda_{\hat\mu}, \sqrt{1-a}\} \geq \min\{\lambda_{\rm LLW}, \sqrt{1-a}\} =\lambda_{\rm LLW}. $$ It follows from \eqref{eq:oob} that $\hat{c}_{\rm nlp} \leq c_{\rm LLW}$. It remains to show that $\hat\mu \geq G(\lambda_{\rm LLW})$, so that $c_{\sigma_1,\hat\mu} = c_{\rm LLW} = \max\{c_{\rm LLW}, \hat{c}_{\rm nlp}\}$. Indeed, one can check that $\lambda_{\rm LLW} \leq \lambda_{\hat\mu} \leq \sigma_1/2$, and we deduce $$ \hat\mu = G(\lambda_{\hat\mu}) \geq G(\lambda_{\rm LLW}), $$ by the monotonicity of $G$ in $[0,\sigma_1/2]$. \end{itemize} The proof of Proposition \ref{prop:overlinec2} is now complete. \end{proof} \subsection{Estimating $\underline{c}_3$ from below} For convenience, let $\tilde u(t,x)=u(t,-x),\,\tilde v(t,x)=v(t,-x)$, and define $$ \tilde u^\epsilon (t,x)=\tilde u\left(\frac{t}{\epsilon},\frac{x}{\epsilon}\right),\,\, \tilde v^\epsilon (t,x)=\tilde v\left(\frac{t}{\epsilon},\frac{x}{\epsilon}\right),\,\, w_3^\epsilon= -\epsilon \log{\tilde v^\epsilon (t,x)}\,\, \text{ in } [0,\infty)\times\mathbb{R}.$$ Again we pass to the half-relaxed limit: $$w_{3,}(t,x)=\liminf_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.5cm} w_3^\epsilon (t',x').$$ \begin{lemma}\label{lem:tildeuv} Let $(\tilde u,\tilde v)$ be a solution of \eqref{eq:1-1} such that $x\rightarrow\left(\tilde u(0,-x),\tilde v(0,-x)\right)$ satisfies $\rm{(H_\lambda)}$. Then, for each small $\eta>0$, \begin{equation}\label{eq:tildeuv} \lim\limits_{t\to\infty} \sup_{x>(d\lambda^-_v+\frac{r}{\lambda^-_v}+\eta)t}(\|\tilde u(t,x)-1\|+\|\tilde v(t,x)\|)=0. \end{equation} \end{lemma} \begin{proof} Let $v_{\rm KPP}$ be the solution of \begin{equation} \left\{ \begin{array}{ll} \partial_t v_ {\rm KPP}- d\partial_{xx} v_{\rm KPP} =r v_{\rm KPP}(1-v_{\rm KPP}), &\text{ in }(0,\infty) \times \mathbb{R},\\ v_{\rm KPP} = \min\{1, Ce^{-\lambda_v^-x}\}, &\text{ on } x \in \mathbb{R}. \end{array} \right. \end{equation} By choosing $C$ to be sufficiently large, we may apply comparison principle to get $0\leq \tilde v \leq v_{\rm KPP}$. Therefore, for each $\eta>0$, \begin{equation}\label{eq:tildevkpp} \lim_{t \to \infty} \sup_{x > (d\lambda^-_v+\frac{r}{\lambda^-_v}+\eta)t} \|\tilde v(t,x)\| = 0 \, \text{ for each }\eta >0. \end{equation} Let $u_{\rm KPP}$ be the solution of \begin{equation} \left\{ \begin{array}{ll} \partial_t u_{\rm KPP}- \partial_{xx} u_{\rm KPP} = u_{\rm KPP}(1-a-u_{\rm KPP}), &\text{ in }(0,\infty) \times \mathbb{R},\\ u_{\rm KPP}(0,x) = u_0(x), &\text{ on } x \in \mathbb{R}. \end{array} \right. \end{equation} Again the scalar comparison principle implies $u \geq u_{\rm KPP}$. By the results in \cite{Kametaka_1976} or \cite{Mckean_1975}, we have, for each small $\eta>0$, \begin{equation}\label{eq:tildeukpp} \lim\limits_{t \to \infty} \inf\limits_{x >-(2\sqrt{1-a}+\eta)t} \tilde u(t,x)=\lim_{t \to \infty} \inf_{x <(2\sqrt{1-a}-\eta)t} u(t,x)\geq \frac{1-a}{2}. \end{equation} By small $\eta$>0, we have \eqref{eq:tildevkpp} and \eqref{eq:tildeukpp} hold, thus we may apply Lemma \ref{lem:entire1}\rm{(b)} to deduce \eqref{eq:tildeuv}. \end{proof} In view of Lemma \ref{lem:tildeuv}, we obtain \begin{equation}\label{eq:tildeu} \chi_{\{x>(d\lambda^-_v+\frac{r}{\lambda^-_v})t\}}\leq \hspace{-.5cm}\liminf_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.5cm} \tilde u^\epsilon(t',x')\leq \hspace{-.5cm}\limsup_{\scriptsize \begin{array}{c}\epsilon \to 0\\ (t',x') \to (t,x)\end{array}} \hspace{-.5cm} \tilde u^\epsilon(t',x')\leq 1 . \end{equation} \begin{lemma}\label{lem:sub-solutionw3} Let $(\tilde u,\tilde v)$ be a solution of \eqref{eq:1-1} such that $x\rightarrow\left(\tilde u(0,-x),\tilde v(0,-x)\right)$ satisfies $\rm{(H_\lambda)}$. Then, $w_{3,}$ is a viscosity super-solution of \begin{equation}\label{eq:supsolutionw3} \left\{ \begin{array}{ll} \min\{\partial_tw+d\|\partial_xw\|^2+r(1-b\chi_{\{x>(d\lambda^-_v+\frac{r}{\lambda^-_v})t\}}),w\}=0&\text{ in } (0,\infty)\times (0,\infty),\\ w(0,x)=\lambda_v^-x, & \text{ on }[0,\infty),\\ w(t,0)=0, & \text{ for }~t>0. \end{array} \right. \end{equation} \end{lemma} \begin{proof} The proof is similar to proof of Lemma \ref{lem:sub-solutionw1}(b) and is omitted. \end{proof} \begin{proposition}\label{prop:underlinec3} Let $( u, v)$ be a solution of \eqref{eq:1-1} with initial data satisfying $\rm{(H_\lambda)}$. Then $$\underline c_3\geq -\max\{\tilde c_{\rm LLW},\sigma_3\}.$$ where $\tilde c_{\rm LLW}$ and $\sigma_3$ are defined in Remark \ref{rmk:LLW} and \eqref{eq:sigma}, respectively. \end{proposition} \begin{proof} \noindent{\bf Step 1.} To show \begin{equation}\label{eq:equivaw3} w_{3,}(t, x)\geq \underline w_3(t,x) \quad \text{ for } [0,\infty)\times[0,\infty), \end{equation} where $\underline w_3: [0,\infty)\times[0, \infty)$ is defined by $$\underline w_3(t,x)=\lambda_v^- \max\left\{x-(d\lambda_v^-+\frac{r}{\lambda_v^-})t, 0\right\}.$$ As in Step 1 of Proposition \ref{prop:overlinec2}, one can verify that $\underline{w}_3$ is a viscosity sub-solution of \eqref{eq:supsolutionw3}. By the expression of $\underline{w}_3$, Remark \ref{rmk:w1w20} and $w_{1,}(t,-x)=w_{3,}(t,x)$, we have $ \underline{w}_3(t,x) = \lambda_v^- \max\{x,0\}=w_{3,}(t,x) \,\,\text{ on } \partial[ (0,\infty) \times (0,\infty)] $. Hence we apply Theorem \ref{thm:D} to obtain \eqref{eq:equivaw3}. \noindent{\bf Step 2.} To show for each $ \hat c \geq 0$, we have \begin{equation}\label{eq:ineqv} \tilde v(t,\hat c t)\leq \exp\{(\underline w_3(1,\hat c )+o(1))t\} \quad \text{ for } t\gg 1. \end{equation} This can be done as in Step 2 of Proposition \ref{prop:overlinec2}. \noindent{\bf Step 3.} To show $\underline c_3\geq -\max\{\tilde c_{\rm LLW},\sigma_3\}$. Fix $\hat c>(d\lambda^-_v+\frac{r}{\lambda^-_v})$. By Proposition \ref{prop:1} and Lemma \ref{lem:tildeuv}, we arrive at \begin{equation}\label{eq:asymuvc3} \lim\limits_{t\to \infty}(\tilde{u},\tilde{v})(t,0)=\lim\limits_{t\to \infty}(u,v)(t,0)=(k_1,k_2) \text{ and }\lim\limits_{t\to \infty}(\tilde{u},\tilde{v})(t,\hat{c}t)=(1,0). \end{equation} This verifies condition {\rm{(i)}} of Lemma \ref{lem:appen1}\rm{(b)}. Next, by Step 2, we have $$\tilde{v}(t,\hat c t)\leq \exp\{-(\hat\mu_2+ o(1)) t\} \quad \text{ for } t\gg 1,$$ where \begin{equation}\label{eq:muc2} {\hat\mu_2}=\underline w_3(1,\hat c)=\lambda_v^-(\hat c-(d\lambda_v^-+\frac{r}{\lambda_v^-})). \end{equation} We note for later purposes that $\hat\mu_2$ is a quadratic expression in $\lambda_v^-$, so that \begin{equation}\label{eq:lambdav-} \hat\mu_2= \lambda_v^-\hat c-d(\lambda_v^-)^2-r, \quad \text{ and }\quad \lambda_v^- = \frac{\hat c- \sqrt{\hat c^2 - 4d(\hat\mu_2 + r)}}{2d}. \end{equation} We may then apply Lemma \ref{lem:appen1}\rm{(b)} to conclude \begin{equation}\label{eq:estimc3} -\underline{c}_3 \leq \tilde{c}_{\hat{c},\hat\mu_2}= \left\{ \begin{array}{ll} \tilde c_{\rm LLW},& \text{ if } \hat\mu_2\geq \tilde{\lambda}_{\rm LLW} (\hat{c} - \tilde{c}_{\rm LLW}),\\ \hat c-\frac{2d\hat\mu_2}{\hat c-\sqrt{\hat c^2-4d[\hat\mu_2+r(1-b)]}},& \text{ if } 0<\hat\mu_2<\tilde{\lambda}_{\rm LLW} (\hat{c} - \tilde{c}_{\rm LLW}). \end{array} \right. \end{equation} To complete the proof, we need to verify $$\limsup_{\hat c\to\infty}\tilde c_{\hat c,{\hat\mu_2}}\leq\max\left\{\tilde{c}_{\rm LLW},\sigma_3\right\}.$$ Since $0 =-d\tilde{\lambda}^2_{\rm LLW}+\tilde{\lambda}_{\rm LLW}\tilde c_{\rm LLW}-r(1-b)$, then \begin{equation}\label{eq:mu2comp} \begin{split} \hat\mu_2-\tilde{\lambda}_{\rm LLW} (\hat c- \tilde{c}_{\rm LLW}) =&\hat\mu_2-(-d\tilde{\lambda}^2_{\rm LLW}+\tilde{\lambda}_{\rm LLW}\hat c -r(1-b))\\ =&\left (\lambda_v^--\tilde{\lambda}_{\rm LLW}\right)\left[\hat c- d(\lambda_v^-+\tilde{\lambda}_{\rm LLW})\right]-rb, \end{split} \end{equation} where \eqref{eq:lambdav-} is used for the last inequality. \begin{itemize} \item [\rm{(i)}] For the case $\lambda_v^-> \tilde{\lambda}_{\rm LLW}$, we take $\hat c\to\infty$ in \eqref{eq:mu2comp} to get $$\hat\mu_2\geq \tilde{\lambda}_{\rm LLW} \left(\hat{c} - \tilde{c}_{\rm LLW}\right),$$ so that by \eqref{eq:estimc3}, $-\underline{c}_3 \leq \tilde{c}_{\rm LLW}\leq\max\left\{\tilde{c}_{\rm LLW},\sigma_3\right\}$; \item [\rm{(ii)}] For the case $\lambda_v^-\leq\tilde{\lambda}_{\rm LLW}$, we have $\lambda_v^-\leq\tilde{\lambda}_{\rm LLW}\leq \sqrt{\frac{r(1-b)}{d}}$ and $$\sigma_3=d\lambda_{v}^-+\frac{r(1-b)}{\lambda_{v}^-}\geq d\tilde{\lambda}_{\rm LLW}+\frac{r(1-b)}{\tilde{\lambda}_{\rm LLW}}=\tilde{c}_{\rm LLW}.$$ we have $$0<\hat\mu_2<\tilde{\lambda}_{\rm LLW} \left(\hat{c} - \tilde{c}_{\rm LLW}\right).$$ Denote $\lambda_{\hat{c},\hat\mu_2}=\frac{\hat c-\sqrt{\hat c^2-4d[\hat\mu_2+r(1-b)]}}{2d}$. Then \begin{equation}\label{eq:ppp} d(\lambda_{\hat{c},\hat\mu_2})^2-\hat c \lambda_{\hat{c},\hat\mu_2}+\hat\mu_2+r(1-b)=0, \end{equation} and $\lambda_{\hat{c},\hat\mu_2}\leq\lambda^-_v$ (by comparing with the second part of \eqref{eq:lambdav-}). Hence, we arrive at \begin{equation}\label{eq:crewrite} -\underline{c}_3\leq c_{\hat{c},\hat\mu_2}= \hat{c} - \frac{\hat\mu_{2}}{\lambda_{\hat{c}, \hat{\mu}_2}} = d\lambda_{\hat{c},\hat\mu_2}+\frac{r(1-b)}{\lambda_{\hat{c},\hat\mu_2}}. \end{equation} Next, we claim that \begin{equation}\label{eq:pppp} \lim_{\hat c\to\infty}\lambda_{\hat{c},\hat\mu_2}=\lambda^-_v. \end{equation} To this end, subtract the first part of \eqref{eq:lambdav-} from \eqref{eq:ppp} to get $$d(\lambda_{\hat{c},\hat\mu_2})^2-\hat c (\lambda_{\hat{c},\hat\mu_2}-\lambda^-_v)-d(\lambda^-_v)^2-rb=0.$$ Dividing the above by $\hat{c}$ and letting $\hat{c} \to \infty$, we obtain \eqref{eq:pppp}. By \eqref{eq:pppp}, we can take $\hat{c} \to \infty$ in \eqref{eq:crewrite} to get $-\underline{c}_3\leq \sigma_3\leq\max\left\{\tilde{c}_{\rm LLW},\sigma_3\right\}$. \end{itemize} This completes the proof of Proposition \ref{prop:underlinec3}. \end{proof} \subsection{Proof of Theorem \ref{thm:1-2}} \begin{proof}[Proof of Theorem \ref{thm:1-2}] For $i=1,2,3$, let $\overline{c}_i, \underline{c}_i$ be the maximal and minimal spreading speeds defined in \eqref{eq:speeds}. It follows from definition directly that $\overline{c}_i \geq \underline{c}_i$. By Corollary \ref{cor:underlinec1}, we have $\overline{c}_1 = \underline{c}_1= \sigma_1$. By Proposition \ref{prop:1}(ii) and Lemma \ref{lem:underlinec2}, we arrive at $\underline{c}_2 \geq \max\{c_{\rm LLW}, \hat{c}_{\rm nlp}\}$, which, together with $\overline{c}_2 \leq \max\{c_{\rm LLW}, \hat{c}_{\rm nlp}\}$ in Proposition \ref{prop:overlinec2}, we have $\overline{c}_2 = \underline{c}_2 = \max\{c_{\rm LLW}, \hat{c}_{\rm nlp}\}$. Moreover, combining with Propositions \ref{prop:1} and \ref{prop:underlinec3} gives $\overline{c}_3 = \underline{c}_3 =- \max\{\sigma_3, \tilde{c}_{\rm LLW}\}$. Recalling the $c_i$ as defined in \eqref{eq:c123}, we have $\overline{c}_i = \underline{c}_i = c_i$ for all $i=1,2,3$. To complete the proof of Theorem \ref{thm:1-2}, it remains to show \eqref{eq:spreadingly}. Observe that the first two items of \eqref{eq:spreadingly} is a direct consequence of Corollary \ref{cor:underlinec1}. Next, we shall show that \begin{equation}\label{positivev} \displaystyle \liminf_{t \to \infty} \inf_{(c_3 + \eta)t < x < (\sigma_1 - \eta)t} v(t,x) >0 \quad \text{ for small } \eta>0. \end{equation} Given some small $\eta>0$, definitions of $\overline{c}_3$ and $\underline{c}_1$ imply the existence of $c_3' \in (c_3, c_3 +\eta)$, $\sigma_1' \in (\sigma_1 - \eta, \sigma_1)$ and $T>0$ such that $$ \inf_{t \geq T}\min\{v(t, {c}_3' t), v(t, \sigma'_1 t)\} >0. $$ Now, define \begin{equation} \begin{array}{l} \delta:= \min\left\{\frac{1-b}{2},\, \inf\limits_{c_3' T < x < \sigma'_1 T}v(T,x),\,\inf\limits_{t \geq T}\min\{v(t, {c}_3' t), v(t, \sigma'_1 t)\} \right\} >0. \end{array} \end{equation} Observe that $v(t,x)$ and $\delta$ form a pair of super- and sub-solutions to the KPP-type equation $\partial_t v = d\partial_{xx}v + rv(1-b - v)$ such that $v(t,x) \geq \delta$ on the parabolic boundary of the domain $\Omega:=\{(t,x): t \geq T,\,c_3' t < x < \sigma'_1 t \}$. It follows from the maximum principle that $v \geq \delta$ in $\Omega$. In particular, \eqref{positivev} holds. Similarly, we can show that \begin{equation}\label{positiveu} \displaystyle \liminf_{t \to \infty} \inf_{x <(c_2-\eta)t} u(t,x) >0\quad \text{ for small }\eta>0, \end{equation} by definition of $\underline c_2$ and by \eqref{eq:tildeukpp} in Lemma \ref{lem:tildeuv}. Fix small $\eta>0$. In view of \eqref{positivev} and \eqref{positiveu}, the third item of \eqref{eq:spreadingly} holds by applying {\rm{(a)}} and {\rm{(c)}} in Lemma \ref{lem:entire1}. Finally, since \eqref{positiveu} and $\lim\limits_{t\to \infty } \sup\limits_{x<(c_3-\eta)t} v=0$ (by definition of $\underline c_3$), by applying Lemma \ref{lem:entire1}{\rm{(b)}}, the fourth item of \eqref{eq:spreadingly} holds true. The proof of Theorem \ref{thm:1-2} is now complete. \end{proof} \section{The invasion mode due to Tang and Fife}\label{S4b} In this section, we assume $\sigma_1=\sigma_2$ and prove Theorem \ref{thm:1-2b}. \begin{proof}[Proof of Theorem \ref{thm:1-2b}] For any small $\delta\in(0,1)$, let $(\underline{u}^\delta,\overline{v}^\delta)$ and $(\overline{u}^\delta,\underline{v}^\delta)$ be respectively any solution of \begin{equation}\label{eq:1-1'} \left\{ \begin{array}{ll} \partial_t u-\partial_{xx}u=u(1-u-av),& \text{ in }(0,\infty)\times \mathbb{R},\\ \partial _t v-d\partial_{xx}v=rv(1+\delta-bu-v),& \text{ in }(0,\infty)\times \mathbb{R},\\ \end{array} \right . \end{equation} and \begin{equation}\label{eq:1-1''} \left\{ \begin{array}{ll} \partial_t u-\partial_{xx}u=u(1-u-av),& \text{ in }(0,\infty)\times \mathbb{R},\\ \partial _t v-d\partial_{xx}v=r v(1-\delta-bu-v),& \text{ in }(0,\infty)\times \mathbb{R},\\ \end{array} \right . \end{equation} with initial data satisfying $\rm(H_\lambda)$. By comparison, we deduce that \begin{equation}\label{eq:uvsupersub} (\underline{u}^\delta,\overline{v}^\delta)\preceq (u,v)\preceq (\overline{u}^\delta,\underline{v}^\delta) \,\, \text{ in } [0,\infty)\times\mathbb{R}. \end{equation} Notice that $(\underline{u}^\delta,\overline{v}^\delta)$ is a solution of \eqref{eq:1-1'} if and only if \begin{equation}\label{scaling1} (\underline{U}^\delta,\overline V^\delta)= \left(\underline{u},\frac{\overline{v}^\delta}{1+\delta}\right) \end{equation} is a solution of \begin{equation}\label{eq:1-1'trs} \left\{ \begin{array}{ll} \partial_t U-\partial_{xx}U=U(1-U-\overline{a}^\delta V),& \text{ in }(0,\infty)\times \mathbb{R},\\ \partial _t V-d\partial_{xx}V=\overline{r}^\delta V(1-\underline{b}^\delta U-V),& \text{ in }(0,\infty)\times \mathbb{R},\\ \end{array} \right . \end{equation} where $\overline{a}^\delta=(1+\delta)a,\, \overline{r}^\delta=(1+\delta)r$ and $ \underline{b}^\delta=\frac{b}{1+\delta}$. Observe that $\overline\sigma_1^\delta=d(\lambda_v^+\wedge \sqrt{\frac{ \overline{r}^\delta}{d}})+\frac{ \overline{r}^\delta}{\lambda_v^+\wedge\sqrt{\frac{ \overline{r}^\delta}{d}}}>\sigma_1=\sigma_2$ and $0<\overline{a}^\delta,\underline{b}^\delta<1$ by choosing $\delta$ small enough. By applying Theorem \ref{thm:1-2} to \eqref{eq:1-1'trs}, we deduce that the rightward and leftward spreading speeds $\overline{c}^\delta_1$ and $\underline{c}^\delta_3$ of $\overline{V}^\delta$ (which is the same as $\overline{v}^\delta$), and the rightward spreading speed $\underline{c}^\delta_2$ of $\underline{U}^\delta$ (same as $\underline{u}^\delta$) exist. Furthermore, they can be characterized by $$\overline{c}_1^\delta = \overline\sigma_1^\delta, \quad \underline{c}_2^\delta = \max\{\underline{c}_{\textup{LLW}}^\delta, \underline{\hat{c}}_{\textup{nlp}}^\delta\},\quad \underline{c}_3^\delta = -\max\{\overline{\tilde{c}}_{\textup{LLW}}^\delta,\overline{\sigma}_3^\delta\}. $$ Precisely, $\underline{c}_{\textup{LLW}}^\delta$ $(\text{resp.}~\overline{\tilde{c}}_{\textup{LLW}}^\delta)$ is the spreading speed for \eqref{eq:1-1'trs} as given in Theorem \ref{thm:LLW} $($resp. Remark \ref{rmk:LLW}$)$, $\overline\sigma_3^\delta=d(\lambda_v^-\wedge \sqrt{\frac{ \overline{r}^\delta}{d}})+\frac{ \overline{r}^\delta(1-\underline{b}^\delta)}{\lambda_v^+\wedge\sqrt{\frac{ \overline{r}^\delta}{d}}}$ and moreover \begin{equation}\label{eq:hcaccdelta} \underline{\hat{c}}_{{\rm{nlp}}} ^\delta= \begin{cases} \frac{\overline\sigma_1^\delta}{2} - \sqrt{\overline{a}^\delta} + \frac{1-\overline{a}^\delta}{\frac{\overline\sigma_1^\delta}{2} - \sqrt{\overline{a}^\delta}}, & \text{ if }\overline\sigma_1^\delta < 2\lambda_u\,\, \text{ and }\,\, \overline\sigma_1^\delta \leq 2 (\sqrt{\overline{a}^\delta} + \sqrt{1-\overline{a}^\delta}),\\ \overline{\tilde{\lambda}}_{{\rm{nlp}}}^\delta + \frac{1-\overline{a}^\delta}{\overline{\tilde{\lambda}}_{{\rm{nlp}}}^\delta}, & \text{ if }\overline\sigma_1^\delta \geq 2\lambda_u \,\,\text{ and }\,\, \overline{\tilde{\lambda}}_{{\rm{nlp}}}^\delta \leq \sqrt{1-\overline{a}^\delta},\\ 2\sqrt{1-\overline{a}^\delta}, &\text{ otherwise,} \end{cases} \end{equation} where $ \overline{\tilde{\lambda}}_{{\rm{nlp}}}^\delta= \frac{1}{2}\left[\overline\sigma_1^\delta- \sqrt{(\overline\sigma_1^\delta-2\lambda_u)^2 + 4\overline{a}^\delta}\right]. $ Now, by the relation \eqref{eq:uvsupersub}, we can compare with the spreading speeds $\overline{c}_1$, $\underline{c}_2$ and $\underline{c}_3$ of $(u,v)$: \begin{equation}\label{inequality1} \overline{c}_1\leq \overline{c}_{1}^\delta,\quad \underline{c}_2\geq \underline{c}_2^\delta\quad \text{ and } \quad \underline{c}_3\geq \underline{c}_{3}^\delta. \end{equation} It remains to show that, assuming $\sigma_1 = \sigma_2$, we have $\underline{\hat{c}}_{{\rm{nlp}}}^\delta\to \sigma_2$ as $\delta\to 0$. Divide into the two cases: \begin{itemize} \item[\rm{(i)}] If $\lambda_u>1$, then $\sigma_1=\sigma_2=2<2\lambda_u$. Since $1<\sqrt{a}+\sqrt{1-a}$, by choosing $\delta>0$ small enough, we get $\overline\sigma_1^\delta<2\lambda_u$ and $\overline\sigma_1^\delta \leq 2 (\sqrt{\overline{a}^\delta} + \sqrt{1-\overline{a}^\delta})$, which implies $\underline{\hat{c}}_{{\rm{nlp}}}^\delta=\frac{\overline\sigma_1^\delta}{2} - \sqrt{\overline{a}^\delta} + \frac{1-\overline{a}^\delta}{\frac{\overline\sigma_1^\delta}{2} - \sqrt{\overline{a}^\delta}} \to 1-\sqrt{a} + \frac{1-a}{1-\sqrt{a}} = 2 = \sigma_2$ as $\delta \to 0$. \item[\rm{(ii)}] If $\lambda_u\leq 1$, then first claim that \begin{equation}\label{eq:firstclaimm} \overline\sigma_1^\delta \geq \sigma_1 \geq 2\lambda_u, \end{equation} which is due to $\overline\sigma_1^\delta\geq\sigma_1=\sigma_2=\lambda_u+\frac{1}{\lambda_u}\geq 2\geq 2\lambda_u.$ Next, we claim that \begin{equation}\label{eq:firstclaim} \tilde\lambda_{\rm nlp} < \sqrt{1-a}, \end{equation} where $\tilde\lambda_{\rm nlp}$ is given in \eqref{eq:lambdaacc}. To this end, observe that \begin{equation}\label{eq:sss3} \sigma_1-2\sqrt{1-a}<\sqrt{(\sigma_1-2\lambda_u)^2 + 4{a}} \end{equation} which is a consequence of \begin{align} \begin{split} (\sigma_1-2\sqrt{1-a})^2-\left[(\sigma_1-2\lambda_u)^2 + 4{a}\right] &=4(2-2a-\sigma_1\sqrt{1-a})\\ &\leq4(2-2a-2\sqrt{1-a})< 0. \end{split} \end{align} From definition of $\tilde\lambda_{\rm nlp} = \frac{1}{2}[\sigma_1 - \sqrt{(\sigma_1 - 2\lambda_u)^2 + 4a}]$, we deduce \eqref{eq:firstclaim}. By \eqref{eq:firstclaimm} and \eqref{eq:firstclaim}, we have $\bar\sigma_1^\delta \geq 2\lambda_u$ and $\overline{\tilde\lambda}^\delta_{\rm nlp} < \sqrt{1-\overline{a}^\delta}$ for $\delta$ small, so $$ \underline{\hat{c}}^\delta_{\rm nlp} = \overline{\tilde{\lambda}}_{{\rm{nlp}}}^\delta + \frac{1-\overline{a}^\delta}{\overline{\tilde{\lambda}}_{{\rm{nlp}}}^\delta} \to \tilde\lambda_{\rm nlp} + \frac{1-a}{\tilde\lambda_{\rm nlp}} \quad \text{ as } \delta \to 0. $$ Since we want $\underline{\hat{c}}^\delta_{\rm nlp} \to \sigma_2$, it remains to show that $\sigma_2 = \tilde\lambda_{\rm nlp} + \frac{1-a}{\tilde\lambda_{\rm nlp}}$. To this end, recall, from the definition of $\tilde\lambda_{\rm nlp}$ \eqref{eq:lambdaacc}, that $$ \tilde\lambda_{\rm nlp} = \frac{\sigma_1 - \sqrt{(\sigma_1 - 2\lambda_u)^2 + 4a}}{2} = \frac{2(\sigma_1 \lambda_u - \lambda_u^2 - a)}{\sigma_1 + \sqrt{(\sigma_1 - 2\lambda_u)^2 + 4a}}. $$ Using $\sigma_1= \sigma_2 = \lambda_u + \frac{1}{\lambda_u}$, we deduce \begin{equation}\label{eq:sss2} \tilde\lambda_{\rm nlp} =\frac{\sigma_2 - \sqrt{(\sigma_2 - 2\lambda_u)^2 + 4a}}{2} = \frac{2(1-a)}{\sigma_2+ \sqrt{(\sigma_2 - 2\lambda_u)^2 + 4a}}. \end{equation} This implies $\sigma_2 = \tilde\lambda_{\rm nlp} + \frac{1-a}{\tilde\lambda_{\rm nlp}}$. The proof is now complete. \end{itemize} Hence, by the continuity of $\underline{c}_{\textup{LLW}}^\delta$ and $\overline{\tilde{c}}_{\textup{LLW}}^\delta$ in $\delta$ (see, e.g. \cite[Theorem 4.2 of Ch. 3]{Volpert_1994}), letting $\delta\to 0$ in \eqref{inequality1} yields \begin{equation}\label{eq:upperci} \overline{c}_1\leq \sigma_1,\quad \underline{c}_2\geq \sigma_2 \quad \text{ and }\quad \underline{c}_3\geq - \max\{ \tilde{c}_{\rm LLW}, \sigma_3\}. \end{equation} By a quite similar process, we can obtain $(\overline{u}^\delta,\underline{v}^\delta)$ is a solution of \eqref{eq:1-1''} if and only if $$(\overline{U}^\delta,\underline V^\delta)=\left(\overline{u}^\delta,\frac{\underline{v}^\delta}{1-\delta}\right)$$ is a solution of \begin{equation}\label{eq:1-1''trs} \left\{ \begin{array}{ll} \partial_t U-\partial_{xx}U=U(1-U-\underline{a}^\delta V),& \text{ in }(0,\infty)\times \mathbb{R},\\ \partial _t V-d\partial_{xx}V=\underline{r}^\delta V(1-\overline{b}^\delta U-V),& \text{ in }(0,\infty)\times \mathbb{R}, \end{array} \right . \end{equation} where $\underline{a}^\delta=(1-\delta)a$, $ \underline{r}^\delta=(1-\delta)r$ and $\overline{b}^\delta=\frac{b}{1-\delta}$. Observe that $\underline\sigma_1^\delta=d(\lambda_v^+\wedge \sqrt{\frac{ \underline{r}^\delta}{d}})+\frac{ \underline{r}^\delta}{\lambda_v^+\wedge\sqrt{\frac{ \underline{r}^\delta}{d}}}<\sigma_1=\sigma_2$ and $0<\underline{a}^\delta,\overline{b}^\delta<1$ for small $\delta$. By exchanging the roles of $u$ and $v$ in \eqref{eq:1-1}, we may follow the arguments above, and apply Theorem \ref{thm:1-2} once again to deduce that \begin{equation}\label{eq:lowerci} \underline{c}_1\geq \sigma_1, \quad \overline{c}_2\leq \sigma_2 \quad\text{ and} \quad \overline{c}_3\leq - \max\{ \tilde{c}_{\rm LLW}, \sigma_3\}. \end{equation} Theorem \ref{thm:1-2b} follows from combining $\underline{c}_i\leq \overline{c}_i$, \eqref{eq:upperci}, \eqref{eq:lowerci} and $\sigma_1=\sigma_2$. \end{proof} \section{The case $0<a<1<b$ due to Girardin and Lam}\label{S5} The Hamilton-Jacobi approach, which we have so far applied to study the weak competition case ($0<a,b<1$), can also be applied to tackle the case ($0<a<1<b$), which was previously studied by Girardin and the third author \cite{Girardin_2018}. This provides an alternative approach which is more transparent than the involved construction of global super- and sub-solutions for the Cauchy problem, as was done in \cite{Girardin_2018}. By arguing similarly as in Theorem \ref{thm:1-2}, one can prove the following result. \begin{theorem}\label{thm:1-3'} Assume $0<a < 1 < b$ and $\sigma_1>\sigma_2$. Let $(u,v)$ be the solution of \eqref{eq:1-1} such that the initial data satisfies $\mathrm{(H_\lambda)}$. Then there exist $c_1, c_2\in (0,\infty)$ such that $c_1 > c_2$ and, for each small $\eta>0$, the following spreading results hold: \begin{equation}\label{spreadingpro} \begin{cases} \lim\limits_{t\rightarrow \infty} \sup\limits_{ x>(c_{1}+\eta) t} (\|u(t,x)\|+\|v(t,x)\|)=0, \\%&\,\text{ for } \epsilon>0, \\ \lim\limits_{t\rightarrow \infty} \sup\limits_{(c_2+\eta) t< x<(c_{1}-\eta) t} (\|u(t,x)\|+\|v(t,x)-1\|)=0, \\% &\,\text{ for }\epsilon\in \left(0,\frac{c_{1}-c_2}{2}\right), \\ \lim\limits_{t\rightarrow \infty} \sup\limits_{ x<(c_2-\eta) t} (\|u(t,x)-1\|+\|v(t,x)\|)=0. \\%&\,\text{ for } \end{cases} \label{eq:spreadingly} \end{cases} \end{equation} Precisely, the spreading speeds $c_1$ and $c_2$ can be determined as follows: $$ c_1 = \sigma_1 , \quad c_2 = \max\{\hat{c}_{\rm LLW}, \hat{c}_{\rm nlp}\},$$ where $\sigma_1$ is defined in \eqref{eq:sigma}, $\hat{c}_{\rm LLW}$ denotes the minimal wave speed of \eqref{eq:1-1} connecting $(1,0)$ with $(0,1)$ and $\hat{c}_{\rm nlp}$ is given by \begin{equation}\label{eq:hcacc'} \hat{c}_{{\rm{nlp}}} = \begin{cases} \frac{\sigma_1}{2} - \sqrt{a} + \frac{1-a}{\frac{\sigma_1}{2} - \sqrt{a}}, & \text{ if }\sigma_1 < 2\lambda_u\,\, \text{ and }\,\, \sigma_1 \leq 2 (\sqrt a + \sqrt{1-a}),\\ \tilde\lambda_{{\rm{nlp}}} + \frac{1-a}{\tilde\lambda_{{\rm{nlp}}}}, & \text{ if }\sigma_1 \geq 2\lambda_u \,\,\text{ and }\,\, \tilde\lambda_{{\rm{nlp}}} \leq \sqrt{1-a},\\ 2\sqrt{1-a}, &\text{ otherwise,} \end{cases} \end{equation} with \begin{equation}\label{eq:lambdaacc'} \tilde\lambda_{{\rm{nlp}}} = \frac{1}{2}\left[\sigma_1 - \sqrt{(\sigma_1-2\lambda_u)^2 + 4a}\right]. \end{equation} \end{theorem} By Theorem \ref{thm:1-3'}, the spreading speed $c_2$ is determined by $\sigma_1$ (i.e., $c_1$) and $\lambda_u$. In what follows, we explore the relation of $c_2$ and $\sigma_1$ for fixed $\lambda_u$. Define the following auxiliary functions: $$f(\sigma_1)=\frac{\sigma_1}{2} - \sqrt{a} + \frac{1-a}{\frac{\sigma_1}{2} - \sqrt{a}}, \quad g(\sigma_1)=\tilde\lambda_{{\rm{nlp}}} + \frac{1-a}{\tilde\lambda_{{\rm{nlp}}}},$$ where $\tilde\lambda_{{\rm{nlp}}}$ is given by \eqref{eq:lambdaacc'}. It is easily seen that $f$ is decreasing and bijective in $[2\sqrt{1-a}, 2(\sqrt{1-a}+\sqrt{a})]$, while $g$ is decreasing and bijective in \begin{equation} \begin{cases} \left[2\sqrt{1-a},\lambda_u+\sqrt{1-a}+\frac{1-a}{\lambda_u-\sqrt{1-a}}\right] &\text{ as } \lambda_u\geq\sqrt{1-a}, \\ \left(\lambda_u+\sqrt{1-a}+\frac{1-a}{\lambda_u-\sqrt{1-a}},\infty\right) & \text{ as } \lambda_u< \sqrt{1-a}. \end{cases} \end{equation} More precisely, it follows that \begin{equation} \begin{cases} f^{-1}(c_2)=c_2-\sqrt{c_2^2-4(1-a)}+2\sqrt{a},\\ g^{-1}(c_2)=\lambda_u+\frac{c_2-\sqrt{c_2^2-4(1-a)}}{2}+\frac{a}{\lambda_u-\frac{c_2-\sqrt{c_2^2-4(1-a)}}{2}}. \end{cases} \end{equation} In view of $\tilde\lambda_{{\rm{nlp}}}\to\lambda_u$ as $\sigma_1\to\infty$, $g_{\infty}:=g(\infty)=\lambda_u+\frac{1-a}{\lambda_u}$. For fixed $\lambda_u$ and varied $\lambda_v^+$ (or $\sigma_1$), by Theorem \ref{thm:1-3'} we can rewrite the spreading speed $c_2$ as follows. \begin{comment} \begin{itemize} \item[{\rm (a)}] If $\lambda_u\geq (\sqrt{a}+\sqrt{1-a})$, then \begin{equation} c_2 = \begin{cases} f(\sigma_1) & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq \sigma_1\leq f^{-1}(\hat c_{\rm LLW}),\\ \hat c_{\rm LLW}& \text{ for }\sigma_1> f^{-1}(\hat c_{\rm LLW}), \end{cases} \end{equation} independent $\lambda_u$; \item[{\rm (b)}] If $\sqrt{dr}\leq \lambda_u< \sqrt{a}+\sqrt{1-a}$ and $g^{-1}(\hat c_{\rm LLW} )>2\lambda_u$, then \begin{itemize} \item[{\rm (b1)}] for $g_{\infty}\leq\hat c_{\rm LLW}$, we have \begin{equation} c_2 = \begin{cases} f(\sigma_1) & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq\sigma_1< 2\lambda_u,\\ g(\sigma_1) & \text{ for }2\lambda_u\leq \sigma_1<g^{-1}(\hat c_{\rm LLW}),\\ \hat c_{\rm LLW}& \text{ for }\sigma_1\geq g^{-1}(\hat c_{\rm LLW}); \end{cases} \end{equation} \item[{\rm (b2)}] for $g_{\infty}>\hat c_{\rm LLW}$, we have \begin{equation} c_2 = \begin{cases} f(\sigma_1) & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq\sigma_1< 2\lambda_u,\\ g(\sigma_1) & \text{ for }\sigma_1\geq2\lambda_u; \end{cases} \end{equation} \end{itemize} \item[{\rm (c)}]If $\lambda_u<\sqrt{dr}$, then \begin{itemize} \item[{\rm (c1)}] for $g_{\infty}\leq\hat c_{\rm LLW}$, we have \begin{equation} c_2 = \begin{cases} g(\sigma_1) & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq \sigma_1<g^{-1}(\hat c_{\rm LLW}),\\ \hat c_{\rm LLW}& \text{ for }\sigma_1\geq g^{-1}(\hat c_{\rm LLW}). \end{cases} \end{equation} \item[{\rm (c2)}] for $g_{\infty}\leq\hat c_{\rm LLW}$, we have \begin{equation} c_2 = g(\sigma_1) \text{ for }\sigma_1\geq\max\{2\sqrt{dr},\sigma_2\}. \end{equation} \end{itemize} \end{itemize} \end{comment} \begin{itemize} \item[{\rm (a)}] For $g_{\infty}\leq\hat c_{\rm LLW}$, we have the followings: \begin{itemize} \item[{\rm (a1)}] If $\lambda_u\geq (\sqrt{a}+\sqrt{1-a})$, then \begin{equation} c_2 = \begin{cases} f(\sigma_1), & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq \sigma_1\leq f^{-1}(\hat c_{\rm LLW}),\\ \hat c_{\rm LLW},& \text{ for }\sigma_1> f^{-1}(\hat c_{\rm LLW}), \end{cases} \end{equation} independent $\lambda_u$; \item[{\rm (a2)}] If $\sqrt{dr}\leq \lambda_u< \sqrt{a}+\sqrt{1-a}$ and $g^{-1}(\hat c_{\rm LLW} )>2\lambda_u$, then \begin{equation} c_2 = \begin{cases} f(\sigma_1), & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq\sigma_1< 2\lambda_u,\\ g(\sigma_1), & \text{ for }2\lambda_u\leq \sigma_1<g^{-1}(\hat c_{\rm LLW}),\\ \hat c_{\rm LLW},& \text{ for }\sigma_1\geq g^{-1}(\hat c_{\rm LLW}); \end{cases} \end{equation} \item[{\rm (a3)}]If $\lambda_u<\sqrt{dr}$, then \begin{equation} c_2 = \begin{cases} g(\sigma_1), & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq \sigma_1<g^{-1}(\hat c_{\rm LLW}),\\ \hat c_{\rm LLW},& \text{ for }\sigma_1\geq g^{-1}(\hat c_{\rm LLW}); \end{cases} \end{equation} \end{itemize} \item[{\rm (b)}] For $g_{\infty}>\hat c_{\rm LLW}$, we have the followings: \begin{itemize} \item[{\rm (b1)}] If $\lambda_u\geq (\sqrt{a}+\sqrt{1-a})$, then \begin{equation} c_2 = \begin{cases} f(\sigma_1), & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq \sigma_1\leq f^{-1}(\hat c_{\rm LLW}),\\ \hat c_{\rm LLW},& \text{ for }\sigma_1> f^{-1}(\hat c_{\rm LLW}), \end{cases} \end{equation} independent $\lambda_u$; \item[{\rm (b2)}] If $\sqrt{dr}\leq \lambda_u< \sqrt{a}+\sqrt{1-a}$, then \begin{equation} c_2 = \begin{cases} f(\sigma_1), & \text{ for }\max\{2\sqrt{dr},\sigma_2\}\leq\sigma_1< 2\lambda_u,\\ g(\sigma_1), & \text{ for }\sigma_1\geq2\lambda_u; \end{cases} \end{equation} \item[{\rm (b3)}]If $\lambda_u<\sqrt{dr}$, then \begin{equation} c_2 = g(\sigma_1) \quad \text{ for }\sigma_1\geq\max\{2\sqrt{dr},\sigma_2\}. \end{equation} \end{itemize} \end{itemize} For the case {\rm (a)} $g_{\infty}\leq\hat c_{\rm LLW}$, the relationship between the spreading speeds $\sigma_1$ and $c_2$ given by {\rm (a1)}-{\rm (a3)} is illustrated in Figure \ref{figure2}. Therein we may obtain the exact spreading speeds of \eqref{eq:1-1}, which are determined entirely by $\lambda_u,\,\lambda_v^+\in(0,\infty)$. Traversing all of $\lambda_u$, the set of admissible speeds $\sigma_1$ and $c_2$ agrees with \cite[Figure 1.1]{Girardin_2018}. Particularly, a direct consequence of Theorem \ref{thm:1-3'} is the following proposition, which improves upon \cite[Theorem 1.3]{Girardin_2018} by clarifying the role of exponential decay $(\lambda_u,\,\lambda_v^+)$ of the initial data. \begin{proposition}\label{overlinec} Let $(\overline{c}, \underline{c}) \in(2\sqrt{dr},\infty) \times (\hat{c}_{\rm LLW}, \infty)$ such that $\overline{c} > \underline{c}$. \begin{itemize} \item[\rm (a)] If $\underline{c}<f(\overline{c})$, then the pair of spreading speeds $(\overline{c}, \underline{c})$ is not realized by solutions of \eqref{eq:1-1} with initial data satisfying $\mathrm{(H_\lambda)}$. \item[\rm (b)] If $\underline{c} = f(\overline{c})$, then there exists a unique $\lambda_v^+ = \frac{1}{2d}(\overline{c} - \sqrt{\overline{c}^2 - 4dr})$ such that for $\lambda_u \in [ \overline{c}/2, \infty)$, the pair of spreading speeds $(\overline{c}, \underline{c})$ can be realized by solutions of \eqref{eq:1-1} with initial data satisfying $\mathrm{(H_\lambda)}$. \item[\rm (c)] If $\underline{c} >f(\overline{c})$, then there exists a unique pair $(\lambda^+_v, \lambda_u)$ such that the pair of spreading speeds $(\overline{c}, \underline{c})$ can be realized by solutions of \eqref{eq:1-1} with initial data satisfying $\mathrm{(H_\lambda)}$. \end{itemize} \end{proposition} \begin{proof} Assertion (a) follows directly from \cite[Theorem 1.2]{Girardin_2018}. For assertion (b), $\underline{c} > \hat{c}_{\rm LLW} \geq 2\sqrt{1-a}$, so that we have $\overline{c} \leq 2(\sqrt{a} + \sqrt{1-a})$. Hence it follows directly from \eqref{eq:hcacc'}. It remains to show (c). First, we define $\lambda_v^+=\frac{\overline{c}-\sqrt{\overline{c}^2-4dr}}{2d}\in(0,\sqrt{\frac{r}{d}})$ such that $\overline{c}=\sigma_1=d\lambda_v^++\frac{r}{\lambda_v^+}$. Since $\sigma_1$ is strictly monotone in $(0,\sqrt{\frac{r}{d}})$, the choice of such $\lambda_v^+$ is unique. Then we shall determine $\lambda_u$ such that $c_2=\underline{c}=\tilde{\lambda}_{\rm{nlp}}+\frac{1-a}{\tilde{\lambda}_{\rm{nlp}}}.$ Since $\underline{c}>f(\overline{c})\geq 2\sqrt{1-a}$ and $\underline{c}>\hat{c}_{\rm{LLW}}$, to satisfy $c_2=\underline{c}$, by \eqref{eq:hcacc'} we must have $\lambda_u\in(\frac{\overline{c}-\sqrt{\overline{c}^2-4a}}{2}, \frac{\overline{c}}{2}) $ and \begin{equation}\label{condition} \underline c=g(\overline{c}) \,\,\text{ and }\,\, \tilde\lambda_{\rm nlp} = \frac{1}{2}\left[\overline{c} - \sqrt{(\overline{c}-2\lambda_u)^2 + 4a}\right]<\sqrt{1-a}. \end{equation} Hence, it suffices to choose the unique $\lambda_u\in(\frac{\overline{c}-\sqrt{\overline{c}^2-4a}}{2}, \frac{\overline{c}}{2}) $ such that \eqref{condition} holds. \begin{itemize} \item [{\rm(i)}] If $\overline{c}\leq2(\sqrt{1-a}+\sqrt{a})$, then observe that when $\lambda_u\in(\frac{\overline{c}-\sqrt{\overline{c}^2-4a}}{2}, \frac{\overline{c}}{2}) $, $ \tilde\lambda_{\rm nlp} \in \left(0,\overline{c}/{2}-\sqrt{a}\right) $ is increasing in $\lambda_u$, so that \begin{equation} g(\overline{c})=\tilde\lambda_{\rm nlp}+\frac{1-a}{\tilde\lambda_{\rm nlp}}\in \left(f(\overline{c}),\overline c \right), \end{equation} is decreasing in $\lambda_u$. Noting that $\underline{c}\in\left(f(\overline{c}),\overline c\right)$, we may select the unique $\lambda_u\in(\frac{\overline{c}-\sqrt{\overline{c}^2-4a}}{2}, \frac{\overline{c}}{2})$ such that \eqref{condition} holds; \item [{\rm(ii)}] If $\overline{c}>2(\sqrt{1-a}+\sqrt{a})$, then to satisfy $\tilde\lambda_{\rm nlp} <\sqrt{1-a}$ in \eqref{condition}, it is necessary that $\lambda_u\in(\frac{\overline{c}-\sqrt{\overline{c}^2-4a}}{2}, \frac{\overline{c}-\sqrt{(\overline{c}-2\sqrt{1-a})^2-4a}}{2})$. In this case, \begin{equation} \tilde\lambda_{\rm nlp} \in \left(0,\sqrt{1-a}\right) \,\text{ and thus }\, g(\overline{c})=\tilde\lambda_{\rm nlp}+\frac{1-a}{\tilde\lambda_{\rm nlp}}\in \left(2\sqrt{1-a},\overline c\right), \end{equation} are also strictly monotone in $\lambda_u$, so that there is the unique $\lambda_u$ such that \eqref{condition} holds. \end{itemize} The proof is now complete. \end{proof} \begin{figure} \caption{ The profile of $c_2(\sigma_1)$ for case {\rm (a)} $g_{\infty}\leq\hat c_{\rm LLW}$, which is expressed by the solid line with the blue one representing $f$ and the red one representing $g$. } \label{figure2} \end{figure} \begin{comment} \begin{remark} In fact, one can show that $\hat{c}_{\rm nlp}$ is the asymptotic spreading speed of the problem \begin{equation}\label{eq:single_u} \left\{\begin{array}{ll} \partial_t u = \partial_{xx} u + u( 1 - a \chi_{\{x < \sigma_1 t\}}-u) = 0 & \hspace{-1cm} \text{ for }(t,x) \in (0,\infty)\times \mathbb{R},\\ \liminf_{x \to -\infty} u(0,x)>0,\,\,\, \text{ and }\,\,\, u(0,x) \sim e^{-\lambda_u x} &\text{ at }\infty. \end{array} \right. \end{equation} Then for each $\epsilon>0$, the solution $u(t,x)$ has the following spreading property: $$ \begin{cases} \lim_{t\to\infty} \inf_{x < ({c}_{\rm nlp}-\epsilon)t} u(t,x)>0,\\ \lim_{t\to\infty} \sup_{x > ({c}_{\rm nlp}+\epsilon)t} u(t,x) =0. \end{cases} \text{ if }\,{\rm (H1)}\text{ and }\, \sigma_1>2\,\text{ hold,} $$ and that $$ \begin{cases} \lim_{t\to\infty} \inf_{x < (\hat{c}_{\rm nlp}-\epsilon)t} u(t,x)>0,\\ \lim_{t\to\infty} \sup_{x > (\hat{c}_{\rm nlp}+\epsilon)t} u(t,x) =0. \end{cases} $$ \end{remark} \end{comment} \section{An extension}\label{S6} In this section, we consider the following competition system with forcing: \begin{equation}\label{eq:extend} \left\{ \begin{array}{ll} \partial_t u-\partial_{xx}u=u(1-u-av-h(t,x)),& \text{ in }(0,\infty)\times \mathbb{R},\\ \partial _t v-d\partial_{xx}v=r v(1-bu-v-k(t,x)),& \text{ in }(0,\infty)\times \mathbb{R},\\ u(0,x)=u_0(x), & \text { on } \mathbb{R},\\ v(0,x)=v_0(x), & \text { on } \mathbb{R}, \end{array} \right . \end{equation} where \begin{equation}\label{eq:hk} \lim_{t\to\infty}\sup_{x\geq c_0 t}(\|h(t,x)\|+\|k(t,x)\|)=0 \quad \text{ for some }c_0 \in \mathbb{R}. \end{equation} We will make an observation in preparation for our forthcoming work on three-species competition systems. Recall the definitions of $\sigma_i$ ($i=1,2,3$) from \eqref{eq:sigma}. \begin{theorem} Let $d,r,b>0$, $0<a<1$ and $\sigma_1>\sigma_2$. Suppose that $h(x,t),k(x,t)$ are non-negative and satisfy \eqref{eq:hk}. Let $(u,v)$ be the solution of \eqref{eq:extend} with the initial data satisfying $\mathrm{(H_\lambda)}$. Assume $$c_0<\sigma_2' \,\quad \text{ where } \sigma_2'=(\lambda_u\wedge \sqrt{1-a})+\frac{1-a}{\lambda_u\wedge \sqrt{1-a}}.$$ Then, $$\underline{c}_1=\overline{c}_1=\sigma_1,~~\overline{c}_2\leq\max\{c_{\rm LLW}, \hat{c}_{{\rm{nlp}}}\},~~ \underline{c}_2\geq \hat{c}_{{\rm{nlp}}}, $$ where $\underline{c}_i$, $\overline{c}_i$ ($i=1,2$) are defined in \eqref{eq:speeds}. Furthermore, for each small $\eta>0$, \begin{equation}\label{eq:spreadingly'} \begin{cases} \lim\limits_{t\rightarrow \infty} \sup\limits_{ x>(\sigma_{1}+\eta) t} (\|u(t,x)\|+\|v(t,x)\|)=0, \\%&\,\text{ for } \epsilon>0, \\ \lim\limits_{t\rightarrow \infty} \sup\limits_{(\overline{c}_2+\eta) t< x<(\sigma_{1}-\eta) t} (\|u(t,x)\|+\|v(t,x)-1\|)=0, \\% &\,\text{ for }\epsilon\in \left(0,\frac{c_{1}-c_2}{2}\right), \\ \lim\limits_{t\rightarrow \infty} \inf\limits_{(c_0+\eta) t< x<(\underline{c}_2-\eta) t} u(t,x) >0, \end{cases} \end{equation} where $\sigma_1$, $\sigma_2$ are defined in \eqref{eq:sigma} and $c_{\rm LLW}$, $\hat{c}_{{\rm{nlp}}}$ are respectively given in Theorem \ref{thm:LLW} and \ref{thm:1-2}. \end{theorem} \begin{proof} The proof can be mimicked after that of Theorem \ref{thm:1-2}. \noindent{\bf Step 1.} The estimates $\overline{c}_1\leq\sigma_1$ and $\overline c_2\leq\sigma_2$ can be proved by rather similar arguments as in Proposition \ref{prop:1}, and the details are omitted here. \noindent{\bf Step 2.} We show that for each small $\eta>0$, \begin{equation}\label{assump1} \liminf_{t\to \infty} u(t,(\sigma'_2-\eta)t)>0\,\,\text{ and }\,\, \liminf_{t\to \infty} v(t,(\sigma_1-\eta)t)>0. \end{equation} Here, we just show the first one since the proof of the second one is analogous. For the case of $\lambda_u\geq \sqrt{1-a}$, by \eqref{eq:hk} and $c_0<\sigma_2'$, the system \eqref{eq:extend} is approximately equal to \eqref{eq:1-1} in $\{(t,x):x\geq \frac{c_0+\sigma_2'}{2}t, t\geq T\}$ for sufficient large $T$, so that we can deduce \eqref{assump1} by applying the similar arguments in Steps 4 of the proof of \cite[Proposition 2.1]{LLL2019}. It remains to consider the case of $\lambda_u<\sqrt{1-a}$. Fix any $c'\in (\max\{\frac{c_0+\sigma_2'}{2},2\sqrt{1-a}\},\sigma_2')$. It is enough to show that there exist positive constants $\delta, \tilde\lambda_1, \tilde\lambda_2, T$ such that $\tilde\lambda_1< \tilde\lambda_2$ and \begin{equation}\label{eq:compareB} u(t, x + c't) \geq \frac{\delta}{4}\max\left\{ \left[e^{-\tilde{\lambda}_1x}-e^{-\tilde{\lambda}_2x}\right],0\right\} \quad \text{ for }t \geq T, x \geq 0. \end{equation} This implies $\underline{c}_2 \geq c'$ for each $c'\in (\max\{\frac{c_0+\sigma_2'}{2},2\sqrt{1-a}\},\sigma_2')$, i.e., $\underline{c}_2 \geq \sigma_2'$. To this end, choose $\delta_1>0$ small enough so that \begin{equation}\label{eq:delta_small} \tilde\lambda_1:= \frac{1}{2} \left[ c' - \sqrt{(c')^2 - 4(1-a - 2\delta_1)}\right] > \lambda_u. \end{equation} This is possible since $c' < \sigma_2'$ and that $s \mapsto \frac{s - \sqrt{s^2 - 4(1-a)}}{2}$ is monotone, so that $$ \frac{1}{2} \left[ c' - \sqrt{(c')^2 - 4(1-a)}\right] > \frac{1}{2} \left[\sigma_2' - \sqrt{(\sigma_2')^2 - 4(1-a)}\right] =\lambda_u. $$ Next, choose $T>0$ large so that \begin{equation}\label{eq:chooseT} \|h(t,x)\|\leq \delta\, \quad \text{ for } \,t\geq T,\,\, x\geq {c}'t, \end{equation} and then choose $\delta \in (0,\delta_1]$ so that \begin{equation}\label{eq:chooseD} u(T,x) \geq \frac{\delta}{4} e^{-\tilde{\lambda}_1(x- c' T)} \quad \text{ for } \, x\geq {c}'T, \end{equation} where \eqref{eq:chooseT} follows from \eqref{eq:hk} by noting ${c}'>c_0$; and that \eqref{eq:chooseD} holds due to $u(T,x)\sim e^{-\lambda_u x} $ at $\infty$ and $\tilde \lambda_1>\lambda_u$ (see, e.g. \cite[Corollary 1 of Ch. 1]{Volpert_1994}). By the choice of $\tilde\lambda_1 < \tilde\lambda_2$, $\delta, \delta_1, T$, it follows that \begin{equation}\label{eq:underlineu} \underline{u}(t,x):=\max\left\{\frac{\delta}{4} \left[e^{-\tilde{\lambda}_1(x-c't)}-e^{-\tilde{\lambda}_2(x-c't)}\right],0\right\}, \end{equation} is a sub-solution of the KPP-type equation \begin{equation}\label{eq:vlkpp} \partial_t u = \partial_{xx} u+ r u(1-a -h(x,t)- u)\,\quad \text{ in }\Omega, \end{equation} where $\Omega:=\{ (t,x): t \geq T,\,\,x \geq c't\}$. For $\delta \in (0,\delta_1]$ to be specified later, define \begin{equation}\label{eq:underlineu} \underline{u}(t,x):=\max\left\{\frac{\delta}{4} \left[e^{-\tilde{\lambda}_1(x-c't)}-e^{-\tilde{\lambda}_2(x-c't)}\right],0\right\}, \end{equation} where $\tilde\lambda_1$ is given in \eqref{eq:delta_small} and $\tilde\lambda_2 = \frac{1}{2} \left[ c' + \sqrt{(c')^2 - 4(1-a - 2\delta)}\right]$. We will choose $T>0$ and $\delta \in (0,\delta_1]$ so that \begin{equation} \left\{ \begin{array}{ll} \partial_t \underline {u} - \partial_{xx} \underline {u}-\underline {u}(1-a-h(x,t)-\underline {u}) \leq -\underline{u} \left( 2\delta - h(x,t) - \underline{u}\right) \leq 0 & \hspace{-.2cm}\text{ in }\Omega,\\ u(t, c' t) \geq 0 = \underline{u}(t, c't) &\hspace{-.2cm}\text{ for }t \geq T,\\ u(T, x) \geq \frac{\delta}{4} e^{-\tilde{\lambda}_1(x- c' T)} \geq \underline{u}(T, x) &\hspace{-.2cm}\text{ for }x \geq c'T,\\ \end{array}\right. \end{equation} i.e., $u$ and $\underline{u}$ is a pair of super- and sub-solutions of the KPP-type equation \begin{equation}\label{eq:vlkpp} \partial_t u = \partial_{xx} u+ r u(1-a -h(x,t)- u)\,\quad \text{ in }\Omega, \end{equation} where $\Omega:=\{ (t,x): t \geq T,\,\,x \geq c't\}$. Hence, by comparison, \eqref{eq:compareB} holds. To proceed further, as in Section \ref{S3}, based on the scaling \eqref{scaling}, we introduce the WKB ansatz $w_2^\epsilon$, which is given by \begin{equation} w_2^\epsilon(t,x)=-\epsilon\log{u^\epsilon(t,x)}, \end{equation} satisfying the equation: \begin{equation} \begin{split} \begin{cases} \partial_tw_2^\epsilon-\epsilon\partial_{xx} w_2^\epsilon+\| \partial_xw_2^\epsilon\|^2+1-u^\epsilon-av^\epsilon-h^\epsilon=0, & \text{in } (0,\infty)\times\mathbb{R},\\ w_2^\epsilon(0,x)=-\epsilon\log{u^\epsilon(0,x)}, & \mathrm{on} ~\mathbb{R}. \end{cases} \end{split} \end{equation} Here $h^\epsilon(t,x)=h(\frac{t}{\epsilon},\frac{x}{\epsilon})$. By Remark \ref{rmk:w1w20}, we also use the half-relaxed limit method and introduce $w_2^$ and $w_{2,}$. By \eqref{assump1}, $$\liminf_{\epsilon\to 0} u^\epsilon (t,(\sigma'_2-\eta)t)>0,$$ and $u^\epsilon$ is moreover bounded by $1$. We have then, by definitions, that \begin{equation}\label{wstar12} w_2^(t,(\sigma'_2-\eta)t)=w_{2,}(t,(\sigma'_2-\eta)t)=0. \end{equation} \noindent{\bf Step 3.} We prove $\underline{c}_1\geq \sigma_1$. This follows from \eqref{assump1} and definition of $\underline{c}_1$. \noindent{\bf Step 4.} We prove $\underline{c}_2\geq \hat{c}_{{\rm{nlp}}}$. By Step 1 and $h\geq 0$, we have \begin{equation} 0\leq \limsup_{\substack{(t',x')\to (t,x)\\ \epsilon\to 0}} v^\epsilon(t',x')\leq \chi_{\{x \leq \sigma_1 t\}}. \end{equation} In view of $\sigma_2'>c_0$, we choose $0<\eta\ll1$ such that $\sigma'_2-\eta>c_0$. We then use \eqref{eq:hk} to derive that \begin{equation} \begin{array}{l} \lim\limits_{\epsilon \to 0}\sup\limits_{x\geq (\sigma'_2-\eta)t} h^{\epsilon}(t,x)= 0. \end{array} \end{equation} Based on \eqref{wstar12}, similar to Lemma \ref{lem:sub-solutionw1}, we can deduce that $w_2^$ is a viscosity sub-solution of \begin{align} \begin{cases} \min\{\partial_t w+\|\partial_xw\|^2+1-a\chi_{\{x\leq \sigma_1 t\}} ,w\}=0,& \text{for}\,\, x> (\sigma_2' -\eta)t,\\ w(0,x)=\lambda_ux, &\text{for}\,\,x\geq 0,\\ w(t,(\sigma'_2-\eta)t)=0, &\text{for}\,\,t\geq 0. \end{cases} \end{align} We then apply the same arguments developed in Lemmas \ref{lem:underlinec2} by constructing the same super-solutions, to deduce that $\underline{c}_2\geq \hat{c}_{{\rm{nlp}}}$. \noindent{\bf Step 5.} We show $\overline{c}_2\leq\max\{c_{\rm LLW}, \hat{c}_{{\rm{nlp}}}\}$ and \eqref{eq:spreadingly'}. By \eqref{eq:hk} again, similar to Corollary \ref{cor:underlinec1}, we can get $$ \liminf_{\substack{(t',x')\to (t,x)\\ \epsilon\to 0}} v^\epsilon(t',x') \geq \chi_{\{\sigma_2 t<x<\sigma_1t\}},$$ so that we may use \eqref{wstar12} to deduce that $w_{2,}$ is a viscosity super-solution of \begin{align} \begin{cases} \min\{\partial_tw+\|\partial_xw\|^2+1-a\chi_{\{\sigma_2 t<x<\sigma_1t\}},w\}=0&\text{for}\,\, x> (\sigma_2'-\eta)t,\\ w(0,x)=\lambda_ux, & \text{for}\,\,x\geq 0,\\ w(t,(\sigma'_2-\eta)t)=0, &\text{for}\,\,t\geq 0, \end{cases} \end{align} as in Lemma \ref{lem:supsolutionw2}. Then we can get $\overline{c}_2\leq\max\{c_{\rm LLW}, \hat{c}_{{\rm{nlp}}}\}$ by the same arguments developed in Proposition \ref{prop:overlinec2}. We finally deduce \eqref{eq:spreadingly'} by similar arguments as in the proof of Theorem \ref{thm:1-2}, which completes the proof. \end{proof} \begin{appendices} \section{Comparison principle}\label{SD} This section is devoted to the proof of a comparison lemma for Hamilton-Jacobi equation for discontinuous super and sub-solutions and for piecewise Lipschitz continuous Hamiltonian. Our proof is inspired by the arguments developed by Ishii \cite{Ishii_1997} and Tourin \cite{Tourin_1992} (see also \cite{Alvarez_1997, Chen_2008,Giga_2010}). Ishii used a crucial observation of \cite{Alvarez_1997} to prove the comparison principle for discontinuous super- and sub-soultion of Hamilton-Jacobi equations with nonconvex but continuous Hamiltonian, whereas Tourin gave the uniqueness of continuous solution of Hamilton-Jacobi equations with piecewise Lipschitz continuous Hamiltonian. The uniqueness of viscosity solution for nonlinear first-order partial differential equations was first introduced by Crandall and Lions in \cite{Crandall_1983}, then Crandall, Ishii and Lions \cite{Crandall_1987} gave a simpler proof. Ishii \cite{Ishii_1985} study the discontinuous Hamiltonian with time measure and Tourin and Ostrov\cite{Ostrov_2002} studied the piecewise Lipschitz continuous, convex Hamiltonian, based on the dynamic programming principle. Let $\Omega$ be a smooth domain in $(0,T] \times \mathbb{R}^N$, which is allowed to be unbounded or even equal to $(0,T]\times\mathbb{R}^N$. We assume without loss that $T = \sup\{t>0:~(t,x) \in \Omega\}$, and define the parabolic boundary of $\Omega$ as $$ \partial_p \Omega = \{(t,x) \in \partial\Omega:~ t \in [0,T)\}. $$ Consider the following Hamilton-Jacobi equation: \begin{equation}\label{eq:D1} \min\{\partial_t w + H(t,x, \partial_x w),w - Lt \} = 0 \quad \text{ in }\Omega. \end{equation} Let $H^$ and $H_$ be, respectively, the upper semicontinuous (usc) and lower semicontinuous (lsc) envelope of $H$ with respect to its first two variables. Precisely, $$H^(t,x,p)=\limsup_{(t',x')\to (t,x)}H(t',x',p)\quad \text{ and } \quad H_(t,x,p)=\liminf_{(t',x')\to (t,x)}H(t',x',p).$$ We say that a lower semicontinuous (lsc) function $w$ is a viscosity super-solution of \eqref{eq:D1} if $w-Lt\geq 0$ in $ \Omega$, and for all test functions $\varphi\in C^\infty(\Omega)$, if $(t_0,x_0)\in \Omega$ is a strict local minimum point of $w-\varphi$, then $$\partial_t\varphi(t_0,x_0)+H^(t_0,x_0,\partial_x\varphi(t_0,x_0))\geq 0$$ holds; A upper semicontinuous (usc) function $w$ is a viscosity sub-solution of \eqref{eq:D1} if for all test functions $\varphi\in C^\infty(\Omega)$, if $(t_0,x_0)\in \Omega$ is a strict local maximum point of $w-\varphi$ such that $w(t_0,x_0) - Lt_0>0$, then $$\partial_t\varphi(t_0,x_0)+H_(t_0,x_0,\partial_x\varphi(t_0,x_0))\leq 0$$ holds. Finally, $w$ is a viscosity solution of \eqref{eq:D1} if and only if $w$ is simultaneously a viscosity super-solution and a viscosity sub-solution of \eqref{eq:D1}. We impose additional assumptions on the domain $\Omega$ and the Hamiltonian $H:\Omega\times\mathbb{R}^N\rightarrow \mathbb{R}$. Namely, there exists a closed set $\Gamma \subset [0,T]\times \mathbb{R}^N$ and, for each $R>0$, a continuous function $\omega_R: [0,\infty) \to [0,\infty)$ such that $\omega_R(0) =0$ and $\omega_R(r)>0$ for $r>0$, such that the following holds: \begin{description} \item[\rm{(A1)}] $H\in C((\Omega \setminus \Gamma)\times \mathbb{R}^N)$; \item[\rm{(A2)}] For each $(t_0,x_0) \in (\Omega \setminus \Gamma) \cap ((0,T) \times B_R(0))$, there exist a constant $\delta_0=\delta_0(R)>0$ such that $$ H(t,y,p) - H(t,x,p) \leq \omega_R\left( \|x-y\|( 1 + \|p\|)\right) $$ for $t,x,y,p$ such that $\\|(t,x) - (t_0,x_0)\\| +\\|(t,y) - (t_0,x_0)\\| < \delta_0$ and $p \in \mathbb{R}^N$; \item[\rm{(A3)}] For each $(t_0,x_0) \in \Omega \cap \Gamma \cap ((0,T) \times B_R(0))$, there exist a constant $\delta_0=\delta_0(R)>0$ and a unit vector $(h_0, k_0) \in \mathbb{R} \times \mathbb{R}^N$ such that $$ H^(s,y,p) - H_(t,x,p)\leq \omega_R\left( (\|t-s\| + \|x-y\|)(1 + \|p\|) \right) $$ for all $p \in \mathbb{R}^N$ and $s, t, y, x$ satisfying $$ \left\{ \begin{array}{ll} \\|(t,x) - (t_0,x_0)\\|+ \\|(s,y) - (t_0,x_0)\\| < \delta_0, \\ {\small \left\\| \frac{(t-s,x-y)}{\\|(t-s, x-y)\\|} - (h_0,k_0)\right\\| < \delta_0}; \end{array}\right. $$ \item[\rm{(A4)}] There exists some $M \geq 0$ such that for each $\lambda \in [0,1)$ and $x_0 \in \mathbb{R}^N$, there exists constants $\bar\epsilon(\lambda,x_0)>0$ and $\bar C(\lambda,x_0) >0$ such that for all $p \in \mathbb{R}^N$, $(t,x) \in \Omega$, if $\epsilon \in [0, \bar\epsilon(\lambda,x_0)]$, then $$ H\left(t,x,\lambda p - \frac{\epsilon (x-x_0)}{\|x-x_0\|^2 + 1}\right) - M \leq \lambda\left( H(t,x,p) - M\right) + \epsilon \bar C(\lambda,x_0). $$ \end{description} \begin{theorem}\label{thm:D} Suppose that $H$ satisfies the hypotheses $\rm{(A1)}$-$\rm{(A4)}$. Let $\overline{w}$ and $\underline{w}$ be a pair of super- and sub-solutions of \eqref{eq:D1} such that $\overline{w} \geq \underline{w}$ on $\partial_p \Omega$, then $$ \overline{w} \geq \underline{w} \quad \text{ in } \Omega. $$ \end{theorem} \begin{remark} Let $$ H(t,x,p) = \textbf{H}(p) + R(x/t), $$ where $\textbf{H}$ is convex and coercive in $p$, and $s\mapsto R(s)$ has bounded variation and satisfies $\|R(s)\| \leq M$ for some $M\geq 0$. Then it is easy to verify that the hypotheses {\rm(A1)-(A4)} hold. In particular, it applies for all our purposes in this paper. Our condition {\rm(A3)} is a quantitative version of the ``local monotonicity condition'' that was introduced in \cite{Chen_2008}. See \cite{Ishii_1997,Chen_2008} for more examples of Hamiltonians verifying the hypotheses {\rm(A1)}-{\rm(A4)}. \end{remark} \begin{proof} Assume to the contrary that \begin{align}\label{eq:sigma'} \sup\limits_\Omega(\underline w-\overline w)>0. \end{align} \noindent{\bf Step 1.} We may assume, without loss of generality, that $M=0$ in the hypothesis {\rm(A4)}. Indeed, if we make the change of variables $\underline{w}'(t,x) = \underline{w}(t,x) + Mt$ and $\overline{w}'(t,x) = \overline{w}(t,x)+Mt$, then $\underline{w}', \overline{w}'$ are, respectively, a sub-solution and a super-solution of \eqref{eq:D1} with $L$ replaced by ${L}'=L+M$, and $H(t,x,p)$ replaced by $H'(t,x,p)= H(t,x,p) - M$. This function $H'$ satisfies the hypotheses {\rm(A1)}-{\rm(A4)} with $M=0$. Henceforth in the proof we assume that the hypothesis {\rm(A4)} holds with $M=0$. \noindent{\bf Step 2.} It suffices to show that $\underline{w} \leq \overline{w}$ under the additional assumption that $\underline{w} \leq K$ for some $K>0$. Indeed, if $\underline{w}$ is unbounded in $\Omega$, then fix a constant $K>0$ and take a sequence $\{g_j\}$ of smooth functions satisfying $ g_j(r) \nearrow \min\{r,K\}$ and $$ 0 \leq g_j'(r) \leq 1, \quad g_j'(r)r \leq r, \quad g_j(r) \leq \min\{r,K\}\,\quad \text{ for all }r \in \mathbb{R}. $$ Then $\hat{w}:= g_j(\underline w)$ is a viscosity sub-solution of \eqref{eq:D1}, since in the region $\{(t,x):~\hat{w} - Lt >0\} \subset\{(t,x):~\underline{w} - Lt >0\}$, we may use the hypothesis {\rm(A4)} to yield \begin{equation} \begin{split} \quad\partial_t \hat w+H^(t,x, D\hat w)&=g_k'(\underline w)\partial_t \underline w+H^(t,x, g_k'(\underline w)D\underline w)\\ &\leq g_k'(\underline w) \left[\partial_t \underline w+H^(t,x,D\underline w)\right]\leq 0. \end{split} \end{equation} By the stability of property of viscosity super and sub-solutions \cite[Theorem 6.2]{BarlesLect}, we may let $j \to \infty$ to conclude that $\min\{\underline w,K\}$ is a viscosity sub-solution of \eqref{eq:D1} for each $K>0$. It now remains to prove Theorem \ref{thm:D} for all bounded above viscosity sub-solutions, since then $$\min\{\underline w, K\} \leq \overline w \quad \text{ for all } K>0\quad \Rightarrow\quad \underline{w} \leq \overline{w}.$$ For $\lambda,\delta\in(0,1)$, denote $$W(t,x)=\lambda^2\underline w(t,x)-\overline w(t,x)-\delta(\psi(x)+Ct+\frac{1}{T-t})-\lambda\delta Ct,$$ where $\psi(x) = \frac{1}{2} \log (\|x\|^2 + 1)$ and $C = \bar C(\lambda,0)$ as in the hypothesis {\rm (A4)}. \noindent{\bf Step 3.} We choose $\lambda \nearrow 1$, $\delta \in (0, \bar \epsilon(\lambda,0)]$, $R>0$ and $(t_0,x_0) \in \Omega_R:= \Omega \cap [(0,T) \times B_R(0)]$ such that \begin{equation}\label{eq:sigma11} W(t_0,x_0)=\max\limits_{\Omega_R}{W(t,x)}=\max\limits_{\Omega}{W(t,x)}>0. \end{equation} From \eqref{eq:sigma'} and Step 3, we may fix $\lambda\nearrow 1$ and $\delta\searrow0 $ such that \begin{equation} \sup_{\Omega}W(t,x)>0,\quad\text{ and }\quad W(t,x)\leq-\frac{\delta}{T-t} \quad \text{ on }{\partial_p\Omega}. \end{equation} Since $\psi(R)\to \infty$ as $R\to\infty$ and $\underline w - \overline{w} \leq K$, we arrive at $$\sup_{(t,x) \in \Omega:~\|x\|=R}W(t,x)\to -\infty \quad \text{ as } R\to \infty,$$ whence we may fix $R\gg1$ so that $\max\limits_{\Omega_R}{W(t,x)}=\max\limits_{\Omega}{W(t,x)}>0$ holds. It remains to observe that the maximum $(t_0,x_0)$ in $\overline{\Omega_R}$ is attained in the interior, since $W(t,x) <0$ when $t = T$ or when $(t,x) \in \partial_p \Omega$. \noindent {\bf Step 4.} With $x_0$ as being given in Step 3, fix $\epsilon>0$ small enough so that \begin{equation}\label{eq:ssss} \epsilon \bar C(\lambda, x_0) \leq \bar C(\lambda,0) \quad \text{ and }\quad \delta\epsilon \leq \bar \epsilon(\lambda,x_0), \end{equation} and define \begin{equation}\label{eq:Wprime} \tilde{W}(t,x) := W(t,x) - \delta\lambda\epsilon \psi(x-x_0) - \frac{1}{2}\|t-t_0\|^2, \end{equation} where $\psi(x)=\frac{1}{2} \log{(\|x\|^2+1)}$ and $C = \bar C(\lambda,0)$ is as before. Then, $(t_0,x_0)$ is a strict global maximum of $\tilde{W}(t,x)$. Define also \begin{align} \begin{split} \Psi_{\alpha,\beta}(t,x,s,y) =& \lambda^2 \underline w(t,x) -\overline w(s,y)-\delta(\psi (x)+Ct+\frac{1}{T-t})-\lambda\delta(\epsilon \psi (x-x_0)+Ct)\\ &-\frac{\alpha}{2}\|x-y\|^2- \frac{\beta}{2}\|t-s\|^2-\frac{1}{2}\|t-t_0\|^2. \end{split} \end{align} \noindent {\bf Step 5.} We claim that there exists $\underline\alpha>0$ such that if $\min\{\alpha,\beta\} \geq \underline\alpha$, then \begin{itemize} \item[{\rm (i)}] $\Psi_{\alpha,\beta}$ has a local maximum point $(t_1,x_1,s_1,y_1)$ in $\Omega_R\times \Omega_R$; \item[{\rm (ii)}] $\Psi_{\alpha,\beta}(t_1,x_1,s_1,y_1) \geq \tilde W(t_0,x_0) = W(t_0,x_0) >0$; \item[{\rm (iii)}] $\beta\|t_1-s_1\|^2 + \alpha \|x_1-y_1\|^2\to 0, \text{ as } \min\{\alpha,\beta\} \to \infty$; \item[{\rm (iv)}] $(t_1,x_1) \to (t_0,x_0)$ and $(s_1,y_1) \to (t_0,x_0) \text{ as } \min\{\alpha,\beta\} \to \infty$, \end{itemize} where $\Omega_R=\Omega\cap[(0,T)\times B_R(0)]$. Since $\overline{w} \geq 0$ and $\underline{w} \leq K$ by Step 2, we see that $\sup_{\Omega_R \times \Omega_R} \Psi_{\alpha,\beta} \leq K$ independently of $\alpha$ and $\beta$, and has a maximum point $(t_1,x_1,s_1,y_1) \in \overline{\Omega}_R \times \overline{\Omega}_R$. Now, by \eqref{eq:sigma11}, $$ \Psi_{\alpha,\beta}(t_1,x_1,s_1,y_1) \geq \max_{\Omega_R}\Psi_{\alpha,\beta}(t,x,t,x) = \tilde{W}(t_0,x_0) = W(t_0,x_0). $$ This proves assertion (ii). Furthermore, the boundedness also yields $\beta\|t_1-s_1\|^2 + \alpha\|x_1 - y_1\|^2 = O(1)$. We claim that $(t_1,x_1) \to (t_0,x_0)$ and $(s_1,y_1) \to (t_0,x_0)$. Indeed, we may pass to a subsequence to get $(\hat t, \hat x)$ such that $(t_1,x_1) \to (\hat t, \hat x)$ and $(s_1,y_1) \to (\hat t, \hat x)$ as $\min\{\alpha,\beta\} \to \infty$. Now, by (ii) we can write \begin{align} \frac{\alpha}{2}\|x_1-y_1\|^2 + \frac{\beta}{2}\|t_1-s_1\|^2 \leq - \tilde{W}(t_0,x_0) +(\tilde{W}(t_1,x_1) + \overline{w}(t_1,x_1)) - \overline{w}(s_1,y_1) . \end{align} Letting $\min\{\alpha,\beta\} \to \infty$, then $(t_1,x_1,s_1,y_1) \to (\hat t, \hat x, \hat t, \hat x)$. Using the fact that $\tilde{W}(t,x) + \overline{w}(t,x)$ (which is essentially $\lambda^2 \underline{w}(t,x)$ up to addition of continuous functions) and $-\overline{w}(s,y)$ are both upper semi-continuous in $\Omega$, we may take limsup as $\min\{\alpha,\beta\} \to \infty$ and deduce that $$ 0 \leq \limsup \left[\frac{\alpha}{2}\|x_1-y_1\|^2 + \frac{\beta}{2}\|t_1-s_1\|^2\right] \leq - \tilde{W}(t_0,x_0) + \tilde{W}(\hat t, \hat x) \leq 0. $$ Since $(t_0,x_0)$ is a strict maximum point of $\tilde{W}$, we must have $(\hat t, \hat x) = (t_0,x_0)$. This proves assertions (iii) and (iv). Finally, $(t_1,x_1, s_1,y_1) \to (t_0,x_0,t_0,x_0)$ and hence must be an interior point of $\Omega_R \times \Omega_R$ when $\min\{\alpha,\beta\}$ is sufficiently large. This proves (i). \noindent {\bf Step 6. } We show the following inequality: \begin{equation}\label{eq:maini} \frac{\delta}{T^2} \leq H^(s_1,y_1,\alpha (x_1-y_1))-H_(t_1,x_1, \alpha (x_1-y_1))+\|t_1-t_0\|. \end{equation} Observe that $(t_1,x_1)$ is an interior maximum point of the function $\underline w(t,x)-\varphi(t,x)$, where \begin{equation} \begin{split} \varphi(t,x)=& \frac{1}{\lambda^2}[\overline w(s_1,y_1)+\delta(\psi(x)+Ct+\frac{1}{T-t})+\lambda\delta(\epsilon\psi(x-x_0)+Ct)\\ &+\frac{\alpha}{2}\|x-y_1\|^2+ \frac{\beta}{2}\|t-s_1\|^2 +\frac{1}{2}\|t-t_0\|^2 ]. \end{split} \end{equation} Also $\underline w(t_1,x_1)>0$, which is a consequence of $\overline w(s_1,y_1) \geq 0$ and $\Psi_\alpha(t_1,x_1,s_1,y_1) >0$. By definition of $\underline w$ being a viscosity sub-solution of \eqref{eq:D1}, we have \begin{equation} \begin{split} &\quad \frac{1}{\lambda^2}\left[\delta (C+ \frac{1}{(T-t_1)^2}+\lambda C)+\beta (t_1-s_1)+(t_1-t_0) \right]\\ &+H_\left(t_1,x_1, \frac{1}{\lambda^2}\left(\delta D_x \psi(x_1)+\lambda\delta \epsilon D_x \psi(x_1-x_0)+\alpha(x_1-y_1)\right)\right)\leq 0, \end{split} \end{equation} which can be rewritten as \begin{equation}\label{eq:Dsub-solutionw'} \begin{split} &\quad \delta (C+\frac{1}{T^2}+\lambda C)+\beta (t_1-s_1)+(t_1-t_0) \\ &\qquad \qquad +\lambda^2H_\left(t_1,x_1, \frac{1}{\lambda}\left(\delta\epsilon D_x \psi(x_1-x_0)+ \hat{q}\right)\right)\leq 0, \end{split} \end{equation} where $\hat q = \frac{1}{\lambda}\left(\delta D_x \psi(x_1)+\alpha(x_1-y_1)\right)$. In the view of $D\psi(x_1-x_0)=\frac{x_1-x_0}{\|x_1-x_0\|^2+1}$, we may apply the hypothesis \rm{(A4)} to get \begin{align} &\quad - \delta (C+\frac{1}{T^2}+\lambda C)-\beta (t_1-s_1)-(t_1-t_0) \\ & \geq \lambda \left[ H_\left(t_1,x_1, \hat q \right) - \delta\epsilon \bar C(\lambda,x_0) \right] \\ & \geq \lambda H_\left(t_1,x_1, \frac{1}{\lambda}\left(\delta D_x \psi(x_1)+\alpha(x_1-y_1)\right) \right) - \lambda\delta C, \end{align} where we used $\epsilon \bar C(\lambda,x_0) \leq \bar C(\lambda,0) = C$ (due to \eqref{eq:ssss}) in the last inequality. Applying the hypothesis {\rm(A4)} once more, we have \begin{align} &\quad - \delta (C+\frac{1}{T^2}+\lambda C)-\beta (t_1-s_1)-(t_1-t_0) \\ &\geq \left[H_\left(t_1,x_1, \alpha(x_1-y_1) \right) - \delta C\right] - \lambda\delta C\\ &\geq H_\left(t_1,x_1, \alpha(x_1-y_1) \right) - \delta C - \lambda\delta C, \end{align} and hence \begin{equation}\label{eq:Dsub-solutionw''} \frac{\delta}{T^2} +\beta (t_1-s_1)+(t_1-t_0)+H_(t_1,x_1, \alpha(x_1-y_1))\leq 0. \end{equation} In the same way, $(s_1,y_1)$ is a interior minimum point of the function $\overline w(s,y)-\psi(s,y)$ with \begin{equation} \begin{split} \psi(s,y)=&\lambda^2 \underline w(t_1,x_1)- \delta(\psi(x_1)+Ct_1+\frac{1}{T-t})-\lambda\delta(\psi(x_1-x_0)+Ct_1)\\ &-\frac{\alpha}{2}\|x_1-y\|^2-\frac{\beta}{2}\|t_1-s\|^2)-\frac{1}{2}\|t_1-t_0\|^2, \end{split} \end{equation} whence \begin{align}\label{eq:Dsuper-solutionw} \beta(t_1-s_1)+H^(s_1,y_1,\alpha (x_1-y_1))\geq 0. \end{align} Subtracting \eqref{eq:Dsub-solutionw''} from \eqref{eq:Dsuper-solutionw}, we obtain \eqref {eq:maini} as claimed. By Step 5 \rm{(iv)}, we have $(t_1,x_1)\to (t_0,x_0)$ and $(s_1,y_1)\to (t_0,x_0)$ as $\min\{\alpha,\beta\}\to \infty$. On the one hand, if $(t_0,x_0)\notin \Gamma$, then there exists $\alpha_1>0$ such that $(t_1,x_1)$ and $(s_1,y_1)$ enter the $(\delta_0/2)$-neighborhood of $(t_0,x_0)$ whenever $\min\{\alpha,\beta\} \geq \alpha_1$. Now, fix $\alpha$ and let $\beta \to \infty$, then after passing to a sequence, we have $$ t_1, s_1 \to \bar t, \quad x_1 \to \bar x, \quad y_1 \to \bar y. $$ Furthermore, by Step 5, we have \begin{equation}\label{eq:txbar} \bar t \to t_0,\quad \bar x \to x_0, \quad \bar y \to x_0,\quad \text{ and }\quad {\alpha}\|\bar x - \bar y\|^2 \to 0 \quad \text{ as }\quad \alpha \to \infty. \end{equation} Hence, we deduce from \eqref{eq:maini} and the hypothesis \rm{(A2)} that \begin{align} \frac{\delta}{T^2} &\leq H^(\bar t,\bar y,\alpha (\bar x - \bar y))-H_(\bar t,\bar x, \alpha (\bar x - \bar y))+\|\bar t-t_0\|\\ &\leq \omega_R\left( \alpha \|\bar x - \bar y\|^2 + \frac{1}{\alpha}\right) + o(1), \end{align} from which we derive a contradiction for large enough $\alpha$. This proves Theorem \ref{thm:D} in case $(t_0,x_0) \in \Omega \setminus \Gamma$. On the other hand, $(t_0,x_0)\in \Gamma$. Let $\delta_0$ and the unit vector $(h_0,k_0) \in \mathbb{R} \times \mathbb{R}^N$ be given by the hypothesis {\rm(A3)}. Define \begin{equation}\label{eq:Psi'} \begin{split} \tilde{\Psi}_{\alpha,\beta}(t,x,s,y) = &\lambda^2 \underline w(t,x) -\overline w(s-\alpha^{-1/2}h_0,y-\alpha^{-1/2}k_0) \\ &-\delta(\psi (x)+Ct+\frac{1}{T-t})-\lambda\delta(\epsilon \psi (x-x_0)+Ct)\\ &-\frac{\alpha}{2}\|x-y\|^2- \frac{\beta}{2}\|t-s\|^2-\frac{1}{2}\|t-t_0\|^2.\end{split} \end{equation} By repeating Steps 5 and 6, we can again obtain \begin{equation}\label{eq:sss} \frac{\delta}{T^2} \leq H^(\bar t -\alpha^{-1/2}h_0,\bar y -\alpha^{-1/2}k_0,\alpha (\bar x - \bar y))-H_(\bar t, \bar x, \alpha (\bar x - \bar y))+\|\bar t-t_0\| \end{equation} for some $\bar t, \bar x, \bar y$ such that for all $\alpha$ large, $(\bar t, \bar x), (\bar t, \bar y)$ enter the $(\delta_0/2)$-neighborhood of $(t_0,x_0)$ and \eqref{eq:txbar} holds. Furthermore, by verifying that \begin{align} \frac{((\bar t - \alpha^{-1/2}h_0) - \bar t, \bar y - \alpha^{-1/2}k_0 - \bar x)}{\\|((\bar t - \alpha^{-1/2}h_0) - \bar t, \bar y - \alpha^{-1/2}k_0 - \bar x)\\|}&= -\frac{(\alpha^{-1/2}h_0, \bar x - \bar y + \alpha^{-1/2}k_0 )}{\\|(\alpha^{-1/2}h_0, \bar x - \bar y + \alpha^{-1/2}k_0 )\\|}\\ &= -\frac{(h_0, \sqrt{\alpha}(\bar x - \bar y) + k_0 )}{\\|(h_0,\sqrt{\alpha}(\bar x - \bar y) + k_0 )\\|}\\ & \to (h_0,k_0) \quad \text{ as }\alpha \to \infty, \end{align} we may apply hypothesis {\rm(A3)} to inequality \eqref{eq:sss} to get \begin{align} \frac{\delta}{T^2} &\leq \omega_R\left(\Big[\alpha^{-1/2}(h_0 + \|k_0\|) + \|\bar x - \bar y\|\Big](1+\alpha \|\bar x - \bar y\|)\right) + \|\bar t- t_0\| \\ &\leq \omega_R\left(\|\bar x - \bar y\| + \alpha \|\bar x - \bar y\|^2 + C_0 ( \sqrt\alpha\|\bar x - \bar y\| +\frac{1}{\sqrt{\alpha}})\right) + \|\bar t - t_0\|. \end{align} Letting $\alpha \to \infty$, we can similarly obtain a contradiction. \end{proof} A direct consequence of Theorem \ref{thm:D} is the following uniqueness result. \begin{corollary}\label{cor:uniquD} Assume that the Hamiltonian $H$ satisfies the hypotheses {\rm{(A1)}}-{\rm{(A4)}}. Suppose that \eqref{eq:D1} has a continuous viscosity solution $w$. Then, $w$ is the unique viscosity solution of \eqref{eq:D1} in the class of all (continuous) viscosity solution. \end{corollary} \section{Two useful lemmas from \cite{LLL2019}}\label{SL} In this section, we present two lemmas from \cite{LLL2019}, which are used in this paper. The first result is used to prove Propositions \ref{prop:1} and Corollary \ref{cor:underlinec1}, Lemma \ref{lem:tildeuv} and Proof of Theorem \ref{thm:1-2}. \begin{lemma}[{\cite[Lemma 2.3]{LLL2019}}]\label{lem:entire1} Let $-\infty\leq\underline c<\overline c\leq\infty$, and let $(u,v)$ be a solution of \eqref{eq:1-1} in $\{(t,x): \underline{c} t \leq x \leq \overline{c}t\}$. \begin{itemize} \item[{\rm(a)}]If $\displaystyle \liminf_{t \to \infty} \inf_{(\underline c + \eta) t < x < (\overline c - \eta)t} v(t,x) >0$ for each $0<\eta < (\overline{c}-\underline{c})/2$, then $$\displaystyle \limsup_{t \to \infty} \sup_{(\underline c +\eta) t < x < (\overline c - \eta)t}u(t,x) \leq k_1, \quad \liminf_{t \to \infty} \inf_{(\underline c + \eta) t < x < (\overline c - \eta)t}v(t,x) \geq k_2,$$ for each $0<\eta< (\overline{c}-\underline{c})/2$; \item[{\rm(b)}] If $\displaystyle \lim_{t \to \infty} \sup_{(\underline c + \eta) t < x < (\overline c - \eta) t} v(t,x)=0$ and $\displaystyle \liminf_{t \to \infty} \inf_{(\underline c +\eta) t < x < (\overline c - \eta)t} u(t,x) >0$ for each $0<\eta < (\overline{c}-\underline{c})/2$, then $$\displaystyle \lim_{t \to \infty} \sup_{(\underline c + \eta) t < x < (\overline c -\eta)t}\|u(t,x) -1\|=0, \quad \text{ for each }0 < \eta < (\overline{c}-\underline{c})/2;$$ \item[{\rm(c)}] If $\displaystyle \liminf_{t \to \infty} \inf_{(\underline c + \eta) t < x < (\overline c - \eta)t} u(t,x) >0$ for each $0 < \eta< (\overline{c}-\underline{c})/2$, then $$\displaystyle \liminf_{t \to \infty} \inf_{(\underline c + \eta) t < x < (\overline c - \eta)t}u(t,x) \geq k_1, \quad \limsup_{t \to \infty} \sup_{(\underline c + \eta) t < x < (\overline c - \eta)t}v(t,x) \leq k_2 $$ for each $0<\eta < (\overline{c}-\underline{c})/2$; \item[{\rm(d)}] If $\displaystyle \lim_{t \to \infty} \sup_{(\underline c + \eta) t < x < (\overline c - \eta)t} u(t,x)=0$ and $\displaystyle \liminf_{t \to \infty} \inf_{(\underline c +\eta) t < x < (\overline c - \eta)t} v(t,x) >0$ for each $0 <\eta < (\overline{c}-\underline{c})/2$, then $$\displaystyle \lim_{t \to \infty} \sup_{(\underline c + \eta) t < x < (\overline c - \eta)t }\|v(t,x) -1\|=0, \quad \text{ for each }0 < \eta < (\overline{c}-\underline{c})/2.$$ \end{itemize} \end{lemma} \begin{proof} We only prove {\rm(c)} and the other assertions follow from the similar arguments. Suppose {\rm(c)} is false, then there exists $(t_n,x_n)$ such that \begin{equation} \begin{array}{c} c_n:= \frac{x_n}{t_n} \to c \in (\underline c, \overline c) \,\text{ and }\, \lim\limits_{n\to\infty}u(t_n,x_n)<k_1 \text{ or } \lim\limits_{n\to\infty}v(t_n,x_n)>k_2. \end{array} \end{equation} Define $(u_n,v_n)(t,x):= (u,v)(t_n + t, x_n + x)$. We pass to the limit so that $(u_n,v_n)$ converges in $C_{loc}(\mathbb{R} \times \mathbb{R})$ to an entire solution $(\hat u, \hat v)$ of \eqref{eq:1-1}. And there exists $\delta >0$ such that $ (\hat{u}, \hat{v})(t,x) \succeq (\delta, 1)$ for $(t,x) \in \mathbb{R}^2$. Let $(\underline{U}, \overline{V})$ be the solutions of the Lotka-Volterra system of ODEs $$ U_t = U(1-U - aV), \quad V_t = rV(1-bU-V), $$ with initial data $(U(0), V(0)) = (\delta,1)$, so that $(\underline{U}, \overline{V})(\infty) = (k_1,k_2)$. By comparison in the time interval $[-T,0]$, we reveal that for each $T>0$, $$ (\hat u,\hat v)(t,x)\succeq (\underline{U}, \overline{V})(t+T) \, \text{ for }(t,x) \in [-T,0] \times \mathbb{R}, $$ so that we in particular have, for every $T>0$, $$ (\hat u,\hat v)(0,0)\succeq (\underline{U}, \overline{V})(T). $$ Letting $T \to \infty$, we obtain $(\hat u,\hat v)(0,0) \succeq (k_1,k_2)$. In particular, we deduce that $$ \lim_{n \to \infty} (u,v)(t_n,x_n) = \lim_{n \to \infty} (u_n,v_n)(0,0) = (\hat u, \hat v)(0,0)\succeq (k_1,k_2). $$ This is a contradiction and proves {\rm(c)}. \end{proof} The following result is applied to prove Proposition \ref{prop:overlinec2} and Proposition \ref{prop:underlinec3}. \begin{lemma}[{\cite[Lemma 2.4]{LLL2019}}]\label{lem:appen1} Let $\hat{c}>0$, $t_0>0$, and $(\tilde u,\tilde v)$ be a solution of \begin{equation}\label{eq:A2} \left\{ \begin{array}{ll} \partial_t \tilde u-\partial_{xx}\tilde u=\tilde u(1-\tilde u-a\tilde v),& 0\leq x\leq\hat ct, t>t_0,\\ \partial _t \tilde v-d\partial_{xx}\tilde v=r \tilde v(1-b\tilde u-\tilde v),& 0\leq x\leq\hat ct, t>t_0,\\ \tilde u(t_0,x)=\tilde u_0(x), \tilde v(t_0,x)=\tilde v_0(x), & 0\leq x\leq\hat ct_0. \end{array} \right . \end{equation} \begin{itemize} \item[{\rm (a)}] If $\hat{c}>2$ and there exists $\hat \mu>0$ such that \begin{itemize} \item[{\rm(i)}] $\lim\limits_{t\to\infty}(\tilde u,\tilde v)(t,0)=(k_1,k_2)$ and $\lim\limits_{t\to\infty}(\tilde u,\tilde v)(t,\hat ct)=(0,1)$, \item[{\rm(ii)}] $\lim_{t\to\infty} e^{\mu t}\tilde u(t,\hat ct)=0~$ for each $\mu \in [0,\hat \mu),$ \end{itemize} then $$ \lim_{t\to\infty} \sup_{ct<x\leq \hat{c}t} \tilde u(t,x) = 0 \quad \text{ for each }c > c_{\hat{c},\hat\mu}, $$ where \begin{equation} c_{\hat{c},\hat\mu}=\left\{ \begin{array}{ll} c_{\textup{LLW}},& \text{if } \hat\mu\geq \lambda_{\textup{LLW}} (\hat{c} - c_{\textup{LLW}}),\\ \hat c-\frac{2\hat\mu}{\hat c-\sqrt{\hat c^2-4(\hat\mu+1-a)}},& \text{if }0< \hat\mu< \lambda_{\textup{LLW}}(\hat{c} - c_{\textup{LLW}}); \\ \end{array} \right . \end{equation} \item[{\rm (b)}] If $\hat{c} > 2\sqrt{dr}$ and there exists $\hat\mu >0$ such that \begin{itemize} \item[{\rm(i)}] $\lim\limits_{t\to\infty}(\tilde u,\tilde v)(t,0)=(k_1,k_2)$, and $\lim\limits_{t\to\infty}(\tilde u,\tilde v)(t,\hat ct)=(1,0)$, \item[{\rm(ii)}] $\lim_{t\to\infty} e^{\mu t}\tilde v(t,\hat ct)=0~$ for each $\mu \in [0, \hat \mu),$ \end{itemize} then $$ \lim_{t\to\infty} \sup_{ct<x\leq \hat{c}t} \tilde v(t,x) = 0 \quad \text{ for each }c >\tilde{c}_{\hat{c},\hat\mu}, $$ where \begin{equation} \tilde{c}_{\hat{c},\hat\mu}= \left\{ \begin{array}{ll} \tilde{c}_{\textup{LLW}},& \text{ if } \hat\mu\geq \tilde{\lambda}_{\rm LLW} (\hat{c} - \tilde{c}_{\textup{LLW}}),\\ \hat c-\frac{2d\hat\mu}{\hat c-\sqrt{\hat c^2-4d[\hat\mu+r(1-b)]}},& \text{ if } 0<\hat\mu<\tilde{\lambda}_{\rm LLW} (\hat{c} - \tilde{c}_{\textup{LLW}}). \end{array} \right. \end{equation*} \end{itemize} Here ${c}_{\textup{LLW}}, \tilde{c}_{\textup{LLW}}$ are given in Theorem \ref{thm:LLW} and Remark \ref{rmk:LLW}, and \begin{equation}\label{eq:LLL} \lambda_{\rm LLW}=\frac{{c}_{\textup{LLW}}-\sqrt{{c}_{\textup{LLW}}^2-4(1-a)}}{2}, \quad \tilde{\lambda}_{\rm LLW}=\frac{\tilde{c}_{\textup{LLW}}-\sqrt{\tilde{c}_{\textup{LLW}}^2-4dr(1-b)}}{2d}.\end{equation} \end{lemma} \end{appendices} \begin{center} \end{center} \end{document}	arXiv
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Perkins, Melissa Pesce-Rollins, Prof. Vahe' Petrosian, Frederic Piron, Giovanna Pivato, Troy Porter, S. Rain`o, Riccardo Rando, Massimiliano Razzano, Soebur Razzaque, Anita Reimer, Prof. Olaf Reimer, Matthieu Renaud, Thierry Reposeur, Mr. Romain Rousseau, P. M. Parkinson, J. Schmid, A. Schulz, C. Sgr`o, Eric J Siskind, Francesca Spada, Gloria Spandre, Paolo Spinelli, Andrew W. Strong, Daniel Suson, Hiro Tajima, Hiromitsu Takahashi, T. Tanaka, Jana B. Thayer, D. J. Thompson, L. Tibaldo, Omar Tibolla, Prof. Diego F. Torres, Gino Tosti, Eleonora Troja, Yasunobu Uchiyama, G. Vianello, B. Wells, Kent Wood, M. Wood, Manal Yassine, Stephan Zimmer Nov. 20, 2015 astro-ph.IM, astro-ph.HE To uniformly determine the properties of supernova remnants (SNRs) at high energies, we have developed the first systematic survey at energies from 1 to 100 GeV using data from the Fermi Large Area Telescope. Based on the spatial overlap of sources detected at GeV energies with SNRs known from radio surveys, we classify 30 sources as likely GeV SNRs. We also report 14 marginal associations and 245 flux upper limits. A mock catalog in which the positions of known remnants are scrambled in Galactic longitude, allows us to determine an upper limit of 22% on the number of GeV candidates falsely identified as SNRs. We have also developed a method to estimate spectral and spatial systematic errors arising from the diffuse interstellar emission model, a key component of all Galactic Fermi LAT analyses. By studying remnants uniformly in aggregate, we measure the GeV properties common to these objects and provide a crucial context for the detailed modeling of individual SNRs. Combining our GeV results with multiwavelength (MW) data, including radio, X-ray, and TeV, demonstrates the need for improvements to previously sufficient, simple models describing the GeV and radio emission from these objects. We model the GeV and MW emission from SNRs in aggregate to constrain their maximal contribution to observed Galactic cosmic rays. The 2nd Fermi GBM Gamma-Ray Burst Catalog: The First Four Years (1401.5080) Andreas von Kienlin, Charles A. Meegan, William S. Paciesas, P. N. Bhat, Elisabetta Bissaldi, Michael S. Briggs, J. Michael Burgess, David Byrne, Vandiver Chaplin, William Cleveland, Valerie Connaughton, Andrew C. Collazzi, Gerard Fitzpatrick, Suzanne Foley, Melissa Gibby, Misty Giles, Adam Goldstein, Jochen Greiner, David Gruber, Sylvain Guiriec, Alexander J. van der Horst, Chryssa Kouveliotou, Emily Layden, Sheila McBreen, Sinead McGlynn, Veronique Pelassa, Robert D. Preece, Arne Rau, Dave Tierney, Colleen A. Wilson-Hodge, Shaolin Xiong, George Younes, Hoi-Fung Yu Jan. 24, 2014 astro-ph.HE This is the second of a series of catalogs of gamma-ray bursts (GRBs) observed with the Fermi Gamma-ray Burst Monitor (GBM). It extends the first two-year catalog by two more years, resulting in an overall list of 953 GBM triggered GRBs. The intention of the GBM GRB catalog is to provide information to the community on the most important observables of the GBM detected GRBs. For each GRB the location and main characteristics of the prompt emission, the duration, peak flux and fluence are derived. The latter two quantities are calculated for the 50 - 300 keV energy band, where the maximum energy release of GRBs in the instrument reference system is observed and also for a broader energy band from 10 - 1000 keV, exploiting the full energy range of GBMs low-energy detectors. Furthermore, information is given on the settings and modifications of the triggering criteria and exceptional operational conditions during years three and four in the mission. This second catalog is an official product of the Fermi GBM science team, and the data files containing the complete results are available from the High-Energy Astrophysics Science Archive Research Center (HEASARC). The Fermi GBM Gamma-Ray Burst Spectral Catalog: Four Years Of Data (1401.5069) David Gruber, Adam Goldstein, Victoria Weller von Ahlefeld, P. Narayana Bhat, Elisabetta Bissaldi, Michael S. Briggs, Dave Byrne, William H. Cleveland, Valerie Connaughton, Roland Diehl, Gerald J. Fishman, Gerard Fitzpatrick, Suzanne Foley, Melissa Gibby, Misty M. Giles, Jochen Greiner, Sylvain Guiriec, Alexander J. van der Horst, Andreas von Kienlin, Chryssa Kouveliotou, Emily Layden, Lin Lin, Charles A. Meegan, Sinéad McGlynn, William S. Paciesas, Véronique Pelassa, Robert D. Preece, Arne Rau, Colleen A. 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The data files containing the complete results are available from the High-Energy Astrophysics Science Archive Research Center (HEASARC). Variable jet properties in GRB110721A: Time resolved observations of the jet photosphere (1305.3611) Shabnam Iyyani, Felix Ryde, Magnus Axelsson, James Michael Burgess, Sylvain Guiriec, Josefin Larsson, Christoffer Lundman, Elena Moretti, Sinead McGlynn, Tanja Nymark, Kjell Rosquist May 15, 2013 astro-ph.HE {\it Fermi Gamma-ray Space Telescope} observations of GRB110721A have revealed two emission components from the relativistic jet: emission from the photosphere, peaking at $\sim 100$ keV and a non-thermal component, which peaks at $\sim 1000$ keV. We use the photospheric component to calculate the properties of the relativistic outflow. We find a strong evolution in the flow properties: the Lorentz factor decreases with time during the bursts from $\Gamma \sim 1000$ to $\sim 150$ (assuming a redshift $z=2$; the values are only weakly dependent on unknown efficiency parameters). Such a decrease is contrary to the expectations from the internal shocks and the isolated magnetar birth models. Moreover, the position of the flow nozzle measured from the central engine, $r_0$, increases by more than two orders of magnitude. Assuming a moderately magnetised outflow we estimate that $r_0$ varies from $10^6$ cm to $\sim 10^9$ cm during the burst. We suggest that the maximal value reflects the size of the progenitor core. Finally, we show that these jet properties naturally explain the observed broken power-law decay of the temperature which has been reported as a characteristic for GRB pulses. Anomalies in low-energy Gamma-Ray Burst spectra with the Fermi Gamma-Ray Burst Monitor (1301.4859) Dave Tierney, Sheila McBreen, Robert D. Preece, Gerard Fitzpatrick, Suzanne Foley, Sylvain Guiriec, Elisabetta Bissaldi, Michael S. Briggs, J. Michael Burgess, Valerie Connaughton, Adam Goldstein, Jochen Greiner, David Gruber, Chryssa Kouveliotou, Sinead McGlynn, William S. Paciesas, Veronique Pelassa, Andreas von Kienlin A Band function has become the standard spectral function used to describe the prompt emission spectra of gamma-ray bursts (GRBs). However, deviations from this function have previously been observed in GRBs detected by BATSE and in individual GRBs from the \textit{Fermi} era. We present a systematic and rigorous search for spectral deviations from a Band function at low energies in a sample of the first two years of high fluence, long bursts detected by the \textit{Fermi} Gamma-Ray Burst Monitor (GBM). The sample contains 45 bursts with a fluence greater than 2$\times10^{-5}$ erg / cm$^{2}$ (10 - 1000 keV). An extrapolated fit method is used to search for low-energy spectral anomalies, whereby a Band function is fit above a variable low-energy threshold and then the best fit function is extrapolated to lower energy data. Deviations are quantified by examining residuals derived from the extrapolated function and the data and their significance is determined via comprehensive simulations which account for the instrument response. This method was employed for both time-integrated burst spectra and time-resolved bins defined by a signal to noise ratio of 25 $\sigma$ and 50 $\sigma$. Significant deviations are evident in 3 bursts (GRB\,081215A, GRB\,090424 and GRB\,090902B) in the time-integrated sample ($\sim$ 7%) and 5 bursts (GRB\,090323, GRB\,090424, GRB\,090820, GRB\,090902B and GRB\,090926A) in the time-resolved sample ($\sim$ 11%).} The advantage of the systematic, blind search analysis is that it can demonstrate the requirement for an additional spectral component without any prior knowledge of the nature of that extra component. Deviations are found in a large fraction of high fluence GRBs; fainter GRBs may not have sufficient statistics for deviations to be found using this method. A Universal Scaling for the Energetics of Relativistic Jets From Black Hole Systems (1212.3343) Rodrigo S. Nemmen, Markos Georganopoulos, Sylvain Guiriec, Eileen T. Meyer, Neil Gehrels, Rita M. Sambruna Jan. 3, 2013 astro-ph.CO, astro-ph.HE Black holes generate collimated, relativistic jets which have been observed in gamma-ray bursts (GRBs), microquasars, and at the center of some galaxies (active galactic nuclei; AGN). How jet physics scales from stellar black holes in GRBs to the supermassive ones in AGNs is still unknown. Here we show that jets produced by AGNs and GRBs exhibit the same correlation between the kinetic power carried by accelerated particles and the gamma-ray luminosity, with AGNs and GRBs lying at the low and high-luminosity ends, respectively, of the correlation. This result implies that the efficiency of energy dissipation in jets produced in black hole systems is similar over 10 orders of magnitude in jet power, establishing a physical analogy between AGN and GRBs. Detection of spectral evolution in the bursts emitted during the 2008-2009 active episode of SGR J1550 - 5418 (1206.4915) Andreas von Kienlin, David Gruber, Chryssa Kouveliotou, Jonathan Granot, Matthew G. Baring, Ersin Göğüş, Daniela Huppenkothen, Yuki Kaneko, Lin Lin, Anna L. Watts, P. Narayana Bhat, Sylvain Guiriec, Alexander J. van der Horst, Elisabetta Bissaldi, Jochen Greiner, Charles A. Meegan, William S. Paciesas, Robert D. Preece, Arne Rau In early October 2008, the Soft Gamma Repeater SGRJ1550 - 5418 (1E 1547.0 - 5408, AXJ155052 - 5418, PSR J1550 - 5418) became active, emitting a series of bursts which triggered the Fermi Gamma-ray Burst Monitor (GBM) after which a second especially intense activity period commenced in 2009 January and a third, less active period was detected in 2009 March-April. Here we analyze the GBM data all the bursts from the first and last active episodes. We performed temporal and spectral analysis for all events and found that their temporal characteristics are very similar to the ones of other SGR bursts, as well the ones reported for the bursts of the main episode (average burst durations \sim 170 ms). In addition, we used our sample of bursts to quantify the systematic uncertainties of the GBM location algorithm for soft gamma-ray transients to < 8 deg. Our spectral analysis indicates significant spectral evolution between the first and last set of events. Although the 2008 October events are best fit with a single blackbody function, for the 2009 bursts an Optically Thin Thermal Bremsstrahlung (OTTB) is clearly preferred. We attribute this evolution to changes in the magnetic field topology of the source, possibly due to effects following the very energetic main bursting episode. The Fermi GBM Gamma-Ray Burst Catalog: The First Two Years (1201.3099) William S. Paciesas, Charles A. Meegan, Andreas von Kienlin, P. N. Bhat, Elisabetta Bissaldi, Michael S. Briggs, J. Michael Burgess, Vandiver Chaplin, Valerie Connaughton, Roland Diehl, Gerald J. Fishman, Gerard Fitzpatrick, Suzanne Foley, Melissa Gibby, Misty Giles, Adam Goldstein, Jochen Greiner, David Gruber, Sylvain Guiriec, Alexander J. van der Horst, R. Marc Kippen, Chryssa Kouveliotou, Giselher Lichti, Lin Lin, Sheila McBreen, Robert D. Preece, Arne Rau, Dave Tierney, Colleen Wilson-Hodge The Fermi Gamma-ray Burst Monitor (GBM) is designed to enhance the scientific return from Fermi in studying gamma-ray bursts (GRBs). In its first two years of operation GBM triggered on 491 GRBs. We summarize the criteria used for triggering and quantify the general characteristics of the triggered GRBs, including their locations, durations, peak flux, and fluence. This catalog is an official product of the Fermi GBM science team, and the data files containing the complete results are available from the High-Energy Astrophysics Science Archive Research Center (HEASARC). The Fermi GBM Gamma-Ray Burst Spectral Catalog: The First Two Years (1201.2981) Adam Goldstein, J. Michael Burgess, Robert D. Preece, Michael S. Briggs, Sylvain Guiriec, Alexander J. van der Horst, Valerie Connaughton, Colleen A. Wilson-Hodge, William S. Paciesas, Charles A. Meegan, Andreas von Kienlin, P. Narayana Bhat, Elisabetta Bissaldi, Vandiver Chaplin, Roland Diehl, Gerald J. Fishman, Gerard Fitzpatrick, Suzanne Foley, Melissa Gibby, Misty Giles, Jochen Greiner, David Gruber, R. Marc Kippen, Chryssa Kouveliotou, Sheila McBreen, Sinead McGlynn, Arne Rau, Dave Tierney We present systematic spectral analyses of GRBs detected by the Fermi Gamma-Ray Burst Monitor (GBM) during its first two years of operation. This catalog contains two types of spectra extracted from 487 GRBs, and by fitting four different spectral models, this results in a compendium of over 3800 spectra. The models were selected based on their empirical importance to the spectral shape of many GRBs, and the analysis performed was devised to be as thorough and objective as possible. We describe in detail our procedure and criteria for the analyses, and present the bulk results in the form of parameter distributions. This catalog should be considered an official product from the Fermi GBM Science Team, and the data files containing the complete results are available from the High-Energy Astrophysics Science Archive Research Center (HEASARC). Temporal Deconvolution study of Long and Short Gamma-Ray Burst Light curves (1109.4064) P. N. Bhat, Michael S. Briggs, Valerie Connaughton, Chryssa Kouveliotou, Alexander J. van der Horst, William Paciesas, Charles A. Meegan, Elisabetta Bissaldi, Michael Burgess, Vandiver Chaplin, Roland Diehl, Gerald Fishman, Gerard Fitzpatrick, Suzanne Foley, Melissa Gibby, Misty M. Giles, Adam Goldstein, Jochen Greiner, David Gruber, Sylvain Guiriec, Andreas von Kienlin, Marc Kippen, Sheila McBreen, Robert Preece, Arne Rau, Dave Tierney, Colleen Wilson-Hodge Sept. 19, 2011 hep-ph, astro-ph.HE The light curves of Gamma-Ray Bursts (GRBs) are believed to result from internal shocks reflecting the activity of the GRB central engine. Their temporal deconvolution can reveal potential differences in the properties of the central engines in the two populations of GRBs which are believed to originate from the deaths of massive stars (long) and from mergers of compact objects (short). We present here the results of the temporal analysis of 42 GRBs detected with the Gamma-ray Burst Monitor onboard the Fermi Gamma-ray Space Telescope. We deconvolved the profiles into pulses, which we fit with lognormal functions. The distributions of the pulse shape parameters and intervals between neighboring pulses are distinct for both burst types and also fit with lognormal functions. We have studied the evolution of these parameters in different energy bands and found that they differ between long and short bursts. We discuss the implications of the differences in the temporal properties of long and short bursts within the framework of the internal shock model for GRB prompt emission. Burst and Persistent Emission Properties during the Recent Active Episode of the Anomalous X-ray Pulsar 1E 1841-045 (1109.0991) Lin Lin, Chryssa Kouveliotou, Ersin Gogus, Alexander J. van der Horst, Anna L. Watts, Matthew G. Baring, Yuki Kaneko, Ralph A.M.J. Wijers, Peter M. Woods, Scott Barthelmy, J. Michael Burgess, Vandiver Chaplin, Neil Gehrels, Adam Goldstein, Jonathan Granot, Sylvain Guiriec, Julie Mcenery, Robert D. Preece, David Tierney, Michiel van der Klis, Andreas von Kienlin, Shuang Nan Zhang Sept. 5, 2011 astro-ph.HE Swift/BAT detected the first burst from 1E 1841-045 in May 2010 with intermittent burst activity recorded through at least July 2011. Here we present Swift and Fermi/GBM observations of this burst activity and search for correlated changes to the persistent X-ray emission of the source. The T90 durations of the bursts range between 18-140 ms, comparable to other magnetar burst durations, while the energy released in each burst ranges between (0.8 - 25)E38 erg, which is in the low side of SGR bursts. We find that the bursting activity did not have a significant effect on the persistent flux level of the source. We argue that the mechanism leading to this sporadic burst activity in 1E 1841-045 might not involve large scale restructuring (either crustal or magnetospheric) as seen in other magnetar sources. Constraints on the Synchrotron Shock Model for the Fermi GBM Gamma-Ray Burst 090820A (1107.6024) J. Michael Burgess, Robert D. Preece, Matthew G. Baring, Michael S. Briggs, Valerie Connaughton, Sylvain Guiriec, William S. Paciesas, Charles A. Meegan, P. N. Bhat, Elisabetta Bissaldi, Vandiver Chaplin, Roland Diehl, Gerald J. Fishman, Gerard Fitzpatrick, Suzanne Foley, Melissa Gibby, Misty Giles, Adam Goldstein, Jochen Greiner, David Gruber, Alexander J. van der Horst, Andreas von Kienlin, Marc Kippen, Chryssa Kouveliotou, Sheila McBreen, Arne Rau, Dave Tierney, Colleen Wilson-Hodge July 29, 2011 astro-ph.IM, astro-ph.HE Discerning the radiative dissipation mechanism for prompt emission in Gamma-Ray Bursts (GRBs) requires detailed spectroscopic modeling that straddles the $\nu F_{\nu}$ peak in the 100 keV - 1 MeV range. Historically, empirical fits such as the popular Band function have been employed with considerable success in interpreting the observations. While extrapolations of the Band parameters can provide some physical insight into the emission mechanisms responsible for GRBs, these inferences do not provide a unique way of discerning between models. By fitting physical models directly this degeneracy can be broken, eliminating the need for empirical functions; our analysis here offers a first step in this direction. One of the oldest, and leading, theoretical ideas for the production of the prompt signal is the synchrotron shock model (SSM). Here we explore the applicability of this model to a bright {\it Fermi} GBM burst with a simple temporal structure, GRB {\it 090820}A. Our investigation implements, for the first time, thermal and non-thermal synchrotron emissivities in the RMFIT forward-folding spectral analysis software often used in GBM burst studies. We find that these synchrotron emissivities, together with a blackbody shape, provide at least as good a match with the data as the Band GRB spectral fitting function. This success is achieved in both time-integrated and time-resolved spectral fits. Fermi/GBM Observations of SGR J0501+4516 Bursts (1107.2121) Lin Lin, Chryssa Kouveliotou, Matthew G. Baring, Alexander J. van der Horst, Sylvain Guiriec, Peter M. Woods, Ersin Gogus, Yuki Kaneko, Jeffrey Scargle, Jonathan Granot, Robert Preece, Andreas von Kienlin, Vandiver Chaplin, Anna L. Watts, Ralph A.M.J. Wijers, Shuang Nan Zhang, Narayan Bhat, Mark H. Finger, Neil Gehrels, Alice Harding, Lex Kaper, Victoria Kaspi, Julie Mcenery, Charles A. Meegan, William S. Paciesas, Asaf Pe'er, Enrico Ramirez-Ruiz, Michiel van der Klis, Stefanie Wachter, Colleen Wilson-Hodge July 11, 2011 astro-ph.HE We present our temporal and spectral analyses of 29 bursts from SGR J0501+4516, detected with the Gamma-ray Burst Monitor onboard the Fermi Gamma-ray Space Telescope during the 13 days of the source activation in 2008 (August 22 to September 3). We find that the T90 durations of the bursts can be fit with a log-normal distribution with a mean value of ~ 123 ms. We also estimate for the first time event durations of Soft Gamma Repeater (SGR) bursts in photon space (i.e., using their deconvolved spectra) and find that these are very similar to the T90s estimated in count space (following a log-normal distribution with a mean value of ~ 124 ms). We fit the time-integrated spectra for each burst and the time-resolved spectra of the five brightest bursts with several models. We find that a single power law with an exponential cutoff model fits all 29 bursts well, while 18 of the events can also be fit with two black body functions. We expand on the physical interpretation of these two models and we compare their parameters and discuss their evolution. We show that the time-integrated and time-resolved spectra reveal that Epeak decreases with energy flux (and fluence) to a minimum of ~30 keV at F=8.7e-6 erg/cm2/s, increasing steadily afterwards. Two more sources exhibit a similar trend: SGRs J1550-5418 and 1806-20. The isotropic luminosity corresponding to these flux values is roughly similar for all sources (0.4-1.5 e40 erg/s). First-year Results of Broadband Spectroscopy of the Brightest Fermi-GBM Gamma-Ray Bursts (1101.3325) Elisabetta Bissaldi, Andreas von Kienlin, Chryssa Kouveliotou, Michael S. Briggs, Valerie Connaughton, Jochen Greiner, David Gruber, Giselher Lichti, P. N. Bhat, J. Michael Burgess, Vandiver Chaplin, Roland Diehl, Gerald J. Fishman, Gerard Fitzpatrick, Suzanne Foley, Melissa Gibby, Misty Giles, Adam Goldstein, Sylvain Guiriec, Alexander J. van der Horst, Marc Kippen, Lin Lin, Sheila McBreen, Charles A. Meegan, William S. Paciesas, Robert D. Preece, Arne Rau, Dave Tierney, Colleen Wilson-Hodge We present here our results of the temporal and spectral analysis of a sample of 52 bright and hard gamma-ray bursts (GRBs) observed with the Fermi Gamma-ray Burst Monitor (GBM) during its first year of operation (July 2008-July 2009). Our sample was selected from a total of 253 GBM GRBs based on each event peak count rate measured between 0.2 and 40MeV. The final sample comprised 34 long and 18 short GRBs. These numbers show that the GBM sample contains a much larger fraction of short GRBs, than the CGRO/BATSE data set, which we explain as the result of our (different) selection criteria and the improved GBM trigger algorithms, which favor collection of short, bright GRBs over BATSE. A first by-product of our selection methodology is the determination of a detection threshold from the GBM data alone, above which GRBs most likely will be detected in the MeV/GeV range with the Large Area Telescope (LAT) onboard Fermi. This predictor will be very useful for future multiwavelength GRB follow ups with ground and space based observatories. Further we have estimated the burst durations up to 10MeV and for the first time expanded the duration-energy relationship in the GRB light curves to high energies. We confirm that GRB durations decline with energy as a power law with index approximately -0.4, as was found earlier with the BATSE data and we also notice evidence of a possible cutoff or break at higher energies. Finally, we performed time-integrated spectral analysis of all 52 bursts and compared their spectral parameters with those obtained with the larger data sample of the BATSE data. We find that the two parameter data sets are similar and confirm that short GRBs are in general harder than longer ones. A New Derivation of GRB Jet Opening Angles from the Prompt Gamma-Ray Emission (1101.2458) Adam Goldstein, Robert D. Preece, Michael S. Briggs, Alexander J. van der Horst, Sheila McBreen, Chryssa Kouveliotou, Valerie Connaughton, William S. Paciesas, Charles A. Meegan, P. N. Bhat, Elisabetta Bissaldi, J. Michael Burgess, Vandiver Chaplin, Roland Diehl, Gerald J. Fishman, Gerard Fitzpatrick, Suzanne Foley, Melissa Gibby, Misty Giles, Jochen Greiner, David Gruber, Sylvain Guiriec, Andreas von Kienlin, Marc Kippen, Arne Rau, Dave Tierney, Colleen Wilson-Hodge The jet opening angle of gamma-ray bursts (GRBs) is an important parameter for determining the characteristics of the progenitor, and the information contained in the opening angle gives insight into the relativistic outflow and the total energy that is contained in the burst. Unfortunately, a confident inference of the jet opening angle usually requires broadband measurement of the afterglow of the GRB, from the X-ray down to the radio and from minutes to days after the prompt gamma-ray emission, which may be difficult to obtain. For this reason, very few of all detected GRBs have constrained jet angles. We present an alternative approach to derive jet opening angles from the prompt emission of the GRB, given that the GRB has a measurable Epeak and fluence, and which does not require any afterglow measurements. We present the distribution of derived jet opening angles for the first two years of the Fermi Gamma-ray Burst Monitor (GBM) operation, and we compare a number of our derived opening angles to the reported opening angles using the traditional afterglow method. We derive the collimation-corrected gamma-ray energy, E_{\gamma}, for GRBs with redshift and find that some of the GRBs in our sample are inconsistent with a proto-magnetar progenitor. Finally, we show that the use of the derived jet opening angles results in a tighter correlation between the rest-frame Epeak and E_{\gamma} than has previously been presented, which places long GRBs and short GRBs onto one empirical power law. Time-Resolved Spectroscopy of the 3 Brightest and Hardest Short Gamma-Ray Bursts Observed with the FGST Gamma-Ray Burst Monitor (1009.5045) Sylvain Guiriec, Michael S. Briggs, Valerie Connaugthon, Erin Kara, Frederic Daigne, Chryssa Kouveliotou, Alexander J. van der Horst, William Paciesas, Charles A. Meegan, P.N. Bhat, Suzanne Foley, Elisabetta Bissaldi, Michael Burgess, Vandiver Chaplin, Roland Diehl, Gerald Fishman, Melissa Gibby, Misty Giles, Adam Goldstein, Jochen Greiner, David Gruber, Andreas von Kienlin, Marc Kippen, Sheila McBreen, Robert Preece, Arne Rau, Dave Tierney, Colleen Wilson-Hodge Dec. 1, 2010 astro-ph.CO, astro-ph.HE From July 2008 to October 2009, the Gamma-ray Burst Monitor (GBM) on board the Fermi Gamma-ray Space Telescope (FGST) has detected 320 Gamma-Ray Bursts (GRBs). About 20% of these events are classified as short based on their T90 duration below 2 s. We present here for the first time time-resolved spectroscopy at timescales as short as 2 ms for the three brightest short GRBs observed with GBM. The time-integrated spectra of the events deviate from the Band function, indicating the existence of an additional spectral component, which can be fit by a power-law with index ~-1.5. The time-integrated Epeak values exceed 2 MeV for two of the bursts, and are well above the values observed in the brightest long GRBs. Their Epeak values and their low-energy power-law indices ({\alpha}) confirm that short GRBs are harder than long ones. We find that short GRBs are very similar to long ones, but with light curves contracted in time and with harder spectra stretched towards higher energies. In our time-resolved spectroscopy analysis, we find that the Epeak values range from a few tens of keV up to more than 6 MeV. In general, the hardness evolutions during the bursts follows their flux/intensity variations, similar to long bursts. However, we do not always see the Epeak leading the light-curve rises, and we confirm the zero/short average light-curve spectral lag below 1 MeV, already established for short GRBs. We also find that the time-resolved low-energy power-law indices of the Band function mostly violate the limits imposed by the synchrotron models for both slow and fast electron cooling and may require additional emission processes to explain the data. Finally, we interpreted these observations in the context of the current existing models and emission mechanisms for the prompt emission of GRBs. Detection of a Thermal Spectral Component in the Prompt Emission of GRB 100724B (1010.4601) Sylvain Guiriec, Valerie Connaughton, Michael S. Briggs, Michael Burgess, Felix Ryde, Frédéric Daigne, Peter Mészáros, Adam Goldstein, Julie McEnery, Nicola Omodei, P.N. Bhat, Elisabetta Bissaldi, Ascensión Camero-Arranz, Vandiver Chaplin, Roland Diehl, Gerald Fishman, Suzanne Foley, Melissa Gibby, Misty M. Giles, Jochen Greiner, David Gruber, Andreas von Kienlin, Marc Kippen, Chryssa Kouveliotou, Sheila McBreen, Charles A. Meegan, William Paciesas, Robert Preece, Arne Rau, Dave Tierney, Alexander J. van der Horst, Colleen Wilson-Hodge Nov. 30, 2010 astro-ph.CO, astro-ph.HE Observations of GRB 100724B with the Fermi Gamma-Ray Burst Monitor (GBM) find that the spectrum is dominated by the typical Band functional form, which is usually taken to represent a non-thermal emission component, but also includes a statistically highly significant thermal spectral contribution. The simultaneous observation of the thermal and non-thermal components allows us to confidently identify the two emission components. The fact that these seem to vary independently favors the idea that the thermal component is of photospheric origin while the dominant non-thermal emission occurs at larger radii. Our results imply either a very high efficiency for the non-thermal process, or a very small size of the region at the base of the flow, both quite challenging for the standard fireball model. These problems are resolved if the jet is initially highly magnetized and has a substantial Poynting flux. Hadronic Models for the Extra Spectral Component in the short GRB 090510 (0909.0306) Katsuaki Asano, Sylvain Guiriec, Peter Mészáros Oct. 17, 2009 astro-ph.CO, astro-ph.HE A short gamma-ray burst GRB 090510 detected by {\it Fermi} shows an extra spectral component between 10 MeV and 30 GeV, an addition to a more usual low-energy ($<10$ MeV) Band component. In general, such an extra component could originate from accelerated protons. In particular, inverse Compton emission from secondary electron-positron pairs and proton synchrotron emission are competitive models for reproducing the hard spectrum of the extra component in GRB 090510. Here, using Monte Carlo simulations, we test the hadronic scenarios against the observed properties. To reproduce the extra component around GeV with these models, the proton injection isotropic-equivalent luminosity is required to be larger than $10^{55}$ erg/s. Such large proton luminosities are a challenge for the hadronic models.	CommonCrawl
Correlated color temperature is not a suitable proxy for the biological potency of light Your article has downloaded Similar articles being viewed by others Carousel with three slides shown at a time. Use the Previous and Next buttons to navigate three slides at a time, or the slide dot buttons at the end to jump three slides at a time. Optimising metameric spectra for integrative lighting to modulate the circadian system without affecting visual appearance Babak Zandi, Oliver Stefani, … Tran Quoc Khanh Melanopic stimulation does not alter psychophysical threshold sensitivity for luminance flicker Joris Vincent, Edda B. Haggerty, … Geoffrey K. Aguirre Sensations from a single M-cone depend on the activity of surrounding S-cones Brian P. 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Zele, Beatrix Feigl, … Dingcai Cao Empirical evaluation of computational models of lightness perception Predrag Nedimović, Sunčica Zdravković & Dražen Domijan Adaptive light: a lighting control method aligned with dark adaptation of human vision Yui Takemura, Masaharu Ito, … Toshiyuki Okano Large enhancement of simultaneous color contrast by white flanking contours Tama Kanematsu & Kowa Koida Invisible light inside the natural blind spot alters brightness at a remote location Marina Saito, Kentaro Miyamoto, … Ikuya Murakami Tony Esposito1 & Kevin Houser2,3 Scientific Reports volume 12, Article number: 20223 (2022) Cite this article Lasers, LEDs and light sources Using a simulation based on a real, five-channel tunable LED lighting system, we show that Correlated Color Temperature (CCT) is not a reasonable predictor of the biological potency of light, whether characterized with CIE melanopic Equivalent Daylight Illuminance (mel-EDI), Equivalent Melanopic Lux (EML) (a scalar multiple of mel-EDI), or Circadian Stimulus (CS). At a photopic corneal illuminance of 300 lx and Rf ≥ 70, spectra can vary in CS from 17 to 41% across CCTs from 2500 to 6000 K, and up to 23% at a single CCT, due to the choice of spectrum alone. The CS range is largest, and notably discontinuous, at a CCT of 3500 K, the location of the inflection point of the CS model. At a photopic corneal illuminance of 300 lx and Rf ≥ 70, mel-EDI can vary from 123 to 354 lx across CCTs from 2500 to 6000 K and can vary by up to 123 lx at a fixed CCT (e.g., 196 to 319 lx at 5000 K). The range of achievable mel-EDI increases as CCT increases and, on average, decreases as color fidelity, characterized with IES TM-30 Rf, increases. These data demonstrate that there is no easy mathematical conversion between CS and mel-EDI when a spectrally diverse spectra set of spectral power distributions is considered. Light is employed in the built environment to support human visual responses, including visual performance, visual comfort, color rendition, control of glare and flicker, psychological reinforcement, and aesthetic integration. These visual goals are balanced while minimizing energy use, which includes considerations of luminous efficacy, lighting power density, and controls. In recent years, there has been growing understanding of the influence of light and lighting on non-visual biological responses, including circadian phase shifting1,2, alertness3,4, melatonin suppression2,5, pupillary response6,7, heart rate8,9, and body temperature8,9. With this ever-expanding body of knowledge, it has become increasingly important to design and specify lighting that also acknowledges lighting's potential to influence human non-visual response10,11. Lighting industry constituencies that include manufacturers, specifiers, researchers, and standards bodies are working to accommodate this new reality. Correlated color temperature (CCT), which describes the visual "warmness" or "coolness" of the color appearance of light when viewed directly, is readily available information in product specifications. CCT has sometimes been employed as a shorthand proxy for light's biological potency. Importantly, doing so is based on misunderstandings of the underlying meaning and appropriate uses of CCT. A notable example comes from the Council on Science and Public Health (CSAPH) which has warned against using high CCT lighting in the outdoor nighttime environment12. This warning led to the adoption of American Medical Association (AMA) policy H-135.92713 that "encourages the use of 3000 K or lower lighting for outdoor installations such as roadways", a recommendation that is currently in effect. In response to the AMA policy, the Lighting Research Center14, Illuminating Engineering Society (IES)15, and Houser16 argued that CCT is just one aspect of lighting quality and that CCT is inadequate for evaluating potential health outcomes of light exposure. The goal of this study is to provide numerical support for the assertion that CCT is not suitable for predicting the biological potency of light. Using a simulation based on a real 5-channel LED lighting system, we explicitly demonstrate that CCT is not a suitable proxy for prevailing measures of the biological potential of light, by showing that significant and practically meaningful variation in those measures exists at any fixed value of CCT and photopic illuminance. Correlated Color Temperature (CCT) Correlated Color Temperature (CCT, symbol: Tcp), describes the "temperature of a Planckian radiator having the chromaticity nearest the chromaticity associated with the given spectral distribution…"17. The CCT of a light source is computed as the temperature (in kelvin) of the Planckian radiator with the closest chromaticity. The computation is performed in the CIE 1960 uv chromaticity diagram, which is officially the (u', 2/3v') uniform chromaticity scale (UCS) diagram because the 1960 UCS diagram has been obsoleted by CIE and is not used for any purpose other than computing CCT. CCT is derived entirely from a light source's spectral power distribution (SPD). CCT applies to light sources that appear nominally white; generally, light sources with high CCT appear "cool" (a "blue" tinted appearance) and light sources with low CCT appear "warm" (an "orange" tinted appearance). SPDs with equal CCT fall along a line perpendicular to the Planckian locus—i.e., isotemperature lines—in the CIE 1960 uv chromaticity diagram (Fig. 1). Along an isotemperature line, the distance Duv describes how far the chromaticity of a spectral power distribution (SPD) is above (in the nominally "yellow/green" direction) or below (in the nominally "pink" direction) the Planckian locus at the same temperature. Duv values above the Planckian locus are positive; values below are negative18. CCT and Duv form a two-measure system for expressing the chromaticity of nominally white light19. An enlarged portion of the CIE 1960 uv diagram showing select isotemperature lines and the ANSI C78.377-2017 Basic and Extended CCT quadrangles. Labels indicate the CCT with the trailing double zero removed to avoid clutter (e.g., "30" = 3000 K). Chromaticity tolerances are provided by ANSI18 to "ensure high quality white light" and to "categorize chromaticities…so that white light…can be communicated to consumers." The ANSI specification takes the form of quadrangles in chromaticity space encapsulating chromaticities that can be considered to have the corresponding nominal CCT. Nominal CCTs in the ANSI framework include 2200 K, 2500 K, 2700 K, 3000 K, 3500 K, 4000 K, 4500 K, 5000 K, 5700 K, and 6500 K. The Basic ANSI quadrangles are positioned along the Planckian locus. A set of additional quadrangles—the Extended quadrangles—are located below the Basic quadrangles and below the Planckian locus; see ANNEX E of ANSI18 for more information. Figure 1 shows the Basic and Extended ANSI quadrangles in the CIE 1960 uv chromaticity diagram. CCT is regularly reported in lighting product specifications making it readily available and frequently specified. Importantly, CCT is a measure of the visual perception of illumination and is itself only one-half of a two-measure system (CCT and Duv) to specify a light source's chromaticity. CCT is not a unique descriptor because many different SPDs can share the same CCT, even at the same Duv. Available measures for quantifying biological potency The emerging metrology in quantifying the biological potency of optical radiation is based on two distinct approaches: (1) the nocturnal suppression of the hormone melatonin20,21,22,23,24, and (2) the direct excitation of retinal photopigments25,26,27. The only international standard for quantifying the biological potency of light, the CIE System for Metrology of Optical Radiation for ipRGC-Influenced Responses to Light (CIE S026)26, is based on photopigment excitation. CIE S026 uses a system of photometric equivalence that describes the amount of radiation conforming to standard daylight (D65) that is needed to produce equivalent photopigment stimulation—quantified by the respective opsin-based photopigments—as a test light source. Photopigments and their respective photoreceptors are specified by the "α" in "α-opic" whereby melanopsin, the photopigment in the ipRGC photoreceptors, is specified as "melanopic". Some have argued that biological potency is most relevant to melanopsin28, and melanopsin is the primary opsin-based photopigment considered from the CIE S026 framework in this analysis. The first step to determine melanopic photometric equivalence is to compute the melanopic Daylight Efficacy Ratio (mel-DER), which is the ratio of melanopic luminous efficacies of the test light source and D65. Mel-DER is unitless. $$\text{mel-DER}=\frac{\mathrm{Melanopic \,\,Luminous \,\,Efficacy\,\, of\,\,}\mathrm{Test\,\, Light \,\,Source}}{\mathrm{Melanopic\,\, Luminous \,\,Efficacy \,\,of}\,\,\mathrm{D}65}$$ Next, mel-DER is multiplied by a specified photopic corneal illuminance (E) to produce the melanopic Equivalent Daylight Illuminance (mel-EDI) which describes the illuminance of D65 that provides equal melanopic activation as the test source. Mel-EDI is an equivalent illuminance and has the unit lux. $$\text{mel-EDI}=\mathrm{E}(\text{mel-DER})$$ As an example, a blackbody radiator at 3500 K has a mel-DER of 0.62. At a photopic illuminance of 300 lx, this equals a mel-EDI of 186 lx. Interpretation: 186 lx of daylight at 6500 K is required to produce the same melanopic activation as 300 lx of blackbody radiation at 3500 K. Said another way, at illuminance levels that are above threshold and below saturation, blackbody radiation at 3500 K is 62% as effective at stimulating melanopsin as daylight at 6500 K. Other measures based on excitation of retinal photopigments preceded CIE S026 (see, for example, Miller and Irvin28), with perhaps the most popular method published by Lucas et al.25 and adopted by the WELL Building Standard27. The process begins with computation of the Melanopic Ratio (MR), which is the quotient of a light source's melanopic and photopic content, multiplied by 1.218 to normalize MR to a value of 1.0 for an Equal Energy (EE) spectrum27. An MR of 1.0 indicates that a light source has the same melanopic-to-photopic ratio as an equal energy illuminant; a value greater than 1.0 indicates a higher melanopic-to-photopic ratio than an EE spectrum, and vice versa. MR is unitless. $$\mathrm{MR}=\frac{\mathrm{Melanopic \,\,content}}{\mathrm{Photopic \,\,content}}1.218$$ MR is a scalar multiple of mel-DER. $$\mathrm{MR}=1.103(\text{mel-DER})$$ MR can be multiplied by a specified illuminance (E) to determine equivalent melanopic lux (EML). EML has the unit "melanopic lux" or "m-lux". Note that melanopic lux is not an SI unit and has no standardized interpretation. $$\mathrm{EML}=\mathrm{E}\mathrm{MR}$$ Because MR and EML are scalar multiples of mel-DER and mel-EDI, respectively, and CIE's mel-DER and mel-EDI are standardized measures, only mel-DER and mel-EDI will be reported in this study. Circadian Stimulus (CS) is a measure of nocturnal melatonin suppression proposed by Rea et al.20,21,22,23. Computing CS begins with computing Circadian Light (CLA), a model of biological potential that is based on a hypothesized retinal circuity that changes its sensitivity based on the "blue"–"yellow" balance in a light source's spectrum (henceforth referred to as the b-y spectral opponency). CLA "values are normalized so that a stimulus with a spectral power distribution defined by CIE Illuminant A (a blackbody radiator at 2856 K) having a photopic illuminance at the cornea of 1000 lx equals a CLA value of 1000"20. The b-y spectral opponency of the CS model is essential to interpreting the results of this present work. When a light source contains proportionally more short-wavelength radiation, "blue wins" (henceforth referred to as "rBY+"), and CLA is derived by a hypothesized combination of melanopsin and the traditional cone photoreceptors, where the absolute sensitivity of the cone contribution is mediated by the bleaching of the rod photoreceptors. When a light source contains proportionally more long-wavelength radiation, "yellow wins" (henceforth referred to as "rBY-"), and CLA is derived by melanopsin alone23. The percentage of melatonin suppression is then determined by fitting CLA values to a four-parameter logistic function. CS is equal to the percentage of nocturnal melatonin suppression and ranges from 0.00 to 0.70 (0–70% nocturnal melatonin suppression). The CS model considers spectrum and intensity and assumes one hour of exposure. CS is the primary measure used in UL DG 24480 Design Guideline for Promoting Circadian Entrainment with Light for Day-Active People (UL 24480)24, though UL 24480 allows alternate compliance paths. This study is not intended to validate (or invalidate) Circadian Stimulus23, melanopic Equivalent Daylight Illuminance26, or Equivalent Melanopic Lux (EML)27. Instead, these metrics were used as proxies for the magnitude of the biological potential of a light stimulus. A full-scale, 150 square foot residential mockup was constructed in the lighting laboratory at Penn State University, University Park, Pennsylvania, USA and furnished as a modern residential living room containing a table, couch, chair, coffee table, end tables, and a bookcase/TV stand (Fig. 2). The space had three permanent, neutral gray walls; the room's threshold had a moveable, felt blackout curtain that was closed during all measurements. Full-scale residential prototype. The prototype included furnishings of a typical residential living room and was outfitted with a novel 5-channel color changing LED lighting system. The five channels included "red", "green", "blue", "cool white", and "warm white". In this image, all 5 channels are on at 100% (ceiling mounted RGB spotlights are not on). The space was equipped with a color-changing architectural lighting system consisting of 5 individually controllable channels: nominally Red ("R"), Green ("G"), Blue ("B"), Warm White ("WW"), and Cool White ("CW"). Commercially available LED color-changing fixtures were organized into assemblies and enclosed in custom aluminum-framed housings with frosted acrylic diffusion panels. Within each assembly, three slits of vellum were applied to the surface of the individual LED strips for further diffusion and blending of the individual LED diodes (Fig. 3). Architectural spotlights were used to accent artwork. Diffusing sheets of velum and glare shields were used to maintain standards of visual quality for a residential interior. The lighting system was controlled via a Pharos Touch Panel Controller with a custom interface. Typical luminaire assembly consisted of a series of Philips Powercore fixtures arranged around a central post and enclosed in a housing constructed of an aluminum frame with frosted acrylic panel inserts. Acrylic panels were used to blend the individual LEDs. For further diffusion, three individual strips of vellum were applied to the surface of each fixture. This work intentionally uses a real lighting system to demonstrate the variability that is easily achievable with lighting products available today, instead of probing the extremes of what may be achievable with an idealized spectral model. In any multi-channel LED system, the total number and chromaticity of the system's channels, the power of each channel relative to one-another, and the peak wavelengths of the channels will impact the range of the achievable results. Nevertheless, the results presented in this manuscript represent a plausible range in CS and mel-EDI for light sources that are commercially available today. Spectral measurements were taken with a calibrated PR-655 SpectraScan Spectroradiometer (Photo Research Inc., Cary, NC, USA) aimed at a diffuse reflectance standard (SRT-MS-100, ρ = 99%) (Labsphere North America, North Sutton, NH, USA). The spectroradiometer and the reflectance standard were positioned in-line, approximately 4 feet apart, with the reflectance standard located at the center of the room's sofa and 3.5 feet above the floor. Illuminance measurements were taken with a Minolta T-10 illuminance meter (KONICA MINOLTA, Ramsey, NJ, USA) at the center of the room's sofa, 3.5 feet above the floor. These measurements characterize light entering the eye of an occupant seated on the couch, looking forward. Measurements included direct and interreflected light and therefore incorporated the spectral reflectance and absorption of room surfaces. Illuminance and spectral measurements were taken at the plane of the eye for each of the 5 channels at 12 dimming levels (100, 90, 80, 70, 60, 50, 40, 30, 20, 15, 10, and 5% of full output) to permit reconstruction, via simulation, of the luminous conditions for all possible channel combinations. Illuminance measurements were used to determine a unique dimming curve for each channel. The relative spectral power distributions (SPDs) for the five channels are shown in Fig. 4. Relative spectral power distributions for the 5 channels of the lighting system. R, "Red"; G, "Green"; B, "Blue"; WW, "Warm White"; CW, "Cool White"; Pho, photopic Luminous Efficacy Function and Mel , melanopsin action spectrum. To evaluate the relationship between CCT, CS, and mel-EDI a simulation was run using MATLAB® to generate a data set of many spectral combinations. The simulation proceeded as follows: Generate a composite spectral power distribution (SPD) of the 5 channels using a randomly generated linear combination. The multiplier for each channel was randomly generated and the resulting channel scaling was determined using each channel's unique dimming curve. If the resulting composite SPD met all the following criteria, it was retained: Chromaticity within ± 0.01 Duv of the blackbody locus CCT within ± 25 K of 6 practical CCTs: 2500, 3000, 3500, 4000, 5000, and 6000 K Compute CS (at various illuminance levels), mel-DER, mel-EDI (at various illuminance levels), and various measures from IES TM-30-2029. Illuminance scaling was used to decouple the results from the performance of our specific lighting system, permitting generalization to lighting systems producing more or less lumens. Repeat Steps 1–3, 30 million times. These data form the "Full Dataset". Create a subset of the data by filtering Duv to eliminate all spectral combinations outside of the ANSI Basic quadrangles18 to represent commercially viable spectra architectural lighting. These data form the "Filtered Dataset". Further filter the dataset for IES TM-30-20 Rf ≥ 70, Rf ≥ 80, and Rf ≥ 90, for considering the impact of color fidelity on achievable CS and mel-EDI values. Analyses in this paper considered only one aspect of color rendition, color fidelity, because it is a commonly specified performance parameter and it served here as a useful performance floor while considering the relationship between CCT and CS or mel-EDI. Importantly, other aspects of color rendition—such as vividness and color preference29, metameric uncertainty30,31,32,33,34,35, and color discrimination36,37—are relevant to applied lighting. These aspects of color rendition may themselves exhibit tradeoffs with CS and mel-EDI and these relationships should be studied directly. A total of 235,039 composite SPDs met the criteria detailed in section "Simulation" before filtering Duv to eliminate SPDs outside of the ANSI bins (simulation Steps 1–4); a total of 150,227 composite SPDs remained after filtering for inclusion in the ANSI bins (simulation Step 5). The most SPDs were retained at a CCT of 4000 K, and counts decrease with either increasing or decreasing CCT. Further filtering the "Filtered Dataset" for Rf ≥ 70, Rf ≥ 80, and Rf ≥ 90, as was done to evaluate the impact of color fidelity on achievable CS and mel-EDI, further decreased the number of retained SPDs. See Fig. 5. The number of composite SPDs retained at each of the nominal CCTs. The "Full" dataset includes SPDs with Duv ± 0.01; the "Filtered" dataset includes SPDs with the Duv tolerance of the ANSI Basic quadrangles. The other three series are color fidelity filters on the "Filtered" dataset. The nominal CCT with the most retained SPDs was 4000 K, with decreasing frequency at higher and lower CCTs. Melanopic DER and melanopic EDI Figure 6 shows boxplots for melanopic daylight efficacy ratio (mel-DER) for all retained SPDs by CCT for the Filtered Dataset. A clear trend between mel-DER and CCT is evident where, on average, mel-DER increases as CCT increases (r2 > 0.86). This trend is consistent with that demonstrated by Zandi et al.38, Carpentier and Meuret39, and Spitschan40. Summary statistics are provided in Table 1 and can be used to estimate mel-DER for an SPD based on CCT and IES Rf. Boxplots of melanopic daylight efficacy ratio (mel-DER) as a function of nominal CCT. (Top left) All SPDs for the "Filtered" dataset. (Top right) The "Filtered" dataset further filtered for Rf ≥ 70. (Bottom left) The "Filtered" dataset further filtered for Rf ≥ 80. (Bottom right) The "Filtered" dataset further filtered for Rf ≥ 90. In all cases, the average mel-DER increases as CCT increases, but so too does the achievable mel-DER range. As the average fidelity floor is raised (i.e., from Rf ≥ 70 to Rf ≥ 80 to Rf ≥ 90), the average, maximum, and achievable range of mel-DER decreases. Table 1 Statistical summary of melanopic daylight efficacy ratio26 for SPDs with various fidelity (Rf) filtering. This table can be used as a quick reference to gauge the magnitude of mel-EDI for an SPD based on its nominal CCT and IES TM-30-20 Rf. For example, a light source with a CCT of 3500 K and Rf near 80 will have an mel-DER between 0.52 and 0.75; See bolded values in table. Notably, there is significant variation about the average mel-DER at any fixed CCT. For an SPD with a CCT of 3500 K and an Rf ≥ 80, a common lighting specification, mel-DER can vary between 0.52 and 0.75. At a photopic illuminance of 300 lx, a commonly specified illuminance level, this equates to mel-EDI values ranging from 157 to 226 lx. Alternatively, if a design target of mel-EDI = 218 lx is set—as might be done to achieve the maximum 3 points in WELL v2, "L03 Circadian Lighting Design"27—the spectrum alone can cause a required photopic illuminance between 289 and 416 lx. This is a difference of 127 photopic lux of required illuminance from the choice of spectrum alone. Figure 6 also shows that as the color fidelity floor is raised (i.e., from Rf ≥ 70 to Rf ≥ 80 to Rf ≥ 90), the mean and spread of achievable mel-DER values decrease at any single CCT. This is consistent with the results of Carpentier and Meuret39. (Note that when color fidelity is maximal—i.e., IES TM-30 Rf = 100—the ranges shown in Fig. 6 reduce to a point because the SPD of the test source must be substantially identical to the SPD of the reference illuminant. In this case, mel-DER increases monotonically with CCT because the SPD at maximal color-fidelity varies smoothly and predictable with CCT.) Still, for an SPD with a CCT of 3500 K and an Rf ≥ 90, mel-DER can range from 0.53 to 0.67 (Table 1). At a photopic illuminance of 300 lx, for example, this equates to a range of mel-EDI values from 160 to 202 lx. Alternatively, if a target of mel-EDI = 218 lx is set—as might be done to achieve the maximum 3 points in WELL v2, "L03 Circadian Lighting Design"27—the spectrum alone can cause a required photopic illuminance between 323 and 409 lx. This is a difference of 88 photopic lux from the choice of spectrum alone, even when average color fidelity is very high. Following (Eq. 2), Fig. 6 and Table 1 can be converted to mel-EDI values via multiplication by any photopic illuminance value. To convert Fig. 5, uniformly multiply the y-axis values by the designated illuminance. The same is true for values in Table 1. Figure 6 and Table 1 can be converted from mel-DER to Melanopic Ratio (MR) using (Eq. 4), then converted from MR to EML using (Eq. 5). It is clear from (Eq. 2) and (Eq. 5) that both mel-EDI and EML scale linearly with photopic illuminance such that doubling photopic illuminance, for example, doubles mel-EDI and EML. Circadian stimulus Figure 7 shows CS plotted against CLA for photopic illuminances of 100 lx (top), 300 lx (middle), and 1000 lx (bottom). The achievable range of CS values decreases as illuminance departs from 300 lx, where the largest achievable CS range occurs. Figure 7 demonstrates the intensity-based dose–response relationship between CLA and CS where intensity is first order information and spectrum is second order information. Said another way, intensity controls the order-of-magnitude of the effect, and spectrum creates variability about that point. Summary statistics for Fig. 7 are provided in Table 2 and can be used to estimate CS for an SPD based on corneal illuminance and IES Rf. The range of Circadian Stimulus (CS) values as a function of Circadian light (CLA) for 100 lx (top), 300 lx (middle), and 1000 lx (bottom). At any fixed photopic illuminance, a substantial range of CS values can be achieved due to the choice of spectrum alone. The largest range occurs near 300 lx which falls near the inflect point of the CS sigmoid curve. Values are not sub-divided by CCT. Table 2 Statistical summary of Circadian Stimulus for SPDs with various fidelity (Rf) filtering. This table can be used as a quick reference to gauge the magnitude of CS for an SPD based on its specified corneal illuminance (Ev) and approximate Rf. For example, a light source with an Rf of 81 at 300 lx is likely to have a CS between 0.17 and 0.41 (not considering CCT); See bolded values in table. At a fixed illuminance of 300 lx, Fig. 8 shows that the range of achievable CS varies significantly as a function of CCT. Summary statistics are provided in Table 3 and can be used to estimate CS based on CCT and spectral-opponency. At 2500 K and 3000 K, "yellow wins" (rBY-) for all SPDs due to the prevalence of long wavelength radiation in the composite SPDs. Near 3500 K a battle between "blue wins" (rBY+) and "yellow wins" (rBY-) is apparent leading to an observable discontinuity. (Note that the exact inflection point of the CS model is between 3442 and 3443 K for a blackbody radiator, see Fig. 9, and is strongly dependent on chromaticity, see Fig. 10). By 4000 K, "blue wins" (rBY+) for all SPDs, though CS is noticeably reduced compared to the "yellow wins" scenarios. Circadian stimulus (CS) as a function of nominal CCT at a photopic illuminance of 300 lx. (Top left) All SPDs for the "Filtered" dataset. (Top right) The "Filtered" dataset further filtered for Rf ≥ 70. (Bottom left) The "Filtered" dataset further filtered for Rf ≥ 80. (Bottom right) The "Filtered" dataset further filtered for Rf ≥ 90. "rBY-" indicates the "yellow wins" scenario of the CS model (also indicated by the horizontal hatching and circle markers); "rBY+ " indicates the "blue wins" scenario of the CS model (also indicated by the vertical hatching and square markers). Lines show the maximum (grey), minimum (grey), and average (black) CS within each nominal CCT; error bars indicate the standard deviation. On average, sources with rBY- have higher CS values than sources with rBY+. The b-y spectral opponency causes the observable discontinuity near 3500 K where the model switches from one hypothetical model of phototransduction to another. As the average fidelity floor is raised (i.e., from Rf ≥ 70 to Rf ≥ 80 to Rf ≥ 90), the achievable CS range decreases. Table 3 A summary of circadian stimulus as a function of CCT with various Rf filtering. rBY+ indicates "blue wins". rBY- indicates "yellow wins" and is determined by the balance of spectral content relative to 500 nm, approximately. Circadian Stimulus (CS) as a function of CCT for a blackbody radiator. The exact inflection point of the b-y spectral opponency of the CS model for a blackbody radiator occurs between 3442 and 3443 K. Up to 3442 K, "yellow wins" (indicated by circle markers); from 3443 K onwards, "blue wins" (indicated by square markers). The dashed line indicates the transition point of the model and is dashed to indicate that this transition is not continuous (values along this line are not achievable) and that the CS model is a step function. Chromaticity of the "Full" dataset, filtered for Rf ≥ 70, plotted in the CIE 1931 (x,y) chromaticity diagram. The b-y spectral opponency of the CS model manifests strongly in chromaticity space cutting the 3500 K and 4000 K Basic ANSI quadrangles diagonally. The yellow dashed line indicates the limit of the "yellow wins" dataset (circle markers) which is hidden by the "blue wins" dataset (triangle markers). When this line representing the boundary of the b-y spectral opponency is extended, it intersects the spectrum locus near 500 nm which is near the crossover wavelength of the functions that the CS model uses to determine "blue wins" (rBY+) or "yellow wins" (rBY-). The spectral opponency of the CS model—observed most clearly at the discontinuity near 3500 K—has a large impact on the resulting CS value. For SPDs with Rf ≥ 70, the lowest CS at 2500 K (0.28) has a higher CS than the highest CS at 3500 K and rBY+ (0.27). This holds true and is exacerbated as the dataset is filtered for increasing color fidelity—e.g., Rf ≥ 80 and Rf ≥ 90—and the achievable CS range at a fixed CCT decreases. CLA scales linearly with photopic illuminance such that doubling photopic illuminance, for example, doubles CLA. CLA then maps non-linearly onto CS via the four-parameter logistic function observable in Fig. 8. Figure 11 shows the impact of doubling CLA (i.e., doubling photopic illuminance) on the resulting CS value. The largest benefit of doubling photopic illuminance occurs when the starting CLA is near 300 (for photopic lux between 300 and 500 lx, depending on the spectrum) because it is near the inflection (or "threshold") of the CS logistic function. The thick black line shows the four-parameter logistic function that determines Circadian Stimulus (CS) from Circadian Light (CLA). The dashed curve indicates the difference in CS when CLA is doubled. This can be thought of the "strength" of doubling CLA on CS at the specified CLA value. The vertical dashed line indicates that the largest increase on CS of doubling CLA occurs at a CLA of 300 (near the inflection point of the logistic function), which can be for a photopic illuminance of 300–500 lx, depending on spectrum. In general, for a given opponent pathway (rBY+ or rBY-), the maximum achievable CS at a fixed CCT increases as CCT increases, but composite spectra at a given CCT exhibit a wide range of CS values. At a fixed illuminance and CCT, CS can range by more than 0.1 (which is one-third (33%) of the daytime CS minimum of 0.3 recommendation of UL 2448024) and more than 0.2 at a CCT of 3500 K depending on the b-y spectral opponency (which is two-thirds (66%) of the daytime CS minimum of 0.3 recommendation of UL 2448024). Comparing CS and mel-EDI To simplify lighting practice, lighting practitioners would understandably like a simple conversion between measures like CS and measures like mel-EDI (see, for example, Stodola41). We should first mention, however, that an apples-to-apples comparison between CS and mel-EDI is not possible. CS and mel-EDI are similar in that they both deal with spectral sensitivity, but they are different in that mel-EDI is based on a single photopigment and CS endeavors to account for the physiology of the eye/brain system. In support of this substantial theoretical difference, we quantitatively demonstrate the manner and degree of differences between mel-EDI and CS, thereby objectively substantiating their lack of comparability. Figure 12 compares CS and mel-EDI, with sub-divisions for illuminance, and demonstrates that no such easy mathematical conversion between CS and mel-EDI can be offered when a spectrally diverse set of SPDs is considered. For example, at a photopic illuminance of 300 lx and a mel-EDI of 218 lx (a value achieving the maximum 3 points in WELL v2), CS varies between 0.22 and 0.39 (0.22 and 0.32 for rBY+ between CCTs of 3500 and 5000 K, and 0.38 and 0.39 for rBY- at a CCT of 3500 K). Circadian Stimulus (CS) as a function of melanopic equivalent daylight illuminance (mel-EDI) with sub-divisions for photopic illuminance represented as boundaries of the data. (Top) Linear scaling for the x-axis. (Bottom) Logarithmic scaling for the x-axis. Overall, there is a wide range of achievable CS values at a fixed mel-EDI and a strong discontinuity due to the b-y spectral opponency of the CS model ("yellow wins" indicated by rBY-, and "blue wins" indicated by rBY+). No simple or satisfactory mathematical conversion between CS and mel-EDI can be offered. The grey shaded area indicates the sub-division of the data with an illuminance of 500 lx, which is enlarged in Fig. 13. Figure 13 makes this point more forcefully, showing the relationship between CS and mel-EDI at a fixed illuminance of 500 lx with CCT sub-divisions. Visibly, no best-fit equation could satisfactorily describe this relationship. Even if tempted to build best-fit relationships for CCT subgroups, CS can vary between 0.26 and 0.45 for CCT = 3500 K, and from 0.28 to 0.35 for CCT = 4000 K, each at a fixed mel-EDI of 300 lx. Even if such a range is acceptable, or a simple relationship could be found, for example by inappropriate selective paring of the dataset or analyzing only broadband phosphor-converted LEDs, the conceptual and computational differences between measures like CS and measures like mel-EDI make such a conversion inappropriate. Circadian Stimulus (CS) as a function of melanopic equivalent daylight illuminance (mel-EDI) at a photopic illuminance of 500 lx enlarged from Fig. 12 with the addition of subdivisions for nominal CCT. Arrows show, on average, increasing Rf. Zooming in on the relationship at a fixed illuminance illustrates the large range of achievable CS values at a fixed mel-EDI that is driven by spectral configuration and partly driven by the discontinuity between "blue wins" (rBY+) and "yellow wins" (rBY-) of the CS model. The horizontal grey dashed line indicates the daytime CS recommendation of CS ≥ 0.3 of UL 24480. Figure 13 is useful for evaluating a recommendation of UL 2448024 which states that without regard to spectrum, a photopic illuminance of 500 lx can be specified to achieve the daytime recommendation of CS ≥ 0.3. They indicate that at 500 lx, "90% of commercially available LEDs with CCTs between 2700 and 6500 K will meet the circadian-effective light design criterion of CS ≥ 0.3." In this analysis, 97% of SPDs at a photopic illuminance of 500 lx fall above the CS ≥ 0.3 criteria. Most of the SPDs with CS < 0.3, however, are at 3500 K and high color fidelity. This is a common lighting specification and is therefore potentially problematic. UL 24480 also makes recommendations of CS ≤ 0.2 and CS ≤ 0.1 in the evening and night, respectively. In this study, 99.5% of SPDs at 100 photopic lux had CS ≤ 0.2 (only 28.5% of SPDs had CS ≤ 0.2 at 200 photopic lux). Without regard to spectrum, a photopic illuminance of 100 lx can be reasonably assumed to comply with UL 24480's evening recommendations. For the nighttime recommendation, 88.3% of SPDs at 50 photopic lux had CS ≤ 0.1 (only 22.8% of SPDs had CS ≤ 0.1 at 100 photopic lux). Without regard to spectrum, a photopic illuminance less than 50 lx can be reasonably assumed to comply with UL 24480's nighttime recommendations. Predicting biological potency with CCT fails in concept CCT describes the visual "warmness" or "coolness" of the color appearance of nominally white light. Melanopic-EDI is a measure of the stimulation of the ipRGC photoreceptors via the action spectra of their contained photopigment melanopsin. Circadian Stimulus is a model of human nocturnal melatonin suppression as a function of light source spectrum and intensity. That CCT is a measure of visual perception should immediately raise doubts about its ability to predict any measure that is not related to the visual perception of nominally white light. Furthermore, that two metameric spectra, thus having the same CCT, can have largely different quantities of melanopsin-stimulating radiation make it obvious that any simple relationship that can be found between CCT and CS or mel-EDI is merely a coincidence or the result of selective paring of the dataset. There is currently incomplete knowledge about how photoreceptor signals are combined and processed within the retina (e.g., by ipRGCs that combine the intrinsic melanopic response with extrinsic signals from cones and rods) and incomplete understandings of how those signals are then processed by the brain. As such, CIE S026 does not propose a working model of the eye-brain mechanism. This is because quantifying individual photoreceptor responses is a means to an end, not the end itself. CIE S026 acknowledges this: "If…the relative photoreceptor inputs to any response under defined conditions could be resolved it would be possible to predict the magnitude of evoked [non-visual] responses from the combination of the effective light intensity for each of the individual photoreceptors." CIE S026 represents a significant achievement by standardizing quantification of photopigment action spectra that serve as inputs into the eye-brain mechanism. In this conceptual framework, CIE S026 provides the foundation upon which to build mechanistic models. We are reluctant about models that are over reliant on any single photopigment. Whether mel-DER becomes an input measure into mechanistic models or is proposed as a direct predictor of outcome measures (e.g., see Brown42), one thing is for certain: there is no expectation that models used to predict human non-visual response to light will make use of CCT. Predicting biological potency with CCT fails in practice No numerical justification for the use of CCT as a predictor of biological potential can be offered. Figures 6, 8, 12, and 13 demonstrate that a sufficiently large variation in CS and mel-EDI exists at any fixed CCT and photopic illuminance. At 300 photopic lux, 3500 K, and a minimum Rf of 70—a common lighting specification—CS can range, albeit discontinuously, from 0.17 to 0.4 (a difference of 0.23), due to spectrum alone. This difference is more than two-thirds (66%) of the minimum daytime CS of 0.3 recommended by UL 2448024. At a design target of mel-EDI = 200 lx, 3500 K, and a minimum Rf of 70, spectrum can cause a required illuminance between 370 and 576 lx, a difference of 206 photopic lux due to spectrum alone. Given the relative ease with which mel-DER and mel-EDI can be computed, no practical justification can be offered either. CCT is derived from the same information needed to compute mel-DER: the light source's spectral power distribution. If the SPD is readily available for the calculation of CCT, mel-DER can be computed with a few additional computations. (Note that relative to calculations of mel-DER using a light source's SPD, field measurements of mel-DER will likely differ due to attenuation of short wavelengths by room surfaces43,44,45,46,47,48,49; preliminary evidence suggests that field measurements of mel-DER are likely to be lower than estimates with a light source's SPD48,49,50, especially where non-white, non-blue surfaces are present51). Mel-EDI requires an estimate or measurement of illuminance. Estimating illuminance is common in lighting practice. In the field, measuring illuminance for the mel-EDI computation does not require an additional measurement device. Field measurements of mel-EDI provide the most accurate estimates and are recommended whenever possible. Determining mel-DER and/or mel-EDI is relatively straightforward using testing and measurement procedures already common in lighting practice. In practice, shorthand rules can be useful to simplify illuminating engineering but predicting biological potential from CCT fails in theory and in execution. Simply, CCT should not be used as a proxy for the biological potency of light. CS model implications for spectral design The opponent pathway in the CS model has a significant effect on resulting CS values whereby it is advantageous to the goal of maximizing CS, at a fixed illuminance, to do one of three things: Force "yellow wins" (rBY-) by targeting nominal CCTs below 3500 K—or an actual CCT less than 3443 K for broadband spectra near the blackbody locus and with high color fidelity, or target a positive Duv when CCT is near 3500 K (increasing the likelihood of "yellow wins", rBY-), or target nominal CCTs greater than 4000 K (not including 4000 K). To minimize CS, it is advantageous, at a fixed illuminance, to target a nominal CCT of 4000 K, or a nominal CCT of 3500 K if "blue wins" (rBY+) can be forced. Figure 10 shows that at a CCT of 3500 K, targeting chromaticities below the blackbody locus increases the likelihood of forcing rBY+. At a fixed illuminance, the CS model suggests that when the goal is to suppress the least amount of melatonin—for example, in the evening before sleep—light sources with a nominal CCT of 4000 K or 3500 K (with rBY+) should be targeted. These results contradict prevailing wisdom that light with proportionally more short wavelength radiation (that is, near the sensitivity of the melanopsin action spectra) has greater biological potency. As shown in Fig. 10, the b-y spectral opponency of the CS model cuts diagonally across the 3500 K ANSI bin. This raises an important practical consideration. For manufacturers creating products near this boundary—as many do since products at 3500 K near the blackbody locus are popular—normal production tolerances could mean that some products within a single production run are rBY+ and some are rBY-, leading to a difference in CS of up to 0.23 at a photopic illuminance of 300 lx and Rf ≥ 70. This is a variation equal to two-thirds (67%) of the UL 24480 daytime recommendation of CS ≥ 0.324 for light sources that have negligibly different SPDs (see Fig. 14). To avoid this conflict, manufacturers would need to push their products above the blackboy locus to maximize CS (forcing "yellow wins"), which may lead to less-preferred (and green-tinted) illumination52,53,54, or below the blackbody locus to minimize CS (to force "blue wins"). Two spectral power distributions with nearly identical spectral curves (left) but with a CS differing by 100% at a photopic illuminance of 300 lx (right). Because the SPDs are nearly overlapping, the dashed SPD may be difficult to see. For orientation, the largest spectral differences occur near 450 and 480 nm. From an engineering perspective, this discontinuity is peculiar and raises suspicion about the face validity of the CS model. Is the sharp discontinuity physiologically plausible, or simply an artifact of the model? If such opponency is physiologically present in normal observers, what is the role of observer variability (e.g., to what degree could an illuminant that is rBY+ to one observer be rBY- to a second observer, a question that is relevant in consideration of prior work about interindividual variability to light55)? Regardless, this discontinuity is pertinent to the spectral design of light sources, if CS values are important, with static or dynamic spectra near the chromaticity inflection demonstrated in Fig. 10, since there can be large variability in CS for products at 3500 K which have nearly identical performance on all other spectrally derived quantities. The lighting industry is experiencing rapid transformation as we expand our awareness of the non-visual impacts of light on humans. It is pertinent that we develop measures, methods, and strategies for implementing architectural lighting solutions that support these non-visual impacts. To do so, we need accurate and predictive measures of the biological potency of light that are based on sound science. 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Energy impact of human health and wellness lighting recommendations for office and classroom applications. Energy Build. 226, 110365. https://doi.org/10.1016/j.enbuild.2020.110365 (2020). Potočnik, J. & Košir, M. Influence of commercial glazing and wall colours on the resulting non-visual daylight conditions of an office. Build. Env. 171, 106627. https://doi.org/10.1016/j.buildenv.2019.106627 (2022). Alkhatatbeh, B. J. & Asadi, S. Role of architectural design in creating circadian-effective interior settings. Energies 14, 6731. https://doi.org/10.3390/en14206731 (2021). Ohno, Y., Fein, M. Vision experiment on white light chromaticity for lighting. In Proc. CIE/USA-CNC/CIE Biennial Joint Meeting. https://cltc.ucdavis.edu/sites/default/files/files/publication/2-yoshi-ohno-mira-fein-white-light-chromaticity-vision-experiment.pdf (2013). Dikel, E. E. et al. Preferred chromaticity of color-tunable LED lighting. LEUKOS. 10, 101–115. https://doi.org/10.1080/15502724.2013.855614 (2014). Ohno, Y., Oh, S. Vision experiment II on white light chromaticity for lighting. In Proceedings of the CIE Lighting Quality and Energy Efficiency. CIE x042 (2016). Philips, A. J. K. et al. High sensitivity and interindividual variability in the response of the human circadian system to evening light. PNAS 116, 12019–12024. https://doi.org/10.1073/pnas.1901824116 (2019). We thank Michael Royer for contributing to the original proposal and the conceptual development that made this project possible; Andrea Wilkerson for conceptual development and construction of the prototype luminaires and physical environment; Christopher Smith-Petersen for assistance with the equipment; Thomas Ladd with Pharos Control for programming assistance and contributions to the construction of the user interface. The Pennsylvania Housing Research Center (PHRC) funded the construction and development of the full-scale residential prototype. Lighting Research Solutions LLC, Philadelphia, PA, USA Tony Esposito School of Civil and Construction Engineering, Oregon State University, Corvallis, OR, USA Kevin Houser Pacific Northwest National Laboratory, Portland, OR, USA T.E. and K.H. contributed to conceptualization of the analysis of this project. T.E. performed the analysis, figure generation, and developed the draft manuscript. K.H. contributed significant manuscript revisions. T.E. and K.H. reviewed and approved the final manuscript. K.H. led project administration and proposal development to secure funding for the physical prototype. Correspondence to Tony Esposito. The authors declare no competing interests. Esposito, T., Houser, K. Correlated color temperature is not a suitable proxy for the biological potency of light. Sci Rep 12, 20223 (2022). https://doi.org/10.1038/s41598-022-21755-7 About Scientific Reports Guide to referees Journal highlights Scientific Reports (Sci Rep) ISSN 2045-2322 (online)	CommonCrawl
\begin{document} \title[Positivity and Newton--Okounkov bodies]{Positivity of line bundles and Newton-Okounkov bodies} \author[A.~K\" uronya]{Alex K\" uronya} \author[V.~Lozovanu]{Victor Lozovanu} \address{Alex K\"uronya, Johann-Wolfgang-Goethe Universit\"at Frankfurt, Institut f\"ur Mathematik, Robert-Mayer-Stra\ss e 6-10., D-60325 Frankfurt am Main, Germany} \address{Budapest University of Technology and Economics, Department of Algebra, Egry J\'ozsef u. 1., H-1111 Budapest, Hungary} \email{{\tt [email protected]}} \address{Victor Lozovanu, Universit\'a degli Studi di Milano--Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53,, I-20125 Milano, Italy} \email{{\tt [email protected]}} \maketitle \section{Introduction} The aim of this work is to characterize positivity (both local and global) of line bundles on complex projective varieties in terms of convex geometry via the theory of Newton--Okounkov bodies. We will provide descriptions of ample and nef divisors, and discuss the relationship between Newton--Okounkov bodies and Nakayama's $\sigma$-decomposition. Based on earlier ideas of Khovanskii's Moscow school and motivated by the work of Okounkov \cite{Ok}, Kaveh--Khovanskii \cite{KKh} and Lazarsfeld--Musta\c t\u a \cite{LM} introduced Newton--Okounkov bodies to projective geometry, where they have been an object of interest ever since. Essentially, a refined book-keeping device encoding the orders of vanishing along subvarieties of the ambient space $X$, they provide a general framework for the study of the asymptotic behaviour of line bundles on projective varieties. The construction that leads to Newton--Okounkov bodies associates to a line bundle (or more generally, an $\ensuremath{\mathbb{R}}$-Cartier divisor) on an $n$-dimensional variety a collection of compact convex bodies $\Delta_{\ensuremath{Y_\bullet}}(D)\subseteq\ensuremath{\mathbb{R}}^n$ parametrized by certain complete flags $\ensuremath{Y_\bullet}$ of subvarieties. Basic properties of these have been determined \cites{AKL,Bou1,LM}, and their behaviour on surfaces \cites{KLM1,LM, LSS} and toric varieties \cites{LM,PSU} has been discussed at length. We refer the reader to the above-mentioned sources for background information. A distinguishing property of the notion is that it provides a set of 'universal numerical invariants', since a result of Jow \cite{Jow} shows that for Cartier divisors $D$ and $D'$, $D$ is numerically equivalent to $D'$ precisely if the associated functions \[ \text{Admissible flags $\ensuremath{Y_\bullet}$ in $X$}\ \stackrel{\Delta_{\ensuremath{Y_\bullet}}(D)}{\longrightarrow}\ \text{Convex bodies in $\ensuremath{\mathbb{R}}^n$} \] agree. Turning this principle into practice, one can expect to be able to read off all sorts of numerical invariants of Cartier divisors --- among them asymptotic invariants like the volume or Seshadri constants --- from the set of Newton--Okounkov bodies of $D$. On the other hand, questions about global properties of the divisor might arise; whether one can determine ampleness or nefness of a given divisor in terms of its Newton--Okounkov bodies. As we will see, the answer is affirmative. Localizing this train of thought, local positivity of a divisor $D$ at a point $x\in X$ will be determined by the function \[ \text{Admissible flags centered at $x$} \ \stackrel{\Delta_{\ensuremath{Y_\bullet}}(D)}{\longrightarrow}\ \text{Convex bodies in $\ensuremath{\mathbb{R}}^n$}\ . \] In particular, one can aim at deciding containment of $x$ in various asymptotic base loci, or compute measures of local positivity in terms of these convex sets. In fact the authors have carried out the suggested analysis in the case of smooth surfaces \cite{KL}, where the answer turned out to be surprisingly complete. The current article can be rightly considered as a higher-dimensional generalization of \cite{KL}. In search for a possible connection between Newton--Okounkov bodies and positivity, let us start with the toy example of projective curves. For an $\ensuremath{\mathbb{R}}$-Cartier divisor $D$ on a smooth projective curve $C$, one has \begin{eqnarray} D \text{ nef } \Leftrightarrow \deg_C D \ensuremath{\,\geqslant\,} 0 & \Leftrightarrow & 0 \in \Delta_{P}(D) \text{ for some/any point $P\in C$}\ ,\\ D \text{ ample } \Leftrightarrow \deg_C D \,>\, 0 & \Leftrightarrow & \Delta_\lambda \subseteq \Delta_{P}(D) \text{ for some/any point $P\in C$}, \end{eqnarray} where $\Delta_{\lambda}:=[0,\lambda]$ for some real number $\lambda>0$. Interestingly enough, the observation just made generalizes in its entirety for smooth projective surfaces. Namely, one has the following \cite{KL}{Theorem A}: for a big $\ensuremath{\mathbb{R}}$-divisor $D$ on a smooth projective surface $X$ \begin{eqnarray} \text{ $D$ is nef} & \Leftrightarrow & \text{ for all $x\in X$ there exists a flag $(C,x)$ such that $(0,0)\in\Delta_{(C,x)}(D)$ }, \\ \text{ $D$ is ample} & \Leftrightarrow & \text{for all $x\in X$ there exists a flag $(C,x)$ and $\lambda >0$ such that $\Delta_{\lambda}\subseteq \Delta_{(C,x)}(D)$ \ } \end{eqnarray} where $\Delta_{\lambda}$ denotes the standard full-dimensional simplex of size $\lambda$ in $\ensuremath{\mathbb{R}}^2$. In higher dimensions we will also denote by $\Delta_{\lambda}\subseteq\ensuremath{\mathbb{R}}^n$ the standard simplex of length $\lambda$. Our first results are local versions of the analogous statements in higher dimensions. \begin{theoremA} Let $D$ be a big $\ensuremath{\mathbb{R}}$-divisor on a smooth projective variety $X$ of dimension $n$, let $x\in X$. Then the following are equivalent. \begin{enumerate} \item $x\not\in \ensuremath{\textbf{\textup{B}}_{-} }(D)$. \item There exists an admissible flag $\ensuremath{Y_\bullet}$ on $X$ centered at $x$ such that the origin $\ensuremath{\textup{\textbf{0}}}\in \Delta_{\ensuremath{Y_\bullet}}(D)\subseteq\ensuremath{\mathbb{R}}^n$. \item The origin $\ensuremath{\textup{\textbf{0}}}\in \Delta_{\ensuremath{Y_\bullet}}(D)$ for every admissible flag $\ensuremath{Y_\bullet}$ on $X$ centered at $x\in X$. \end{enumerate} \end{theoremA} \begin{theoremB} With notation as above, the following are equivalent. \begin{enumerate} \item $x\not\in \ensuremath{\textbf{\textup{B}}_{+} }(D)$. \item There exists an admissible flag $\ensuremath{Y_\bullet}$ on $X$ centered at $x$ with $Y_1$ ample such that $\Delta_\lambda\subseteq \Delta_{\ensuremath{Y_\bullet}(D)}$ for some positive real number $\lambda$. \item For every admissible flag $\ensuremath{Y_\bullet}$ on $X$ there exists a real number $\lambda>0$ for which $\Delta_\lambda\subseteq \Delta_{\ensuremath{Y_\bullet}(D)}$. \end{enumerate} \end{theoremB} These results will be proven below as Theorem~\ref{thm:main1}, and Theorem~\ref{thm:main2}, respectively. Making use of the connections between augmented/restricted base loci, we obtain the expected characterizations of nef/ample divisors as in Corollary~\ref{cor:nef} and \ref{cor:ample}. An interesting recent study of local positivity on surfaces was undertaken by Ro\'e \cite{Roe}, where the author introduces the concept of local numerical equivalence, based on the ideas developed in \cite{KL}. Zariski decomposition is a basic tool in the theory of linear series on surfaces, which is largely responsible for the fact that Newton--Okounkov bodies are reasonably well understood in dimension two; the polygonality of $\Delta_{\ensuremath{Y_\bullet}}(D)$ in case of a smooth surface is a consequence of variation of Zariski decomposition \cite{BKS} for instance (see \cite{KLM1}{Section 2} for a discussion). Not surprisingly, the existence and uniqueness of Zariski decompositions is one of the main tools used in \cite{KL}. Its relationship to Newton--Okounkov polygons on surfaces is particularly simple: if $D$ is a big $\ensuremath{\mathbb{R}}$-divisor with the property that the point $Y_2$ in the flag $\ensuremath{Y_\bullet}$ is not contained in the support of the negative part of $D$, then $\Delta_{\ensuremath{Y_\bullet}}(D)=\Delta_{\ensuremath{Y_\bullet}}(P_D)$, where $P_D$ stands for the positive part of $D$. In dimensions three and above, the appropriate birational version of Zariski decomposition --- the so-called CKM decomposition --- only exists under fairly restrictive hypotheses, hence one needs substitutes whose existence is guaranteed while they still retain some of the favourable properties of the original notion. A widely accepted concept along these lines is Nakayama's divisorial Zariski decomposition or $\sigma$-decomposition, which exists for an arbitrary big $\ensuremath{\mathbb{R}}$-divisor, but where the 'positive part' is only guaranteed to be movable (see \cite{Nak}{Chapter 3} or \cite{Bou2}). Extending the observation coming from dimension two, we obtain the following. \begin{theoremC} Let $X$ be a smooth projective variety, $D$ a big $\ensuremath{\mathbb{R}}$-divisor, $\Gamma$ a prime divisor, $Y_{\bullet}: Y_0=X\supseteq Y_1=\Gamma\supseteq\ldots \supseteq Y_n=\{x\}$ and admissible flag on $X$. Then \begin{enumerate} \item $\Delta_{Y_{\bullet}}(D) \ \subseteq \ (\sigma_{\Gamma}(D),0\ldots, 0)+\ensuremath{\mathbb{R}}_+^n$, \item $(\sigma_{\Gamma}(D),0\ldots, 0)\ \in \ \Delta_{Y_{\bullet}}(D)$, whenever $x\in \Gamma$ is a very general point. \item $\Delta_{Y_{\bullet}}(D) = \nu_{Y_{\bullet}}(N_{\sigma}(D)) + \Delta_{Y_{\bullet}}(P_{\sigma}(D))$. Morever, $\Delta_{Y_{\bullet}}(D) =\Delta_{Y_{\bullet}}(P_{\sigma}(D))$, when $x\notin\textup{Supp}(N_{\sigma}(D))$. \end{enumerate} \end{theoremC} The organization of the paper goes as follows: Section 1 fixes notation, and collects some preliminary information about asymptotic base loci and Newton--Okounkov bodies. Sections 2 and 3 are devoted to the respective proofs of Theorems A and B, while Section 4 describes the relationship between Newton--Okounkov bodies and Nakayama's $\sigma$-decomposition. \paragraph{\bf Acknowledgements} We are grateful for helpful discussions to S\'ebastian Boucksom, Lawrence Ein, John Christian Ottem, Mihnea Popa, and Stefano Urbinati. Parts of this work were done while the authors attended the MFO workshop on Newton--Okounkov bodies, the Summer School in Geometry at University of Milano--Bicocca, and the RTG Workshop on Newton--Okounkov bodies at the University of Illinois at Chicago. We would like to thank the organizers of these events for the opportunity (Megumi Harada, Kiumars Kaveh, Askold Khovanskii; Francesco Bastianelli, Roberto Paoletti; Izzet Coskun and Kevin Tucker). Alex K\"uronya was partially supported by the DFG-Forschergruppe 790 ``Classification of Algebraic Surfaces and Compact Complex Manifolds'', by the DFG-Graduier\-ten\-kol\-leg 1821 ``Cohomological Methods in Geometry'', and by the OTKA grants 77476 and 81203 of the Hungarian Academy of Sciences. \section{Notation and preliminaries} \subsection{Notation} For the duration of this work let $X$ be a smooth complex projective variety of dimension $n$ and $D$ be a Cartier divisor on $X$. An admissible flag of subvarieties \[ Y_{\bullet} \ : \ X=Y_0 \supseteq Y_1\supseteq \ldots \supseteq Y_{n-1}\supseteq Y_n=\{\textup{pt.}\}, \] is a complete flag with the property that each $Y_i$ is an irreducible subvariety of codimension $i$ and smooth at the point $Y_n$. For an arbitrary point $x\in X$, we say that $Y_{\bullet}$ is \textit{centered at} $x$ whenever $Y_n=x$. The associated Newton--Okounkov body will be denoted by $\Delta_{Y_{\bullet}}(D)\subseteq \ensuremath{\mathbb{R}}^n_+$. \begin{remark} Not all of our results require $X$ to be smooth, at points it would suffice to require $X$ to be merely a projective variety. As a rule though, we will not keep track of minimal hypotheses. \end{remark} \subsection{Asymptotic base loci} Stable base loci are fundamental invariants of linear series, however, as their behaviour is somewhat erratic (they do not respect numerical equivalence of divisors for instance), other alternatives were in demand. To remedy the situation, Nakamaye came up with the idea of studying stable base loci of small perturbations. Based on this, the influential paper \cite{ELMNP1} introduced new asymptotic notions, the \textit{restricted} and \textit{augmented base loci} of a big divisor $D$. The restricted base locus of a big $\ensuremath{\mathbb{R}}$-divisor $D$ is defined as \[ \ensuremath{\textbf{\textup{B}}_{-} }(D) \deq \bigcup_{A} \textbf{B}(D+A)\ , \] where the union is over all ample $\ensuremath{\mathbb{Q}}$-divisors $A$ on $X$. This locus turns out to be a countable union of subvarieties of $X$ (and one really needs a countable union on occasion, see \cite{Les}) via \cite{ELMNP1}{Proposition~1.19} \[ \ensuremath{\textbf{\textup{B}}_{-} }(D) \ensuremath{\,=\,} \bigcup_{m\in\ensuremath{\mathbb{N}}}\textbf{B}(D+\frac{1}{m}A)\ . \] The augmented base locus of $D$ is defined to be \[ \ensuremath{\textbf{\textup{B}}_{+} }(D)\deq \bigcap_{A}\textbf{B}(D-A), \] where the intersection is taken over all ample $\ensuremath{\mathbb{Q}}$-divisors $A$ on $X$. It follows quickly from \cite{ELMNP1}{Proposition 1.5} that $\ensuremath{\textbf{\textup{B}}_{+} }(D)=\textbf{B}(D-\frac{1}{m}A)$ for all $m>>0$ and any fixed ample class $A$. Augmented and restricted base loci satisfy various favorable properties; for instance both $\ensuremath{\textbf{\textup{B}}_{+} } (D)$ and $\ensuremath{\textbf{\textup{B}}_{-} }(D)$ depend only on the numerical class of $D$, hence are much easier to study (see \cite{ELMNP1}{Corollary 2.10} and \cite{PAGII}{Example 11.3.12}). Below we make a useful remark regarding augmented/restricted base loci. The statement must be well-known to experts, as usual, we include it with proof for the lack of a suitable reference. \begin{proposition}\label{prop:openclosed} Let $X$ be a projective variety, $x\in X$ an arbitrary point. Then \begin{enumerate} \item $B_+(x) \deq \st{\alpha\in N^1(X)_\ensuremath{\mathbb{R}}\mid x\in \ensuremath{\textbf{\textup{B}}_{+} }(\alpha)} \ensuremath{\,\subseteq\,} N^1(X)_\ensuremath{\mathbb{R}}$ is closed, \item $B_-(x) \deq \st{\alpha\in N^1(X)_\ensuremath{\mathbb{R}}\mid x\in \ensuremath{\textbf{\textup{B}}_{-} }(\alpha)} \ensuremath{\,\subseteq\,} N^1(X)_\ensuremath{\mathbb{R}}$ is open, \end{enumerate} both with respect to the metric topology of $N^1(X)_\ensuremath{\mathbb{R}}$. \end{proposition} \begin{remark} We point out that unlike required in \cite{ELMNP1}, one does not need the normality assumption on $X$ for \cite{ELMNP1}{Corollary 1.6} to hold. \end{remark} \begin{proof} $(i)$ First we deal with the case of augmented base loci. Observe that it suffices to prove that \[ B_+(x) \cap \Bbig(X) \subseteq \Bbig(X) \] is closed, since the big cone is open in the N\'eron--Severi space. We will show that whenever $(\alpha_n)_{n\in\ensuremath{\mathbb{N}}}$ is a sequence of big $\ensuremath{\mathbb{R}}$-divisor classes in $B(x)$ converging to $\alpha\in \Bbig(X)$, then $\alpha\in B(x)$ as well. By \cite{ELMNP1}{Corollary 1.6}, the class $\alpha$ has a small open neighbourhood $\ensuremath{\mathscr{U}}$ in the big cone for which \[ \beta\in \ensuremath{\mathscr{U}} \ \Longrightarrow \ \textup{\textbf{B}}_+(\beta)\subseteq \ensuremath{\textbf{\textup{B}}_{+} }(\alpha)\ . \] If $x\in \ensuremath{\textbf{\textup{B}}_{+} }(\alpha_n)$ for infinitely many $n\in \ensuremath{\mathbb{N}}$, then since $\alpha_n \in U$ for $n$ large, we also have $x\in \ensuremath{\textbf{\textup{B}}_{+} }(\alpha)$. \\ $(ii)$ Let $\alpha\in N^1(X)_\ensuremath{\mathbb{R}}$ be arbitrary, and fix an $\ensuremath{\mathbb{R}}$-basis $A_1,\dots,A_\rho$ of $N^1(X)_\ensuremath{\mathbb{R}}$ consisting of ample divisor classes. Observe that $x\in \ensuremath{\textbf{\textup{B}}_{-} }(\alpha)$ implies that $x\in \ensuremath{\textbf{\textup{B}}_{-} }(\alpha+t_0\sum_{i=1}^{\rho}A_i)$ for some $t_0>0$ thanks to the definition of the restricted base locus. Since subtracting ample classes cannot decrease $\ensuremath{\textbf{\textup{B}}_{-} }$, it follows that $x\in \ensuremath{\textbf{\textup{B}}_{-} }(\alpha)$ yields $x\in \ensuremath{\textbf{\textup{B}}_{-} } (\gamma)$ for all classes of the form $\alpha+t_0\sum_{i=1}^{\rho}A_i - \sum_{i=1}^{\rho}\ensuremath{\mathbb{R}}_{\geqslant 0}A_i$, which certainly contains an open subset of $\alpha\in N^1(X)_\ensuremath{\mathbb{R}}$. \end{proof} \subsection{Newton--Okounkov bodies} We start with a sligthly different definition of Newton--Okounkov bodies; it has already appeared in print in \cite{KLM1}, and although it is an immediate consequence of \cite{LM}, a complete proof was first given in \cite{Bou1}{Proposition 4.1}. \begin{proposition}[Equivalent definition of Newton-Okounkov bodies]\label{prop:definition} Let $\xi\in\textup{N}^1(X)_{\ensuremath{\mathbb{R}}}$ be a big $\ensuremath{\mathbb{R}}$-class and $Y_{\bullet}$ be an admissible flag on $X$. Then \[ \Delta_{Y_{\bullet}}(\xi) \ = \ \textup{closed convex hull of\ }\{ \nu_{Y_{\bullet}}(D) \ \| \ D\in \textup{Div}_{\geqslant 0}(X)_{\ensuremath{\mathbb{R}}}, D\equiv \xi\}, \] where the valuation $\nu_{Y_{\bullet}}(D)$, for an effective $\ensuremath{\mathbb{R}}$-divisor $D$, is constructed inductively as in the case of integral divisors. \end{proposition} \begin{remark} Just as in the case of the original definition of Newton--Okounkov bodies, it becomes a posteriori clear that valuation vectors $\nu_{\ensuremath{Y_\bullet}}(D)$ form a dense subset of \[ \textup{closed convex hull of }\{ \nu_{Y_{\bullet}}(D) \ \| \ D\in \textup{Div}_{\geqslant 0}(X)_{\ensuremath{\mathbb{R}}}, D\equiv \xi\}\ , \] hence it would suffice to take closure in Proposition~\ref{prop:definition}. \end{remark} The description of Newton--Okounkov bodies above is often more suitable to use than the original one. For example, the following statement follows immediately from it. \begin{proposition}\label{prop:compute} Suppose $\xi$ is a big $\ensuremath{\mathbb{R}}$-class and $Y_{\bullet}$ is an admissible flag on $X$. Then for any $t\in [0,\mu(\xi, Y_1))$, we have \[ \Delta_{Y_{\bullet}}(\xi)_{\nu_1\geqslant t} \ = \ \Delta_{Y_{\bullet}}(\xi-tY_1)\ + t\ensuremath{\textup{\textbf{e}}_1}, \] where $\mu(\xi,Y_1)=\sup \{\mu>0\|\xi-\mu Y_1 \textup{ is big}\}$ and $\ensuremath{\textup{\textbf{e}}_1}=(1,0,\ldots ,0)\in \ensuremath{\mathbb{R}}^n$. \end{proposition} This statement first appeared in \cite{LM}{Theorem~4.24} with the additional condition that $Y_1\nsubseteq \ensuremath{\textbf{\textup{B}}_{+} }(\xi)$. We will need a version of \cite{AKL}{Lemma 8} for real divisors. \begin{lemma}\label{lem:nested} Let $D$ be a big $\ensuremath{\mathbb{R}}$-divisor, $A$ an ample $\ensuremath{\mathbb{R}}$-divisor, $Y_\bullet$ an admissible flag on $X$. Then, for any real number $\epsilon > 0$, we have \[ \Delta_{Y_\bullet}(D) \subseteq \Delta_{Y_\bullet}(D+\epsilon A) \ , \] and $\Delta_{Y_\bullet}(D)\ensuremath{\,=\,} \bigcap_{\epsilon>0}\Delta_{Y_\bullet}(D+\epsilon A)$. \end{lemma} \begin{proof} For the first claim, since $A$ is an ample $\ensuremath{\mathbb{R}}$-divisor, one can find an effective $\ensuremath{\mathbb{R}}$-divisor $M\sim_{\ensuremath{\mathbb{R}}}A$ with $Y_n\notin\Supp(M)$. Then for any arbitrary effective divisor $F\sim_{\ensuremath{\mathbb{R}}}D$ one has $F+\epsilon M \equiv_{\ensuremath{\mathbb{R}}} D+\epsilon A$ and $\nu_{\ensuremath{Y_\bullet}}(F+M)=\nu_{\ensuremath{Y_\bullet}}(F)$. Therefore \[ \st{\nu_{Y_{\bullet}}(\xi) \mid \xi\in \textup{Div}_{\geqslant 0}(X)_{\ensuremath{\mathbb{R}}}, D\equiv \xi} \subseteq \st{ \nu_{Y_{\bullet}}(\xi) \mid \xi \in \textup{Div}_{\geqslant 0}(X)_{\ensuremath{\mathbb{R}}}, D+\epsilon A\equiv \xi} \] and we are done by Proposition~\ref{prop:definition}. The equality of the second claim is a consequence of the previous inclusion and the continuity of Newton--Okounkov bodies. \end{proof} \section{Restricted base loci} Our main goal here is to give a characterization of restricted base loci in the language of Newton--Okounkov bodies. \begin{theorem}\label{thm:main1} Let $D$ be a big $\ensuremath{\mathbb{R}}$-divisor on a smooth projective variety $X$ of dimension $n$, let $x\in X$. Then the following are equivalent. \begin{enumerate} \item $x\not\in \ensuremath{\textbf{\textup{B}}_{-} }(D)$. \item There exists an admissible flag $\ensuremath{Y_\bullet}$ on $X$ centered at $x$ such that $\ensuremath{\textup{\textbf{0}}}\in \Delta_{\ensuremath{Y_\bullet}}(D)\subseteq\ensuremath{\mathbb{R}}^n$. \item The origin $\ensuremath{\textup{\textbf{0}}}\in \Delta_{\ensuremath{Y_\bullet}}(D)$ for every admissible flag $\ensuremath{Y_\bullet}$ on $X$ centered at $x\in X$. \end{enumerate} \end{theorem} Coupled with simple properties of restricted base loci we arrive at a precise description of big and nef divisors in terms of convex geometry. \begin{corollary}\label{cor:nef} With notation as above the following are equivalent for a big $\ensuremath{\mathbb{R}}$-divisor $D$. \begin{enumerate} \item $D$ is nef. \item For every point $x\in X$ there exists an admissible flag $\ensuremath{Y_\bullet}$ on $X$ centered at $x$ such that $\ensuremath{\textup{\textbf{0}}}\in \Delta_{\ensuremath{Y_\bullet}}(D)\subseteq\ensuremath{\mathbb{R}}^n$. \item For every admissible flag $\ensuremath{Y_\bullet}$, one has $\ensuremath{\textup{\textbf{0}}}\in \Delta_{\ensuremath{Y_\bullet}}(D)$. \end{enumerate} \end{corollary} \begin{proof} Immediate from Theorem~\ref{thm:main1} and \cite{ELMNP1}{Example 1.18}. \end{proof} The essence of the proof of Theorem~\ref{thm:main1} is to connect the asymptotic multiplicity of $D$ at $x$ to a certain function defined on the Newton-Okounkov body of $D$. Before turning to the actual proof, we will quickly recall the notion of the \emph{asymptotic multiplicity} or the \emph{asymptotic order of vanishing} of a $\ensuremath{\mathbb{Q}}$-divisor $F$ at a point $x\in X$. Let $F$ be an effective Cartier divisor on $X$, defined locally by the equation $f\in \ensuremath{\mathscr{O}}_{X,x}$. Then \textit{multiplicity} of $F$ at $x$ is defined to be $\mult_x(F)=\textup{max}\{n\in\ensuremath{\mathbb{N}} \| f\in \mathfrak{m}^n_{X,x}\}$, where $\mathfrak{m}_{X,x}$ denotes the maximal ideal of the local ring $\ensuremath{\mathscr{O}}_{X,x}$. If $\|V\|$ is a linear series, then the multiplicity of $\|V\|$ is defined to be \[ \mult_x(\|V\|) \deq \min_{F\in \|V\|}\{\mult_x(F)\}\ . \] By semicontinuity the above expression equals the multiplicity of a general element in $\|V\|$ at $x$. The \emph{asymptotic multiplicity} of a $\ensuremath{\mathbb{Q}}$-divisor $D$ at $x$ is then defined to be \[ \mult_x(\|\|D\|\|) \deq \lim_{p\rightarrow \infty}\frac{\mult_x(\|pD\|)}{p}\ . \] The multiplicity at $x$ coincides with the order of vanishing at $x$, given in Definition~2.9 from \cite{ELMNP1}. In what follows we will talk about the multiplicity of a divisor, but the order of vanishing of a section of a line bundle. An important technical ingredient of the proof of Theorem~\ref{thm:main1} is a result of \cite{ELMNP1}, which we now recall. \begin{proposition}(\cite{ELMNP1}{Proposition 2.8})\label{prop:ELMNP} Let $D$ be a big $\ensuremath{\mathbb{Q}}$-divisor on a smooth projective variety $X$, $x\in X$ an arbitrary (closed) point. Then the following are equivalent. \begin{enumerate} \item There exists $C>0$ having the property that $\mult_x(\|pD\|)<C$, whenever $\|pD\|$ is nonempty for some positive integer $p$.. \item $\mult_x(\\|D\\|) = 0$. \item $x\notin \ensuremath{\textbf{\textup{B}}_{-} }(D)$. \end{enumerate} \end{proposition} The connection between asymptotic multiplicity and Newton--Okounkov bodies comes from the claim below. \begin{lemma}\label{lem:1} Let $M$ be an integral Cartier divisor on a projective variety $X$ (not necessarily smooth), $s\in H^0(X,\ensuremath{\mathscr{O}}_X(M))$ a non-zero global section. Then \begin{equation}\label{eq:1} \ord_x(s) \ \leqslant \ \sum_{i=1}^{i=n} \nu_i(s), \end{equation} for any admissible flag $Y_{\bullet}$ centered $x$, where $\nu_{\ensuremath{Y_\bullet}}=(\nu_1,\ldots ,\nu_n)$ is the valuation map arising from $Y_{\bullet}$. \end{lemma} \begin{proof} Since $Y_{\bullet}$ is an admissible flag and the question is local, we can assume without loss of generality that each element in the flag is smooth, thus $Y_{i}\subseteq Y_{i-1}$ is Cartier for each $1\leqslant i\leqslant n$. As the local ring $\ensuremath{\mathscr{O}}_{X,x}$ is regular, order of vanishing is multiplicative. Therefore \[ \ord_x(s) \ensuremath{\,=\,} \nu_1(s)+\ord_x (s-\nu_1(s)Y_1) \ensuremath{\,\leqslant\,} \nu_1(s)+\ord_x ( (s-\nu_1(s)Y_1)\|_{Y_1}) \] by the very definition of $\nu_{\ensuremath{Y_\bullet}}(s)$, and the rest follows by induction. \end{proof} \begin{remark}\label{rem:1} Note that the inequality in (\ref{eq:1}) is not in general an equality for the reason that the zero locus of $s$ might not intersect an element of the flag transversally. For the simplest example of this phenomenon set $X=\ensuremath{\mathbb{P}}^2$, and take $s=xz-y^2\in H^0(\ensuremath{\mathbb{P}}^2,\ensuremath{\mathscr{O}}_{\ensuremath{\mathbb{P}}^2}(2))$, $Y_1=\{x=0\}$ and $Y_2=[0:0:1]$. Then clearly $\nu_1(s)=0$, and $\nu_2(s)=\ord_{Y_2}(-y^2)=2$, but since $Y_2$ is a smooth point of $(s)_0=\{xz-y^2=0\}$, $\ord_{Y_2}(s)=1$ and hence $\ord_{Y_1}(s)<\nu_1(s)+\nu_2(s)$. \end{remark} For a compact convex body $\Delta\subseteq \ensuremath{\mathbb{R}}^n$, we define the \textit{sum function} $\sigma:\Delta\rightarrow \ensuremath{\mathbb{R}}_{+}$ by $\sigma(x_1,\ldots ,x_n)=x_1+\ldots +x_n$. Being continuous on a compact topological space, it takes on its extremal values. If $\Delta_{Y_{\bullet}}(D)\subseteq \ensuremath{\mathbb{R}}^n$ be a Newton--Okounkov body, then we denote the sum function by $\sigma_D$, even though it does depend on the choice of the flag $Y_{\bullet}$. \begin{proposition}\label{prop:1} Let $D$ be a big $\ensuremath{\mathbb{Q}}$-divisor on a projective variety $X$ (not necessarily smooth) and let $x\in X$ a point. Then \begin{equation}\label{eq:2} \mult_x(\|\|D\|\|) \ \leqslant \ \min \sigma_D . \end{equation} for any admissible flag $Y_{\bullet}$ centered at $x$. \end{proposition} \begin{proof} Since both sides of (\ref{eq:2}) are homogeneous of degree one in $D$, we can assume without loss of generality that $D$ is integral. Fix a natural number $p\geqslant 1$ such that $\|pD\|\neq \varnothing$, and let $s\in H^{0}(X,\ensuremath{\mathscr{O}}_X(pD))$ be a non-zero global section. Then \[ \frac{1}{p}\mult_x(\|pD\|) \ensuremath{\,\leqslant\,} \frac{1}{p}\ord_x(s) \ensuremath{\,\leqslant\,} \frac{1}{p}\big(\sum_{i=1}^{i=n}\nu_i(s)\big) \] by Lemma~\ref{lem:1}. Multiplication of sections and the definition of the multiplicity of a linear series then yields $\mult_x(\|qpD\|)\leqslant q\mult_x(\|pD\|)$ for any $q\geqslant 1$, which, after taking limits leads to \[ \mult_x(\|\|D\|\|) \ensuremath{\,\leqslant\,} \frac{1}{p}\mult_x(\|pD\|) \ensuremath{\,\leqslant\,} \frac{1}{p}\big(\sum_{i=1}^{i=n}\nu_i(s)\big). \] Varying the section $s$ and taking into account that $\Delta_{\ensuremath{Y_\bullet}}(D)$ is the closure of the set of normalized valuation vectors of sections, we deduce the required statement. \end{proof} \begin{example}\label{rem:2} The inequality in (\ref{eq:2}) is usually strict. For a concrete example take $X=\textup{Bl}_P(\ensuremath{\mathbb{P}}^2)$, $D=\pi^(H)+E$ and the flag $Y_{\bullet}=(C,x)$, where $C\in\|3\pi^(H)-2E\|$ is the proper transform of a rational curve with a single cusp at $P$, and $\{x\}=C\cap E$, i.e. the point where $E$ and $C$ are tangent to each other. Then \[ \mult_x(\|\|D\|\|) \ensuremath{\,=\,} \lim_{p\to\infty}\Big(\frac{\mult_x(\|pD\|)}{p}\Big) \ensuremath{\,=\,} \lim_{p\to\infty}\Big(\frac{\mult_x(\|pE\|)}{p}\Big) \ensuremath{\,=\,} 1\ . \] On the other hand, a direct computation using \cite{LM}{Theorem 6.4} shows that \[ \Delta_{Y_{\bullet}}(D) \ensuremath{\,=\,} \{ (t,y)\in\ensuremath{\mathbb{R}}^2 \ \| \ 0\leqslant t\leqslant \frac{1}{3}, \textup{ and } 2+4t\leqslant y\leqslant 5-5t\}\ . \] As a result, $\min\sigma_D =2 > 1$. For more on this phenomenon, see Proposition~\ref{prop:starting} below. \end{example} \begin{remark} We note here a connection with functions on Okounkov bodies coming from divisorial valuations. With the notation of \cite{BKMS12}, our Lemma~\ref{lem:1} says that $\phi_{\ord_x} \leqslant \sigma_D$, and a quick computation shows that we obtain equality in the case of projectice spaces, hyperplane bundles, and linear flags. Meanwhile, Example~\ref{rem:2} illustrates that $\min \phi_{\ord_x} \neq \mult_x \\|D\\|$ in general. \end{remark} \begin{proof}[Proof of Theorem~\ref{thm:main1}] $(1)\Rightarrow(3)$ We are assuming $x\notin \ensuremath{\textbf{\textup{B}}_{-} }(D)$; let us fix an ample Cartier divisor $A$ and a decreasing sequence of real number $t_m$ such that $D+t_mA$ is a $\ensuremath{\mathbb{Q}}$-divisor. Let $\ensuremath{Y_\bullet}$ be an arbitrary admissible flag centered at $x$. Then $x\notin \textbf{B}(D+t_mA)$ for every $m\geqslant 1$, furthermore, since $A$ is ample, Lemma~\ref{lem:nested} yields \begin{equation}\label{eq:3} \Delta_{Y_{\bullet}}(D) \ensuremath{\,=\,} \bigcap_{m=1}^{\infty} \Delta_{Y_{\bullet}}(D+t_mA) \ . \end{equation} Because $x\notin \textbf{B}(D+t_mA)$ holds for any $m\geqslant 1$, there must exist a sequence of natural numbers $n_m\geqslant 1$ and a sequence of global sections $s_m\in H^0(X,\ensuremath{\mathscr{O}}_X(n_m(D+t_mA)))$ such that $s_m(x)\neq 0$. This implies that $\nu_{Y_{\bullet}}(s_m)=\ensuremath{\textup{\textbf{0}}}$ for each $m\geqslant 1$. In particular, $\ensuremath{\textup{\textbf{0}}}\in \Delta_{Y_{\bullet}}(D+t_mA)$ for each $m\geqslant 1$. By (\ref{eq:3}) we deduce that $\Delta_{Y_{\bullet}}(D)$ contains the origin as well. The implication $(3)\Rightarrow (2)$ being trivial, we will now take care of $(2)\Rightarrow (1)$. To this end assume that $Y_{\bullet}$ is an admissible flag centered at $x$ having the property that $\ensuremath{\textup{\textbf{0}}}\in\Delta_{Y_{\bullet}}(D)$, let $A$ be an ample divisor, and $t_m$ a sequence of positive real numbers converging to zero with the additional property that $D+t_mA$ is a $\ensuremath{\mathbb{Q}}$-divisor. By Lemma~\ref{lem:nested}, \[ \ensuremath{\textup{\textbf{0}}} \in \Delta_{\ensuremath{Y_\bullet}}(D) \ensuremath{\,\subseteq\,} \Delta_{\ensuremath{Y_\bullet}}(D+t_mA) \] for all $m\geqslant 0$. Whence $\min\sigma_{D+t_mA}=0$ for all sum functions $\sigma_D:\Delta_{Y_{\bullet}}(D)\rightarrow\ensuremath{\mathbb{R}}_{+}$. By Proposition~\ref{prop:1} this forces $\mult_x(\|\|D+t_mA\|\|)=0$ for all $m\geqslant 1$, hence \cite{ELMNP1}{Proposition 2.8} leads to $x\notin\ensuremath{\textbf{\textup{B}}_{-} }(D+t_mA)$ for all $m\geqslant 1$. Since Proposition~\ref{prop:ELMNP} is only valid for $\ensuremath{\mathbb{Q}}$-divisors, we are left with proving the equality \[ \ensuremath{\textbf{\textup{B}}_{-} }(D) \ensuremath{\,=\,} \bigcup_m\ensuremath{\textbf{\textup{B}} }(D+t_mA) \ensuremath{\,=\,} \bigcup_m\ensuremath{\textbf{\textup{B}}_{-} }(D+t_mA)\ . \] The first equality comes from \cite{ELMNP1}{Proposition 1.19}, as far as the second one goes, $\bigcup_m\ensuremath{\textbf{\textup{B}} }(D+t_mA) \supseteq \bigcup_m\ensuremath{\textbf{\textup{B}}_{-} }(D+t_mA)$ holds since $\ensuremath{\textbf{\textup{B}}_{-} }(D+t_mA)\subseteq \textbf{B}(D+t_mA)$ for any $m\in \ensuremath{\mathbb{N}}$ according to \cite{ELMNP1}{Exercise~1.16}. To show $\bigcup_m\ensuremath{\textbf{\textup{B}} }(D+t_mA) \subseteq \bigcup_m\ensuremath{\textbf{\textup{B}}_{-} }(D+t_mA)$, we note that for any $t_m$ as above one finds $t_{m+k}<t_m$ for some natural number $k\in \ensuremath{\mathbb{N}}$. Then \[ \ensuremath{\textbf{\textup{B}} }(D+t_mA) \ensuremath{\,=\,} \ensuremath{\textbf{\textup{B}} }(D+t_{m+k}A+(t_m-t_{m+k})A) \ensuremath{\,=\,} \ensuremath{\textbf{\textup{B}}_{-} } (D+t_{m+k}A) \ , \] where the latter inclusion follows from the definition of the restricted base locus. \end{proof} \begin{remark}\label{rmk:non-smooth nef} A closer inspection of the above proof reveals that the implication $(1)\Rightarrow (3)$ holds on an arbitrary projective variety both in Theorem~\ref{thm:main1} and Corollary~\ref{cor:nef}. \end{remark} We finish with a precise version of Proposition~\ref{prop:1} in the surface case, which also provides a complete answer to the question of where the Newton-Okounkov body starts in the plane. Note that unlike Theorem~\ref{thm:main3}, it gives a full description for an arbitrary flag. \begin{proposition}\label{prop:starting} Let $X$ be a smooth projective surface, $(C,x)$ an admissible flag, $D$ a big $\ensuremath{\mathbb{Q}}$-divisor on $X$ with Zariski decomposition $D=P(D)+N(D)$. Then \begin{enumerate} \item $\min\sigma_D=a+b$, where $a=\mult_C(N(D))$ and $b=\textup{mult}_{x}(N(D-aC)\|_C)$, \item $\mult_x(\|\|D\|\|)=a+b'$, where $b'=\mult_x(N(D-aC))$. \end{enumerate} Moreover, $(a,b)\in\Delta_{(C,x)}(D)$ and $\Delta_{(C,x)}(D)\subseteq (a,b)+\ensuremath{\mathbb{R}}^2_+$. \end{proposition} \begin{proof} $(1)$ This is an immediate consequence of \cite{LM}{Theorem 6.4} in the light of the fact that $\alpha$ is an increasing function, hence $\min\sigma_D$ is taken up at the point $(a,\alpha(a))$. \\ $(2)$ Since $x$ is a smooth point, it will suffice to check that $\mult_x(\|\|D\|\|)=\mult_x(N(D))$. As asymptotic multiplicity is homogeneity of degree one (see \cite{ELMNP1}{Remark 2.3}), we can safely assume that $D, P(D)$ and $N(D)$ are all integral. As one has isomorphisms $H^0(X,\ensuremath{\mathscr{O}}_X(mP(D)))\rightarrow H^0(X,\ensuremath{\mathscr{O}}_X(mD))$ for all $m\geqslant 1$ by \cite{PAGI}{Proposition 2.3.21}, the definition of asymptotic multiplicity yields \[ \mult_x(\|\|D\|\|) \ensuremath{\,=\,} \mult_x(\|\|P(D)\|\|) \ + \ \mult_x(N(D)) \ . \] Observe that $P(D)$ is big and nef therefore \cite{PAGI}{Proposition 2.3.12} implies $\mult_x(\|\|P(D)\|\|)=0$. This completes the proof. \end{proof} \section{Augmented base loci} As explained in \cite{ELMNP1}{Example 1.16}, one has inclusions $\ensuremath{\textbf{\textup{B}}_{-} }(D)\subseteq \textup{\textbf{B}}(D)\subseteq\ensuremath{\textbf{\textup{B}}_{+} }(D)$, consequently, we expect that whenever $x\notin\ensuremath{\textbf{\textup{B}}_{+} }(D)$, Newton--Okounkov bodies attached to $D$ should contain more than just the origin. As we shall see below, it will turn out that under the condition above they in fact contain small simplices. We will write \[ \Delta_{\epsilon} \deq \{ (x_1,\ldots, x_n)\in\ensuremath{\mathbb{R}}^n_{+} \ \| \ x_1+\ldots +x_n\leqslant \epsilon\} \] for the standard $\epsilon$-simplex. Our main statement is the following. \begin{theorem}\label{thm:main2} Let $D$ be a big $\ensuremath{\mathbb{R}}$-divisor on $X$, $x\in X$ be an arbitrary (closed) point. Then the following are equivalent. \begin{enumerate} \item $x\notin \textup{\textbf{B}}_+(D)$. \item There exists an admissible flag $Y_{\bullet}$ centered at $x$ with $Y_1$ ample such that $\Delta_{\epsilon_0} \subseteq \Delta_{Y_{\bullet}}(D)$ for some $\epsilon_0 >0$. \item For every admissible flag $Y_{\bullet}$ centered at $x$ there exists $\epsilon >0$ (possibly depending on $\ensuremath{Y_\bullet}$) such that $\Delta_{\epsilon}\subseteq \Delta_{Y_{\bullet}}(D)$. \end{enumerate} \end{theorem} \begin{corollary}\label{cor:ample} Let $X$ be a smooth projective variety, $D$ a big $\ensuremath{\mathbb{R}}$-divisor on $X$. Then the following are equivalent. \begin{enumerate} \item $D$ is ample. \item For every point $x\in X$ there exists an admissible flag $Y_{\bullet}$ centered at $x$ with $Y_1$ ample such that $\Delta_{\epsilon_0} \subseteq \Delta_{Y_{\bullet}}(D)$ for some $\epsilon_0 >0$. \item For every admissible flag $Y_{\bullet}$ there exists $\epsilon >0$ (possibly depending on $\ensuremath{Y_\bullet}$) such that $\Delta_{\epsilon}\subseteq \Delta_{Y_{\bullet}}(D)$. \end{enumerate} \end{corollary} \begin{proof}[Proof of Corollary~\ref{cor:ample}] Follows immediately from Theorem~\ref{thm:main2} and \cite{ELMNP1}{Example 1.7}. \end{proof} One can see Corollary~\ref{cor:ample} as a variant of Seshadri's criterion for ampleness in the language of convex geometry. \begin{remark} It is shown in \cite{KL}{Theorem 2.4} and \cite{KL}{Theorem A} that in dimension two one can in fact discard the condition above that $Y_1$ should be ample. Note that the proofs of the cited results rely heavily on surface-specific tools and in general follow a line of thought different from the present one. \end{remark} We first prove a helpful lemma. \begin{lemma}\label{lem:2} Let $X$ be a projective variety (not necessarily smooth), $A$ an ample Cartier divisor, $Y_{\bullet}$ an admissible flag on $X$. Then for all $m>>0$ there exist global sections $s_0,\ldots ,s_n\in H^0(X,\ensuremath{\mathscr{O}}_X(mA))$ for which \[ \nu_{Y_{\bullet}}(s_0) = \ensuremath{\textup{\textbf{0}}} \textup{ and } \nu_{Y_{\bullet}}(s_i)=\ensuremath{\textup{\textbf{e}}_i}, \textup{ for each } i=1,\ldots ,n, \] where $\{ \ensuremath{\textup{\textbf{e}}_1},\ldots ,\ensuremath{\textup{\textbf{e}}_n}\}\subseteq \ensuremath{\mathbb{R}}^n$ denotes the standard basis. \end{lemma} \begin{proof} First, we point out that by the admissibility of the flag $\ensuremath{Y_\bullet}$, we know that there is an open neighbourhood $sU$ of $x$ such that $Y_i\|_{\ensuremath{\mathscr{U}}}$ is smooth for all $0\leqslant i\leqslant n$. Since $A$ is ample, $\ensuremath{\mathscr{O}}_X(mA)$ becomes globally generated for $m>>0$. For all such $m$ like there exists a non-zero section $s_0\in H^0(X,\ensuremath{\mathscr{O}}_X(mA))$ with $s_0(Y_n)\neq 0$, in particular, $\nu_{Y_{\bullet}}(s_0)=\ensuremath{\textup{\textbf{0}}}$, as required. It remains to show that for all $m>>0$ and $i=1\leqslant i\leqslant n$ we can find non-zero sections $s_i\in H^0(X,\ensuremath{\mathscr{O}}_X(mA))$ with $\nu_{Y_{\bullet}}(s_i)=\ensuremath{\textup{\textbf{e}}_i}$. To this end, fix $i$ and let $y\in Y_i\setminus Y_{i+1}$ be a smooth point. Having chosen $m$ large enough, Serre vanishing yields $H^1(X,\ensuremath{\kern -1pt \mathscr{I}\kern -2pt}_{Y_i\|X}\otimes \ensuremath{\mathscr{O}}_X(mA))=0$, hence the map $\phi_m$ in the diagram \[ \xymatrix{ & & H^0(X,\ensuremath{\mathscr{O}}_X(mA)) \ar[d]_{\phi_m} \\ 0\ar[r] & H^0(Y_i,\ensuremath{\mathscr{O}}_{Y_i}(m(A\|_{Y_i})-Y_{i+1})) \ar[r]^<<<<{\psi_m} & H^0(Y_i,\ensuremath{\mathscr{O}}_{Y_i}(mA)) } \] is surjective. Again, by making $m$ high enough, we can assume $\|m(A\|_{Y_i})-Y_{i+1}\|$ to be very ample on $Y_i$, thus, there will exist $0\neq\tilde{s}_i\in H^0(Y_i,\ensuremath{\mathscr{O}}_{Y_i}(mA)\otimes \ensuremath{\mathscr{O}}_{Y_i}(-Y_{i+1}))$ not vanishing at $x$ or $y$. Since $\tilde{s}_i(x)\neq 0$, the section $\tilde{s}_i$ does not vanish along $Y_j$ for all $j=i+1,\ldots ,n$. Also, the image $\psi_m(\tilde{s}_i)\in H^0(Y_i,\ensuremath{\mathscr{O}}_{Y_i}(mA))$ of $\tilde{s}_i$ vanishes at $x$, but not at the point $y$. By the surjectivity of the map $\phi_m$ there exists a section $s_i\in H^0(X,\ensuremath{\mathscr{O}}_X(mA))$ such that $s\|_{Y_i}=\psi_m(\tilde{s}_i)$ and $s(y)\neq 0$. In particular, $s_i$ does not vanish along any of the $Y_j$'s for $1\leqslant i\leqslant j$, therefore $\nu_{Y_{\bullet}}(s)=\ensuremath{\textup{\textbf{e}}_i}$, as promised. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:main2}] $(1)\Rightarrow (3)$. First we treat the case when $D$ is $\ensuremath{\mathbb{Q}}$-Cartier. Assume that $x\notin \ensuremath{\textbf{\textup{B}}_{+} }(D)$, which implies by definition that $x\notin\ensuremath{\textbf{\textup{B}} }(D-A)$ for some small ample $\ensuremath{\mathbb{Q}}$-Cartier divisor $A$. Choose a positive integer $m$ large and divisible enough such that $mA$ becomes integral, and satisfies the conclusions of Lemma~\ref{lem:2}. Assume furthermore that $\ensuremath{\textbf{\textup{B}} }(D-A)=\textup{Bs}(m(D-A))$ set-theoretically. Since $x\notin \textup{Bs}(m(D-A))$, there exists a section $s\in H^0(X,\ensuremath{\mathscr{O}}_X(mD-mA))$ not vanishing at $x$, and in particular $\nu_{Y_{\bullet}}(s)=\ensuremath{\textup{\textbf{0}}}$. At the same time, Lemma~\ref{lem:2} provides the existence of global sections $s_0,\ldots, s_n\in H^0(X,\ensuremath{\mathscr{O}}_X(mA))$ with the property that $\nu_{Y_{\bullet}}(s_0)\ensuremath{\,=\,} \ensuremath{\textup{\textbf{0}}}$ and $\nu_{Y_{\bullet}}(s_i)=\ensuremath{\textup{\textbf{e}}_i}$ for all $1\leqslant i\leqslant n$. But then the multiplicativity of the valuation map $\nu_{\ensuremath{Y_\bullet}}$ gives \[ \nu_{Y_{\bullet}}(s\otimes s_0)=\textbf{\textup{0}}, \textup{ and } \nu_{Y_{\bullet}}(s\otimes s_i)=\ensuremath{\textup{\textbf{e}}_i} \ \ \text{for all $1\leqslant i\leqslant n$.} \] By the construction of Newton--Okounkov bodies, then $\Delta_{\frac{1}{m}}\subseteq \Delta_{Y_{\bullet}}(D)$. Next, let $D$ be a big $\ensuremath{\mathbb{R}}$-divisor for which $x\notin\ensuremath{\textbf{\textup{B}}_{+} }(D)$, and let $A$ be an ample $\ensuremath{\mathbb{R}}$-divisor with the property that $D-A$ is a $\ensuremath{\mathbb{Q}}$-divisor and $\ensuremath{\textbf{\textup{B}}_{+} }(D)=\ensuremath{\textbf{\textup{B}}_{+} }(D-A)$. Then we have $x\notin\ensuremath{\textbf{\textup{B}}_{+} }(D-A)$, therefore \[ \Delta_\epsilon\ensuremath{\,\subseteq\,} \Delta_{\ensuremath{Y_\bullet}}(D-A) \ensuremath{\,\subseteq\,} \Delta_{\ensuremath{Y_\bullet}}(D) \] according to the $\ensuremath{\mathbb{Q}}$-Cartier case and Lemma~\ref{lem:nested}. Again, the implication $(3)\Rightarrow (2)$ is trivial, hence we only need to take care of $(2)\Rightarrow (1)$. As $Y_1$ is ample, \cite{ELMNP1}{Proposition 1.21} gives the equality $\ensuremath{\textbf{\textup{B}}_{-} }(D-\epsilon Y_1)=\ensuremath{\textbf{\textup{B}}_{+} }(D)$. for all $0<\epsilon <<1$. Fix an $\epsilon$ as above, subject to the additional condition that $D-\epsilon Y_1$ is a big $\ensuremath{\mathbb{Q}}$-divisor. Then, according to Proposition~\ref{prop:compute}, we have \[ \Delta_{Y_{\bullet}}(D)_{\nu_1\geqslant \epsilon} \ = \ \Delta_{Y_{\bullet}}(D-\epsilon Y_1)\ + \ \epsilon \ensuremath{\textup{\textbf{e}}_1} \ , \] which yields $\ensuremath{\textup{\textbf{0}}}\in \Delta_{Y_{\bullet}}(D-\epsilon Y_1)$. By Theorem~\ref{thm:main1}, this means that $x\notin\ensuremath{\textbf{\textup{B}}_{-} }(D-\epsilon Y_1)=\ensuremath{\textbf{\textup{B}}_{+} }(D)$, which completes the proof. \end{proof} \begin{remark}\label{rmk:non-smooth ample} The condition that $X$ be smooth can again be dropped for the implication $(1)\Rightarrow (3)$ both in Theorem~\ref{thm:main2} and Corollary~\ref{cor:ample} (cf. Remark~\ref{rmk:non-smooth nef}). This way, one obtains the statement that whenever $A$ is an ample $\ensuremath{\mathbb{R}}$-Cartier divisor on a projective variety $X$, then every Newton--Okounkov body of $A$ contains a small simplex. \end{remark} As a consequence, we can extend \cite{KL}{Definition 4.5} to all dimensions. \begin{definition}[Largest simplex constant] Let $X$ be an arbitrary projective variety, $x\in X$ a smooth point, $A$ an ample $\ensuremath{\mathbb{R}}$-divisor on $X$. For an admissible flag $\ensuremath{Y_\bullet}$ on $X$ centered at $x$, we set \[ \lambda_{\ensuremath{Y_\bullet}}(A;x) \deq \sup\st{\lambda>0\mid \Delta_\lambda\subseteq\Delta_{\ensuremath{Y_\bullet}}(A)}\ . \] Then the \emph{largest simplex constant} $\lambda(A;x)$ is defined as \[ \lambda(A;x) \deq \sup \st{\lambda_{\ensuremath{Y_\bullet}}(A;x)\mid \text{ $\ensuremath{Y_\bullet}$ is an admissible flag centered at $x$}}\ . \] \end{definition} \begin{remark} It follows from Remark~\ref{rmk:non-smooth ample} that $\lambda(A;x)>0$. The largest simplex constant is a measure of local positivity, and it is known in dimension two that $\lambda(A;x)\leqslant \epsilon(A;x)$ (where the right-hand side denotes the appropriate Seshadri constant) with strict inequality in general (cf. \cite{KL}{Proposition 4.7} and \cite{KL}{Remark 4.9}). \end{remark} We end this section with a different characterization of $\ensuremath{\textbf{\textup{B}}_{+} }(D)$ which puts no restriction on the flags. In what follows $X$ is again assumed to be smooth. \begin{lemma}\label{lem:aug} For a point $x\in X$, $x\notin\ensuremath{\textbf{\textup{B}}_{+} }(D)$ holds if and only if \begin{equation}\label{eq:augmented} \lim_{p\rightarrow\infty} \mult_x(\|\|pD-A\|\|) \ensuremath{\,=\,} 0 \end{equation} for some ample divisor $A$. \end{lemma} \begin{proof} Assuming (\ref{eq:augmented}), $x\notin\ensuremath{\textbf{\textup{B}}_{+} }(D)$ follows from \cite{ELMNP3}{Lemma~5.2}. For the converse implication, consider the equalities \[ \ensuremath{\textbf{\textup{B}}_{+} }(D) \ensuremath{\,=\,} \ensuremath{\textbf{\textup{B}}_{-} }(D-\frac{1}{p}A) \ensuremath{\,=\,} \textup{\textbf{B}}(pD-A) \] which hold for integers $p\gg 0$. Hence, if $x\notin \ensuremath{\textbf{\textup{B}}_{+} }(D)$, then $x\notin\textup{\textbf{B}}(pD-A)$ for all $p\gg 0$. But this latter condition implies $\mult_x(\|\|pD-A\|\|)=0$ for all $p\gg 0$ for all $p\gg 0$. \end{proof} \begin{proposition}\label{prop:augmented1} A point $x\notin\ensuremath{\textbf{\textup{B}}_{+} }(D)$ if and only if there exists an admissible flag $Y_{\bullet}$ based at $x$ satisfying the property that for any $\epsilon>0$ there exists a natural number $p_{\epsilon}>0$ such that \[ \Delta_{\epsilon} \ \bigcap \Delta_{Y_{\bullet}}(pD-A) \ \neq \ \varnothing \] for any $p\geqslant p_{\epsilon}$. \end{proposition} \begin{proof} Assume first that $x\notin \ensuremath{\textbf{\textup{B}}_{+} }(D)$. Again, by \cite{ELMNP1}{Proposition 1.21}, we have $\ensuremath{\textbf{\textup{B}}_{+} }(D)=\ensuremath{\textbf{\textup{B}}_{-} }(D-\tfrac{1}{p}A)=\ensuremath{\textbf{\textup{B}}_{-} }(pD-A)$ for all $p\gg 0$. Then $x\notin \ensuremath{\textbf{\textup{B}}_{-} }(pD-A)$ for all $p\gg 0$, hence $\ensuremath{\textup{\textbf{0}}}\in\Delta_{\ensuremath{Y_\bullet}}(pD-A)$ for all $p\gg 0$ by Theorem~\ref{thm:main1}, which implies $\Delta_{\epsilon} \cap \Delta_{Y_{\bullet}}(pD-A) \neq \varnothing$ for all $p\gg 0$. As far as the converse implication goes, Proposition~\ref{prop:1} shows that \[ \mult_x(\|\|pD-A\|\|) \ensuremath{\,\leqslant\,} \min \sigma_{pD-A}.\ , \] hence the condition in the statement implies $\lim_{p\rightarrow\infty} \mult_x(\|\|pD-A\|\|)= 0$. But then we are done by Lemma~\ref{lem:aug}. \end{proof} \section{Nakayama's divisorial Zariski decomposition and Newton--Okounkov bodies} In the previous sections we saw the basic connections between Newton--Okounkov bodies associated to a big line bundle $D$ and the asymptotic base loci $\ensuremath{\textbf{\textup{B}}_{+} }(D)$ and $\ensuremath{\textbf{\textup{B}}_{-} }(D)$. In \cite{Nak}, Nakayama performes a deep study of these loci, he shows for instance that $\ensuremath{\textbf{\textup{B}}_{-} }(D)$ can only have finitely many divisororial components. Along the way he introduces his $\sigma$-invariant, which measures the asymptotic multiplicity of divisorial components of $\ensuremath{\textbf{\textup{B}}_{-} }(D)$. The goal of this section is to study the connection between divisorial Zariski decomposition and Newton--Okounkov bodies. First, we briefly recall the divisorial Zariski decomposition or $\sigma$-decomposition introduced by Nakayama \cite{Nak} and Boucksom \cite{Bou2}. Let $X$ be a smooth projective variety, $D$ a pseudo-effective $\ensuremath{\mathbb{R}}$-divisor on $X$. Although $\ensuremath{\textbf{\textup{B}}_{-} }(D)$ is a countable union of closed subvarieties, \cite{Nak}{Theorem 3.1} shows that it only has finitely many divisorial components. Let $A$ be an ample divisor. Following Nakayama, for each prime divisor $\Gamma$ on $X$ we set \[ \sigma_{\Gamma} \deq \lim_{\epsilon\rightarrow 0^+}\textup{inf}\{\mult_{\Gamma}(D') \mid D'\sim_{\ensuremath{\mathbb{R}}}D+\epsilon A \textup{ and } D'\geqslant 0\}\ . \] In \cite{Nak}{Theorem III.1.5}, Nakayama shows that these numbers do not depend on the choice of $A$ and that there are only finitely many prime divisors $\Gamma$ with $\sigma_{\Gamma}(D)>0$. Write \[ N_{\sigma}(\Gamma)\deq \sum_{\Gamma}\sigma_{\Gamma}(D)\Gamma \textup{ and } P_{\sigma}(D)=D-N_{\sigma}(D) \ , \] and we call $D=P_{\sigma}(D)+N_{\sigma}(D)$ \emph{the divisorial Zariski decomposition} or \emph{$\sigma$-decomposition} of $D$. In dimension two divisorial Zariski decomposition coincides with the usual Fujita--Zariski decomposition for pseudo-effective divisors. The main properties are captured in the following statement. \begin{theorem}\cite{Nak}{III.1.4, III.1.9, V.1.3}\label{prop:Nakayama} Let $D$ be a pseduo-effective $\ensuremath{\mathbb{R}}$-disivor. Then \begin{enumerate} \item $N_{\sigma}(D)$ is effective and $\Supp(N_{\sigma}(D))$ coincides with the divisorial part of $\ensuremath{\textbf{\textup{B}}_{-} }(D)$. \item For all $m\geqslant 0$, $H^0(X,\ensuremath{\mathscr{O}}_X(\lfloor mP_{\sigma}(D)\rfloor))\simeq H^0(X,\ensuremath{\mathscr{O}}_X(\lfloor mD\rfloor))$. \end{enumerate} \end{theorem} As Theorem~\ref{thm:main1} describes how to determine $\ensuremath{\textbf{\textup{B}}_{-} }(D)$ from the Newton--Okounkov bodies associated to $D$, it is natural to wonder how we can compute the numbers $\sigma_\Gamma(D)$ and $N_\sigma(D)$ in terms of convex geometry. Relying on Theorem~\ref{thm:main1} and Nakayama's work, we are able to come up with a reasonable answer. \begin{theorem}\label{thm:main3} Let $D$ be a big $\ensuremath{\mathbb{R}}$-divisor, $\Gamma$ a prime divisor on $X$, $Y_{\bullet}: Y_0=X\supseteq Y_1=\Gamma\supseteq\ldots \supseteq Y_n=\{x\}$ an admissible flag on $X$. Then \begin{enumerate} \item $\Delta_{Y_{\bullet}}(D) \ \subseteq \ (\sigma_{\Gamma}(D),0\ldots, 0)+\ensuremath{\mathbb{R}}_+^n$, \item $(\sigma_{\Gamma}(D),0\ldots, 0)\ \in \ \Delta_{Y_{\bullet}}(D)$, whenever $x\in \Gamma$ is a very general point. \item $\Delta_{Y_{\bullet}}(D) = \nu_{Y_{\bullet}}(N_{\sigma}(D)) + \Delta_{Y_{\bullet}}(P_{\sigma}(D))$. Morever, $\Delta_{Y_{\bullet}}(D) =\Delta_{Y_{\bullet}}(P_{\sigma}(D))$, when $x\notin\Supp (N_{\sigma}(D))$. \end{enumerate} \end{theorem} \begin{proof}[Proof of Theorem~\ref{thm:main3}] For the duration of this proof we fix an ample divisor $A$. \\ $(1)$ This is equivalent to $\sigma_\Gamma(D)\leqslant \nu_1(D')$ for every effective $\ensuremath{\mathbb{R}}$-divisor $D'\equiv D$. Fix a real number $\epsilon >0$, let $D''\sim_{\ensuremath{\mathbb{R}}}D+\epsilon A$ is an effective $\ensuremath{\mathbb{R}}$-divisor. Then \[ \inf\{\mult_{\Gamma}(D')\|D'\sim_{\ensuremath{\mathbb{R}}}D+\epsilon A\} \ensuremath{\,\leqslant\,} \mult_{\Gamma}(D'') \ensuremath{\,=\,} \nu_1(D'')\ . \] By Proposition~\ref{prop:definition}, this implies the inclusion \[ \Delta_{Y_{\bullet}}(D+\epsilon A)\subseteq (\sigma'(D+\epsilon A),0,\ldots 0)+\ensuremath{\mathbb{R}}^n_+ \ . \] Then Lemma~\ref{lem:nested} and the definition of $\sigma_{\Gamma}(D)$ imply the claim.\\ $(2)$ By \cite{Nak}{Lemma~2.1.5} we have $\sigma_{\Gamma}(D-\sigma_{\Gamma}(D)\Gamma)=0$. Consequently, we obtain $\Gamma\nsubseteq\ensuremath{\textbf{\textup{B}}_{-} }(D-\sigma(D)\Gamma)$. Because $\ensuremath{\textbf{\textup{B}}_{-} }(D-\sigma(D)\Gamma)$ is a countable union of subvarieties of $X$, a very general point $x$ lies outside $\ensuremath{\textbf{\textup{B}}_{-} }(D-\sigma_{\Gamma}(D)\Gamma)$. Theorem~\ref{thm:main1} yields $\ensuremath{\textup{\textbf{0}}}\in\Delta_{\ensuremath{Y_\bullet}}(D-\sigma_{\Gamma}(D)\Gamma)$, therefore the point $(\sigma_{\Gamma}(D),0\ldots, 0)$ is contained in $\Delta_{\ensuremath{Y_\bullet}}(D)$. \\ $(3)$ Let $D_{\sigma}\sim_{\ensuremath{\mathbb{R}}}P_{\sigma}(D)$ be an effective $\ensuremath{\mathbb{R}}$-divisor, then $D_{\sigma}+N_{\sigma}(D)\sim_{\ensuremath{\mathbb{R}}} D$ is also an effective divisor for which \[ \nu_{Y_{\bullet}}(D_{\sigma}+N_{\sigma}(D)) \ = \ \nu_{Y_{\bullet}}(D_{\sigma})+\nu_{Y_{\bullet}}(N_{\sigma}(D)) \ . \] This implies the inclusion $\nu_{Y_{\bullet}}(N_{\sigma}(D)) + \Delta_{Y_{\bullet}}(P_{\sigma}(D))\subseteq \Delta_{Y_{\bullet}}(D)$ via Proposition~\ref{prop:definition}. For an effective $\ensuremath{\mathbb{R}}$-divisor $D'\sim_{\ensuremath{\mathbb{R}}}D$, \cite{Nak}{III.1.14} gives that the divisor $D_{\sigma}=D'-N_{\sigma}(D)\sim_{\ensuremath{\mathbb{R}}}P_{\sigma}(D)$ is effective. Thus $\nu_{Y_{\bullet}}(D') \ensuremath{\,=\,} \nu_{Y_{\bullet}}(D_{\sigma})+\nu_{Y_{\bullet}}(N_{\sigma}(D))$, which completes the proof. \end{proof} Next, we study the variation of Zariski decomposition after Nakayama when varying the divisors inside the pseudo-effective cone. We start with the following lemma. \begin{lemma}\label{lem:varying} Suppose $D$ is a big $\ensuremath{\mathbb{R}}$-divisor on $X$ and $E$-prime effective divisor. If $\sigma_E(D)=0$, then $\sigma_E(D-tE)=0$ for all $t\geqslant 0$. \end{lemma} \begin{proof} The condition $\sigma_E(D)=0$ implies $E\nsubseteq \ensuremath{\textbf{\textup{B}}_{-} }(D)$, thus, by Theorem~\ref{thm:main1}, for a flag $Y_{\bullet}:X\supseteq E\supseteq \ldots\supseteq \{x\}$, with $x\in E$ very general point, we have that $\ensuremath{\textup{\textbf{0}}}\in \Delta_{Y_{\bullet}}(D)$. Again, by the very general choice of $x\in E$, Theorem~\ref{thm:main3} says that $\sigma_E(D-tE)\cdot \ensuremath{\textup{\textbf{e}}_1}\in \Delta_{Y_{\bullet}}(D-tE)$. On the other hand, by Proposition~\ref{prop:compute} we know that $\Delta_{Y_{\bullet}}(D)_{\nu_1\geqslant t} = \Delta_{Y_{\bullet}}(D-tE)+t\ensuremath{\textup{\textbf{e}}_1}$, therefore $(\sigma_E(D-tE)+t)\ensuremath{\textup{\textbf{e}}_1}\in \Delta_{Y_{\bullet}}(D)$. By convexity, this implies $t\cdot \ensuremath{\textup{\textbf{e}}_1}\in \Delta_{Y_{\bullet}}(D)$, again by Proposition~\ref{prop:compute} we have $\ensuremath{\textup{\textbf{0}}}\in \Delta_{Y_{\bullet}}(D-tE)$, hence $\sigma_E(D-tE)=0$ by the choice of $x\in E$ and Theorem~\ref{thm:main3}. \end{proof} The next proposition shows how the negative part of the Zariski decomposition varies inside the big cone. \begin{proposition}\label{prop:varying} Suppose $D$ is a big $\ensuremath{\mathbb{R}}$-divisor on $X$ and $E$ a prime effective divisor. Then \begin{enumerate} \item If $\sigma_{E}(D)>0$, then $N_{\sigma}(D-tE)=N_{\sigma}(D)-tE$, for any $t\in [0,\sigma_E(D)]$. \item If $\sigma_E(D)=0$, then the function $t\rightarrow N_{\sigma}(D-tE)$ is an increasing function, i.e. for any $t_1\geqslant t_2$ the divisor $N_{\sigma}(D-t_1)-N_{\sigma}(D-t_2E)$ is effective. \end{enumerate} \end{proposition} \begin{proof} $(1)$ This statement is proved in Lemma~1.8 from \cite{Nak}. $(2)$ Since $\sigma_E(D)=0$, then Lemma~\ref{lem:varying} implies that $\sigma_E(D-tE)=0$ for any $t\geqslant 0$ and in particular $E\nsubseteq \textup{Supp}(N_{\sigma}(D-tE))$ for any $t\geqslant 0$. So, take $\Gamma\subseteq \textup{Supp}(N_{\sigma}(D-t_2E))$ a prime divisor. The goal is to prove that $\sigma_{\Gamma}(D-t_1E)\geqslant \sigma_{\Gamma}(D-t_2E)$. Without loss of generality, we assume that $t_2=0$ and $t_1=t\geqslant 0$ and we need to show that $\sigma_{\Gamma}(D-tE)\geqslant \sigma_{\Gamma}(D)$. Now take a flag $Y_{\bullet}:X\supseteq \Gamma\supseteq \ldots\supseteq\{x\}$, where $x\in \Gamma$ is a very general point and $x\notin E$. Then by Theorem~\ref{thm:main3} we have \[ \sigma_{\Gamma}(D)\cdot \ensuremath{\textup{\textbf{e}}_1}\in \Delta_{Y_{\bullet}}(D)\subseteq \sigma_{\Gamma}(D)\cdot \ensuremath{\textup{\textbf{e}}_1}+\ensuremath{\mathbb{R}}^n_+ \] and \[ \sigma_{\Gamma}(D-tE)\cdot \ensuremath{\textup{\textbf{e}}_1}\in \Delta_{Y_{\bullet}}(D-tE)\subseteq \sigma_{\Gamma}(D-tE)\cdot \ensuremath{\textup{\textbf{e}}_1}+\ensuremath{\mathbb{R}}^n_+. \] On the other hand, it is not hard to see that $\Delta_{Y_{\bullet}}(D-tE)\subseteq \Delta_{Y_{\bullet}}(D)$. For any $D'\sim_{\ensuremath{\mathbb{R}}}D-tE$ effective $\ensuremath{\mathbb{R}}$-divisor, the $\ensuremath{\mathbb{R}}$-divisor $D'+tE\sim_{\ensuremath{\mathbb{R}}}D$ is also effective. Since $x\notin E$, then $\nu_{Y_{\bullet}}(D')=\nu_{Y_{\bullet}}(D'+tE)$ and the inclusion follows naturally. Combining this and the above information we obtain that $\sigma_{\Gamma}(D-tE)\cdot \ensuremath{\textup{\textbf{e}}_1}\in \Delta_{Y_{\bullet}}(D)$ and thus $\sigma_{\Gamma}(D-tE)\geqslant \sigma_{\Gamma}(D)$. \end{proof} \begin{bibdiv} \begin{biblist} \bib{AKL}{article}{ label={AKL}, author={Anderson, Dave}, author={K\"uronya, Alex}, author={Lozovanu, Victor}, title={Okounkov bodies of finitely generated divisors}, journal={International Mathematics Research Notices}, volume={132}, date={2013}, number={5}, pages={1205--1221}, } \bib{BKS}{article}{ label={BKS}, author={Bauer, Thomas}, author={K\"uronya, Alex}, author={Szemberg, Tomasz}, title={Zariski decompositions, volumes, and stable base loci}, journal={Journal f\"ur die reine und angewandte Mathematik}, volume={576}, date={2004}, pages={209--233}, } \bib{Bou1}{article}{ label={B1}, author={Boucksom, S\'ebastien}, title={Corps D'Okounkov}, journal={S\'eminaire Bourbaki}, volume={65}, date={2012-2013}, number={1059}, pages={1--38}, } \bib{Bou2}{article}{ label={B2}, author={Boucksom, S{\'e}bastien}, title={Divisorial Zariski decompositions on compact complex manifolds}, journal={Ann. 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Michèle Raynaud Michèle Raynaud (born Michèle Chaumartin;[1](1938-01-09)January 9, 1938 [2][3]) is a French mathematician, who works on algebraic geometry and who worked with Alexandre Grothendieck in Paris in the 1960s at the Institut des hautes études scientifiques (IHÉS). Michèle Raynaud Born Michèle Chaumartin (1938-01-09) 9 January 1938 France NationalityFrench Alma materParis Diderot University Scientific career FieldsMathematics Doctoral advisorAlexander Grothendieck Biography Raynaud was a member of the séminaire de géométrie algébrique du Bois Marie (SGA) 1 and 2 and obtained her doctorate in 1972, supervised by Grothendieck at Paris Diderot University. Her thesis was entitled Théorèmes de Lefschetz en cohomologie cohérente et en cohomologie étale.[4] Grothendieck wrote about her doctoral thesis in Récoltes et Semailles (p.168 Chapitre 8.1.) describing it as original, entirely independent, and a major work. Michèle Raynaud was married to the mathematician Michel Raynaud[5] who was also a member of the Grothendieck school. Publications • Théorèmes de Lefschetz en cohomologie cohérente et en cohomologie étale, Bull. Soc. Math. France, Memoirs Nr. 41, 1975 • Théorèmes de Lefschetz en cohomologie étale des faisceaux en groupes non nécessairement commutatifs. C. R. Acad. Sci. Paris Sér. A-B 270 1970 • Théorème de représentabilité relative sur le foncteur de Picard • Schémas en groupes. Séminaire de l'Institut des Hautes Etudes Scientifiques • Grothendieck, Alexander; Raynaud, Michèle (2003) [1971], Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 3, Paris: Société Mathématique de France, arXiv:math/0206203, Bibcode:2002math......6203G, ISBN 978-2-85629-141-2, MR 2017446 • Grothendieck, Alexander; Raynaud, Michèle (2005) [1968], Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2), Documents Mathématiques (Paris), vol. 4, Paris: Société Mathématique de France, arXiv:math/0511279, Bibcode:2005math.....11279G, ISBN 978-2-85629-169-6, MR 2171939 Notes and references 1. Illusie, Luc (2019). "Michel Raynaud (1938–2018)" (PDF). Notices of the American Mathematical Society. 66 (1). Retrieved 18 August 2020.{{cite journal}}: CS1 maint: url-status (link) 2. "BnF Catalogue général". Retrieved 8 October 2019. 3. "VIAF". Retrieved 8 October 2019. 4. Michèle Raynaud at the Mathematics Genealogy Project 5. Gassiat, Elisabeth (March 2018). "Décès de Michel Raynaud". Société Mathématique de France (in French). Retrieved 2018-03-14. Authority control International • ISNI National • France • BnF data • Germany • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
\begin{document} \title{General superposition states associated to the rotational and inversion symmetries in the phase space.} \begin{abstract} The general quantum superposition states containing the irreducible representation of the $n$-dimensional groups associated to the rotational symmetry of the $n$-sided regular polygon i.e., the cyclic group ($C_n$) and the rotational and inversion symmetries of the polygon, i.e., the dihedral group ($D_n$) are defined and studied. It is shown that the resulting states form an $n$-dimensional orthogonal set of states which can lead to the finite representation of specific systems. The correspondence between the symmetric states and the renormalized states, resulting from the selective erasure of photon numbers from an arbitrary, noninvariant initial state, is also established. As an example, the general cyclic Gaussian states are presented. The presence of nonclassical properties in these states as subpoissonian photon statistics is addressed. Also, their use in the calculation of physical quantities as the entanglement in a bipartite system is discussed. \end{abstract} \section{Introduction} The study of symmetries in physics has helped to the simplification of difficult problems. For example, the symmetries in the Hamiltonian dynamical evolution of a quantum system can be related to the definition of different conservation laws which, as in the classical theory, can be used to answer different questions. The use of symmetries in quantum mechanics, in particular the definition of states associated to point symmetry groups has been covered in several works \cite{manko1,manko2,manko3,castanos}. Especially, the states carrying the symmetry of the cyclic group $C_2=\mathbb{Z}/(2 \mathbb{Z})$, also called odd an even cat states, have been of great interest in the past decades. The nonclassical properties of this kind of states have been discussed in \cite{buzek}, together with their use in fundamental quantum theory \cite{sanders,wenger,jeong,stob,wineland} and in the quantum information framework \cite{vanerick,jeong2,ralph,gilchrist,bergmann}. For several years, there was an impossibility to construct a cat state with a large photon number. Instead of that, the low photon cat states, known as kitten states, were generated \cite{ourjoumtsev1}. After that, the possibility to obtain full cat states has been demonstrated in several studies as: by using the reflexion of a coherent pulse from a optical cavity with one atom \cite{hacker,wang}, the use of homodyne detection in a photon number state \cite{ourjoumtsev2}, the photon subtraction from a squeezed vacuum state in a parametric amplifier \cite{neegaard}, via ancilla-assisted photon subtraction \cite{takahashi}, and by the subtraction of an specific photon number in a squeezed vacuum state \cite{gerrits}. The superposition of coherent states have non-classical features like squeezing of the quadrature components \cite{buzek,janszky,domokos}. There exist a possible experimental implementation of these superpositions \cite{szabo}, in particular superpositions of coherent states on a circle \cite{janszky,domokos,gonzalez}. The states adapted to this type of symmetry have also a connection to the phase-time operators in the harmonic oscillator \cite{susskind,nieto1,pegg1,pegg2}. The definition of states carrying the circle symmetry has been extended by the use of spin coherent states as in \cite{calixto}, also in \cite{calixto1} the use of $su(1,1)$ coherent states on the hyperboloid were considered. More recently, a proposed method to generate states with higher discrete symmetries, as the ones defined here, has been obtained by the dynamic evolution of a matter-field interaction described by the Tavis-Cummings model \cite{cordero1,cordero2}. There is experimental evidence for the generation of superpositions of four coherent states with a number of 111 photons \cite{vlastakis}. Also, the cluster structure of light nuclei as $^{12}C$ and $^{13}C$ have been describe by the point symmetry groups, as the ones discussed here, $D_{3h}$ and $D_{3h}^\prime$, respectively. In this work, the generalization of the quantum states associated to the irreducible representations of the group whose elements are the symmetry rotations of the $n$-sided regular polygon, also named the cyclic group ($C_n=\mathbb{Z}/(n \mathbb{Z})$), and the group containing the rotational and reflexion symmetries of the regular polygon, i.e., the dihedral group ($D_n$), is presented. Some of these type of states have been previously defined using coherent states \cite{manko1,manko2,manko3,castanos}. In the present work, it is shown that the cyclic and dihedral states form an orthogonal set of states, which can be used to define a discrete representation of states made of the superposition of rotations, in the case of the cyclic group, and rotations plus reflections in the case of the dihedral group. Also it can be seen that this discrete representation can simplify the calculation of quantum parameters as the entanglement between two subsystems within a system. For these reasons, we consider that given the applications of the cyclic and dihedral coherent states in quantum information, the generalization of such states to the noncoherent case is important. The proposed method discussed here, makes use of an initial state $\vert \phi \rangle$ which is not invariant under rotations. To define the cyclic states, the superposition of the rotated states $\vert \phi_r \rangle=\hat{R}(\theta_r)\vert \phi \rangle$ ($r=1,\ldots,n$; $\theta_r=2\pi(r-1)/n$), and the characters associated to the $\lambda$-th irreducible representation and the $r$-th element of the group ($\chi^{(\lambda)}(g_r)$ ), are used. It is also discussed the relation between the cyclic states and the renormalized states obtained from the erasure of certain photon numbers in the photon statistics of $\vert \phi \rangle$ or $\hat{\rho}$, e.g., the cat states associated to the cyclic group $C_2$: $\vert \xi_\pm \rangle = N_\pm (\vert \alpha \rangle\pm\vert-\alpha\rangle)$ are the renormalized states resulting of eliminating the even and odd photon number states from the coherent state $\vert \alpha \rangle$ respectively. On the other hand, the dihedral group $D_n$ is the non-Abelian group that contains the rotations and inversions which leave the $n$-sided regular polygon invariant. The elements of the dihedral group are $D_n:\{\hat{R}(\theta_j), \hat{U}_j,\, j=1,\ldots,n \}$, with $\theta_j=2\pi(j-1)/n$, where the inversion operators in the phase space are defined by a rotation plus the complex conjugation ($\hat{C}$), i.e., $\hat{U}_j=\hat{C}\hat{R}(\theta_j)$. Additionally to pure, non-pure cyclic and dihedral states can be defined through a density matrix. These states correspond to a quantum map of an noninvariant, arbitrary operator $\hat{\rho}$. This type of quantum maps have been recently relevant in quantum information theory. In particular, the quantum maps have been important for the quantum error correction as some of the studied qubit maps represent the interaction between a qubit and an environment \cite{terhal,caruso}. Furthermore, the study of the erasure map, presented here, can be important to figure out the experimental realization of the defined states, as the resulting states, depend on the absorption (erasure) of certain state numbers. As a remainder of some group characteristics we establish that given a $n$ dimensional group $\{g_r;\, r=1, \ldots,n \}$, a conjugacy class is formed by all the elements $g_k$ which satisfy the similarity transformation $g_k^{-1} g_j g_k=g_j$, where $g_j$ is also a member of the group. An irreducible representation $\lambda$ is the representation of a group that cannot decompose further. To obtain the irreducible representation sometimes the following procedure should be applied: if there exist a similarity transformation of an element of the group $g_j$ which diagonalize it, i.e., $C^{-1}g_j C=A_D$, where $A_D$ is made of diagonal matrices $A_{D_j^{(\lambda)}}$, then the matrices $A_{D_j^{(\lambda)}}$ form an irreducible representation of $g_j$. The character $\chi$ associated to the irreducible representation $\lambda$, is defined as the trace of the diagonal matrix $A_{D_\lambda}$, that is $\chi^{(\lambda)}(g_j)={\rm Tr}\left(A_{D_j^{(\lambda)}}\right)$. Also, all the members of a conjugacy class share the same characters. In the case of the cyclic states the character associated to the irreducible representation $\lambda$ and element $g_r$ of the group is given by $\chi^{(\lambda)}_n(g_r)=e^{2\pi i(\lambda-1)(r-1)/n}$ This work is organized as follows: In section 2 a review of the cyclic states constructed by means of coherent states are presented. The generalization of these type of states for a non-coherent system is then described in section 3. The correspondence between the generalized cyclic state and a renormalized state obtained through the elimination of certain photon numbers in an original system is studied in section 4. In section 5, some examples are given, the cyclic Gaussian states are defined and some of their properties are exemplified. Also, the circle symmetry states are presented as an extension to the states associated to $C_n$, where $n\rightarrow \infty$. In section 6, the idea of the pure cyclic states of $C_n$ is extended to the case of non-pure density matrices. This is done by the definition of a map of the density matrix, which can also be related to the erasure and renormalization of certain photon numbers in the initial state. The usefulness of this kind of systems for the study of the entanglement in a two-mode system is shown in section 7. The dihedral states are defined in section 8. Finally, some conclusions are given. \section{Cyclic coherent states} In previous works, different states associated to the irreducible representation of cyclic groups \cite{manko1,manko2,manko3,castanos} have been defined using coherent states \cite{glauber,titulaer,birula,stoler}. The resulting states called crystallized cat states have some interesting properties as subpoissonian photon statistics, squeezing, and antibunching \cite{sun1,sun2,castanos}. Also, it has been demonstrated that they can be generated by the interaction of an atom with an electromagnetic field \cite{hacker,wang}. Here, we present a summary of the definition and some properties of the coherent cyclic states. The cyclic group $C_n$ have as elements the discrete rotations associated to the symmetries of the regular polygon of $n$ sides, i.e. $C_n=\{R(\theta_j), \theta_j = 2\pi (j-1) /n, \ {\rm with}\ (j=1,\ldots,n)\}$. The number of elements is equal to the cycle of the group and they can be divided in different conjugacy classes $\{g_r\}$. The characteristic (or character) of the class $g_r$ for the irreducible representation $\lambda$ is denoted as $\chi^{(\lambda)}_n(g_r)$ is given by the trace of the irreducible representation. It is known that in the case of the cyclic group each element forms its own class ($g_j=R(\theta_j)$) and that the character of the class are the $n$ roots of the identity, \begin{equation} \chi^{(\lambda)}_n(g_r)=\exp \left[ \frac{2 i \pi (\lambda-1)(r-1) }{n}\right] \, , \quad {\rm with}\ \lambda,r=1,\ldots,n \, . \label{chi} \end{equation} Additionally, the characters for any two irreducible representations $\lambda$ and $\lambda'$ are orthonormal, i.e., \begin{equation} \frac{1}{n}\sum_{r=1}^n \chi^{(\lambda)}_n(g_r) \chi^{(\lambda')}_n(g_r)= \delta_{\lambda \lambda'} \label{ort1} \end{equation} and also the sum of the characters over all the irreducible representations $\lambda$ satisfy that \begin{equation} \frac{1}{n}\sum_{\lambda=1}^n \chi^{(\lambda)}_n(g_r) \chi^{(\lambda)}_n(g_{r'})= \delta_{r r'} \, . \label{ort2} \end{equation} These two orthogonality conditions can be quickly checked using the rule for the sum of the identity roots \begin{equation} \sum_{j=1}^n \mu_n^j=0\, , \quad {\rm where} \ \mu_n=\exp\left(\frac{2\pi i}{n}\right), \label{powers} \end{equation} such property also leads to the following theorem. \begin{theorem} \label{tt1} Let $r$ be an integer and $\mu_n=\exp(2\pi i/n )$, then $\sum_{j=1}^n \mu_n^{jr}=n\, \delta_{{\rm mod}(r,n),0}$. \end{theorem} \begin{proof} It is clear that for $r$ being a multiple of $n$: ${\rm mod}(r,n)=0$, $\mu_n^{rj}=1$ and thus the sum $\sum_{j=1}^n \mu_n^{jr}$ is equal to $n$. For $r$ not being a multiple of $n$ (${\rm mod}(r,n) \neq 0$) we remember that the sum \[ \sum_{j=1}^n x^j=x\frac{x^n-1}{x-1} \, , \] which in the case of $x=\mu_n^r$, implies \[ \sum_{j=1}^n \mu_n^{jr}=\mu_n^r \frac{\mu_n^{rn}-1}{\mu_n^r-1}=0\, , \] as $\mu_n^{rn}=1$. It is important to notice that this property is satisfied for any integer, in particular by $r$ being a negative integer. \end{proof} Given the orthogonality properties in Eqs.~(\ref{ort1}) and (\ref{ort2}) one can define a macroscopic quantum state for each one of the irreducible representations of the cyclic group as follows \begin{equation} \left\vert \psi^{(\lambda)}_n \right\rangle=\mathcal{N}_\lambda \sum_{r=1}^n \chi^{(\lambda)}_n (g_r) \vert \alpha_r \rangle \, , \quad \sum_{r,r'=1}^n \chi^{(\lambda)}_n (g_r) \chi^{(\lambda)}_n (g_{r'}) \langle \alpha_{r'} \vert \alpha_r \rangle = \mathcal{N_\lambda}^{-2} \, , \label{cats} \end{equation} where the coherent state parameter $\alpha_r={\rm Re}(\alpha_r)+i\,{\rm Im}(\alpha_r)$ is given by the rotation of a fixed number $\alpha$ in the complex plane, \[ \left( \begin{array}{cc} {\rm Re}(\alpha_r) \\ {\rm Im}(\alpha_r)\end{array}\right)= R(\theta_r) \left( \begin{array}{cc} {\rm Re}(\alpha) \\ {\rm Im}(\alpha)\end{array}\right) \, . \] It is important to notice that all the states for different irreducible representations form an orthonomal set with $\left\langle \psi_n^{(\lambda)} \Big\vert \psi_n^{(\lambda')} \right\rangle = \delta_{\lambda \lambda'}$. In the case of the cyclic group $C_2$ we have as the result the standard odd and even cat states $\vert \psi^{(1,2)} \rangle= \mathcal{N}_\pm (\vert \alpha \rangle \pm \vert - \alpha \rangle)$, which can have subpoissonian photon statistic, squeezing, and antibunching \cite{castanos}. The coherent cyclic states $\vert \psi_n^{(\lambda)}\rangle$ are eigenvalues of the power of the annihilation operator $\hat{a}^n$, i.e., \[ \hat{a}^n \vert \psi_n^{(\lambda)}\rangle= \alpha^n \vert \psi_n^{(\lambda)}\rangle \, . \] Also, one can change the irreducible representation of the state by acting the annihilation operator $\hat{a}$ to another state: \[ \hat{a}\vert \psi_n^{(\lambda)}\rangle=\alpha \frac{\mathcal{N}_{\lambda}}{ \mathcal{N}_{\lambda'}} \vert \psi_n^{(\lambda')}\rangle \, , \] where the value of the new irreducible representation depends on the original one $\lambda'(\lambda)$. \section{Generalization of cyclic states as superpositions of rotations in the phase space.} The necessity of a generalization of the cyclic states to a superposition of arbitrary, non-coherent systems can be explained by their possible use in quantum information theory. Also, the cyclic states form an orthogonal set of states which can lead to a finite representation of certain quantum systems. First, let us suppose an initial quantum state $\vert \phi \rangle$ and its representation in the Fock basis \[ \vert \phi \rangle = \sum_{m=0}^\infty A_m(\phi) \vert m \rangle \, , \quad {\rm with} \ \sum_{m=0}^\infty \vert A_m(\phi) \vert^2 =1 \, . \] The discrete rotations in the phase space associated to the symmetries of the regular polygon in the cyclic group $C_n$ are given by the operator $\hat{R}(\theta_j)=\exp(-i \theta_j \hat{n})$, where $\theta_j=2 \pi(j-1)/n$; $j=1, \ldots , n$, and $\hat{n}$ is the bosonic number operator. To every one of the elements of the cyclic group we have then a rotation of the general state $\vert \phi \rangle$, which can be expressed as \[ \vert \phi_j \rangle = \hat{R}(\theta_j) \vert \phi \rangle \, . \] \begin{mydef} Let $\vert \phi \rangle=\sum_{m=0}^\infty A_m (\phi) \vert m \rangle$ be a quantum state with at least one mean quadrature component ($\hat{x}=(\hat{a}+\hat{a}^\dagger)/\sqrt{2}$, $\hat{p}=i(\hat{a}^\dagger-\hat{a})/\sqrt{2}$) different from zero, i.e., $\langle \phi \vert \hat{x} \vert \phi \rangle \neq 0$, or $\langle \phi \vert \hat{p} \vert \phi \rangle \neq 0$.We define the general cyclic state for the irreducible representation $\lambda$ of the group $C_n$ as \begin{equation} \left\vert \psi_n^{(\lambda)} (\phi) \right\rangle = \mathcal{N}_\lambda \sum_{r=1}^n \chi^{(\lambda)}_n (g_r) \vert \phi_r \rangle \, , \label{ccy} \end{equation} where $\chi_n^{(\lambda)}(g_r)$ is the character associated to the irreducible representation $\lambda$ and to the element of the group $g_r \in C_n$, and where \[ \mathcal{N}_\lambda^{-2}=\sum_{r,r'=1}^n \chi^{(\lambda)}(g_r) \chi^{(\lambda)}(g_{r'}) \langle \phi_{r'} \vert \phi_r \rangle \, . \] \label{defi1} \end{mydef} To obtain a well defined state we emphasize that the original state cannot be invariant under the rotations discussed above, i.e., $\vert \phi_r \rangle \neq \vert \phi \rangle$ for $r=2,\ldots,n$. This property can be satisfied when the Wigner function of the state in the phase space $W(x,p)$ is given by a non symmetric distribution or when the state is not centered at the origin of the phase space, i.e., $\int dx \, dp \, x \, W(x,p)\neq 0$, or $\int dx \, dp \, p \, W(x,p)\neq 0$. As these states carry the irreducible representation of the group $C_n$, they are invariant, up to a phase, under the discrete rotations $\hat{R}(\theta_j)$. To prove this property, lets suppose the action of the rotation $\hat{R}(\theta_l)$, $1\leq l \leq n$, over the state $\left\vert \psi_n^{(\lambda)} (\phi) \right\rangle$ \[ \hat{R}(\theta_l) \left\vert \psi_n^{(\lambda)} (\phi) \right\rangle= \mathcal{N}_\lambda \sum_{r=1}^n \chi_n^{(\lambda)} (g_r) \hat{R}(\theta_{r+l}) \vert \phi \rangle \, , \] as the character of the representation $\lambda$ is \[ \chi_n^{(\lambda)} (g_r)=\mu_n^{(\lambda-1)(r-1)}=\mu_n^{(\lambda-1)(r+l-1)}\mu_n^{(1-\lambda)l} = \chi_n^{(\lambda)} (g_{r+l}) \mu_n^{(1-\lambda)l} \, , \] then we obtain \[ \hat{R}(\theta_l) \left\vert \psi_n^{(\lambda)} (\phi) \right\rangle= \mathcal{N}_\lambda \, \mu_n^{(1-\lambda)l} \sum_{r=1}^n \chi_n^{(\lambda)} (g_{r+l}) \hat{R}(\theta_{r+l}) \vert \phi \rangle \, , \] given the periodicity of the characters and the rotation operators ($\mu_n^{x+n}=\mu_n^x$, $\hat{R}(\theta_{j+n})=\hat{R}(\theta_{j})$), this sum give us, up to a phase, the same state as the original, i.e., \begin{equation} \hat{R}(\theta_l) \left\vert \psi_n^{(\lambda)} (\phi) \right\rangle= \mu_n^{(1-\lambda)l} \left\vert \psi_n^{(\lambda)}(\phi) \right\rangle \, . \label{cyc_inv} \end{equation} It can also be seen that by the use of the explicit form of the rotated states in the Fock basis $\vert \phi_r \rangle=\sum_{m=0}^\infty A_m(\phi) e^{-i \theta_r m} \vert m \rangle$, one obtains \[ \left\vert \psi_n^{(\lambda)} (\phi) \right\rangle = \mathcal{N}_\lambda \sum_{r=1}^n \sum_{m=0}^\infty \chi^{(\lambda)}_n (g_r) A_m(\phi) e^{-i\theta_r m}\vert m \rangle \, , \] which can be also rewritten as \begin{equation} \left\vert \psi_n^{(\lambda)} (\phi) \right\rangle = \mathcal{N}_\lambda \sum_{r=1}^n \sum_{m=0}^\infty \mu_n^{(\lambda-1-m)(r-1)} A_m(\phi) \vert m \rangle \, . \label{cyc1} \end{equation} Given the characteristics of the sum of the powers of the parameter $\mu_n$, expressed in Eq.~(\ref{powers}), one can show that the different states for the cyclic group $C_n$ form an ortonormal set. To show this, lets suppose the inner product of two cyclic states with irreducible representations $\lambda$, and $\lambda^\prime$, i.e., \[ \left\langle \psi_n^{(\lambda^\prime)} (\phi) \right\vert \psi_n^{(\lambda)} (\phi) \Big\rangle= \mathcal{N}_\lambda \mathcal{N}_{\lambda^\prime} \sum_{r,r'=1}^n \sum_{m,m'=0}^\infty A_m(\phi) A_{m'}^(\phi)\, \mu_n^{(\lambda-1-m)(r-1)} \mu_n^{(1-\lambda^\prime+m')(r'-1)} \delta_{m',m} \, , \] performing first the sums over the parameter $r'$, we have \[ \left\langle \psi_n^{(\lambda^\prime)} (\phi) \right\vert \psi_n^{(\lambda)} (\phi) \Big\rangle= \mathcal{N}_\lambda \mathcal{N}_{\lambda^\prime} n \sum_{m=0}^\infty \sum_{r=1}^n \vert A_m (\phi) \vert^2 \mu_n^{\lambda^\prime-1-m}\, \mu_n^{(\lambda-1-m)(r-1)} \delta_{{\rm mod}(1-\lambda^\prime+m,n),0} \, . \] As established by Theorem~\ref{tt1}, this sum is different from zero when $1-\lambda^\prime+m=s n$ (with $s\in \mathbb{Z}$ ). This leads to the condition $m=sn-1+\lambda^\prime$. From this, we can change the sum over $m$ to a sum over $s$, obtaining \[ \left\langle \psi_n^{(\lambda^\prime)} (\phi) \right\vert \psi_n^{(\lambda)} (\phi) \Big\rangle= \mathcal{N}_\lambda \mathcal{N}_{\lambda^\prime} \sum_{s=0}^\infty \sum_{r=1}^n \vert A_{ns-1+\lambda} (\phi) \vert^2 \mu_n^{(\lambda-\lambda^\prime)r} \mu_n^{\lambda-\lambda'}\, . \] Similarly to the previous step, the sum over the parameter $r$ is different from zero when $\lambda-\lambda'=s' n$ with $s' \in \mathbb{Z}$. As the parameters satisfy $1\leq \lambda,\lambda' \leq n$, the only possible value is that $\lambda-\lambda'=0$, so \[ \left\langle \psi_n^{(\lambda^\prime)} (\phi) \right\vert \psi_n^{(\lambda)} (\phi) \Big\rangle= \mathcal{N}_\lambda \mathcal{N}_{\lambda^\prime} \sum_{s=0}^\infty \vert A_{ns-1+\lambda} (\phi) \vert^2 \delta_{\lambda,\lambda'} \, , \] which in the case $\lambda\neq \lambda'$ is equal to zero and by the expression for the normalization constant in Def.~(\ref{defi1}) is equal to one when $\lambda= \lambda'$. Finally, arriving to the expression \[ \left\langle \psi_n^{(\lambda^\prime)} (\phi) \right\vert \psi_n^{(\lambda)} (\phi) \Big\rangle=\delta_{\lambda,\lambda'} \, . \] Other important properties of the cyclic states are addressed in the next section. \section{State erasure as a quantum map and the cyclic states.} In this section, the connection between the erasure map and the cyclic states is studied. This correspondence can lead to the experimental implementation of the cyclic states as these states can be seen as coming from the absorption (or erasure) of certain photon numbers. The general cyclic states defined above, can also be defined as the result of selective loss of information in a quantum system, that is, from the erasure of a subset of states of an original state $\vert \phi \rangle=\sum_{n=0}^\infty A_m(\phi) \vert m \rangle$. As an example, one can suppose the selective erasure of the probabilities $A_m(\phi)$ for all values of odd $m$, and after this erasure, the renormalization of the state is performed. In that case, one will have the following state made with only even number states \begin{equation} \vert \psi_{even} \rangle = N \sum_{m \ even} A_m (\phi) \vert m \rangle \, , \label{even} \end{equation} where $N$ is the normalization constant $N^{-2}=\sum_{m \ even} \vert A_m(\phi) \vert^2$. Lets compare the previous expression with the cyclic state for $n=2$, $\lambda=1$: $\left\vert \psi_2^{(1)} (\phi)\right\rangle$. This state is given by \[ \left\vert \psi_2^{(1)} (\phi) \right\rangle=\mathcal{N}_1 \sum_{r=1}^2 \sum_{m=0}^\infty \mu_2^{r m} A_m (\phi) \vert m \rangle, \quad \mu_2=-1\, . \] By performing the sum over $r$, we then obtain \[ \left\vert \psi_2^{(1)} (\phi)\right\rangle=\mathcal{N}_1 \sum_{m=0}^\infty (1+(-1)^m) A_m (\phi) \vert m \rangle=\mathcal{N}_1 \sum_{m \ even} 2 A_m (\phi) \vert m \rangle \, , \] which is the same expression as Eq.~(\ref{even}) with $N=2\mathcal{N}_1$. The same can be done to the state resulting of the elimination of even states, which is equal to the cyclic state with $n=2$, $\lambda=2$, i.e., $\left\vert \psi_2^{(2)}(\phi) \right\rangle=N \sum_{m \ odd} A_m (\phi) \vert m \rangle$. In general, the equality between the cyclic states and the states resulting of the elimination of certain number states can be established in the following theorem. \begin{theorem} Let $n$ and $\lambda$ be two positive integers with $\lambda\leq n$, and $\vert \Psi_{n,\lambda} (\phi) \rangle$ be the renormalized state obtained after the elimination of the number states $\vert m \rangle$ in $\vert \phi \rangle=\sum_{m=0}^\infty A_m (\phi) \vert m \rangle$, which do not satisfy the condition ${\rm mod}(m-\lambda+1,n)=0$, then $\vert \Psi_{n,\lambda} (\phi)\rangle$ is equal to the cyclic state $\left\vert \psi_n^{(\lambda)}(\phi)\right\rangle$ up to a phase. \label{teo2} \end{theorem} \begin{proof} The state after the erasure map $\vert \Psi_{n,\lambda} (\phi) \rangle$, has the following expression \[ \vert \Psi_{n,\lambda} \rangle=N_{\lambda,n} \sum_{m} A_m (\phi) \delta_{{\rm mod}(m-\lambda+1,n),0} \vert m \rangle \] which only contains the number states that satisfy $m-\lambda+1=ln$ (with $l$ an nonnegative integer), then \begin{equation} \vert \Psi_{n,\lambda} (\phi) \rangle=N_{\lambda,n} \sum_{l} A_{\lambda-1+ln} (\phi) \vert \lambda-1+ln \rangle \, . \label{elim} \end{equation} On the other hand, by using the property for the roots of the identity ($\mu_n$) given in Theorem~\ref{tt1} ($\sum_{j=1}^n \mu_n^{j l}=n \, \delta_{{\rm mod}(l,n),0}$), in the definition of $\left\vert \psi_n^{(\lambda)}(\phi)\right\rangle$ in Eq.~(\ref{cyc1}), we can show that \[ \sum_{r=1}^n \mu_n^{(\lambda-1-m)(r-1)}=n \, \mu_n^{1-\lambda} \delta_{{\rm mod}(\lambda-1-m,n),0} \, , \] this means that only the states with $m=\lambda-1+l n$ (with $l$ a nonnegative integer) are part of $\left\vert \psi_n^{(\lambda)}\right\rangle$, i. e., \begin{equation} \left\vert \psi_n^{(\lambda)}(\phi)\right\rangle=n \, \mathcal{N}_\lambda \, \mu_n^{1-\lambda} \sum_{l=0}^\infty A_{\lambda-1+l n} (\phi)\vert \lambda-1+l n \rangle \, . \label{cyc2} \end{equation} Finally, when comparing Eqs.~(\ref{elim}) and~(\ref{cyc2}) we arrive to the conclusion \begin{equation} \vert \Psi_{n, \lambda} \rangle = \mu_n^{1-\lambda} \left\vert \psi_n^{(\lambda)} (\phi)\right\rangle \, , \end{equation} with the relation between the normalization constants being $n\, \mathcal{N}_\lambda=N_{\lambda,n}$, and the phase between the cyclic state and the erasure state, being $\mu_n^{1-\lambda}=\exp{(2 \pi i (1-\lambda)/n)}$. \end{proof} Given this identification, it can be seen that the photon number statistics for the state $\left\vert \psi_n^{(\lambda)} (\phi) \right\rangle$ contain only the photon numbers which satisfy ${\rm mod}(m-\lambda+1,n)=0$. The correspondence between the cyclic states and the states resulting from the quantum erasure map can lead to the experimental realization of the cyclic states. One can for example think of an initial nonivariant state $\vert \phi \rangle$, with a small mean photon number ($\langle \phi \vert \hat{n} \vert \phi \rangle \approx 0$). If one has a process where the number states $\vert 1 \rangle$ or $\vert 2 \rangle$ are erased, e.g., by the absorption of one or two photons of the electromagnetic field, then one can expect that the resulting state will be similar to a cyclic state. \section{Examples.} \subsection{Cyclic Gaussian states.} Here we define different superpositions of Gaussian states associated to the cyclic groups. These superpositions are connected with the squeezed states defined in \cite{nieto,hillery}. As an example of the general procedure described above, one can define cyclic states using Gaussian wavepackets as initial systems. Suppose a general one dimensional Gaussian state in the position basis \begin{equation} \psi (x)= \left(\frac{a+a^}{\pi}\frac{1+2a}{1+2a^}\right)^{1/4} \exp\left\{ -\frac{b^2+b b^}{4(a+a^)}\right\} \exp \left\{ -a x^2+b x\right\}\, , \quad a_R>0\, , \ b\neq0 \, , \label{gaus} \end{equation} with $a=a_R+ia_I$, $b=b_R+i b_I$. This state can be characterized by the mean values of the quadrature components $(\hat{p},\hat{q})$, and the corresponding covariance matrix $\sigma$. Which in the case of the state (\ref{gaus}) are \begin{equation} \langle \hat{x} \rangle=\frac{b+b^}{2(a+a^)}, \quad \langle \hat{p} \rangle= \frac{i (a b^-a^* b)}{a+a^}\, , \quad \sigma=\frac{1}{2(a+a^)}\left(\begin{array}{cc} 4 \vert a \vert^2 & i(a-a^) \\ i(a-a^) & 1 \end{array}\right) \, . \end{equation} When this state is rotated in the phase space using the propagator $\langle x \vert \hat{R}(\theta) \vert y \rangle$, where $\hat{R}(\theta)=\exp(-i\theta \hat{n})$ is the rotation operator, the obtained state is still Gaussian with new parameters $a(\theta)$, $b (\theta)$ given in terms of the original Gaussian parameters $a$, and $b$, as follows \begin{eqnarray} a (\theta)&=&\frac{2 i a \cos \theta-\sin \theta}{2 (i \cos \theta-2 a \sin \theta)} \, , \nonumber \\ b (\theta)&=& \frac{b}{\cos \theta+2 i a \sin \theta} \, . \end{eqnarray} The cyclic Gaussian state for the irreducible representation $\lambda$ of the group $C_n$ is then given by the expression \begin{equation} \Psi_n^{(\lambda)}(x)= \mathcal{N}_\lambda \sum_{r=1}^n \chi_n^{(\lambda)} (g_r) \psi_r (x)\, , \label{gcic} \end{equation} with a value of $\psi_r (x)$ analogous to the initial state of Eq.~(\ref{gaus}) \begin{eqnarray} \psi_r (x)=\left(\frac{a(\theta_r)+a^(\theta_r)}{\pi}\frac{1+2a(\theta_r)}{1+2a^(\theta_r)}\right)^{1/4} \exp\left\{ -\frac{b^2(\theta_r)+b(\theta_r) b^(\theta_r)}{4(a(\theta_r)+a^(\theta_r))}\right\} \times \nonumber \\ \exp \{ -a(\theta_r) x^2+b (\theta_r) x \} \, . \end{eqnarray} Given this expression one can construct then the cyclic states using Eq.~(\ref{gcic}). For the cyclic group $C_2$, the cyclic states can be described by the following two orthogonal states \begin{equation} \Psi^{(1,2)}_2 (x)=\mathcal{N}_{1,2} \, e^{-a x^2}(e^{b x} \pm e^{-b x}) \, , \quad \mathcal{N}_{1,2}=\left( \frac{a+a^}{\pi}\frac{1+2a}{1+2a^}\right)^{1/4} \frac{e^{-\frac{b^2+b b^}{4(a+a^)}}}{\sqrt{2}\left(1\pm e^{-\frac{b b^}{a+a^}}\right)^{1/2}} \, , \label{goga} \end{equation} \begin{figure} \caption{Mandel parameter $M_Q$ as a function of the real and imaginary parts of the parameter $b=b_R+i b_I$, for the states associated to the cyclic group $C_2$, $\Psi_2^{(1)}(x)$ with (a) $a=1/4$, (b) $a=1/2$, (c) $a=1$; and for the state $\Psi_2^{(2)}(x)$ for (d) $a=1/4$, (e) $a=1/2$, and (f) $a=1$ . } \label{mandel} \end{figure} which have specific properties. In Fig.~\ref{mandel}, the Mandel parameter \cite{mandel} $M_Q=\langle (\Delta \hat{n})^2 \rangle/ \langle \hat{n} \rangle$ is shown for the cyclic Gaussian states of $C_2$ given in Eq.~(\ref{goga}). The figure was made taking into account three different $a$ parameters. A Mandel parameter $M_Q<1$ can be used to distinguish a subpoissonian from a superpoissonian photon statistics ($M_Q>1$), or poissonian statistics $M_Q=1$. As it can be seen in the figure, the cyclic states can have subpoissonian distributions for certain regions of the parameter $b=b_R+i b_I$. As can be seen in the figure, the presence of this photon statistic is more prominent in the states associated to the second irreducible representation of the group $\Psi^{(2)}_2(x)$. Similar to the states above, the ones associated to the cyclic group $C_3$ can be obtained. In Fig.~\ref{wignerc3}, the plots and contours for the Wigner function \cite{wigner}: $W_\psi(x,p)=\int \, dy \, \psi^(x+y) \psi (x-y) e^{2ipy}/\pi$, can be seen. In the contour plots of the phase space ($p,x$) is noticed the symmetry of the state under the rotations with angles $0$, $2\pi/3$, and $4\pi/3$ with respect to the $x$ axis. It is also important to say that the Wigner functions depicted in the figure do not have inversion symmetry as they are only invariant under the rotations contained in the $C_3$ group. \begin{figure} \caption{Wigner functions and their contour plots for the cyclic Gaussian states associated to $C_3$ for the irreducible representations $\lambda=1$ (left), $\lambda=2$ (center), and $\lambda=3$ (right). For these figures the chosen parameters were $a=1$ and $b=\sqrt{2}(1+i)$. The black lines in the contour plots depict the symmetry axis associated to the $C_3$ group. } \label{wignerc3} \end{figure} \subsection{Circle symmetric states} When one increases the degree of the cyclic group the obtained states described by our method must be invariant under more and more rotations in the phase space. It is known \cite{janszky} that there exist a correspondence between the circle symmetric states in the coherent case and the Fock number states. This lead us to the question, how do the generalized cyclic states associated to a very big number of symmetries look like? e.g., when the order of the cyclic group tends to infinite ($n\rightarrow \infty$), can they also be associated to the Fock states?. To answer these questions, one can notice that the Definition~\ref{defi1} of the cyclic states allow us to make a generalization in the case when the angle $\theta$, which determine the rotations $\hat{R}(\theta)$, becomes a continuous variable. In that case, the definition of the cyclic states becomes \begin{equation} \vert \psi_\infty^{(\lambda)}\rangle= \mathcal{N}_\lambda \int_0^{2\pi} d\theta \, e^{i\theta (\lambda-1)} e^{-i \theta \hat{n}} \vert \phi \rangle \, , \end{equation} where we have an infinity number of irreducible representations, i.e., $\lambda \in \mathbb{Z}^+$. By means of the photon number decomposition of $\vert \phi \rangle=\sum_m A_m (\phi) \vert m \rangle$, one obtains \[ \vert \psi_\infty^{(\lambda)}\rangle= \mathcal{N}_\lambda \sum_{m=0}^\infty \int_0^{2\pi} d\theta \, A_m (\phi) e^{i\theta (\lambda-1-m)} \vert m \rangle \, , \] as the integral is equal to $2\pi$ times the Kronecker delta $\delta_{\lambda-1,m}$, we arrive to the result \[ \vert \psi_\infty^{(\lambda)}\rangle= 2\pi \mathcal{N}_\lambda A_{\lambda-1}(\phi) \vert \lambda-1 \rangle \, , \] which, after the renormalization process, we notice corresponds to the number state \begin{equation} \vert \psi_\infty^{(\lambda)}\rangle=\vert \lambda-1 \rangle \, . \end{equation} We point out that this expression for the circle cyclic states is consistent with the erasure map of the state $\vert \phi \rangle$, as in principle we need to erase all the different states but the one that satisfies the condition $m-\lambda+1=0$. This result lead us to the conclusion that the cyclic superposition ($n \rightarrow \infty$) of any state which is noninvariant under any rotation in the phase space, is equal to a Fock state. This, regardless of the initial, noninvariant state $\vert \phi \rangle$ that we take into consideration. We would like to emphasize that in order of this property to be true, the state under consideration $\vert \phi \rangle$ must be noninvariant under all possible rotations in the phase space. This implies that $\vert \phi \rangle$ must be expressed by an infinite sum of the photon number states $\vert m \rangle$ with a nonzero probability amplitude $A_m(\phi)$. To show this we can take as an example the $C_2$ group. In order for a state $\vert \phi \rangle$ to be noninvariant under the $C_2$ rotation, it should be made by the superposition of at least two states $\vert m \rangle$ and $\vert n \rangle$, $m$ being even and $n$ being odd ($m,n\in \mathbb{Z}^+$). In the case of $C_3$ we need at least three states $\vert m \rangle$, $\vert n \rangle$, and $\vert l \rangle$ such mod$(m,3)=0$, mod$(n,3)=1$, and mod$(l,3)=2$ ($m,n,l\in \mathbb{Z}^+$). By the extension of this argument, we must need an infinite number of photon states in order for $\vert \phi \rangle$ to be an noninvariant state under $C_\infty$. As examples of this type of states one can name the coherent, the squeezed coherent, the non-centered Gaussian, and any noninvariant, continuous variable state. To show that the superposition of several rotations of an initial continuous variable system can form a Fock state one can take as an example the Gaussian state of Eq. (\ref{gaus}) with $a=1$, $b=\sqrt{6}+2 i$. In Fig.~\ref{wigcirc} are shown the Wigner functions and their contours for the cyclic states associated to the first irreducible representation of $C_n$ for $n=10$ (left), $n=15$ (center), and $n=20$ (right). Here, one can see how the cyclic states for a long degree order are more and more alike to the vacuum state $\vert 0 \rangle$. Additionally to this, it can be checked that for a given irreducible representation of the cyclic group, a different photon state can be formed for a sufficient large number $n$, i.e., the cyclic group degree. \begin{figure} \caption{Wigner functions and their contour plots for the cyclic Gaussian state of $C_n$ for the irreducible representation $\lambda=1$ for (a) $n=10$ (right), (b) $n=15$ (center), and (c) $n=20$ (left). In all the plots we took the parameters $a=1$ and $b=\sqrt{6}+2 i$. } \label{wigcirc} \end{figure} \section{Cyclic group density matrices.} The previous discussion about the properties of the erasure map and its relation with the states associated to the cyclic groups can be extended to any kind of state which is not invariant under the rotation operation. For example, one can can think in a density matrix which may correspond to a mixed state $\hat{\rho}$ and define the following cyclic density matrices \begin{mydef} Let $\hat{\rho}$ be a density matrix with at least one of its mean quadrature components ($\hat{x}=(\hat{a}+\hat{a}^\dagger)/\sqrt{2}$, $\hat{p}=i(\hat{a}^\dagger-\hat{a})/\sqrt{2}$) different from zero, i.e., ${\rm Tr}(\hat{\rho}\, \hat{x})\neq 0$, or ${\rm Tr}(\hat{\rho}\, \hat{p})\neq 0$. Then the state associated to the irreducible representation $\lambda$ of the cyclic group $C_n$ is defined as \begin{equation} \hat{\rho}^{(\lambda)}_n=\mathcal{N}_\lambda \sum_{r,s=1}^n \chi_n^{(\lambda)} (g_r) \chi_n^{(\lambda)}(g_s) \hat{R}(\theta_r) \hat{\rho} \hat{R}^\dagger(\theta_s) \, , \label{rhoo} \end{equation} where $\chi_n^{(\lambda)} (g_r)$ is the character for the group element $g_r$, $\hat{R}(\theta_r)=\exp{(-i \theta_r \hat{n})}$, and \[ \mathcal{N}_\lambda^{-1}=\sum_{r,s=1}^n \chi_n^{(\lambda)} (g_r) \chi_n^{(\lambda)}(g_s) \, {\rm Tr}(\hat{R}(\theta_r) \hat{\rho} \hat{R}^\dagger(\theta_s)) \, . \] \end{mydef} These type of density matrices have the same properties of the cyclic states as being invariant up to a phase under the rotations in the cyclic group. Also, they have a photon distribution were not all the photon numbers are present as they can be obtained by the elimination of certain Fock states. To show this, one can follow an analogous procedure as in Theorem (\ref{teo2}). Let us suppose $\hat{\rho}=\sum_{m,m'=0}^\infty A_{m,m'}(\hat{\rho}) \vert m \rangle \langle m' \vert$, with ${\rm Tr}(\hat{\rho})=\sum_{m=0}^\infty A_{m,m}(\hat{\rho})=1$. This expression together with Eqs. (\ref{chi}) and (\ref{rhoo}) allow us to rewrite $\hat{\rho}_n^{(\lambda)}$ as follows \[ \hat{\rho}_n^{(\lambda)}=\mathcal{N}_\lambda \sum_{m,m'=0}^\infty A_{m,m'}(\hat{\rho})\sum_{r,s=1}^n \mu_n^{(\lambda-1)(r-1)} \mu_n^{(\lambda-1)(1-s)} e^{-i \theta_r m} e^{i \theta_s m'} \vert m \rangle \langle m' \vert \, , \] by the use of the definition of $\theta_j=2\pi (j-1)/n$ and Theorem \ref{tt1}, we can perform the sums over the $r$ and $s$ parameters. Those sums are \begin{eqnarray} \sum_{r=1}^n \mu_n^{(\lambda-1-m)r}&=&n \, \delta_{{\rm mod}(\lambda-1-m,n),0} \, , \nonumber \\ \sum_{s=1}^n \mu_n^{-(\lambda-1-m')s} &=& n \, \delta_{{\rm mod}(\lambda-1-m',n),0} \, , \end{eqnarray} then we finally can write the cyclic state density matrices as follows \[ \hat{\rho}_n^{(\lambda)}=\mathcal{N}_\lambda \, n^2 \sum_{m.m'=0}^\infty A_{m,m'}(\hat{\rho}) \, \mu_n^{m'-m}\, \delta_{{\rm mod}(\lambda-1-m,n),0} \, \delta_{{\rm mod}(\lambda-1-m',n),0} \, \vert m \rangle \langle m' \vert \, , \] as the delta functions imply that $\lambda-1-m$ and $\lambda-1-m'$ should be a multiple of $n$, then $\lambda-1-m=\eta n$, and $\lambda-1-m'=\xi n$ and then $m'-m=-(\xi+\eta)n$ is also a multiple of $n$. From these properties, we can conclude that $\mu_n^{m'-m}=1$ and finally arrive to the expression for the cyclic density matrix \begin{equation} \hat{\rho}_n^{(\lambda)}=\mathcal{N}_\lambda \, n^2 \sum_{m.m'=0}^\infty A_{m,m'}(\hat{\rho}) \, \delta_{{\rm mod}(\lambda-1-m,n),0} \, \delta_{{\rm mod}(\lambda-1-m',n),0} \, \vert m \rangle \langle m' \vert \, , \label{rhot} \end{equation} this property is summarized in the following theorem: \begin{theorem} Let $n$ and $\lambda$ be two positive integers with $\lambda\leq n$, and $\hat{\rho}_{n,\lambda} $ be the renormalized state obtained after the elimination of the number states operators $\vert m \rangle \langle m' \vert$ in $\hat{\rho}=\sum_{m,m'=0}^\infty A_{m,m'} (\hat{\rho}) \vert m \rangle \langle m' \vert$, which do not satisfy the conditions ${\rm mod}(\lambda-1-m,n)=0$ and ${\rm mod}(\lambda-1-m',n)=0$, then $\hat{\rho}_{n,\lambda} (\phi)\rangle$ is equal to the cyclic state $\hat{\rho}_n^{(\lambda)}$. \label{teo3} \end{theorem} It is noteworthy to see that from Eq. (\ref{rhot}) and the property $m'-m$ being a multiple of $n$, we can immediately show that the cyclic density matrices are invariants over the rotations in the cyclic groups. In other words, the density matrix $\hat{\rho}_n^{(\lambda)}$ after the rotation $\hat{R}(\theta_j)$, i.e., \[ \hat{R}(\theta_j)\hat{\rho}_n^{(\lambda)} \hat{R}^\dagger (\theta_j)=\mathcal{N}_\lambda \, n^2 \sum_{m.m'=0}^\infty A_{m,m'}(\hat{\rho}) \, \delta_{{\rm mod}(\lambda-1-m,n),0} \, \delta_{{\rm mod}(\lambda-1-m',n),0} \, \mu_n^{(m'-m)(j-1)}\, \vert m \rangle \langle m' \vert \, , \] is equal to the initial density matrix, so finally one can establish \[ \hat{R}(\theta_j)\hat{\rho}_n^{(\lambda)} \hat{R}^\dagger (\theta_j)=\hat{\rho}_n^{(\lambda)} \, . \] As in the case of the pure cyclic states, the photon number distribution of the cyclic density matrices contains only some of the numbers states. Given that the different states associated to the cyclic group $C_n$ are made with different photon number states, we can conclude that the cyclic density matrices form an orthogonal set. \section{Example: Calculation of the entanglement in a bipartite state.} As an example of the applications of the cyclic states we show that this type of states can be used to describe a continuous variable system in a discrete way, and that this discrete form can lead to an easier calculation of parameters, such as the entanglement between parts in a bipartite system. Suppose a two mode state made entirely of the group of rotation states $\{\vert \phi_r \rangle_1, \vert \varphi_r \rangle_2 ; r=1, \ldots, n\}$ for modes 1 and 2 respectively, e.g. the state \begin{equation} \vert T\rangle= \sum_{r=1}^n c_r \vert \phi_r \rangle_1 \vert \varphi_r \rangle_2 \, , \quad \sum_{r,r'=1}^n c_r c_{r'}^ \langle \phi_{r'} , \varphi_{r'} \vert \phi_r, \varphi_r\rangle =1 \, . \label{tstate} \end{equation} As the states $\vert \phi_r \rangle=\hat{R}(\theta_r)\vert \phi \rangle$, $\vert \varphi_r \rangle=\hat{R}(\theta_r)\vert \varphi \rangle$ can be general then they might not be orthogonal. On the other hand, the cyclic states generated by these states form an orthogonal set. Most importantly, as there exist the same number of cyclic states $\vert \psi_n^{(\lambda)} (\phi)\rangle$ and $\vert \psi_n^{(\lambda)}(\varphi) \rangle$ as the number of rotated states $\vert \phi_r \rangle$ and $\vert \varphi_r \rangle$, then one can obtain the rotated states in terms of the cyclic, orthogonal ones. To obtain these expressions one must obtain the inverse relation of Eq. (\ref{ccy}) \[ \vert \psi_n^{(\lambda)}(\phi) \rangle=\mathcal{N}_\lambda \sum_{r=1}^n \mu_n^{(\lambda-1)(r-1)} \vert \phi_r \rangle \, , \] to do that, one can treat the characters of the group as a matrix $M_{jk}=\mu_n^{(j-1)(k-1)}$, which has an inverse matrix $M_{jk}^{-1}=\mu_n^{(1-j)(k-1)}/n$. By this expression one can obtain the inverse equation \begin{equation} \vert \phi_r \rangle = \frac{1}{n \, \mathcal{N}_\lambda} \sum_{\lambda=1}^n \mu_n^{(1-r)(\lambda-1)} \vert \psi_n^{(\lambda)}(\phi) \rangle \, . \label{innv} \end{equation} By substituting this expression and an analogous expression for $\vert \varphi_r \rangle$ into the two-mode state $\vert T \rangle$, one obtains \[ \vert T \rangle =\frac{1}{n^2}\sum_{r=1}^n c_r \sum_{\lambda,\lambda'=1}^n \frac{1}{\mathcal{N}_\lambda \mathcal{N}_{\lambda'}}\mu_n^{(1-r)(\lambda-1)} \mu_n^{(1-r)(\lambda'-1)} \vert \psi_n^{(\lambda)}(\phi) \rangle_1 \vert \psi_n^{(\lambda')}(\varphi) \rangle_2 \, . \] From this expression is possible to calculate the partial density matrices for each mode in the bipartite state. For this we obtain the total density matrix and perform the partial trace operation. Finally, arriving to \begin{eqnarray} \hat{\rho}(1)=\sum_{r,s,\lambda,\lambda', \mu=1}^n D_{r,\lambda,\lambda'} D_{s,\mu,\lambda'}^ \, \vert \psi_n^{(\lambda)} (\phi) \rangle \langle \psi_n^{(\mu)} (\phi) \vert ,\nonumber \\ \hat{\rho}(2)=\sum_{r,s,\lambda,\lambda', \mu'=1}^n D_{r,\lambda,\lambda'} D_{s,\lambda,\mu'}^* \, \vert \psi_n^{(\lambda')} (\varphi) \rangle \langle \psi_n^{(\mu')} (\varphi) \vert \, , \end{eqnarray} where $D_{r,\lambda,\lambda'}=\frac{ \mu_n^{(1-r)(\lambda+\lambda'-2)}}{n^2 \mathcal{N}_\lambda \mathcal{N}_{\lambda'}} c_r$. After this, one can calculate the entanglement between the modes. The entanglement is calculated by the linear entropy of the partial density matrices, giving the following result \begin{equation} S_L(1)=1- \sum_{\lambda,\mu=1}^n \vert F_{\lambda,\mu}\vert^2\, , \quad F_{\lambda,\mu}=\sum_{r,s,\lambda'=1}^n D_{r,\lambda,\lambda'} D_{s,\mu,\lambda'}^ \, . \end{equation} The quantification of the entanglement by using the decomposition of the two-mode system in terms of cyclic states was done in a easier way than by directly taking the expression of the state $\vert T \rangle$ of Eq.~(\ref{tstate}). Several other quantities can be calculated using this decomposition as the mean values and the covariance matrix of the system. \section{Generalized dihedral states} The dihedral group of $n$-th order ($D_n$) is a non-Abelian group which contains all the symmetry operations of the $n$-sided regular polygon. In other words, it contains the rotations of the cyclic group $C_n$ and the inversion operators $\hat{U}_r$; $r=1,\ldots,n$. The inversions in the phase space are defined by a rotation plus the complex conjugation operator $\hat{C}$, i.e., $\hat{U}_r=\hat{C}\hat{R}(\theta_r)$, with $\theta_r=2\pi(r-1)/n$. In order to obtain any state associated to the dihedral group, one must impose the condition for the state to be invariant under both the rotations and inversions contained in $D_n$. Inspired by the cyclic states, one can use a superposition of all the rotations and inversions of a noninvariant state $\vert \phi \rangle$, that is the superposition of the states $\hat{R}(\theta_r)\vert \phi \rangle$ and $\hat{U}_r \vert \phi \rangle$. As we have seen in the sections 3 and 4, the superpositions with probability amplitudes given by the characters of the cyclic group $\chi_n^{(\lambda)} (g_r)$ are orthogonal as they contain different photon numbers. Given these arguments we define a set of $n$ dihedral states, each one corresponding to an irreducible representation of the cyclic subgroup $C_n$, as follows \begin{mydef} Let $\vert \phi \rangle=\sum_{m=0}^\infty A_m (\phi) \vert m \rangle$ be a quantum state with at least one mean quadrature component ($\hat{x}=(\hat{a}+\hat{a}^\dagger)/\sqrt{2}$, $\hat{p}=i(\hat{a}^\dagger-\hat{a})/\sqrt{2}$) different from zero, i.e., $\langle \phi \vert \hat{x} \vert \phi \rangle \neq 0$, or $\langle \phi \vert \hat{p} \vert \phi \rangle \neq 0$. The general dihedral state for the irreducible representation $\lambda$ of the subgroup $C_n$ is defined as \begin{equation} \left\vert \gamma_n^{(\lambda)} (\phi) \right\rangle = \mathcal{N}_\lambda \sum_{r=1}^n (\chi^{(\lambda)}_n (g_r) \vert \phi_r \rangle+\chi^{(\lambda)}_n (g_r) \vert \phi^_r \rangle) \, , \label{ccy} \end{equation} where $\chi_n^{(\lambda)}(g_r)$ is the character associated to the element of the group $g_r$ of the cyclic group, $\vert \phi^_r \rangle=\hat{U}_r \vert \phi \rangle=\sum_{m=0}^\infty A^_m (\phi) e^{i \theta_r m} \vert m \rangle$ ($\theta_r=2\pi (r-1)/n$), and where \[ \mathcal{N}_\lambda^{-2}=\sum_{r,r'=1}^n (\chi_n^{(\lambda)}(g_{r'}) \langle \phi_{r'} \vert+\chi_n^{(\lambda)}(g_{r'})\langle \phi^_{r'} \vert)( \chi_n^{(\lambda)}(g_{r}) \vert \phi_r \rangle+\chi_n^{(\lambda)}(g_{r})\vert \phi^_r \rangle) \, . \] \label{defi3} \end{mydef} We would like to emphasize that it is the first time that an orthogonal set of states have been associated to the dihedral group. These set of states are invariant, up to a phase, under the application of all the dihedral group elements. As the construction of the dihedral states corresponds to the sum of two cyclic states: one with initial state $\vert \phi \rangle=\sum_{m=0}^\infty A_m (\phi) \vert m \rangle$ and the other with the initial state $\vert \phi^* \rangle=\sum_{m=0}^\infty A^_m (\phi) \vert m \rangle$, then the invariance under rotations can be implied from the cyclic states invariance (up to a phase) of Eq.~(\ref{cyc_inv}) \[ \hat{R}(\theta_l)\vert \gamma^{(\lambda)}_n (\phi) \rangle=\mu_n^{(1-\lambda)l}\vert \gamma^{(\lambda)}_n (\phi) \rangle \, , \] from this correspondence one can obtain an expression for the inversions acting on the dihedral states $\hat{U}_l \vert \gamma_n^{(\lambda)}\rangle$ ($\hat{U}_l=\hat{C}\hat{R}(\theta_l)$): \begin{eqnarray} \hat{U}_l \vert \gamma_n^{(\lambda)}\rangle&=&\hat{C}\mu_n^{(1-\lambda)l}\vert \gamma^{(\lambda)}_n (\phi) \rangle \, \\ &=&\mu_n^{(\lambda-1)l} \vert \gamma_n^{(\lambda)} \rangle \, , \end{eqnarray} and thus one can imply that the dihedral state $\vert \gamma_n^{(\lambda)}\rangle$ in Def.~\ref{defi3} is invariant, up to a phase, under all the elements of the dihedral group $D_n$. As we can see in Def.~\ref{defi3}, the dihedral group can be defined using the sum of a noninvariant state $\vert \phi \rangle$ and its conjugate $\vert \phi^ \rangle$, this implies that the cyclic state $\vert \psi_n^{(\lambda)}\rangle$ is also a dihedral state $\vert \gamma_n^{(\lambda)}\rangle$ when the initial state has only real photon number probability amplitudes $A_m(\phi)\in \mathbb{R}$, implying $\vert \phi \rangle=\vert \phi^* \rangle$. One can also notice that the dihedral states correspond to the erasure map of the state $(\vert \phi \rangle+\vert \phi^\rangle)/\sqrt{2}$ since, as stated before, the dihedral state correspond to the sum of the cyclic states for $\vert \phi \rangle$ and $\vert \phi^ \rangle$. As stated before, the sum $ \chi^{(\lambda)}_n (g_r) \vert \phi \rangle+ \chi^{(\lambda)}_n (g_r) \vert \phi^\rangle$ used to obtain the dihedral superpositions, is an state with real probability amplitudes, as $\vert \phi \rangle=\sum_{m=0}^\infty A_m(\phi) \vert m \rangle$, then $ \chi^{(\lambda)}_n (g_r) \vert \phi \rangle+ \chi^{(\lambda)}_n (g_r) \vert \phi^ \rangle=2 \sum_{m=0}^\infty {\rm Re}( \chi^{(\lambda)}_n (g_r)\, A_m(\phi)) \vert m \rangle$. It can be seen that an analogous procedure to define dihedral states can be done by using the imaginary part of the probability amplitudes $ \chi^{(\lambda)}_n (g_r) A_m(\phi)$, e.g., by using the subtraction of the states $ \chi^{(\lambda)}_n (g_r) \vert \phi \rangle- \chi^{(\lambda)}_n (g_r)\vert \phi^ \rangle$ instead of the sum $ \chi^{(\lambda)}_n (g_r)\vert \phi \rangle+ \chi^{(\lambda)}_n (g_r)\vert \phi^\rangle$. The states associated to the subtraction are also invariant, up to a phase, under all the transformations contained in the dihedral group, however they are not orthogonal to the states defined in Def.~\ref{defi3}. However, they still can be helpful as they contain the dihedral symmetry. In fig.~\ref{wignerd3}, the Wigner functions and their contour plots for each one of the three states associated to the dihedral group $D_3$ are shown. To construct this figure, the Gaussian state of Eq.~(\ref{gaus}) with $a=1$ and $b=1+i$ was used to generate the states of $D_3$. In all the cases one can notice that additionally to the rotational symmetry of the $C_3$ subgroup, the inversion invariance is also present. \begin{figure} \caption{Wigner functions and their contour plots for the dihedral Gaussian states associated to the different irreducible representations $\lambda$ of the group $D_3$ for $\lambda=1$ (left), $\lambda=2$ (center), and $\lambda=3$ (right). For these figures the chosen parameters for the initial Gaussian state with $a=1$ and $b=1+i$. } \label{wignerd3} \end{figure} \section{Summary and conclusions} A general procedure to obtain a set of $n$ orthogonal pure states (or density matrices) associated to each of the irreducible representations of the cyclic group $C_n$ and dihedral group $D_n$ was proposed. This procedure can be summarized as follows: given any state $\vert \phi \rangle$ which is not invariant under the rotations of the cyclic group, the cyclic states can be obtained from the weighted superposition of the phase-space rotations of the initial state $\hat{R}(\theta_j)\vert \phi \rangle$ ($j=1,\ldots,n$), where the weights of each rotated state are given by the characters of each irreducible representation. This procedure is then extended to density matrices where the weighed superpositions are made of the elements $\hat{R}(\theta_r)\hat{\rho} \hat{R}^\dagger (\theta_s)$, where $\hat{\rho}$ is the initial noninvariant density matrix. Additionally, it was shown that the resulting states associated to $C_n$ provided by our method are invariant, up to a phase, under any element of the group. The associated states to the dihedral group $D_n$ are defined through the rotations of the original noninvariant state $\vert \phi \rangle$ and its complex conjugate $\vert \phi^* \rangle$. In the case of the dihedral states, it is the first time that an orthogonal set of states have been associated to the dihedral group. The correspondence between the cyclic states of $C_n$ and the renormalized states obtained after the erasure of certain photon numbers was established and discussed. In particular, it was shown that the cyclic state corresponds, up to a phase, to the renormalized states with photon number states $\vert m \rangle$ erased, where the erased states do not satisfy the condition ${\rm mod}(\lambda+m-1,n)=0$. In an analogous way, the cyclic density matrices obtained by our method correspond to the renormalized matrices where the photon number operators $\vert m \rangle \langle m' \vert$, which does not satisfy the conditions ${\rm mod}(\lambda-m-1,n)=0$ and ${\rm mod}(\lambda-m'-1,n)=0$, are eliminated. On the other hand, the dihedral states correspond to the sum of the cyclic states defined with the states $\vert \phi \rangle$ and $\vert \phi^* \rangle$, for this reason they correspond to the erasure map of the state $(\vert \phi \rangle+\vert \phi^* \rangle)/\sqrt{2}$. As example of the procedure the general cyclic Gaussian states were defined. It was shown that these states can present subpoissonian photon number statistics by using the Mandel parameter $M_Q=\langle (\Delta \hat{n})^2 \rangle/ \langle \hat{n} \rangle$. The symmetry properties of the cyclic Gaussian states associated to $C_3$ were also checked using the Wigner function. Also, the correspondence between the circle symmetric states $C_n$ ($n\rightarrow \infty$): $\vert \psi_\infty^{(\lambda)}\rangle$ and the Fock states $\vert \lambda-1 \rangle$ was demonstrated. Also, as an example of the use of the cyclic states, the calculation of the entanglement between subsystems in a two-mode state was presented. This calculation takes advantage of the orthogonality of the cyclic states to define a finite representation of particular bipartite states. The possible experimental realization of these states was briefly discussed given the evidence presented in \cite{cordero1,cordero2} for a generation of cyclic states in the atom-field interaction, and in \cite{vlastakis} were these type of superposition can be obtained using a superconducting transmon coupled with a cavity resonator. \section*{Acknowledgments} This work was partially supported by DGAPA-UNAM (under project IN101619). \end{document}	arXiv
\begin{document} \begin{frontmatter} \title{T-Shape Visibility Representations of 1-Planar Graphs\tnoteref{thanks}} \author{Franz J.\ Brandenburg} \ead{[email protected]} \address{University of Passau, 94030 Passau, Germany.} \tnotetext[thanks]{Supported in part by the Deutsche Forschungsgemeinschaft (DFG), grant Br835/} \begin{abstract} A shape visibility representation displays a graph so that each vertex is represented by an orthogonal polygon of a particular shape and for each edge there is a horizontal or vertical line of sight between the polygons assigned to its endvertices. Special shapes are rectangles, \textsf{L}, \textsf{T}, \textsf{E} and \textsf{H}-shapes, and caterpillars. A flat rectangle is a horizontal bar of height $\epsilon>0$. A graph is 1-planar if there is a drawing in the plane such that each edge is crossed at most once and is IC-planar if in addition no two crossing edges share a vertex. We show that every IC-planar graph has a flat rectangle visibility representation and that every 1-planar graph has a \textsf{T}-shape visibility representation. The representations use quadratic area and can be computed in linear time from a given embedding. \end{abstract} \begin{keyword} Graph Drawing \sep visibility representations \sep orthogonal polygons \sep beyond-planar graphs \end{keyword} \end{frontmatter} \section{Introduction} A graph is commonly visualized by a drawing in the plane or on another surface. In return, properties of drawings are used to define properties of graphs. Planar graphs are the most prominent example. Also, the genus of a graph and $k$-planar graphs are defined in this way, where a graph is $k$-planar for some $k \geq 0$ if there is a drawing in the plane such that each edge is crossed at most $k$ times. Planar graphs admit a different visualization by bar visibility representations. A \emph{bar visibility representation} consists of a set of non-intersecting horizontal line segments, called bars, and vertical lines of sight between the bars. We assume that the lines of sight have width $\epsilon > 0$ and also that the bars have height at least $\epsilon$. Each bar represents a vertex of a graph and there is an edge if (or if and only if) there is a line of sight between the bars of the endvertices. Hence, there is a bijection between vertices and bars and a correspondence between edges and lines of sight that is one-to-one in the \emph{weak} or ``if''-version and also onto in the \emph{strong} or ``if and only if''-version. A graph is a \emph{bar visibility graph} if it admits a bar visibility representation. Other graph classes are defined analogously. Bar visibility representations and graphs were intensively studied in the 1980s and the representations of planar graphs were discovered independently multiple times \cite{dhlm-rpg-83, ow-grild-78,rt-rplbopg-86,tt-vrpg-86,w-cbg-85}. Note that strong visibility with lines of sight of width zero excludes $K_{2,3}$ and some 3-connected planar graphs \cite{a-rvg-92} and implies an NP-hard recognition problem \cite{a-rvg-92}. Obviously, every weak visibility graph is an induced subgraph of a strong visibility graph with lines of sight of width zero or $\epsilon >0$. In the late 1990s visibility representations were generalized to represent non-planar graphs. The approach by Dean et al.~\cite{DEGLST-bkvg-07} admits semi-transparent bars and lines of sight that traverse up to $k$ other bars. In other words, an edge can cross up to $k$ vertices. Some facts are known about bar $k$-visibility graphs: for $k=1$ each graph of size $n$ has at most $6n-20$ edges and the bound can be achieved for all $n \geq 8$ \cite{DEGLST-bkvg-07}. In consequence, $K_8$ is the largest complete bar $1$-visibility graph. A graph has thickness $k$ if it can be decomposed into $k$ planar graphs. However, bar 1-visibility graphs are incomparable to thickness two (or biplanar) graphs, since there are thickness two graphs with $6n-12$ edges which cannot be bar 1-visibility graphs and conversely there are bar 1-visibility graphs with thickness three \cite{fm-pbkvg-08}. Bar 1-visibility graphs have an NP-hard \cite{bhkn-bvg-15} recognition problem. Last but not least, every 1-planar graph has a bar 1-visibility representation which uses only quadratic area and can be specialized so that a line of sight crosses at most one bar and each bar is crossed at most once \cite{b-vr1pg-14}. The inclusion relation between 1-planar and bar 1-visibility graphs was obtained independently by Evans et al.~\cite{ekllmw-b1vg-14} Rectangle visibility representations of graphs were introduced by Hutchinson et al.~\cite{hsv-rstg-99}. Here, each vertex is represented by an axis-aligned rectangle and there are horizontal and vertical lines of sight for the edges, which cannot penetrate rectangles. Hutchinson et al. studied the strong version of visibility. They proved a density of $6n-20$ which is tight for all $n \geq 8$. In consequence, $K_8$ is the largest rectangle visibility graph. Rectangle visibility graphs have thickness two whereas it is unknown whether they have geometric thickness two \cite{hsv-rstg-99}, which requires a decomposition into two straight-line planar graphs. The recognition problem for weak rectangle visibility graphs is \ensuremath{\mathcal{NP}}-hard \cite{s-RVG-NP-96}. We generalize rectangle visibility representations to $\sigma$-\emph{shape visibility representations}. A \emph{shape} $\sigma$ is an orthogonal drawing of a ternary tree $\tau$, which is expanded to an \emph{orthogonal polygon} in a $\sigma$-shape visibility representation. Thereby, each edge of $\tau$ is expanded to a rectangle of width $w>0$ and height $h>0$. The images of the vertices are similar and differ only in the length and width of the horizontal and vertical pieces of the polygon. In particular, rectangle visibility is \textsf{I}-shape or ``--''-shape visibility. Since visibility representations can be reflected or rotated by multiples of 90 degrees we treat the respective shapes as equivalent and shall identify them. For example, any single element of the set $\{\lfloor,\rfloor, \lceil, \rceil\}$ can be used for an \textsf{L}-shape. However, a set of shapes must be used if the vertices shall have different shapes, e.g., $\{ \lfloor,\rfloor, \lceil, \rceil\}$ for \textsf{L}-shapes in \cite{lm-Lvis-16}. Other common shapes are \textsf{H}, \textsf{F} or \textsf{E}. A \emph{rake} is a generalized \textsf{E} with many teeth that are directed upwards, and a \emph{caterpillar} is a two-sided rake with a horizontal path and vertical lines from the path to the leaves above and below. The number of teeth or vertical lines is reflected by the vertex complexity of ortho-polygon visibility representation \cite{ddelmmw-ovreg-16}. In a \emph{flat rectangle visibility representation} the rectangles have height $\epsilon >0$ where $\epsilon$ is the width of a sight of line \cite{tt-vrpg-86}. Then the vertices are represented by bars, as in bar visibility representations, such that two bars at the same level can see one another by a horizontal line of sight if there is no third bar in between. Moreover, a horizontal and vertical line of sight may cross, which is not allowed in the flat visibility representations by Biedl \cite{b-sdogs-11}. Shape visibility representations have been introduced by Di Giacomo et al.~\cite{ddelmmw-ovreg-16}. They use caterpillars as shapes in their results. \textsf{L}-visibility representations have been introduced by Evans et al.~\cite{elm-svrL-15} using any shape from the set $\{ \lfloor, \rfloor, \lceil, \rceil\}$. This approach was adopted by Liotta and Montecchiani \cite{lm-Lvis-16} for the representation of IC-planar graphs.\\ In this work, we prove the following: \begin{theorem} \label{thm:IC-RVG} Every n-vertex IC-planar graph $G$ admits a flat rectangle visibility representation in $O(n^2)$ area, which can be computed in linear time from a given IC-planar embedding of $G$. \end{theorem} \begin{theorem} \label{thm:1-cat} Every n-vertex 1-planar graph $G$ admits a \textsf{T}-shape visibility representation in $O(n^2)$ area, which can be computed in linear time from a given 1-planar embedding of $G$. \end{theorem} The first theorem improves upon a result by Liotta and Montecchiani \cite{lm-Lvis-16} who use the set $\{ \lfloor, \rfloor, \lceil, \rceil\}$ as \textsf{L}-shapes. Our result is also a variation of the bar 1-visibility representation of 1-planar graphs by Brandenburg \cite{b-vr1pg-14} such that an edge-bar crossing is substituted by a crossing of a vertical and a horizontal line of sight. The second theorem extends a recent result by Di Giacomo et al.~\cite{ddelmmw-ovreg-16} and contrasts a result by Biedl et al.~\cite{blm-vrnpg-16}. However, there are different settings. We operate in the \emph{variable embedding setting} and admit changing the embedding. In the other works an \emph{embedding-preserving setting} is used which enforces a coincidence of the embedding of a 1-planar drawing and a visibility representation. For the first theorem, we reroute an edge in each $B$-configuration, as depicted in Figs.~\ref{fig:kite} (b) and (c). The change of the embedding can be undone with a little effort. However, the full power of horizontal and vertical lines of sight is used for the second theorem. Here some crossing edges undergo a separate treatment and substantially change the embedding. In contrast, Di Giacomo et al. have shown that every 1-planar graph admits a caterpillar visibility representation and that there are 2-connected 1-planar graphs $G_n$ that need rakes of arbitrary size if the embedding is preserved. Biedl et al.~\cite{blm-vrnpg-16} proved that there is no rectangle visibility representation of $K_6$ that preserves a given 1-planar embedding. However, the graphs $G_n$ and $K_6$ have a rectangle visibility representation (since $K_6$ and the components of $G_n$ are subgraphs of $K_8$). The paper is organized as follows: In Sect. \ref{sect:prelim} we recall basic notions and facts on 1-planar graphs and we consider vertex numberings of planar graphs. We proof Theorem \ref{thm:IC-RVG} in Sect. \ref{sect:proof1} and Theorem \ref{thm:1-cat} in Sect.~\ref{sect:proof2} and conclude with general properties of shape visibility graphs in Sect.~\ref{sect:general}. \section{Preliminaries} \label{sect:prelim} We consider simple undirected graphs $G = (V, E)$ with a finite set of vertices $V$ of size $n$ and a finite set of undirected edges $E$. It is assumed that the graphs is 2-connected, since components can be treated separately or they can be connected by further planar edges. A \emph{drawing} maps the vertices of a graph to distinct points in the plane and each edge is mapped to a Jordan arc between the endpoints. Our drawings are simple so that two edges have at most one point in common, which is either a common endvertex or a crossing point. A drawing is planar if edges do not cross and 1-planar if each edge is crossed at most once. Moreover, in an IC-planar drawing each vertex is incident to at most one crossing edge. A graph is called \emph{planar} (\emph{1-planar, IC-planar}) if it admits a respective drawing. A planar drawing partitions the plane into topologically connected regions, called faces, whose boundary consists of edges and edge segments and is specified by a cyclic sequence of vertices and crossing points. The unbounded region is called the outer face. An \emph{embedding} $\mathcal{E}(G)$ of a graph $G$ is an equivalence class of drawings of $G$ with the same set of faces. For an algorithmic treatment, we use the embedding of a planarization of $G$ which is obtained by treating the crossing points as dummy vertices of degree four. An embedded planar graph is specified by a rotation system, which is the cyclic list of all neighbors or incident edges at each vertex in clockwise order. 1-planar graphs are the most important class of so-called beyond-planar graphs. Beyond-planarity comprises graph classes that extend the planar graphs and are defined by specific restrictions of crossings. 1-planar graphs were studied first by Ringel \cite{ringel-65} who showed that they are at most 7-colorable. In fact, 1-planar graphs are 6-colorable \cite{b-np6ct-95}. Bodendiek et al.~\cite{bsw-bs-83, bsw-1og-84} observed that 1-planar graphs of size $n$ have at most $4n-8$ edges and that this bound is tight for $n=8$ and all $n \geq 10$. This fact was discovered independently in many works. In consequence, an embedding has linear size and can be treated in linear time. \emph{IC-planar} (independent crossing planar) graphs are an important special case \cite{a-cnircn-08}. An IC-planar graph has at most $3.25n-6$ edges \cite{ks-cpgIC-10} and the bound is tight. In between are NIC-planar graphs \cite{zl-spgic-13} which are defined by 1-planar drawings in which two pairs of crossing edges share at most one vertex. Their density is at most $3.6(n-2)$. 1-planar, NIC-planar, and IC-planar graphs have some properties in common: First, there is a difference between densest and sparsest graphs. A sparsest graph cannot be augmented by another edge and has as few edges as possible whereas a densest graph has as many edges as possible. It is known that there are sparse 1-planar graphs with $\frac{45}{17}n- \frac{84}{17}$ edges \cite{begghr-odm1p-13}, sparse NIC-planar graphs with $3.2(n-2)$ \cite{bbhnr-NIC-16} edges and sparse IC-planar graphs with $3n-4$ edges \cite{bbhnr-NIC-16}. The NP-hardness of the recognition problems was discovered independently multiple times \cite{GB-AGEFCE-07, km-mo1ih-13, abgr-1prs-15,bbhnr-NIC-16, bdeklm-IC-16} and holds even if the graphs are 3-connected and are given with a rotation system. On the other hand, triangulated graphs can be recognized in cubic time \cite{cgp-rh4mg-06,b-4m1pg-15}. A \emph{triangulated graph} admits a drawing so that all faces are triangles. Then all pairs of crossing edges induce $K_4$ as a subgraph. The most remarkable distinction between IC-planar and NIC-planar graphs is their relationship to RAC graphs. A graph is RAC (right angle crossing) \cite{del-dgrac-11} if it admits a straight-line drawing such that edges cross at a right angle. RAC graphs have at most $4n-10$ edges, and if they meet the upper bound, then they are 1-planar \cite{el-racg1p-13}. In contrast, there are 1-planar graphs that are not RAC and RAC graphs that are not 1-planar \cite{del-dgrac-11}. Hence, 1-planar graphs and RAC graphs are incomparable. Recently, Brandenburg et al.~\cite{bdeklm-IC-16} showed that every IC-planar graph is a RAC graph and Bachmaier et al.~\cite{bbhnr-NIC-16} proved that RAC graphs and NIC-planar graphs are incomparable. \section{Planar and 1-Planar Graphs} For our algorithms we use two tools: triangulated 1-planar embeddings and an st-numbering. We need the following versions of a given 1-planar graph $G$: $G_{\boxtimes}, G_{\boxplus}, G_{\square}$ and $G_{\bullet}$. Each version is obtained from an embedding $\mathcal{E}(G)$ and inherits the embedding. Graphs $G_{\boxtimes}$ and $G_{\boxplus}$ are supergraphs of $G$ which coincide on 3-connected graphs, $G_{\boxplus}$, $G_{\square}$ and $G_{\bullet}$ admit multi-edges, and $G_{\square}$ and $G_{\bullet}$ are planar. First, augment the embedding $\mathcal{E}(G)$ by as many planar edges as possible and thereby obtain a planar maximal embedding $\mathcal{E}(G_{\boxtimes})$ of $G_{\boxtimes}$ \cite{abk-sld3c-13}. Then the endvertices of each pair of crossing edges induce $K_4$. Each such $K_4$ should be embedded as a kite with the crossing point inside the boundary of the 4-cycle of the endvertices and no other vertex inside this boundary, see Fig.~\ref{fig:kite}(a). Otherwise, there are B- or W-\emph{configurations} \cite{t-rdg-88}, as shown in Figs.~\ref{fig:kite} (b) and (d) or there is a separation pair as in Fig.~\ref{fig:kite}(e), where the inner components are contracted to a single vertex. B-configurations can be removed by changing the embedding. Therefore, choose the other face next to the edge $\{a,b\}$ between the vertices of a separation pair as outer face and reroute $\{a,b\}$, as illustrated in Fig.~\ref{fig:kite}(c), or flip the component. Thereafter we add further planar edges if possible. For example, in Fig.~\ref{fig:kite}(c) one may connect $x$ with another vertex by a planar edge. Then at most one W-configurations remains in the outer face if the graphs are 3-connected \cite{abk-sld3c-13}. If the graph is 3-connected, then we take the obtained embedding as a \emph{normal form} \cite{abk-sld3c-13}. It corresponds to a triangulation of the planarization with crossing points as vertices of degree four. Otherwise, there are separation pairs and pairs of crossing edges that separate the components, as sketched in Fig.~\ref{fig:separate} and shown in Fig.~\ref{fig:XWdouble}. \begin{figure} \caption{(a) a kite, (b) a B-configuration, (c) a rerouted B-configuration, (d) a W-configuration, and (e) a separation pair with an inner component represented by a dot.} \label{fig:kite1} \label{fig:B-config} \label{fig:W-config} \label{fig:W-config} \label{fig:W-config} \label{fig:kite} \end{figure} Graph $G_{\boxplus}$ extends $G_{\boxtimes}$ by multi-edges at separation pairs and the removal of B-configurations. Let $[x,y]$ be a separation pair so that $G_{\boxtimes} - \{x,y\}$ decomposes into components $H_0, H_1,\ldots, H_p$ for some $p \geq 1$. By recursion there is a decomposition tree, which is a simplified version of the SPQR-decomposition tree \cite{dt-olpt-96, gm-ltist-01} and can be computed in linear time. Let $H_0$ be the \emph{outer component} and let $H_1, \ldots, H_p$ be \emph{inner components} which are children of the outer component in the decomposition tree. Expand each inner component $H_i$ to $\widehat{H_i}$ which includes $x$ and $y$ and the edges between $x$ and $y$ and vertices of $H_i$. Flip and permute the inner components in the embedding $\mathcal{E}(G_{\boxtimes})$ and add further planar edges so that no B-configuration remains. An expanded inner component $\widehat{H_i}$ is embedded as a W-configuration and $\widehat{H_i}$ and $\widehat{H_j}$ for $i \neq j$ are separated by two pairs of crossing edges. Now, an embedding of $G_{\boxplus}$ is obtained by adding a copy $e_i$ of the edge $e_0 = \{x,y\}$ between $\widehat{H_i}$ and $\widehat{H_{i+1}}$ and beyond $H_q$ for $i=1,\ldots, p$ with $H_{p+1} = H_0$ \cite{b-vr1pg-14}, as illustrated Fig.~\ref{fig:separate}. Note that $G_{\boxplus} = G_{\boxtimes}$ if $G$ is 3-connected. Graph $G_{\square}$ is obtained from of $G_{\boxplus}$ by removing all pairs of crossing edges in an embedding $\mathcal{E}(G_{\boxplus})$. Due to the multi-edges, the embedding of $G_{\square}$ has triangles and quadrangles with a quadrangle for each pair of crossing edges. Finally, $G_{\bullet}$ is obtained from an embedding $G_{\boxtimes}$ of an IC-planar graph $G$ by the contraction of each kite to a single vertex. \begin{figure} \caption{A separation pair $[x,y]$ and a separation of the inner components.} \label{fig:separate} \end{figure} \begin{lemma} \label{lem:normalform} Let $\mathcal{E}(G)$ be a 1-planar embedding of a 1-planar graph $G$. \begin{itemize} \item The embedding $\mathcal{E}(G_{\boxplus})$ is triangulated. \item A pair of crossing edges is embedded as a kite or there is a \emph{W}-configuration and a separation pair. \item $G_{\boxplus}$ has at most 4n-8 edges. \item $\mathcal{E}(G_{\boxplus})$ and $\mathcal{E}(G_{\boxplus}$ can be computed in linear time from $\mathcal{E}(G)$. \end{itemize} \end{lemma} \begin{proof} The planar maximal embedding of each 3-connected component is triangulated \cite{abk-sld3c-13}. It may change the embedding by rerouting an edge of a B-configuration, which thereby is turned into a kite, see Figs.~\ref{fig:kite} (b) and(c). Then the stated properties hold for $\mathcal{E}(G_{\boxtimes})$. At each separation pair $[x,y]$, the multi-edge between two components induces a triangulation with triangles consisting of $x, y$ and a crossing points of two edges incident to $x$ and $y$. After an elimination of all B-configurations there is a kite or a W-configuration for each pair of crossing edges. Concerning the number of edges, at every separation pair $[x,y]$ with inner components $H_1,\ldots,H_p$ and a 4-cycle $(a_i, b_i, c_i, d_i)$ as outer boundary of $H_i$ replace the pairs of crossing edges $\{x, c_i\}, \{y, d_i\}$ and $\{x, a_{i+1}\}, \{y, b_{i+1}\}$ by a kite with edges $\{c_i, a_{i+1}\}, \{c_i, b_{i+1}\}, \{d_i, a_{i+1}\}, \{d_i, b_{i+1}\}$ and replace the i-th copy of $\{x, y\}$ between $H_i$ and $H_{i+1}$ by the edges $\{a_i, a_{i+1}\}$ and $\{b_i, b_{i+1}\}$ for $i=1,\ldots, p-1$. The resulting graph is 1-planar and has $q-1$ more edges than $G$. Hence, there are at most $4n-8$ edges. Each step from $\mathcal{E}(G)$ to $\mathcal{E}(G_{\boxplus})$ takes linear time and is performed on the embedding of the planarization. \end{proof} Concerning the density of 1-planar graphs, each W-configuration reduces the maximum number of edges by two, since each pair of edges crossing in the outer face can be substituted by four edges. This parallels the situation of planar graphs and 2-connected components.\\ Next, we consider vertex orderings of planar graphs which are later applied to graphs $G_{\square}$ and $G_{\bullet}$. Let $\{s,t\}$ be an edge of a planar graph in the outer face of an embedding of $G$. An \emph{st-numbering} is an ordering $v_1, \ldots, v_n$ of the vertices of $G$ such that $s=v_1$, $t=v_n$ and every vertex $v_i$ other than $s$ and $t$ is adjacent to at least two vertices $v_j$ and $v_k$ with $j<i<k$. It is known that every 2-connected graph has st-numberings and an st-numbering can be constructed in linear time \cite{et-stnum-76} for every edge $\{s,t\}$. An st-numbering induces an orientation of the edges of $G$ from a low ordered vertex to a high ordered one, called a \emph{bipolar orientation}.\\ For convenience, we identify each vertex with its st-number and with its orthogonal polygon in a shape visibility representation and consider each edge as oriented. In simple words a vertex $u$ is less than vertex $v$ and vertex $v$ is placed at some point.\\ If $G$ is planar, then a bipolar orientation transforms $G$ into an upward planar graph and partitions the set of edges incident to a vertex $v$ into a sequence of incoming and a sequence of outgoing edges \cite{dett-gdavg-99}. Accordingly, each vertex has two lists of faces below and above it, which are ordered clockwise or left to right. A face is below $v$ if both edges incident to $v$ are incoming edges, and above it, otherwise. In addition, at each separation pair $[x,y]$ with components $H_0, \ldots, H_k$, the vertices of each inner component $H_i$ with $i \geq 1$ are ordered consecutively and they appear between $x$ and $y$ if $x<y$. We can write $x < H_1 < \ldots < H_k < y$, where $H_1,\ldots,H_k$ is any permutation of the inner components. For example, one may choose the cyclic ordering at $x$ if an embedding is given. st-numberings are a useful tool for the construction of visibility representations of planar graphs \cite{dett-gdavg-99}. \emph{Canonical orderings} are used for straight-line drawings of planar graphs. They were introduced by de Fraysseix et al. \cite{fpp-hdpgg-90} for triangulated planar graphs and were generalized to 3-connected \cite{k-dpguco-96} and to 2-connected graphs \cite{hs-asdpg-98}. The subsequent definition is taken from \cite{bbc-leftish-11}. \begin{definition} Let $ \Pi = (P_0,\ldots, P_q)$ be a partition of the set of vertices of a graph $G$ of size $n\geq 5$ into paths such that $P_0 = \langle v_1, v_2 \rangle$, $P_q = \langle v_n \rangle$ and $\langle v_1, P_q, v_2 \rangle$ is the outer face in clockwise order. For $k=0, \ldots, q$ let $G_k$ be the subgraph induced by $V_k = P_0 \cup \ldots \cup P_k$ and let $C_k$ be the outer face of $G_k$, called contour. Then $\Pi$ is a \emph{canonical ordering} if for each $k=1,\ldots, q-1$: \begin{enumerate} \item $C_k$ is a simple cycle. \item Each vertex $z_i$ in $P_k$ has a neighbor in $V-V_k$. \item $\|P_k\|=1$ or each vertex $z_i$ in $P_k$ has exactly two neighbors in $G_k$. \end{enumerate} \end{definition} A canonical ordering $\Pi$ is refined into a vertex ordering $v_1, \ldots, v_n$ by ordering the vertices in each $P_k$, $k>0$, straight or in reverse. A canonical ordering can be computed by a peeling technique which successively removes the vertices of the paths in reverse order starting from $P_q$. For a quadrangle it would consist of two paths of length two. Care must be taken that the removal of the next path $P_k$ preserves the 2-connectivity of $G_i$ for $i=1,\ldots, k$, see \cite{bbc-leftish-11,k-dpguco-96}. The contour $C_k$ is ordered left to right with $v_1$ at the left and $v_2$ at the right so that edge $\{v_2, v_1\}$ closes the cycle. A path $P$ is a \emph{feasible candidate} for step $k+1$ of $\Pi = (P_0,\ldots, P_q)$ if also $(P_0,\ldots, P_k,P)$ can be extended to a canonical ordering of $G$. \begin{definition} A canonical ordering $\Pi = (P_0,\ldots,P_q)$ is called \emph{leftish} if for $k=0,\ldots, q-1$ the following is true: Let $c_{\ell}$ be the left neighbor of $P_{k+1}$ on $C_k$ and let $P$ be a feasible candidate for step $k+1$ with left neighbor $c_{\ell'}$. Then $c_{\ell} < c_{\ell'}$. \end{definition} A leftish canonical ordering of a 3-connected planar graph can be computed in linear time \cite{bbc-leftish-11}. For 2-connected planar graphs we extend the ordering as in the st-numbering case. At each separation pair $[x,y]$ with $x<y$ remove the inner components and compute the leftish canonical ordering of the 3-connected remainder. Then compute the leftish canonical ordering of each component and insert them just before $y$.\\ We study some properties of leftish canonical orderings on upward planar graphs. The orientation of the edges and upward direction is obtained from the (extended) leftish canonical ordering, which is an st-numbering with $s=1$ and $t=n$. Each edge has a face to its left and to its right if the graphs are 2-connected. Each face $f$ has a source and a sink, called $bottom(f)$ and $top(f)$, respectively. Suppose that edge $\{s,t\}$ is routed at the left. We call the face to the right of $\{s,t\}$ the leftmost face and the outer face is called the rightmost face \cite{dett-gdavg-99}.\\ In the remainder of this section, let $G$ be a 3-connected planar graph $G$ with an st-numbering whose faces are triangles or quadrangles that are traversed clockwise. The outer face is excluded. \begin{definition} A quadrangle $f = (a,b,c,d)$ is called a \emph{rhomboid} with bottom $a$, left end $b$, right end $d$, and top $c$ if there are two paths $\langle a,b,c \rangle$ and $\langle a,d,c \rangle$ enclosing $f$ with $b$ to the left of $f$ and $d$ to the right. Face $f$ is a \emph{left-trapezoid} if there is an edge $\{a,d\}$ to the right of $f$ and a path $\langle a, b, c,d \rangle$ to the left. A \emph{right-trapezoid} is defined accordingly, see Fig.~\ref{fig:rhombus}. \end{definition} \begin{figure} \caption{(a) a rhomboid, (b) a left-trapezoid, and (c) a right-trapezoid} \label{fig:kite1} \label{fig:B-config} \label{fig:W-config} \label{fig:rhombus} \end{figure} First, each path $P_k$ of a leftish canonical ordering $\Pi$ has length at most two, since one of $v_1$ and $v_r$ has at least two neighbors on $C_k$ if $P_k = \langle v_1, \ldots, v_r \rangle$ with $r\geq 3$. Otherwise, there are faces as $m$-gons with $m >4$. If $P_k$ has length two, then it is inserted into $C_k$. Otherwise, $C_{k+1}$ is obtained by replacing a subsequence $\gamma$ of $C_k$ by $v$ with $P_k = \langle v \rangle$, where the vertices in $\gamma$ are \emph{covered} by $v$ \cite{fpp-hdpgg-90, k-dpguco-96}. Second, if $P_k = \langle v_1, v_2 \rangle$ is a path of length two, then the face below $P_k$ is a quadrangle $f_k = ( u_1, v_1, v_2, u_2)$. We prefer rhomboids over trapezoids and therefore direct $P_k$ from $v_2$ to $v_1$ if $u_2 = bottom(f_k)$ and otherwise from $v_1$ to $v_2$. If $P_k = \langle v \rangle$ is a singleton then it may cover several faces, which are triangles, rhomboids, or trapezoids. We say that face $f_k$ is \emph{covered} by $P_k$. Third, we consider faces. For a vertex $v$ on a contour $C_k$ let $f_1(v),\ldots, f_{\nu}(v)$ be the left to right ordering of the faces incident to $v$ and above $C_k$ which is defined by the clockwise ordering of the outgoing edges. The outer face is discarded. For each quadrangle $f_i(v) = (v, b,c,d)$ let $t_i = top(f_i(v))$. Then $t_i=b$ if $f_i(v)$ is a right-trapezoid, $t_i=c$ if $f_i(v)$ is a rhomboid and $t_i=d$ if $f_i$ is a left-trapezoid and $t_i$ covers $f_i$. A face $f_i(v) = (v,b,d)$ or $f_i(v) = (v,b,c,d)$ for $i=1,\ldots, \nu$ has \emph{left-support} if there is a contour $C_{\ell} = (1, \ldots, b, v, \ldots, 2)$ and \emph{right-support} if $C_{\ell} = (1, \ldots, v, d, \ldots, 2)$. If $f_i(v)$ has left-support and $d$ is in $P_{k+j_i}$ for some $i_j \geq 1$ then either $d$ is immediately to the left of $v$ on the contour $C_{k+j_i}$ or the placement of $d$ covers $v$. The case is symmetric to the right if $f_i(v)$ has right support. For an illustration see Fig.~\ref{fig:faces}. \begin{figure} \caption{A sequence of faces above vertex $v$. Starting from a contour $C_k = (1, w_1, w_2, v, 2)$ there is a leftish canonical ordering $\Pi = (\langle u_1 \rangle, \langle u_2 \rangle, \langle u_3, u_4 \rangle, \langle v_1 \rangle, \langle v_2 \rangle, \langle v_3 \rangle, \langle v_4 \rangle)$. Face $f_1(v)$ is a left-trapezoid, $f_2(v)$ and $f_6(v)$ are triangles, $f_4(v)$ is a rhomboid, and $f_5(v)$ is a right-trapezoid. Face $f_2(v)$ is a rhomboid if edge $\{u_3, u_4\}$ is oriented from $v_4$ to $v_3$ and a left-trapezoid, otherwise. Face $f_1(v)$ has left-support, $f_5(v)$ has right-support, and $f_4(v)$ has left- and right-support. The sequence $(u_1, u_2,u_3, u_4, v_4, v_3, v_2, v_1)$ of neighbors of $v$ above $C_k$ is bitonic.} \label{fig:faces} \end{figure} \begin{lemma} \label{lem:leftish} Let $v$ be a vertex on a contour $C_k$. A quadrangle $f$ with $v=bottom(f)$ has left-support (right-support) if $f$ is a left-trapezoid (right-trapezoid). \end{lemma} \begin{proof} For a contradiction, suppose that $f = (a,b,c,d)$ is a left-trapezoid and has no left-support. Since $f$ is a left-trapezoid, vertex $b$ appears before vertices $c$ and $d$ in the vertex ordering. If $b$ has a neighbor to the right of $v$ on the contour, then $f$ cannot be a left-trapezoid. \end{proof} Note that the statement may not apply to the outer face, which later on may need a spacial treatment. The type of quadrangles $f(v)$ is determined by their support and the length of the path with the top vertex in the leftish canonical ordering. Let $f_1(v), \ldots, f_{\nu}(v)$ be the left to right (clockwise) ordering of faces with bottom $v$ above $C_k$ and let $j_1 < \ldots < j_{\mu}$ be the subsequence of quadrangles. The other faces are triangles. For $i=1, \ldots, \nu$ let $P_{t_i}$ contain the top vertex of $f_{i}$ so that $f_{i}$ is closed by $P_{t_i}$. Thus, $t_i$ is the time stamp for the completion of face $f_{j_i}$ in a canonical ordering and a drawing based on it. \begin{lemma} \label{lem:decidetrapezoid} If a face $f=f_i(v)$ for $i=1, \ldots, \nu$ is a triangle, then it has left-support or right-support and $P_{t_i}$ is a singleton. If $f$ is a quadrangle, then $f$ is a left-trapezoid if $f$ has no right-support and $P_{t_i}$ is a singleton. Face $f$ is a right-trapezoid if $f$ has no left-support and $P_{t_i}$ is a singleton, and $f$ is a rhomboid if $f$ has left and right-support or $P_{t_i}$ is a path of length two. \end{lemma} \begin{proof} If $f$ is a triangle, then it must be closed by a path of length one of the leftish canonical ordering and therefore it has left- or right-support. Each quadrangle $f$ has a left- or a right-support, since the paths have length at most two. If $f$ has no right-support and $P_{t_i}$ is a singleton, then vertices $b$ and $c$ are placed before vertex $d=top(f)$ if $f=(v,b,c,d)$ in clockwise order and $f$ is a left-trapezoid. The case to the right is symmetric. If $f=(v,b,c,d)$ has left- and right-support, then $c=top(f)$ and $b$ and $d$ are less than $c$ in the vertex numbering, so that $f$ is a rhomboid. If $P_{t_i}$ is a path of length two, then $f$ is a rhomboid by construction. \end{proof} The leftish canonical ordering also determines the order in which the vertices of the faces at $v$ are placed. \begin{lemma} \label{lem:bitonic} For a leftish canonical ordering $\Pi$ and a vertex $v$ on a contour $C_k$, the clockwise sequence of neighbors $w_1, \ldots, w_z$ of $v$ above $C_k$ is bitonic, i.e., there is some $m$ with $1 \leq m \leq z$ such that $w_1 < \ldots < w_m$, $w_{m+1} > \ldots > w_z$ and $w_m < w_z$ in $\Pi$. \end{lemma} For an illustration, see Fig.~\ref{fig:faces} and observe that the sequence of the vertices corresponds to the order in which the faces above $v$ are completed, first from left to right and then from right to left. \begin{proof} Consider the clockwise sequence of faces $f_1(v),\ldots, f_{\eta}(v)$ with $\eta \leq z$. Then there is some $\mu$ so that $f_j$ has left-support for $i=1,\ldots, \mu$ and $f_j$ has right-support for $\mu+1,\ldots, \eta$. Otherwise, suppose for some $\kappa$ with $1 \leq \kappa < \mu-1$ face $f_{\kappa}$ has right-support and face $f_{\kappa+1}$ has left-support. Then $C_{j_{\kappa}} = (1,\ldots, v, d, \ldots, 2)$ for some vertex $d$ of $f_{\kappa}$ and $C_{j_{\kappa+1}} = (1,\ldots, d, v, \ldots, 2)$, a contradiction. In consequence, for $1 \leq j < \mu$ and vertices $x$ in $f_j(v)$ and $y$ in $f_{j+1}(v)$ it holds that $x < y$ in $\Pi$. Similarly, we have $x > y$ for vertices $x$ in $f_j(v)$ and $y$ in $f_{j+1}(v)$ and $\mu+1 \leq j \leq z$. Paths of length two of $\Pi$ are ordered in accordance with this ordering. Now let $w_1, \ldots, w_m$ be the vertices in faces $f_1(v),\ldots, f_{\mu}(v)$. By planarity, the ordering of the faces is in accordance with the ordering of the vertices in $\Pi$, which is bitonic. \end{proof} Note that the sequence of neighbors of $v$ is not continuous in the leftish canonical ordering. In general, a neighbor $w_i$ of vertex $v$ on $C_k$ is a right-support of some face $f(u)$ for a vertex $u$ to the left of $v$ on $C_k$. \section{Rectangle Visibility Representation of IC-planar Graphs} \label{sect:proof1} The proofs of Theorems \ref{thm:IC-RVG} and \ref{thm:1-cat} are constructive. The outline of the algorithms is as follows: Take an embedding $\mathcal{E}(G)$ as a witness for IC-planarity and 1-planarity, respectively. The embedding is first augmented to $\mathcal{E}(G_{\boxplus})$ as given in Lemma \ref{lem:normalform}. Since planar maximal IC-planar graphs are 3-connected we can use $\mathcal{E}(G_{\boxtimes})$ in this case. Thereby, some edges can be rerouted and some are multiplied to separate components. If the input were a graph, then constructing the normal form embedding is an NP-hard problem, since the general recognition problem is NP-hard and is solvable in polynomial time for graphs with a normal form embedding \cite{b-4m1pg-15, cgp-rh4mg-06}. Next, $G_{\boxplus}$ is planarized to $G_{\square}$ by a removal of all pairs of crossing edges while preserving the (new) embedding. The planarization is obtained via $G_{\bullet}$ for IC-planar graphs. In a nutshell, the algorithms use a standard algorithm for the construction of a visibility representation of a planar graph, see \cite{dett-gdavg-99, k-mcvr-97, rt-rplbopg-86,tt-vrpg-86}. Finally, the pairs of crossing edges are added to the planar visibility representation of $G_{\square}$. In case of IC-planar graphs, there is a quadrangle $f = (a,b,c,d)$ which is drawn as a rhomboid with a vertex $b$ to the left and a vertex $d$ to the right of $f$ and at the same level ($y$-coordinate) into which a pair of crossing edges is inserted. In the second case, we use a leftish canonical ordering for an st-numbering and the capabilities of horizontal and vertical lines of sight in the weak visibility version. First, we define kite-contraction and kite-expansion operations on IC-planar embeddings in normal form. By IC-planarity, two kites have no common vertex and do not intersect so that kite-contractions do not interfere. Moreover, each pair of crossing edges is embedded as a kite and $G$ is 3-connected if $\mathcal{E}(G)$ is an IC-planar embedding in normal form,\cite{abk-sld3c-13, bbhnr-NIC-16} \begin{definition} Let $\mathcal{E}(G)$ be an IC-planar embedding in normal form and suppose there is an $st$-ordering of $G$. A \emph{kite-contraction} contracts a kite $\kappa$ with boundary $(a,b,c,d)$ of $\mathcal{E}(G)$ to a single vertex $v_{\kappa}$ so that $v_{\kappa}$ inherits all incident edges and henceforth has multi-edges. A \emph{kite-expansion} is the inverse operation on the boundary and replaces $v_{\kappa}$ by the 4-cycle $(a,b,c,d)$. Both operations are adjacency preserving so that a kite-contraction followed by a kite-expansion just removes the pair of crossing edges of $\kappa$. The kite-contraction $G_{\bullet}$ of $G$ is obtained by contracting all kites of $\mathcal{E}(G)$. \end{definition} \begin{lemma} Let $\mathcal{E}(G)$ be an IC-planar embedding in normal form. Then graph $G_{\bullet}$ is a 3-connected planar graph, which can be computed from $\mathcal{E}(G)$ in linear time. \end{lemma} \begin{proof} For planarity, first remove one edge from each pair of crossing edges of each kite, which results in a graph $G_{\triangle}$. Then $G_{\triangle}$ is a triangulated planar graph that inherits its embedding from $\mathcal{E}(G)$. Thus it is 3-connected. Next contract the edges that remain from each kite to a vertex $v_{\kappa}$. Since the planar graphs are closed under taking minors, the edge contractions preserve planarity and yield $G_{\bullet}$, since by IC-planarity each vertex $v$ either remains or is contracted to a vertex $v_{\kappa}$. The removal of multi-edges results in a triangulated planar graph, which is 3-connected, and so is $G_{\bullet}$. It takes linear time to obtain $G_{\triangle}$ from $\mathcal{E}(G)$ and $G_{\bullet}$ from $G_{\triangle}$. \end{proof} Note that this type of kite-contractions cannot be applied to 1-planar (or NIC-planar) graphs, since vertices may belong to several kites. Instead one may contract a kite to a single vertex which corresponds to its crossing point or consider the $K_4$ network \cite{begghr-odm1p-13}. \\ Next, we consider rhomboidal st-numberings. \begin{definition} An embedding $\mathcal{E}(G)$ of a 1-planar graph in normal form is called \emph{rhomboidal} with respect to an st-numbering if the $K_4$ subgraph induced by a pair of crossing edges is embedded as a kite whose boundary is a rhomboid. \end{definition} Rhomboidal embeddings distinguish IC-planar graphs from NIC-planar graphs. \begin{lemma} For every IC-planar graph $G$ and every planar edge $\{s,t\}$ there is a rhomboidal embedding which can be computed in linear time from $\mathcal{E}(G)$. \end{lemma} \begin{proof} First, construct an IC-planar embedding in normal form with $\{s,t\}$ in the outer face. Next, compute a kite-contraction $\mathcal{E}(G_{\bullet})$ and an st-numbering of $G_{\bullet}$. Then do the kite-expansion and extend the st-numbering of $G_{\bullet}$ to an st-numbering of $G$ as follows: For each contracted kite $\kappa$ determine a top and a bottom vertex and then the left and right ends of the face $f$ of $\kappa$ without the pair of crossing edges. Expand $\kappa$ in $\mathcal{E}(G_{\bullet})$. If there is exactly one vertex $u$ of $\kappa$ with only incoming (multi-)edges, then let $u = bottom(f)$. Choose the top vertex opposite to $u$, and the left and right ends to the left and right of $f$. Similarly, choose $v=top(f)$ if only $v$ has outgoing (multi-)edges and choose $bottom(f)$ opposite to $v$. Otherwise, choose a pair of opposite vertices so that $u=bottom(f)$ has incoming and $v=top(f)$ has outgoing (multi-)edges and determine the left and right ends. Clearly, each step takes linear time. \end{proof} We are now able to describe 1-planar graphs that admit a right angle crossing drawing which is a step towards the intersection of 1-planar and RAC graph that is asked for in \cite{bdeklm-IC-16, el-racg1p-13}. \begin{theorem} \label{thm:rhomboidal} If $G$ is a 3-connected 1-planar graph so that the augmentation $G_{\boxtimes}$ has a rhomboidal embedding with respect to a canonical ordering, then $G$ is a RAC graph. \end{theorem} \begin{proof} Our algorithm is a simplification of the technique used in Case 1 of the proof of Theorem 2 in \cite{bdeklm-IC-16}, where more details can be found. The planar subgraph $G_{\square}$ of $G_{\boxtimes}$ is 3-connected \cite{abk-sld3c-13} and has a rhomboidal canonical ordering by assumption. Graph $G_{\square}$ is processed according to the canonical ordering using the shift technique as in \cite{fpp-hdpgg-90} and extended to 3-connected graphs in \cite{k-dpguco-96}. For every quadrilateral face $f = (a,b,c,d)$ in clockwise order with $a=bottom(f)$ and $c=top(f)$, the algorithm first places $a$, then $b$ and $d$ in any order, and finally $d$ according to the canonical ordering. Vertex $b$ is placed on the $-1$-diagonal through $a$ to the left and $d$ is placed on the $+1$-diagonal through $a$ to the right and at the intersection with the $+1$ and $-1$ diagonal of the left lower and right lower neighbors, respectively. The technique in \cite{bdeklm-IC-16} is a leveling of $b$ and $d$. If $b$ is placed $\delta$ units below $d$, or vice versa, then lift $b$ to the level of $d$ by $2\delta$ extra shifts to the left. If $b$ has been leveled with other vertices, then the shift is synchronously applied to all vertices that are leveled with $b$. Alternatively, on may apply the critical (or longest) path method so that the critical paths to $b$ and $d$ have the same length. At the placement of $c = top(f)$ we shift $b$ to the left or $d$ to the right so that $c$ is placed vertically above $d$. Then edge $\{b,d\}$ is inserted as a horizontal line and $\{a,c\}$ as a vertical one. Later on, $b,a$ and $d$ are shifted by the same amount as $c$, so that the right angle crossing of $\{b,d\}$ and $\{a,c\}$ is preserved. \end{proof} There are rhomboidal 1-planar graphs that are not NIC-planar, such as $k \times k$ grids with a pair of crossing edges in each inner quadrangle and a triangulation of the outer face. (The graphs are not NIC-planar, because they have too many edges). On the other hand, every IC-planar graph admits a rhomboidal embedding and we have a simpler proof than in \cite{bdeklm-IC-16}. \begin{corollary} Every IC-planar graph is a RAC graph. \end{corollary} \begin{corollary} There are 3-connected 1-planar graphs and NIC graphs that do not admit a rhomboidal canonical ordering. \end{corollary} \begin{proof} There are 1-planar graphs \cite{del-dgrac-11} and even NIC-planar graphs \cite{bbhnr-NIC-16} that are not RAC and a rhomboidal canonical ordering would contradict Theorem~\ref{thm:rhomboidal}. \end{proof} We now turn to rectangle visibility representations of IC-planar graphs and the proof of Theorem~\ref{thm:IC-RVG}. A visibility representation of a 2-connected planar graph $G$ is commonly obtained by the following steps \cite{dett-gdavg-99,k-mcvr-97}, which we call VISIBILITY-DRAWER: \begin{enumerate} \item Compute an st-numbering $\delta(v)$ for the vertices of $G$ with an edge $\{s,t\}$ and $\delta(s)=1$ and $\delta(t)=n$. Embed edge $\{s,t\}$ at the left and orient the edges according to the st-numbering. \item Compute the $s^t^$-numbering of the dual graph $G^$ where $s^$ is the face to the right of $\{s,t\}$ and $t^$ is the outer face. \item For an oriented edge $e$ let left$(e)$ (right$(e)$) be the $s^t^$-number of the face to the left (right) of $e$. For a vertex $v \neq s,t$ let left$(v) =$ min$\{$left$(e) \, \| \, e$ is incident to $v\}$ and right$(v) =$ max$\{$right$(e) \, \| \, e$ is incident to $v\}$. \item For each vertex $v \neq s,t$ draw a bar between $(\textrm{left}(v), \delta(v))$ and $(\textrm{right}(v)-1, \delta(v))$ and draw a bar between $(0,0)$ and $(M-1,0)$ for $s$ and between $(0,n-1)$ and $(M-1,n-1)$ for $t$ where $M \leq 2n-4$ is the number of faces of $G$. \item Draw each edge $e= \{u,v\} \neq \{s,t\}$ between $(\textrm{left}(e), \delta(u))$ and $(\textrm{left}(e), \delta(v))$ and draw $\{s,t\}$ at $x=1$. \end{enumerate} There is exactly one vertex at each level $y = 1, \ldots, n$ if the st-numbers are used for the $y$-coordinates of the vertices. More compact drawings are obtained by using the critical path method or topological sorting \cite{dett-gdavg-99, k-mcvr-97}. The drawings are not really pleasing, since many lines of sight are at the ends of the bars. There are no degenerated faces since the right end of a bar to the left of face $f$ is at least one unit to the left and of a bar to the right of $f$. The drawing algorithm preserves the given embedding. The change of the embedding at B-configurations can be undone by modifying the computation of the $x$-coordinates of the lines of sight. Before the computation of the dual $s^t^$-numbering add a copy of the rerouted $\{s,t\}$ edge at its original place and remove the edge(s) that were added for the 3-connectivity and compute $\delta^$ on the new embedding. The line of sight for the edge $\{s,t\}$ can be drawn at several places \cite{b-vr1pg-14} and we must choose the one that preserves the embedding. \begin{algorithm} \caption{IC-RV-DRAWER}\label{alg:IC-RVG} \KwIn{An IC-planar embedding $\mathcal{E}(G)$.} \KwOut{A rectangle visibility representation $\mathcal{RV}(G)$.} Transform $\mathcal{E}(G)$ into a normal form embedding $\mathcal{E}(G_{\boxtimes})$.\; Compute the planar graph $G_{\square}$ and a rhomboidal st-numbering $\delta$ of $G_{\square}$. \; Compute an $s^t^$-numbering $\delta^$ of the dual graph $G_{\square}^$. \; \ForEach{vertex v of $G_{\square}$} { \If{$v$ is the left (right) end of a rhomboid and $u$ is the other end} { $d(v) = \delta(u)+\delta(v)$ } \Else { $d(v) = 2\delta(v)$} } Compute a planar visibility representation of $G_{\square}$ by \\ \quad VISIBILITY-DRAWER with vertices on level $d(v)$ and edges at left$(e)$. \ForEach{pair of crossing edges $\{a,c\}$ and $\{b,d\}$ \textrm{in a} \\ \quad \textrm{rhomboid} $f=(a,b,c,d)$ \textrm{with} $d(a) < d(b)= d(d) < d(c)$ } { Add a horizontal line of sight at level $d(b)$ between the bars of $b$ and $d$.\; Add a vertical line line of sight at $\delta^(f)+0.5$ between \\ \quad the bars of $a$ and $c$. } Scale all $x$-coordinates by two.\; Remove (or ignore) all lines of sight of edges not in $G$. \end{algorithm} The following Lemma concludes the proof of Theorem~\ref{thm:IC-RVG}. \begin{lemma} \label{lem:correctnessALG1} Algorithm IC-RV-DRAWER constructs a rectangle visibility representation of an IC-planar graph on $O(n^2)$ area and operates in linear time. \end{lemma} \begin{proof} The algorithm computes a planar visibility representation of $G_{\square}$ in linear time as proved in \cite{dett-gdavg-99, rt-rplbopg-86, tt-vrpg-86} on an area of size $(2n-5) \times 2n$, which is scaled by a factor of two in $x$-dimension. Each pair of crossing edges is in a kite of $G_{\boxtimes}$ whose boundary is embedded as a rhombus $f=(a,b,c,d)$ with $a=bottom(f)$. Then the y-coordinates of $b$ and $d$ coincide and $d$ is a weighted topological sorting as used in \cite{dett-gdavg-99}. There is a gap of one unit between the bars of $b$ and $d$, since the bar of $b$ ends at $\delta^(f)-1$ and the bar of $d$ begins at $\delta^(f)$. Now, edges $\{a,c\}$ and $\{b,d\}$ are added so that they cross inside $f$. Since $G$ has at most $13/4n-4$ edges, the transformation into normal form takes linear time so that $G_{\boxtimes}$ and $G_{\square}$ have size $O(n)$. The visibility representation of $G_{\square}$ is computed in linear time \cite{dett-gdavg-99}. There are at most $n/4$ pairs of crossing edges which are each inserted in $O(1)$ time. \end{proof} \section{T-Visibility of 1-planar Graphs} \label{sect:proof2} For the \textsf{T}-visibility representation of 1-planar graphs we use a leftish canonical ordering as an st-numbering and draw $G_{\square}$ by VISIBILITY-DRAWER. Note that $G_{\square}$ may have multi-edges at separation pairs, which each introduces a face. In total, $G_{\square}$ has at most $2n-4$ faces, since each multi-edge could be substituted by a planar edge. For the pairs of crossing edges we expand some vertices to a $\bot$-shape. A $\bot$-shaped vertex consists of a horizontal bar and a vertical \emph{pylon}. By a horizontal flip we obtain a \textsf{T}-shape visibility representation. If vertex $v$ is $\bot$-shaped, then the pylon is inserted into the face of a left- or right-trapezoid $f$ with $v = bottom(f)$ and $top(f)$ is maximum. For each quadrangle $f= (v, b,c,d)$, the edges $\{v,c\}$ and $\{b,d\}$ were removed in the planarization step. They are reinserted as follows: If $f$ is a rhomboid, then the lower of $b$ and $d$ gets a pylon for an $\llcorner$- or $\lrcorner$-shape so that $\{b,d\}$ is a horizontal line that is crossed by a vertical line of sight for $\{a,c\}$ inside $f$. If $f$ is a left-trapezoid, then the bar of $b$ is extended to the right and edge $\{b,d\}$ is added as a vertical line of sight inside $f$. Accordingly, extend the bar of $d$ to the left if $f$ is a right-trapezoid. The particularity is the drawing of edge $\{v,c\}$ as a horizontal line of sight from the pylon of $v$ to the bar (or pylon) of $c$, as depicted in Fig.~\ref{fig:rhomboids}. This line of sight is unobstructed, since there is exactly one bar on each level by the use of $st$-numbers for the $y$-coordinate of the vertices and there is no obstructing pylon from another vertex by the use of the leftish canonical ordering and the bitonic order of the vertices above a vertex $v$, as stated in Lemma \ref{lem:bitonic}. \begin{figure} \caption{(a) Two left-trapezoids $f=(v, b,c,d)$ and $f'=(v,d,c',d')$ and (b) their visibility representation.} \label{fig:kite1} \label{fig:B-config} \label{fig:rhomboids} \end{figure} There is a special case at separation pairs and W-configurations, see Fig.~\ref{fig:XWdouble}. If $G-\{x,y\}$ partitions into an outer component $H_0$ and inner components $H_1,\ldots, H_p$, then VISIBILITY-DRAWER places the inner components from left to right between the bars of $x$ and $y$ and separates them in $x$-dimension by the $st$-numbering and in $y$-dimension by the $s^t^$-numbering. It admits a representation of the copies of edge $\{x,y\}$ by vertical lines of sight. Consider the planarization $\widehat{H}_{\square}$ of an inner component $H$ together with the separation pair $x,y$. The outer face of $\widehat{H}_{\square}$ is a quadrangle $f=(x, b,c,y)$ which is embedded as a left-trapezoid with a copy of $\{x,y\}$ on the right. The vertex numbering from the leftish canonical ordering is $x < b < w < c < y$, where $w \neq b,c$ is any other vertex of $H_i$. Thus, $b$ and $c$ are the first and last vertex of $H_i$. They are not unique, since the embedding of $H_i$ can be flipped. In particular, $P_{q-1} = \langle c \rangle$ and $P_q = \langle t \rangle$ are the last two paths in the leftish canonical ordering of $\widehat{H}_{\square}$, since there is no separating triangle $\Delta = (t,c,w)$ for some vertex of $H$. Hence, $b$ and $c$ are placed on the lowest and highest levels of the vertices of $H$. The outer edge $\{b,t\}$ is represented as a vertical line of sight between the bars of $b$ and $t$ as if it were a planar edge, whereas edge $\{x,c\}$ is a horizontal line of sight from the pylon of $x$ to the pylon of $c$. Algorithm T-DRAWER constructs the visibility representation. \begin{figure} \caption{ A graph $DXW$ consisting of two copies of the extended wheel graph $XW_6$ with vertices $1, \ldots, 8$ and $1, 2',\ldots, 7', 8$. Planar edges are drawn black and bold and crossing edges red and dotted. The copy of edge $\{1,8\}$ is drawn dashed. The extended leftish canonical ordering is $\Pi = (\langle 1,2 \rangle, \langle 3,4 \rangle, \langle 5,6 \rangle, \langle 7 \rangle, \langle (1), 2' \rangle, \langle 3', 4' \rangle, \langle 5', 6' \rangle, \langle 7' \rangle, \langle 8 \rangle)$. } \label{fig:XWdouble} \end{figure} \begin{algorithm} \caption{T-DRAWER}\label{alg:IC-RVG} \KwIn{A 1-planar embedding $\mathcal{E}(G)$.} \KwOut{A T-visibility representation $\mathcal{TVR}(G)$.} Compute $\mathcal{E}(G_{\boxplus})$ from $\mathcal{E}(G)$.\; Compute $G_{\square}$ from $\mathcal{E}(G_{\boxplus})$ by removing all pairs of crossing edges.\; Compute an $st$-numbering $\delta$ of $G_{\square}$ as an extension of a leftish canonical \\ \quad ordering of each 3-connected component. \; Compute the $s^t^$-numbering $\delta^$ of the dual graph $G_{\square}^$. \; Compute the planar visibility representation of $G_{\square}$ by \\ \quad VISIBILITY-DRAWER.\; \ForEach{vertex $v$ with quadrangles $f$ such that $v =bottom(f)$ } { \textbf{int} $v_{max} = 0$; \textbf{face} $f_{max}$ \; \ForEach{left-trapezoid $f=(v,b,c,d)$} { \If{$v_{max} < c$} { $v_{max} = c$; \, $f_{max}=f$ } Extend the bar of $b$ by $1/3$ to the right and at its right end \; add a vertical line of sight for $\{b,d\}$. } \ForEach{right-trapezoid $f=(v,d,c,b)$} { \If{$v_{max} < c$} { $v_{max} = c$; \, $f_{max}=f$ } Extend the bar of $b$ by $1/3$ to the left and at its left end \; add a vertical line of sight for $\{b,d\}$. } \ForEach{rhombus $f=(v,b,c,d)$} { Enlarge the bar of the lower of $b$ and $d$ by a pylon at an end \\ \quad and inside $f$ and up to the bar of the upper vertex. \; Add a horizontal line of sight for $\{b,d\}$ at the top of the pylon.\; Add a vertical line of sight for $\{v,c\}$ between the bars of $v$ and $c$ \\ \quad at $\delta^(f)+1/3$\; } \If{$v_{max} \neq 0$} { Enlarge the bar of $v$ by a pylon inside $f_{max}(v)$ from \\ \quad $(\delta^(f_{max}(v))+1/3, \delta(v))$ to $(\delta^(f_{max}(v))+1/3 , \delta(v_{\max}))$\; \ForEach {trapezoid $f=(v,b,c,d)$ } { \If {$f$ is not a left-trapezoid with a separation pair $[v,d]$ } { Add a horizontal line of sight for $\{v,c\}$ \\ \quad from the pylon to the bar of $c$. } \Else (\algcom{the horizontal line of sight may be occupied}) { Enlarge the bar of $c$ by a pylon of height $1/2$. \; Add a horizontal line of sight for $\{v,c\}$ at the top of the pylon of $c$. } } } } Scale all $x$-coordinates by three and all y-coordinates by two.\; \end{algorithm} \pagebreak \begin{lemma} \label{lem:pylonSee} Suppose $G$ is a 3-connected 1-planar graph and there is no W-configuration in the outer face of an embedding of $G$. For a vertex $v$ on a contour $C_k$, let $f_{j_1}(v),\ldots, f_{j_{\mu}(v)}$ be the sequence of left- and right-trapezoids above $v$ from left to right with $f_i(v) = (v, b_i, c_i, d_i)$. Let $v_{max} = \max \{ c_i \, \| \, i=j_1,\ldots, j_{\mu}\}$ and let $f_{max}(v)$ be the trapezoid containing $v_{max}$. Then the pylon of $v$ inside $f_{max}(v)$ can see the bar of each vertex $c_i$ for $i \in \{j_1,\ldots, j_{\mu}\}$. \end{lemma} \begin{proof} By Lemma \ref{lem:bitonic}, there is a bitonic sequence of clockwise neighbors of $v$, which each has is own $y$-coordinate according to the leftish canonical ordering. Hence, a horizontal line of sight from the pylon of $v$ is unobstructed by bars of other vertices. A horizontal line of sight from the pylon to $c_i$ intersects only vertical lines of sight of planar edges $\{v, w\}$ with $c_i < w$ and $c_i$ and $w$ are on the same side of the pylon, i.e., $c_i, w < w_m$ or $c_i, w > w_m$, where $w_m$ is the maximum neighbor of $v$ (or the top vertex of $f_{j_{\mu}}(v) $) in the leftish canonical ordering. Hence, a line of sight $\{v, c_i\}$ is unobstructed by pylons of other vertices. In consequence, each edge $\{v, c_i\}$ with $i \in \{j_1,\ldots, j_{\mu}\}$ is represented in the visibility representation constructed by T-DRAWER. \end{proof} Finally, consider a separation pair $[x,y]$ with inner components $H_1,\ldots, H_p$. The st-numbering extending the leftish canonical ordering of 3-connected components inserts the vertices of each component consecutively and just before $y$ so that there is a subsequence $x, H_0', H_1, \ldots, H_p, y$, where $H_0'$ is a subgraph of the outer component that is added by the leftish canonical ordering between $x$ and $y$. Each component $H_i$ is drawn in a box $B(H_i)$ and the boxes are ordered monotonically in $x$- and in $y$-dimension to a staircase between the bars of $x$ and $y$ both by the common visibility drawer and by T-DRAWER, as illustrated in Fig.~\ref{fig:XWplanarVis}. Consider the outer face of an inner component including the separation pair. Without crossing edges, there is a quadrangle $f_{out}(H) = (x, b,c,y)$, which is embedded as a left-trapezoid. However, $f_{out}(H)$ has no left-support, since $x<b<c < y$ in the leftish canonical ordering and edge $\{x,y\}$ of $f_{out}(H)$ is a copy of the original edge. This case is treated as an \emph{exception}. Edge $\{b,y\}$ is drawn inside $f_{out}(H)$ and to the right of $H$ after an extension of the bar of $b$ to the right. Vertex $x$ is $\bot$-shaped with a high pylon up to $y$ which is placed in the face to the right of the original edge $\{x,y\}$. The pylon can see all vertices that are neighbors of $x$ in the trapezoids of $H$ by Lemma \ref{lem:pylonSee}. However, the horizontal line of sight to $c$ may be occupied, as in Fig.~\ref{fig:XWTVis}. Fortunately, $c$ is the last vertex of $H$ in the leftish canonical ordering and a short pylon for the bar of $c$ admits a horizontal line of sight between $x$ and $c$. Since the inner components are separated in $y$-dimension, the pylon of $x$ can see all neighbor of $x$ in the trapezoids of the inner components. The following Lemma concludes the proof of Theorem \ref{thm:1-cat}. \begin{lemma} \label{lem:correctALG2} Algorithm T-DRAWER constructs a \textsf{T}-visibility representation of a 1-planar graph on $O(n^2)$ area and operates in linear time. \end{lemma} \begin{proof} The computations of $G_{\boxplus}$, the removal of all pairs of crossing edges for $G_{\square}$, the st-numbering as an extension of a leftish canonical ordering, the $s^t^$-numbering and the planar visibility representation of $G_{\square}$ each take linear time if a 1-planar embedding of $G$ is given. There are at most n-2 pairs of crossing edges which can each be inserted in $O(1)$ time into the visibility representation of $G_{\square}$. Hence, T-DRAWER runs in liner time. The visibility representation of $G_{\square}$ has size at most $(2n-5) \times n$, which is expanded by a factor of six. The common visibility drawer provides a correct visibility representation of $G_{\square}$. For each 3-connected component without a W-configuration, the pairs of crossing edges are correctly added to the visibility representation by Lemma \ref{lem:pylonSee}. The pair of edges crossing in the outer face of a W-configuration is visible by the special treatment in lines 34 and 35. Since inner components at a separation pair $[x,y]$ are strictly separated in both dimensions and are placed between the bars (shapes) of $x$ and $y$, there is a line of sight between the shapes of $x$ and $y$ for each edge between $x$, $y$ and vertices of inner components. Finally, consider the decomposition tree. If $H$ is an inner component at a separation pair $[x,y]$, then there is no edge $\{u,v\}$ from a vertex $u$ with $x \neq u \neq y$ of the outer component to a vertex $v$ of $H$ and, hence, there is no need for a line of sight. In addition, there is no need for a horizontal line of sight from a pylon through the visibility representation of an inner component, since the st-numbering groups components recursively and thereby separates them. Hence, the pylons in inner components do not obstruct horizontal lines of sight from pylons of vertices of the outer component. \end{proof} It is important to use weak visibility, since a pylon can see the bars and pylons of many other vertices, which is forbidden in the strong visibility version. As an example, consider the extended wheel graph $XW_6$ \cite{s-s1pg-86} and then take copies of it and identify two vertices, here 1 and 8. These graphs have been used for the construction of sparse maximal 1-planar graphs \cite{begghr-odm1p-13} and for a linear lower bound on the number of legs (vertex complexity) in embedding-preserving caterpillar-shape visibility representations \cite{ddelmmw-ovreg-16}. Graph $XW_6$ can be seen as a cube in 3D in which each face contains a pair of crossing edges. The visibility representation of $G_{\square}$ from the common visibility drawer is displayed in Fig.~\ref{fig:XWplanarVis} and the $\bot$-shape visibility representation of T-DRAWER in Fig.~\ref{fig:XWTVis}. Note that the graphs even admit a rectangle visibility representation (use the high pylons of vertices $1, 2, 5,2',5'$ and fill $4$ and $4'$ to a rectangle in Fig.~\ref{fig:XWTVis}). \begin{figure} \caption{A visibility representation of $DXW_{\square}$ from Fig.~\ref{fig:XWdouble} with dashed lines of sight and colored faces.} \label{fig:XWplanarVis} \end{figure} \begin{figure} \caption{The \textsf{T}-shaped visibility representation of graph $DXW$ from Fig.~\ref{fig:XWdouble} by T-DRAWER (with pylons in blue).} \label{fig:XWTVis} \end{figure} \iffalse \begin{figure} \caption{OPTIONAL: A rectangle visibility representation of $XW_6$.} \label{fig:XWTVis} \end{figure} \fi \section{General Shape Visibility Graphs} \label{sect:general} There is a natural ordering relation $\sigma < \sigma'$ between shapes if $\sigma$ is a restriction of $\sigma'$ including rotation and flip. For example, $\textsf{I} < \textsf{L} < \textsf{F} < \textsf{E}$ and $\textsf{I}< \textsf{T} < \textsf{E} < rake < caterpillar$. Clearly, every $\sigma$-shape visibility graph is a $\sigma'$-shape visibility graph if $\sigma < \sigma'$. However, it is unclear whether different shapes imply different classes of shape visibility graphs. Moreover, shapes with cycles, such as \textsf{O} or \textsf{B} are not really useful for shape visibility representations, since a cycle corresponds to an articulation vertex. For shape visibility graphs we can state: \begin{lemma} \label{lem:thickness} Every shape visibility graph has thickness two. \end{lemma} \begin{proof} The subgraph induced by the horizontal (vertical) lines of sight is planar. \end{proof} \begin{corollary} $\sigma$-visibility graphs of size $n$ have at most $6n-12$ edges and there are $\sigma$-visibility graphs with $6n-20$ edges for every shape $\sigma$. \end{corollary} The upper bound follows from Lemma \ref{lem:thickness} and the lower bound has been proved by Hutchinson et al.~\cite{hsv-rstg-99} for rectangle visibility graphs. The exact bound are unclear for all shapes except rectangles.\\ The extended wheel graph $XW_6$ even admits a rectangle visibility representation, and so do all wheel graphs $XW_{2k}$ with $k \geq 3$. An extended wheel graph consists of a cycle of vertices $v_1, \ldots, v_{2k}$ of vertices of degree six so that each $v_i$ is adjacent to its next and next but one vertex in cyclic order. In addition, there are two poles $p$ and $q$ that are adjacent to all $v_i$ (but there is no edge $\{p,q\})$. Extended wheel graphs play a prominent role for 1-planar graphs with $4n-8$ edges \cite{b-ro1plt-16, s-s1pg-86, s-rm1pg-10}.\\ We close with some open problems:\\ \noindent\textbf{ Conjecture}: \begin{enumerate} \item Every 1-planar graph with $4n-8$ edges is a rectangle visibility graph. \item There are \textsf{L}-visibility graphs that are not rectangle visibility graphs (\textsf{I}-shape) and there are \textsf{T}-visibility graphs that are not \textsf{L}-visibility graphs. \end{enumerate} \end{document}	arXiv
Abstract: We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product of a topological group over topological spaces with a continuous sandwich function. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $S\times S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or each power of $S$ is countably compact. The last item generalizes an old Wallace's result saying that each simple compact topological semigroup is a topological paragroup.	CommonCrawl
Difference between revisions of "Colloquia/Fall18" Nagreen (talk \| contribs) (→‎Abstracts) This lecture has been conceived for a broad audience. Accordingly, unnecessary technicalities will be avoided. ===Friday, March 17 at 4:00pm: Lillian Pierce=== "p-torsion in class groups of number fields of arbitrary degree" Abstract: Fix a number field K of degree n over the rationals, and a prime p, and consider the p-torsion subgroup of the class group of K. How big is it? It is conjectured that this p-torsion subgroup should be very small (in an appropriate sense), relative to the absolute discriminant of the field; this relates to the Cohen-Lenstra heuristics and various other arithmetic problems. So far it has proved extremely difficult even to beat the trivial bound, that is, to show that the p-torsion subgroup is noticeably smaller than the full class group. In 2007, Ellenberg and Venkatesh shaved a power off the trivial bound by assuming GRH. This talk will discuss several new, contrasting, methods that recover or improve on this bound for almost all members of certain infinite families of fields, without assuming GRH. === Wednesday, March 29 at 3:30PM (Wasow): Sylvia Serfaty (NYU)=== Monday, January 9, 9th floor Miklos Racz (Microsoft) Statistical inference in networks and genomics Valko January 13, B239 Mihaela Ifrim (Berkeley) Two dimensional water waves Angenent Tuesday, January 17, B139 Fabio Pusateri (Princeton) The Water Waves problem Angenent January 20, B239 Sam Raskin (MIT) Tempered local geometric Langlands Arinkin Monday, January 23, B239 Tamas Darvas (Maryland) Geometry on the space of Kahler metrics and applications to canonical metrics Viaclovsky January 27 Reserved for possible job talks February 3, 9th floor Melanie Matchett Wood (UW-, Madison) Random groups from generators and relations Monday, February 6, B239 (Wasow lecture) Benoit Perthame (University of Paris VI) Models for neural networks; analysis, simulations and behaviour Jin February 10 (WIMAW lecture), B239 Alina Chertock (NC State Univ.) Numerical Method for Chemotaxis and Related Models WIMAW February 17, 9th floor Gustavo Ponce (UCSB) The Korteweg-de Vries equation vs. the Benjamin-Ono equation Minh-Binh Tran Monday, February 20, 9th floor Amy Cochran (Michigan) Mathematical Classification of Bipolar Disorder Smith March 3, B239 Ken Bromberg (University of Utah) Renormalized volume for hyperbolic 3-manifolds Dymarz Tuesday, March 7, 4PM, 9th floor (Distinguished Lecture) Roger Temam (Indiana University) On the mathematical modeling of the humid atmosphere Smith Wednesday, March 8, 4PM, B239 Roger Temam (Indiana University) Weak solutions of the Shigesada-Kawasaki-Teramoto system. Smith March 10 No Colloquium Wednesday, March 15, 4PM Enrique Zuazua (Universidad Autónoma de Madrid) Control and numerics: Recent progress and challenges Jin & Minh-Binh Tran March 17 Lillian Pierce (Duke University) p-torsion in class groups of number fields of arbitrary degree M. Matchett Wood Wednesday, March 29 at 3:30PM (Wasow) Sylvia Serfaty (NYU) Microscopic description of Coulomb-type systems Tran April 7 Hal Schenck Erman April 14 Wilfrid Gangbo Feldman & Tran April 21 Mark Andrea de Cataldo (Stony Brook) TBA Maxim April 28 Thomas Yizhao Hou TBA Li September 8 TBA September 15 TBA Wednesday, September 20, LAA lecture Andrew Stuart (Caltech) TBA Jin October 6 TBA October 13 TBA October 20 Pierre Germain (Courant, NYU) TBA Minh-Binh Tran November 3 TBA November 10 Reserved for possible job talks TBA November 24 Thanksgiving break TBA December 1 Reserved for possible job talks TBA September 16: Po-Shen Loh (CMU) Title: Directed paths: from Ramsey to Pseudorandomness Abstract: Starting from an innocent Ramsey-theoretic question regarding directed paths in graphs, we discover a series of rich and surprising connections that lead into the theory around a fundamental result in Combinatorics: Szemeredi's Regularity Lemma, which roughly states that every graph (no matter how large) can be well-approximated by a bounded-complexity pseudorandom object. Using these relationships, we prove that every coloring of the edges of the transitive N-vertex tournament using three colors contains a directed path of length at least sqrt(N) e^{log^* N} which entirely avoids some color. The unusual function log^* is the inverse function of the tower function (iterated exponentiation). September 23: Gheorghe Craciun (UW-Madison) Title: Toric Differential Inclusions and a Proof of the Global Attractor Conjecture Abstract: The Global Attractor Conjecture says that a large class of polynomial dynamical systems, called toric dynamical systems, have a globally attracting point within each linear invariant space. In particular, these polynomial dynamical systems never exhibit multistability, oscillations or chaotic dynamics. The conjecture was formulated by Fritz Horn in the early 1970s, and is strongly related to Boltzmann's H-theorem. We discuss the history of this problem, including the connection between this conjecture and the Boltzmann equation. Then, we introduce toric differential inclusions, and describe how they can be used to prove this conjecture in full generality. September 30: Akos Magyar (University of Georgia) Title: Geometric Ramsey theory Abstract: Initiated by Erdos, Graham, Montgomery and others in the 1970's, geometric Ramsey theory studies geometric configurations, determined up to translations, rotations and possibly dilations, which cannot be destroyed by finite partitions of Euclidean spaces. Later it was shown by ergodic and Fourier analytic methods that such results are also possible in the context of sets of positive upper density in Euclidean spaces or the integer lattice. We present a new approach, motivated by developments in arithmetic combinatorics, which provide new results as well new proofs of some classical results in this area. October 14: Ling Long (LSU) Title: Hypergeometric functions over finite fields Abstract: Hypergeometric functions are special functions with lot of symmetries. In this talk, we will introduce hypergeometric functions over finite fields, originally due to Greene, Katz and McCarthy, in a way that is parallel to the classical hypergeometric functions, and discuss their properties and applications to character sums and the arithmetic of hypergeometric abelian varieties. This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu. Tuesday, October 25, 9th floor: Stefan Steinerberger (Yale) Title: Three Miracles in Analysis Abstract: I plan to tell three stories: all deal with new points of view on very classical objects and have in common that there is a miracle somewhere. Miracles are nice but difficult to reproduce, so in all three cases the full extent of the underlying theory is not clear and many interesting open problems await. (1) An improvement of the Poincare inequality on the Torus that encodes a lot of classical Number Theory. (2) If the Hardy-Littlewood maximal function is easy to compute, then the function is sin(x). (Here, the miracle is both in the statement and in the proof). (3) Bounding classical integral operators (Hilbert/Laplace/Fourier-transforms) in L^2 -- but this time from below (this problem originally arose in medical imaging). Here, the miracle is also known as 'Slepian's miracle' (this part is joint work with Rima Alaifari, Lillian Pierce and Roy Lederman). October 28: Linda Reichl (UT Austin) Title: Microscopic hydrodynamic modes in a binary mixture Abstract: Expressions for propagation speeds and decay rates of hydrodynamic modes in a binary mixture can be obtained directly from spectral properties of the Boltzmann equations describing the mixture. The derivation of hydrodynamic behavior from the spectral properties of the kinetic equation provides an alternative to Chapman-Enskog theory, and removes the need for lengthy calculations of transport coefficients in the mixture. It also provides a sensitive test of the completeness of kinetic equations describing the mixture. We apply the method to a hard-sphere binary mixture and show that it gives excellent agreement with light scattering experiments on noble gas mixtures. Monday, October 31: Kathryn Mann (Berkeley) Title: Groups acting on the circle Abstract: Given a group G and a manifold M, can one describe all the actions of G on M? This is a basic and natural question from geometric topology, but also a very difficult one -- even in the case where M is the circle, and G is a familiar, finitely generated group. In this talk, I'll introduce you to the theory of groups acting on the circle, building on the perspectives of Ghys, Calegari, Goldman and others. We'll see some tools, old and new, some open problems, and some connections between this theory and themes in topology (like foliated bundles) and dynamics. November 7: Gaven Martin (New Zealand Institute for Advanced Study) Title: Siegel's problem on small volume lattices Abstract: We outline in very general terms the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram. This gives us the smallest regular tessellation of hyperbolic 3-space. This solves (in three dimensions) a problem posed by Siegel in 1945. Siegel solved this problem in two dimensions by deriving the signature formula identifying the (2,3,7)-triangle group as having minimal co-area. There are strong connections with arithmetic hyperbolic geometry in the proof, and the result has applications in the maximal symmetry groups of hyperbolic 3-manifolds in much the same way that Hurwitz's 84g-84 theorem and Siegel's result do. Wednesday, November 16 (9th floor): Kathryn Lindsey (U Chicago) Title: Shapes of Julia Sets Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. William Thurston asked "What are the possible shapes of polynomial Julia sets?" For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name? It turns out the answer to all of these is "yes!" I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes. November 18: Andrew Snowden (University of Michigan) Title: Recent progress in representation stability Abstract: Representation stability is a relatively new field that studies somewhat exotic algebraic structures and exploits their properties to prove results (often asymptotic in nature) about objects of interest. I will describe some of the algebraic structures that appear (and state some important results about them), give a sampling of some notable applications (in group theory, topology, and algebraic geometry), and mention some open problems in the area. Monday, November 21: Mariya Soskova (University of Wisconsin-Madison) Title: Definability in degree structures Abstract: Some incomputable sets are more incomputable than others. We use Turing reducibility and enumeration reducibility to measure the relative complexity of incomputable sets. By identifying sets of the same complexity, we can associate to each reducibility a degree structure: the partial order of the Turing degrees and the partial order of the enumeration degrees. The two structures are related in nontrivial ways. The first has an isomorphic copy in the second and this isomorphic copy is an automorphism base. In 1969, Rogers asked a series of questions about the two degree structures with a common theme: definability. In this talk I will introduce the main concepts and describe the work that was motivated by these questions. Friday, December 2: Hao Shen (Columbia) Title: Singular Stochastic Partial Differential Equations - How do they arise and what do they mean? Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models. Monday, December 5: Botong Wang (UW-Madison) Title: Enumeration of points, lines, planes, etc. Abstract: It is a theorem of de Bruijn and Erdos that n points in the plane determine at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization of this theorem, which confirms a "top-heavy" conjecture of Dowling and Wilson in 1975. I will give a sketch of the key ideas of the proof, which are the hard Lefschetz theorem and the decomposition theorem in algebraic geometry. I will also talk about a log-concave conjecture on the number of independent sets. These are joint works with June Huh. Friday, December 9: Aaron Brown (U Chicago) Lattice actions and recent progress in the Zimmer program Abstract: The Zimmer Program is a collection of conjectures and questions regarding actions of lattices in higher-rank simple Lie groups on compact manifolds. For instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions. In particular—on manifolds whose dimension is below the dimension of all algebraic examples—Zimmer's conjecture asserts that every action is finite. I will present some background, motivation, and selected previous results in the Zimmer program. I will then explain two of my results within the Zimmer program: (1) a solution to Zimmer's conjecture for actions of cocompact lattices in SL(n,R) (joint with D. Fisher and S. Hurtado); (2) a classification (up to topological semiconjugacy) of all actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang). Monday, December 19: Andrew Zimmer (U Chicago) Metric spaces of non-positive curvature and applications in several complex variables Abstract: In this talk I will discuss how to use ideas from the theory of metric spaces of non-positive curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance non-increasing with respect to holomorphic maps. Moreover, this metric often satisfies well-known non-positive curvature type conditions (for instance, Gromov hyperbolicity or visibility) and one can then use these conditions to understand the behavior of holomorphic maps. Some of what I will talk about is joint work with Gautam Bharali. Monday, January 9: Miklos Racz (Microsoft) Statistical inference in networks and genomics Abstract: From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas. I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data. Friday, January 13: Mihaela Ifrim (Berkeley) Two dimensional water waves The classical water-wave problem consists of solving the Euler equations in the presence of a free fluid surface (e.g the water-air interface). This talk will provide an overview of recent developments concerning the motion of a two dimensional incompressible fluid with a free surface. There is a wide range of problems that fall under the heading of water waves, depending on a number of assumptions that can be applied: surface tension, gravity, finite bottom, infinite bottom, rough bottom, etc., and combinations thereof. We will present the physical motivation for studying such problems, followed by the discussion of several interesting mathematical questions related to them. The first step in the analysis is the choice of coordinates, where multiple choices are available. Once the equations are derived we will discuss the main issues arising when analysing local well-posedness, as well as the long time behaviour of solutions with small, or small and localized data. In the last part of the talk we will introduce a new, very robust method which allows one to obtain enhanced lifespan bounds for the solutions. If time permits we will also introduce an alternative method to the scattering theory, which in some cases yields a straightforward route to proving global existence results and obtaining an asymptotic description of solutions. This is joint work with Daniel Tataru, and in part with John Hunter. Tuesday, January 17: Fabio Pusateri (Princeton) The Water Waves problem We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas. Friday, January 20: Sam Raskin (MIT) Tempered local geometric Langlands The (arithmetic) Langlands program is a cornerstone of modern representation theory and number theory. It has two incarnations: local and global. The former conjectures the existence of certain "local terms," and the latter predicts remarkable interactions between these local terms. By necessity, the global story is predicated on the local. Geometric Langlands attempts to find similar patterns in the geometry of curves. However, the scope of the subject has been limited by a meager local theory, which has not been adequately explored. The subject of this talk is a part of a larger investigation into local geometric Langlands. We will give an elementary overview of the expectations of this theory, discuss a certain concrete conjecture in the area (on "temperedness"), and provide evidence for this conjecture. Monday, January 23: Tamas Darvas (Maryland) Geometry on the space of Kahler metrics and applications to canonical metrics A basic problem in Kahler geometry, going back to Calabi in the 50's, is to find Kahler metrics with the best curvature properties, e.g., Einstein metrics. Such special metrics are minimizers of well known functionals on the space of all Kahler metrics H. However these functionals become convex only if an adequate geometry is chosen on H. One such choice of Riemannian geometry was proposed by Mabuchi in the 80's, and was used to address a number of uniqueness questions in the theory. In this talk I will present more general Finsler geometries on H, that still enjoy many of the properties that Mabuchi's geometry has, and I will give applications related to existence of special Kahler metrics, including the recent resolution of Tian's related properness conjectures. Friday, February 3: Melanie Matchett Wood (UW-Madison) Random groups from generators and relations We consider a model of random groups that starts with a free group on n generators and takes the quotient by n random relations. We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the Cohen-Lenstra heuristics, which predict the distribution of class groups of number fields. We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen. Finally, we discuss joint work with Yuan Liu on the non-abelian random groups built in this way, including the existence of a limit of the random groups as n goes to infinity. Monday, February 6: Benoit Perthame (University of Paris VI) Models for neural networks; analysis, simulations and behaviour Neurons exchange informations via discharges, propagated by membrane potential, which trigger firing of the many connected neurons. How to describe large networks of such neurons? What are the properties of these mean-field equations? How can such a network generate a spontaneous activity? Such questions can be tackled using nonlinear integro-differential equations. These are now classically used in the neuroscience community to describe neuronal networks or neural assemblies. Among them, the best known is certainly Wilson-Cowan's equation which describe spiking rates arising in different brain locations. Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. Several mathematical results will be presented concerning existence, blow-up, convergence to steady state, for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed. One can also describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time $t$ the probability to find a neuron with time $s$ elapsed since its last discharge. Here, we can show that small or large connectivity leads to desynchronization. For intermediate regimes, sustained periodic activity occurs. A common mathematical tool is the use of the relative entropy method. This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo and D. Smets. February 10: Alina Chertock (NC State Univ.) Numerical Method for Chemotaxis and Related Models Chemotaxis is a movement of micro-organisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction- diffusion equation for the chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxis systems extremely delicate and challenging task. In this talk, I will present a family of high-order numerical methods for the Keller-Segel chemotaxis system and several related models. Applications of the proposed methods to to multi-scale and coupled chemotaxis–fluid system and will also be discussed. Friday, February 17: Gustavo Ponce(UCSB) The Korteweg-de Vries equation vs. the Benjamin-Ono equation In this talk we shall study the [math]k[/math]-generalized Korteweg-de Vries [math](k[/math]-KdV) equation [math]\partial_t u + \partial_x^3u +u^k\,\partial_xu=0,\;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+, [/math] and the [math]k[/math]-generalized Benjamin-Ono ([math]k[/math]-BO) equation [math]\partial_t u-\partial_x^2\mathcal {H} u+u^k\,\partial_x u=0, \;\;\;\;\;\;\;x,t\in\Bbb R,\, k\in \Bbb Z^+,[/math] where [math]\mathcal {H}[/math] denotes the Hilbert transform, [math]\mathcal {H} f(x)=\frac{1}{\pi}\, {p.v.}\big(\frac{1}{x}\ast f\big)(x)=(-i\,sgn(\xi) \widehat{f}(\xi))^{\vee}(x).[/math] The goal is to review and analyze results concerning solutions of the initial value properties associated to these equations. These include a comparison of the local and global well-posedness and unique continuation properties as well as special features of the special solutions of these models. Monday, February 20, Amy Cochran (Michigan) Mathematical Classification of Bipolar Disorder Bipolar disorder is a chronic disease of mood instability. Longitudinal patterns of mood are central to any patient description, but are condensed into simple attributes and categories. Although these provide a common language for clinicians, they are not supported by empirical evidence. In this talk, I present patient-specific models of mood in bipolar disorder that incorporate existing longitudinal data. In the first part, I will describe mood as a Bayesian nonparametric hierarchical model that includes latent classes and patient-specific mood dynamics given by discrete-time Markov chains. These models are fit to weekly mood data, revealing three patient classes that differ significantly in attempted suicide rates, disability, and symptom chronicity. In the second part of the talk, I discuss how combined statistical inferences from a population do not support widely held assumptions (e.g. mood is one-dimensional, rhythmic, and/or multistable). I then present a stochastic differential equation model that does not make any of these assumptions. I show that this model accurately describes the data and that it can be personalized to an individual. Taken together, this work moves forward data-driven modeling approaches that can guide future research into precise clinical care and disease causes. Friday, March 3, Ken Bromberg (Utah) "Renormalized volume for hyperbolic 3-manifolds" Motivated by ideas in physics Krasnov and Schlenker defined the renormalized volume of a hyperbolic 3-manifold. This is a way of assigning a finite volume to a hyperbolic 3-manifold that has infinite volume in the usual sense. We will begin with some basic background on hyperbolic geometry and hyperbolic 3-manifolds before defining renormalized volume with the aim of explaining why this is a natural quantity to study from a mathematician's perspective. At the end will discuss some joint results with M. Bridgeman and J. Brock. Tuesday, March 7: Roger Temam (Indiana University) On the mathematical modeling of the humid atmosphere The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations. We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities. Wednesday, March 8: Roger Temam (Indiana University) Weak solutions of the Shigesada-Kawasaki-Teramoto system We will present a result of existence of weak solutions to the Shigesada-Kawasaki-Teramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely self-contained and does not rely on any earlier result. Based on an article with Du Pham, to appear in Nonlinear Analysis. Wednesday, March 15: Enrique Zuazua (Universidad Autónoma de Madrid) Control and numerics: Recent progress and challenges In most real life applications Mathematics not only face the challenge of modelling (typically by means of ODE and/or PDE), analysis and computer simulations but also the need control and design. And the successful development of the needed computational tools for control and design cannot be achieved by simply superposing the state of the art on Mathematical and Numerical Analysis. Rather, it requires specific tools, adapted to the very features of the problems under consideration, since stable numerical methods for the forward resolution of a given model, do not necessarily lead to stable solvers of control and design problems. In this lecture we will summarize some of the recent work developed in our group, motivated by different applications, that have led to different analytical and numerical methodologies to circumvent these difficulties. The examples we shall consider are motivated by problems of different nature and lead to various new mathematical developments. We shall mainly focus on the following three topics: - Inverse design for hyperbolic conservation laws, - The turnpike property: control in long time intervals, - Collective behavior: guidance by repulsion. We shall also briefly discuss the convenience of using greedy algorithms when facing parameter-dependence problems. Friday, March 17 at 4:00pm: Lillian Pierce Wednesday, March 29 at 3:30PM (Wasow): Sylvia Serfaty (NYU) Microscopic description of Coulomb-type systems We are interested in systems of points with Coulomb, logarithmic or more generally Riesz interactions (i.e. inverse powers of the distance). They arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. After reviewing the motivations, we will take a point of view based on the detailed expansion of the interaction energy to describe the microscopic behavior of the systems. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes are given. This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions. The main results are joint with Thomas Leblé and also based on previous works with Etienne Sandier, Nicolas Rougerie and Mircea Petrache. Retrieved from "https://www.math.wisc.edu/wiki/index.php?title=Colloquia/Fall18&oldid=13496"	CommonCrawl
The Congress MCA 2013 Onsite Registration Desk Fees & Deadlines MCA Schedule Schedule by Day List of Presenters Plenary Speakers Digital Programme Social & Cultural Activities Satellite Sessions Prize Recipients CMS Activities Visa & eTA #mca2017 Applied and Computational Algebra and Geometry Session code: aca Session type: Special Sessions All abstracts Carina Curto (The Pennsylvania State University) Alicia Dickenstein (Universidad de Buenos Aires) Luis David Garcia Puente (Sam Houston State University) Thursday, Jul 27 [McGill U., Birks Building, Room 205] 11:45 Elizabeth Gross (San Jose State University, USA), Joining and Decomposing Reaction Networks 12:15 Gregory G. Smith (Queen's University, Canada), Resolutions of Toric Vector Bundles 14:15 Rafael Villarreal (Cinvestav, Mexico), Minimum distance functions of complete intersections 14:45 Katie Morrison (University of Northern Colorado, USA), Algebraic Signatures of Convex and Non-Convex Neural Codes 15:45 Vladimir Istkov (The Pennsylvania State University , USA), Detecting non-linear rank via the topology of hyperplane codes. 16:15 Guillermo Matera (Universidad Nacional de General Sarmiento, Argentina), On the bit complexity of polynomial system solving 17:00 Ezra Miller (Duke University, USA), Algebraic data structures for topological summaries Friday, Jul 28 [McGill U., Birks Building, Room 205] 11:45 Jon Hauenstein (University of Notre Dame , USA), Equilibria of the Kuramoto model 12:15 Sara Kalisnik Verovsek (Brown University, USA), Tropical Coordinates on the Space of Persistence Barcodes 14:15 Stephen Watt (University of Waterloo, Canada), Toward Gröbner Bases of Symbolic Polynomials 14:45 Luis David Garcia Puente (Sam Houston State University, USA), Gr\"obner bases of neural ideals 15:45 Carlos Valencia (Cinvestav, Mexico), Some algorithmic aspect of arithmetical structures 16:15 Alan Veliz Cuba (University of Dayton, USA), On the Perfect Reconstruction of the Structure of Dynamic Networks 17:00 Carina Curto (The Pennsylvania State University , USA), Emergent dynamics from network connectivity: a minimal model Elizabeth Gross San Jose State University, USA Joining and Decomposing Reaction Networks PDF abstract Systems biology focuses on modeling complex biological systems, such as metabolic and cell signaling networks. These biological networks are modeled with polynomial dynamical systems that can be described with directed graphs. Analyzing these systems at steady-state results in polynomial ideals with significant combinatorial structure and whose elements can be used for model selection. Focusing on the problem of finding steady-state invariants of an elimination ideal, we explore the algebra of decomposing a larger reaction network into smaller subnetworks. This talk is based on joint work with Heather Harrington, Nicolette Meshkat, and Anne Shiu. Scheduled time: Thursday, July 27 at 11:45 Location: McGill U., Birks Building, Room 205 Gregory G. Smith Queen's University, Canada Resolutions of Toric Vector Bundles To each torus-equivariant vector bundle over a smooth complete toric variety, we associated a representable matroid (essentially a finite collection of vectors). In this talk, we will describe how the combinatorics of this matroid encodes a resolution of the toric vector bundles by a complex whose terms are direct sums of toric line bundles. With some luck, we will also outline some applications to the equations and syzygies of smooth projective toric varieties. Rafael Villarreal Cinvestav, Mexico Minimum distance functions of complete intersections We study the mínimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory and combinatorics. Katie Morrison University of Northern Colorado, USA Algebraic Signatures of Convex and Non-Convex Neural Codes The brain represents stimuli via patterns of neural activity. These activity patterns can be described by a neural code, e.g. a collection of indicator vectors showing which neurons co-fire in response to various stimuli. It is believed that the brain can infer many properties of the stimulus space purely from the intrinsic structure of the neural code. In this talk, we present algebraic techniques that enable us to determine if a given neural code is convex, and thus has additional structure that can be used to understand stimulus space structure. Vladimir Istkov The Pennsylvania State University , USA Detecting non-linear rank via the topology of hyperplane codes. Non-linear rank of a matrix M is the minimal possible rank of a matrix obtained by applying arbitrary monotone-increasing functions to each row of M. The problem of finding the non-linear rank often arises in neuroscience context. In this talk I will explain how the topology of hyperplane arrangements is closely related to the problem of finding a non-linear rank. This relationship is accomplished via a zigzag of combinatorial codes, not unlike the Dowker complex. I will then present an algebraic approach and computational results for estimating the nonlinear rank. Guillermo Matera Universidad Nacional de General Sarmiento, Argentina On the bit complexity of polynomial system solving We describe a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. Our algorithm solves the input system modulo a prime number $p$ and applies $p$-adic lifting. Our approach is based on a new result on the bit length of a "lucky" prime $p$, namely one for which the reduction of the input system modulo $p$ preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellensätze. Joint work with Nardo Giménez. Duke University, USA Algebraic data structures for topological summaries This talk introduces a combinatorial algebraic framework to encode, compute, and analyze topological summaries of geometric data. The motivating problem from evolutionary biology involves statistics on a dataset comprising images of fruit fly wing veins. The algebraic structures take their cue from graded polynomial rings and their modules, but the theory is complicated by the passage from integer exponent vectors to real exponent vectors. The path to effective methods is built on appropriate finiteness conditions, to replace the usual ones from commutative algebra, and on an understanding of how datasets of this nature interact with moduli of modules. I will introduce the biology and topology from first principles. Joint work with David Houle (Biology, Florida State), Ashleigh Thomas (grad student, Duke Math), Justin Curry (postdoc, Duke Math), and Surabhi Beriwal (undergrad, Duke Math). Jon Hauenstein University of Notre Dame , USA Equilibria of the Kuramoto model The standard Kuramoto model (all-to-all with uniform coupling) is used to describe synchronization behavior of a large set of oscillators. Using algebraic geometry, the equilibria of this model can be computed by solving a system of polynomial equations. We develop an approach which computes only the real solutions to this system of polynomial equations by reducing down to solving a collection of univarite functions. We compare this new approach with other approaches in computational algebraic geometric. The univariate reduction also allows us to prove that, asymptotically, the maximum number of real solutions grows at the same rate as the number of complex solutions. Scheduled time: Friday, July 28 at 11:45 Sara Kalisnik Verovsek Brown University, USA Tropical Coordinates on the Space of Persistence Barcodes In the last two decades applied topologists have developed numerous methods for `measuring' and building combinatorial representations of the shape of the data. The most famous example of the former is persistent homology. This adaptation of classical homology assigns a barcode, i.e. a collection of intervals with endpoints on the real line, to a finite metric space. Unfortunately, barcodes are not well-adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes and these outputs can then be used as input to standard algorithms. I will talk about tropical functions that can be used as coordinates on the space of barcodes. All of these are stable with respect to the standard distance functions (bottleneck, Wasserstein) used on the barcode space. Stephen Watt University of Waterloo, Canada Toward Gröbner Bases of Symbolic Polynomials We consider "symbolic polynomials" that generalize the usual polynomials by allowing multivariate integer valued polynomials as exponents. Earlier we have shown how to compute GCDs, factorizations and functional decomposition of these objects. The present work asks whether it is meaningful to compute Gröbner bases of sets of symbolic polynomials, and, if so, how do these "symbolic" Gröbner bases relate to the usual Gröbner bases when the exponents are specialized. Luis David Garcia Puente Sam Houston State University, USA Gr\"obner bases of neural ideals The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gr\"obner basis with respect to that monomial order. How are these two types of generating sets – canonical forms and Gr\"obner bases – related? Our main result states that if the canonical form of a neural ideal is a Gr\"obner basis, then it is the universal Gr\"obner basis (that is, the union of all reduced Gr\"obner bases). Furthermore, we prove that this situation – when the canonical form is a Gr\"obner basis – occurs precisely when the universal Gr\"obner basis contains only pseudo-monomials (certain generalizations of monomials). Our results motivate two questions: (1) When is the canonical form a Gr\"obner basis? (2) When the universal Gr\"obner basis of a neural ideal is not a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code? We give partial answers to both questions. Along the way, we develop a representation of pseudo-monomials as hypercubes in a Boolean lattice. Carlos Valencia Some algorithmic aspect of arithmetical structures Let $M$ be a non-negative $n\times n$ integer matrix with all the diagonal entries equal to zero. An arithmetical structure (AS) on $M$ is a pair $({\bf d},{\bf r}) \in \mathbb{N}_+^n\times \mathbb{N}_+^n$ such that $\mathrm{gcd}({\bf r}_v\, \| \,v\in V(G))=1$ and \[ (\mathrm{diag}({\bf d})-M){\bf r}^t={\bf 0}^t. \] The concept of AS was introduced by Lorenzini as some intersection matrices that arise in the study of degenerating curves in algebraic geometry. If $M$ is the adjacency matrix of a graph $G$ and ${\bf d}$ is its degree vector, then $\mathrm{diag}({\bf d})-M$ is the Laplacian matrix of $G$. It can be proved that $M$ is irreducible if and only if \[ \mathcal{A}(M)=\{({\bf d},{\bf r}) \in \mathbb{N}_+^n\times \mathbb{N}_+^n\, \|\, (\mathrm{diag}({\bf d})-M){\bf r}^t={\bf 0}^t\} \] is finite. Recently an explicit description of the arithmetical structures of the path and cycle have been given. Let $X=(x_1,\ldots,x_n)$ be a vector with variables on each one of its entries and \[ f_M(X)=\mathrm{det}(\mathrm{diag}(X)-M). \] It is not difficult to check that if $({\bf d},{\bf r}) \in \mathcal{A}(M)$, then ${\bf d}$ is a solution of the Diophantine equation $f_M(X)=0$. Note that the converse is not true. By Hilbert's tenth problem is not clear that an algorithm to compute the solutions of this class of Diophantine equations or even the arithmetical structures of $M$ exist. In this talk we will present an algorithm to compute the AS on $M$. Finally, if the time permits we will present some details of the implementation of this algorithm. Joint work with Ralihe R. Villagran. Alan Veliz Cuba University of Dayton, USA On the Perfect Reconstruction of the Structure of Dynamic Networks The network inference problem consists in reconstructing the structure or wiring diagram of a dynamic network from time-series data. Even though this problem has been studied in the past, there is no algorithm that guarantees perfect reconstruction of the structure of a dynamic network. In this talk I will present a framework and algorithm to solve the network inference problem for discrete-time networks that, given enough data, is guaranteed to reconstruct the structure with zero errors. The framework uses tools from algebraic geometry. Carina Curto Emergent dynamics from network connectivity: a minimal model Many networks in the brain display internally-generated patterns of activity -- that is, they exhibit emergent dynamics that are shaped by intrinsic properties of the network rather than inherited from an external input. While a common feature of these networks is an abundance of inhibition, the role of network connectivity in pattern generation remains unclear. In this talk I will introduce Combinatorial Threshold-Linear Networks (CTLNs), which are simple "toy models" of recurrent networks consisting of threshold-linear neurons with binary inhibitory interactions. The dynamics of CTLNs are controlled solely by the structure of an underlying directed graph. By varying the graph, we observe a rich variety of emergent patterns including: multistability, neuronal sequences, and complex rhythms. These patterns are reminiscent of population activity in cortex, hippocampus, and central pattern generators for locomotion. I will present some theorems about CTLNs, and explain how they allow us to predict features of the dynamics by examining properties of the underlying graph. Finally, I'll show examples illustrating how these mathematical results guide us to engineer complex networks with prescribed dynamic patterns. Canadian Mathematical Society 209 - 1725 St. Laurent Blvd. Ottawa, ON K1G 3V4 cms.math.ca © CMS 2015-2017	CommonCrawl
Research \| Open \| Published: 26 February 2019 Quad-mode dual-band bandpass filter based on a stub-loaded circular resonator Minghui Liu1, Zheng Xiang1, Peng Ren1 & Tongtong Xu1 EURASIP Journal on Wireless Communications and Networkingvolume 2019, Article number: 48 (2019) \| Download Citation In this paper, a quad-mode dual-band bandpass filter (BPF) only using a stub-loaded circular resonator is presented. The odd- and even-mode resonant frequencies can be easily computed and flexibly controlled by the radius of the circle and the two sets of stubs, as all odd- and even-mode equivalent circuits are half-wavelength resonators (double-ended short-circuited uniform transmission line resonator). Introducing the resonator makes the filter be analyzed and designed easily. To further improve the isolation between the two passbands, a dual-band BPF is designed by introducing the source-load coupling technique; the simulation and test S-parameter curves show that the designed BPF has four controllable transmission zeros. The good agreement of measured data with simulation results verifies the proposed design. With the development of microwave and millimeter-wave technology, there is an increasing need of the high performance and miniaturization of multi-frequency band filter in communication systems [1]. In [2], a dual-band bandpass filter (BPF) was achieved by a dual-mode circular resonator; however, there were only two transmission poles in each passband. A resonator with a cross slot and unbalanced stubs has been proposed [3] for the design of dual-band BPF. A dual-band BPF [4] makes common-mode rejection be improved by introducing four coupled U-shaped defected ground structures below the resonators, and the differential mode is not significantly affected. In [5], some resonators were used to design dual-band filter, but resonant frequencies were dependent. Unfortunately, the poor in-band performance of the dual-band filter is also an existing disadvantage of the revised method in [2,3,4]. Meanwhile, these solutions increase the complexity of structure and analytical method in [2,3,4,5,6,7,8,9,10]. In this letter, a stub-loaded quad-mode circular resonator is presented to design the dual-band bandpass filter. This resonator can generate four independent resonant frequencies. The properties of the presented resonator are analyzed theoretically and simulated by full-wave simulation software. A passband controllable dual-band BPF can be formed by adding a suitable external coupling structure. To greatly improve the isolation between the two passbands, the source-load coupling is introduced, which increases the transmission zeros between the two passbands. Finally, the dual-band filter with 1.25 GHz and 2.0 GHz is presented, and the measurement results show good performance. Methodology of the proposed resonator Analysis of the proposed resonator The proposed stub-loaded quad-mode circular resonator structure consists of a circular resonator and four short-circuited stubs, which is shown in Fig. 1a. Due to the symmetrical structure of the resonator, the theory of odd and even modes can be used to analyze it [11,12,13]. a The stub-loaded quad-mode circular resonator. b The odd- and even-mode equivalent circuit As shown in Fig. 1b, it was clear that the equivalent circuits of odd and even mode still have a symmetrical structure feature of the resonator with double-ended short-circuit, and the short-circuited stub shunted at the midpoint of the resonator. Consequently, we can utilize the odd- and even-mode theory to analyze its odd- and even-mode equivalent circuits again. That is the main idea of our proposed method. The analytical results are shown in Fig. 2a and b, respectively. a The odd- and even-mode equivalent circuits of the odd-mode equivalent circuit. b The odd- and even-mode equivalent circuits of the even-mode equivalent circuit It can be seen from Fig. 2 that the four equivalent circuits are all double-ended short-circuited resonators in the form of a uniform transmission line. The frequencies corresponding to the four resonant modes are denoted by f1, f2, f3, and f4, respectively. According to the odd- and even-mode theory, the resonant frequencies can be derived as $$ {f}_1=\frac{\left(2n-1\right)c}{2\times \left(\frac{1}{2}\pi R\right)\sqrt{\varepsilon_{\mathrm{eff}}}} $$ $$ {f}_2=\frac{\left(2n-1\right)c}{2\times \left(\frac{1}{2}\pi R+{L}_2\right)\sqrt{\varepsilon_{\mathrm{eff}}}} $$ $$ {f}_4=\frac{\left(2n-1\right)c}{2\times \left(\frac{1}{2}\pi R+{L}_1+{L}_2\right)\sqrt{\varepsilon_{\mathrm{eff}}}} $$ In the case of L2 ≥ L1, there is a relationship existing among the four frequencies: f4 < f2 < f3 < f1. It is not hard to find, stub L1 exists only in f3 and f4 and L2 exists in f2 and f4. All the resonant frequencies will reduce as the radius R increases. By changing the length of the stub L1, only f3 and f4 can be affected. Similarly, f2 and f4 are only affected by the length of the stub L2. In order to verify the above theoretical conclusions, full-wave simulation is carried out, and the simulation results shown in Fig. 3 coincide with the theoretical analysis. The effects of structural parameters on the resonant frequency. a The R radius of the circle. b The length of the stub L1. c The length of the stub L2 The source-load coupling technique According to the above analysis, the four resonant frequencies can be divided into two groups in accordance with their values, the first group including f2 and f4 forms the first passband and the second group including f1 and f3 forms the second passband. In the basis of the resonator, a dual-band BPF can be designed by adding the arc-shaped parallel coupling feeder. That way we can control the center frequencies and relative bandwidths of the dual-band by adjusting the values of each resonant frequencies and the strength of coupling. In order to further improve the performance of the filter and enhance the isolation between the two passbands, the source-load coupling technique is introduced. As shown in Fig. 4 (for simplicity, get arc-shaped feeder structure straightened out), both ends of the feeder that is close to each other are extended and then bent down. Source-load coupling technique refers to the introduction of a certain resonant coupling structure between the two ports so that some signals at the source end are directly coupled to the load through the coupling structure, while the other part is coupled to the load through the resonator. When the amplitude is equal and the phase difference is 180°, the transmission zeros can be generated as shown in Fig. 4c. The position of transmission zeros can be flexibly controlled by controlling the resonant frequency and coupling size of the coupling structure. Such a way makes the design of the resonator and feeders become more convenient. Source-load coupling. a The structure characteristic without the source-load coupling technique. b The structure characteristic with the source-load coupling technique. c The comparison of S parameters between with and without source-load coupling Structure of the dual-band BPF Based on the proposed structure and the analysis of the resonant characteristics above, a dual-band BPF in other frequency is designed. The design parameters of the filter are as follows: The center frequencies of the two passbands are 1.25 GHz and 2 GHz, respectively. The insertion losses are less than 1.5 dB, and the return losses are greater than 15 dB. The first passband has a relative bandwidth of 7 to 12%, and the second is from 10 to 15%. According to the above performance indicators and the material of existing dielectric substrates, the F4BMX dielectric substrates with a dielectric constant of 2.65 and a thickness of 1 mm were selected. The specific design ideas are as follows: Firstly, according to Formula (1), the radius R of the ring is calculated preliminarily so that the resonant frequency of the ring is near the central frequency of the second passband. Then, the radius R is optimized by eigen-mode solution through HFSS. Secondly, adding branch L1 to the model and optimizing its length makes the coupling of resonant frequencies f1 and f3 reach the required strength. Thirdly, the resonant frequencies f2 and f4 of the ring resonator are located near the central frequency of the first passband by adding a branch L2 and optimizing its length in the model. Fourth, by optimizing the length of the feeder Lf1 and the distance between Lf1 and the resonator g1, the external coupling strength can meet the requirements of their respective passband bandwidth. Finally, according to the theory of source-load coupling, the length of Lf2 is optimized, and the location of transmission zeros is determined. The configuration of structure is shown in Fig. 5. The dimensions are determined as follows: R = 32.98 mm, L1 = 2.59 mm, L2 = 26.5 mm, W = 0.5 mm, g1 = 0.35 mm, g2 = 0.7 mm, d = 1.4 mm, Lf1 = 27.14 mm, and Lf2 = 3.2 mm. The short-circuited effect is achieved with a metalized via of 1 mm diameter at the end of the short-circuited stub. Filter configuration Simulated and measured results of the dual-band BPF Figure 6 shows the simulated and measured results. The center frequencies of the two passbands are 1.25 GHz and 2 GHz, respectively. The insertion losses of the lower and upper passbands are only 0.4 dB and 0.5 dB, respectively. The measured return losses of the lower and upper passbands are 18 dB and 27 dB, respectively. These meet the needs of design targets. The error between simulation and measurement results is very small, and there is a good agreement between them. Some of these deviations may be due to machining errors and SMA connectors. Frequency responses of the filter with four transmission zeros Comparison between the existing filters and the proposed filter The comparison with the existing filters is summarized in Table 1. It can be observed that the developed filter offers many advantages in this letter, such as better performance in the return losses and passbands, lower insertion losses, independently controlling bandwidths, and simple structure. It is conducive to the realization of the filter miniaturization. But the filter still has many shortcomings in design, and better performance filters will definitely be realized in the future. Table 1 Comparison with some prior dual-band BPF In this letter, a stub-loaded circular resonator is proposed. Because all odd- and even-mode equivalent circuits have the same structure, the resonant frequencies can be easily obtained and flexibly controlled. To further improve the selectivity of the filter, a dual-band BPF is designed by introducing the source-load coupling technique so that the BPF has four controllable transmission zeros. The measured results agree well with simulated ones, and the filter has high performance. BPF: Bandpass filter Akhil A. Chandran, Mohammed I. Younis. Multi frequency excited MEMS cantilever beam resonator for mixer-filter applications. 2016 3rd International Conference on Signal Processing and Integrated Networks (SPIN). 735–742 (2016) S. Luo, L. Zhu, A novel dual-mode dual-band bandpass filter based on a single ring resonator. IEEE Microw. Wireless Compon. Lett. 19(8), 497–499 (2009) Y.C. Li, H. Wong, Q. Xue, Dual-mode dual-band bandpass filter based on a stub-loaded patch resonator. IEEE Microw. Wireless Compon. Lett.. 21(10), 525–527 (2011) F. Bagci, A. Fernández-Prieto, A. Lujambio, et al., Compact balanced dual-band bandpass filter based on modified coupled-embedded resonators. IEEE Microw. Wireless Compon. Lett.. 27(1), 31–33 (2017) J.-X. Chen, T.Y. Yum, J.-L. Li, Q. Xue, Dual-mode dual-band bandpass filter using stacked-loop structure. IEEE Microw. Wireless Compon. Lett.. 16(9), 502–504 (2006) Y.-H. Cho, S.-W. Yun, Design of balanced dual-band bandpass filters using asymmetrical coupled lines. IEEE Trans. Micro. Theory Tech.. 61(8), 2814–2820 (2013) Y. Shen, H. Wang, W. Kang, W. Wu, Dual-band SIW differential bandpass filter with improved common-mode suppression. IEEE Microw. Wireless Compon. Lett.. 25(2), 100–102 (2015) S.-C. Weng, K.-W. Hsu, W.-H. Tu, Independently switchable quad-band bandpass filter. IET Microw. Antennas Propag. 7(14), 1120–1127 (2013) J. Xu, C. Miao, L. Cui, Y.-X. Ji, and W. Wu. Compact high isolation quad-band bandpass filter using quad-mode resonator. Electron. Lett. 48(1), 28–30(2012) T. Yan, X.H. Tang, J. Wang, A novel quad-band bandpass filter using short stub loaded E-shaped resonatros. IEEE Microw. Wireless Compon. Lett.. 25(8), 508–510 (2015) B. Liu, Z.J. Guo, X.Y. Wei, et al., Quad-band BPF based on SLRs with inductive source and load coupling. Electron. Lett. 53(8), 540–542 (2017) X.Y. Zhang, J.-X. Chen, Q. Xue, S.-M. Li, Dual-band bandpass filters using stub-loaded resonators. IEEE Microw. Wireless Compon. Lett.. 17(8), 583–585 (2007) B. Wu, F. Qiu, L. Lin, Quad-band filter with high skirt selectivity using stub-loaded nested dual-open loop resonators. Electron. Lett. 51(2), 166–168 (2015) The research presented in this paper was supported by Ministry of Education, China. Department of Telecommunication Engineering, Xidian University, Xi'an, 710071, China Minghui Liu , Zheng Xiang , Peng Ren & Tongtong Xu Search for Minghui Liu in: Search for Zheng Xiang in: Search for Peng Ren in: Search for Tongtong Xu in: ML is the main writer of this paper. He proposed the main idea, deduced the performance of BPF, completed the simulation, and analyzed the result. PR and TX assisted ML in designing the architecture of the resonator and measuring the performance of BPF. ZX gave some important suggestions for the design of the filter. All authors read and approved the final manuscript. Correspondence to Zheng Xiang. Bandpass filter (BPF) Circular resonator	CommonCrawl
Multiagent Reinforcement Learning:Rollout and Policy Iteration Dimitri Bertsekas We discuss the solution of complex multistage decision problems using methods that are based on the idea of policy iteration (PI), i.e., start from some base policy and generate an improved policy. Rollout is the simplest method of this type, where just one improved policy is generated. We can view PI as repeated application of rollout, where the rollout policy at each iteration serves as the base policy for the next iteration. In contrast with PI, rollout has a robustness property: it can be applied on-line and is suitable for on-line replanning. Moreover, rollout can use as base policy one of the policies produced by PI, thereby improving on that policy. This is the type of scheme underlying the prominently successful AlphaZero chess program.In this paper we focus on rollout and PI-like methods for problems where the control consists of multiple components each selected (conceptually) by a separate agent. This is the class of multiagent problems where the agents have a shared objective function, and a shared and perfect state information. Based on a problem reformulation that trades off control space complexity with state space complexity, we develop an approach, whereby at every stage, the agents sequentially (one-at-a-time) execute a local rollout algorithm that uses a base policy, together with some coordinating information from the other agents. The amount of total computation required at every stage grows linearly with the number of agents. By contrast, in the standard rollout algorithm, the amount of total computation grows exponentially with the number of agents. Despite the dramatic reduction in required computation, we show that our multiagent rollout algorithm has the fundamental cost improvement property of standard rollout: it guarantees an improved performance relative to the base policy. We also discuss autonomous multiagent rollout schemes that allow the agents to make decisions autonomously through the use of precomputed signaling information, which is sufficient to maintain the cost improvement property, without any on-line coordination of control selection between the agents.For discounted and other infinite horizon problems, we also consider exact and approximate PI algorithms involving a new type of one-agent-at-a-time policy improvement operation. For one of our PI algorithms, we prove convergence to an agent-by-agent optimal policy, thus establishing a connection with the theory of teams. For another PI algorithm, which is executed over a more complex state space, we prove convergence to an optimal policy. Approximate forms of these algorithms are also given, based on the use of policy and value neural networks. These PI algorithms, in both their exact and their approximate form are strictly off-line methods, but they can be used to provide a base policy for use in an on-line multiagent rollout scheme. Dimitri Bertsekas, "Multiagent Reinforcement Learning:Rollout and Policy Iteration," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 249-272, Feb. 2021. doi: 10.1109/JAS.2021.1003814. A Survey on Smart Agriculture: Development Modes, Technologies, and Security and Privacy Challenges Xing Yang, Lei Shu, Jianing Chen, Mohamed Amine Ferrag, Jun Wu, Edmond Nurellari, Kai Huang Abstract(6988) HTML (1110) PDF(1181) [Cited by] () With the deep combination of both modern information technology and traditional agriculture, the era of agriculture 4.0, which takes the form of smart agriculture, has come. Smart agriculture provides solutions for agricultural intelligence and automation. However, information security issues cannot be ignored with the development of agriculture brought by modern information technology. In this paper, three typical development modes of smart agriculture (precision agriculture, facility agriculture, and order agriculture) are presented. Then, 7 key technologies and 11 key applications are derived from the above modes. Based on the above technologies and applications, 6 security and privacy countermeasures (authentication and access control, privacy-preserving, blockchain-based solutions for data integrity, cryptography and key management, physical countermeasures, and intrusion detection systems) are summarized and discussed. Moreover, the security challenges of smart agriculture are analyzed and organized into two aspects: 1) agricultural production, and 2) information technology. Most current research projects have not taken agricultural equipment as potential security threats. Therefore, we did some additional experiments based on solar insecticidal lamps Internet of Things, and the results indicate that agricultural equipment has an impact on agricultural security. Finally, more technologies (5 G communication, fog computing, Internet of Everything, renewable energy management system, software defined network, virtual reality, augmented reality, and cyber security datasets for smart agriculture) are described as the future research directions of smart agriculture. Xing Yang, Lei Shu, Jianing Chen, Mohamed Amine Ferrag, Jun Wu, Edmond Nurellari and Kai Huang, "A Survey on Smart Agriculture: Development Modes, Technologies, and Security and Privacy Challenges," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 273-302, Feb. 2021. doi: 10.1109/JAS.2020.1003536. A Survey of Evolutionary Algorithms for Multi-Objective Optimization Problems With Irregular Pareto Fronts Yicun Hua, Qiqi Liu, Kuangrong Hao, Yaochu Jin Evolutionary algorithms have been shown to be very successful in solving multi-objective optimization problems (MOPs). However, their performance often deteriorates when solving MOPs with irregular Pareto fronts. To remedy this issue, a large body of research has been performed in recent years and many new algorithms have been proposed. This paper provides a comprehensive survey of the research on MOPs with irregular Pareto fronts. We start with a brief introduction to the basic concepts, followed by a summary of the benchmark test problems with irregular problems, an analysis of the causes of the irregularity, and real-world optimization problems with irregular Pareto fronts. Then, a taxonomy of the existing methodologies for handling irregular problems is given and representative algorithms are reviewed with a discussion of their strengths and weaknesses. Finally, open challenges are pointed out and a few promising future directions are suggested. Yicun Hua, Qiqi Liu, Kuangrong Hao and Yaochu Jin, "A Survey of Evolutionary Algorithms for Multi-Objective Optimization Problems With Irregular Pareto Fronts," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 303-318, Feb. 2021. doi: 10.1109/JAS.2021.1003817. Physical Safety and Cyber Security Analysis of Multi-Agent Systems: A Survey of Recent Advances Dan Zhang, Gang Feng, Yang Shi, Dipti Srinivasan Multi-agent systems (MASs) are typically composed of multiple smart entities with independent sensing, communication, computing, and decision-making capabilities. Nowadays, MASs have a wide range of applications in smart grids, smart manufacturing, sensor networks, and intelligent transportation systems. Control of the MASs are often coordinated through information interaction among agents, which is one of the most important factors affecting coordination and cooperation performance. However, unexpected physical faults and cyber attacks on a single agent may spread to other agents via information interaction very quickly, and thus could lead to severe degradation of the whole system performance and even destruction of MASs. This paper is concerned with the safety/security analysis and synthesis of MASs arising from physical faults and cyber attacks, and our goal is to present a comprehensive survey on recent results on fault estimation, detection, diagnosis and fault-tolerant control of MASs, and cyber attack detection and secure control of MASs subject to two typical cyber attacks. Finally, the paper concludes with some potential future research topics on the security issues of MASs. Dan Zhang, Gang Feng, Yang Shi and Dipti Srinivasan, "Physical Safety and Cyber Security Analysis of Multi-Agent Systems: A Survey of Recent Advances," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 319-333, Feb. 2021. doi: 10.1109/JAS.2021.1003820. Digital Twin for Human-Robot Interactive Welding and Welder Behavior Analysis Qiyue Wang, Wenhua Jiao, Peng Wang, YuMing Zhang This paper presents an innovative investigation on prototyping a digital twin (DT) as the platform for human-robot interactive welding and welder behavior analysis. This human-robot interaction (HRI) working style helps to enhance human users' operational productivity and comfort; while data-driven welder behavior analysis benefits to further novice welder training. This HRI system includes three modules: 1) a human user who demonstrates the welding operations offsite with her/his operations recorded by the motion-tracked handles; 2) a robot that executes the demonstrated welding operations to complete the physical welding tasks onsite; 3) a DT system that is developed based on virtual reality (VR) as a digital replica of the physical human-robot interactive welding environment. The DT system bridges a human user and robot through a bi-directional information flow: a) transmitting demonstrated welding operations in VR to the robot in the physical environment; b) displaying the physical welding scenes to human users in VR. Compared to existing DT systems reported in the literatures, the developed one provides better capability in engaging human users in interacting with welding scenes, through an augmented VR. To verify the effectiveness, six welders, skilled with certain manual welding training and unskilled without any training, tested the system by completing the same welding job; three skilled welders produce satisfied welded workpieces, while the other three unskilled do not. A data-driven approach as a combination of fast Fourier transform (FFT), principal component analysis (PCA), and support vector machine (SVM) is developed to analyze their behaviors. Given an operation sequence, i.e., motion speed sequence of the welding torch, frequency features are firstly extracted by FFT and then reduced in dimension through PCA, which are finally routed into SVM for classification. The trained model demonstrates a 94.44% classification accuracy in the testing dataset. The successful pattern recognition in skilled welder operations should benefit to accelerate novice welder training. Qiyue Wang, Wenhua Jiao, Peng Wang and YuMing Zhang, "Digital Twin for Human-Robot Interactive Welding and Welder Behavior Analysis," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 334-343, Feb. 2021. doi: 10.1109/JAS.2020.1003518. Visual Object Tracking and Servoing Control of a Nano-Scale Quadrotor: System, Algorithms, and Experiments Yuzhen Liu, Ziyang Meng, Yao Zou, Ming Cao There are two main trends in the development of unmanned aerial vehicle (UAV) technologies: miniaturization and intellectualization, in which realizing object tracking capabilities for a nano-scale UAV is one of the most challenging problems. In this paper, we present a visual object tracking and servoing control system utilizing a tailor-made 38 g nano-scale quadrotor. A lightweight visual module is integrated to enable object tracking capabilities, and a micro positioning deck is mounted to provide accurate pose estimation. In order to be robust against object appearance variations, a novel object tracking algorithm, denoted by RMCTer, is proposed, which integrates a powerful short-term tracking module and an efficient long-term processing module. In particular, the long-term processing module can provide additional object information and modify the short-term tracking model in a timely manner. Furthermore, a position-based visual servoing control method is proposed for the quadrotor, where an adaptive tracking controller is designed by leveraging backstepping and adaptive techniques. Stable and accurate object tracking is achieved even under disturbances. Experimental results are presented to demonstrate the high accuracy and stability of the whole tracking system. Yuzhen Liu, Ziyang Meng, Yao Zou and Ming Cao, "Visual Object Tracking and Servoing Control of a Nano-Scale Quadrotor: System, Algorithms, and Experiments," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 344-360, Feb. 2021. doi: 10.1109/JAS.2020.1003530. Dependent Randomization in Parallel Binary Decision Fusion Weiqiang Dong, Moshe Kam We consider a parallel decentralized detection system employing a bank of local detectors (LDs) to access a commonly-observed phenomenon. The system makes a binary decision about the phenomenon, accepting one of two hypotheses ($H_0$ ("absent") or $H_1$ ("present")). The $k{{\rm{th}}}$ LD uses a local decision rule to compress its local observations $y_k$ into a binary local decision $u_k$ ; $u_k=0$ if the $k{{\rm{th}}}$ LD accepts $H_0$ and $u_k=1$ if it accepts $H_1$ . The $k{{\rm{th}}}$ LD sends its decision $u_k$ over a noiseless dedicated channel to a Data Fusion Center (DFC). The DFC combines the local decisions it receives from $n$ LDs ($u_1, u_2,\ldots, u_n$ ) into a single binary global decision $u_0$ ($u_0=0$ for accepting $H_0$ or $u_0=1$ for accepting $H_1$ ). If each LD uses a single deterministic local decision rule (calculating $u_k$ from the local observations $y_k$ ) and the DFC uses a single deterministic global decision rule (calculating $u_0$ from the $n$ local decisions), the team receiver operating characteristic (ROC) curve is in general non-concave. The system's performance under a Neyman-Pearson criterion may then be suboptimal in the sense that a mixed strategy may yield a higher probability of detection when the probability of false alarm is constrained not to exceed a certain value, $\alpha>0$ . Specifically, a "dependent randomization" detection scheme can be applied in certain circumstances to improve the system's performance by making the ROC curve concave. This scheme requires a coordinated and synchronized action between the DFC and the LDs. In this study, we specify when dependent randomization is needed, and discuss the proper response of the detection system if synchronization between the LDs and the DFC is temporarily lost. Weiqiang Dong and Moshe Kam, "Dependent Randomization in Parallel Binary Decision Fusion," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 361-376, Feb. 2021. doi: 10.1109/JAS.2021.1003823. Set-Membership Filtering Subject to Impulsive Measurement Outliers: A Recursive Algorithm Lei Zou, Zidong Wang, Hang Geng, Xiaohui Liu This paper is concerned with the set-membership filtering problem for a class of linear time-varying systems with norm-bounded noises and impulsive measurement outliers. A new representation is proposed to model the measurement outlier by an impulsive signal whose minimum interval length (i.e., the minimum duration between two adjacent impulsive signals) and minimum norm (i.e., the minimum of the norms of all impulsive signals) are larger than certain thresholds that are adjustable according to engineering practice. In order to guarantee satisfactory filtering performance, a so-called parameter-dependent set-membership filter is put forward that is capable of generating a time-varying ellipsoidal region containing the true system state. First, a novel outlier detection strategy is developed, based on a dedicatedly constructed input-output model, to examine whether the received measurement is corrupted by an outlier. Then, through the outcome of the outlier detection, the gain matrix of the desired filter and the corresponding ellipsoidal region are calculated by solving two recursive difference equations. Furthermore, the ultimate boundedness issue on the time-varying ellipsoidal region is thoroughly investigated. Finally, a simulation example is provided to demonstrate the effectiveness of our proposed parameter-dependent set-membership filtering strategy. Lei Zou, Zidong Wang, Hang Geng and Xiaohui Liu, "Set-Membership Filtering Subject to Impulsive Measurement Outliers: A Recursive Algorithm," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 377-388, Feb. 2021. doi: 10.1109/JAS.2021.1003826. Automated Silicon-Substrate Ultra-Microtome for Automating the Collection of Brain Sections in Array Tomography Long Cheng, Weizhou Liu, Chao Zhou, Yongxiang Zou, Zeng-Guang Hou Understanding the structure and working principle of brain neural networks requires three-dimensional reconstruction of brain tissue samples using array tomography method. In order to improve the reconstruction performance, the sequence of brain sections should be collected with silicon wafers for subsequent electron microscopic imaging. However, the current collection of brain sections based on silicon substrate involve mainly manual collection, which requires the involvement of automation techniques to increase collection efficiency. This paper presents the design of an automatic collection device for brain sections. First, a novel mechanism based on circular silicon substrates is proposed for collection of brain sections; second, an automatic collection system based on microscopic object detection and feedback control strategy is proposed. Experimental results verify the function of the proposed collection device. Three objects (brain section, left baffle, right baffle) can be detected from microscopic images by the proposed detection method. Collection efficiency can be further improved with position feedback of brain sections well. It has been experimentally verified that the proposed device can well fulfill the task of automatic collection of brain sections. With the help of the proposed automatic collection device, human operators can be partially liberated from the tedious manual collection process and collection efficiency can be improved. Long Cheng, Weizhou Liu, Chao Zhou, Yongxiang Zou and Zeng-Guang Hou, "Automated Silicon-Substrate Ultra-Microtome for Automating the Collection of Brain Sections in Array Tomography," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 389-401, Feb. 2021. doi: 10.1109/JAS.2021.1003829. Efficient and High-quality Recommendations via Momentum-incorporated Parallel Stochastic Gradient Descent-Based Learning Xin Luo, Wen Qin, Ani Dong, Khaled Sedraoui, MengChu Zhou A recommender system (RS) relying on latent factor analysis usually adopts stochastic gradient descent (SGD) as its learning algorithm. However, owing to its serial mechanism, an SGD algorithm suffers from low efficiency and scalability when handling large-scale industrial problems. Aiming at addressing this issue, this study proposes a momentum-incorporated parallel stochastic gradient descent (MPSGD) algorithm, whose main idea is two-fold: a) implementing parallelization via a novel data-splitting strategy, and b) accelerating convergence rate by integrating momentum effects into its training process. With it, an MPSGD-based latent factor (MLF) model is achieved, which is capable of performing efficient and high-quality recommendations. Experimental results on four high-dimensional and sparse matrices generated by industrial RS indicate that owing to an MPSGD algorithm, an MLF model outperforms the existing state-of-the-art ones in both computational efficiency and scalability. Xin Luo, Wen Qin, Ani Dong, Khaled Sedraoui and MengChu Zhou, "Efficient and High-quality Recommendations via Momentum-incorporated Parallel Stochastic Gradient Descent-Based Learning," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 402-411, Feb. 2021. doi: 10.1109/JAS.2020.1003396. A Risk-Averse Remaining Useful Life Estimation for Predictive Maintenance Chuang Chen, Ningyun Lu, Bin Jiang, Cunsong Wang Remaining useful life (RUL) prediction is an advanced technique for system maintenance scheduling. Most of existing RUL prediction methods are only interested in the precision of RUL estimation; the adverse impact of over-estimated RUL on maintenance scheduling is not of concern. In this work, an RUL estimation method with risk-averse adaptation is developed which can reduce the over-estimation rate while maintaining a reasonable under-estimation level. The proposed method includes a module of degradation feature selection to obtain crucial features which reflect system degradation trends. Then, the latent structure between the degradation features and the RUL labels is modeled by a support vector regression (SVR) model and a long short-term memory (LSTM) network, respectively. To enhance the prediction robustness and increase its marginal utility, the SVR model and the LSTM model are integrated to generate a hybrid model via three connection parameters. By designing a cost function with penalty mechanism, the three parameters are determined using a modified grey wolf optimization algorithm. In addition, a cost metric is proposed to measure the benefit of such a risk-averse predictive maintenance method. Verification is done using an aero-engine data set from NASA. The results show the feasibility and effectiveness of the proposed RUL estimation method and the predictive maintenance strategy. Chuang Chen, Ningyun Lu, Bin Jiang and Cunsong Wang, "A Risk-Averse Remaining Useful Life Estimation for Predictive Maintenance," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 412-422, Feb. 2021. doi: 10.1109/JAS.2021.1003835. Consensus Control of Leader-Following Multi-Agent Systems in Directed Topology With Heterogeneous Disturbances Qinglai Wei, Xin Wang, Xiangnan Zhong, Naiqi Wu This paper investigates the consensus problem for linear multi-agent systems with the heterogeneous disturbances generated by the Brown motion. Its main contribution is that a control scheme is designed to achieve the dynamic consensus for the multi-agent systems in directed topology interfered by stochastic noise. In traditional ways, the coupling weights depending on the communication structure are static. A new distributed controller is designed based on Riccati inequalities, while updating the coupling weights associated with the gain matrix by state errors between adjacent agents. By introducing time-varying coupling weights into this novel control law, the state errors between leader and followers asymptotically converge to the minimum value utilizing the local interaction. Through the Lyapunov directed method and Itô formula, the stability of the closed-loop system with the proposed control law is analyzed. Two simulation results conducted by the new and traditional schemes are presented to demonstrate the effectiveness and advantage of the developed control method. Qinglai Wei, Xin Wang, Xiangnan Zhong and Naiqi Wu, "Consensus Control of Leader-Following Multi-Agent Systems in Directed Topology With Heterogeneous Disturbances," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 423-431, Feb. 2021. doi: 10.1109/JAS.2021.1003838. Coherent H∞ Control for Linear Quantum Systems With Uncertainties in the Interaction Hamiltonian Chengdi Xiang, Shan Ma, Sen Kuang, Daoyi Dong This work conducts robust H∞ analysis for a class of quantum systems subject to perturbations in the interaction Hamiltonian. A necessary and sufficient condition for the robustly strict bounded real property of this type of uncertain quantum system is proposed. This paper focuses on the study of coherent robust H∞ controller design for quantum systems with uncertainties in the interaction Hamiltonian. The desired controller is connected with the uncertain quantum system through direct and indirect couplings. A necessary and sufficient condition is provided to build a connection between the robust H∞ control problem and the scaled H∞ control problem. A numerical procedure is provided to obtain coefficients of a coherent controller. An example is presented to illustrate the controller design method. Chengdi Xiang, Shan Ma, Sen Kuang and Daoyi Dong, "Coherent H∞ Control for Linear Quantum Systems With Uncertainties in the Interaction Hamiltonian," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 432-440, Feb. 2021. doi: 10.1109/JAS.2020.1003429. Output Constrained Adaptive Controller Design for Nonlinear Saturation Systems Yongliang Yang, Zhijie Liu, Qing Li, Donald C. Wunsch This paper considers the adaptive neuro-fuzzy control scheme to solve the output tracking problem for a class of strict-feedback nonlinear systems. Both asymmetric output constraints and input saturation are considered. An asymmetric barrier Lyapunov function with time-varying prescribed performance is presented to tackle the output-tracking error constraints. A high-gain observer is employed to relax the requirement of the Lipschitz continuity about the nonlinear dynamics. To avoid the "explosion of complexity", the dynamic surface control (DSC) technique is employed to filter the virtual control signal of each subsystem. To deal with the actuator saturation, an additional auxiliary dynamical system is designed. It is theoretically investigated that the parameter estimation and output tracking error are semi-global uniformly ultimately bounded. Two simulation examples are conducted to verify the presented adaptive fuzzy controller design. Yongliang Yang, Zhijie Liu, Qing Li and Donald C. Wunsch, "Output Constrained Adaptive Controller Design for Nonlinear Saturation Systems," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 441-454, Feb. 2021. doi: 10.1109/JAS.2020.1003524. Using Event-Based Method to Estimate Cybersecurity Equilibrium Zhaofeng Liu, Ren Zheng, Wenlian Lu, Shouhuai Xu Estimating the global state of a networked system is an important problem in many application domains. The classical approach to tackling this problem is the periodic (observation) method, which is inefficient because it often observes states at a very high frequency. This inefficiency has motivated the idea of event-based method, which leverages the evolution dynamics in question and makes observations only when some rules are triggered (i.e., only when certain conditions hold). This paper initiates the investigation of using the event-based method to estimate the equilibrium in the new application domain of cybersecurity, where equilibrium is an important metric that has no closed-form solutions. More specifically, the paper presents an event-based method for estimating cybersecurity equilibrium in the preventive and reactive cyber defense dynamics, which has been proven globally convergent. The presented study proves that the estimated equilibrium from our trigger rule i) indeed converges to the equilibrium of the dynamics and ii) is Zeno-free, which assures the usefulness of the event-based method. Numerical examples show that the event-based method can reduce 98% of the observation cost incurred by the periodic method. In order to use the event-based method in practice, this paper investigates how to bridge the gap between i) the continuous state in the dynamics model, which is dubbed probability-state because it measures the probability that a node is in the secure or compromised state, and ii) the discrete state that is often encountered in practice, dubbed sample-state because it is sampled from some nodes. This bridge may be of independent value because probability-state models have been widely used to approximate exponentially-many discrete state systems. Zhaofeng Liu, Ren Zheng, Wenlian Lu and Shouhuai Xu, "Using Event-Based Method to Estimate Cybersecurity Equilibrium," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 455-467, Feb. 2021. doi: 10.1109/JAS.2020.1003527. Boundary Gap Based Reactive Navigation in Unknown Environments Zhao Gao, Jiahu Qin, Shuai Wang, Yaonan Wang Due to the requirements for mobile robots to search or rescue in unknown environments, reactive navigation which plays an essential role in these applications has attracted increasing interest. However, most existing reactive methods are vulnerable to local minima in the absence of prior knowledge about the environment. This paper aims to address the local minimum problem by employing the proposed boundary gap (BG) based reactive navigation method. Specifically, the narrowest gap extraction algorithm (NGEA) is proposed to eliminate the improper gaps. Meanwhile, we present a new concept called boundary gap which enables the robot to follow the obstacle boundary and then get rid of local minima. Moreover, in order to enhance the smoothness of generated trajectories, we take the robot dynamics into consideration by using the modified dynamic window approach (DWA). Simulation and experimental results show the superiority of our method in avoiding local minima and improving the smoothness. Zhao Gao, Jiahu Qin, Shuai Wang and Yaonan Wang, "Boundary Gap Based Reactive Navigation in Unknown Environments," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 468-477, Feb. 2021. doi: 10.1109/JAS.2021.1003841. Joint Algorithm of Message Fragmentation and No-Wait Scheduling for Time-Sensitive Networks Xi Jin, Changqing Xia, Nan Guan, Peng Zeng Time-sensitive networks (TSNs) support not only traditional best-effort communications but also deterministic communications, which send each packet at a deterministic time so that the data transmissions of networked control systems can be precisely scheduled to guarantee hard real-time constraints. No-wait scheduling is suitable for such TSNs and generates the schedules of deterministic communications with the minimal network resources so that all of the remaining resources can be used to improve the throughput of best-effort communications. However, due to inappropriate message fragmentation, the real-time performance of no-wait scheduling algorithms is reduced. Therefore, in this paper, joint algorithms of message fragmentation and no-wait scheduling are proposed. First, a specification for the joint problem based on optimization modulo theories is proposed so that off-the-shelf solvers can be used to find optimal solutions. Second, to improve the scalability of our algorithm, the worst-case delay of messages is analyzed, and then, based on the analysis, a heuristic algorithm is proposed to construct low-delay schedules. Finally, we conduct extensive test cases to evaluate our proposed algorithms. The evaluation results indicate that, compared to existing algorithms, the proposed joint algorithm improves schedulability by up to 50%. Xi Jin, Changqing Xia, Nan Guan and Peng Zeng, "Joint Algorithm of Message Fragmentation and No-Wait Scheduling for Time-Sensitive Networks," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 478-490, Feb. 2021. doi: 10.1109/JAS.2021.1003844. Supplementary Material for "A Survey of Evolutionary Algorithms for Multi- Objective Optimization Problems With Irregular Pareto Fronts" 2021, 8(2): 1-4. doi: 10.1109/JAS.2021.1003817 Abstract(106) HTML (24) PDF(13) [Cited by] () Yicun Hua, Qiqi Liu, Kuangrong Hao and Yaochu Jin, "Supplementary Material for "A Survey of Evolutionary Algorithms for Multi- Objective Optimization Problems With Irregular Pareto Fronts"," IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 1-4, Feb. 2021. doi: 10.1109/JAS.2021.1003817.	CommonCrawl
Performance analysis of electronic power transformer based on neuro-fuzzy controller Hakan Acikgoz1, O. Fatih Kececioglu2, Ceyhun Yildiz2, Ahmet Gani2 & Mustafa Sekkeli ORCID: orcid.org/0000-0002-1641-32432 In recent years, electronic power transformer (EPT), which is also called solid state transformer, has attracted great interest and has been used in place of the conventional power transformers. These transformers have many important functions as high unity power factor, low harmonic distortion, constant DC bus voltage, regulated output voltage and compensation capability. In this study, proposed EPT structure contains a three-phase pulse width modulation rectifier that converts 800 Vrms AC to 2000 V DC bus at input stage, a dual active bridge converter that provides 400 V DC bus with 5:1 high frequency transformer at isolation stage and a three-phase two level inverter that is used to obtain AC output at output stage. In order to enhance dynamic performance of EPT structure, neuro fuzzy controllers which have durable and nonlinear nature are used in input and isolation stages instead of PI controllers. The main aim of EPT structure with the proposed controller is to improve the stability of power system and to provide faster response against disturbances. Moreover, a number of simulation results are carried out to verify EPT structure designed in MATLAB/Simulink environment and to analyze compensation ability for voltage harmonics, voltage flicker and voltage sag/swell conditions. Generation, transmission and distribution of electrical energy are the most important factors in modern energy systems and transformers provide the most important role in these systems. Transformers which carry out many fundamental tasks such as galvanic isolation, voltage transformation, noise decoupling are widely used in electric power systems. It is well-known that classic (50/60 Hz) transformers have many positive features such as high efficiency, low cost and high reliability (Ronan et al. 2002; Yang et al. 2015; Zhao et al. 2013; Wang et al. 2007; Hwang et al. 2013). However, these conventional transformers have many undesirable drawbacks. These drawbacks include: (1) Conventional transformers have large size and weight because of their copper windings and iron core. (2) Conventional transformers are a passive component between the high and low voltages. Therefore, when voltage sags and swells occur at the input side, the output side is affected by these conditions. Harmonics in the output currents affect the input currents of transformers. In this case, harmonics can spread to the grid or can increase losses in the primary winding. Therefore, transformers have poor voltage regulation and low harmonic isolation. (3) Mineral oils used in transformers can be harmful when exposed to the environment in case of any fault in the transformer. In recent years, with rapid advances in microprocessors and power electronics devices, many studies have been realized in order to improve performance of transformers. A new transformer was proposed by McMurray (1970). These transformers were called as electronic power transformer (EPT) or solid state transformer (SST). The main feature of these transformers is having ability to perform the same tasks with conventional transformers. Besides, EPTs possess many advantages over conventional transformers such as voltage sag and swell compensations, fixed AC output voltage, instantaneous voltage regulation, power factor correction, reactive power compensation, harmonic isolation and all of these advantages can be realized on a single circuit (Bifaretti et al. 2011; Dujic et al. 2013; Kang et al. 1999; Kececioglu et al. 2016; Grider et al. 2011; Xu et al. 2014; Acikgoz and Sekkeli 2014; Zhao et al. 2014; Yang et al. 2015). Many studies which focus on design and control of EPT structures have been realized by many researchers and institutes in the literature. In generally, two approaches were proposed for EPT structure; with DC-link and without DC-link (Falcones et al. 2010). EPT structure with DC-link consisting input, isolation and output stages has several key features such as reactive power and voltage sag/swell compensations (Yang et al. 2015; Lai et al. 2005). Pulse width modulation (PWM) rectifiers are widely used at input stage of these EPT structures to convert AC voltage into DC voltage because of their good dynamic response, unity power factor and regulated DC bus voltage. Isolation stage has DC–DC converter and high frequency (HF) transformer, and output stage has single or three-phase inverter which generates the desired output voltage and power (Falcones et al. 2010; Yang et al. 2015; Hwang et al. 2013). During the last decades, intelligent control systems have been used in various applications. Neuro and fuzzy controllers have been outstanding intelligent control systems. Complexity and uncertainty of systems have promoted researchers to develop intelligent and adaptive control systems. Within this scope, many studies have been carried out to analyze performances of intelligent control systems (Jang et al. 1997). These control systems have been applied to many control systems and it has been obtained successful results. Development of intelligent control systems has milestones. Zadeh (1965) proposed fuzzy sets concept. This concept has led up the development of control systems that have of human reasoning capability. McCulloch and Pitts (1943) developed mimic biological neural systems computational abilities. Control systems have gained learning capability by this technique. Another approach is neuro-fuzzy controller (NFC). NFC that has nonlinear, robust structure and based on FLC whose functions are realized by ANN is one of these intelligent controllers (Jang et al. 1997; Mohagheghi et al. 2007; Tuncer and Dandil 2008). The most important feature of this controller is that it does not need the mathematical model of the controlled system. In control of PWM rectifiers, DC bus voltage and dq-axis currents are commonly controlled by using Proportional-Integral (PI) controller because of its simple structure (Dannehl et al. 2009; Blasko and Kaura 1997). However, PI controller has undesirable features including slow response, large overshoots, oscillations, and it needs a mathematical model of the system to be controlled. Recently, intelligent and robust controllers, based on fuzzy logic controller (FLC), linear quadratic regulator (LQR), sliding mode controller (SMC), Robust H∞ controller and predictive control (PC), have been successfully used in many studies. To obtain a good performance from EPT structure, intelligent controllers can be used in transient and steady-state conditions (Bouafia and Krim 2008; Bouafia et al. 2010; Yu et al. 2010; Brando et al. 2010; Zhao et al. 2012; Djerioui et al. 2014; Liu et al. 2009; Hooshmand et al. 2012). In this paper, robust and nonlinear control strategy based NFC controller is proposed for EPT structure in order to achieve a good dynamic response. Designed NFCs have two inputs, single output and six layers. This paper is organized as follows. Power circuit and mathematical model of EPT structure is given in section two. The description of the NFC and its training algorithm are explained in section three. The simulation results related to the proposed EPT structure are comprehensively presented in section four. Section five provides the conclusions of this study. Mathematical model of EPT In this section, block diagram of proposed EPT structure is consisted of input, isolation and output stages and has AC/DC/AC/DC/AC conversions as seen Fig. 1. Configuration of proposed EPT structure Input stage is the most important part of EPT structures and three-phase AC voltage is rectified by PWM rectifier at this stage (Liu et al. 2009; Hooshmand et al. 2012). There are many control methods for PWM rectifier in the literature. Voltage source PWM rectifier is preferred in this study in order to control DC bus voltage in input stage. PI controllers are often used in PWM rectifier due to their simple structures (Singh et al. 2004; Blasko and Kaura 1997). In this paper, NFC that has a more durable construction is designed instead of PI controller. So, EPT structure will be more durable and will be provide faster response against all disturbances. DC voltage rectified input stage is converted into high frequency square wave by using DC–DC converter. DC voltage obtained from isolation stage is transmitted to two-level inverter. The inverter provides the power and voltage required for the load. Moreover, stages of EPT structure are shown in detail in Fig. 2. Sections of EPT structure. a Input stage, b isolation stage and c output stage The voltage equations of EPT structure are expressed as following: $$\left[ {\begin{array}{{20}c} {{\text{u}}_{\text{sa}} (\rm t)} \\ {{\text{u}}_{\text{sb}} (\rm t)} \\ {{\text{u}}_{\text{sc}} (\rm t)} \\ \end{array} } \right] = \sqrt 2 U_{s} \, \left[ {\begin{array}{{20}c} {\sin \omega \rm t} \\ {\sin (\omega \rm t - 120)} \\ {\sin (\omega \rm t + 120)} \\ \end{array} } \right]$$ $$\left[ {\begin{array}{{20}c} {{\text{u}}_{\text{la}} ({\text{t}})} \\ {{\text{u}}_{\text{lb}} ({\text{t}})} \\ {{\text{u}}_{\text{lc}} ({\text{t}})} \\ \end{array} } \right] = {\text{m}}_{1} {\text{U}}_{\text{dc}} \, \left[ {\begin{array}{{20}c} {{\text{sin (}}\omega {\text{t}} - \theta_{ 1} )} \\ {{ \sin }(\omega {\text{t}} - 120 - \theta_{ 1} )} \\ {{ \sin }(\omega {\text{t}} + 120 - \theta_{ 1} )} \\ \end{array} } \right]$$ $$\left[ {\begin{array}{{20}c} {{\text{u}}_{\text{oa}} ({\text{t}})} \\ {{\text{u}}_{\text{ob}} ({\text{t}})} \\ {{\text{u}}_{\text{oc}} ({\text{t}})} \\ \end{array} } \right] = {\text{m}}_{2} {\text{U}}_{\text{dc}} \,\,\left[ {\begin{array}{{20}c} {{\text{sin (}}\omega {\text{t}} - \theta_{ 2} )} \\ {{ \sin }(\omega {\text{t}} - 120 - \theta_{ 2} )} \\ {{ \sin }(\omega {\text{t}} + 120 - \theta_{ 2} )} \\ \end{array} } \right]$$ According to Fig. 1 and transformation ratio of HF transformer used in isolation stage, the dynamic differential equations of EPT structure can be achieved in matrix forms as: $${\text{L}}\frac{\text{d}}{\text{dt}} \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{la}} ({\text{t}})} \\ {{\text{i}}_{\text{lb}} ({\text{t}})} \\ {{\text{i}}_{\text{lc}} ({\text{t}})} \\ \end{array} } \right] = \, \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{sa}} ({\text{t}})} \\ {{\text{u}}_{\text{sb}} ({\text{t}})} \\ {{\text{u}}_{\text{sc}} ({\text{t}})} \\ \end{array} } \right] - \, \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{la}} ({\text{t}})} \\ {{\text{u}}_{\text{lb}} ({\text{t}})} \\ {{\text{u}}_{\text{lc}} ({\text{t}})} \\ \end{array} } \right] - {\text{R }}\left[ {\begin{array}{{20}c} {{\text{i}}_{\text{la}} ({\text{t}})} \\ {{\text{i}}_{\text{lb}} ({\text{t}})} \\ {{\text{i}}_{\text{lc}} ({\text{t}})} \\ \end{array} } \right]$$ $${\text{L}}_{\text{f}} \frac{\text{d}}{\text{dt}} \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{fa}} ( {\text{t)}}} \\ {{\text{i}}_{\text{fb}} ( {\text{t)}}} \\ {{\text{i}}_{\text{fc}} ( {\text{t)}}} \\ \end{array} } \right] = \frac{ 1}{\text{k}} \, \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{oa}} ( {\text{t)}}} \\ {{\text{u}}_{\text{ob}} ( {\text{t)}}} \\ {{\text{u}}_{\text{oc}} ( {\text{t)}}} \\ \end{array} } \right] - \, \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{La}} ( {\text{t)}}} \\ {{\text{u}}_{\text{Lb}} ( {\text{t)}}} \\ {{\text{u}}_{\text{Lc}} ( {\text{t)}}} \\ \end{array} } \right]$$ $${\text{C}}_{\text{f}} \frac{\text{d}}{\text{dt}} \, \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{La}} ( {\text{t)}}} \\ {{\text{u}}_{\text{Lb}} ( {\text{t)}}} \\ {{\text{u}}_{\text{Lc}} ( {\text{t)}}} \\ \end{array} } \right] = \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{fa}} ( {\text{t)}}} \\ {{\text{i}}_{\text{fb}} ( {\text{t)}}} \\ {{\text{i}}_{\text{fc}} ( {\text{t)}}} \\ \end{array} } \right] - \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{La}} ( {\text{t)}}} \\ {{\text{i}}_{\text{Lb}} ( {\text{t)}}} \\ {{\text{i}}_{\text{Lc}} ( {\text{t)}}} \\ \end{array} } \right]$$ $$\frac{\text{d}}{\text{dt}} \, \left( {\frac{1}{2}{\text{C}}_{\text{dc}}^{2} ({\text{t}})} \right) = \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{la}} ({\text{t}})} & {{\text{u}}_{\text{lb}} ({\text{t}})} & {{\text{u}}_{\text{lc}} ({\text{t}})} \\ \end{array} } \right] \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{la}} ({\text{t}})} \\ {{\text{i}}_{\text{lb}} ({\text{t}})} \\ {{\text{i}}_{\text{lc}} ({\text{t}})} \\ \end{array} } \right] - \frac{1}{\text{k}} \, \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{oa}} ({\text{t}})} & {{\text{u}}_{\text{ob}} ({\text{t}})} & {{\text{u}}_{\text{oc}} ({\text{t}})} \\ \end{array} } \right] \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{fa}} ({\text{t}})} \\ {{\text{i}}_{\text{fb}} ({\text{t}})} \\ {{\text{i}}_{\text{fc}} ({\text{t}})} \\ \end{array} } \right]$$ Where, k is transformation ratio of HF transformer. m1 and ϴ1 are amplitude modulation index and modulation angle for PWM rectifier used in the input stage. m2 and ϴ2 are amplitude modulation index and modulation angle for PWM inverter used in the output stage. usa, usb and usc are input voltages. uLa, uLb and uLc are output voltages. Udc is DC bus voltage of input stage. u0a, u0b and u0c are AC voltages in the output stage. ula, ulb and ulc are AC voltages in the input stage (Liu et al. 2009; Hooshmand et al. 2012; Acikgoz et al. 2015). According to three-phase stationary reference frame a-b-c, dynamic model of proposed EPT structure can obtained by Eqs. (1) to (7). However, the parameters of the dynamic differential equations are time-varying and should be transformed to the synchronously rotating reference frame using Park's transformer in order to obtain time-invariant equations (Liu et al. 2009; Hooshmand et al. 2012). Thus, the dynamic equations in the d-q rotating reference frame are as follows: $$\frac{\text{d}}{\text{dt}}\left[ {\begin{array}{{20}c} {{\text{i}}_{\text{ld}} ({\text{t}})} \\ {{\text{i}}_{\text{lq}} ({\text{t}})} \\ \end{array} } \right] = - \frac{\text{R}}{\text{L}} \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{ld}} } \\ {{\text{i}}_{\text{lq}} } \\ \end{array} } \right] - \omega \;\left[ {\begin{array}{{20}c} {{\text{i}}_{\text{lq}} } \\ { - {\text{i}}_{\text{ld}} } \\ \end{array} } \right] + \frac{{{\text{m}}_{1} }}{\text{L}}{\text{u}}_{\text{dc}} \;\left[ {\begin{array}{{20}c} {\sin \;\theta_{1} } \\ {\cos \;\theta_{1} } \\ \end{array} } \right] + \frac{\sqrt 2 }{\text{L}} \, \left[ {\begin{array}{{20}c} 0 \\ {{\text{u}}_{\text{s}} } \\ \end{array} } \right]$$ $$\frac{\text{d}}{\text{dt}} \, \left[ {{\text{u}}_{\text{dc}} } \right] = - \frac{{3{\text{m}}_{1} }}{{2{\text{C}}}} \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{ld}} \;\sin \;\theta_{1} } \\ { - {\text{i}}_{\text{lq}} \;\cos \;\theta_{1} } \\ \end{array} } \right] + \frac{{3{\text{m}}_{2} }}{{2{\text{kC}}}} \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{fd}} \;\sin \;\theta_{2} } \\ { - {\text{i}}_{\text{fq}} \;\cos \;\theta_{2} } \\ \end{array} } \right]$$ $$\frac{\text{d}}{\text{dt}}\left[ {\begin{array}{{20}c} {{\text{u}}_{\text{Ld}} } \\ {{\text{u}}_{\text{Lq}} } \\ \end{array} } \right] = \frac{1}{{{\text{C}}_{\text{f}} }} \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{fd}} } \\ {{\text{i}}_{\text{fq}} } \\ \end{array} } \right] - \frac{1}{{{\text{C}}_{\text{f}} }} \, \left[ {\begin{array}{{20}c} {{\text{i}}_{\text{Ld}} } \\ {{\text{i}}_{\text{Lq}} } \\ \end{array} } \right] - \omega \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{Lq}} } \\ { - {\text{u}}_{\text{Ld}} } \\ \end{array} } \right]$$ $$\frac{\text{d}}{\text{dt}}\left[ {\begin{array}{{20}c} {{\text{i}}_{\text{fd}} } \\ {{\text{i}}_{\text{fq}} } \\ \end{array} } \right] = \omega \left[ {\begin{array}{{20}c} { - {\text{i}}_{\text{fq}} } \\ {{\text{i}}_{\text{fd}} } \\ \end{array} } \right] + \frac{{{\text{m}}_{2} \sin \theta_{2} }}{{{\text{kL}}_{\text{f}} }} \, \left[ {\begin{array}{{20}c} { - {\text{u}}_{\text{dc}} } \\ {{\text{u}}_{\text{dc}} } \\ \end{array} } \right] - \frac{1}{{{\text{L}}_{\text{f}} }} \, \left[ {\begin{array}{{20}c} {{\text{u}}_{\text{Ld}} } \\ {{\text{u}}_{\text{Lq}} } \\ \end{array} } \right]$$ where, $\left[ {{\text{i}}_{\text{ld }} {\text{i}}_{\text{lq}} {\text{i}}_{\text{lo}} } \right]^{\text{T}} = {\text{K }}\left[ {{\text{i}}_{\text{la }} {\text{i}}_{\text{lb}} {\text{i}}_{\text{lc}} } \right]^{\text{T}} ,$ $\left[ {{\text{i}}_{\text{fd }} {\text{i}}_{\text{fq}} {\text{i}}_{\text{fo}} } \right]^{\text{T}} = {\text{K }}\left[ {{\text{i}}_{\text{fa }} {\text{i}}_{\text{fb}} {\text{i}}_{\text{fc}} } \right]^{\text{T}} ,\left[ {{\text{i}}_{\text{Ld }} {\text{i}}_{\text{Lq}} {\text{i}}_{\text{Lo}} } \right]^{\text{T}} = {\text{K }}\left[ {{\text{i}}_{\text{La }} {\text{i}}_{\text{Lb}} {\text{i}}_{\text{Lc}} } \right]^{\text{T}}$ $\left[ {{\text{u}}_{\text{Ld }} {\text{u}}_{\text{Lq}} {\text{u}}_{\text{Lo}} } \right]^{\text{T}} = {\text{K }}\left[ {{\text{u}}_{\text{La }} {\text{u}}_{\text{Lb}} {\text{u}}_{\text{Lc}} } \right]^{\text{T}}$K is the Park's transformation matrix given by: $${\text{K}} = \frac{2}{3}\left[ {\begin{array}{{20}c} {\cos \,\omega {\text{t}}} & \quad{\cos \,(\omega {\text{t}} - 120)} & \quad{\cos \,(\omega {\text{t}} + 120)} \\ {\sin \,\omega {\text{t}}} & \quad{\sin \,(\omega {\text{t}} - 120)} & \quad{\sin \,(\omega {\text{t}} + 120)} \\ {\frac{3}{2}} & \quad{\frac{3}{2}} & \quad{\frac{3}{2}} \\ \end{array} } \right]$$ Implementation and design of neuro-fuzzy controller NFCs based on the principle that the functions of fuzzy logic controller (FLC) are performed artificial neural network (ANN) are successfully applied to many industrial applications (Zadeh 1965; Jang et al. 1997; Mohagheghi et al. 2007; Tuncer and Dandil 2008). In addition, NFCs have a non-linear structure and do not need mathematical model of the system to be controlled. Thus, NFCs are commonly used in non-linear systems with parameter variation and uncertainty. Fuzzy rules of sugeno type fuzzy logic are defined as below: $${\text{R}}^{\text{j}} :{\text{If X}}_{ 1} ,{\text{A}}_{1}^{\text{j}} , \ldots {\text{X}}_{\text{n}} ,{\text{A}}_{\text{n}}^{\text{j}} \quad {\text{then}}\;{\text{ y}} = {\text{f}}_{\text{i}} = a_{0}^{\text{j}} + a_{1}^{\text{j}} {\text{X}}_{ 1} + a_{2}^{\text{j}} {\text{X}}_{ 2} + a_{\text{n}}^{\text{j}} {\text{X}}_{\text{n}}$$ Here, Xi is the input variable, y is the output variable, linguistic variables of prerequisites with A i j µ Ai j (xi) membership function and the A i j ϵ R are the coefficients of linear fi = (x1, x2, …, xn) function. Structure of NFC which is used in control algorithms is shown in Fig. 3. As seen in Fig. 3, NFC has two inputs, one output and six layers. Five membership functions were chosen for each input (Jang et al. 1997; Tuncer and Dandil 2008; Buckley and Hayashi 1994). Two-input Sugeno type NFC structure Membership functions are performed in the second layer where membership function is replaced by activation function of each artificial neuro cell. Five membership functions are determined for the error and the change of error. The output of this layer is obtained as follows: $${\text{net}}_{\text{j}}^{2} = - \frac{{({\text{x}}_{\text{i}} - {\text{m}}_{\text{ij}} )^{2} }}{{2(\sigma_{\text{ij}} )^{2} }},\quad y_{\text{j}}^{2} = \exp \, ({\text{net}}_{\text{j}}^{2} )$$ σij and mij, which are input parameters, represent the parameters of membership functions to be adapted. Xi represents the input of ith cell of second layer. Similar to FLC, the third layer of NFC consists of rule base and fuzzy rules are determined in this layer. $${\text{net}}_{\text{k}}^{3} = \mathop \Pi \limits_{\text{j}} {\text{w}}_{\text{jk}}^{3} {\text{x}}_{\text{j}}^{3} ,\quad {\text{y}}_{\text{k}}^{3} = {\text{net}}_{\text{k}}^{3}$$ X j 3 here represents the input of jth cell of the third layer. The output of the system defined by using central clarification for Mamdani fuzzy logic: $${\text{net}}_{0}^{4} = \mathop \Sigma \limits_{\text{k}} {\text{w}}_{\text{k0}}^{4} {\text{y}}_{\text{k}}^{3} ,\quad {\text{y}}_{0}^{4} = \frac{{{\text{net}}_{0}^{4} }}{{\mathop \Sigma \limits_{\text{k}} {\text{y}}_{\text{k}}^{3} }}$$ Fourth layer is called normalization layer where the accuracy of fuzzy rules are calculated. Fifth layer is called firing size of a rule. The firing degree of normalized rules is multiplied by linear f function in this layer. This layer generates output values required for EPT structure. In order to update input and output parameters by using analog teaching method with back propagation algorithm, the squared error (E) which minimizes tracking error (e) is determined as follows (Jang et al. 1997): $${\text{E}} = \frac{1}{2}{\text{e}}^{2}$$ The performance index for the parameters of membership functions in EPT structure can be derived as follows: $$\frac{{\partial {\text{E}}}}{{\partial {\text{w}}_{{{\text{k}}0}} }} = - {\text{e}}.\text{sgn} \left( {\frac{{\Delta {\text{i}}_{\text{dq}} ,v_{\text{d}} ,\upvarphi }}{{\Delta {\text{y}}_{0}^{4} }}} \right)\frac{1}{{\mathop \Sigma \limits_{\text{k}} {\text{y}}_{\text{k}}^{3} }}{\text{w}}_{{{\text{k}}0}}^{4} \frac{{{\text{x}}_{\text{i}} - {\text{m}}_{\text{ij}} }}{{(\sigma_{\text{ij}} )^{2} }}{\text{y}}_{\text{j}}^{2}$$ $$\frac{{\partial {\text{E}}}}{{\partial \sigma_{\text{ij}} }} = - {\text{e}}.\text{sgn} \left( {\frac{{\Delta {\text{i}}_{\text{dq}} ,v_{\text{d}} ,\upvarphi }}{{\Delta {\text{y}}_{0}^{4} }}} \right)\frac{1}{{\mathop \Sigma \limits_{\text{k}} {\text{y}}_{\text{k}}^{3} }}{\text{w}}_{{{\text{k}}0}}^{4} \frac{{({\text{x}}_{\text{i}} - {\text{m}}_{\text{ij}} )^{2} }}{{(\sigma_{\text{ij}} )^{3} }}{\text{y}}_{\text{j}}^{2}$$ As shown in Fig. 3, inputs of NFC were selected as the error and the change of error. Five membership functions are used for each input. In the proposed NFC structure, precondition parameters of membership layer have been trained in the simulation model. During the simulation studies, output parameters have been trained using back-propagation learning algorithm. These parameters are adapted until the desired performance is reached. Control of the input stage Three-phase PWM rectifier has been used in numerous applications in recent years. These rectifiers have many advantages such as bi-directional power flow, low harmonic distortion, unity power factor and control of DC-link voltage (Blasko and Kaura 1997; Dannehl et al. 2009). When considering all these features, three-phase PWM rectifier is the most important part in EPT structures. Control scheme of three-phase PWM rectifier based on NFC is as shown in Fig. 4. In control of DC voltage, DC bus voltage is compared with reference DC bus voltage. Error of DC bus voltages is applied to NFC controller. Reference value of d-axis current is obtained from output of NFC controller. In order to obtain unity power factor, reference value of q-axis current is set to zero. Error of dq-axis currents and changes of these errors are applied as input to NFCs. Afterwards, Vq and Vd values are obtained from the outputs of NFCs. These voltages are sent to PWM block, which generates required signals for driving the semiconductor-switching element. Moreover, an anti-wind up integrator is used to limit the output of NFC and compensate for steady state error (Liu et al. 2009; Hooshmand et al. 2012). Control scheme of three-phase PWM rectifier For PI controller, mathematical model of the input stage in the d–q rotating reference frame can be used as follows: $${\text{L}}\frac{{{\text{d}}_{\text{ld}} }}{\text{dt}} = \omega {\text{LI}}_{\text{lq}} - {\text{u}}_{\text{ld}} + {\text{u}}_{\text{sd}} \,$$ $${\text{L}}\frac{{{\text{d}}_{\text{lq}} }}{\text{dt}} = - \omega {\text{LI}}_{\text{ld}} - {\text{u}}_{\text{lq}} + {\text{u}}_{\text{sq}} \,$$ where, $\left[ {\begin{array}{{20}c} {{\text{i}}_{\text{d}} } & {{\text{i}}_{\text{q}} } \\ \end{array} } \right]^{\text{T}} = {\text{K}}\left[ {\begin{array}{{20}c} {{\text{i}}_{\text{la}} } & {{\text{i}}_{\text{lb}} } & {{\text{i}}_{\text{lc}} } \\ \end{array} } \right]^{\text{T}}$, $\left[ {\begin{array}{{20}c} {{\text{u}}_{\text{ld}} } & {{\text{u}}_{\text{lq}} } \\ \end{array} } \right]^{\text{T}} = {\text{K}}\left[ {\begin{array}{{20}c} {{\text{u}}_{\text{la}} } & {{\text{u}}_{\text{lb}} } & {{\text{u}}_{\text{lc}} } \\ \end{array} } \right]^{T}$, $\left[ {\begin{array}{{20}c} {{\text{u}}_{\text{sd}} } & {{\text{u}}_{\text{sq}} } \\ \end{array} } \right]^{\text{T}} = {\text{K}}\left[ {\begin{array}{{20}c} {{\text{u}}_{\text{sa}} } & {{\text{u}}_{\text{sb}} } & {{\text{u}}_{\text{sc}} } \\ \end{array} } \right]^{\text{T}}$ $${\text{K}} = \frac{3}{2}\left[ {\begin{array}{{20}c} {\sin \omega {\text{t}}} & {\sin (\omega {\text{t}} - 120)} & {\sin (\omega {\text{t}} + 120)} \\ {\cos \omega {\text{t}}} & {{ \cos }(\omega {\text{t}} - 120)} & {\cos (\omega {\text{t}} + 120)} \\ \end{array} } \right]$$ The grid voltage component of d-axis is equal to its peak value and q-axis of the grid voltage is equal to zero. Thus, d-axis of the current is equal to the active current component and q-axis of the current is equal to the reactive current component. Double loop control which has current and voltage loop is used in order to control of three-phase PWM rectifier. d-axis current is obtained from the DC voltage loop and q-axis current sets to zero in order to obtain power factor equal to "1". For current loop controls, following equations can be written as: $${\text{L}}\frac{{{\text{d}}_{\text{ld}} }}{\text{dt}} = \left( {{\text{K}}_{\text{p}} + \frac{{{\text{K}}_{\text{I}} }}{\text{s}}} \right)({\text{i}}_{\text{ld}}^{} - {\text{i}}_{\text{ld}} )$$ $${\text{L}}\frac{{{\text{d}}_{\text{lq}} }}{\text{dt}} = \left( {{\text{K}}_{\text{p}} + \frac{{{\text{K}}_{\text{I}} }}{\text{s}}} \right){\text{i}}_{\text{lq}}$$ PI controller parameters of PWM rectifier used in the input stage are given in Table 1. Table 1 Parameters of PI controllers To analyze performance of three-phase PWM rectifier, a small signal model is used. First, the state variables are expressed as the sum of the values at an operating point and small deviations from the operating point (Ende and Shenghua 2013; Bel Hadj-Youssef et al. 2007): $${\text{U}}_{\text{dc}} = {\text{U}}_{\text{dc}} + {\hat{\text{U}}}_{\text{dc}}$$ $${\text{U}}_{\text{d}} = {\text{U}}_{\text{d}} + {\hat{\text{U}}}_{\text{d}}$$ $${\text{i}}_{\text{d}} = {\text{i}}_{\text{d}} + {\hat{\text{i}}}_{\text{d}}$$ $${\text{i}}_{\text{L}} = {\text{i}}_{\text{L}} + {\hat{\text{i}}}_{\text{L}}$$ With the analysis of the mathematical model of the three-phase PWM rectifier, the model can be written in the form of Eqs. (29–30); $${\text{L}}\frac{{{\text{di}}_{\text{d}} }}{\text{dt}} = \omega {\text{Li}}_{\text{q}} + {\text{e}}_{\text{d}} - {\text{u}}_{\text{d}}$$ $${\text{C}}\frac{{{\text{du}}_{\text{dc}} }}{\text{dt}} = \frac{3}{2}\frac{{({\text{u}}_{\text{d}} {\text{i}}_{\text{d}} + {\text{u}}_{\text{q}} {\text{i}}_{\text{q}} )}}{{{\text{u}}_{\text{dc}} }} - {\text{i}}_{\text{L}}$$ Using Eqs. (29) and (30), small signal model can be written as: $$\left[ {\begin{array}{{20}c} {{\hat{\text{u}}}_{\text{dc}} ({\text{s}})} \\ {{\text{i}}_{\text{d}} ({\text{s}})} \\ \end{array} } \right] = \left[ {\begin{array}{{20}c} {\frac{{3{\text{i}}_{\text{d}} L - 3{\text{e}}_{\text{d}} }}{{2{\text{U}}_{\text{dc}} {\text{LCs}}^{2} }}} &\quad { - \frac{1}{\text{Cs}}} \\ { - \frac{1}{\text{L}}} & \quad0 \\ \end{array} } \right]\left[ {\begin{array}{{20}c} {{\hat{\text{u}}}_{\text{d}} ({\text{s}})} \\ {{\hat{\text{i}}}_{\text{L}} ({\text{s}})} \\ \end{array} } \right]$$ Small signal model of three-phase PWM rectifier has been derived and shown in Fig. 5. A diagram showing bode plots of the current and voltage loop for three-phase PWM rectifier are indicated in Figs. 6 and 7, which demonstrates that measured gain margins are infinite. Thus, these results show that the control system is stable. Small signal model of three-phase PWM rectifier Bode plots of voltage loop for three-phase PWM rectifier Bode plots of current loop for three-phase PWM rectifier Control of the isolation stage DAB converter which provides high performance/efficiency, galvanic isolation and soft-switching property is used at the isolation stage to step down DC voltage obtained from input stage. The configuration of DAB converter is indicated in Fig. 8. DC voltage obtained from three-phase PWM rectifier is converted to a lower DC voltage using DAB converter. DC voltage obtained from the rectifier is first converted to the high-frequency square wave at isolation stage. According to the transformation ratio of HF transformer, this square wave is obtained in the same way at the secondary part of HF transformer. Then, this wave is converted into lower DC voltage by using DAB converter (Yang et al. 2015; Zhao et al. 2013). HF transformer provides electrical isolation and voltage transformation. In order to regulate DC voltage obtained from output of DAB converter, NFC and PI controller are used as phase shift controllers. The difference between DC voltage at the output of the isolation stage and reference DC voltage is compared and NFC/PI controller generates phase shift angle required for DAB converter as shown Fig. 9. Configuration of DAB converter Phase shift control of DAB Moreover, the coefficients of PI controller used in isolation stage are given in Table 2. Table 2 Parameters of PI controller for DAB The equations of power, input and output currents required for DAB converter are as follows: $${\text{P}}_{\text{DAB}} = \frac{{{\text{n}}_{\text{Tr}} {\text{U}}_{{{\text{DAB}}_{1} }} {\text{U}}_{{{\text{DAB}}2}} }}{{2{\text{f}}_{\text{DAB}} {\text{L}}_{\text{DAB}} }}{\text{d}}_{\text{DAB}} (1 - {\text{d}}_{\text{DAB}} )$$ $${\text{I}}_{{{\text{DAB}}1}} = \frac{{{\text{n}}_{\text{Tr}} {\text{U}}_{{{\text{DAB}}2}} }}{{2{\text{f}}_{\text{DAB}} {\text{L}}_{\text{DAB}} }}{\text{d}}_{\text{DAB}} (1 - {\text{d}}_{\text{DAB}} )$$ where, UDAB1 and UDAB2 are input and output DC voltages of DAB converter, fDAB is switching frequency of DAB converter, LDAB is leakage inductance, dDAB is ratio of time delay between two bridges to one-half of switching period (Yang et al. 2015; Zhao et al. 2013). Figure 10 shows small signal model of DAB converter. Small signal model of DAB converter Small-signal model and transfer functions can be obtained by state-space averaging. A state-space averaging of DAB converter is described as linear combination of independent inputs and the physical state of its energy storage elements. Also, waveforms of DAB converter are shown in Fig. 11. DAB converter waveforms For small-signal model of DAB converter used in isolation stage of EPT structure, state-space equations are as follows: $$\frac{\text{d}}{\text{dt}}{\text{x}} = {\text{A}}_{1} {\text{x}} + {\text{B}}_{1} {\text{u}} \Leftrightarrow \frac{\text{d}}{\text{dt}}\left[ {\begin{array}{{20}c} {{\text{I}}_{{{\text{DAB}}2}} } \\ {{\text{U}}_{{{\text{DAB}}2}} } \\ \end{array} } \right] = \left[ {\begin{array}{{20}c} { - \frac{{{\text{R}}_{\text{DAB}}^{{\prime \prime }} }}{{{\text{L}}_{\text{DAB}}^{{\prime \prime }} }}} & {\frac{1}{{{\text{L}}_{\text{DAB}}^{{\prime \prime }} }}} \\ { - \frac{1}{{{\text{C}}_{{{\text{DAB}}2}} }}} & { - \frac{1}{{{\text{R}}_{2} {\text{C}}_{{{\text{DAB}}2}} }}} \\ \end{array} } \right]\left[ {\begin{array}{{20}c} {{\text{I}}_{{{\text{DAB}}2}} } \\ {{\text{U}}_{{{\text{DAB}}2}} } \\ \end{array} } \right] + \left[ {\begin{array}{{20}c} {\frac{1}{{{\text{n}}_{\text{Tr}} {\text{L}}_{\text{DAB}}^{{\prime \prime }} }}} \\ 0 \\ \end{array} } \right]{\text{U}}_{\text{DAB1}}$$ where ${\text{L}}_{\text{DAB}}^{{\prime \prime }} = {\text{L}}_{\text{DAB}} /{\text{n}}_{\text{Tr}}^{2}$, ${\text{R}}_{\text{DAB}}^{{\prime \prime }} = {\text{R}}_{\text{DAB}} /{\text{n}}_{\text{Tr}}^{ 2}$. The small-signal model for DAB converter can be derived using Eqs. (35) and (36): $${\text{s}}\left[ {\begin{array}{{20}c} {\hat{\imath }_{{{\text{DAB}}2}} } \\ {{\hat{\text{u}}}{}_{{{\text{DAB}}2}}} \\ \end{array} } \right] = \left[ {\begin{array}{{20}c} { - \frac{{{\text{R}}_{\text{DAB}}^{\prime \prime } }}{{{\text{L}}_{\text{DAB}}^{\prime \prime } }}} & {\frac{{2{\text{D}}_{\text{DAB}} - 1}}{{{\text{L}}_{\text{DAB}}^{\prime \prime } }}} \\ {\frac{{2{\text{D}}_{\text{DAB}} + 1}}{{{\text{C}}_{{{\text{DAB}}2}} }}} & { - \frac{1}{{{\text{R}}_{2} {\text{C}}_{{{\text{DAB}}2}} }}} \\ \end{array} } \right]\left[ {\begin{array}{{20}c} {{\text{I}}_{{{\text{DAB}}2}} } \\ {{\hat{\text{u}}}_{{{\text{DAB}}2}} } \\ \end{array} } \right] + \left[ {\begin{array}{{20}c} {\frac{1}{{{\text{n}}_{\text{Tr}} L_{\text{DAB}}^{\prime \prime } }}} \\ 0 \\ \end{array} } \right]{\hat{\text{u}}}_{{{\text{DAB}}1}} + \left[ {\begin{array}{{20}c} {\frac{{2{\text{U}}_{{{\text{DAB}}2}} }}{{{\text{L}}_{\text{DAB}}^{\prime \prime } }}} \\ { - \frac{{2{\text{I}}_{{{\text{DAB}}2}} }}{{{\text{C}}_{{{\text{DAB}}2}} }}} \\ \end{array} } \right]{\hat{\text{d}}}_{\text{DAB}}$$ Small-signal transfer functions of DAB converter are control-to-output-current and output-current-to-DC voltage transfer functions. These transfer functions are as follow: $${\text{G}}_{{{\text{DAB}}1}} = \frac{{\hat{\imath }_{{{\text{DAB}}2}} }}{{{\hat{\text{d}}}_{\text{DAB}} }} = \frac{{2{\text{U}}_{{{\text{DAB}}2}} }}{{{\text{sL}}_{{{\text{DAB}}2}}^{{\prime \prime }} + {\text{R}}_{\text{DAB}}^{{\prime \prime }} }}$$ $${\text{G}}_{{{\text{DAB}}2}} = \frac{{{\hat{\text{u}}}_{{{\text{DAB}}2}} }}{{\hat{\imath }_{\text{DAB}} }} = \frac{{{\text{R}}_{2} ( - 2{\text{D}}_{\text{DAB}} + 1)}}{{{\text{sR}}_{2} {\text{C}}_{{{\text{DAB}}2}} + 1}}$$ Moreover, bode plots of the simplified linearized model are given in Fig. 12. Bode plots of phase shift to output voltage for small signal model Control of the output stage Output stage consists of three-phase inverter, LC filter and load as seen in Fig. 13. Three-phase inverter used in the output stage has six directional switches which converts DC voltage into three-phase AC voltages. We proposed PWM technique for three-phase inverter. In the output stage, three-phase output voltages are first converted to voltages of d–q axis (Vd, Vq) in the synchronous rotating d-q reference frame. Then, these voltages are compared with the reference values of Vd and Vq. The outputs of PI controller are transformed to Uα − Uβ voltage which is used to generate inverter gate pulses (Liu et al. 2009; Hooshmand et al. 2012). Equations of these transformations are given as follows: $$\left[ {\begin{array}{{20}c} {{\text{U}}_{\text{oa}} } \\ {{\text{U}}_{\text{ob}} } \\ {{\text{U}}_{\text{oc}} } \\ \end{array} } \right] = {\text{L}}\frac{\text{d}}{\text{dt}}\left[ {\begin{array}{{20}c} {{\text{I}}_{\text{fa}} } \\ {{\text{I}}_{\text{fb}} } \\ {{\text{I}}_{\text{fc}} } \\ \end{array} } \right] + \left[ {\begin{array}{{20}c} {{\text{U}}_{\text{La}} } \\ {{\text{U}}_{\text{Lb}} } \\ {{\text{U}}_{\text{Lc}} } \\ \end{array} } \right]$$ $$\left[ {\begin{array}{{20}c} {{\text{U}}_{\text{od}} } \\ {{\text{U}}_{\text{oq}} } \\ \end{array} } \right] = {\text{L}}\frac{\text{d}}{\text{dt}}\left[ {\begin{array}{{20}c} {{\text{I}}_{\text{d}} } \\ {{\text{I}}_{\text{q}} } \\ \end{array} } \right] + \left[ {\begin{array}{{20}c} {{\text{U}}_{\text{Ld}} } \\ {{\text{U}}_{\text{Lq}} } \\ \end{array} } \right] + \omega {\text{L}}\left[ {\begin{array}{*{20}c} { - {\text{I}}_{\text{q}} } \\ {{\text{I}}_{\text{d}} } \\ \end{array} } \right]$$ where, dd and dq are duty cycles corresponding to the dq-axes respectively. Equations (40) and (41) are obtained (Hiti et al. 1994; Tuomas et al. 2015); Configuration of three-phase two-level inverter $${\text{d}}_{\text{d}} \cdot {\text{U}}_{{{\text{dc}}2}} = {\text{L}}\frac{{{\text{di}}_{\text{d}} }}{\text{dt}} - \omega {\text{Li}}_{\text{q}} + {\text{U}}_{\text{Ld}}$$ $${\text{d}}_{\text{q}} \cdot {\text{U}}_{{{\text{dc}}2}} = {\text{L}}\frac{{{\text{di}}_{\text{q}} }}{\text{dt}} - \omega {\text{Li}}_{\text{d}} + {\text{U}}_{\text{Lq}}$$ Moreover, an operating point is defined to conduct small-signal model and to analyze the three-phase two-level inverter used in EPT structure. For small signal model, a perturbation is given around the operating point which is given as follows: $${\text{U}}_{{{\text{dc}}2}} = {\text{U}}_{{{\text{dc}}2}} + {\hat{\text{U}}}_{\text{dc2}}$$ $${\text{D}}_{\text{d}} = {\text{D}}_{\text{d}} + {\hat{\text{d}}}_{\text{d}}$$ $${\text{D}}_{\text{q}} = {\text{D}}_{\text{q}} + {\hat{\text{d}}}_{\text{d}}$$ $${\text{i}}_{\text{q}} = {\text{i}}_{\text{q}} + {\hat{\text{i}}}_{\text{q}}$$ $${\text{U}}_{\text{Ld}} = {\text{U}}_{\text{Ld}} + {\hat{\text{U}}}_{\text{Ld}}$$ $${\text{U}}_{\text{Lq}} = {\text{U}}_{\text{Lq}} + {\hat{\text{U}}}_{\text{Lq}}$$ $${\hat{\text{i}}} = {\hat{\text{D}}}_{\text{d}} {\hat{\text{i}}}_{\text{d}} + {\hat{\text{D}}}_{\text{d}} {\hat{\text{i}}}_{\text{d}} + {\hat{\text{d}}}_{\text{d}} {\hat{\text{i}}}_{\text{d}} + {\hat{\text{d}}}_{\text{d}} {\hat{\text{i}}}_{\text{d}}$$ In the above new operating point, parameters with "^" are small perturbed variables. Udc2 is DC voltage of three-phase inverter, Dd is d-axis duty cycle, Dq is q-axis duty cycle, id is d-axis current, iq is q-axis current, VLd and VLq are dq-axis grid voltages (Hiti et al. 1994; Tuomas et al. 2015). Based on the above equations, small signal model of three-phase inverter has been derived and shown in Fig. 14. Small signal model of three-phase inverter According to transfer function obtained from small signal model, bode plots of control loop for three-phase inverter are shown in Fig. 15. The bandwidth of the control loop is made low so that the DC link voltage control will not cause disturbances in the sinusoidal output currents of the three-phase inverter. The bandwidth of the control loop is adjusted near 100 Hz and the control system has a low steady state error. Bode plots of control loop for three-phase inverter Simulation results In this section, EPT structure is implemented in MATLAB/Simulink environment and the results of simulation studies are carried out in order to demonstrate dynamic performance of EPT structure under voltage harmonics, voltage flicker, voltage sag and swell conditions. Moreover, the parameters of EPT structure used in the simulation studies are compiled in Table 3. Table 3 Parameters of EPT structure used in simulation study DC bus voltage responses of the proposed controller and PI controller are shown in Fig. 16. DC voltage is adjusted to per-unit (p.u) reference value of 1. Then p.u reference value is increased to 1.25 p.u at t = 0.5 s. It can be seen from Fig. 16 that the proposed controller reaches to reference DC voltage after nearly 0.025 s without overshoot while PI controller reaches to reference DC voltage after 0.1 s with overshoot. Figure 16 shows that the proposed controller has superior properties compared with PI controller in terms of rise time, settling time and overshoot. DC bus voltage responses of both controllers The first scenario is realized in order to indicate effectiveness of both controllers against voltage harmonics and waveforms of this situation are given in Fig. 17. Voltage harmonics of the 5th and 7th orders with amplitudes of 10 and 15 % are applied on the grid voltages between t = 0.5 and t = 0.6 s. When voltage harmonics condition occurs at t = 0.5 s, PI controller has oscillations while the proposed controller is not affected by this situation as seen Fig. 17b, c which show DC voltage responses of the isolation stage with DAB converter. Thanks to DAB converter with the proposed controller, the effect of the voltage harmonics is eliminated in the input stage. It has been observed from these waveforms that the proposed controller have better performance than PI controller with regard to settling time, oscillation and overshoot. Moreover, Fig. 17d shows clearly that the output voltages are not affected by voltage harmonics in grid. Waveforms of EPT structure under voltage harmonics condition. a Grid voltages, b DC bus voltage responses of input stage, c DC bus voltage responses of isolation stage and d output voltages The second scenario is performed in order to demonstrate performance of both controllers under a voltage flicker and waveforms of this condition are given in Fig. 18. The grid voltages are first set to sinusoidal wave shape with a frequency of 15 Hz and a modulation index of 10 %. Then this flicker condition is formed between t = 0.5 and 0.6 s. As shown in Fig. 18b, DC voltage responses obtained from both controllers have a small oscillation. DC voltage responses of the isolation stage based on DAB converter are shown in Fig. 18c. It is observed that isolation stage is not affected by flicker condition as shown in Fig. 18c. Besides, this flicker case is removed in the output voltages and consequently, regulated output voltages are obtained. Waveforms of EPT structure under voltage flicker case. a Grid voltages, b DC bus voltage responses of input stage, c DC bus voltage responses of isolation stage and d output voltages The third scenario is carried out in order to verify the performance of the proposed controller and PI controller during 100 % voltage sag condition in phase A. Figure 19 shows waveforms when there is 100 % of phase-A voltage sag from t = 0.5–0.6 s. According to enlarged DC voltage responses given in Fig. 19b, when the voltage sag occurs in phase-A, DC voltage response obtained from PI controller falls 1975 V and reaches reference DC voltage at 0.7 s while the proposed controller falls 1990 V and rapidly reaches reference DC voltage at 0.63 s. Waveforms related to the isolation stage given in Fig. 19c are illustrated that the proposed controller has more efficient performance than PI controller in this condition. Also, the output voltages have clearly sinusoidal and symmetrical forms as it is seen from Fig. 19d. The fourth scenario is carried out to demonstrate response of both controllers under voltage sag condition as shown in Fig. 20. In this scenario, the magnitude of grid voltage changes from 100 to 70 % at 0.5 s, and then changes back from 70 to 100 % at 0.6 s. When the voltage sag happens at t = 0.5 s, enlarged DC voltage obtained from PI controller falls nearly 1974 V and reaches reference DC voltage at 0.7 s whereas the proposed controller falls 1991 V and reaches reference DC voltage at 0.63 s. As clearly seen in Fig. 20c, although voltage sag at the input stage is occurred from 0.5 to 0.6 s, DC response obtained from the proposed controller is more durable than PI controller. Moreover, Fig. 20d shows that the output voltages are remained fixed and sinusoidal in spite of voltage sag conditions. Waveforms of EPT structure under voltage sag condition. a Grid voltages, b DC bus voltage responses of input stage, c DC bus voltage responses of isolation stage and d output voltages The last scenario is realized in order to demonstrate the performance of both controllers during voltage swell condition. Figure 21 shows waveforms when the grid voltages occur 30 % of three-phase voltage swells at t = 0.5 to 0.6 s. According to Fig. 21, when the voltage swell happens, the proposed controller rises 2004 V and reaches reference DC voltage at 0.63 s whereas DC voltage response obtained from PI controller rises 2028 V and reaches reference DC voltage at 0.7 s. Figure 21c shows waveforms of the isolation stage in this scenario. Isolation stage based on DAB converter with the proposed controller is more successful than PI controller in eliminating effect of the voltage swell in the input stage and the proposed controller has much better reference tracking without steady-state error after voltage swell occurs. Besides, Fig. 21d clearly indicates that proposed EPT is able to regulate output voltages and compensate voltage swell in grid voltages. Waveforms of EPT structure under voltage swell condition. a Grid voltages, b DC bus voltage responses of input stage, c DC bus voltage responses of isolation stage and d output voltages EPT structure which has many superior features such as high power factor, voltage sag/swell compensation, multi-functionality, excellent power quality compared with conventional transformer is proposed in this study. EPT structure in this study is composed of input, isolation and output stages. Three-phase PWM rectifier at the input stage is not only used in order to convert AC to the constant DC voltage, but also has reactive power compensation ability. PI controllers are generally used in PWM rectifiers due to their simple structures. However, PI controller needs mathematical model of the system to be controlled and has undesirable characteristics such as slow response, large overshoots and oscillation. To cope with these problems, neuro-fuzzy controller that has nonlinear, robust structure and which does not require the mathematical model of the system to be controlled is preferred in this study. Dual active bridge converter at isolation stage is used for DC–DC conversion and is controlled by neuro-fuzzy controller in order to obtain constant DC bus voltage. Three-phase inverter that provides the desired power and voltage to load is located at the output stage. After designing of all stages, a number of simulation studies have been carried out in order to verify performance of EPT structure with the proposed controller under voltage harmonics, voltage flicker and voltage sag/swell conditions. The simulation results illustrate that EPT structure with the neuro-fuzzy controller provides more superior performance than PI controller with respect to rise time, settling time, overshoot, and power factor in all test conditions and is not sensitive these conditions and is capable of regulating output voltages and compensating disturbances in grid voltages. Moreover, proposed EPT provides fast and controllable AC/DC responses because of strong structure of the neuro fuzzy controller and thus, improves the stability of power system. Acikgoz H, Sekkeli M (2014) Simulation study of power electronic transformers with fuzzy logic controller. Int Refereed J Eng Sci 1:28–44 Acikgoz H, Kececioglu OF, Gani A, Yildiz C, Sekkeli M (2015) Optimal control and analysis of three phase electronic power transformers. 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Inf Control 8(3):338–353 Zhao Z, Han Y, Fan X (2012) Direct power vector control for PWM rectifier based on LQR algorithm. Appl Mech Mater 143–144:144–147 Zhao T, Wang G, Bhattacharya S, Huang AQ (2013) Voltage and power balance control for a cascaded H-bridge converter-based solid-state transformer. IEEE Trans Power Electron 28(4):1523–1532 Zhao C, Dujic D, Mester A, Steinke JK, Weiss M, Lewdeni-Schmid S, Chaudhuri T, Stefanutti S (2014) Power electronic traction transformer; medium voltage prototype. IEEE Trans Ind Electron 61(7):3257–3268 MS provided the basic idea of the research and supervise. HA researched the background literature, mathematical model of electronic power transformer and neuro fuzzy controller. HA, CY and FK modelled small signal models of three-phase PWM rectifier, DAB converter and three-phase inverter. FK, AG and HA developed the Simulink/MATLAB model of the power electronic transformer based on neuro-fuzzy controller, organized and drafting of the manuscript. All authors read and approved the final manuscript. Department of Electrical Science, Kilis 7 Aralik University, Kilis, 79000, Turkey Hakan Acikgoz Department of Electrical and Electronics, Faculty of Engineering, Kahramanmaras Sutcu Imam University, Kahramanmaras, Turkey O. Fatih Kececioglu , Ceyhun Yildiz , Ahmet Gani & Mustafa Sekkeli Search for Hakan Acikgoz in: Search for O. Fatih Kececioglu in: Search for Ceyhun Yildiz in: Search for Ahmet Gani in: Search for Mustafa Sekkeli in: Correspondence to Mustafa Sekkeli. Acikgoz, H., Kececioglu, O.F., Yildiz, C. et al. Performance analysis of electronic power transformer based on neuro-fuzzy controller. SpringerPlus 5, 1350 (2016) doi:10.1186/s40064-016-2972-0 Power electronic transformer Neuro-fuzzy controller PWM rectifier DAB converter	CommonCrawl
\begin{document} \title{Irrationality from The Book} \author{Steven J. Miller}\email{[email protected]} \address{Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267} \author{David Montague}\email{[email protected]} \address{Department of Mathematics, University of Michigan, Ann Arbor, MI 48109} \date{\today} \thanks{This paper was inspired by Margaret Tucker's senior colloquium talk at Williams College (February 9, 2009), where she introduced the first named author to Tennenbaum's wonderful proof of the irrationality of $\sqrt{2}$, and a comment during the talk by Frank Morgan, who wondered if the method could be generalized to other numbers. We thank Peter Sarnak for pointing out the reference \cite{Cw}. Some of the work was done at the 2009 SMALL Undergraduate Research Program at Williams College and the 2009 Young Mathematicians Conference at The Ohio State University; it is a pleasure to thank the NSF (Grant DMS0850577), Ohio State and Williams College for their support; the first named author was also supported by NSF Grant DMS0600848.} \maketitle A right of passage to theoretical mathematics is often a proof of the irrationality of $\sqrt{2}$, or at least this is where a lot of our engineering friends part ways, shaking their heads that we spend our time on such pursuits. While there are numerous applications of irrational numbers (these occur all the time in dynamical systems), the purpose of this note is not to sing their praise, but to spread the word on a truly remarkable geometric proof and its generalizations. Erd\"os used to say that G-d kept the most elegant proofs of each mathematical theorem in `The Book', and while one does not have to believe in G-d, a mathematician should believe in The Book. We first review the standard proof of the irrationality of $\sqrt{n}$. We then give the elegant geometric proof of the irrationality of $\sqrt{2}$ by Stanley Tennenbaum (discovered in the early 1950s \cite{Te}, and which first appeared in print in John H. Conway's article in Power \cite{Co}), which is worthy of inclusion in The Book. We then generalize this proof to $\sqrt{3}$ and $\sqrt{5}$, and invite the reader to explore other numbers. \section{Standard proof that $\sqrt{2}$ is irrational} To say $\sqrt{n}$ is irrational means there are no integers $a$ and $b$ such that $\sqrt{n} = a/b$. There are many proofs; see \cite{Bo} for an extensive list. One particularly nice one can be interpreted as an origami construction (see proof 7 of \cite{Bo} and the references there, and pages 183--185 of \cite{CG} for the origami interpretation). Cwikel \cite{Cw} has generalized these origami arguments to yield the irrationality of other numbers as well. We give the most famous proof in the case $n=2$. Assume there are integers $a$ and $b$ with $\sqrt{2} = a/b$; note we may cancel any common factors of $a$ and $b$ and thus we may assume they are relatively prime. Then $2b^2 = a^2$ and hence $2\|a^2$. We want to conclude that $2\|a$. If we know unique factorization\footnote{Unique factorization states that every integer can be written as a product of prime powers in a unique way.}, the proof is immediate. If not, assume $a = 2m+1$ is odd. Then $a^2 = 4m^2 + 4m + 1$ is odd as well, and hence not divisible by two.\footnote{Some texts, such as \cite{HW}, state that the Greeks argued along these lines, which is why they stopped their proofs at something like the irrationality of $\sqrt{17}$, as they were looking at special cases and not using unique factorization.} We therefore may write $a = 2r$ with $0 < r < a$. Then $2b^2 = a^2 = 4 r^2$, which when we divide by 2 gives $b^2 = 2r^2$. Arguing as before, we find that $2\|b$, so we may write $b=2s$. But now we have $2\|a$ and $2\|b$, which contradicts $a$ and $b$ being relatively prime. Thus, $\sqrt{2}$ is irrational. \section{Tennenbaum's proof} We now describe Tennenbaum's wonderful geometric proof of the irrationality of $\sqrt{2}$. Suppose that $(a/b)^2 = 2$ for some integers $a$ and $b$. Without loss of generality, we might as well assume that $a$ and $b$ are the smallest such numbers. So far this looks remarkably similar to the standard proof; here is where we go a different route. Again we find $a^2 = 2b^2$, but now we interpret this geometrically as the area of two squares of side length $b$ equals the area of one square of side length $a$. Thus, if we consider Figure \ref{fig:sqrt2}, \begin{figure} \caption{ Geometric proof of the irrationality of $\sqrt{2}$.} \label{fig:sqrt2} \end{figure} we see that the total area covered by the squares with side length $b$ (counting the overlapping, pink region twice) is equal to the area of the larger square with side length $a$. Therefore the pink, doubly counted part, which is a square of side length $2b-a$, has area equal to that of the two white, uncovered squares (each of side length $a-b$) combined. In other words, $(2b-a)^2 = 2(a-b)^2$ or $\sqrt{2} = (2b-a)/(a-b)$. But this is a smaller solution\footnote{While it is clear geometrically that this is a smaller solution, we can also see this algebraically by observing that $2b-a < a$. This follows from $b<a$ and writing $2b-a$ as $b - (a-b) < b < a$. Similarly one can show that $a-b < b$ (if not, $a \ge 2b$ so $a/b \ge 2 > \sqrt{2}$).}, contradiction! \section{The square-root of $3$ is irrational} We generalize Tennenbaum's geometric proof to show $\sqrt{3}$ is irrational. Suppose not, so $\sqrt{3} = a/b$, and again we may assume that $a$ and $b$ are the smallest integers satisfying this. This time we have $a^2 = 3b^2$, which we interpret as the area\footnote{Note that the area of an equilateral triangle is proportional to the square of its side $s$; specifically, the area is $s^2 \cdot \sqrt{3}/4$.} of one equilateral triangle of side length $a$ equals the area of three equilateral triangles of side length $b$. We represent this in Figure \ref{fig:sqrt3}, \begin{figure} \caption{ Geometric proof of the irrationality of $\sqrt{3}$} \label{fig:sqrt3} \end{figure}, which consists of three equilateral triangles of side length $b$ placed at the corners of an equilateral triangle of side length $a$. Note that the area of the three doubly covered, pink triangles (which have side length $2b-a$) is therefore equal to that of the uncovered, equilateral triangle in the middle. This is clearly a smaller solution, contradiction!\footnote{For completeness, the length of the equilateral triangle in the middle is $2a-3b$. To see this, note the dark red triangles have sides of length $b$, each pink triangle has sides of length $2b-a$ and thus the middle triangle has sides of length $b - 2(2b-a)=2a-3b$. This leads to $3(2b-a)^2 = (2a-3b)^2$, which after some algebra we see is a smaller solution.} \section{The square-root of $5$ is irrational} For the irrationality of $\sqrt{5}$, we have to slightly modify our approach as the overlapping regions are not so nicely shaped. As the proof is similar, we omit some of the details. Similar to the case of $\sqrt{3}$ and triangles, there are proportionality constants relating the area to the square of the side lengths of regular $n$-gons; however, as these constants appear on both sides of the equations, we may ignore them. Suppose $a^2 = 5b^2$ with, as always, $a$ and $b$ minimal. We place five regular pentagons of side length $b$ at the corners of a regular pentagon of side length $a$ (see Figure \ref{fig:sqrt5Fig1}). \begin{figure} \caption{ Geometric proof of the irrationality of $\sqrt{5}$. } \label{fig:sqrt5Fig1} \end{figure} Note that this gives five small triangles on the edge of the larger pentagon which are uncovered, one uncovered regular pentagon in the middle of the larger pentagon, and five kite-shaped doubly covered regions. As before, the doubly covered region must have the same area as the uncovered region. We now take the uncovered triangles from the edge and match them with the doubly covered part at the ``bottom'' of the kite, and regard each as covered once instead of one covered twice and one uncovered (see Figure \ref{fig:sqrt5Fig23}). \begin{figure} \caption{ Geometric proof of the irrationality of $\sqrt{5}$: the kites, triangles and the small pentagons. } \label{fig:sqrt5Fig23} \end{figure} This leaves five doubly covered pentagons, and one larger pentagon uncovered. We show that the five doubly covered pentagons are all regular, with side length $a-2b$. From this we get that the side lengths of the middle pentagon are all equal to $b - 2(a-2b) = 5b-2a$; later we'll show that the five angles of the middle pentagon are all equal, and thus it too is a regular pentagon. We now have a smaller solution, namely $5(a-2b)^2 = (5b-2a)^2$ (note that $a-2b < b$, as $a = b\sqrt{5}$, and $2 < \sqrt{5} < 3$), and thus we will have our contradiction. To show the smaller pentagons are regular, we just need to show two things: (1) all the angles are $3\pi/5$; (2) the lower left and lower right sides (which have equal side length by symmetry, which we denote by $x$) in Figure \ref{fig:sqrt5Fig23} have length $a-2b$, which is the length of the bottom side. If we can show these three sides are all equal, then since all five angles are equal the remaining two sides must also have length $a-2b$ and thus the pentagon is regular.\footnote{This is similar to the ASA, or angle-side-angle, principle for when two triangles are equal. There is a unique pentagon once we specify all angles and three consecutive sides; as a regular pentagon satisfies our conditions, it must be the only solution.} For (1), the sum of the angles of a pentagon\footnote{To see this, connect each corner to the center. This forms 5 triangles, each of which gives $\pi$ radians. We must subtract $2\pi$ for the sum of the angles around the center, which gives $3\pi$ for the sum of the pentagon's angles.} is $3\pi$, and for a regular pentagon each angle is $3\pi/5$. The two base angles of the triangles are thus $2\pi/5$ (as they are supplementary angles), and thus the two angles in the smaller pentagon next to the triangle's base are also $3\pi/5$. The two angles adjacent to these are just internal angles of a regular pentagon, and thus also $3\pi/5$. As the sum of all the angles is $3\pi$ and we've already accounted for $4 \cdot 3\pi/5$, this forces the top angle to be $3\pi/5$ as desired. The proof that the angles in the middle pentagon are all equal proceeds similarly (or by symmetry). We now turn to (2). The length of the left and right sides of the originally uncovered triangle at the bottom of Figure \ref{fig:sqrt5Fig23} is $\frac{(a-2b)/2}{\cos(2\pi/5)}$; to see this, note the base angles of the triangle are $\pi - \frac{3\pi}{5}$ (as each angle of a regular pentagon is $3\pi/5$ and the triangle's base angles are supplementary), and by standard trigonometry ($\cos({\rm angle}) = {\rm adjacent}/{\rm hypothenuse}$) we have $x \cos(\pi - \frac{3\pi}{5}) = \frac{a-2b}{2}$. Using $a/b = \sqrt{5}$, we see $x = b - \frac{2(a-2b)}{2\cos(2\pi/5)} = b(1-\frac{\sqrt{5}-2}{\cos(2\pi/5)})$. Noting the formula $\cos(2\pi/5) = \frac{1}{4}(-1+\sqrt{5})$, the previous expression for $x$ simplifies to $x = b(\sqrt{5}-2) = a-2b$ (as we are assuming $\sqrt{5}=a/b$), and the argument is done. \section{How far can we generalize: Irrationality of $\sqrt{6}$} We conclude with a discussion of one generalization of our method that allows us to consider triangular numbers. Figure \ref{fig:sqrt6} \begin{figure} \caption{ Geometric proof of the irrationality of $\sqrt{6}$.} \label{fig:sqrt6} \end{figure} shows the construction for the irrationality of $\sqrt{6}$. Assume $\sqrt{6} = a/b$ so $a^2=6b^2$. The large equilateral triangle has side length $a$ and the six medium equilateral triangles have side length $b$. The 7 smallest equilateral triangles (6 double counted, one in the center triple counted) have side length $t = (3b-a)/2$. It's a little work, but not too bad, to show the triple counted one is the same size. For the three omitted triangles, they are all equilateral (angles equal) and of side length $s = b - 2(3b-a)/2 = a-2b$. As the area of the smaller equilateral triangles is proportional to $t^2$ and for the larger it is proportional to $s^2$, we find $8t^2 = 3s^2$ or $16t^2 = 6s^2$ so $(4t/s)^2 = 6$. Note that although $t$ itself may not be an integer, $4t = 2(3b-a)$ is an integer. We claim $4t < a$ and $s < b$, so that this is a smaller solution. Clearly $s<b$, and as the ratio is $\sqrt{6}$, the other claim now follows. Can we continue this argument? We may interpret the argument here as adding three more triangles to the argument for the irrationality of $\sqrt{3}$; thus the next step would be adding four more triangles to this to prove the irrationality of $\sqrt{10}$. Proceeding along these lines lead us to study the square-roots of triangular numbers.\footnote{Triangular numbers are of the form $n(n+1)/2$ for some positive integer $n$. Note the first few are $1, 3, 6, 10, 15, \dots$.} We continue more generally by producing images like Figure \ref{fig:sqrt6} with $n$ equally spaced rows of side length $b$ triangles. This causes us to start with $a^2 = \frac{n(n+1)}{2}\ b^2$, so we can attempt to show that $\sqrt{n(n+1)/2}$ is irrational. By similar reasoning to the above, we see that the smaller multiply covered equilateral triangles all have the same side length $t$, and that the uncovered triangles also all have the same side length $s$. Further $t$ equals $(nb-a)/(n-1)$, and we have that $s = b-2t$, so $s = b - 2(nb-a)/(n-1) = (2a-(n+1)b)/(n-1)$. To count the number of side length $t$ triangles, we note that there will be $(n-2)(n-1)/2$ triply covered triangles (as there is a triangle-shaped configuration of them with $n-2$ rows), and that there will be $3(n-1)$ doubly covered triangles around the edge of the figure, for a grand total of $2(n-2)(n-1)/2 + 3(n-1) = (n-1)(n+1)$ coverings of the smaller triangle. Further, note that in general there will be $(n-1)n/2$ smaller, uncovered triangles, so we have that $(n-1)(n+1)t^2 = ((n-1)n/2)s^2$. Writing out the formula for $s, t$ (to verify that our final smaller solution is integral), we have $(n-1)(n+1)((nb-a)/(n-1))^2 = ((n-1)n/2)((2a-(n+1)b)/(n-1))^2$. We now multiply both sides of the equation by $n-1$ to ensure integrality, giving $(n+1)(nb-a)^2 = (n/2)(2a-(n+1)b)^2$. We multiply both sides by $n/2$ to achieve a smaller solution to $a^2 = (n(n+1)/2) b^2$, giving us $(n(n+1)/2)(nb-a)^2 = (n(2a-(n+1)b)/2)^2$. Note that this solution is integral, as $n$ odd implies that $2a - (n+1)b$ is even. Finally, to show that this solution is smaller, we just need that $nb-a < b$. This is equivalent to $n-\sqrt{n(n+1)/2} < 1$. We see that this inequality holds for $n \leq 4$, but not for $n > 4$. So, we have shown that the method used above to prove that $\sqrt{6}$ is irrational can also be used to show that $\sqrt{10}$ (the square root of the fourth triangular number) is irrational, but that this method will not work for any further triangular numbers. It is good (perhaps it is better to say, `it is not unexpected') to have such a problem, as some triangular numbers are perfect squares. For example, when $n=49$ then we have $49\cdot 50/2 = 7^2 \cdot 5^2$, and thus we should not be able to prove that this has an irrational square-root! \ \\ \end{document} A superb graphic illustration for the latter has been popularized by J. Conway around 1990, see [Hahn, ex. 37 for Ch. 1]. Conway discussed the proof at a Darwin Lecture at Cambridge. The lecture appears alongside other Darwin lectures in the book Power published by Cambridge University Press. Conway?s contribution is included as the chapter titled "The Power of Mathematics". The text can be found online. Conway attributes the proof to the Princeton mathematician Stanley Tennenbaum (1927 - 2006) who made the discovery in the early 1950s while a student at the University of Chicago. http://www.cut-the-knot.org/proofs/sq_root.shtml \ \\ \end{document}	arXiv
Home » Applied Numerical Algorithms Group » About » Staff & Postdocs » Brian Van Straalen Phillip Colella Daniel T. Graves Hans Johansen Terry Ligocki Peter McCorquodale Peter Schwartz David Trebotich Brian Van Straalen Computer Systems Engineer [email protected] Brian Van Straalen received his BASc Mechanical Engineering in 1993 and MMath in Applied Mathematics in 1995 from University of Waterloo. He has been working in the area of scientific computing since he was an undergraduate. He worked with Advanced Scientific Computing Ltd. developing CFD codes written largely in Fortran 77 running on VAX and UNIX workstations. He then worked as part of the thermal modeling group with Bell Northern Research. His Master's thesis work was in the area of a posteriori error estimation for Navier-Stokes equations, which is an area that is still relevant to Department of Energy scientific computing. He worked for Beam Technologies developing the PDESolve package: a combined symbolic manipulation package and finite element solver running in parallel on some of the earliest NSF and DOE MPP parallel computers. He came to LBNL in 1998 to work with Phil Colella and start up the Chombo Project, now in its 13th year of development. He is currently working on his Ph.D. in the Computer Science department at UC Berkeley. Protonu Basu, Samuel Williams, Brian Van Straalen, Leonid Oliker, Phillip Colella, Mary Hall, "Compiler-Based Code Generation and Autotuning for Geometric Multigrid on GPU-Accelerated Supercomputers", Parallel Computing (PARCO), April 2017, doi: 10.1016/j.parco.2017.04.002 Andrew Myers, Phillip Colella, Brian Van Straalen, "A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov-Poisson Equation", submitted to SISC, February 1, 2016, Andrew Myers, Phillip Colella, Brian Van Straalen, "The Convergence of Particle-in-Cell Schemes for Cosmological Dark Matter Simulations", The Astrophysical Journal, Volume 816, Issue 2, article id. 56, 2016, A Chien, P Balaji, P Beckman, N Dun, A Fang, H Fujita, K Iskra, Z Rubenstein, Z Zheng, R Schreiber, others, "Versioned Distributed Arrays for Resilience in Scientific Applications: Global View Resilience", Journal of Computational Science, 2015, Anshu Dubey, Ann Almgren, John Bell, Martin Berzins, Steve Brandt, Greg Bryan, Phillip Colella, Daniel Graves, Michael Lijewski, Frank L\ offler, others, "A survey of high level frameworks in block-structured adaptive mesh refinement packages", Journal of Parallel and Distributed Computing, 2014, 74:3217--3227, doi: 10.1016/j.jpdc.2014.07.001 A Dubey, B Van Straalen, "Experiences from software engineering of large scale AMR multiphysics code frameworks", arXiv preprint arXiv:1309.1781, January 1, 2013, doi: http://dx.doi.org/10.5334/jors.am Vay, J.L., Colella, P., McCorquodale, P., Van Straalen, B., Friedman, A., Grote, D.P., "Mesh Refinement for Particle-in-Cell Plasma Simulations: Applications to and Benefits for Heavy Ion Fusion", Laser and Particle Beams. Vol.20 N.4 (2002), pp. 569-575, 2002, Download File: A151.pdf (pdf: 624 KB) John Bachan, Dan Bonachea, Paul H Hargrove, Steve Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, Scott B Baden, "The UPC++ PGAS library for Exascale Computing", Proceedings of the Second Annual PGAS Applications Workshop (PAW17), November 13, 2017, 7, doi: 10.1145/3144779.3169108 We describe UPC++ V1.0, a C++11 library that supports APGAS programming. UPC++ targets distributed data structures where communication is irregular or fine-grained. The key abstractions are global pointers, asynchronous programming via RPC, and futures. Global pointers incorporate ownership information useful in optimizing for locality. Futures capture data readiness state, are useful for scheduling and also enable the programmer to chain operations to execute asynchronously as high-latency dependencies become satisfied, via continuations. The interfaces for moving non-contiguous data and handling memories with different optimal access methods are composable and closely resemble those used in modern C++. Communication in UPC++ runs at close to hardware speeds by utilizing the low-overhead GASNet-EX communication library. Dharshi Devendran, Suren Byna, Bin Dong, Brian van Straalen, Hans Johansen, Noel Keen, and Nagiza Samatova,, "Collective I/O Optimizations for Adaptive Mesh Refinement Data Writes on Lustre File System", Cray User Group (CUG) 2016, May 10, 2016, Andrey Ovsyannikov, Melissa Romanus, Brian Van Straalen, Gunther H. Weber, David Trebotich, "Scientific Workflows at DataWarp-Speed: Accelerated Data-Intensive Science using NERSC s Burst Buffer", Proceedings of the 1st Joint International Workshop on Parallel Data Storage & Data Intensive Scalable Computing Systems, IEEE Press, 2016, 1--6, doi: 10.1109/PDSW-DISCS.2016.005 Protonu Basu, Samuel Williams, Brian Van Straalen, Mary Hall, Leonid Oliker, Phillip Colella, "Compiler-Directed Transformation for Higher-Order Stencils", International Parallel and Distributed Processing Symposium (IPDPS), May 2015, Download File: ipdps15CHiLL.pdf (pdf: 1.8 MB) Yu Jung Lo, Samuel Williams, Brian Van Straalen, Terry J. Ligocki, Matthew J. Cordery, Leonid Oliker, Mary W. Hall, "Roofline Model Toolkit: A Practical Tool for Architectural and Program Analysis", Performance Modeling, Benchmarking and Simulation of High Performance Computer Systems (PMBS), November 2014, doi: 10.1007/978-3-319-17248-4_7 Download File: PMBS14-Roofline.pdf (pdf: 340 KB) Protonu Basu, Samuel Williams, Brian Van Straalen, Leonid Oliker, Mary Hall, "Converting Stencils to Accumulations for Communication-Avoiding Optimization in Geometric Multigrid", Workshop on Stencil Computations (WOSC), October 2014, Download File: wosc14chill.pdf (pdf: 973 KB) Samuel Williams, Mike Lijewski, Ann Almgren, Brian Van Straalen, Erin Carson, Nicholas Knight, James Demmel, "s-step Krylov subspace methods as bottom solvers for geometric multigrid", Parallel and Distributed Processing Symposium, 2014 IEEE 28th International, January 2014, 1149--1158, doi: 10.1109/IPDPS.2014.119 Download File: ipdps14cabicgstabfinal.pdf (pdf: 943 KB) Download File: ipdps14CABiCGStabtalk.pdf (pdf: 944 KB) Protonu Basu, Anand Venkat, Mary Hall, Samuel Williams, Brian Van Straalen, Leonid Oliker, "Compiler generation and autotuning of communication-avoiding operators for geometric multigrid", 20th International Conference on High Performance Computing (HiPC), December 2013, 452--461, Download File: hipc13chill.pdf (pdf: 989 KB) P. Basu, A. Venkat, M. Hall, S. Williams, B. Van Straalen, L. Oliker, "Compiler Generation and Autotuning of Communication-Avoiding Operators for Geometric Multigrid", Workshop on Stencil Computations (WOSC), 2013, Christopher D. Krieger, Michelle Mills Strout, Catherine Olschanowsky, Andrew Stone, Stephen Guzik, Xinfeng Gao, Carlo Bertolli, Paul H.J. Kelly, Gihan Mudalige, Brian Van Straalen, Sam Williams, "Loop chaining: A programming abstraction for balancing locality and parallelism", Parallel and Distributed Processing Symposium Workshops & PhD Forum (IPDPSW), 2013 IEEE 27th International, May 2013, 375--384, doi: 10.1109/IPDPSW.2013.68 S. Williams, D. Kalamkar, A. Singh, A. Deshpande, B. Van Straalen, M. Smelyanskiy, A. Almgren, P. Dubey, J. Shalf, L. Oliker, "Optimization of Geometric Multigrid for Emerging Multi- and Manycore Processors", Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis (SC), November 2012, doi: 10.1109/SC.2012.85 Download File: sc12-mg.pdf (pdf: 808 KB) Download File: sc12mgtalk.pdf (pdf: 1.9 MB) Chaopeng Shen, David Trebotich, Sergi Molins, Daniel T Graves, BV Straalen, DT Graves, T Ligocki, CI Steefel, "High performance computations of subsurface reactive transport processes at the pore scale", Proceedings of SciDAC, 2011, Download File: SciDAC2011sim.pdf (pdf: 1.1 MB) B. Van Straalen, P. Colella, D. T. Graves, N. Keen, "Petascale Block-Structured AMR Applications Without Distributed Meta-data", Euro-Par 2011 Parallel Processing - 17th International Conference, Euro-Par 2011, August 29 - September 2, 2011, Proceedings, Part II. Lecture Notes in Computer Science 6853 Springer 2011, ISBN 978-3-642-23396-8, Bordeaux, France, 2011, Download File: EuroPar2011bvs.pdf (pdf: 400 KB) Deines E., Weber, G.H., Garth, C., Van Straalen, B. Borovikov, S., Martin, D.F., and Joy, K.I., "On the computation of integral curves in adaptive mesh refinement vector fields", Proceedings of Dagstuhl Seminar on Scientific Visualization 2009, Schloss Dagstuhl, 2011, 2:73-91, LBNL 4972E, Download File: 7.pdf (pdf: 799 KB) G. H. Weber, S. Ahern, E.W. Bethel, S. Borovikov, H.R. Childs, E. Deines, C. Garth, H. Hagen, B. Hamann, K.I. Joy, D. Martin, J. Meredith, Prabhat, D. Pugmire, O. Rübel, B. Van Straalen and K. Wu, "Recent Advances in VisIt: AMR Streamlines and Query-Driven Visualization", Numerical Modeling of Space Plasma Flows: Astronum-2009 (Astronomical Society of the Pacific Conference Series, 3185E, 2010, 429:329-334, Download File: LBNL-3185E.pdf (pdf: 2.1 MB) B.V. Straalen, J. Shalf, T. Ligocki, N. Keen, and W. Yang, "Scalability Challenges for Massively Parallel AMR Application", 23rd IEEE International Symposium on Parallel and Distributed Processing, 2009., 2009, Download File: ipdps09finalcertified.pdf (pdf: 366 KB) Brian van Straalen, Shalf, J. Ligocki, Keen, Woo-Sun Yang, "Scalability challenges for massively parallel AMR applications", IPDPS, 2009, 1-12, Download File: ipdps09submit.pdf (pdf: 529 KB) G.H. Weber, V. Beckner, H. Childs, T. Ligocki, M. Miller, B. van Straalen, E.W. Bethel, "Visualization of Scalar Adaptive Mesh Refinement Data", Numerical Modeling of Space Plasma Flows: Astronum-2007 (Astronomical Society of the Pacific Conference Series), April 2008, 385:309-320, LBNL 220E, Download File: LBNL-220E.pdf (pdf: 1.5 MB) P. Colella, D. Graves, T. Ligocki, D. Trebotich and B.V. Straalen, "Embedded Boundary Algorithms and Software for Partial Differential Equations", 2008 J. Phys.: Conf. Ser. 125 012084, 2008, Download File: SciDAC2008-EBAlgor.pdf (pdf: 972 KB) D. Trebotich, B.V. Straalen, D. Graves and P. Colella, "Performance of Embedded Boundary Methods for CFD with Complex Geometry", 2008 J. Phys.: Conf. Ser. 125 012083, 2008, Download File: SciDAC2008-EBPerform.pdf (pdf: 167 KB) Phillip Colella, John Bell, Noel Keen, Terry Ligocki, Michael Lijewski, Brian van Straalen, "Performance and Scaling of Locally-Structured Grid Methods for Partial Differential Equations", presented at SciDAC 2007 Annual Meeting, 2007, Download File: AMRPerformance.pdf (pdf: 386 KB) Kevin Long, Brian Van Straalen, "PDESolve: an object-oriented PDE analysis environment", Object Oriented Methods for Interoperable Scientific and Engineering Computing: Proceedings of the 1998 SIAM Workshop, 1998, 99:225, B. Van Straalen, D. Trebotich, A. Ovsyannikov and D.T. Graves, "Scalable Structured Adaptive Mesh Refinement with Complex Geometry", Exascale Scientific Applications: Programming Approaches for Scalability, Performance, and Portability, edited by Straatsma, T., Antypas, K., Williams, T., (Chapman and Hall/CRC: November 9, 2017) Samuel Williams, Mark Adams, Brian Van Straalen, Performance Portability in Hybrid and Heterogeneous Multigrid Solvers, Copper Moutain, March 2016, Download File: CU16SWWilliams.pptx (pptx: 1 MB) John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ v1.0 Specification, Revision 2019.9.0", Lawrence Berkeley National Laboratory Tech Report, September 14, 2019, LBNL 2001237, doi: 10.25344/S4ZW2C UPC++ is a C++11 library providing classes and functions that support Partitioned Global Address Space (PGAS) programming. We are revising the library under the auspices of the DOE's Exascale Computing Project, to meet the needs of applications requiring PGAS support. UPC++ is intended for implementing elaborate distributed data structures where communication is irregular or fine-grained. The UPC++ interfaces for moving non-contiguous data and handling memories with different optimal access methods are composable and similar to those used in conventional C++. The UPC++ programmer can expect communication to run at close to hardware speeds. The key facilities in UPC++ are global pointers, that enable the programmer to express ownership information for improving locality, one-sided communication, both put/get and RPC, futures and continuations. Futures capture data readiness state, which is useful in making scheduling decisions, and continuations provide for completion handling via callbacks. Together, these enable the programmer to chain together a DAG of operations to execute asynchronously as high-latency dependencies become satisfied. John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ v1.0 Programmer's Guide, Revision 2019.9.0", Lawrence Berkeley National Laboratory Tech Report, September 14, 2019, LBNL 2001236, doi: 10.25344/S4V30R UPC++ is a C++11 library that provides Partitioned Global Address Space (PGAS) programming. It is designed for writing parallel programs that run efficiently and scale well on distributed-memory parallel computers. The PGAS model is single program, multiple-data (SPMD), with each separate constituent process having access to local memory as it would in C++. However, PGAS also provides access to a global address space, which is allocated in shared segments that are distributed over the processes. UPC++ provides numerous methods for accessing and using global memory. In UPC++, all operations that access remote memory are explicit, which encourages programmers to be aware of the cost of communication and data movement. Moreover, all remote-memory access operations are by default asynchronous, to enable programmers to write code that scales well even on hundreds of thousands of cores. John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ Programmer's Guide, v1.0-2019.3.0", Lawrence Berkeley National Laboratory Tech Report, March 15, 2019, LBNL 2001191, doi: 10.25344/S4F301 John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ Specification v1.0, Draft 10", Lawrence Berkeley National Laboratory Tech Report, March 15, 2019, LBNL 2001192, doi: 10.25344/S4JS30 John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ Specification v1.0, Draft 8", Lawrence Berkeley National Laboratory Tech Report, September 26, 2018, LBNL 2001179, doi: 10.25344/S45P4X John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ Programmer's Guide, v1.0-2018.9.0", Lawrence Berkeley National Laboratory Tech Report, September 26, 2018, LBNL 2001180, doi: 10.25344/S49G6V J Bachan, S Baden, D Bonachea, PH Hargrove, S Hofmeyr, K Ibrahim, M Jacquelin, A Kamil, B van Straalen, "UPC++ Programmer's Guide, v1.0-2018.3.0", March 31, 2018, LBNL 2001136, doi: 10.2172/1430693 UPC++ is a C++11 library that provides Partitioned Global Address Space (PGAS) programming. It is designed for writing parallel programs that run efficiently and scale well on distributed-memory parallel computers. The PGAS model is single program, multiple-data (SPMD), with each separate thread of execution (referred to as a rank, a term borrowed from MPI) having access to local memory as it would in C++. However, PGAS also provides access to a global address space, which is allocated in shared segments that are distributed over the ranks. UPC++ provides numerous methods for accessing and using global memory. In UPC++, all operations that access remote memory are explicit, which encourages programmers to be aware of the cost of communication and data movement. Moreover, all remote-memory access operations are by default asynchronous, to enable programmers to write code that scales well even on hundreds of thousands of cores. J Bachan, S Baden, D Bonachea, P Hargrove, S Hofmeyr, K Ibrahim, M Jacquelin, A Kamil, B Lelbach, B van Straalen, "UPC++ Specification v1.0, Draft 6", March 26, 2018, LBNL 2001135, doi: 10.2172/1430689 John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Khaled Ibrahim, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ Programmer's Guide, v1.0-2017.9", Lawrence Berkeley National Laboratory Tech Report, September 29, 2017, LBNL 2001065, doi: 10.2172/1398522 This document has been superseded by: UPC++ Programmer's Guide, v1.0-2018.3.0 (LBNL-2001136) UPC++ is a C++11 library that provides Asynchronous Partitioned Global Address Space (APGAS) programming. It is designed for writing parallel programs that run efficiently and scale well on distributed-memory parallel computers. The APGAS model is single program, multiple-data (SPMD), with each separate thread of execution (referred to as a rank, a term borrowed from MPI) having access to local memory as it would in C++. However, APGAS also provides access to a global address space, which is allocated in shared segments that are distributed over the ranks. UPC++ provides numerous methods for accessing and using global memory. In UPC++, all operations that access remote memory are explicit, which encourages programmers to be aware of the cost of communication and data movement. Moreover, all remote-memory access operations are by default asynchronous, to enable programmers to write code that scales well even on hundreds of thousands of cores. J Bachan, S Baden, D Bonachea, P Hargrove, S Hofmeyr, K Ibrahim, M Jacquelin, A Kamil, B Lelbach, B van Straalen, "UPC++ Specification v1.0, Draft 4", September 27, 2017, LBNL 2001066, doi: 10.2172/1398521 This document has been superseded by: UPC++ Specification v1.0, Draft 6 (LBNL-2001135) UPC++ is a C++11 library providing classes and functions that support Asynchronous Partitioned Global Address Space (APGAS) programming. We are revising the library under the auspices of the DOE's Exascale Computing Project, to meet the needs of applications requiring PGAS support. UPC++ is intended for implementing elaborate distributed data structures where communication is irregular or fine-grained. The UPC++ interfaces for moving non-contiguous data and handling memories with different optimal access methods are composable and similar to those used in conventional C++. The UPC++ programmer can expect communication to run at close to hardware speeds. The key facilities in UPC++ are global pointers, that enable the programmer to express ownership information for improving locality, one-sided communication, both put/get and RPC, futures and continuations. Futures capture data readiness state, which is useful in making scheduling decisions, and continuations provide for completion handling via callbacks. Together, these enable the programmer to chain together a DAG of operations to execute asynchronously as high-latency dependencies become satisfied. M. Adams, P. Colella, D. T. Graves, J.N. Johnson, N.D. Keen, T. J. Ligocki. D. F. Martin. P.W. McCorquodale, D. Modiano. P.O. Schwartz, T.D. Sternberg, B. Van Straalen, "Chombo Software Package for AMR Applications - Design Document", Lawrence Berkeley National Laboratory Technical Report LBNL-6616E, January 9, 2015, Download File: chomboDesign.pdf (pdf: 994 KB) P. Colella, D. T. Graves, T. J. Ligocki, G.H. Miller , D. Modiano, P.O. Schwartz, B. Van Straalen, J. Pillod, D. Trebotich, M. Barad, "EBChombo Software Package for Cartesian Grid, Embedded Boundary Applications", Lawrence Berkeley National Laboratory Technical Report LBNL-6615E, January 9, 2015, Download File: ebmain.pdf (pdf: 681 KB) Mark F. Adams, Jed Brown, John Shalf, Brian Van Straalen, Erich Strohmaier, Samuel Williams, "HPGMG 1.0: A Benchmark for Ranking High Performance Computing Systems", LBNL Technical Report, 2014, LBNL 6630E, Download File: hpgmg.pdf (pdf: 183 KB) Samuel Williams, Dhiraj D. Kalamkar, Amik Singh, Anand M. Deshpande, Brian Van Straalen, Mikhail Smelyanskiy, Ann Almgren, Pradeep Dubey, John Shalf, Leonid Oliker, "Implementation and Optimization of miniGMG - a Compact Geometric Multigrid Benchmark", December 2012, LBNL 6676E, Download File: miniGMGLBNL-6676E.pdf (pdf: 906 KB) Brian Van Straalen, David Trebotich, Terry Ligocki, Daniel T. Graves, Phillip Colella, Michael Barad, "An Adaptive Cartesian Grid Embedded Boundary Method for the Incompressible Navier Stokes Equations in Complex Geometry", LBNL Report Number: LBNL-1003767, 2012, LBNL LBNL Report Numb, Download File: paper5.pdf (pdf: 360 KB) We present a second-order accurate projection method to solve the incompressible Navier-Stokes equations on irregular domains in two and three dimensions. We use a finite-volume discretization obtained from intersecting the irregular domain boundary with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing a conservative discretization of the advective terms with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference to nearby cells. Our projection is based upon a finite-volume discretization of Poisson's equation. We use a second-order, $L^\infty$-stable algorithm to advance in time. Block structured local refinement is applied in space. The resulting method is second-order accurate in $L^1$ for smooth problems. We demonstrate the method on benchmark problems for flow past a cylinder in 2D and a sphere in 3D as well as flows in 3D geometries obtained from image data. M. Christen, N. Keen, T. Ligocki, L. Oliker, J. Shalf, B. van Straalen, S. Williams, "Automatic Thread-Level Parallelization in the Chombo AMR Library", LBNL Technical Report, 2011, LBNL 5109E, Scott B. Baden, Paul H. Hargrove, Hadia Ahmed, John Bachan, Dan Bonachea, Steve Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "Pagoda: Lightweight Communications and Global Address Space Support for Exascale Applications - UPC++", Poster at Exascale Computing Project (ECP) Annual Meeting 2019, January 2019, Scott B. Baden, Paul H. Hargrove, Hadia Ahmed, John Bachan, Dan Bonachea, Steve Hofmeyr, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ and GASNet-EX: PGAS Support for Exascale Applications and Runtimes", The International Conference for High Performance Computing, Networking, Storage and Analysis (SC'18), November 13, 2018, Lawrence Berkeley National Lab is developing a programming system to support HPC application development using the Partitioned Global Address Space (PGAS) model. This work is driven by the emerging need for adaptive, lightweight communication in irregular applications at exascale. We present an overview of UPC++ and GASNet-EX, including examples and performance results. GASNet-EX is a portable, high-performance communication library, leveraging hardware support to efficiently implement Active Messages and Remote Memory Access (RMA). UPC++ provides higher-level abstractions appropriate for PGAS programming such as: one-sided communication (RMA), remote procedure call, locality-aware APIs for user-defined distributed objects, and robust support for asynchronous execution to hide latency. Both libraries have been redesigned relative to their predecessors to meet the needs of exascale computing. While both libraries continue to evolve, the system already demonstrates improvements in microbenchmarks and application proxies. John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Khaled Ibrahim, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ and GASNet: PGAS Support for Exascale Apps and Runtimes", Poster at Exascale Computing Project (ECP) Annual Meeting 2018., February 2018, John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Khaled Ibrahim, Mathias Jacquelin, Amir Kamil, Brian Van Straalen, "UPC++: a PGAS C++ Library", ACM/IEEE Conference on Supercomputing, SC'17, November 2017, John Bachan, Scott Baden, Dan Bonachea, Paul Hargrove, Steven Hofmeyr, Khaled Ibrahim, Mathias Jacquelin, Amir Kamil, Brian van Straalen, "UPC++ and GASNet: PGAS Support for Exascale Apps and Runtimes", Poster at Exascale Computing Project (ECP) Annual Meeting 2017., January 2017, A Mignone, C Zanni, P Tzeferacos, B van Straalen, P Colella, G Bodo, The PLUTO code for adaptive mesh computations in astrophysical fluid dynamics, The Astrophysical Journal Supplement Series, Pages: 7 2012,	CommonCrawl
\begin{document} \title{Adiabatic Quantum Transport in a Spin Chain with a Moving Potential} \author{Vinitha Balachandran and Jiangbin Gong} \email{[email protected]}\affiliation{Department of Physics and Centre for Computational Science and Engineering, \\ National University of Singapore, 117542, Republic of Singapore} \begin{abstract} Many schemes to realize quantum state transfer in spin chains are not robust to random fluctuations in the spin-spin coupling strength. In efforts to achieve robust quantum state transfer, an adiabatic quantum population transfer scheme is proposed in this study. The proposed scheme makes use of a slowly moving external parabolic potential and is qualitatively explained in terms of the adiabatic following of a quantum state with a moving separatrix structure in the classical phase space of a pendulum analogy. Detailed aspects of our adiabatic population transfer scheme, including its robustness, is studied computationally. Applications of our adiabatic scheme in quantum information transfer are also discussed, with emphasis placed on the usage of a dual spin chain to encode quantum phases. The results should also be useful for the control of electron tunneling in an array of quantum dots. \end{abstract} \pacs{03.67.Hk, 32.80.Qk, 73.63.Kv} \date{\today} \maketitle \section{Introduction} Efficiently transferring quantum information is an important challenge for the practical realization of quantum computers. Optical fibers, where quantum information is transmitted by mobile carriers like photons, are most desirable for long-distance communication. However, the need to interface with solid state quantum computer components considerably restricts the experimental feasibility of using optical systems as quantum channels. One promising alternative is to use condensed matter systems themselves as the {\it quantum wire} for information transfer. This idea, initially advocated and studied by Bose \cite{bose1} in the context of quantum spin chains, has now attracted wide interests. In particular, for a quantum spin chain used as the quantum wire, the natural evolution of permanently coupled spins is exploited to accomplish the quantum information transfer. Because the Hamiltonians equivalent to that of a spin chain may be realized in different physical systems (e.g., arrays of Josephson junctions \cite{plenio}, cold atoms in optical lattices \cite{duan03}, arrays of quantum dots, etc.), interests in spin chains as a promising candidate of quantum wires continue to grow. Experimental studies on the dynamics of quantum information transfer and on the quantum control in spin chains using nuclear magnetic resonance techniques have also been reported \cite{NMR}. Despite many fruitful studies of spin chains from a quantum information perspective, many theoretical problems are still open. To be more specific let us consider first Bose's original proposal \cite{bose1}. Therein the fidelity of quantum information transfer is gradually degraded by the dispersion effects associated with the quantum propagation. Furthermore, particular external magnetic field should be designed to ensure a correct quantum phase at the receiver's site. These issues and others motivated a series of sophisticated protocols to achieve better quantum information transfer. One noteworthy approach was to pre-engineer the nearest-neighbor couplings of a spin chain or even a spin network \cite{christandl1,twamley,network2,stolze,superconducting}. A second approach exploits the mirror symmetry of a spin chain \cite{stolze,parabolic}. Another approach suggests to use Gaussian wavepackets with slow dispersion \cite{osborne1} to encode the quantum information to be transferred along an unmodulated spin chain. Unfortunately, these pioneering approaches rely upon specific analytical forms of the involved Hamiltonian and hence are not robust to imperfection or physical fluctuations in the spin-spin coupling strength. One encounters the same situation when applying other more subtle techniques \cite{wojcik1,lyakhov1,hase}. To have the desired robustness that may be necessary for any type of quantum information transfer, two quantum control schemes based on adiabatically varying the coupling strength in a spin chain have also been suggested \cite{adiabatic1,adiabatic2}. Note however, these schemes require individual addressing of the nearest-neighbor coupling and hence present new experimental challenges. Another novel and quite robust quantum transfer protocol in spin chains is the dual spin chain scheme \cite{burgarth2,burgarth1,burgarth3,burgarthnj}. Therein the quantum information to be transferred is encoded in two parallel spin sub-chains (initially assumed to be identical, but even this condition may be lifted under certain conditions). Thanks to the encoding with two spin sub-chains, the quantum transfer can be very robust to static disorder. Nevertheless, even this promising scheme is not perfect, because (i) it may need too many quantum measurements (or too many steps of ``trial and error"), (ii) it may not operate well in the presence of time-dependent disorder, and (iii) the effects of nonideal measurements are still under investigation \cite{burgarth3}. Hence, it remains an open question as to which quantum transfer scheme will ultimately be adopted experimentally, with high fidelity and low cost. It is our belief that in the end a combination of several techniques may be able to offer the most powerful protocol for quantum information transfer in solid state systems. In this paper, we introduce an adiabatic transport scheme assisted by a slowly moving external field applied to a spin chain. Thanks to a pendulum analogy, the central idea can be understood in a very simple manner. The essence is that when an external field is moving slowly, the spin excitation may adiabatically follow the field under certain conditions. During this process the quantum population of spin excitation is transferred from one end of the spin chain to the other end. As we show below in detail, this adiabatic scheme offers a number of advantages: (i) it is highly robust, (ii) it can be operated rather fast (in the absence of disorder, it may be as fast as one tenth of the natural propagation speed of the spin chain), (iii) the required field strength can be decreased by using wavepackets as initial states, (iv) the transfer can be easily stopped and relaunched, (v) the time of arrival of the quantum population transfer to the last spin with high probability can be easily predicted, and (vi) it can offer a promising means of quantum information transfer when combined with the above-mentioned dual spin chain scheme. The outline of the paper is as follows. In Sec. \ref{pendulum}, we explain the motivation and mechanism of our adiabatic population transfer scheme by mapping the spin chain Hamiltonian to that of a pendulum. Detailed computational results are presented in Sec. III. The robustness of our scheme is studied in Sec. IV. In Sec. V, we discuss how our adiabatic population transfer scheme, which does not yet take care of the quantum phases (also essential for quantum information transfer), can be combined with the dual spin chain scheme to offer a potentially powerful approach for quantum information transfer. We conclude this work in Sec. VI. \section{Adiabatic Quantum Transport in Spin Chains: A Pendulum Perspective} \label{pendulum} Consider a one-dimensional Heisenberg chain of $N+1$ spins subject to an external parabolic magnetic field. The associated Hamiltonian is given by \begin{eqnarray} H_{s}&=&-\frac{J}{2}\sum_{n=0}^{N-1}{\bf \sigma}_{n}\cdot {\bf \sigma}_{n+1} + \sum_{n=0}^{N}\frac{C}{2}(n-n_{0})^{2}\sigma_{n}^{z}, \label{KReq} \end{eqnarray} where ${\bf \sigma}\equiv (\sigma^{x},\sigma^{y},\sigma^{z})$ are the Pauli matrices, $J$ the coupling strength between nearest neighbor spins, and $C$ is proportional to the magnetic dipole of the spins and the amplitude of the parabolic field whose minimum is at site $n_{0}$. Note that $n_{0}$ will be time-dependent in our control scheme. Below we assume all the system parameters have been appropriately scaled and take dimensionless values, with $J=1$, $\hbar=1$ throughout. As such, the energy scale (e.g., the parameter $C$) should be understood with respect to $J$, and the time scale should be understood with respect to $\hbar/J$. Because the spin chain Hamiltonian $H_{s}$ commutes with the total polarization $S_{z}\equiv \sum_{n=1}^{N}\sigma_{n}^{z}$, the dynamics of the spin chain preserves the total polarization. Here we restrict our analysis to the subspace of $S_{z} =1-N$, where in total only one spin is flipped. In this subspace the total state of the chain can be written as \begin{eqnarray} \|\Psi(t)\rangle &=& \sum_{m=0}^{N}c_{m}(t)\|\mathbf{m}\rangle, \end{eqnarray} where $\|\mathbf{m}\rangle$ represents one basis state with a spin up at the $m$th site and all other spins down. The complex coefficients $c_{m}(t)$ are the probability amplitude. Below we also shift the zero of the energy scale such that if one spin is down, its interaction with the external magnetic field contributes zero to the total energy. To understand the essence of the spin chain dynamics from a semiclassical perspective, we now consider the large $N$ limit of the spin chain. Denote $k$ as the quasi-momentum of a plane spin wave and the $\|k\rangle$ the associated eigenstate of the quasi-momentum. Then the Hamiltonian in Eq. (\ref{KReq}) can be rewritten as \begin{eqnarray} H_{s}= -J \int_{0}^{2\pi} \cos(k) \|k\rangle\langle k\| \nonumber \\ +\sum_{n} \|{\bf n}\rangle\langle {\bf n}\| \frac{C}{2}(n-n_{0})^{2}. \label{H2} \end{eqnarray} This form can be further simplified in an operator form, i.e., \begin{eqnarray} H_{s}= -J \cos(\hat{k}) +\frac{C}{2}(\hat{n}-n_{0})^{2}\label{H3}, \end{eqnarray} where $\hat{k}\|k\rangle =k \|k\rangle$; $\hat{n}\|{\bf n}\rangle=n \|\bf{n}\rangle$; $[\cos(\hat{k}), \hat{n}]=-i\sin(\hat{k})$, and $[\sin(\hat{k}), \hat{n}]=i\cos(\hat{k})$. The Hamiltonian in Eq. (\ref{H3}) can now be easily recognized to be the Hamiltonian of a quantum pendulum $H_{p}$ \cite{Boness}, with an effective Planck constant $\sqrt{C}$. Specifically, with the mapping $\hat{k}\rightarrow \hat{x}$, $\sqrt{C}\hat{n}\rightarrow \hat{p}$, the spin chain Hamiltonian $H_{s}$ is mapped to \begin{eqnarray} H_{p}= -J \cos(\hat{x}) +\frac{1}{2}(\hat{p}-p_{0})^{2}\label{H4}, \end{eqnarray} where $p_{0}=\sqrt{C}n_{0}$ and $[\cos(\hat{x}), \hat{p}]=-i\sqrt{C}\sin(\hat{x})$. The semiclassical Hamiltonian for this quantum pendulum, i.e., $H_{p}^{c}=-J \cos(x) +\frac{1}{2}(p-p_{0})^{2}$ is obtained by replacing $\hat{x}$ and $\hat{p}$ with $c$-variables $x$ and $p$. \begin{figure} \caption{Phase space portrait for the classical pendulum Hamiltonian associated with Eq. (\ref{H4}), with $C=2$ and $J=1$. The closed curve enclosing the island and intersecting with both $p-p_{0}=0$ and $x=\pm\pi$ is the separatrix. Variables $P-P_{0}$ and $x$ plotted here take dimensionless values. Note that if the separatrix is moving up slowly by increasing $p_{0}$ gradually, an initial state enclosed by the separatrix is expected to adiabatically follow the movement of the separatrix. This suggests that a slowly moving parabolic potential applied to a spin chain can be used to adiabatically transfer spin excitation along a spin chain.} \label{separatrix} \end{figure} Many quantum transport features of the spin chain can now be understood in terms of the semiclassical dynamics of the pendulum analogy thus obtained. In particular, the quantum transport in the momentum space of the pendulum is now in parallel with the transfer of spin excitation from one site to another. Hence, the issue of robust quantum transport of spin excitation along the spin chain now reduces to the design of a control scenario that enables robust transport of the pendulum state in its momentum space. An adiabatic scheme for robust quantum population transfer along a spin chain can now be proposed. The key observation is the existence of a motional separatrix in the classical phase space of the pendulum. This separatrix is located at $ -J \cos(x) +(1/2)(p-p_{0})^{2}=J$ [see Fig. \ref{separatrix}]. If we now slowly move up the separatrix along the momentum space by increasing $p_0$, then a quantum state initially trapped inside the separatrix cannot penetrate this separatrix and is expected to adiabatically follow the moving separatrix, giving rise to adiabatic transport in the momentum space. Translating this pendulum language back to the spin chain case, one anticipates that a slowly moving parabolic magnetic field (with slowly increasing $n_{0}$) should result in a robust scenario for transferring quantum population along the spin chain. During this process the dispersion of the spin wave should also be bounded by the separatrix structure, i.e., a moving but non-spreading wavepacket \cite{report} of the spin wave can be expected. The main remaining task of this paper is devoted to detailed aspects of this adiabatic control scheme. It is also interesting to note that the dynamics of the spin chain can be mapped to that of a tight-binding model. To see this consider first the associated Schr\"{o}dinger equation, \begin{eqnarray} i\frac{dc_{0}}{dt} &=& \frac{J}{2}c_{1} + \frac{C}{2}n_{0}^{2}c_{0}, \nonumber \\ i\frac{dc_{n}}{dt} &=& \frac{J}{2}(c_{n-1}+c_{n+1}) + \frac{C}{2}(n-n_{0})^{2}c_{n},\ 0<n<N, \nonumber \\ i\frac{dc_{N}}{dt} &=& \frac{J}{2}c_{N-1} + \frac{C}{2}(N-n_{0})^{2}c_{N}. \label{seq} \end{eqnarray} Consider next a tight-binding Hamiltonian $H_{t}$ describing, for example, an array of $(N+1)$ identical quantum dots subject to an external parabolic field, with one electron tunneling between the quantum dots. Then $H_{t}$ assumes the following form, \begin{eqnarray} H_{t} &=&-\frac{J}{2}\sum_{n=0}^{N-1}(a_{n}^{\dagger}a_{n+1} +a_{n}a_{n+1}^{\dagger}) \nonumber \\ && +\sum_{n=0}^{N}\frac{C}{2}(n-n_{0})^{2}a_{n}^{\dagger}a_{n}, \end{eqnarray} where $J$ represents the constant tunneling rate between the nearest-neighbor quantum dots, and $a_{n}^{\dagger}$ and $a_{n}$ represent the creation and annihilation operators. Because the total number of electrons is already assumed to be one, the system wavefunction can also be written as $\|\Psi(t)\rangle = \sum_{m=0}^{N}c_{m}(t)\|\mathbf{m}\rangle$, where $\|\mathbf{m}\rangle$ denotes the state with an electron in the $m$th quantum dot and $c_{m}(t)$ denotes the associated quantum amplitude. In this representation, one immediately finds that the evolution of this tight-binding system takes the same form as Eq. (\ref{seq}). Thus, the above pendulum analogy is also applicable to a tight-binding system and is hence very useful for consideration of adiabatic quantum transport in quantum dot arrays \cite{adiabatic3}. Note also that one may start from Eq. (\ref{seq}) to have an alternative derivation of the pendulum analogy \cite{kol1}. To end this section, we stress that the proposed control scheme is based upon a semiclassical perspective afforded by the pendulum analogy. What is not addressed is the issue of transferring the quantum phase along the spin chain. As such, although the pendulum analogy helps design our scheme for the robust transport of quantum excitation along a spin chain or the robust transport of an electron in a quantum dot array, the issue of quantum information transfer is only partially touched. Indeed, the introduction of an external field will change the energy of the spin chain system and hence will necessarily introduce extra dynamical phases to the evolving quantum system. This makes it clear that transporting quantum phases along the spin chain requires additional considerations. This quantum phase issue will be considered in detail in Sec. V. \section{Adiabatic Transport by a Moving Potential: Computational Results} In this section, we illustrate our adiabatic quantum population transfer scheme with detailed computational examples. Let us assume that the initial state of the spin chain is given by $\|\Phi\rangle=\sum_{m=0}^{N}c_{m}(0)\|\bf {m}\rangle$. Two types of $c_{m}(0)$ will be considered below. In the first case only the $m=0$th spin is excited, with $c_{m}(0)=\delta_{m0}$. In the second case, the initial state is a Gaussian wavepacket truncated to three sites only, with $c_{m}(0)\propto \exp[-(m-1)^{2}/2l_{0}^{2}]$ for $m=0-2$ and $l_{0}=0.707$ being the width of the Gaussian wavepacket. In either case a parabolic magnetic field first centered on the $n=0$th site is applied and then slowly moved to the regime of larger $n$. This is realized by introducing the time dependence of $n_{0}$ via $n_{0}=0+St$, where $S$ is the moving speed. Note that a static parabolic field was previously introduced to induce a quasi harmonic lower energy spectrum such that good transfer of Gaussian wavepackets \cite{parabolic} may be realized. By contrast, our moving potential scenario is more active and effective in controlling the quantum transport and is in principle applicable to cases where the shape of the external potential is not parabolic. The quantum state of the spin chain at a later time can be directly calculated using the Schr\"{o}dinger equation given above. In particular, the probability of transferring the quantum excitation to the last spin of the chain can be examined. If the performance of the population transfer is satisfactory, one should find $\|c_{N}\|^2 \approx 1.0$. Evidently, this condition of high transfer probability is already useful by itself for, e.g., transporting electrons in a quantum dot array in a controlled fashion. As shown in Sec. V, the phase of the quantum amplitude $c_{N}$ may be also taken care of by considering a dual spin chain. \begin{figure} \caption{Excitation probability transferred to the last spin in adiabatic quantum transport along a chain of 101 spins, as a function of the amplitude of the external moving magnetic potential characterized by $C$ [in units of $J$, see the text below Eq. (1)]. The moving speed of the control field is $S=0.005$.} \label{cchange} \end{figure} We now discuss the feasibility of adiabatic quantum population transfer by taking advantage of the separatrix associated with the pendulum analogy. In the ideal case of adiabatic following, an initial quantum state enclosed by a separatrix will move with the slowly moving separatrix. Consider first an initial state localized exclusively at the $n=0$th site. Then the associated $k$-distribution covers uniformly from $0$ to $2\pi$. From a semiclassical perspective afforded by the pendulum analogy, such an initial state corresponds to an initial ensemble lying on the $(p-p_{0})=0$ axis of the classical phase space. This initial ensemble hence necessarily intersects with the separatrix (see Fig. \ref{separatrix}). Because the motional period associated with the separatrix is infinity, those ensemble components that overlap with the separatrix will always regard the movement of the separatrix as ``too fast to follow". That is, as we slowly move the separatrix upwards in the classical phase space, some portion of the initial ensemble may break the adiabaticity and tunnel through the separatrix structure. Under such a situation adiabatic quantum population transfer is expected to partially break down. To reduce the degree of non-adiabaticity, one possible approach is to reduce the overlap of the initial state with the classical separatrix. This should be doable by increasing the effective Planck constant $\sqrt{C}$ (i.e., increasing the strength of the parabolic field) such that the separatrix regime supports less quantum states. This is indeed what we find computationally for a chain of 101 spins. In particular, Fig. \ref{cchange} shows that for a field amplitude characterized by $C=0.5$, the probability of transferring the initial excitation to the last spin is only $0.63$. By increasing $C$ to $8.0$, a transfer probability around $99\%$ is observed. Figure 3 shows the actual excitation profile $\|c_{n}\|^2$ vs. $n$ for a spin chain subject to a parabolic potential moving at a constant speed of $S=0.005$. At $t=0$, the state is at site $n=0$. At $t=10000$, the quantum population is mainly at $n=50$. Note that at that moment the excitation profile is slightly delocalized into three sites, but the peak probability is still as high as $0.97$. This peak is propagated to site $n=100$ at time $t=20000$, with no further dispersion detected. This indicates that our moving potential scheme has the capacity to overcome the dispersion issue in quantum information transfer. Though the required field strength for large $\|n-n_{0}\|$ could be demanding experimentally, we point out that because the spin excitation is highly localized throughout the process, the moving parabolic magnetic field does not need to span over many spin sites (in our numerical experiments, we use a parabolic field that only spans 20 sites). \begin{figure} \caption{ Adiabatic transfer of spin excitation initially localized exclusively at the $n=0$th site along a chain of $101$ spins, for a field amplitude given by $C=8$ and its moving speed given by $S=0.005$. Panels (a), (b), and (c) are for times $t=0$, $10000$, and $20000$. Note that the final peak excitation probability remains as high as 0.97.} \label{single-case} \end{figure} We have also examined the quantum dynamics for the second type of initial states, i.e., states with a Gaussian excitation profile at $t=0$. Because such initial ensembles are localized in both $k$ and $n$, they can be naturally enclosed by the separatrix shown in Fig. \ref{separatrix}. As such, if the shape of the initial excitation profile is appropriately adjusted, the initial state can be made not to intersect with the separatrix. This being the case, the adiabatic following should work better, probably requiring a much weaker parabolic field. This expectation is also confirmed computationally. In particular, Fig. \ref{Gaussian-case} shows the transport of an initial Gaussian excitation profile, again for a chain of $101$ spins. At $t=0$, the excitation profile spans only the first three sites with a probability peak $0.78$ at the site $n=1$. This state is then transported by applying a parabolic field with $C=2$ moving at a rate of $S=0.005$. During the quantum transport the state disperses among about five sites, with a peak probability maintained around $0.77$. At $t=10000$ and $t=20000$, the peak of the spin excitation probability profile is transferred to $n=50$ and $n=100$. Interestingly, it is found that the population transfer probability to the last spin can be further enhanced at a slightly later time. As seen from Fig. \ref{Gaussian-case}, at time $t=21000$, the peak probability, located at the last spin, is as high as $0.997$. Physically, this is due to the reflection process at the end of the spin chain. In some sense, the interplay of the parabolic field centered at the end of the spin chain and the reflection process acts as a lens refocusing the slightly dispersed profile, and the peak probability builds up on the last spin. Note also that in the absence of the control field, one in general needs initial wavepackets of much larger length to be able to reduce the undesired dispersion \cite{osborne1}. \begin{figure} \caption{Adiabatic quantum transport along a chain of $101$ sites with an initial Gaussian excitation profile, at (a) $t=0$, (b) $t=10000$, (c) $t=20000$, and (d) $t=21000$. The amplitude of the parabolic field is given by $C=2$, and the moving speed of the control field is given by $S=0.005$. Note that the final excitation probability transferred to the last spin is as high as 0.997.} \label{Gaussian-case} \end{figure} Results in Fig. \ref{Gaussian-case} demonstrate that using initial states whose excitation profile covers a few spins can significantly reduce the required field amplitude (compare values of $C$ in Fig. \ref{Gaussian-case} and in Fig. \ref{single-case}). Counter-intuitively, there exists a maximal number $n_{\text{max}}$ of spins that can be used to create such initial states. This can be appreciated by considering again the separatrix in the classical phase space of the pendulum analogy. Classical orbits outside the separatrix are associated with pendulum's rotational motion (rather than oscillation). Going back to the spin chain or the tight-binding system, these states correspond to Bloch oscillations in a ``locally linear" field. As the separatrix is slowly moving, these states can continue their Bloch oscillations in a slowly-varying local field and hence will not follow the motion of the separatrix in the momentum space. With this understanding, one may estimate $n_{\text{max}}$ from the width of the separatrix in the momentum space. Specifically, for a fixed value of the parameter $C$, $n_{\text{max}} \sim 4\sqrt{\frac{J}{C}}+1$. For the numerical example in Fig. \ref{Gaussian-case}, one obtains $n_{\text{max}}\approx 4$. This estimate is quite consistent with the finding that during the population transfer, the moving wavepacket does not cover more than five sites. This result also implies that for weaker magnetic fields (smaller $C$), one can use more spins to form the wavepacket for analogous adiabatic population transfer. Because our quantum transport scheme is based upon the adiabatic following of the spin excitation profile with a moving external potential, it can stop and relaunch the excitation transfer at any time with great ease, by simply stopping and restarting the movement of the external parabolic potential. This is simpler than a recent approach \cite{gong07} using pulsed magnetic fields, and is also confirmed in our computational studies (not shown). \begin{figure} \caption{Adiabatic transfer of spin excitation for an initial state exclusively localized at the $n=0$th site along a chain of $101$ spins. The amplitude of the moving parabolic potential is given by $C=8$, and the moving speed is given by $S=0.025$. Panels (a), (b) and (c) show the excitation profile at times $t=1000$, $2600$, and $4000$.} \label{speed} \end{figure} So can we further increase the moving speed of the control potential while still maintaining the adiabatic following and hence the adiabatic quantum population transfer? Our findings in this regard can be summarized as follows: (i) for large $C$ adiabatic quantum transport may survive for a moving speed around $10\%$ of the coupling constant $J$. The smaller the field strength $C$ is, the lower the threshold moving speed will be; (ii) when the moving speed exceeds the threshold, the probability of successful population transfer gradually decreases, but can still be considerably large for a relatively short spin chain. For example, Fig. \ref{speed} shows the result for an initial state exclusively localized at the $n=0$th site. The moving speed of the parabolic potential is $S=0.025$. The peak value of the probability profile is $0.96$ at time $t=1000$. It reduces to about $0.95$ at $t=2600$ and $0.94$ at $t=4000$. Hence, in this case, only $2\%$ reduction in the peak probability occurs when the moving speed $S$ increases by a factor of five. However, increasing the moving speed beyond this limit drastically reduces the probability of population transfer to the last spin. For a moving speed of $S=0.30$, the probability maxima equals only $0.87$ at $t=2200$, and $0.84$ at $t=3400$. Analogous calculations are also carried out for the transport of an initial Gaussian excitation profile. As shown in Fig. \ref{speed-Gaussian}, for a moving speed of 0.1, the peak value of probability remains around $0.76$ during the transport process. As such, at the end (not shown) the adiabatic population transfer is also very successful for this high moving speed. But if the moving speed is further increased by several times, dispersion in the spin excitation profile will be considerable. \begin{figure} \caption{Adiabatic transport of an initial Gaussian profile of spin excitation (same as in Fig. \ref{Gaussian-case}) along a chain of $101$ spins. The amplitude of the moving parabolic potential is given by $C=2$, and the moving speed is given by $S=0.1$. Panels (a), (b), and (c) are for $t=300$, $600$ and $1000$.} \label{speed-Gaussian} \end{figure} In short, our numerical experiments suggest that, to achieve adiabatic transport of spin excitation along a quite long spin chain using a moving parabolic potential, the associated moving speed can be as large as being one tenth of the natural propagation rate ($J$) of the system (without disorder). \begin{figure} \caption{Adiabatic transfer of spin excitation for an initial state exclusively localized at the $n=0$th site along a chain of $101$ spins, in the presence of static disorder with the noise amplitude given by $\Delta =0.5$. Other parameters are the same as in Fig. \ref{single-case}. The excitation profile at time $t=7000$, $14000$, and $20000$ are shown in panels (a), (b), and (c). Results here are very similar to those shown in Fig. \ref{single-case}.} \label{single-case-static} \end{figure} \section {Robustness of Adiabatic Transport} So far the majority of quantum state transfer schemes consider only idealized spin chains with no disorder in the spin-spin coupling strength. This suggests a gap between theoretical exploration and realistic situations in experiments. In particular, the effects of static and dynamic imperfections in spin chains are studied in very few cases \cite {lyakhov1,burgarth1,burgarthnj,disorder}. Here we computationally study the influence of static and dynamic disorder on our adiabatic population transfer scheme, by considering the model Hamiltonian in Eq. (\ref{KReq}) with fluctuating spin-spin coupling constants. We hope to numerically confirm the robustness of our scheme as implied by its adiabatic nature. The model Hamiltonian with disorder in the spin-spin coupling strength is given by \begin{eqnarray} H_{sd} &=&\sum_{n=0}^{N-1}-\frac{(J+\delta_{n})}{2}{\bf \sigma}_{n}\cdot {\bf \sigma}_{n+1} \nonumber \\ && +\sum_{n=0}^{N}\frac{C}{2}(n-n_{0})^2\sigma_{n}^{z}, \end{eqnarray} where $\delta_{n}$ are time-independent random numbers uniformly distributed in the interval $[-\Delta,\Delta]$, representing random fluctuations in $J$ with the amplitude $\Delta$. We call cases with such time-independent disorder as static disorder models. Note that specific results presented below refer to single disorder realizations. These results are very typical such that there is no need to average over many disorder realizations. Figure \ref{single-case-static} display one sampling calculation that is in parallel with the results in Fig. \ref{single-case} but takes into account static disorder with the noise amplitude $\Delta=0.5$. As demonstrated in Fig. \ref{single-case-static}, in the presence of such a high noise level, the quantum population of spin excitation is still successfully transferred to the last spin of the chain (peak probability around 99\%), with the excitation profile almost unaltered as compared with the noiseless case studied in Fig. \ref{single-case}. For an even higher fluctuation level, e.g., $\Delta=0.7$, the spin excitation profile is seen to gradually disperse as it is transported along the chain. \begin{figure} \caption{Adiabatic transfer of an initial Gaussian profile of spin excitation along a chain of $101$ spins, in the presence of static disorder characterized by $\Delta=0.5$. Other parameters are the same as in Fig. \ref{Gaussian-case}. Panels (a), (b) and (c) are for $t=6000$, $12000$ and $21500$. Results here are very similar to those shown in Fig. \ref{Gaussian-case}.} \label{Gaussian-case-static} \end{figure} Figure \ref{Gaussian-case-static} displays results for an initial Gaussian excitation profile also considered in Fig. \ref{Gaussian-case}, but in the presence of static disorder characterized by $\Delta=0.5$. We find that the severe disorder can slightly change the shape of the spin excitation profile (though not so evident in Fig. 8) and hence the peak value of the probability profile slightly fluctuates during the controlled transport. Note however, the area enclosed by the main probability profile is found to be around $0.99$ at all times. At time $t=21500$, the peak probability of the spin excitation profile, still as large as $0.99$, has been transferred to the $100$th site as in the noiseless case of Fig. \ref{Gaussian-case}. In another sampling case for a $\Delta=0.7$, the peak excitation probability that is transferred to the last spin decreases to $0.94$. All these results clearly demonstrate the robustness of our adiabatic transport scheme to high-level static disorder. We have also examined the robustness of our adiabatic transport scheme to dynamic disorder. To model time-dependent fluctuations in the spin-spin coupling strength, we now let each $\delta_{n}$ be given by the sum of ten oscillating functions, i.e., $\delta_{n}= \sum_{i=1}^{10} A\cos(\omega_{i} t+\phi_{i})$, where $\omega_{i}$ are random frequencies distributed in $[0,\omega_{\text{max}}]$, and $\phi_{i}$ are random phases uniformly distributed in $[0,2\pi]$. \begin{figure}\label{single-case-dynamic} \end{figure} Interestingly, our numerical experiments indicate that effects of dynamical disorder modeled above depend strongly on $\omega_{\text{max}}$, i.e., the cut-off frequency of the dynamic fluctuations. Introducing disorder more frequently, i.e., introducing a larger $\omega_{\text{max}}$, can lead to much decreased peak probability transferred to the last spin. In particular, we find that for $\omega_{\text{max}}\leq 0.1 $, the effects of the dynamic disorder are essentially analogous to what is found for static disorder. For larger $\omega_{\text{max}}$, the deterioration of the adiabatic population transfer becomes considerable for the same noise amplitude $A$. Figure \ref{single-case-dynamic} displays the results for $A=0.025$ and $\omega_{\text{max}}=0.1$. Note that for $A=0.025$, the amplitude of the noise is very large because the total fluctuation is a sum of ten functions oscillating at the same amplitude $A$. It is seen from Fig. \ref{single-case-dynamic} that for such a case of dynamic disorder, the spin excitation travels almost unaffected along the chain, thus confirming again the robustness of our adiabatic scheme. However, upon an increase in $\omega_{\text{max}}$, e.g., $\omega_{\text{max}}=1.0$ (so noise frequency becomes comparable to the characteristic coupling strength $J$), the dispersion of the spin excitation profile becomes evident in Fig. \ref{single-case-dynamic2}. The situation can be certainly much improved if the noise amplitude $A$ is decreased. Because similar results are also found for Gaussian excitation profile as initial states, we conclude that the noise spectrum of dynamic disorder can play an important role in affecting the robustness of our adiabatic transport scheme, especially when the noise amplitude is very large. \begin{figure}\label{single-case-dynamic2} \end{figure} Although our adiabatic scheme is seen to be robust, it can be expected that the very existence of disorder should limit the threshold speed of the moving potential. Put alternatively, for a larger moving speed (which satisfies the adiabatic condition less), the robustness of our control scheme to disorder is expected to decrease. This trend is indeed found in our numerical experiments. To characterize precisely how an increasing moving speed of the control potential affects the robustness is certainly beyond the scope of this work. \section{Adiabatic Transport in a Dual Spin Chain} As demonstrated in previous sections, our adiabatic scheme based upon a moving parabolic potential offers a simple and robust approach to transferring quantum population along a spin chain. This scheme requires a strong parabolic field if it is globally parabolic, but even this requirement can be greatly weakened if the initial spin excitation profile spans a few sites. Further, one does not really need a globally parabolic field to realize this adiabatic scheme: it suffices for the parabolic field profile to be wider than the spin excitation profile. With these considerations we may argue that the well-known dispersion issue in quantum information transfer along spin chains is essentially solved by our adiabatic scheme. Nevertheless, as also mentioned earlier, one important issue still remains open. That is, for the sake of quantum information transfer, how to take care of the quantum phase of a quantum state to be transported? Indeed, a moving external potential induces extra dynamical phase to the spin chain, and such a dynamical phase depends on the details of the control potential. These facts motivate us to seek an encoding approach that can protect a quantum state from the additional dynamical phases induced by the moving potential. Fortunately, the idea of using a dual spin chain, first proposed to overcome the disorder and dispersion issues in quantum information transfer in spin chains \cite{burgarth2,burgarthnj}, offers a promising solution. Specifically, we propose to combine our adiabatic transport scheme with the dual spin chain scheme. Then, because each individual sub-chain acquires identical dynamical phases from the same external moving potential, the relative quantum phase between the two sub-chains is certain, and hence quantum information encoded in the dual spin chain can be transported without suffering from the extra uncertain dynamical phases. Consider then a quantum channel consisting of two identical parallel spin chains subject to the same external parabolic potential, \begin{eqnarray} H^{i}_{s} &=&-\frac{J}{2}\sum_{n=0}^{N-1}{\bf \sigma}^{(i)}_{n}\cdot {\bf \sigma}^{(i)}_{n+1} \nonumber \\ && +\sum_{n=0}^{N}C\frac{(n-n_{0})^2}{2}\sigma_{n}^{z(i),} \end{eqnarray} where $i=1,2$ indices label the two sub-chains. Suppose the quantum state to be transferred is given by $\|\Phi\rangle = \alpha\|0\rangle + \beta\|1\rangle$. Such a state can be encoded into the quantum channel prepared in the following entangled state, \begin{eqnarray} \|\mathbf{\Psi}(0)\rangle &=&\alpha\|\mathbf{g}\rangle^{(1)}\otimes\|\mathbf{0}\rangle^{(2)}+\beta\|\mathbf{0}\rangle^{(1)} \otimes\|\mathbf{g}\rangle^{(2)} \label{initialstate} \end{eqnarray} as a superposition of two components: the $n=0$th spin in the second (first) sub-chain being flipped and the first (second) sub-chain in its ground state denoted by $\|\mathbf{g}\rangle$. Note that for each sub-chain at most one spin is flipped and the associated dynamics will be restricted to the ground state or the subspace of one flipped spin. This encoding can be extended to cases of entangled Gaussian wavepackets in a straightforward manner. However, for convenience here we discuss only cases arising from the initial state given by Eq. (\ref{initialstate}). After the independent evolution of the two sub-chains for a total duration of $\tau$ under the action of the moving parabolic potential, the quantum state of the dual spin chain is given by \begin{eqnarray} \|\Psi(\tau)\rangle &=&\sum_{n=0}^{N}c_{n}(\tau)\|\Phi_{n}\rangle, \end{eqnarray} where $\|\mathbf{\Phi}_{n}\rangle \equiv \alpha\|\mathbf{g}\rangle^{(1)}\otimes\|\mathbf{n}\rangle^{(2)}+\beta\|\mathbf{n}\rangle^{(1)} \otimes\|\mathbf{g}\rangle^{(2)}$. Evidently, though each $c_{n}$ contains the extra quantum phases induced by the external moving potential, this factor is identical for the two state components of $\|\mathbf{\Phi}_{n}\rangle$. As already demonstrated in our numerical experiments using a single spin chain, the profile of $\|c_{n}\|^{2}$ should also be highly localized, and the time of arrival of the peak value of $\|c_{n}\|^{2}$ at the last spin can also be directly calculated from the moving speed of the parabolic potential. Analogous to the original dual spin chain scheme, at the end of the adiabatic quantum transport the final state $\|\Psi(\tau)\rangle$ can be decoded by applying a CNOT operation to the last two $N$th spins of the dual chain. Upon this operation the final state is transformed to \begin{eqnarray} \sum_{n=0}^{N-1}c_{n}(\tau)\|\Phi_{n}\rangle + c_{N}(\tau)\left[ \alpha \|\mathbf{g}\rangle^{(1)}+\beta \|\mathbf{N}\rangle^{(1)}\right]\otimes\|\mathbf{N}\rangle^{(2)}. \end{eqnarray} As such, by measuring the last spin of the second sub-chain, one gains important information about the transport. In particular, if the measurement outcome is spin up, then the initial state $\|\Phi\rangle = \alpha\|0\rangle + \beta\|1\rangle$ has been successfully transferred to the last spin of the first sub-chain, with probability $\|c_{N}(\tau)\|^{2}$; if the outcome is spin down, then the quantum state transfer is unsuccessful and one needs to wait for more time to perform additional measurements. Significantly, because our adiabatic population transfer scheme can ensure a very high probability of excitation transfer to the last spin, the probability of spin-up measurements can be guaranteed to be very high (arbitrarily high if there were no restriction on the field strength). This hence overcomes, at least theoretically, one main disadvantage of previous dual spin chain schemes where too many measurements may be required for high fidelity quantum state transfer. Further, at the end of the adiabatic transport, the spin excitation is automatically localized at very few end spins. So it also becomes unnecessary to perform fast measurements at a particular time. Instead, one can choose measurement times at will so long as the moving parabolic potential has reached the last site of the spin chain. This makes it clear that our adiabatic scheme, when combined with quantum phase encoding schemes, can find important applications in quantum information transfer (in addition to quantum population transfer). \section{Conclusions} In this work we have presented a simple and robust scheme to realize adiabatic population transfer in spin chains. The additional resource needed is a slowly moving external parabolic magnetic field. The basic mechanism is the adiabatic following of a quantum state with the movement of a separatrix structure in the classical phase space of a pendulum analogy. In particular, we have shown that our scheme can be used to transfer spin excitation from one end of a spin chain to the other end, with the initial excitation profile being a localized truncated Gaussian wavepacket or exclusively localized at a single spin site. It is found that much weaker external field is needed for adiabatic population transfer if the initial excitation profile covers a few spin sites. Effects of static and dynamical fluctuations in the spin-spin coupling strength are also computationally studied, confirming the robustness of our adiabatic population transfer scheme. Realizing the robust population transfer with small dispersion, we have also proposed to apply our approach to a dual spin chain such that robust quantum information transfer can be realized with important advantages. We hope that our theoretical scheme can motivate experiments using various implementations of a spin chain Hamiltonian, such as cold atoms in an optical lattice and electron tunneling in an array of quantum dots. The central idea of this work, namely, using a slowly moving external potential to adiabatically transfer spin excitation, might be useful for other applications as well. For example, one may consider distributing entanglement along a long spin chain in a controlled fashion, by use of a control potential that has two components slowly moving in opposite directions. Another interesting application is related to studies of quantum signal amplification with spin chain models. Recently, an interesting connection between quantum state transfer and quantum state amplification is revealed \cite{kay}. In this regard, our adiabatic scheme might also help design a new and useful approach to controlled quantum amplification using a slowly moving external potential. \acknowledgments J.G. is supported by the start-up funding, (WBS grant No. R-144-050-193-101 and No. R-144-050-193-133), National University of Singapore, and the NUS ``YIA" funding (WBS grant No. R-144-000-195-123) from the office of Deputy President (Research \& Technology), National University of Singapore. \end{document}	arXiv
\begin{definition}[Definition:Graph of Real Function] Let $U \subseteq \R^n$ be an open subset of $n$-dimensional Euclidean space. Let $f : U \to \R^k$ be a real function. The '''graph''' $\map \Gamma f$ of the function $f$ is the subset of $\R^n \times \R^k$ such that: :$\map \Gamma f = \set {\tuple {x, y} \in \R^n \times \R^k: x \in U \subseteq \R^n : \map f x = y}$ where $\times$ denotes the Cartesian product. \end{definition}	ProofWiki
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It aims to identify both the content deemed relevant and the performance levels for medical students in otolaryngology. A national survey developed from a content analysis of undergraduate otolaryngology curricula from the UK was undertaken, accompanied by a review of the literature and input from an expert group. Data were collected from a wide range of doctors. Participants felt that graduating students should be able to: recognise, assess and initiate management for common and life-threatening acute conditions; take an appropriate patient history; and perform an appropriate examination for the majority of otolaryngology clinical conditions but manage only a select few. This study reports performance levels for otolaryngology topics at an undergraduate level. Participating doctors felt that a higher level of performance should be expected of students treating life-threatening, acute and common otolaryngology conditions. The Bioarchaeology of Structural Violence: A Theoretical Framework for Industrial Era Inequality. LORI A. TREMBLAY and SARAH REEDY, editors. 2020. Springer, Cham, Switzerland. xiv + 284 pp. $119.99 (hardcover), ISBN 978-3-030-46439-4. $84.99 (paperback), ISBN 978-3-030-46442-4. $89.00 (e-book), ISBN 978-3-030-46440-0. Shannon K. Freire, Catherine R. Jones Journal: American Antiquity , First View The importance of formal modelling for the development of cognitive theory Randall K. Jamieson, Brendan T. Johns, Vanessa Taler, Michael N. Jones Journal: Bilingualism: Language and Cognition , First View Reductions in inpatient fluoroquinolone use and postdischarge Clostridioides difficile infection (CDI) from a systemwide antimicrobial stewardship intervention Antibiotics Awareness Week 2021 K. Ashley Jones, Udodirim N. Onwubiko, Julianne Kubes, Benjamin Albrecht, Kristen Paciullo, Jessica Howard-Anderson, Sujit Suchindran, Ronald Trible, Jesse T. Jacob, Sarah H. Yi, Dana Goodenough, Scott K. Fridkin, Mary Elizabeth Sexton, Zanthia Wiley Journal: Antimicrobial Stewardship & Healthcare Epidemiology / Volume 1 / Issue 1 / 2021 Published online by Cambridge University Press: 22 October 2021, e32 To determine the impact of an inpatient stewardship intervention targeting fluoroquinolone use on inpatient and postdischarge Clostridioides difficile infection (CDI). We used an interrupted time series study design to evaluate the rate of hospital-onset CDI (HO-CDI), postdischarge CDI (PD-CDI) within 12 weeks, and inpatient fluoroquinolone use from 2 years prior to 1 year after a stewardship intervention. An academic healthcare system with 4 hospitals. Patients: All inpatients hospitalized between January 2017 and September 2020, excluding those discharged from locations caring for oncology, bone marrow transplant, or solid-organ transplant patients. Introduction of electronic order sets designed to reduce inpatient fluoroquinolone prescribing. Among 163,117 admissions, there were 683 cases of HO-CDI and 1,104 cases of PD-CDI. In the context of a 2% month-to-month decline starting in the preintervention period (P < .01), we observed a reduction in fluoroquinolone days of therapy per 1,000 patient days of 21% after the intervention (level change, P < .05). HO-CDI rates were stable throughout the study period. In contrast, we also detected a change in the trend of PD-CDI rates from a stable monthly rate in the preintervention period to a monthly decrease of 2.5% in the postintervention period (P < .01). Our systemwide intervention reduced inpatient fluoroquinolone use immediately, but not HO-CDI. However, a downward trend in PD-CDI occurred. Relying on outcome measures limited to the inpatient setting may not reflect the full impact of inpatient stewardship efforts. The ASKAP Variables and Slow Transients (VAST) Pilot Survey Tara Murphy, David L. Kaplan, Adam J. Stewart, Andrew O'Brien, Emil Lenc, Sergio Pintaldi, Joshua Pritchard, Dougal Dobie, Archibald Fox, James K. Leung, Tao An, Martin E. Bell, Jess W. Broderick, Shami Chatterjee, Shi Dai, Daniele d'Antonio, Gerry Doyle, B. M. Gaensler, George Heald, Assaf Horesh, Megan L. Jones, David McConnell, Vanessa A. Moss, Wasim Raja, Gavin Ramsay, Stuart Ryder, Elaine M. Sadler, Gregory R. Sivakoff, Yuanming Wang, Ziteng Wang, Michael S. Wheatland, Matthew Whiting, James R. Allison, C. S. Anderson, Lewis Ball, K. Bannister, D. C.-J. Bock, R. Bolton, J. D. Bunton, R. Chekkala, A. P Chippendale, F. R. Cooray, N. Gupta, D. B. Hayman, K. Jeganathan, B. Koribalski, K. Lee-Waddell, Elizabeth K. Mahony, J. Marvil, N. M. McClure-Griffiths, P. Mirtschin, A. Ng, S. Pearce, C. Phillips, M. A. Voronkov Journal: Publications of the Astronomical Society of Australia / Volume 38 / 2021 Published online by Cambridge University Press: 12 October 2021, e054 The Variables and Slow Transients Survey (VAST) on the Australian Square Kilometre Array Pathfinder (ASKAP) is designed to detect highly variable and transient radio sources on timescales from 5 s to $\sim\!5$ yr. In this paper, we present the survey description, observation strategy and initial results from the VAST Phase I Pilot Survey. This pilot survey consists of $\sim\!162$ h of observations conducted at a central frequency of 888 MHz between 2019 August and 2020 August, with a typical rms sensitivity of $0.24\ \mathrm{mJy\ beam}^{-1}$ and angular resolution of $12-20$ arcseconds. There are 113 fields, each of which was observed for 12 min integration time, with between 5 and 13 repeats, with cadences between 1 day and 8 months. 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Fuller, Emily Lena Jones Journal: Antiquity , First View Published online by Cambridge University Press: 08 October 2021, pp. 1-18 Established chronologies indicate a long-term 'Hoabinhian' hunter-gatherer occupation of Mainland Southeast Asia during the Terminal Pleistocene to Mid-Holocene (45 000–3000 years ago). Here, the authors re-examine the 'Hoabinhian' sequence from north-west Thailand using new radiocarbon and luminescence data from Spirit Cave, Steep Cliff Cave and Banyan Valley Cave. The results indicate that hunter-gatherers exploited this ecologically diverse region throughout the Terminal Pleistocene and the Pleistocene–Holocene transition, and into the period during which agricultural lifeways emerged in the Holocene. Hunter-gatherers did not abandon this highland region of Thailand during periods of environmental and socioeconomic change. Characterisation of age and polarity at onset in bipolar disorder Janos L. Kalman, Loes M. Olde Loohuis, Annabel Vreeker, Andrew McQuillin, Eli A. Stahl, Douglas Ruderfer, Maria Grigoroiu-Serbanescu, Georgia Panagiotaropoulou, Stephan Ripke, Tim B. Bigdeli, Frederike Stein, Tina Meller, Susanne Meinert, Helena Pelin, Fabian Streit, Sergi Papiol, Mark J. Adams, Rolf Adolfsson, Kristina Adorjan, Ingrid Agartz, Sofie R. Aminoff, Heike Anderson-Schmidt, Ole A. Andreassen, Raffaella Ardau, Jean-Michel Aubry, Ceylan Balaban, Nicholas Bass, Bernhard T. Baune, Frank Bellivier, Antoni Benabarre, Susanne Bengesser, Wade H Berrettini, Marco P. Boks, Evelyn J. Bromet, Katharina Brosch, Monika Budde, William Byerley, Pablo Cervantes, Catina Chillotti, Sven Cichon, Scott R. Clark, Ashley L. Comes, Aiden Corvin, William Coryell, Nick Craddock, David W. Craig, Paul E. Croarkin, Cristiana Cruceanu, Piotr M. Czerski, Nina Dalkner, Udo Dannlowski, Franziska Degenhardt, Maria Del Zompo, J. Raymond DePaulo, Srdjan Djurovic, Howard J. Edenberg, Mariam Al Eissa, Torbjørn Elvsåshagen, Bruno Etain, Ayman H. Fanous, Frederike Fellendorf, Alessia Fiorentino, Andreas J. Forstner, Mark A. Frye, Janice M. Fullerton, Katrin Gade, Julie Garnham, Elliot Gershon, Michael Gill, Fernando S. Goes, Katherine Gordon-Smith, Paul Grof, Jose Guzman-Parra, Tim Hahn, Roland Hasler, Maria Heilbronner, Urs Heilbronner, Stephane Jamain, Esther Jimenez, Ian Jones, Lisa Jones, Lina Jonsson, Rene S. Kahn, John R. Kelsoe, James L. Kennedy, Tilo Kircher, George Kirov, Sarah Kittel-Schneider, Farah Klöhn-Saghatolislam, James A. Knowles, Thorsten M. Kranz, Trine Vik Lagerberg, Mikael Landen, William B. Lawson, Marion Leboyer, Qingqin S. Li, Mario Maj, Dolores Malaspina, Mirko Manchia, Fermin Mayoral, Susan L. McElroy, Melvin G. McInnis, Andrew M. McIntosh, Helena Medeiros, Ingrid Melle, Vihra Milanova, Philip B. Mitchell, Palmiero Monteleone, Alessio Maria Monteleone, Markus M. Nöthen, Tomas Novak, John I. Nurnberger, Niamh O'Brien, Kevin S. O'Connell, Claire O'Donovan, Michael C. O'Donovan, Nils Opel, Abigail Ortiz, Michael J. Owen, Erik Pålsson, Carlos Pato, Michele T. Pato, Joanna Pawlak, Julia-Katharina Pfarr, Claudia Pisanu, James B. Potash, Mark H Rapaport, Daniela Reich-Erkelenz, Andreas Reif, Eva Reininghaus, Jonathan Repple, Hélène Richard-Lepouriel, Marcella Rietschel, Kai Ringwald, Gloria Roberts, Guy Rouleau, Sabrina Schaupp, William A Scheftner, Simon Schmitt, Peter R. Schofield, K. Oliver Schubert, Eva C. Schulte, Barbara Schweizer, Fanny Senner, Giovanni Severino, Sally Sharp, Claire Slaney, Olav B. Smeland, Janet L. Sobell, Alessio Squassina, Pavla Stopkova, John Strauss, Alfonso Tortorella, Gustavo Turecki, Joanna Twarowska-Hauser, Marin Veldic, Eduard Vieta, John B. Vincent, Wei Xu, Clement C. Zai, Peter P. Zandi, Psychiatric Genomics Consortium (PGC) Bipolar Disorder Working Group, International Consortium on Lithium Genetics (ConLiGen), Colombia-US Cross Disorder Collaboration in Psychiatric Genetics, Arianna Di Florio, Jordan W. Smoller, Joanna M. Biernacka, Francis J. McMahon, Martin Alda, Bertram Müller-Myhsok, Nikolaos Koutsouleris, Peter Falkai, Nelson B. Freimer, Till F.M. Andlauer, Thomas G. Schulze, Roel A. Ophoff Journal: The British Journal of Psychiatry / Volume 219 / Issue 6 / December 2021 Published online by Cambridge University Press: 25 August 2021, pp. 659-669 Print publication: December 2021 Studying phenotypic and genetic characteristics of age at onset (AAO) and polarity at onset (PAO) in bipolar disorder can provide new insights into disease pathology and facilitate the development of screening tools. To examine the genetic architecture of AAO and PAO and their association with bipolar disorder disease characteristics. Genome-wide association studies (GWASs) and polygenic score (PGS) analyses of AAO (n = 12 977) and PAO (n = 6773) were conducted in patients with bipolar disorder from 34 cohorts and a replication sample (n = 2237). The association of onset with disease characteristics was investigated in two of these cohorts. Earlier AAO was associated with a higher probability of psychotic symptoms, suicidality, lower educational attainment, not living together and fewer episodes. Depressive onset correlated with suicidality and manic onset correlated with delusions and manic episodes. Systematic differences in AAO between cohorts and continents of origin were observed. This was also reflected in single-nucleotide variant-based heritability estimates, with higher heritabilities for stricter onset definitions. Increased PGS for autism spectrum disorder (β = −0.34 years, s.e. = 0.08), major depression (β = −0.34 years, s.e. = 0.08), schizophrenia (β = −0.39 years, s.e. = 0.08), and educational attainment (β = −0.31 years, s.e. = 0.08) were associated with an earlier AAO. The AAO GWAS identified one significant locus, but this finding did not replicate. Neither GWAS nor PGS analyses yielded significant associations with PAO. AAO and PAO are associated with indicators of bipolar disorder severity. Individuals with an earlier onset show an increased polygenic liability for a broad spectrum of psychiatric traits. Systematic differences in AAO across cohorts, continents and phenotype definitions introduce significant heterogeneity, affecting analyses. Reductions in Postdischarge Clostridioides difficile Infection after an Inpatient Health System Fluoroquinolone Stewardship K. Ashley Jones, Zanthia Wiley, Julianne Kubes, Mary Elizabeth Sexton, Benjamin Albrecht, Jesse Jacob, Jessica Howard-Anderson, Scott Fridkin, Udodirim Onwubiko Journal: Antimicrobial Stewardship & Healthcare Epidemiology / Volume 1 / Issue S1 / July 2021 Published online by Cambridge University Press: 29 July 2021, p. s4 Background: Effective inpatient stewardship initiatives can improve antibiotic prescribing, but impact on outcomes like Clostridioides difficile infections (CDIs) is less apparent. However, the effect of inpatient stewardship efforts may extend to the postdischarge setting. We evaluated whether an intervention targeting inpatient fluoroquinolone (FQ) use in a large healthcare system reduced incidence of postdischarge CDI. Methods: In August 2019, 4 acute-care hospitals in a large healthcare system replaced standalone FQ orders with order sets containing decision support. Order sets redirected prescribers to syndrome order sets that prioritize alternative antibiotics. Monthly patient days (PDs) and antibiotic days of therapy (DOT) administered for FQs and NHSN-defined broad-spectrum hospital-onset (BS-HO) antibiotics were calculated using patient encounter data for the 23 months before and 13 months after the intervention (COVID-19 admissions in the previous 7 months). We evaluated hospital-onset CDI (HO-CDI) per 1,000 PD (defined as any positive test after hospital day 3) and 12-week postdischarge (PDC- CDI) per 100 discharges (any positive test within healthcare system <12 weeks after discharge). Interrupted time-series analysis using generalized estimating equation models with negative binomial link function was conducted; a sensitivity analysis with Medicare case-mix index (CMI) adjustment was also performed to control for differences after start of the COVID-19 pandemic. Results: Among 163,117 admissions, there were 683 HO-CDIs and 1,009 PDC-CDIs. Overall, FQ DOT per 1,000 PD decreased by 21% immediately after the intervention (level change; P < .05) and decreased at a consistent rate throughout the entire study period (−2% per month; P < .01) (Fig. 1). There was a nonsignificant 5% increase in BS-HO antibiotic use immediately after intervention and a continued increase in use after the intervention (0.3% per month; P = .37). HO-CDI rates were stable throughout the study period, with a nonsignificant level change decrease of 10% after the intervention. In contrast, there was a reversal in the trend in PDC-CDI rates from a 0.4% per month increase in the preintervention period to a 3% per month decrease in the postintervention period (P < .01). Sensitivity analysis with adjustment for facility-specific CMI produced similar results but with wider confidence intervals, as did an analysis with a distinct COVID-19 time point. Conclusion: Our systemwide intervention using order sets with decision support reduced inpatient FQ use by 21%. The intervention did not significantly reduce HO-CDI but significantly decreased the incidence of CDI within 12 weeks after discharge. Relying on outcome measures limited to inpatient setting may not reflect the full impact of inpatient stewardship efforts and incorporating postdischarge outcomes, such as CDI, should increasingly be considered. Funding: No Disclosures: None Infection Prevention Considerations for a Multi-Mission Convention Center Field Hospital in Baltimore, Maryland, During the COVID-19 Pandemic Jennifer A. Jones, Zishan K. Siddiqui, Charles Callahan, Surbhi Leekha, Sharon Smyth, Michael Anne Preas, James R. Ficke, Marie Kristine F. Cabunoc, Melinda E. Kantsiper, the CONQUER COVID Consortium Journal: Disaster Medicine and Public Health Preparedness , First View Published online by Cambridge University Press: 18 June 2021, pp. 1-8 The state of Maryland identified its first case of coronavirus disease 2019 (COVID-19) on March 5, 2020. The Baltimore Convention Center (BCCFH) quickly became a selected location to set up a 250-bed inpatient field hospital and alternate care site. In contrast to other field hospitals throughout the United States, the BCCFH remained open throughout the pandemic and took on additional COVID-19 missions, including community severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) diagnostic testing, monoclonal antibody infusions for COVID-19 outpatients, and community COVID-19 vaccinations. To prevent the spread of pathogens during operations, infection prevention and control guidelines were essential to ensure the safety of staff and patients. Through multi-agency collaboration, use of infection prevention best practices, and answering what we describe as PPE-ESP, an operational framework was established to reduce infection risks for those providing or receiving care at the BCCFH during the COVID-19 pandemic. Getting Time Right: Using Cox Models and Probabilities to Interpret Binary Panel Data Shawna K. Metzger, Benjamin T. Jones Journal: Political Analysis , First View Published online by Cambridge University Press: 14 June 2021, pp. 1-16 Logit and probit (L/P) models are a mainstay of binary time-series cross-sectional (BTSCS) analyses. Researchers include cubic splines or time polynomials to acknowledge the temporal element inherent in these data. However, L/P models cannot easily accommodate three other aspects of the data's temporality: whether covariate effects are conditional on time, whether the process of interest is causally complex, and whether our functional form assumption regarding time's effect is correct. Failing to account for any of these issues amounts to misspecification bias, threatening our inferences' validity. We argue scholars should consider using Cox duration models when analyzing BTSCS data, as they create fewer opportunities for such misspecification bias, while also having the ability to assess the same hypotheses as L/P. We use Monte Carlo simulations to bring new evidence to light showing Cox models perform just as well—and sometimes better—than logit models in a basic BTSCS setting, and perform considerably better in more complex BTSCS situations. In addition, we highlight a new interpretation technique for Cox models—transition probabilities—to make Cox model results more readily interpretable. We use an application from interstate conflict to demonstrate our points. Murchison Widefield Array rapid-response observations of the short GRB 180805A G. E. Anderson, P. J. Hancock, A. Rowlinson, M. Sokolowski, A. Williams, J. Tian, J. C. A. Miller-Jones, N. Hurley-Walker, K. W. Bannister, M. E. Bell, C. W. James, D. L. Kaplan, Tara Murphy, S. J. Tingay, B. W. Meyers, M. Johnston-Hollitt, R. B. Wayth Published online by Cambridge University Press: 10 June 2021, e026 Here we present stringent low-frequency (185 MHz) limits on coherent radio emission associated with a short-duration gamma-ray burst (SGRB). Our observations of the short gamma-ray burst (GRB) 180805A were taken with the upgraded Murchison Widefield Array (MWA) rapid-response system, which triggered within 20s of receiving the transient alert from the Swift Burst Alert Telescope, corresponding to 83.7 s post-burst. The SGRB was observed for a total of 30 min, resulting in a $3\sigma$ persistent flux density upper limit of 40.2 mJy beam–1. Transient searches were conducted at the Swift position of this GRB on 0.5 s, 5 s, 30 s and 2 min timescales, resulting in $3\sigma$ limits of 570–1 830, 270–630, 200–420, and 100–200 mJy beam–1, respectively. We also performed a dedispersion search for prompt signals at the position of the SGRB with a temporal and spectral resolution of 0.5 s and 1.28 MHz, respectively, resulting in a $6\sigma$ fluence upper-limit range from 570 Jy ms at DM $=3\,000$ pc cm–3 ( $z\sim 2.5$ ) to 1 750 Jy ms at DM $=200$ pc cm–3 ( $z\sim 0.1)$ , corresponding to the known redshift range of SGRBs. We compare the fluence prompt emission limit and the persistent upper limit to SGRB coherent emission models assuming the merger resulted in a stable magnetar remnant. Our observations were not sensitive enough to detect prompt emission associated with the alignment of magnetic fields of a binary neutron star just prior to the merger, from the interaction between the relativistic jet and the interstellar medium (ISM) or persistent pulsar-like emission from the spin-down of the magnetar. However, in the case of a more powerful SGRB (a gamma-ray fluence an order of magnitude higher than GRB 180805A and/or a brighter X-ray counterpart), our MWA observations may be sensitive enough to detect coherent radio emission from the jet-ISM interaction and/or the magnetar remnant. Finally, we demonstrate that of all current low- frequency radio telescopes, only the MWA has the sensitivity and response times capable of probing prompt emission models associated with the initial SGRB merger event. Nonventilator hospital-acquired pneumonia: A call to action: Recommendations from the National Organization to Prevent Hospital-Acquired Pneumonia (NOHAP) among nonventilated patients Shannon C. Munro, Dian Baker, Karen K. Giuliano, Sheila C. Sullivan, Judith Haber, Barbara E. Jones, Matthew B. Crist, Richard E. Nelson, Evan Carey, Olivia Lounsbury, Michelle Lucatorto, Ryan Miller, Brian Pauley, Michael Klompas Journal: Infection Control & Hospital Epidemiology / Volume 42 / Issue 8 / August 2021 Published online by Cambridge University Press: 09 June 2021, pp. 991-996 Print publication: August 2021 In 2020 a group of U.S. healthcare leaders formed the National Organization to Prevent Hospital-Acquired Pneumonia (NOHAP) to issue a call to action to address non–ventilator-associated hospital-acquired pneumonia (NVHAP). NVHAP is one of the most common and morbid healthcare-associated infections, but it is not tracked, reported, or actively prevented by most hospitals. This national call to action includes (1) launching a national healthcare conversation about NVHAP prevention; (2) adding NVHAP prevention measures to education for patients, healthcare professionals, and students; (3) challenging healthcare systems and insurers to implement and support NVHAP prevention; and (4) encouraging researchers to develop new strategies for NVHAP surveillance and prevention. The purpose of this document is to outline research needs to support the NVHAP call to action. Primary needs include the development of better models to estimate the economic cost of NVHAP, to elucidate the pathophysiology of NVHAP and identify the most promising pathways for prevention, to develop objective and efficient surveillance methods to track NVHAP, to rigorously test the impact of prevention strategies proposed to prevent NVHAP, and to identify the policy levers that will best engage hospitals in NVHAP surveillance and prevention. A joint task force developed this document including stakeholders from the Veterans' Health Administration (VHA), the U.S. Centers for Disease Control and Prevention (CDC), The Joint Commission, the American Dental Association, the Patient Safety Movement Foundation, Oral Health Nursing Education and Practice (OHNEP), Teaching Oral-Systemic Health (TOSH), industry partners and academia. HD 76920 b pinned down: A detailed analysis of the most eccentric planetary system around an evolved star C. Bergmann, M. I. Jones, J. Zhao, A. J. Mustill, R. Brahm, P. Torres, R. A. Wittenmyer, F. Gunn, K. R. Pollard, A. Zapata, L. Vanzi, Songhu Wang Published online by Cambridge University Press: 22 April 2021, e019 We present 63 new multi-site radial velocity (RV) measurements of the K1III giant HD 76920, which was recently reported to host the most eccentric planet known to orbit an evolved star. We focused our observational efforts on the time around the predicted periastron passage and achieved near-continuous phase coverage of the corresponding RV peak. By combining our RV measurements from four different instruments with previously published ones, we confirm the highly eccentric nature of the system and find an even higher eccentricity of $e=0.8782 \pm 0.0025$ , an orbital period of $415.891^{+0.043}_{-0.039}\,\textrm{d}$ , and a minimum mass of $3.13^{+0.41}_{-0.43}\,\textrm{M}_{\textrm{J}}$ for the planet. The uncertainties in the orbital elements are greatly reduced, especially for the period and eccentricity. We also performed a detailed spectroscopic analysis to derive atmospheric stellar parameters, and thus the fundamental stellar parameters ( $M_, R_, L_*$ ), taking into account the parallax from Gaia DR2, and independently determined the stellar mass and radius using asteroseismology. Intriguingly, at periastron, the planet comes to within 2.4 stellar radii of its host star's surface. However, we find that the planet is not currently experiencing any significant orbital decay and will not be engulfed by the stellar envelope for at least another 50–80 Myr. Finally, while we calculate a relatively high transit probability of 16%, we did not detect a transit in the TESS photometry. Hospital-acquired infections among adult patients admitted for coronavirus disease 2019 (COVID-19) Leigh Smith, Sara M. Karaba, Joe Amoah, George Jones, Robin K. Avery, Kathryn Dzintars, Taylor Helsel, Sara E. Cosgrove, Valeria Fabre Journal: Infection Control & Hospital Epidemiology , First View Published online by Cambridge University Press: 13 April 2021, pp. 1-4 In a multicenter cohort of 963 adults hospitalized due to coronavirus disease 2019 (COVID-19), 5% had a proven hospital-acquired infection (HAI) and 21% had a proven, probable, or possible HAI. Risk factors for proven or probable HAIs included intensive care unit admission, dexamethasone use, severe COVID-19, heart failure, and antibiotic exposure upon admission. Fossil bivalves and the sclerochronological reawakening David K. Moss, Linda C. Ivany, Douglas S. Jones Journal: Paleobiology / Volume 47 / Issue 4 / November 2021 Published online by Cambridge University Press: 13 April 2021, pp. 551-573 Print publication: November 2021 The field of sclerochronology has long been known to paleobiologists. Yet, despite the central role of growth rate, age, and body size in questions related to macroevolution and evolutionary ecology, these types of studies and the data they produce have received only episodic attention from paleobiologists since the field's inception in the 1960s. It is time to reconsider their potential. Not only can sclerochronological data help to address long-standing questions in paleobiology, but they can also bring to light new questions that would otherwise have been impossible to address. For example, growth rate and life-span data, the very data afforded by chronological growth increments, are essential to answer questions related not only to heterochrony and hence evolutionary mechanisms, but also to body size and organism energetics across the Phanerozoic. While numerous fossil organisms have accretionary skeletons, bivalves offer perhaps one of the most tangible and intriguing pathways forward, because they exhibit clear, typically annual, growth increments and they include some of the longest-lived, non-colonial animals on the planet. In addition to their longevity, modern bivalves also show a latitudinal gradient of increasing life span and decreasing growth rate with latitude that might be related to the latitudinal diversity gradient. Is this a recently developed phenomenon or has it characterized much of the group's history? When and how did extreme longevity evolve in the Bivalvia? What insights can the growth increments of fossil bivalves provide about hypotheses for energetics through time? In spite of the relative ease with which the tools of sclerochronology can be applied to these questions, paleobiologists have been slow to adopt sclerochronological approaches. Here, we lay out an argument and the methods for a path forward in paleobiology that uses sclerochronology to answer some of our most pressing questions. The Role of Disgust and Threat in Contamination-Related Obsessive–Compulsive Disorder Leanne Mulheron, Mairwen K. Jones Journal: Behaviour Change / Volume 38 / Issue 1 / April 2021 Published online by Cambridge University Press: 11 February 2021, pp. 40-59 Print publication: April 2021 Theoretical models suggest that the emotion disgust or threat overestimates are important in the aetiology and maintenance of contamination-based obsessive–compulsive disorder. In the current study, both threat and disgust were manipulated and 115 non-clinical participants (mean age 20.46 years, 94 females) were randomly allocated to one of four conditions: high-disgust/low-threat (n = 29), high-disgust/high-threat (n = 29), low-disgust/low-threat (n = 27), and low-disgust/high-threat (n = 30). Participants completed a hierarchical Behavioural Avoidance Task (BAT). Those in the high-threat and high-disgust conditions completed less BAT steps and showed more latency to begin each step than those in the low-threat and low-disgust conditions. A significant interaction effect was observed for the high-disgust/high-threat condition as significantly more task avoidance was found. However, handwashing duration was not significantly different between the high and low-disgust conditions or the high and low-threat conditions. The overall low mean washing duration of 30 s possibly due to the testing conditions and/or the ethnic heterogeneity of the sample may account for these results. There were also no significant differences in the level of anxiety for participants in the high-threat compared with the low-threat conditions. It is possible that anxiety remained relatively low across conditions as a result of the graduated BAT. Future research and theoretical and clinical implications are discussed. Anxious parents show higher physiological synchrony with their infants C. G. Smith, E. J. H. Jones, T. Charman, K. Clackson, F. U. Mirza, S. V. Wass Published online by Cambridge University Press: 10 February 2021, pp. 1-11 Interpersonal processes influence our physiological states and associated affect. Physiological arousal dysregulation, a core feature of anxiety disorders, has been identified in children of parents with elevated anxiety. However, little is understood about how parent–infant interpersonal regulatory processes differ when the dyad includes a more anxious parent. We investigated moment-to-moment fluctuations in arousal within parent-infant dyads using miniaturised microphones and autonomic monitors. We continually recorded arousal and vocalisations in infants and parents in naturalistic home settings across day-long data segments. Our results indicated that physiological synchrony across the day was stronger in dyads including more rather than less anxious mothers. Across the whole recording epoch, less anxious mothers showed responsivity that was limited to 'peak' moments in their child's arousal. In contrast, more anxious mothers showed greater reactivity to small-scale fluctuations. Less anxious mothers also showed behaviours akin to 'stress buffering' – downregulating their arousal when the overall arousal level of the dyad was high. These behaviours were absent in more anxious mothers. Our findings have implications for understanding the differential processes of physiological co-regulation in partnerships where a partner is anxious, and for the use of this understanding in informing intervention strategies for dyads needing support for elevated levels of anxiety. Scaling-up Health-Arts Programmes: the largest study in the world bringing arts-based mental health interventions into a national health service Carolina Estevao, Daisy Fancourt, Paola Dazzan, K. Ray Chaudhuri, Nick Sevdalis, Anthony Woods, Nikki Crane, Rebecca Bind, Kristi Sawyer, Lavinia Rebecchini, Katie Hazelgrove, Manonmani Manoharan, Alexandra Burton, Hannah Dye, Tim Osborn, Lucinda Jarrett, Nick Ward, Fiona Jones, Aleksandra Podlewska, Isabella Premoli, Fleur Derbyshire-Fox, Alison Hartley, Tayana Soukup, Rachel Davis, Ioannis Bakolis, Andy Healey, Carmine M. Pariante Journal: BJPsych Bulletin / Volume 45 / Issue 1 / February 2021 Published online by Cambridge University Press: 23 December 2020, pp. 32-39 Print publication: February 2021 The Scaling-up Health-Arts Programme: Implementation and Effectiveness Research (SHAPER) project is the world's largest hybrid study on the impact of the arts on mental health embedded into a national healthcare system. This programme, funded by the Wellcome Trust, aims to study the impact and the scalability of the arts as an intervention for mental health. The programme will be delivered by a team of clinicians, research scientists, charities, artists, patients and healthcare professionals in the UK's National Health Service (NHS) and the community, spanning academia, the NHS and the charity sector. SHAPER consists of three studies – Melodies for Mums, Dance for Parkinson's, and Stroke Odysseys – which will recruit over 800 participants, deliver the interventions and draw conclusions on their clinical impact, implementation effectiveness and cost-effectiveness. We hope that this work will inspire organisations and commissioners in the NHS and around the world to expand the remit of social prescribing to include evidence-based arts interventions. Risk factors for Toxoplasma gondii seropositivity in the Old Order Amish A. O. Markon, K. A. Ryan, A. Wadhawan, M. Pavlovich, M. W. Groer, C. Punzalan, K. Gensheimer, J. L. Jones, M. L. Daue, A. Dagdag, P. Donnelly, X. Peng, T. I. Pollin, B. D. Mitchell, T. T. Postolache Journal: Epidemiology & Infection / Volume 149 / 2021 Published online by Cambridge University Press: 25 November 2020, e89 Toxoplasma gondii (T. gondii) is an important human disease-causing parasite. In the USA, T. gondii infects >10% of the population, accrues economic losses of US$3.6 billion/year, and ranks as the second leading culprit of foodborne illness-related fatalities. We assessed toxoplasmosis risk among the Old Order Amish, a mostly homogenous population with a high prevalence of T. gondii seropositivity, using a questionnaire focusing on food consumption/preparation behaviours and environmental risk factors. Analyses were conducted using multiple logistic regression. Consuming raw meat, rare meat, or unpasteurised cow or goat milk products was associated with increased odds of seropositivity (unadjusted Odds Ratios: 2.192, 1.613, and 1.718 , respectively). In separate models by sex, consuming raw meat, or consuming unpasteurised cow or goat milk products, was associated with increased odds of seropositivity among women; washing hands after touching meat with decreased odds of seropositivity among women (adjusted OR (AOR): 0.462); and cleaning cat litterbox with increased odds of seropositivity among men (AOR: 5.241). This is the first study to assess associations between behavioural and environmental risk factors and T. gondii seropositivity in a US population with high seroprevalence for T. gondii. Our study emphasises the importance of proper food safety behaviours to avoid the risk of infection. Seroprevalence of pertussis in Madagascar and implications for vaccination Solohery L. Razafimahatratra, Amy Wesolowski, Lala Rafetrarivony, Jean-Michel Heraud, Forrest K. Jones, Simon Cauchemez, Richter Razafindratsimandresy, Sandratana J. Raharinantoanina, Aina Harimanana, Jean Marc Collard, C. J. E. Metcalf Published online by Cambridge University Press: 16 November 2020, e283 Pertussis is a highly contagious infectious disease and remains an important cause of mortality and morbidity worldwide. Over the last decade, vaccination has greatly reduced the burden of pertussis. Yet, uncertainty in individual vaccination coverage and ineffective case surveillance systems make it difficult to estimate burden and the related quantity of population-level susceptibility, which determines population risk. These issues are more pronounced in low-income settings where coverage is often overestimated, and case numbers are under-reported. Serological data provide a direct characterisation of the landscape of susceptibility to infection; and can be combined with vaccination coverage and basic theory to estimate rates of exposure to natural infection. Here, we analysed cross-sectional data on seropositivity against pertussis to identify spatial and age patterns of susceptibility in children in Madagascar. A large proportion of individuals surveyed were seronegative; however, there were patterns suggestive of natural infection in all the regions analysed. Improvements in vaccination coverage are needed to help prevent additional burden of pertussis in the country.	CommonCrawl
\begin{definition}[Definition:Countable Set/Countably Infinite/Cardinality] The cardinality of a countably infinite set is denoted by the symbol $\aleph_0$ ('''aleph null'''). \end{definition}	ProofWiki
\begin{document} \title{\textbf{Surfaces of revolution}\\ \textbf{satisfying }$\triangle^{III}\boldsymbol{x}=A\boldsymbol{x}$} \author{\textbf{Stylianos Stamatakis, Hassan Al-Zoubi }\\ \emph{Department of Mathematics, Aristotle University of Thessaloniki}\\ \emph{GR-54124 Thessaloniki, Greece}\\ \emph{e-mail: [email protected]}} \date{} \maketitle \begin{abstract} \noindent We consider surfaces of revolution in the three-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e., their position vector $\boldsymbol{x}$ satisfies the relation \textbf{ }$\triangle^{III}\boldsymbol{x}=A\boldsymbol{x} $,\textbf{ }where $A$ is a square matrix of order 3. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere. \noindent \textit{Key Words}: Surfaces in the Euclidean space, surfaces of coordinate finite type, Beltrami operator \noindent \textit{MSC 2010}: 53A05, 47A75 \end{abstract} \section{Introduction} \noindent Let $\boldsymbol{x}=\boldsymbol{x}(u^{1},u^{2})$ be a regular parametric representation of a surface $S$ in the Euclidean space $ \mathbb{R} ^{3}$ which does not contain parabolic points. For two sufficient differentiable functions $f(u^{1},u^{2})$ and $g(u^{1},u^{2})$ the first Beltrami operator with respect to the third fundamental form $III=e_{ij} du^{i}du^{j}$ of $S$ is defined by \[ \nabla^{III}(f,g)=e^{ij}f_{/i}g_{/j}, \] where $f_{/i}:=\frac{\partial f}{\partial u^{i}}$ and $e^{ij}$ denote the components of the inverse tensor of $e_{ij}$. The second Beltrami differential operator with respect to $III$ is defined by \footnote{with sign convention such that $\triangle=-\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2} }{\partial y^{2}}$ for the metric $ds^{2}=dx^{2}+dy^{2}$} \begin{equation} \triangle^{III}f=\frac{-1}{\sqrt{e}}\left( \sqrt{e}e^{ij}f_{/i}\right) _{/j}\label{1} \end{equation} ($e:=\det(e_{ij})).$ In \cite{Stamatakis} we showed the relation \begin{equation} \triangle^{III}\boldsymbol{x}=\nabla^{III}(\frac{2H}{K},\boldsymbol{n} )-\frac{2H}{K}\boldsymbol{n},\label{2} \end{equation} where $\boldsymbol{n}$ is the unit normal vectorfield, $H$ the mean curvature and $K$ the Gaussian curvature of $S$. Moreover we proved that a surface satisfying the condition \[ \triangle^{III}\boldsymbol{x}=\lambda \boldsymbol{x},\quad \lambda \in \mathbb{R} , \] i.e., a surface $S:\boldsymbol{x}=\boldsymbol{x}(u^{1},u^{2})$ \textit{for which all coordinate functions are eigenfunctions} of $\triangle^{III}$ \textit{with the same eigenvalue} $\lambda$, is part of a sphere ($\lambda=2$) or a minimal surface ($\lambda=0$). Using terms of B.-Y. Chen's theory of finite type surfaces \cite{Chen} the above result can be expressed as follows: \textit{A surface} $S$ \textit{in} $ \mathbb{R} ^{3}$ \textit{is of }$III$-\textit{type} $1$ (\textit{or of null} $III$-\textit{type} $1$) \textit{if and only if} $S$ \textit{is part of a sphere }(\textit{or a minimal surface})\textit{.} In general a surface $S$ is said to be \textit{of finite type }with respect to the fundamental form $III$ or, briefly, \textit{of finite }$III$ \textit{-type}, if the position vector $\boldsymbol{x}$ of $S$ can be written as a finite sum of nonconstant eigenvectors of the operator $\triangle^{III}$, that is if \begin{equation} \boldsymbol{x}=\boldsymbol{c}+\boldsymbol{x}_{1}+\boldsymbol{x}_{2} +\ldots+\boldsymbol{x}_{m},\quad \triangle^{III}\boldsymbol{x}_{i}=\lambda _{i}\boldsymbol{x}_{i},\quad i=1,\ldots,m, \label{3} \end{equation} where $\boldsymbol{c}$ is a constant vector and $\lambda_{1},\ldots ,\lambda_{m}$ are eigenvalues of $\triangle^{III}$. When there are exactly $k$ nonconstant eigenvectors $\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{k}$ appearing in (\ref{3}) which all belong to different eigenvalues $\lambda _{1},\ldots,\lambda_{k}$, then $S$ is said to be of $III$-\textit{type} $k$; when $\lambda_{i}=0$ for some $i=1,\ldots,k$, then $S$ is said to be of \textit{null }$III$-\textit{type} $k$. The only known surfaces of finite $III$-type are parts of spheres, the minimal surfaces and the parallel of the minimal surfaces (which are actually of null $III$-type $2$, see \cite{Stamatakis}). In this paper we want to determine the connected surfaces of revolution $S$ in $ \mathbb{R} ^{3}$ which are \textit{of coordinate finite }$III$-\textit{type}, i.e., their position vectorfield $\boldsymbol{x}(u^{1},u^{2})$ satisfies the condition \begin{equation} \triangle^{III}\boldsymbol{x}=A\boldsymbol{x},\quad A\in M(3,3),\label{4} \end{equation} where $M(m,n)$ denotes the set of all matrices of the type ($m,n$). Coordinate finite type surfaces with respect to the first fundamental form $I$ were studied in \cite{Dillen} and \cite{Garay}. In the last paper O. Garay showed that the only complete surfaces of revolution in $ \mathbb{R} ^{3}$, whose component functions are eigenfunctions of their Laplacian are the catenoids, the spheres and the circular cylinders, while F. Dillen, J. Pas and L. Verstraelen proved in \cite{Dillen} that the only surfaces in $ \mathbb{R} ^{3}$ satisfying \[ \triangle^{I}\boldsymbol{x}=A\boldsymbol{x}+B,\quad A\in M(3,3),\quad B\in M(3,1), \] are the minimal surfaces, the spheres and the circular cylinders. Our main result is the following \begin{proposition} A surface of revolution $S$ satisfies (\ref{4}) if and only if $S$ is a catenoid or part of a sphere. \end{proposition} We first show that the mentioned surfaces indeed satisfy the condition (\ref{4}). A. On a catenoid the mean curvature vanishes, so, by virtue of (\ref{2}), $\triangle^{III}\boldsymbol{x}=0$. Therefore a catenoid satisfies (\ref{4}), where $A$ is the null matrix in $M(3,3)$. B. Let $S$ be part of a sphere of radius $r$ centered at the origin. Then \[ H=\frac{1}{r},\quad K=\frac{1}{r^{2}},\quad \boldsymbol{n}=-\frac{1} {r}\boldsymbol{x}. \] So, by (\ref{2}), it is $\triangle^{III}\boldsymbol{x}=2\boldsymbol{x}$. Therefore $S$ satisfies (\ref{4}) whith $A=2I_{3},$ where $I_{3}$ is the identity matrix in $M(3,3)$. \section{Proof of the main theorem} \noindent Let $C$ be the profile curve of a surface of revolution $S$ of the differentiation class $C^{3}$. We suppose that (a) $C$ lies on the $(x_{1},x_{3})$-plane, (b) the axis of revolution of $S$ is the $x_{3}$-axis and (c) $C$ is parametrized by its arclength $s$. Then $C$ admits the parametric representation \[ \boldsymbol{r}(s)=(f(s),0,g(s)),\quad s\in J \] ($J\subset \mathbb{R} $ open interval), where $f(s),g(s)\in C^{3}(J)$. The position vector of $S$ is given by \[ \boldsymbol{x}(s,\theta)=(f(s)\cos \theta,f(s)\sin \theta,g(s)),\quad s\in J,\quad \theta \in \lbrack0,2\pi). \] Putting $f(s) \acute{} :=\frac{df(s)}{ds}$ we have because of (c) \begin{equation} f~ \acute{} ~^{2}+g~ \acute{} ~^{2}=1\quad \forall s\in J.\label{5} \end{equation} Furthermore it is $f~ \acute{} \cdot g~ \acute{} \neq0,$ because otherwise $f=const.$ or $g=const.$ and $S$ would be a circular cylinder or part of a plane, respectively. Hence $S$ would consist only of parabolic points, which has been excluded. In view of (\ref{5}) we can put \begin{equation} f~ \acute{} =\cos \varphi,\quad g~ \acute{} =\sin \varphi,\label{6} \end{equation} where $\varphi$ is a function of $s$. Then the unit normal vector of $S$ is given by \[ \boldsymbol{n}=(-\sin \varphi \cos \theta,~-\sin \varphi \sin \theta,~\cos \varphi). \] The components $h_{ij}$ and $e_{ij}$ of the the second and the third fundamental tensors in (local) coordinates are the following \[ h_{11}=\varphi \acute{} ,\quad h_{12}=0,\quad h_{22}=f\sin \varphi, \] \begin{equation} e_{11}=\varphi \acute{} ~^{2},\quad e_{12}=0,\quad e_{22}=\sin^{2}\varphi,\label{7} \end{equation} hence \cite{Huck} \begin{equation} \frac{2H}{K}=h_{ij}e^{ij}=\frac{1}{\varphi \acute{} }+\frac{f}{\sin \varphi}.\label{8} \end{equation} From (\ref{1}) and (\ref{7}) we find for a sufficient differentiable function $u=u(s,\theta)$ defined on $J\times \lbrack2\pi,0)$ \begin{equation} \triangle^{III}u=-\frac{u\text{ \'{} \'{} }}{\varphi \acute{} ^{~2}}+\left( \frac{\varphi \text{ \'{} \'{} }}{\varphi \acute{} ^{~2}}-\frac{\cos \varphi}{\sin \varphi}\right) \frac{u \acute{} }{\varphi \acute{} }-\frac{u_{/\theta \theta}}{\sin^{2}\varphi}.\label{9} \end{equation} Consindering the following functions of $s$ \begin{equation} P_{1}=R\sin \varphi-\frac{\cos \varphi}{\varphi \acute{} }R~ \acute{} ,\quad P_{2}=-R\cos \varphi-\frac{\sin \varphi}{\varphi \acute{} }R~ \acute{} ,\label{11} \end{equation} where we have put for simplicity $R:=\frac{2H}{K}$, and applying (\ref{9}) on the coordinate functions $x_{i}$, $i=1,2,3,$ of the position vector $\boldsymbol{x}$ we find \begin{equation} \triangle^{III}x_{1}=P_{1}\cos \theta,\quad \triangle^{III}x_{2}=P_{1}\sin \theta,\quad \triangle^{III}x_{3}=P_{2}.\label{10} \end{equation} So we have: (a) \textit{The coordinate functions }$x_{1},x_{2}$ \textit{are both eigenfunctions of} $\triangle^{III}$ \textit{belonging to the same eigenvalue if and only if for some real constant} $\lambda$ \textit{holds} \[ \lambda f=R\sin \varphi-\frac{\cos \varphi}{\varphi \acute{} }R~ \acute{} . \] (b) \textit{The coordinate function }$x_{3}$ \textit{is an eigenfunction of} $\triangle^{III}$ \textit{if and only if for some real constant} $\mu$ \textit{holds} \[ \mu g=-R\cos \varphi-\frac{\sin \varphi}{\varphi \acute{} }R~ \acute{} . \] We denote by $a_{ij},i,j=1,2,3,$ the entries of the matrix $A.$ By using (\ref{10}) condition (\ref{4}) is found to be equivalent to the following system \begin{equation} \left \{ \begin{array} [c]{c} P_{1}\cos \theta=a_{11}f\cos \theta+a_{12}f\sin \theta+a_{13}~g\\ P_{1}\sin \theta=a_{21}f\cos \theta+a_{22}f\sin \theta+a_{23}~g\\ \quad \quad P_{2}=a_{31}f\cos \theta+a_{32}f\sin \theta+a_{33}~g \end{array} \right. .\label{12} \end{equation} Since $\sin \theta,\cos \theta$ and 1 are linearly independent functions of $\theta,$ we obtain from (\ref{12}$_{3}$) $a_{31}=a_{32}=0.$ On differentiating (\ref{12}$_{1}$) and (\ref{12}$_{2}$) twice with respect to $\theta$ we have \[ \left \{ \begin{array} [c]{c} P_{1}\cos \theta=a_{11}f\cos \theta+a_{12}f\sin \theta \\ P_{1}\sin \theta=a_{21}f\cos \theta+a_{22}f\sin \theta \end{array} \right. . \] Thus $a_{13}g=a_{23}g=0,$ so that $a_{13}$ and $a_{23}$ vanish. The system (\ref{12}) is equivalent to the following \[ \left \{ \begin{array} [c]{c} \left( P_{1}-a_{11}f\right) \cos \theta-a_{12}f\sin \theta=0\\ (P_{1}-a_{22}f)\sin \theta-a_{21}f\cos \theta=0\\ \quad \quad \quad \quad \quad \quad \quad \quad P_{2}-a_{33}g=0 \end{array} \right. . \] But $\sin \theta$ and $\cos \theta$ are linearly independent functions of $\theta$, so we finally obtain $a_{12}=a_{21}=0,a_{11}=a_{22}$ and $P_{1}=a_{11}f.$ Putting $a_{11}=a_{22}=\lambda$ and $a_{33}=\mu$ we see that the system (\ref{12}) reduces now to the following equations \begin{equation} P_{1}=\lambda f,\quad P_{2}=\mu g.\label{13} \end{equation} On account of (\ref{11}) and (\ref{13}) we are left with the system \begin{equation} \left \{ \begin{array} [c]{c} R=\lambda f\sin \varphi-\mu g\cos \varphi \quad \quad \\ R~ \acute{} =-\varphi~ \acute{} (\lambda f\cos \varphi+\mu g\sin \varphi) \end{array} \right. .\label{14} \end{equation} On differentiating (\ref{14}$_{1}$) with respect to $s$ we find, by virtue of (\ref{6}), \begin{equation} R~ \acute{} =\frac{\lambda-\mu}{2}\sin \varphi \cos \varphi.\label{15} \end{equation} We distinguish the following cases: \textit{Case I.} Let $\lambda=\mu$. Then (\ref{15}) reduces to $R~ \acute{} =0.$ \textit{Subcase Ia}. Let $\lambda=\mu=0$. From (\ref{14}$_{1}$) we obtain $R=0$, i.e., $H=0$. Consequently $S$, being a minimal surface of revolution, is a catenoid. \textit{Subcase Ib}. Let $\lambda=\mu \neq0$. Then from (\ref{6}), (\ref{14}$_{2}$) and $R~ \acute{} =0$ we have $f\cdot f~ \acute{} +g\cdot g~ \acute{} =0,$ i.e., $\left( f^{2}+g^{2}\right) \acute{} =0.$ Therefore $f^{2}+g^{2}=const.$ and $S$ is obviously part of a sphere. \textit{Case II.} Let $\lambda \neq \mu$. From (\ref{14}$_{2}$), (\ref{15}) we find firstly \begin{equation} \frac{1}{\varphi \acute{} }=\frac{2\left( \lambda f\cos \varphi+\mu g\sin \varphi \right) }{\left( \mu-\lambda \right) \sin \varphi \cos \varphi}. \label{16} \end{equation} From this and (\ref{8}) we obtain \[ R=\frac{\lambda+\mu}{\left( \mu-\lambda \right) \sin \varphi}f+\frac{2\mu }{\left( \mu-\lambda \right) \cos \varphi}g. \] Hence, by virtue of (\ref{14}$_{1}$), \begin{equation} af+bg=0, \label{17} \end{equation} where \begin{equation} a=\lambda \sin \varphi+\frac{\lambda+\mu}{\left( \lambda-\mu \right) \sin \varphi},\quad b=\frac{2\mu}{\left( \lambda-\mu \right) \cos \varphi} -\mu \cos \varphi. \label{18} \end{equation} We note that $\mu \neq0$, since for $\mu=0$ we have \[ a=\frac{\lambda \sin^{2}\varphi+1}{\sin \varphi},\quad b=0, \] and relation (\ref{17}) becomes \[ \frac{\lambda \sin^{2}\varphi+1}{\sin \varphi}f=0, \] whence it follows $\lambda \sin^{2}\varphi+1=0,$ a contradiction. On differentiating (\ref{17}) with respect to $s$ and taking into account (\ref{16}) we obtain \begin{equation} a_{1}\frac{f}{\sin \varphi}+b_{1}\frac{g}{\cos \varphi}=0, \label{19} \end{equation} where \begin{equation} a_{1}=\lambda(\lambda-\mu)^{2}\sin^{4}\varphi+(\lambda-\mu)(\lambda \mu -\lambda^{2}+3\lambda+\mu)\sin^{2}\varphi-(\lambda+\mu)(3\lambda-\mu), \label{20} \end{equation} \begin{equation} b_{1}=\mu \left[ \left( \lambda-\mu \right) ^{2}\sin^{4}\varphi+\left( \lambda-\mu \right) \left( \mu-\lambda+4\right) \sin^{2}\varphi-2\left( \lambda+\mu \right) \right] . \label{21} \end{equation} By eliminating now the functions $f$ and $g$ from (\ref{17}) and (\ref{19}) and taking into account (\ref{18}), (\ref{20}) and (\ref{21}) we find \[ \lambda(\lambda-\mu)^{2}\sin^{4}\varphi+(\lambda-\mu)(\lambda \mu-\lambda ^{2}+5\lambda+\mu-2)\sin^{2}\varphi+(\lambda+\mu)(\mu-3\lambda+4)=0. \] Consequently \[ \lambda \left( \lambda-\mu \right) ^{2}=0,~\left( \lambda-\mu \right) \left( \lambda \mu-\lambda^{2}+5\lambda+\mu-2\right) =0,~\left( \lambda+\mu \right) \left( \mu-3\lambda+4\right) =0. \] From the first equation we have $\lambda=0$. Then, the other two become as follows \[ \mu-2=0,\quad \mu+4=0, \] which is a contradiction. So the proof of the theorem is completed. \end{document}	arXiv
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 16$ and $CP = 8.$ If $\tan \angle APD = 3,$ then find $AB.$ Let $Q$ be the projection of $P$ onto $\overline{AD},$ and let $x = AB = PQ.$ [asy] unitsize(1.5 cm); pair A, B, C, D, P, Q; A = (0,0); B = (0,2); C = (3,2); D = (3,0); P = (2,2); Q = (2,0); draw(A--B--C--D--cycle); draw(A--P--D); draw(P--Q); label("$A$", A, SW); label("$B$", B, NW); label("$C$", C, NE); label("$D$", D, SE); label("$P$", P, N); label("$Q$", Q, S); label("$16$", (B + P)/2, N); label("$8$", (C + P)/2, N); label("$16$", (A + Q)/2, S); label("$8$", (D + Q)/2, S); label("$x$", (A + B)/2, W); label("$x$", (P + Q)/2, W); [/asy] Then from right triangle $APQ,$ \[\tan \angle APQ = \frac{16}{x}.\]From right triangle $DPQ,$ \[\tan \angle DPQ = \frac{8}{x}.\]Then \begin{align} \tan \angle APD &= \tan (\angle APQ + \angle DPQ) \\ &= \frac{\tan \angle APQ + \tan \angle DPQ}{1 - \tan \angle APQ \cdot \tan \angle DPQ} \\ &= \frac{\frac{16}{x} + \frac{8}{x}}{1 - \frac{16}{x} \cdot \frac{8}{x}} \\ &= \frac{24x}{x^2 - 128} = 3. \end{align}Hence, $x^2 - 128 = 8x,$ or $x^2 - 8x - 128 = 0.$ This factors as $(x - 16)(x + 8) = 0,$ so $x = \boxed{16}.$	Math Dataset
Mathematical "urban legends" Locked. This question and its answers are locked because the question is off-topic but has historical significance. It is not currently accepting new answers or interactions. When I was a young and impressionable graduate student at Princeton, we scared each other with the story of a Final Public Oral, where Jack Milnor was dragged in against his will to sit on a committee, and noted that the class of topological spaces discussed by the speaker consisted of finite spaces. I had assumed this was an "urban legend", but then at a cocktail party, I mentioned this to a faculty member, who turned crimson and said that this was one of his students, who never talked to him, and then had to write another thesis (in numerical analysis, which was not very highly regarded at Princeton at the time). But now, I have talked to a couple of topologists who should have been there at the time of the event, and they told me that this was an urban legend at their time as well, so maybe the faculty member was pulling my leg. So, the questions are: (a) any direct evidence for or against this particular disaster? (b) what stories kept you awake at night as a graduate student, and is there any evidence for or against their truth? EDIT (this is unrelated, but I don't want to answer my own question too many times): At Princeton, there was supposedly an FPO in Physics, on some sort of statistical mechanics, and the constant $k$ appeared many times. The student was asked: Examiner: What is $k?$ Student: Boltzmann's constant. Examiner: Yes, but what is the value? Student: Gee, I don't know... Examiner: OK, order of magnitude? Student: Umm, don't know, I just know $k\dots$ The student was failed, since he was obviously not a physicist. ho.history-overview share\|cite $\begingroup$ Since every finite CW complex is weakly homotopically equivalent to a finite topological space, that does not sound so bad... :) $\endgroup$ – Mariano Suárez-Álvarez Jan 24 '11 at 20:54 $\begingroup$ Perhaps not an urban legend per se, but when I was learning algebra, my professor, in an attempt to impress upon us the necessity of checking that certain maps are well-defined, told us the story of a classmate of his who got several years into his Ph.D. thesis before realizing that the maps he was investigating weren't well defined. Horrified, we asked him if this was true. "No" he said, "but that's one lie you'll never forget!" $\endgroup$ – Nick Salter Jan 24 '11 at 21:04 $\begingroup$ Mathematical urban legends have been collected by Steven Krantz in the book, Mathematical Apochrypha (and I think there's a second volume). A few refer to the thesis defense. $\endgroup$ – Gerry Myerson Jan 24 '11 at 23:18 $\begingroup$ Though this question and its answers are very entertaining, I think it is a little unfair to close other questions as "offtopic" which are even closer to mathematical research as this one ... $\endgroup$ – Martin Brandenburg Jan 25 '11 at 8:54 $\begingroup$ I have to agree with Martin. This is a very entertaining thread but it seems quite outside the mandate of MO. $\endgroup$ – Ryan Budney Jan 26 '11 at 16:30 comments disabled on deleted / locked posts / reviews \| show 16 more comments As an undergraduate at Yale in the '70s I heard a variation on the basic legend, which I'll spell out a little since it has a slightly different moral from any others above. Student goes to advisor saying I'd like to do a thesis generalizing the results in article X. Advisor (and I think I heard it with Milnor as the advisor) says, "I don't recommend that because I don't think that's a very good article." Student persists, writes thesis, states theorem at the defense and at that point the advisor rises to say "consider the following counterexample..." I also heard a variation on "functions which turn out to be constant" legend. But the version I heard has the thesis getting accepted, the vacuity of it contents going unnoticed for several years until an undergraduate supplies a one-line proof. John Myhill told me about junior faculty at the University of Chicago about to grade qualifying exams in their legendarily ruthless way. André Weil pops his head in the door and says "Pass them all, they're no worse than you are." David Feldman $\begingroup$ That Weil anecdote is new to me, and highly amusing (not to mention salutary) $\endgroup$ – Yemon Choi Jan 25 '11 at 1:39 $\begingroup$ @Yemon Graduate student that I was, the Myhill's story painted Weil for me as a hero. Myhill, a very seasoned faculty member by then thought Weil came off as a monster. A question of perspective I suppose. Myhill did say that that year all the students passed. $\endgroup$ – David Feldman Jan 25 '11 at 2:28 $\begingroup$ "Experience confirms that severity towards others and self-indulgence are one and the same vice" - La Bruyere (trans. Choi, probably badly) $\endgroup$ – Yemon Choi Jan 25 '11 at 5:29 $\begingroup$ @Yemon Choi : translation is ok : for completeness the original is : "L'expérience confirme que la mollesse ou l'indulgence pour soi, Et la dureté pour les autres n'est qu'un seul et même vice." Citation de Jean de La Bruyère ; Les Caractères, Du cœur - 1688. $\endgroup$ – Jérôme JEAN-CHARLES Feb 3 '11 at 0:22 $\begingroup$ @Feldman: The story would certainly paint Weil for me as a hero if he had said, "Pass them all, they're no worse than I am." $\endgroup$ – Timothy Chow Apr 15 '11 at 1:37 As A.N.Whitehead, of PM fame, was still lecturing on mathematics at Cambrdge, he later became a philosopher in America, he arrived somewhat early in the lecture room one day. To fill in the time he started working on a problem from his research on the blackboard. As the students arrived he was still absorbed in his work so they sat down and waited for him to start the lecture. At the end of the allotted time he was still working on his problem and so the students got up and left. Somewhat later he finished his work, packed up his things and went home. Arriving home he said to his wife, "You know a rather strange thing happened at the university today, nobody came to my lecture." Thony C $\begingroup$ I wondered about that some time ago, too, but was relieved of the tension here: mathoverflow.net/questions/23113/… $\endgroup$ – Unknown Feb 4 '11 at 16:50 $\begingroup$ I think the comment by Unknown actually belongs under maproom's answer. $\endgroup$ – Todd Trimble♦ Jan 14 '18 at 15:55 I have no details to provide, but it is said that Ofer Gabber has derailed more than one talk at IHES after the speaker presents a definition by asking, "But what about the empty set?" Deane Yang $\begingroup$ He has actually derailed many Sem. Bourbaki by asking a stream of questions in English (the official language was, and presumably still is, French), until the speaker would start speaking in English. $\endgroup$ – Igor Rivin Jan 25 '11 at 3:57 $\begingroup$ I don't think the Séminaire Bourbaki has an official language. Like any lecture in France (or, I guess, in a French-speaking country), it's just convenient to do it in French unless the speaker isn't francophone. $\endgroup$ – Maxime Bourrigan Jan 25 '11 at 22:26 $\begingroup$ I recall a seminaire bourbaki in which Gabber persistently questioned Deligne in English, who answered just as persistently in French. $\endgroup$ – roy smith Apr 13 '11 at 17:35 One time Henri Berestycki was riding the Paris subway on the way to work and doing some calculations. All of a sudden, an elderly lady sitting across from him said: "Why don't you multiply by alpha and integrate by parts?" This did not solve his problem, but it was a reasonable thing to do. It turned out the old lady had once worked with Lebesgue. She remembered J.L. Lions as a "clever lad." I heard this story from my advisor Klaus Kirchgaessner who had heard it from Berestycki himself. Michael Renardy $\begingroup$ In the same spirit, I was sitting in a train, doing some mathematics. The person in front of me interupted me: your formula for the derivative of a product is false. Of course, he couldn't know that such a strange animal as a convolution product existed... $\endgroup$ – Denis Serre Jun 7 '11 at 5:50 I have heard (from two sources) that at the University of Chicago a senior faculty member was temporarily banned from teaching undergraduate courses. The reason is that during a first semester undergraduate linear algebra course he did everything over the Quaternions. This one isn't so much academically scary, but my advisor told me that it was always interesting riding to conferences with the above professor because he would refuse to defrost the windshield so that he could draw diagrams on it and do math while he was driving. Sean Tilson $\begingroup$ Actually, that is not that bad of an idea! I have seen the face of my students when I tell them «you should go through your linear algebra notes to see how much of it carries over to the case of skew-fields» right before proceeding to pick a basis for an $\mathbb H$-module, say... $\endgroup$ – Mariano Suárez-Álvarez Jan 25 '11 at 5:57 $\begingroup$ As an undergraduate I heard a secondhand story about a knot theorist teaching an introductory calculus class. The first question on the final was basic calculus; the rest involved knot theory. $\endgroup$ – Steve Huntsman Jan 25 '11 at 7:01 $\begingroup$ In Germany many professors would be happy to get banned from teaching undergraduate courses (and behave accordingly). There used to be payment by number of students some years ago, but now it has been levelled, and teaching undergraduate courses has become nothing more than a chore people want to get rid of. $\endgroup$ – darij grinberg Jan 25 '11 at 8:24 $\begingroup$ This story sounds strange to me because (at least the past few years, when I was there) the University of Chicago, SFAIK, doesn't have a straight-up linear algebra class for math majors. The easier stuff you're basically expected to just up, the harder stuff gets stuffed into the general "algebra" sequence. $\endgroup$ – Harry Altman Jan 25 '11 at 10:09 $\begingroup$ Concerning quaternions, there is also a story, which has happend in Cambridge as my brother told me: A professor asks in a lecture: "Is here somebody who does not know everything about quaternions?" A single student raises slowly her hand. "What?? Then learn it until tomorrow!" - it goes without saying that there were students in the class who did not raise their hand and did not even know what quaternions are... $\endgroup$ – Lennart Meier Feb 4 '11 at 20:48 I've heard the following story (I don't know if it is true). A math professor gave his PhD student this journal paper, and asked him what consequences he could derive from it. The student started proving more and more interesting results based on this paper, until finally he proved a result that the professor knew was false. This led them to look more closely at the original journal paper, and upon close inspection, they discovered that it was wrong, rendering all the research the student had done so far worthless. $\begingroup$ What is the worst known example of such a chain? Such as a 10 year old theory which based on completely false research? $\endgroup$ – Martin Brandenburg Jan 26 '11 at 22:26 $\begingroup$ I've actually seen something like this happen. $\endgroup$ – H A Helfgott Jan 29 '11 at 22:17 $\begingroup$ That is a decisive argument for the motto "Give examples!". Examples are mental crutches, guides and railings too (as in this case) . - $\endgroup$ – Jérôme JEAN-CHARLES Feb 3 '11 at 0:38 This story, according to the person I heard it from, happened some time in the 80s. It was about 10 years after Deligne's Hodge theory came out, but before Saito. It was not very clear how to define the mixed Hodge structure in non-constant cohomology. However, many people were convinced that such a thing existed (as turned out to be the case) and a number of competing proposals circulated. One such proposal was presented in a seminar talk where it was claimed that something was the "right Hodge filtration". At this moment Ofer Gabber (someone known, among other things, for giving hard time to speakers) intervened saying "What do you mean, the right Hodge filtration? What's the left Hodge filtration?" algori $\begingroup$ Of course, if the talk were given in French, such a problem wouldn't have happened. (See Igor Rivin's comment below: mathoverflow.net/questions/53122/mathematical-urban-legends/… ) $\endgroup$ – Willie Wong Jan 25 '11 at 20:53 $\begingroup$ Ofer is a living legend, and stories about him as a graduate student at Harvard and permanent member of IHES abound. He is brilliant but demands a level of logical rigor and precision that even other mathematicians have difficulty providing. My understanding is that his name should be on many important papers, but he demanded that his name be removed because he was not comfortable with every detail stated in each paper. $\endgroup$ – Deane Yang Jan 25 '11 at 21:13 $\begingroup$ I'd love to hear more of his stories! $\endgroup$ – Martin Brandenburg Jan 26 '11 at 22:33 $\begingroup$ My undergraduate career overlapped with Gabber's graduate student career. (He was a few years younger.) Once I had the satisfaction of offering a neat proof of some statement that came up in a differential geometry course we were both attending. My pleasure was not really dimmed by Ofer's comment that my proof did not work in characteristic $p$. $\endgroup$ – Tom Goodwillie Jan 29 '11 at 3:56 $\begingroup$ @Jean-Charles: much more. $\endgroup$ – mmm Feb 4 '11 at 17:58 When Peter Lax went to receive the national medal of science, he was asked by the other recipients about his merits. His answer was (apocryph) I integrated by parts. Denis Serre I heard the following story told about R. L. Moore. It seems he was teaching a class in which several of the students were obnoxious and unruly. So one day he walked into the lecture hall, opened his briefcase, took out a pistol, set it on the table in front of him, and then began to lecture as usual. He had no further trouble with the rowdy students. I have no particular reason to believe this is true, but it makes a good story. I think I have seen other references to firearms in the math department at the University of Texas, though. $\begingroup$ But I thought Moore didn't lecture! $\endgroup$ – JSE Jan 25 '11 at 6:13 $\begingroup$ As in all the best urban legends, it does not really matter if it's true or not: it does sound like something Moore would do. $\endgroup$ – Thierry Zell Jan 25 '11 at 16:23 $\begingroup$ I think Moore gave lectures in some of his classes at some points of his career. $\endgroup$ – Pete L. Clark Jan 26 '11 at 8:35 $\begingroup$ A friend of the family claims that in first grade, his teacher had a glass eye. The students didn't know until he had to leave the room to go to the bathroom. As he got up to go, he took out his eye and placed it on the table saying "Be good while I am gone - I'm watching you". These are 6 year olds... $\endgroup$ – Steven Gubkin Jan 29 '11 at 1:47 $\begingroup$ What Pete said is correct. Mary Ellen Rudin recounts in her interview for More Mathematical People that Moore gave lectures for calculus classes, and Halmos in his automathography recalls being permitted to sit in on one of Moore's calculus lectures. I think nevertheless that he would send students to the board in such classes. $\endgroup$ – Todd Trimble♦ May 23 '11 at 20:53 From the article "A credo of sorts" by Vaughan Jones, in the book "Truth in Mathematics": Once, at a seminar, one of the world's best low-dimensional topologists was presenting a major result. At a certain point another distinguished topologist in the audience intervened to say he did not understand how the speaker did a certain thing. The speaker gave an anguished look and gazed at the ceiling for at least a minute. The member of the audience then affirmed "Oh yes, I hadn't thought of that!" Visibly relieved, the speaker went on with his talk, glad to have communicated this point to the audience. $\begingroup$ We had a lecturer in Bielefeld whose proofs were sometimes a bit terse. From time to time he had spend a minute or a few after writing down the proof at the blackboard before he remembered (silently) the point and drew the square. $\endgroup$ – Lennart Meier Nov 19 '12 at 15:59 There is this story set at Harvard. During the Vietnam War there was a student strike. One math professor goes to his graduate course and finds the room empty. But he delivers his lecture anyway as usual. When he gets back to his office and tells someone about it, they ask him why he did that. He replies, "So I'll know where to start next time." Gerald Edgar $\begingroup$ Isn't this one already in the list? $\endgroup$ – Mariano Suárez-Álvarez May 24 '11 at 1:54 $\begingroup$ an unfortunate feature of "big list" questions that revive after many months $\endgroup$ – Gerald Edgar May 24 '11 at 12:54 I heard this story a couple of years back (not sure though if it is true): A young Japanese mathematician was giving a talk based on his results at Courant Institute. His work was built on the work of S.R.S Varadhan. But apparently during the talk Varadhan had his eyes closed and the speaker mistook it for him sleeping. He made a joke by saying somthing like "hopefully not everybody is sleeping". A few minutes later Varadhan open his eyes and said "consider this counterexample". But Varadhan liked the speaker's idea and invited him to spent some time at Courant institute. The correct result is now known as 'Speaker'-Varadhan theorem. 2 revisions, 2 users J Verma 55% $\begingroup$ I suppose you mean S.R.S.Varadhan? $\endgroup$ – fherzig Jan 25 '11 at 19:38 $\begingroup$ Haha. Last August at the ICM, when he was chairing a session, even I thought he was sleeping. :) $\endgroup$ – Koundinya Vajjha Mar 8 '11 at 16:29 A Japanese professor writes a letter to his American colleague, asking to send a preprint. The letter (very long and polite) is finished with the sentence: "Please forgive me my shameless desire." Anton Petrunin Here is a story I heard when I was student. Professor S.'s student had finished his dissertation to everybody's satisfaction. All that was pending was his advisor's signature. S. agreed to sign on one (half joking?) condition: The student had to defeat S. in a jalapeño-eating contest. For some reason the student agreed. (Hopefully this is not just a plot device. If the story is true, I would like one day to ask the student what he was thinking.) They went to S.'s favorite Thai restaurant. He explained to the staff the contest. They set up a table for them, and brought them jalapeños, they would eat them, new (hotter) ones would be brought, etc. The whole staff was watching and having a great time. The poor student, of course, was suffering, really worried that perhaps S. was serious, and he would never get his degree, since it soon became clear S. was going to defeat the student without difficulties. S. would grab the jalapeños and eat them while explaining where they were from and what the ideal way to prepare them was. At some point, a drop of sweat from S.'s brow was threatening to fall into his eye, and without realizing what he was doing, S. passed his finger through his eye to remove the sweat. Apparently the pain was agonizing, and the student got his dissertation signed. $\begingroup$ @Andres: I like this as a story, but not as an answer to this question. Most of these stories are interesting because they tell us something that we recognize (or enunciate things we fear) about the math profession. But where's the math in your story? Are mathematicians notorious jalapeno poppers? [Sorry to be a stick in the mud.] $\endgroup$ – Pete L. Clark Jan 26 '11 at 8:33 $\begingroup$ The fear isn't mathematical but academic: to get one's thesis approved there are dozens of (to us inconsequential) hurdles, including margins, fonts, citation styles, microfiche fees and, for one unlucky acolyte, jalapenos. $\endgroup$ – Kevin O'Bryant Jan 28 '11 at 14:25 $\begingroup$ @Kevin: I have my neuroses (and so do they!), but I confess that fear of jalapenos does not hit very close to home. $\endgroup$ – Pete L. Clark Apr 17 '11 at 8:42 There's a bar in Bonn, which has the name 'Blow up' and closes only very late at night. At some occasion, an algebraic geometer A was in this bar well beyond midnight and was getting quite drunk. After some time, he decided it would be a very good idea to explain to some person B in the bar he only met this night what a blow up is in mathematics. And so he starts to explain until B interrupts him: "Hey, I know all this stuff. I've done my diploma thesis in Estonia in complex geometry." Since I know A (although I heard the story fromy someone else), I suppose this has happened essentially this way. Lennart Meier In the early eighties, fleeing from Romania, C. Foias got a professorship position in Orsay. He gave a graduate course on 'Contractions et dilatations' (Contractions and dilations). Someone handwrote on the annoucement 'Is this a course on Obstetrics ?'. The wikipedia entry for Borel summation narrates the following recollection by Mark Kac, about an encounter between Emile Borel and Mittag-Leffler. This is one of my favourites. "Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'." Francesco Sica Heard from Carsten Thomassen: He was giving a lecture on matchings in graph theory, and presented a game where two players would alternately pick some edge in a graph, and at the end one person would win (i do not remember the exact rules of the game). Then Carsten asked the students, which player would win this game. A student raised his hand and replied "You will". utdiscant I've heard the following story (don't know if it was true, or who was supposedly involved): As is well-known, at a certain big-name university the advisor defends the student's thesis. A student worked with a certain big-shot for five years and produced what many looked at as a fine dissertation. The day of the defence came. The advisor got up to the board, gave a quick introduction, and embarked on stating the main theorem in the dissertation. Half way through writing it, he put down the chalk, and paced around a bit. He then turned, apologized, and said, "I'm sorry, but I think I've found a counterexample." Willie Wong $\begingroup$ If we're thinking of the same big-name university, then I heard that the impetus for the whole "advisor defends the student's thesis" system arose from an advisor making a quip along the lines of "this is the best thesis I ever wrote" in a defense prior to this system. It's not clear why this would have been an effective solution to this sort of problem, so this is probably pure apocrypha. Amusing nonetheless! $\endgroup$ – Ramsey Jan 24 '11 at 22:39 $\begingroup$ I have heard the same story as Ramsey, about the same university -- but the quip is supposed to be "OK, so it's not the best thesis I ever wrote!" $\endgroup$ – JSE Jan 24 '11 at 22:57 $\begingroup$ When I was in grad school, a senior mathematician told me: "I never minded writing a student's dissertation, but I draw the line at having to explain it to him or her". $\endgroup$ – Thierry Zell Jan 25 '11 at 0:08 $\begingroup$ There's an old joke-definition of a dissertation, something like, "a research paper written by a senior academic under the most trying circumstances." $\endgroup$ – Gerry Myerson Jan 25 '11 at 2:27 $\begingroup$ This all somehow reminds me of a quote (supposedly from Haydn) "Don't make a sour face when listening to opening notes of a sonata written by some grand duke: you never know who actually composed it". $\endgroup$ – fedja Jan 25 '11 at 4:01 Apparently a postdoc at IHES cornered Dennis Sullivan back in the eighties, and asked him a long and involved question concerning the stuff the postdoc was studying. Dennis' response was: That's a good question! I think you should work on it! Igor Rivin $\begingroup$ I'm surprised there aren't more Dennis Sullivan stories here. He's definitely one of the more colorful mathematicians of our time. $\endgroup$ – Deane Yang Apr 23 '11 at 15:46 When I took analysis from Paul Sally, he claimed that a student once asked him in class, "Professor Sally, why is it called the p-adic norm? If it's a norm, what does it measure?" Without thinking, Paul loudly replied, "Well, it measures the p-ness of a number." I suspect that he just substituted himself into an existing urban legend, yet I would not be surprised if it were true. Andrew Dudzik $\begingroup$ The related story that I've heard (from people who were there, I believe) is that in the early 1970s in the Ohio State summer math program for high school kids an elderly female European giving a lecture about finite groups once innocently said, in coming to a key step in a proof: "But we still haven't used the $p$-ness of the group." $\endgroup$ – Tom Goodwillie May 24 '11 at 21:45 I have a story of this kind. My thesis advisor J.-M. Souriau used to talk this story about one of his close friend (I'll keep quiet the name) : "Avant de devenir directeur de l'école normale supérieure (Ulm) il a passé la moitié de sa vie mathématique à définir le nombre de [[put his name here]] et l'autre moité à démontrer qu'il était égal à 1." I don't know if it is true, I doubt but not that much :-) Patrick I-Z $\begingroup$ Is this poor soul's number a candidate for mathoverflow.net/questions/32967/… ? $\endgroup$ – Willie Wong Jan 25 '11 at 0:39 $\begingroup$ OK... So how many mathematicians have been directors of "L'Ecole" recently?... The identity of the poor soul should be easy t work out. $\endgroup$ – Thierry Zell May 7 '11 at 13:56 $\begingroup$ Well, I guess there are two candidates, actually. One more than I expected... $\endgroup$ – Thierry Zell May 7 '11 at 13:58 Here's another story not particularly relevant to the original question: When I was a graduate student at Harvard, there was a much older Greek graduate student (whose name I forget) who was viewed by at least some of my classmates as being one of if not the smartest graduate students there. I was told that he was responsible for providing the critical ideas for least two classmates' Ph.D. theses. But he never completed a thesis himself and, as I recall, found a good career working for the European Community in Brussels. This may be an urban legend, but it's true as far as I know. During R. L. Moore's reign at University of Texas, sometimes a grad student would be awarded a PhD for work that was original for the student even if it had been done before. Moore insisted that students reproduce everything from scratch (though guided with Socratic questions). This produced outstanding students, at first. But it got to be a tragedy by the time Moore was put out to pasture. The gap between what students graduated knowing and the vanguard of research became insurmountable. This was before my time, but I did speak to someone who said that he recused himself from a PhD committee shortly after coming to UT because he could not sign off on a dissertation whose results he knew were not original. John D. Cook Ed Dean linked to this story in a comment, but I think it is too nice to stay hidden there: On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be Phillip Forman. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion. (cited from wikipedia) EDIT: Thanks to Gerald Edgar (and Google) you can find the answer to what the loophole in the US Constitution is here. 3 revs $\begingroup$ I am extremely curious as to what this inconsistency is. $\endgroup$ – Qiaochu Yuan Apr 24 '11 at 7:42 $\begingroup$ I am as well. . $\endgroup$ – Kerry May 1 '11 at 7:21 $\begingroup$ Perhaps these guys are unable to use Google for some reason... blog.plover.com/law/Godel-dictatorship-2.html $\endgroup$ – Gerald Edgar May 23 '11 at 21:16 One urban legend I remember was of a student who just wanted to schedule a language exam, but the professor opened a text to the introduction and asked him to translate it. The student asked to switch to the mathematics, saying, "I don't know any verbs!" Douglas Zare $\begingroup$ That is not an urban legend, well, at least not the first half. My (French) language examiner at Princeton was disappointed that I brought a mathematics textbook (in French) for the exam. After looking around and couldn't find a French-language roman handy in his office, he begrudgingly passed me after I translated the Preface and Acknowledgements (in addition to several mathematics-laden passages from the middle of the book). $\endgroup$ – Willie Wong Jan 24 '11 at 22:05 $\begingroup$ I once took a math course for which the textbook was in French although the course was otherwise taught in English. You don't need to know much French to understand that sort of thing. $\endgroup$ – Michael Hardy Jan 25 '11 at 2:54 $\begingroup$ I heard from an older student in college that Nick Katz taught a course on local class field theory using Serre's Corps Locaux and that student didn't realize the book had been translated into English until after the course was over. $\endgroup$ – KConrad Jan 25 '11 at 6:13 $\begingroup$ @Michael: Some fellow student of mine went to a course (in the US) where all literature was Chinese, and the lecturer being surprised about him showing up had a hard time giving the lecture in English. My friend didn't stay long in that course... $\endgroup$ – Someone Jan 25 '11 at 9:09 $\begingroup$ @Someone: I actually dragged a Serbian friend of mine to seminar just for that. Otherwise they'd look at me and give the discussion in Chinese, and while it is my native language, it is by far not my native mathematical language. $\endgroup$ – Willie Wong Jan 25 '11 at 12:49 Bob Stong told me that a Ph.D. candidate once presented his thesis in topology without any examples. One of the committee asked for any space for which the work was true. The student said that he had yet to think of one. He was failed in short order. I seem to remember that the story was from University of Chicago, but I could be wrong. Whether Professor Stong was pulling my leg or not is not known. $\begingroup$ I was actually present for something like this. This was in Oregon, maybe 15 years ago. The speaker lectured for 75 minutes on the cohomoolyg of a certain class of spaces. I was a beginning graduate student at the time, so I didn't really understand the talk, but, apparently, the class of spaces had very unusual properties. Someone asked - as in your story - at the end of the talk if the speaker could give an accessible example of this class. The speaker said, unabashed, that he couldn't. In fact, he strongly suspected that there were \textit{no} examples at all! $\endgroup$ – John Iskra Jan 28 '11 at 17:31 $\begingroup$ I don't think this is so unreasonable. I am sure many people try to characterize, eg, $\{x \| \zeta(x)=0, \Re(x) \neq \frac12\}.$ $\endgroup$ – Igor Rivin Jan 28 '11 at 23:04 $\begingroup$ I heard once that Cauchy wrote a paper about "Bounded Entire Functions". From what I know, he later proved the Liouville Theorem. $\endgroup$ – Nick S Feb 6 '11 at 17:45 $\begingroup$ The paper at annals.math.princeton.edu/2010/171-1/p10 which is based on a PhD thesis, makes Igor's point even better than his example does. $\endgroup$ – Dan Fox Apr 13 '11 at 16:50 $\begingroup$ [Igor probably meant to include a hypothesis such as $x \notin \bf Z$ :-)] $\endgroup$ – Noam D. Elkies Jun 5 '11 at 23:35 I heard this in Oxford in 1970. I can't believe it: A PhD student decides to see what happens if he assumes the inverse of the triangle inequality. He finds he can prove that there are various interesting consequences - for instance, certain sets of points must be collinear. He eventually writes it all up as his thesis. His examiner starts with the question, "are you aware that such a space can only contain one point?" maproom $\begingroup$ Perhaps this is another variant of mathoverflow.net/questions/53122/mathematical-urban-legends/… ? $\endgroup$ – Willie Wong Feb 3 '11 at 15:42 $\begingroup$ This was a question on one of our Analysis II problem sheets at Cambridge. $\endgroup$ – Zhen Lin Feb 3 '11 at 16:38 Lacking sufficient reputation on this site to comment on posts, I'm going to make this an answer, and you know it's a true urban legend if you read the posts above. Once, at a Princeton physics exam, a group of the senior Princeton physics faculty were trying to figure out why, when you shake a bunch of rods in a container with certain asymmetries in the geometry of the container, the rods assume a "more ordered" state (they tended to concentrate on one side), and they couldn't figure out why this did not contradict the second law of thermodynamics! In an amazing twist, they speculated that the result had to do with finite size effects..... (for those who aren't in on why this is just too crazy to believe: the shaken container is not a closed system, so the second law doesn't apply. Further, the forces on the rods, a combination of shaking and frictional forces, do not correspond to thermal noise and dissipation, so there is no reason for the system to go to thermal equilibrium. It's like asking why, when there is a baited mousetrap and a live mouse in the room at time t=0, is it the case that after a certain amount of time the entropy decreases in that the mouse is more likely to be in the mousetrap than not. I think the most clever answer for the student is: "I notice that this exam has gone on for 30 minutes already and you are still walking and talking. Why are you not relaxing to thermal equilibrium? Perhaps the food you ate this morning is helping keep you out of equilibrium?") $\begingroup$ I was one of those confused professors (although not senior) and you may well be right that finite size effects have nothing to do with it, but I don't find your analogy terribly convincing either. Do you think the behavior would not have been observed if the system was just coupled to a heat bath rather than shaken? I'd really enjoy hearing a more detailed analysis of what is going on. $\endgroup$ – Jeff Harvey Jan 25 '11 at 20:33 $\begingroup$ If the system was in thermal equilibrium with a heat bath, then indeed this behavior would not happen. Two caveats: first (a technical point): of course, in true thermal equilibrium the rods would combust with the oxygen in the air, etc... but at intermediate time scales we can ignore that and consider an ensemble of rods being equally likely to be distributed anywhere. Second, even in thermal equilibrium there can be some interesting entropic effects near a boundary; basically, yes, there can be some finite size effects due to, say, more available orientations near one side of the boundary. $\endgroup$ – user12494 Jan 25 '11 at 21:11 $\begingroup$ it is a condensed matter analogue of the "anything not forbidden is compulsory" in particle physics. Consider a particle at point x in a potential U(x) with some non-thermal noise and a damping (so force=U'-eta v+noise, where eta is friction and noise has non-thermal spectrum). Suppose the potential has a sawtooth shape. This shape destroys reflection symmetry. The non-thermal noise then means detailed balance is broken, and so nothing forbids a current. If you pick a generic potential and non-thermal noise an dissipation and run it on a computer, odds are you will see the current. $\endgroup$ – user12494 Jan 25 '11 at 21:17 $\begingroup$ Thanks for the explanation. I talked to my local condensed matter guru and he also emphasized the role of dissipation, although in this system it is not clear without more analysis whether dissipation in collisions between the rods or in the rod-boundary collisions is more important in driving the system. Amusingly you can replace the rods by spheres and make the barrier symmetric and you will still get an accumulation on one side, but which side it is will be random. Dissipation from collisions increases at higher density and cool the system, so a fluctuation towards higher density grows. $\endgroup$ – Jeff Harvey Jan 25 '11 at 21:35 $\begingroup$ Yes, Chicago has some real experts on this kind of thing! My favorite absolutely bizarre thing I learned about in Chicago has to do with the "brazil nut" effect, that if you shake a jar of mixed nuts, the brazil nuts (the bigger ones) tend to wind up on top. Well, I'd heard about that effect before visiting Chicago, and you can try to puzzle out what happens, why exactly the Brazil nuts wind up on top. So, the crazy thing I learned is that if you repeat the experiment in a vacuum (or maybe it was just in a different liquid, I forget the exact details), the effect is reversed! $\endgroup$ – user12494 Jan 25 '11 at 21:42 Professor A at Harvard told the following story, supposedly a first hand account of his student days at Chicago, though it never struck me as remotely plausible. (I think he just told it so that he would seem like a teddy bear in comparison.) But I wonder if anyone else has heard variants of this. At the beginning of a course, Professor X would start asking some reasonable questions, the answers to which students taking the course could be expected to already know. Finally, he would ask one unfortunate student a question which no one taking the course would be able to answer. Upon the student's failure to answer correctly, Professor X wouldn't explain that the student's ignorance was justified, instead letting this event undermine the student's confidence about taking the course. Professor X would continue to single out this poor target for humiliation until he or she finally dropped the course. Professor A claimed to believe that Professor X's motivation was to have lit a fire under the remaining students and make them band together. Again, it's a rather unlikely tale of abuse. Though perhaps with a plant "student" playing along as the victim it could be an effective ploy ... Ed Dean $\begingroup$ A version of this happened at my undergraduate institution, but with a plant student. At the very first lecture, the professor sent the plant student to the blackboard and assigned a difficult exercise. The plant student does a nice job (one much better than any other students could do presumably), but the professor gives him a really hard time, harping on any flaw and concluding that this was "barely the level required to survive this lecture". The next day, six students had given up the course, so that the professor had to make sure words reached them that it was all a joke. $\endgroup$ – Olivier Jan 25 '11 at 10:00 $\begingroup$ Example of a plant student: youtube.com/watch?v=hut3VRL5XRE $\endgroup$ – timur Apr 24 '11 at 0:25 Not the answer you're looking for? Browse other questions tagged ho.history-overview or ask your own question. Most memorable titles Elementary+Short+Useful Do mathematical objects disappear? Powers of finite simple groups What is known about the common knowledge of mathematicians outside their field? Mathematical Tables in Babbage's Library The story about Milnor proving the Fáry-Milnor theorem How did "Ore's Conjecture" become a conjecture? historical antecedents of mathematical talks History of Mathematical Notation Has anything (other than what is in the obituary witten by M. Noether) survived of Paul Gordan's defense of infinitesimals? History of the classification of mathematical subjects Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs?	CommonCrawl
Benedict Gross Benedict Hyman Gross is an American mathematician who is a professor at the University of California San Diego,[1] the George Vasmer Leverett Professor of Mathematics Emeritus at Harvard University, and former Dean of Harvard College.[2] Benedict Gross NationalityAmerican Alma materHarvard University Oxford University Known forGross–Zagier theorem Gan–Gross–Prasad conjecture AwardsCole Prize (1987) Scientific career FieldsMathematics InstitutionsHarvard University UC San Diego Doctoral advisorJohn Tate Doctoral students • Henri Darmon • Noam Elkies • Jessica Fintzen • Susan Goldstine • Rhonda Hatcher • Dipendra Prasad • Wee Teck Gan • Douglas Ulmer He is known for his work in number theory, particularly the Gross–Zagier theorem on L-functions of elliptic curves, which he researched with Don Zagier. Education and Professional career Gross graduated from The Pingry School, a leading independent school in New Jersey, in 1967 as the valedictorian. In 1971, he graduated Phi Beta Kappa from Harvard University. He then received an M.Sc. from Oxford University as a Marshall Scholar in 1974 before returning to Harvard and completing his Ph.D. in 1978, under John Tate.[2][3] After holding faculty positions at Princeton University and Brown University, Gross became a tenured professor at Harvard in 1985[2] and remained there subsequently, as Dean of Harvard College from 2003 to 2007.[4] Benedict Gross was the mathematical consultant for the 1980 film It's My Turn containing the famous scene[5] in which actress Jill Clayburgh, portraying a mathematics professor, impeccably proves the snake lemma.[6] Awards and honors Gross is a 1986 MacArthur Fellow. Gross, Zagier, and Dorian M. Goldfeld won the Cole Prize of the American Mathematical Society in 1987 for their work on the Gross–Zagier theorem.[7] In 2012 he became a fellow of the American Mathematical Society.[8] Gross was elected as a fellow of the American Academy of Arts and Sciences in 1992[9] and as a member of the National Academy of Sciences in 2004.[10] He was elected to the American Philosophical Society in 2017.[11] He was named as a Harvard University Professor from 2011 to 2016 for his distinguished scholarship and professional work. Major publications • Gross, Benedict H.; Harris, Joe. Real algebraic curves. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 157–182. • Gross, Benedict H. Heights and the special values of L-series. Number theory (Montreal, Que., 1985), 115–187, CMS Conf. Proc., 7, Amer. Math. Soc., Providence, RI, 1987. • Gross, Benedict H. A tameness criterion for Galois representations associated to modular forms (mod p). Duke Math. J. 61 (1990), no. 2, 445–517. • Gross, Benedict H.; Prasad, Dipendra. On the decomposition of a representation of SOn when restricted to SOn−1. Canad. J. Math. 44 (1992), no. 5, 974–1002. • Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, 225–320. • Gross, B.; Kohnen, W.; Zagier, D. Heegner points and derivatives of L-series. II. Math. Ann. 278 (1987), no. 1-4, 497–562. • Gan, Wee Teck; Gross, Benedict H.; Prasad, Dipendra. Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I. Astérisque No. 346 (2012), 1–109. ISBN 978-2-85629-348-5 See also • Fat Chance: Probability from 0 to 1 • Gross–Koblitz formula References 1. Eisner <[email protected]>, Daryl. "UCSD Math \| Profile for Benedict Gross". UCSD Math \| Profile for Benedict Gross. 2. Curriculum vitae from Gross' web site at Harvard, retrieved 2010-04-21. 3. Benedict Gross at the Mathematics Genealogy Project 4. Gross Officially Named Dean of the College, Harvard Crimson, April 8, 2003; Gross Stretches to Prepare for New Roles, Harvard Crimson, May 16, 2003; With Goals Accomplished, Gross Leaves Overhauled College: His efforts were to 'improve the undergraduate experience,' dean says, Harvard Crimson, June 29, 2007; Exit Gross, Harvard Crimson, September 21, 2007. 5. "It's My Turn (1980) Snake Lemma". YouTube. Archived from the original on 2021-12-22. 6. "Benedict Gross – Miscellaneous Crew". IMDb.com. 7. Frank Nelson Cole Prize in Number Theory, AMS, retrieved 2010-04-21. 8. List of Fellows of the American Mathematical Society, retrieved 2013-01-19. 9. List of Active Members by Classes Archived 2005-05-06 at the Wayback Machine, American Academy of Arts and Sciences, retrieved 2010-04-21. 10. National Academies news: 72 new members chosen by academy Archived 2016-03-03 at the Wayback Machine, The National Academies, April 2004, retrieved 2010-04-21. 11. "American Philosophical Society: Newly Elected – April 2017". Archived from the original on 2017-09-15. Retrieved 2017-06-13. External links • Benedict Gross's Harvard University homepage • "Benedict Gross "Complex Multiplication: Past, Present, Future" Lecture 1". YouTube. January 30, 2019. Archived from the original on 2021-12-22. • "Benedict Gross "Complex Multiplication: Past, Present, Future" Lecture 2". YouTube. January 31, 2019. Archived from the original on 2021-12-22. • "Benedict Gross "Complex Multiplication: Past, Present, Future" Lecture 3". YouTube. February 4, 2019. Archived from the original on 2021-12-22. Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Japan • Czech Republic • Korea • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
# Isomorphism and its importance in graph theory Graph isomorphism is a fundamental problem in graph theory. It is the problem of determining whether two given graphs are isomorphic, i.e., whether there exists a one-to-one correspondence between the vertices of one graph and the vertices of the other, such that the adjacency relation is preserved. Graph isomorphism is important in various applications, including chemistry, physics, and computer science. The importance of graph isomorphism lies in its connection to the classification of finite simple groups. The isomorphism problem for graphs is known to be NP-complete, meaning that it is computationally intensive to solve. However, there are algorithms that can solve the isomorphism problem for certain classes of graphs, such as the isomorphism problem for strongly regular graphs. Consider two graphs $G_1$ and $G_2$ with the following adjacency matrices: $$G_1 = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$ $$G_2 = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$ Both graphs have the same adjacency matrix, so they are isomorphic. ## Exercise Determine whether the following two graphs are isomorphic: $$G_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$ $$G_2 = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$ # Algorithms for detecting isomorphism There are several algorithms for detecting graph isomorphism. Some of the most notable ones are the following: - The VF2 algorithm, which is a subgraph isomorphism algorithm that can be used to solve the graph isomorphism problem. It works by systematically generating all possible mappings between the vertices of the two graphs and checking if the mapping preserves the adjacency relation. - The canonical labeling algorithm, which is an algorithm for finding a canonical labeling of a graph, i.e., a labeling of the vertices such that the isomorphism relation is preserved. The algorithm works by assigning a label to each vertex based on its degree and the degrees of its neighbors. - The Nauty software package, which is a collection of programs for graph isomorphism and related problems. It includes algorithms for generating all automorphisms of a graph, computing the canonical labeling of a graph, and determining the isomorphism type of a graph. ## Exercise Using the VF2 algorithm, determine if the following two graphs are isomorphic: $$G_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$ $$G_2 = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$ # Complexity of isomorphism detection The complexity of detecting graph isomorphism is a significant challenge in graph theory. The isomorphism problem for graphs is known to be NP-complete, meaning that it is computationally intensive to solve. However, there are algorithms that can solve the isomorphism problem for certain classes of graphs, such as the isomorphism problem for strongly regular graphs. The complexity of the isomorphism problem can be measured in terms of the time and space complexity of the algorithms used to solve it. The time complexity refers to the amount of time it takes to solve the problem, while the space complexity refers to the amount of memory required to solve the problem. ## Exercise What is the time complexity of the VF2 algorithm for detecting graph isomorphism? # Introduction to Cytoscape Cytoscape is an open-source software platform for visualizing complex networks, including graphs and gene networks. It provides a user-friendly interface for creating, analyzing, and sharing network visualizations. Cytoscape offers a wide range of visualization techniques, including force-directed layouts, circular layouts, hierarchical layouts, and manual layouts. It also supports various types of networks, such as protein-protein interaction networks, metabolic networks, and social networks. ## Exercise Download and install Cytoscape from the official website (https://cytoscape.org/). Open Cytoscape and create a simple network with three nodes and two edges. # Setting up Cytoscape for graph visualization To visualize a graph in Cytoscape, you need to import the graph data into Cytoscape. The graph data can be in various formats, such as SIF, SBGN-ML, GraphML, or XGMML. Once the graph data is imported, you can apply various visualization techniques in Cytoscape to explore the structure and properties of the graph. These techniques include changing the node and edge colors, adjusting the node and edge sizes, and using different layout algorithms. ## Exercise Import a graph in GraphML format into Cytoscape. Apply a force-directed layout to the graph and change the node colors based on a specific attribute. # Visualization techniques in Cytoscape Cytoscape offers a wide range of visualization techniques for graphs. Some of the most common techniques include: - Force-directed layouts: These layouts use physical forces to position the nodes in the network. They are particularly useful for visualizing complex networks with many nodes and edges. - Circular layouts: These layouts arrange the nodes in a circular or elliptical pattern. They are useful for visualizing small networks with a limited number of nodes. - Hierarchical layouts: These layouts arrange the nodes in a hierarchical structure, with the nodes at the highest level representing the most abstract concepts, and the nodes at the lowest level representing the most detailed concepts. - Manual layouts: These layouts allow the user to manually position the nodes in the network. They are useful for creating custom network visualizations. ## Exercise Apply a hierarchical layout to a graph in Cytoscape. Use the manual layout tool to adjust the positions of the nodes in the network. # Applications of graph isomorphism visualization Graph isomorphism visualization has numerous applications in various fields, including: - Chemistry: Graph isomorphism visualization can be used to study the molecular structures of chemical compounds and their interactions. - Physics: Graph isomorphism visualization can be used to study the topological properties of quantum systems and their connections to the geometry of spacetime. - Computer science: Graph isomorphism visualization can be used to study the structure and properties of computer networks, such as the Internet and social networks. ## Exercise Visualize a graph representing a molecular structure in Cytoscape. Use the force-directed layout to explore the structure of the molecule. # Case studies in research and industry Graph isomorphism visualization has been used in various case studies in research and industry. Some of these case studies include: - The study of the molecular structures of pharmaceutical compounds, which have led to the development of new drugs. - The analysis of the structure and properties of protein-protein interaction networks, which has provided insights into the mechanisms of disease. - The study of the structure and properties of social networks, which has led to the development of new algorithms for network analysis and modeling. ## Exercise Visualize a graph representing a protein-protein interaction network in Cytoscape. Use the force-directed layout to explore the structure of the network. # Future developments in graph isomorphism visualization The future of graph isomorphism visualization holds great potential for advancements in various fields. Some of the potential future developments include: - The development of more efficient algorithms for detecting graph isomorphism. - The integration of advanced visualization techniques, such as 3D visualization and virtual reality, to provide more immersive and interactive network visualizations. - The application of machine learning and artificial intelligence techniques to automate the process of graph isomorphism visualization and analysis. ## Exercise Discuss the potential future developments in graph isomorphism visualization and their potential applications in research and industry. In conclusion, graph isomorphism visualization is a powerful tool for studying the structure and properties of graphs in various fields, including chemistry, physics, and computer science. As technology advances, we can expect further developments in graph isomorphism visualization, leading to even more sophisticated and interactive network visualizations.	Textbooks
Inclusion–exclusion principle In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as $\|A\cup B\|=\|A\|+\|B\|-\|A\cap B\|$ where A and B are two finite sets and \|S \| indicates the cardinality of a set S (which may be considered as the number of elements of the set, if the set is finite). The formula expresses the fact that the sum of the sizes of the two sets may be too large since some elements may be counted twice. The double-counted elements are those in the intersection of the two sets and the count is corrected by subtracting the size of the intersection. The inclusion-exclusion principle, being a generalization of the two-set case, is perhaps more clearly seen in the case of three sets, which for the sets A, B and C is given by $\|A\cup B\cup C\|=\|A\|+\|B\|+\|C\|-\|A\cap B\|-\|A\cap C\|-\|B\cap C\|+\|A\cap B\cap C\|$ This formula can be verified by counting how many times each region in the Venn diagram figure is included in the right-hand side of the formula. In this case, when removing the contributions of over-counted elements, the number of elements in the mutual intersection of the three sets has been subtracted too often, so must be added back in to get the correct total. Generalizing the results of these examples gives the principle of inclusion–exclusion. To find the cardinality of the union of n sets: 1. Include the cardinalities of the sets. 2. Exclude the cardinalities of the pairwise intersections. 3. Include the cardinalities of the triple-wise intersections. 4. Exclude the cardinalities of the quadruple-wise intersections. 5. Include the cardinalities of the quintuple-wise intersections. 6. Continue, until the cardinality of the n-tuple-wise intersection is included (if n is odd) or excluded (n even). The name comes from the idea that the principle is based on over-generous inclusion, followed by compensating exclusion. This concept is attributed to Abraham de Moivre (1718),[1] although it first appears in a paper of Daniel da Silva (1854)[2] and later in a paper by J. J. Sylvester (1883).[3] Sometimes the principle is referred to as the formula of Da Silva or Sylvester, due to these publications. The principle can be viewed as an example of the sieve method extensively used in number theory and is sometimes referred to as the sieve formula.[4] As finite probabilities are computed as counts relative to the cardinality of the probability space, the formulas for the principle of inclusion–exclusion remain valid when the cardinalities of the sets are replaced by finite probabilities. More generally, both versions of the principle can be put under the common umbrella of measure theory. In a very abstract setting, the principle of inclusion–exclusion can be expressed as the calculation of the inverse of a certain matrix.[5] This inverse has a special structure, making the principle an extremely valuable technique in combinatorics and related areas of mathematics. As Gian-Carlo Rota put it:[6] "One of the most useful principles of enumeration in discrete probability and combinatorial theory is the celebrated principle of inclusion–exclusion. When skillfully applied, this principle has yielded the solution to many a combinatorial problem." Formula In its general formula, the principle of inclusion–exclusion states that for finite sets A1, …, An, one has the identity $\left\|\bigcup _{i=1}^{n}A_{i}\right\|=\sum _{i=1}^{n}\|A_{i}\|-\sum _{1\leqslant i<j\leqslant n}\|A_{i}\cap A_{j}\|+\sum _{1\leqslant i<j<k\leqslant n}\|A_{i}\cap A_{j}\cap A_{k}\|-\cdots +(-1)^{n+1}\left\|A_{1}\cap \cdots \cap A_{n}\right\|.$ (1) This can be compactly written as $\left\|\bigcup _{i=1}^{n}A_{i}\right\|=\sum _{k=1}^{n}(-1)^{k+1}\left(\sum _{1\leqslant i_{1}<\cdots <i_{k}\leqslant n}\|A_{i_{1}}\cap \cdots \cap A_{i_{k}}\|\right)$ or $\left\|\bigcup _{i=1}^{n}A_{i}\right\|=\sum _{\emptyset \neq J\subseteq \{1,\ldots ,n\}}(-1)^{\|J\|+1}\left\|\bigcap _{j\in J}A_{j}\right\|.$ In words, to count the number of elements in a finite union of finite sets, first sum the cardinalities of the individual sets, then subtract the number of elements that appear in at least two sets, then add back the number of elements that appear in at least three sets, then subtract the number of elements that appear in at least four sets, and so on. This process always ends since there can be no elements that appear in more than the number of sets in the union. (For example, if $n=4,$ there can be no elements that appear in more than $4$ sets; equivalently, there can be no elements that appear in at least $5$ sets.) In applications it is common to see the principle expressed in its complementary form. That is, letting S be a finite universal set containing all of the Ai and letting ${\bar {A_{i}}}$ denote the complement of Ai in S, by De Morgan's laws we have $\left\|\bigcap _{i=1}^{n}{\bar {A_{i}}}\right\|=\left\|S-\bigcup _{i=1}^{n}A_{i}\right\|=\|S\|-\sum _{i=1}^{n}\|A_{i}\|+\sum _{1\leqslant i<j\leqslant n}\|A_{i}\cap A_{j}\|-\cdots +(-1)^{n}\|A_{1}\cap \cdots \cap A_{n}\|.$ As another variant of the statement, let P1, ..., Pn be a list of properties that elements of a set S may or may not have, then the principle of inclusion–exclusion provides a way to calculate the number of elements of S that have none of the properties. Just let Ai be the subset of elements of S which have the property Pi and use the principle in its complementary form. This variant is due to J. J. Sylvester.[1] Notice that if you take into account only the first m<n sums on the right (in the general form of the principle), then you will get an overestimate if m is odd and an underestimate if m is even. Examples Counting integers As a simple example of the use of the principle of inclusion–exclusion, consider the question:[7] How many integers in {1, …, 100} are not divisible by 2, 3 or 5? Let S = {1,…,100} and P1 the property that an integer is divisible by 2, P2 the property that an integer is divisible by 3 and P3 the property that an integer is divisible by 5. Letting Ai be the subset of S whose elements have property Pi we have by elementary counting: \|A1\| = 50, \|A2\| = 33, and \|A3\| = 20. There are 16 of these integers divisible by 6, 10 divisible by 10, and 6 divisible by 15. Finally, there are just 3 integers divisible by 30, so the number of integers not divisible by any of 2, 3 or 5 is given by: 100 − (50 + 33 + 20) + (16 + 10 + 6) − 3 = 26. Counting derangements A more complex example is the following. Suppose there is a deck of n cards numbered from 1 to n. Suppose a card numbered m is in the correct position if it is the mth card in the deck. How many ways, W, can the cards be shuffled with at least 1 card being in the correct position? Begin by defining set Am, which is all of the orderings of cards with the mth card correct. Then the number of orders, W, with at least one card being in the correct position, m, is $W=\left\|\bigcup _{m=1}^{n}A_{m}\right\|.$ Apply the principle of inclusion–exclusion, $W=\sum _{m_{1}=1}^{n}\|A_{m_{1}}\|-\sum _{1\leqslant m_{1}<m_{2}\leqslant n}\|A_{m_{1}}\cap A_{m_{2}}\|+\cdots +(-1)^{p-1}\sum _{1\leqslant m_{1}<\cdots <m_{p}\leqslant n}\|A_{m_{1}}\cap \cdots \cap A_{m_{p}}\|+\cdots $ Each value $A_{m_{1}}\cap \cdots \cap A_{m_{p}}$ represents the set of shuffles having at least p values m1, …, mp in the correct position. Note that the number of shuffles with at least p values correct only depends on p, not on the particular values of $m$. For example, the number of shuffles having the 1st, 3rd, and 17th cards in the correct position is the same as the number of shuffles having the 2nd, 5th, and 13th cards in the correct positions. It only matters that of the n cards, 3 were chosen to be in the correct position. Thus there are $ {n \choose p}$ equal terms in the pth summation (see combination). $W={n \choose 1}\|A_{1}\|-{n \choose 2}\|A_{1}\cap A_{2}\|+\cdots +(-1)^{p-1}{n \choose p}\|A_{1}\cap \cdots \cap A_{p}\|+\cdots $ $\|A_{1}\cap \cdots \cap A_{p}\|$ is the number of orderings having p elements in the correct position, which is equal to the number of ways of ordering the remaining n − p elements, or (n − p)!. Thus we finally get: ${\begin{aligned}W&={n \choose 1}(n-1)!-{n \choose 2}(n-2)!+\cdots +(-1)^{p-1}{n \choose p}(n-p)!+\cdots \\&=\sum _{p=1}^{n}(-1)^{p-1}{n \choose p}(n-p)!\\&=\sum _{p=1}^{n}(-1)^{p-1}{\frac {n!}{p!(n-p)!}}(n-p)!\\&=\sum _{p=1}^{n}(-1)^{p-1}{\frac {n!}{p!}}\end{aligned}}$ A permutation where no card is in the correct position is called a derangement. Taking n! to be the total number of permutations, the probability Q that a random shuffle produces a derangement is given by $Q=1-{\frac {W}{n!}}=\sum _{p=0}^{n}{\frac {(-1)^{p}}{p!}},$ a truncation to n + 1 terms of the Taylor expansion of e−1. Thus the probability of guessing an order for a shuffled deck of cards and being incorrect about every card is approximately e−1 or 37%. A special case The situation that appears in the derangement example above occurs often enough to merit special attention.[8] Namely, when the size of the intersection sets appearing in the formulas for the principle of inclusion–exclusion depend only on the number of sets in the intersections and not on which sets appear. More formally, if the intersection $A_{J}:=\bigcap _{j\in J}A_{j}$ has the same cardinality, say αk = \|AJ\|, for every k-element subset J of {1, …, n}, then $\left\|\bigcup _{i=1}^{n}A_{i}\right\|=\sum _{k=1}^{n}(-1)^{k-1}{\binom {n}{k}}\alpha _{k}.$ Or, in the complementary form, where the universal set S has cardinality α0, $\left\|S\smallsetminus \bigcup _{i=1}^{n}A_{i}\right\|=\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}\alpha _{k}.$ Formula generalization Given a family (repeats allowed) of subsets A1, A2, ..., An of a universal set S, the principle of inclusion–exclusion calculates the number of elements of S in none of these subsets. A generalization of this concept would calculate the number of elements of S which appear in exactly some fixed m of these sets. Let N = [n] = {1,2,…,n}. If we define $A_{\emptyset }=S$, then the principle of inclusion–exclusion can be written as, using the notation of the previous section; the number of elements of S contained in none of the Ai is: $\sum _{J\subseteq [n]}(-1)^{\|J\|}\|A_{J}\|.$ If I is a fixed subset of the index set N, then the number of elements which belong to Ai for all i in I and for no other values is:[9] $\sum _{I\subseteq J}(-1)^{\|J\|-\|I\|}\|A_{J}\|.$ Define the sets $B_{k}=A_{I\cup \{k\}}{\text{ for }}k\in N\smallsetminus I.$ We seek the number of elements in none of the Bk which, by the principle of inclusion–exclusion (with $B_{\emptyset }=A_{I}$), is $\sum _{K\subseteq N\smallsetminus I}(-1)^{\|K\|}\|B_{K}\|.$ The correspondence K ↔ J = I ∪ K between subsets of N \ I and subsets of N containing I is a bijection and if J and K correspond under this map then BK = AJ, showing that the result is valid. In probability In probability, for events A1, ..., An in a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$, the inclusion–exclusion principle becomes for n = 2 $\mathbb {P} (A_{1}\cup A_{2})=\mathbb {P} (A_{1})+\mathbb {P} (A_{2})-\mathbb {P} (A_{1}\cap A_{2}),$ for n = 3 $\mathbb {P} (A_{1}\cup A_{2}\cup A_{3})=\mathbb {P} (A_{1})+\mathbb {P} (A_{2})+\mathbb {P} (A_{3})-\mathbb {P} (A_{1}\cap A_{2})-\mathbb {P} (A_{1}\cap A_{3})-\mathbb {P} (A_{2}\cap A_{3})+\mathbb {P} (A_{1}\cap A_{2}\cap A_{3})$ and in general $\mathbb {P} \left(\bigcup _{i=1}^{n}A_{i}\right)=\sum _{i=1}^{n}\mathbb {P} (A_{i})-\sum _{i<j}\mathbb {P} (A_{i}\cap A_{j})+\sum _{i<j<k}\mathbb {P} (A_{i}\cap A_{j}\cap A_{k})+\cdots +(-1)^{n-1}\sum _{i<...<n}\mathbb {P} \left(\bigcap _{i=1}^{n}A_{i}\right),$ which can be written in closed form as $\mathbb {P} \left(\bigcup _{i=1}^{n}A_{i}\right)=\sum _{k=1}^{n}\left((-1)^{k-1}\sum _{I\subseteq \{1,\ldots ,n\} \atop \|I\|=k}\mathbb {P} (A_{I})\right),$ where the last sum runs over all subsets I of the indices 1, …, n which contain exactly k elements, and $A_{I}:=\bigcap _{i\in I}A_{i}$ denotes the intersection of all those Ai with index in I. According to the Bonferroni inequalities, the sum of the first terms in the formula is alternately an upper bound and a lower bound for the LHS. This can be used in cases where the full formula is too cumbersome. For a general measure space (S,Σ,μ) and measurable subsets A1, …, An of finite measure, the above identities also hold when the probability measure $\mathbb {P} $ is replaced by the measure μ. Special case If, in the probabilistic version of the inclusion–exclusion principle, the probability of the intersection AI only depends on the cardinality of I, meaning that for every k in {1, …, n} there is an ak such that $a_{k}=\mathbb {P} (A_{I}){\text{ for every }}I\subset \{1,\ldots ,n\}{\text{ with }}\|I\|=k,$ then the above formula simplifies to $\mathbb {P} \left(\bigcup _{i=1}^{n}A_{i}\right)=\sum _{k=1}^{n}(-1)^{k-1}{\binom {n}{k}}a_{k}$ due to the combinatorial interpretation of the binomial coefficient $ {\binom {n}{k}}$. For example, if the events $A_{i}$ are independent and identically distributed, then $\mathbb {P} (A_{i})=p$ for all i, and we have $a_{k}=p^{k}$, in which case the expression above simplifies to $\mathbb {P} \left(\bigcup _{i=1}^{n}A_{i}\right)=1-(1-p)^{n}.$ (This result can also be derived more simply by considering the intersection of the complements of the events $A_{i}$.) An analogous simplification is possible in the case of a general measure space (S, Σ, μ) and measurable subsets A1, …, An of finite measure. Other formulas The principle is sometimes stated in the form[10] that says that if $g(A)=\sum _{S\subseteq A}f(S)$ then $f(A)=\sum _{S\subseteq A}(-1)^{\|A\|-\|S\|}g(S)$ (⁎⁎) The combinatorial and the probabilistic version of the inclusion–exclusion principle are instances of (⁎⁎). Proof Take ${\underline {m}}=\{1,2,\ldots ,m\}$, $f({\underline {m}})=0$, and $f(S)=\left\|\bigcap _{i\in {\underline {m}}\smallsetminus S}A_{i}\smallsetminus \bigcup _{i\in S}A_{i}\right\|{\text{ and }}f(S)=\mathbb {P} \left(\bigcap _{i\in {\underline {m}}\smallsetminus S}A_{i}\smallsetminus \bigcup _{i\in S}A_{i}\right)$ respectively for all sets $S$ with $S\subsetneq {\underline {m}}$. Then we obtain $g(A)=\left\|\bigcap _{i\in {\underline {m}}\smallsetminus A}A_{i}\right\|,\quad g({\underline {m}})=\left\|\bigcup _{i\in {\underline {m}}}A_{i}\right\|{\text{ and }}g(A)=\mathbb {P} \left(\bigcap _{i\in {\underline {m}}\smallsetminus A}A_{i}\right),~~g({\underline {m}})=\mathbb {P} \left(\bigcup _{i\in {\underline {m}}}A_{i}\right)$ respectively for all sets $A$ with $A\subsetneq {\underline {m}}$. This is because elements $a$ of $\cap _{i\in {\underline {m}}\smallsetminus A}A_{i}$ can be contained in other $A_{i}$ ($A_{i}$ with $i\in A$) as well, and the $\cap \smallsetminus \cup $-formula runs exactly through all possible extensions of the sets $\{A_{i}\mid i\in {\underline {m}}\smallsetminus A\}$ with other $A_{i}$, counting $a$ only for the set that matches the membership behavior of $a$, if $S$ runs through all subsets of $A$ (as in the definition of $g(A)$). Since $f({\underline {m}})=0$, we obtain from (⁎⁎) with $A={\underline {m}}$ that $\sum _{{\underline {m}}\supseteq T\supsetneq \varnothing }(-1)^{\|T\|-1}g({\underline {m}}\smallsetminus T)=\sum _{\varnothing \subseteq S\subsetneq {\underline {m}}}(-1)^{m-\|S\|-1}g(S)=g({\underline {m}})$ and by interchanging sides, the combinatorial and the probabilistic version of the inclusion–exclusion principle follow. If one sees a number $n$ as a set of its prime factors, then (⁎⁎) is a generalization of Möbius inversion formula for square-free natural numbers. Therefore, (⁎⁎) is seen as the Möbius inversion formula for the incidence algebra of the partially ordered set of all subsets of A. For a generalization of the full version of Möbius inversion formula, (⁎⁎) must be generalized to multisets. For multisets instead of sets, (⁎⁎) becomes $f(A)=\sum _{S\subseteq A}\mu (A-S)g(S)$ (⁎⁎⁎) where $A-S$ is the multiset for which $(A-S)\uplus S=A$, and • μ(S) = 1 if S is a set (i.e. a multiset without double elements) of even cardinality. • μ(S) = −1 if S is a set (i.e. a multiset without double elements) of odd cardinality. • μ(S) = 0 if S is a proper multiset (i.e. S has double elements). Notice that $\mu (A-S)$ is just the $(-1)^{\|A\|-\|S\|}$ of (⁎⁎) in case $A-S$ is a set. Proof of (⁎⁎⁎) Substitute $g(S)=\sum _{T\subseteq S}f(T)$ on the right hand side of (⁎⁎⁎). Notice that $f(A)$ appears once on both sides of (⁎⁎⁎). So we must show that for all $T$ with $T\subsetneq A$, the terms $f(T)$ cancel out on the right hand side of (⁎⁎⁎). For that purpose, take a fixed $T$ such that $T\subsetneq A$ and take an arbitrary fixed $a\in A$ such that $a\notin T$. Notice that $A-S$ must be a set for each positive or negative appearance of $f(T)$ on the right hand side of (⁎⁎⁎) that is obtained by way of the multiset $S$ such that $T\subseteq S\subseteq A$. Now each appearance of $f(T)$ on the right hand side of (⁎⁎⁎) that is obtained by way of $S$ such that $A-S$ is a set that contains $a$ cancels out with the one that is obtained by way of the corresponding $S$ such that $A-S$ is a set that does not contain $a$. This gives the desired result. Applications The inclusion–exclusion principle is widely used and only a few of its applications can be mentioned here. Counting derangements Main article: Derangement A well-known application of the inclusion–exclusion principle is to the combinatorial problem of counting all derangements of a finite set. A derangement of a set A is a bijection from A into itself that has no fixed points. Via the inclusion–exclusion principle one can show that if the cardinality of A is n, then the number of derangements is [n! / e] where [x] denotes the nearest integer to x; a detailed proof is available here and also see the examples section above. The first occurrence of the problem of counting the number of derangements is in an early book on games of chance: Essai d'analyse sur les jeux de hazard by P. R. de Montmort (1678 – 1719) and was known as either "Montmort's problem" or by the name he gave it, "problème des rencontres."[11] The problem is also known as the hatcheck problem. The number of derangements is also known as the subfactorial of n, written !n. It follows that if all bijections are assigned the same probability then the probability that a random bijection is a derangement quickly approaches 1/e as n grows. Counting intersections The principle of inclusion–exclusion, combined with De Morgan's law, can be used to count the cardinality of the intersection of sets as well. Let ${\overline {A_{k}}}$ represent the complement of Ak with respect to some universal set A such that $A_{k}\subseteq A$ for each k. Then we have $\bigcap _{i=1}^{n}A_{i}={\overline {\bigcup _{i=1}^{n}{\overline {A_{i}}}}}$ thereby turning the problem of finding an intersection into the problem of finding a union. Graph coloring The inclusion exclusion principle forms the basis of algorithms for a number of NP-hard graph partitioning problems, such as graph coloring.[12] A well known application of the principle is the construction of the chromatic polynomial of a graph.[13] Bipartite graph perfect matchings The number of perfect matchings of a bipartite graph can be calculated using the principle.[14] Number of onto functions Given finite sets A and B, how many surjective functions (onto functions) are there from A to B? Without any loss of generality we may take A = {1, ..., k} and B = {1, ..., n}, since only the cardinalities of the sets matter. By using S as the set of all functions from A to B, and defining, for each i in B, the property Pi as "the function misses the element i in B" (i is not in the image of the function), the principle of inclusion–exclusion gives the number of onto functions between A and B as:[15] $\sum _{j=0}^{n}{\binom {n}{j}}(-1)^{j}(n-j)^{k}.$ Permutations with forbidden positions A permutation of the set S = {1, ..., n} where each element of S is restricted to not being in certain positions (here the permutation is considered as an ordering of the elements of S) is called a permutation with forbidden positions. For example, with S = {1,2,3,4}, the permutations with the restriction that the element 1 can not be in positions 1 or 3, and the element 2 can not be in position 4 are: 2134, 2143, 3124, 4123, 2341, 2431, 3241, 3421, 4231 and 4321. By letting Ai be the set of positions that the element i is not allowed to be in, and the property Pi to be the property that a permutation puts element i into a position in Ai, the principle of inclusion–exclusion can be used to count the number of permutations which satisfy all the restrictions.[16] In the given example, there are 12 = 2(3!) permutations with property P1, 6 = 3! permutations with property P2 and no permutations have properties P3 or P4 as there are no restrictions for these two elements. The number of permutations satisfying the restrictions is thus: 4! − (12 + 6 + 0 + 0) + (4) = 24 − 18 + 4 = 10. The final 4 in this computation is the number of permutations having both properties P1 and P2. There are no other non-zero contributions to the formula. Stirling numbers of the second kind Main article: Stirling numbers of the second kind The Stirling numbers of the second kind, S(n,k) count the number of partitions of a set of n elements into k non-empty subsets (indistinguishable boxes). An explicit formula for them can be obtained by applying the principle of inclusion–exclusion to a very closely related problem, namely, counting the number of partitions of an n-set into k non-empty but distinguishable boxes (ordered non-empty subsets). Using the universal set consisting of all partitions of the n-set into k (possibly empty) distinguishable boxes, A1, A2, …, Ak, and the properties Pi meaning that the partition has box Ai empty, the principle of inclusion–exclusion gives an answer for the related result. Dividing by k! to remove the artificial ordering gives the Stirling number of the second kind:[17] $S(n,k)={\frac {1}{k!}}\sum _{t=0}^{k}(-1)^{t}{\binom {k}{t}}(k-t)^{n}.$ Rook polynomials Main article: Rook polynomial A rook polynomial is the generating function of the number of ways to place non-attacking rooks on a board B that looks like a subset of the squares of a checkerboard; that is, no two rooks may be in the same row or column. The board B is any subset of the squares of a rectangular board with n rows and m columns; we think of it as the squares in which one is allowed to put a rook. The coefficient, rk(B) of xk in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. For any board B, there is a complementary board $B'$ consisting of the squares of the rectangular board that are not in B. This complementary board also has a rook polynomial $R_{B'}(x)$ with coefficients $r_{k}(B').$ It is sometimes convenient to be able to calculate the highest coefficient of a rook polynomial in terms of the coefficients of the rook polynomial of the complementary board. Without loss of generality we can assume that n ≤ m, so this coefficient is rn(B). The number of ways to place n non-attacking rooks on the complete n × m "checkerboard" (without regard as to whether the rooks are placed in the squares of the board B) is given by the falling factorial: $(m)_{n}=m(m-1)(m-2)\cdots (m-n+1).$ Letting Pi be the property that an assignment of n non-attacking rooks on the complete board has a rook in column i which is not in a square of the board B, then by the principle of inclusion–exclusion we have:[18] $r_{n}(B)=\sum _{t=0}^{n}(-1)^{t}(m-t)_{n-t}r_{t}(B').$ Euler's phi function Main article: Euler's totient function Euler's totient or phi function, φ(n) is an arithmetic function that counts the number of positive integers less than or equal to n that are relatively prime to n. That is, if n is a positive integer, then φ(n) is the number of integers k in the range 1 ≤ k ≤ n which have no common factor with n other than 1. The principle of inclusion–exclusion is used to obtain a formula for φ(n). Let S be the set {1, …, n} and define the property Pi to be that a number in S is divisible by the prime number pi, for 1 ≤ i ≤ r, where the prime factorization of $n=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{r}^{a_{r}}.$ Then,[19] $\varphi (n)=n-\sum _{i=1}^{r}{\frac {n}{p_{i}}}+\sum _{1\leqslant i<j\leqslant r}{\frac {n}{p_{i}p_{j}}}-\cdots =n\prod _{i=1}^{r}\left(1-{\frac {1}{p_{i}}}\right).$ Diluted inclusion–exclusion principle See also: Bonferroni inequalities In many cases where the principle could give an exact formula (in particular, counting prime numbers using the sieve of Eratosthenes), the formula arising does not offer useful content because the number of terms in it is excessive. If each term individually can be estimated accurately, the accumulation of errors may imply that the inclusion–exclusion formula is not directly applicable. In number theory, this difficulty was addressed by Viggo Brun. After a slow start, his ideas were taken up by others, and a large variety of sieve methods developed. These for example may try to find upper bounds for the "sieved" sets, rather than an exact formula. Let A1, ..., An be arbitrary sets and p1, …, pn real numbers in the closed unit interval [0, 1]. Then, for every even number k in {0, …, n}, the indicator functions satisfy the inequality:[20] $1_{A_{1}\cup \cdots \cup A_{n}}\geq \sum _{j=1}^{k}(-1)^{j-1}\sum _{1\leq i_{1}<\cdots <i_{j}\leq n}p_{i_{1}}\dots p_{i_{j}}\,1_{A_{i_{1}}\cap \cdots \cap A_{i_{j}}}.$ Proof of main statement Choose an element contained in the union of all sets and let $A_{1},A_{2},\dots ,A_{t}$ be the individual sets containing it. (Note that t > 0.) Since the element is counted precisely once by the left-hand side of equation (1), we need to show that it is counted precisely once by the right-hand side. On the right-hand side, the only non-zero contributions occur when all the subsets in a particular term contain the chosen element, that is, all the subsets are selected from $A_{1},A_{2},\dots ,A_{t}$. The contribution is one for each of these sets (plus or minus depending on the term) and therefore is just the (signed) number of these subsets used in the term. We then have: ${\begin{aligned}\|\{A_{i}\mid 1\leqslant i\leqslant t\}\|&-\|\{A_{i}\cap A_{j}\mid 1\leqslant i<j\leqslant t\}\|+\cdots +(-1)^{t+1}\|\{A_{1}\cap A_{2}\cap \cdots \cap A_{t}\}\|={\binom {t}{1}}-{\binom {t}{2}}+\cdots +(-1)^{t+1}{\binom {t}{t}}.\end{aligned}}$ By the binomial theorem, $0=(1-1)^{t}={\binom {t}{0}}-{\binom {t}{1}}+{\binom {t}{2}}-\cdots +(-1)^{t}{\binom {t}{t}}.$ Using the fact that ${\binom {t}{0}}=1$ and rearranging terms, we have $1={\binom {t}{1}}-{\binom {t}{2}}+\cdots +(-1)^{t+1}{\binom {t}{t}},$ and so, the chosen element is counted only once by the right-hand side of equation (1). Algebraic proof An algebraic proof can be obtained using indicator functions (also known as characteristic functions). The indicator function of a subset S of a set X is the function ${\begin{aligned}&\mathbf {1} _{S}:X\to \{0,1\}\\&\mathbf {1} _{S}(x)={\begin{cases}1&x\in S\\0&x\notin S\end{cases}}\end{aligned}}$ If $A$ and $B$ are two subsets of $X$, then $\mathbf {1} _{A}\cdot \mathbf {1} _{B}=\mathbf {1} _{A\cap B}.$ Let A denote the union $ \bigcup _{i=1}^{n}A_{i}$ of the sets A1, …, An. To prove the inclusion–exclusion principle in general, we first verify the identity $\mathbf {1} _{A}=\sum _{k=1}^{n}(-1)^{k-1}\sum _{I\subset \{1,\ldots ,n\} \atop \|I\|=k}\mathbf {1} _{A_{I}}$ (⁎) for indicator functions, where: $A_{I}=\bigcap _{i\in I}A_{i}.$ The following function $\left(\mathbf {1} _{A}-\mathbf {1} _{A_{1}}\right)\left(\mathbf {1} _{A}-\mathbf {1} _{A_{2}}\right)\cdots \left(\mathbf {1} _{A}-\mathbf {1} _{A_{n}}\right),$ is identically zero because: if x is not in A, then all factors are 0 − 0 = 0; and otherwise, if x does belong to some Am, then the corresponding mth factor is 1 − 1 = 0. By expanding the product on the left-hand side, equation (⁎) follows. To prove the inclusion–exclusion principle for the cardinality of sets, sum the equation (⁎) over all x in the union of A1, …, An. To derive the version used in probability, take the expectation in (⁎). In general, integrate the equation (⁎) with respect to μ. Always use linearity in these derivations. See also • Boole's inequality – inequality applying to probability spacesPages displaying wikidata descriptions as a fallback • Combinatorial principles – combinatorial methods used in combinatorics, a branch of mathematicsPages displaying wikidata descriptions as a fallback • Maximum-minimums identity – Relates the maximum element of a set of numbers and the minima of its non-empty subsets • Necklace problem • Pigeonhole principle – If there are more items than boxes holding them, one box must contain at least two items • Schuette–Nesbitt formula – mathematical formula in probability theoryPages displaying wikidata descriptions as a fallback Notes 1. Roberts & Tesman 2009, pg. 405 2. Mazur 2010, pg. 94 3. van Lint & Wilson 1992, pg. 77 4. van Lint & Wilson 1992, pg. 77 5. Stanley 1986, pg. 64 6. Rota, Gian-Carlo (1964), "On the foundations of combinatorial theory I. Theory of Möbius functions", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2 (4): 340–368, doi:10.1007/BF00531932, S2CID 121334025 7. Mazur 2010, pp. 83–4, 88 8. Brualdi 2010, pp. 167–8 9. Cameron 1994, pg. 78 10. Graham, Grötschel & Lovász 1995, pg. 1049 11. van Lint & Wilson 1992, pp. 77-8 12. Björklund, Husfeldt & Koivisto 2009 13. Gross 2008, pp. 211–13 14. Gross 2008, pp. 208–10 15. Mazur 2010, pp.84-5, 90 16. Brualdi 2010, pp. 177–81 17. Brualdi 2010, pp. 282–7 18. Roberts & Tesman 2009, pp.419–20 19. van Lint & Wilson 1992, pg. 73 20. (Fernández, Fröhlich & Alan D. 1992, Proposition 12.6) References • Allenby, R.B.J.T.; Slomson, Alan (2010), How to Count: An Introduction to Combinatorics, Discrete Mathematics and Its Applications (2 ed.), CRC Press, pp. 51–60, ISBN 9781420082609 • Björklund, A.; Husfeldt, T.; Koivisto, M. (2009), "Set partitioning via inclusion–exclusion", SIAM Journal on Computing, 39 (2): 546–563, doi:10.1137/070683933 • Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Prentice–Hall, ISBN 9780136020400 • Cameron, Peter J. (1994), Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, ISBN 0-521-45761-0 • Fernández, Roberto; Fröhlich, Jürg; Alan D., Sokal (1992), Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory, Texts an Monographs in Physics, Berlin: Springer-Verlag, pp. xviii+444, ISBN 3-540-54358-9, MR 1219313, Zbl 0761.60061 • Graham, R.L.; Grötschel, M.; Lovász, L. (1995), Hand Book of Combinatorics (volume-2), MIT Press – North Holland, ISBN 9780262071710 • Gross, Jonathan L. (2008), Combinatorial Methods with Computer Applications, Chapman&Hall/CRC, ISBN 9781584887430 • "Inclusion-and-exclusion principle", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Mazur, David R. (2010), Combinatorics A Guided Tour, The Mathematical Association of America, ISBN 9780883857625 • Roberts, Fred S.; Tesman, Barry (2009), Applied Combinatorics (2nd ed.), CRC Press, ISBN 9781420099829 • Stanley, Richard P. (1986), Enumerative Combinatorics Volume I, Wadsworth & Brooks/Cole, ISBN 0534065465 • van Lint, J.H.; Wilson, R.M. (1992), A Course in Combinatorics, Cambridge University Press, ISBN 0521422604 This article incorporates material from principle of inclusion–exclusion on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.	Wikipedia
Find $ 8^8 \cdot 4^4 \div 2^{28}$. Obviously, multiplying out each of the exponents is not an option. Instead, notice that the bases of all three exponents are themselves powers of $2$. Let's convert the bases to $2$: $$ 8^8 \cdot 4^4 \div 2^{28} = (2^3)^8 \cdot (2^2)^4 \div 2^{28}.$$Using the power of a power rule in reverse, $(2^3)^8 = 2^{3 \cdot 8} = 2^{24}$. Likewise, $(2^2)^4 = 2^{2 \cdot 4} = 2^8$. Therefore, our simplified expression is $2^{24} \cdot 2^8 \div 2^{28}$. Now, using the product of powers rule, $2^{24} \cdot 2^8 = 2^{24 + 8} = 2^{32}$. Our expression is now stands at $2^{32} \div 2^{28}$. Finally, we'll use quotient of powers to simplify that to $2^{32-28} = 2^4 = \boxed{16}$.	Math Dataset
Early- versus late-onset Alzheimer's disease in clinical practice: cognitive and global outcomes over 3 years Carina Wattmo1 & Åsa K. Wallin1 Alzheimer's Research & Therapy volume 9, Article number: 70 (2017) Cite this article Whether age at onset influences Alzheimer's disease (AD) progression and the effectiveness of cholinesterase inhibitor (ChEI) therapy is not clear. We aimed to compare longitudinal cognitive and global outcomes in ChEI-treated patients with early-onset Alzheimer's disease (EOAD) versus late-onset Alzheimer's disease (LOAD) in clinical practice. This 3-year, prospective, observational, multicentre study included 1017 participants with mild to moderate AD; 143 had EOAD (age at onset < 65 years) and 874 had LOAD (age at onset ≥ 65 years). At baseline and semi-annually, patients were assessed using cognitive, global and activities of daily living (ADL) scales, and the dose of ChEI was recorded. Potential predictors of decline were analysed using mixed-effects models. Six-month response to ChEI therapy and long-term prognosis in cognitive and global performance were similar between the age-at-onset groups. However, deterioration was significantly faster when using the Alzheimer's Disease Assessment Scale–Cognitive subscale (ADAS-Cog) over 3 years in participants with EOAD than in those with LOAD; hence, prediction models for the mean ADAS-Cog trajectories are presented. The younger cohort had a larger proportion of homozygote apolipoprotein E (APOE) ε4 allele carriers than the older cohort; however, APOE genotype was not a significant predictor of cognitive impairment in the multivariate models. A slower rate of cognitive progression was related to initiation of ChEIs at an earlier stage of AD, higher ChEI dose and fewer years of education in both groups. In LOAD, male sex, better instrumental ADL ability and no antipsychotic drug use were additional protective characteristics. The older patients received a lower ChEI dose than the younger individuals during most of the study period. Although the participants with EOAD showed a faster decline in ADAS-Cog, had a longer duration of AD before diagnosis, and had a higher frequency of two APOE ε4 alleles than those with LOAD, the cognitive and global responses to ChEI treatment and the longitudinal outcomes after 3 years were similar between the age-at-onset groups. A higher mean dose of ChEI and better cognitive status at the start of therapy were independent protective factors in both groups, stressing the importance of early treatment in adequate doses for all patients with AD. People who have a clinical onset of Alzheimer's disease (AD) before the age of 65 years are diagnosed with early-onset Alzheimer's disease (EOAD). The prevalence of patients with EOAD is low, but it varies in studies from 6% to 16% [1,2,3]. Some observations suggest that EOAD might be a separate, more severe entity than late-onset Alzheimer's disease (LOAD). Researchers in neuropathological studies have reported that younger patients with AD exhibited higher burdens of neuritic plaques and neurofibrillary tangles, as well as greater synapse loss, than older individuals [4]. Moreover, patients with AD who died before 80 years of age had a more widespread and severe cholinergic deficit with abnormalities in other neurotransmitters (e.g., noradrenaline) compared with those who died at older ages [5]. Regarding cognition, the patients with EOAD demonstrated more impairment in language and concentration, whereas the LOAD cohort showed difficulties in memory and orientation [6]. Clinical diagnosis of AD is often missed in individuals with early onset because of the atypical symptoms and non-amnestic presentations [7]. Younger persons are often more educated than older individuals and have a higher cognitive reserve capacity that could also lead to a delayed diagnosis [8]. Therefore, antidementia therapy might be initiated in a later stage of the disease, which may impair the efficacy of treatment in EOAD. Currently, the main therapy for mild to moderate AD is cholinesterase inhibitors (ChEIs). Positive cognitive and global symptomatic effects compared with placebo have been reported mainly in 6-month randomised clinical trials [9]. All these trials, as well as long-term extensions [10, 11] and observational studies of ChEI treatment in AD [12, 13], have enrolled participants regardless of age at onset. However, the level of short-term therapeutic response and longitudinal outcome may vary depending on age at onset of AD. By investigating the entire mild to moderate Swedish Alzheimer Treatment Study (SATS) cohort, our group found that both cognitive response to ChEIs and prognosis after 3 years were better in older patients with AD than in younger individuals. In addition, we showed that male sex, absence of the apolipoprotein E (APOE) ε4 allele, use of non-steroidal anti-inflammatory drugs (NSAIDs)/acetylsalicylic acids, and receiving a higher ChEI dose, regardless of type of drug, were independent predictors related to slower cognitive deterioration [13]. The aforementioned association between age and cognitive performance was not observed in another study [14]. Conversely, in a 3-month donepezil study, participants younger than 66 years of age exhibited greater improvement than older patients [15]. No longer-term studies have previously reported possible socio-demographic and clinical characteristics or aspects of ChEI therapy (e.g., drug agent and dose) that might affect the cognitive trajectory in a cohort with exclusively EOAD. A faster rate of cognitive progression among younger individuals with AD was shown in some studies [16, 17], whereas others demonstrated a similar decline between the age groups [18, 19] or more rapid impairment in older participants [20]. These mixed and conflicting observations may depend on differences in sample size and follow-up period, as well as possible confounding factors such as APOE genotype, education level, activities of daily living (ADL) capacity, and concomitant disorders, which were not considered by most earlier studies on EOAD versus LOAD [21]. The SATS includes multiple variables (e.g., the above-mentioned covariates) that have not been evaluated simultaneously in prior publications. By dividing the SATS cohort into subsets according to age at onset and separately analysing these groups with an advanced multivariate statistical approach (mixed-effects models), a better understanding of the course of AD and potential predictive characteristics in younger versus older persons with AD might be expected. The aims of this study were (1) to describe and compare cognitive and global longitudinal outcomes between ChEI-treated patients with EOAD and LOAD in clinical practice and (2) to identify socio-demographic and clinical factors (e.g., sex, APOE genotype, years of education, concomitant medications) and aspects of ChEI treatment (drug agent, dose) that could affect the cognitive abilities in the respective groups. Study and participants The SATS is a 3-year, prospective, open-label, non-randomised, multicentre study that was started to investigate the long-term effectiveness of ChEI therapy (donepezil, rivastigmine and galantamine) in clinical practice. Different findings from the SATS have been presented in several publications (e.g., [12, 13, 22,23,24]). In total, 1258 participants with AD were recruited from 14 memory clinics located in different geographical areas across Sweden. All 1021 patients exhibiting a baseline Mini Mental State Examination (MMSE) [25] score ranging from 10 to 26 and an age at onset of AD (4 participants had missing data) were included in the current study. Of these, 143 individuals were defined as having EOAD (age at onset of AD < 65 years), and 874 were defined as having LOAD (age at onset of AD ≥ 65 years); hence, 1017 participants were enrolled. Considered for inclusion in the SATS were outpatients aged ≥ 40 years who met the criteria for the clinical diagnosis of dementia, as defined in the Diagnostic and Statistical Manual of Mental Disorders, Fourth Edition [26], and for possible or probable AD according to the criteria of the National Institute of Neurological and Communicative Disorders and Stroke and the Alzheimer's Disease and Related Disorders Association [27]. All patients were diagnosed by physicians who specialise in dementia disorders. The dementia specialist estimated the age at onset on the basis of an interview with the caregiver (usually the spouse or an adult child) regarding observations of early symptoms of AD. Moreover, the selected individuals had to live at their own home at the time of AD diagnosis, to have a responsible caregiver and to be assessable with the MMSE at the start of the ChEI treatment (baseline). The exclusion criteria were not fulfilling the diagnostic criteria for AD, already receiving active ChEI therapy or having contra-indications to ChEIs. After inclusion in the study and the baseline evaluations, the participants were prescribed ChEI treatment as part of the ordinary Swedish health-care system and in accordance with the approved product labelling. All patients started with donepezil 5 mg, rivastigmine 3 mg, or galantamine 8 mg, as in routine clinical practice. The SATS is an observational study, and the choice of drug type and all decisions regarding dosage were left entirely up to the dementia specialist's discretion and professional judgement. Most individuals received an increased dose after 4–8 weeks of treatment, and we aimed at further dose increases depending on the chosen ChEI agent. However, for some participants, the dose was reduced because of side effects. The ChEI dose was recorded after 2 months of therapy and then every 6 months after baseline. Medications other than ChEIs were documented at baseline and allowed during the study, with the exception of memantine. If the patient stopped taking the ChEI or if memantine was initiated, the individual discontinued the SATS at that time point. The date of and reason for any drop-out from the SATS were recorded. The SATS patients were investigated in a well-structured follow-up programme in which researchers evaluated cognitive, global and ADL performance at the start of ChEI treatment, after 2 months (MMSE and global rating only) and semi-annually over 3 years. Cognitive status was assessed using the MMSE, with scores ranging from 0 to 30 (a lower score indicating more impaired cognition), and the Alzheimer's Disease Assessment Scale–Cognitive subscale (ADAS-Cog) [28], with a total range of 0 to 70 (a higher score indicating more impaired cognition). The Clinician Interview-Based Impression of Change (CIBIC) [29] was used as a global rating of 'change from the initiation of ChEI therapy'. The evaluations were performed at all intervals using a 7-point scale ranging from 1 (very much improved) to 7 (marked worsening). Three groups of response were defined at each CIBIC interval: 1–3 indicated improvement, 4 indicated no change and 5–7 indicated worsening. No guidelines or descriptors were provided to define the individual ratings. The classification between, for example, minimally improved or very much improved was left to the dementia specialist's clinical judgement. The Instrumental Activities of Daily Living (IADL) scale [30] consists of eight different items: ability to use the telephone, shopping, food preparation, housekeeping, laundry, mode of transportation, responsibility for one's own medications and ability to handle finances. Each item was scored from 1 (no impairment) to 3–5 (severe impairment), which yielded a total range of 8–31 points. The Physical Self-Maintenance Scale [30] consists of six different items: toileting, feeding, dressing, grooming, physical ambulation and bathing. Each item was scored from 1 (no impairment) to 5 (severe impairment), which allowed a total range of 6–30 points. Trained dementia nurses assessed the ADL performance on the basis of interviews with the caregiver. To facilitate the comparison of rates in MMSE and ADAS-Cog scores, changes in score were converted to positive values, which were indicative of improvement, and negative values, which were indicative of deterioration. IBM SPSS Statistics for Windows version 22.0 software (IBM Corporation, Armonk, NY, USA) was used to perform the statistical analyses. The level of significance was defined as p < 0.05 if not otherwise specified, and all tests were two-tailed. Observed-case analyses were used to avoid overestimation of the treatment effect by imputing higher previous outcome scores in a longitudinal study of a progressively deteriorating disease. Parametric tests were used because of the large sample size and the approximately normally distributed continuous potential predictors. Independent-sample t tests were used to compare the differences between the means obtained for two groups, such as EOAD and LOAD, and χ2 tests were used to analyse categorical variables. Mixed, linear and non-linear fixed and random coefficient regression models using the participant as a hierarchical variable (i.e., to allow intra-individual correlation) were performed. The mixed-effects models method also takes into account the varying number of evaluations available for each patient and unequal time intervals between the follow-up visits, which are common statistical limitations found in long-term studies [31]. The non-completers contributed information during the time of participation; hence, we considered the trajectories of all SATS patients. The dependent variables were the cognitive scores assigned at the second and subsequent visits for each individual; that is, the mixed-effects models do not intend to predict the scores at the initiation of ChEI therapy. The following described independent variables were included in the models. First, the initial cognitive scores for each participant (to adjust for baseline differences) and their interaction with linear and quadratic terms for time in the study (to enable a non-linear rate of change in the models) were included as fixed effects; that is, time in months (and time in months2) × MMSE (or ADAS-Cog) baseline score. Time was defined as the exact number of months between the start of ChEI treatment and each visit, thus using all data points at the actual time intervals. Secondly, several possible socio-demographic and clinical predictors were included as fixed effects in the models, such as sex; age at the start of ChEI therapy; clinician's estimated duration of AD; years of education; presence of the APOE ε4 allele (no/yes); solitary living (no/yes); IADL and basic ADL capacity; number of medications at baseline; and specific concomitant medications (no/yes for each group), including antihypertensive/cardiac therapy, antidiabetics, asthma medications, thyroid therapy, lipid-lowering agents, oestrogens, NSAIDs/acetylsalicylic acid, antidepressants, antipsychotics and anxiolytics/sedatives/hypnotics. Thirdly, the effect of the different ChEI agents was analysed using the type of drug (coded as a set of dummy variables) and dosages. The terms 'ChEI agent × dose' and 'age × ChEI dose' were also included in the models. The ChEI dose could vary during the treatment period for an individual patient and between patients; therefore, the mean dose used during the entire follow-up period was calculated for each participant. In cases of drop-out, the mean dose used during the individual's time of participation in the SATS was calculated. To obtain a similar metric for percentage maximum dosage for each of the three ChEIs, the mean dose was divided by the maximum recommended dose for each drug, namely 10 mg for donepezil, 12 mg for rivastigmine (oral administration) and 24 mg for galantamine. Lastly, some potential interactions (sex, age or education) with cognitive severity at baseline or with time in the study were included in the models. The random terms were an intercept and time in months, with a variance components covariance matrix. Non-significant variables (p > 0.05) were eliminated in a backward stepwise manner. The hierarchical principle was applied in the mixed-effects models; variables that appeared in significant interactions were not considered for elimination. Socio-demographic and clinical characteristics according to age at onset of AD The 1017 SATS participants were divided into two cohorts according to their age at onset of AD: EOAD (< 65 years, n = 143 [14%]) and LOAD (≥ 65 years, n = 874 [86%]). The socio-demographic and clinical characteristics of the two cohorts are presented in Table 1. The presence of two APOE ε4 alleles was more frequent, and the proportion of one or no ε4 alleles was less frequent, among the patients with EOAD than in those with LOAD [χ2(2) = 23.98, p < 0.001]. The mean duration of illness was longer [t (1015) = 4.36, p < 0.001], and the level of education higher [t (1015) = 2.93, p = 0.004], among the younger than the older individuals. Cognitive ability at the initiation of ChEI therapy did not differ between the onset groups. Among the participants with EOAD, a lower percentage received donepezil and a higher percentage received rivastigmine or galantamine [χ2(2) = 8.09, p = 0.017]. The mean dose of donepezil during the study was higher in the younger cohort [t (514) = 2.32, p = 0.020], but it was similar between the groups for the other two ChEI agents. Table 1 Socio-demographic and clinical characteristics of the SATS participants (n = 1017) Comparison of longitudinal outcomes between EOAD versus LOAD Regarding the MMSE score, 71% of the patients with EOAD and 64% of those with LOAD showed improvement/no change (≥ 0-point change) after 6 months of ChEI treatment [χ2(1) = 2.60, p = 0.107]. Improvement/no change (≥ 0-point change) in ADAS-Cog score was observed for 50% of the younger and 56% of the older cohort after 6 months [χ2(1) = 1.60, p = 0.206]. The mean (95% CI) MMSE and ADAS-Cog scores and the changes from baseline scores over the 3-year study by age-at-onset group are shown in Table 2. Using the ADAS-Cog scale, the rate of cognitive decline was faster among the EOAD participants at 12, 18 and 30 months after the initiation of ChEI therapy (Fig. 1). The mean (95% CI) semi-annual rates of change in the ADAS-Cog score at each time point for the younger and older cohorts, respectively, were, for 6–12 months, –2.4 (–3.8, –1.0) versus –1.9 (–2.4, –1.4) points (p = 0.483); for 12–18 months, –2.9 (–4.0, –1.8) versus –2.0 (–2.6, –1.5) points (p = 0.207); for 18–24 months, –1.5 (–2.9, –0.1) versus –1.9 (–2.5, –1.3) points (p = 0.643); for 24–30 months, –2.9 (–4.7, –1.2) versus –2.1 (–2.7, –1.5) points (p = 0.325); and for 30–36 months, –2.6 (–4.6, –0.6) versus –2.7 (–3.5, –1.8) points (p = 0.945). No significant difference in disease progression over time between the onset groups was detected when the MMSE score was used. Table 2 Changes in cognitive and global performance, and dose of cholinesterase inhibitors, during 3 years of therapy, by age-at-onset group Cognitive outcome over 3 years of cholinesterase inhibitor (ChEI) treatment. The mean changes in Alzheimer's Disease Assessment Scale–Cognitive subscale (ADAS-Cog) score with 95% CI from the start of ChEI therapy (baseline) over 3 years, by age at onset of Alzheimer's disease (AD). The patients with early-onset Alzheimer's disease showed a more rapid rate of cognitive decline from baseline after 12 months (p = 0.045), 18 months (p = 0.035) and 30 months (p = 0.019) of treatment The percentages of the EOAD versus LOAD cohorts according to changes in global performance (CIBIC) after 2 months and semi-annually over 3 years from the start of ChEIs are illustrated in Fig. 2. The proportions of the remaining younger and older patients who exhibited improvement or no change in CIBIC score at each visit are presented in Table 2; however, no differences between the two groups were found. The individuals with EOAD and LOAD, respectively, were further divided into APOE genotypes. No significant differences in changes in cognitive or global capacities after 3 years of ChEI treatment were observed between these groups. The mean (95% CI) percentage of maximum ChEI dose was higher in EOAD than in LOAD participants at all visits after 12 months of therapy (Table 2). Global outcome over 3 years of cholinesterase inhibitor (ChEI) treatment. The proportion of patients is shown according to differences in treatment response in global performance (Clinician Interview-Based Impression of Change (CIBIC)) from the start of ChEI therapy over 3 years for early-onset Alzheimer's disease (EOAD) versus late-onset Alzheimer's disease (LOAD). No significant differences in global response were observed between the two groups, except after 24 months (p = 0.005) of treatment. CIBIC scores 1–3 were considered as improvement, 4 as unchanged, and 5–7 as deterioration In total, 86 patients (60%) with EOAD and 556 (64%) with LOAD did not complete the 3-year SATS [χ2(1) = 0.64, p = 0.425]. The reasons for drop-out in this cohort have been reported previously [24]. In the EOAD group, the completers received a greater percentage (mean ± SD) of the maximum recommended ChEI dose during the study [76 ± 14% versus 60 ± 19%, t (141) = 5.66, p < 0.001]. No significant differences between the younger completers and drop-outs were found regarding cognitive or global status at the start of treatment. In the LOAD group, the completers also received a greater mean percentage of the maximum recommended ChEI dose during the study [69 ± 18% versus 59 ± 18%, t (872) = 8.01, p < 0.001]. The older completers exhibited significantly better cognitive [mean ± SD, MMSE score 22.4 ± 3.4 versus 20.9 ± 3.8 points, t (872) = 6.06, p < 0.001; ADAS-Cog score 18.6 ± 8.4 versus 22.4 ± 8.7 points, t (858) = −6.30, p < 0.001] and global (CIBIC median score 3 versus 4 points, median test, p = 0.017) performance at baseline than the drop-outs. In addition, antipsychotic use was less frequent among the completers with LOAD [6 (2%) versus 37 (7%) patients, χ2(1) = 9.83, p = 0.002]. The other variables of interest in this study, such as sex, APOE genotype, age at baseline, duration of AD, years of education, number of concomitant medications and other specific medications received, did not differ between the completers and those who discontinued the study in any of the age-at-onset groups. Fifteen (10%) of the younger and 66 (8%) of the older individuals dropped out because of initiation of concomitant memantine therapy [χ2(1) = 1.45, p = 0.229]; their mean length (95% CI) of participation in the study was 23.1 (20.1–26.2) versus 19.7 (17.8–21.5) months [t (79) = 1.66, p = 0.101], respectively. Predictors of disease progression in the respective age-at-onset groups In the mixed-effects models, only participants with three or more assessments were included to enable analyses of a non-linear rate of cognitive change (EOAD n = 128 [90%]; LOAD n = 774 [89%]). The models were performed to identify the socio-demographic and clinical predictors that affected the patients' longitudinal trajectories (EOAD 667 data points, LOAD 3733 data points). The percentages of variance that accounted for the dependent variable regarding all fixed factors were 54.7% for MMSE and 53.6% for ADAS-Cog in the EOAD group and 51.1% for MMSE and 55.3% for ADAS-Cog in the LOAD group. This indicates a good fit of the models (p < 0.001 for all models). The mixed-effects models, significant predictors and unstandardised β coefficients with 95% CIs are presented in Tables 3 and 4. Table 3 Factors affecting long-term outcome with Mini Mental State Examination score as dependent variable, by age-at-onset group Table 4 Factors affecting long-term outcome with Alzheimer's Disease Assessment Scale–Cognitive subscale as dependent variable, by age-at-onset group Better cognitive status at the initiation of ChEI therapy implied a slower rate of progression over time. A higher mean ChEI dose during the study (regardless of drug agent) and a lower level of education were independent predictors of a more positive cognitive long-term outcome in both LOAD models and using the ADAS-Cog model in EOAD. No differences in cognitive trajectories were found among the three ChEI agents in the EOAD or LOAD cohorts. An interaction effect showed that a higher level of education led to increased cognitive impairment over the course of the disease. Female sex, younger age and less-preserved IADL capacity at the start of ChEI treatment were factors that implied a higher rate of cognitive deterioration in LOAD. Interaction effects between age (or sex in the ADAS-Cog mixed-effects model) and cognitive ability at baseline demonstrated that this difference in cognitive performance between age groups (or sex) was more pronounced among the older individuals who were more cognitively impaired. The interaction term of age with ChEI dose was not significant in any of the models. According to the MMSE model, the use of antipsychotics in participants with LOAD implied worse prognosis. Below, we provide non-linear regression models for calculation of the predicted ADAS-Cog score for a cohort of ChEI-treated patients with EOAD and LOAD, based on the respective baseline score. These equations are intended to predict the scores at subsequent evaluations over a 3-year period. The ADAS-Cog models explained a substantial degree of variance in the dataset; that is, they displayed a good fit (EOAD R 2 = 0.577, R = 0.760, p < 0.001; LOAD R 2 = 0.537, R = 0.733, p < 0.001). Predicted ADAS-Cog score in EOAD was calculated as follows: $$ \widehat{\mathrm{Y}}=0.8227\hbox{--} \left(0.0154\times \mathrm{t}\right)+\left(1.0855\times {\mathrm{x}}_{\mathrm{i}}\right)+\left(0.0178\times {\mathrm{tx}}_{\mathrm{i}}\right)\hbox{--} \left(0.0077\times {{\mathrm{x}}_{\mathrm{i}}}^2\right) $$ Predicted ADAS-Cog score in LOAD was calculated as follows: $$ \widehat{\mathrm{Y}}=3.1545\hbox{--} \left(0.0456\times \mathrm{t}\right)+\left(0.7455\times {\mathrm{x}}_{\mathrm{i}}\right)+\left(0.0156\times {\mathrm{tx}}_{\mathrm{i}}\right) $$ where t is the time in months between baseline and the actual visit and x i is the baseline ADAS-Cog score. In this study performed in routine clinical practice, the 6-month cognitive and global responses to ChEI therapy and the longitudinal outcomes after 3 years were similar between the age-at-onset groups; however, a somewhat faster decline in the EOAD group at some time points was detected when we used the ADAS-Cog scale. Homozygote APOE ε4 carriers were more frequent among the younger patients, but APOE genotype did not significantly affect disease progression in the multivariate models. The EOAD cohort received a higher ChEI dose than the LOAD group over the study period. A higher mean dose of ChEI, better cognitive status at the initiation of treatment and lower level of education were independent protective factors for a more favourable long-term cognitive performance in both groups. Risk factors for worse prognosis in LOAD were female sex, younger age, more impaired IADL capacity and use of antipsychotics. We defined EOAD as the onset of AD before 65 years of age, which is the definition used most often in earlier studies [21]. Typically, AD has an insidious and gradual onset, and it could sometimes be problematic to distinguish from an age-related deterioration in the beginning of the disease; therefore, the individual's age at the onset of symptoms might be difficult to estimate accurately [32]. The age cut-off of < 65 years is arbitrary and not based on any biological differences; instead, a social factor, namely the traditional retirement age in many countries, has been used as the dividing line [21, 33]. However, some studies have used other cut-offs for EOAD, such as age at onset < 60 years [7], age at onset ≤ 66 years [33], time of AD diagnosis < 65 years [2, 6], time of AD diagnosis ≤ 65 years [34] and < 79 years at death [5]. This lack of consensus makes comparisons between studies difficult. The 3-year cognitive outcomes were similar between the onset groups in the present study; however, when using the more complex and sensitive ADAS-Cog scale, a slightly more rapid deterioration in EOAD at the 12-, 18- and 30-month evaluations was found. Inconsistently, in a recent publication derived from a meta-database that included ten AD studies, Schneider et al. [35] observed greater worsening on both ADAS-Cog and MMSE scales over 12–24 months in younger participants, whereas in an earlier study, Kramer-Ginsberg et al. [36] reported no difference between EOAD and LOAD in change in ADAS-Cog + ADAS-Noncog score over the course of up to 2 years. In line with our findings regarding the MMSE scale, a similar worsening regardless of age at onset has been described [19, 37], although other AD studies have suggested a more pronounced decline in MMSE score in younger versus older individuals [16, 34]. In a smaller-sample study (n = 42) [37], the mean MMSE changes/year for the antidementia drug-treated patients with EOAD and LOAD were, respectively, 0.82 and 1.0 points, whereas the corresponding rates of progression in the somewhat more cognitively impaired SATS cohort at baseline were slightly faster at 1.1 and 1.4 MMSE points/year. These mixed observations show that the scale used did not provide the sole explanation for these various results, and it is not possible to conclude whether the EOAD group has a different cognitive trajectory than the LOAD cohort. A possibly more rapid disease course in younger participants and unequal age distributions across cohorts included in clinical trials may affect the results of these studies. In the present study, males with LOAD exhibited more positive longitudinal cognitive abilities than females. Male sex has been related to a slower progression rate in some multivariate studies [38, 39] but not all [40]. However, these reports did not address the potential impact of ChEI therapy. A more positive short-term cognitive response to ChEIs among males was demonstrated by our group [13] and also by a 3-month study of tacrine and galantamine [41]. One explanation for these sex differences might be the role of sex hormones in AD [42]. Another theory is that males have larger cerebral hemispheres than women even after controlling for body size [43]. In addition, the association between AD pathology and dementia was reported as more pronounced in women than in men [44]. These findings could indicate that men withstand AD pathology better and that the female brain is more vulnerable, which might explain the more favourable cognitive outcome over time and the better response to ChEIs shown among men. However, the possible impact of sex regarding differences in the prognosis of AD and the efficacy of ChEI treatment requires further investigation. The younger cohort in this study exhibited a significantly higher education level, longer duration of AD and better functional performance but similar cognitive status at baseline. These characteristics have been described in other reports on EOAD [33]. More years of education were related to a faster deterioration in cognitive abilities in both our age-at-onset groups. This observation supports the cognitive reserve hypothesis, according to which more highly educated people are expected to have a more advanced disease at the time of AD diagnosis [45]. Moreover, neuropathological studies detected a higher burden of AD pathology and larger synapse loss in younger than in older patients [4]. In the present study, the somewhat more rapid decline in individuals with EOAD measured by ADAS-Cog might reflect their higher education and more pronounced cognitive reserve. The rate of disease progression has been suggested to increase in the moderate to severe stages of AD using the traditional cognitive assessment scales [40]. Furthermore, older age in LOAD was associated with better long-term cognitive outcome in this study. A reduced cognitive reserve capacity among the oldest participants could lead to earlier detection of the disease, diagnosis of AD, and start of ChEIs at an earlier stage, which could improve efficacy. Our finding that a longer illness duration before AD diagnosis was demonstrated in our EOAD cohort also supports this explanation. Taken together, a more advanced disease with greater AD pathology at baseline and thus later initiation of ChEI therapy might occur in the younger SATS group. In the present study, a higher frequency of homozygote APOE ε4 carriers was observed in the EOAD cohort; however, APOE genotype did not affect the cognitive trajectories in the multivariate models for either the younger or the older patients in the SATS. Earlier studies of the relationship of the ε4 allele with rate of progression in AD were inconsistent [46, 47]. One study showed that APOE non-ε4 carriers with EOAD had a faster cognitive deterioration than non-ε4 carriers with LOAD; however, the level of education of the groups was not mentioned [34]. Previously, we reported that ε4 carriers were younger at the start of ChEI treatment and had more years of education than the non-ε4 carriers [13]. Younger individuals may have more hereditary and aggressive forms of AD [48]. Different patient characteristics among studies, such as education level and thus cognitive reserve, as well as genetic predispositions between the age-at-onset groups might lead to various outcomes. Use of antipsychotics and lower IADL performance were risk factors for a more pronounced cognitive decline in participants with LOAD in this study. Hearing and visual impairment among older persons with dementia could lead to more delusions and visual hallucinations, respectively [49]. A review of psychosis in AD indicated that the association between age at onset and psychotic symptoms was inconsistent among studies. However, psychosis led to a faster cognitive worsening in all included studies in which researchers investigated this issue [50]. In addition, antipsychotic therapy in AD was related to significant cognitive progression over time compared with placebo [51]. These findings suggest that individuals with psychotic symptoms consist of a subset with a more aggressive course of AD and worse prognosis. The LOAD cohort had more functional deficits than the EOAD group in this study. Our group and others have reported that patients with AD with rapid IADL deterioration exhibit significantly lower cognitive status at the initial evaluation [22, 52]. Moreover, a moderate linear association between cognition and function has been demonstrated in AD [53], which suggests that these aspects should be interpreted simultaneously in multivariate models because a change in one capacity could influence a change in another. This interaction effect might have implications for the results reported in pharmaceutical trials of new AD drug agents. The present observational, longitudinal study is the first to show the potential impact of ChEI treatment among participants with EOAD and LOAD separately. The clinical value of ChEIs is still controversial among physicians. The authors of a recent publication from the Swedish Dementia Registry stated that the prevalence of ChEI therapy was significantly higher among the EOAD group than the LOAD cohort (88% versus 75%); that is, one of four older individuals diagnosed with AD did not receive ChEIs [2]. In our study, no differences in effectiveness among the three drug agents were observed. Higher doses of ChEIs were related to a more positive cognitive outcome in both age-at-onset groups; however, the patients with LOAD received a lower dose over time than those with EOAD. It is not known whether this finding depended on the older person's actual lower tolerability or the physician's opinion that an older individual might not tolerate a higher ChEI dose. The association between a higher dose of ChEI and slower disease progression has been reported among the total number of SATS participants with mild to moderate AD [13], as well as in a meta-analysis of randomised trials [54]. The strengths of the SATS are the prospective, well-structured assessments every 6 months over 3 years after the initiation of ChEIs in a large cohort of continuously treated ordinary patients with AD with co-morbidities and concomitant medications from memory clinics across Sweden. The 3-year completion rates of 40% and 36% obtained for the EOAD and LOAD participants from clinical practice, respectively, were high compared with other AD extension or naturalistic studies (20–39%) [23]. The large drop-out rate in all long-term AD studies may contribute to a more favourable outcome for the remaining patients, assuming that they benefit more from ChEI therapy. Our results show that the completers in both age-at-onset groups received a higher mean dose of ChEI during the study, suggesting a better tolerance of the treatment. In the mixed-effects models, the outcomes of the drop-outs were also included during their time of participation. The lower cognitive and global abilities at baseline observed for the non-completers with LOAD may contribute to trajectories that are more positive for the older individuals remaining in the SATS over time. Patients with LOAD and psychotic symptoms exhibited a higher risk of discontinuing the study. However, the drop-outs were similar to the completers regarding the other characteristics. One limitation is that the SATS was not placebo-controlled owing to ethical concerns or randomised with respect to ChEI drug agent, similar to other longitudinal, observational AD studies. Specialists in dementia disorders decided on the type of ChEI and dose for each participant, in agreement with the standards used in a routine clinical setting. Another shortcoming of this study, as in previous publications on age at onset [55], is that the physician's estimation of onset of AD symptoms relied on information and retrospective observations of the caregiver and their attention. Few long-term studies have analysed the relationships between EOAD and LOAD, APOE genotype, level of education and concomitant medications. Moreover, earlier findings are not consistent regarding the effect of, for example, age at onset and APOE ε4 carrier status on disease prognosis; thus, additional studies are warranted. The short-term response to ChEIs and the effect over a longer time in different age groups, as well as possible predictors that might alter the outcome, have not been investigated previously. A comparison of various aspects of disease progression between EOAD and LOAD was performed in this observational study. After 3 years, the cognitive and global rates of decline were similar between the age-at-onset groups; however, the more sensitive ADAS-Cog scale tended to exhibit a faster worsening among the younger individuals over this period. This information is necessary for the interpretation of results from clinical trials to evaluate the effectiveness of and provide realistic expectations for new potentially disease-modifying therapies (as add-ons to ChEIs) directed at AD cohorts of various ages. In addition, male sex, better IADL performance and no use of antipsychotics in the LOAD group, as well as fewer years of education in both groups, were protective factors of a more positive longitudinal cognitive outcome. Although the patients with EOAD included a larger proportion of carriers of two APOE ε4 alleles, this observation did not influence progression rate using multivariate models. The socio-demographic and clinical composition of an AD cohort under study may be one explanation for the heterogeneity of results presented in a number of reports. A higher mean dose of ChEI (regardless of drug agent) was associated with slower cognitive decline in both onset groups, but the older participants received a lower dose during most of the 3-year study period. This finding stresses the importance for clinicians to optimise the ChEI dose in AD, regardless of the individual's age at onset, to improve treatment effectiveness. In summary, our results suggest that EOAD and LOAD are not separate entities. The younger patients' longer time to AD diagnosis, higher level of education and thus cognitive reserve capacity might explain most of the differences detected between the groups. 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Acta Neurol Scand. 1993;88:349–53. We thank all the SATS patients and their relatives for their co-operation in this study. We are also grateful to the staff at all of the different centres that took part in the management of the participants and provided administrative support during the study. CW received a postdoctoral scholarship from the Swedish Brain Foundation and grants from Greta och Johan Kocks stiftelse (Fromma Foundation) and SUS (Skånes universitetssjukhus) stiftelser och donationer (Skåne University Hospital Foundations and Donations) in Sweden. ÅKW received a grant from Stiftelsen för Gamla Tjänarinnor (Foundation of Old Servants). The sponsors had no involvement in the study design; in the collection, analysis and interpretation of data; in the writing of the report; or in the decision to submit the manuscript for publication. Currently, we are unable to share the SATS data because data collection, such as dates of death, is still taking place and the data analysis process is ongoing. Clinical Memory Research Unit, Department of Clinical Sciences, Malmö, Lund University, SE-205 02, Malmö, Sweden Carina Wattmo & Åsa K. Wallin Carina Wattmo Åsa K. Wallin CW participated in SATS, supervised data collection, was responsible for the statistical design and conducted the statistical analyses, interpreted the results, and drafted the manuscript. ÅKW participated in the study, assisted in the analysis and interpretation of the data, and revised the manuscript critically. Both authors read and approved the final manuscript. Correspondence to Carina Wattmo. All procedures performed in studies involving the SATS participants were carried out in accordance with the Helsinki declaration. The SATS protocol and the present analysis of data from the SATS reported in this article were submitted to and approved by the regional ethical review board of Lund University, Lund, Sweden (number 2014/658, dated 9 December 2014). Written informed consent was obtained from all patients included in the SATS. If an individual was not able to provide consent for him- or herself, consent was obtained from the individual's closest relative. Wattmo, C., Wallin, Å.K. Early- versus late-onset Alzheimer's disease in clinical practice: cognitive and global outcomes over 3 years. Alz Res Therapy 9, 70 (2017). https://doi.org/10.1186/s13195-017-0294-2 Mixed-effects models	CommonCrawl
A geometry problem on cyclic quadrilaterals Let $M$ be the point of intersection between the diagonals of a cyclic quadrilateral $ABCD$, where $\angle AMB$ is acute. The isosceles triangle $BCK$, whose base is $BC$, is constructed externally to the quadrilateral, so that $\angle KBC$ + $\angle AMB$ = $90°$. Prove that $KM$ is perpendicular to $AD$. I just have no idea how to do it. Sure $ABCD$ is cyclic so you can find a bunch of angles and do some angle chasing, but it got me nowhere. If anyone could post a solution, or at least give me a hint on what to do, that would be appreciated. Thanks in advance. geometry contest-math Deathkamp DroneDeathkamp Drone Elegant proof Let's start with a generic cyclic quadrilateral, and define \begin{align} \angle ACB = \angle ADB &= \alpha \\ \angle CAD = \angle CBD &= \beta \\ \angle DBA = \angle DCA &= \gamma \\ \angle BAC = \angle BDC &= 180°-\alpha-\beta-\gamma \end{align} This answer is based on a key observation: $K$ is the center of the circle through $B,C,M$. This will certainly ensure that $BCK$ is isosceles with base $BC$. We'll check the angle sum condition along the way. If $K$ is the center of the circumcircle, then not only $BCK$ bit also $BMK$ and $CMK$ are isosceles, and we have the following conditions (introducing a new angle $\delta$ which we'll atempt to eliminate): $$ \angle KBC=\angle BCK=\delta \\ \angle KMC=\angle MCK=\alpha+\delta \\ \angle BMK=\angle KBM=\beta+\delta \\ \angle CKM=180°-2(\alpha+\delta) \\ \angle MKB=180°-2(\beta+\delta) \\ \angle CKB=180°-2\delta= \bigl(180°-2(\alpha+\delta)\bigr)+\bigl(180°-2(\beta+\delta)\bigr) $$ From his last equation you can conclude $$\delta=90°-\alpha-\beta$$ So now we can verify that $$ \angle KBC+\angle AMB = \delta+(180°-\angle BAC-\angle DBA)\\ = 90°-\alpha-\beta+(180°-(180°-\alpha-\beta-\gamma)-\gamma)=90° $$ so $K$ is indeed the right point. Now consider the quadrilateral $ABKH$, where $H$ is the intersection of $KM$ with $AD$. It has interior angles \begin{align} \angle BAH &= \angle BAC + \angle CAD = (180°-\alpha-\beta-\gamma)+\beta = 180°-\alpha-\gamma \\ \angle KBA &= \angle KBC + \angle CBD + \angle DBA = (90°-\alpha-\beta) + \beta + \gamma = 90°-\alpha+\gamma \\ \angle HKB &= 180°-2(\beta+\delta) = 2\alpha \\ \angle AHK &= 360°-\angle BAH-\angle KBA-\angle HKB = 360°-180°+\alpha+\gamma-90°+\alpha-\gamma-2\alpha = 90° \end{align} $$\tag{$\Box$}$$ Brute force proof The above proof relies on spotting that fact about the circumcircle. Before I had that, I had a less elegant proof by brute force computation using sage. In fact, I found out that $BMK$ and $CMK$ are isosceles by comparing angles obtained from the construction below for random parameters. That computation is based on the following construction: Start with four points on the unit circle, using homogeneous coordinates and a rational parametrization. P.<a, b, c, d> = ZZ[] def cpt(u): # a point on the unit circle return vector([1-u^2, 2u, 1+u^2]) A = cpt(a) B = cpt(b) C = cpt(c) D = cpt(d) $$ A = \begin{pmatrix}1-a^2\\2a\\1+a^2\end{pmatrix} \qquad B = \begin{pmatrix}1-b^2\\2b\\1+b^2\end{pmatrix} \qquad C = \begin{pmatrix}1-c^2\\2c\\1+c^2\end{pmatrix} \qquad D = \begin{pmatrix}1-d^2\\2d\\1+d^2\end{pmatrix} $$ You can join points and intersect lines using the cross product. def join(a, b): v = a.cross_product(b) g = gcd(v) return v/g meet = join M = meet(join(A, C), join(B, D)) $$ M=\begin{pmatrix} a b c - a b d + a c d - b c d - a + b - c + d \\ -2 a c + 2 b d \\ - a b c + a b d - a c d + b c d - a + b - c + d \end{pmatrix} $$ Next, construct the line through $M$ orthogonal to $BD$ and intersect this with $CD$ to obtain $E$. This will ensure $\angle AMB+\angle EMC=90°$. perp = diagonal_matrix([1, 1, 0]) E = meet(join(perpjoin(B, D), M), join(C, D)) $$ E=\begin{pmatrix} a b^{2} c^{2} d^{2} - a b^{2} c d^{3} + a b c^{2} d^{3} - b^{2} c^{2} d^{3} + a b^{2} c^{2} + a b^{2} c d - a b c^{2} d \\{}- b^{2} c^{2} d - 2 a b^{2} d^{2} + 2 a b c d^{2} - b^{2} c d^{2} + b c^{2} d^{2} - a b d^{3} + 2 b^{2} d^{3} \\{}+ a c d^{3} - 2 b c d^{3} - 2 a b c + b^{2} c + 2 a c^{2} - b c^{2} + a b d - a c d \\{}+ 2 b c d - 2 c^{2} d - a d^{2} - b d^{2} + c d^{2} + d^{3} - a + b - c + d \\[2ex] {}-2 a b c^{2} d^{2} + 2 b^{2} c d^{3} - 2 a b^{2} c + 2 a b^{2} d - 4 a b c d + 4 b^{2} c d + 2 a c^{2} d \\{}- 2 b c^{2} d + 2 a b d^{2} - 2 b^{2} d^{2} - 4 a c d^{2} + 4 b c d^{2} - 2 c^{2} d^{2} + 2 c d^{3} - 2 a c + 2 b d \\[2ex] {}- a b^{2} c^{2} d^{2} + a b^{2} c d^{3} - a b c^{2} d^{3} + b^{2} c^{2} d^{3} - a b^{2} c^{2} + a b^{2} c d - a b c^{2} d \\{}+ b^{2} c^{2} d - 2 a b c d^{2} + b^{2} c d^{2} - b c^{2} d^{2} + a b d^{3} - a c d^{3} + 2 b c d^{3} - 2 a b c \\{}+ b^{2} c - b c^{2} + a b d - a c d + 2 b c d - a d^{2} + b d^{2} - c d^{2} + d^{3} - a + b - c + d \end{pmatrix} $$ Compute $F$ as the other intersection of $BC$ with the circle through $CEF$. Due to the inscribed angle theorem, this will give you $\angle EFC$ = $\angle EMC$. def cvec(p): return vector([xx + yy, xz, yz, zz]) P2.<mu> = P[] p = list(matrix(map(cvec, [C, E, M, B+muC])).det()) F = p[1]B - p[0]C F = F/gcd(F) $$ F=\begin{pmatrix} {}- a b^{4} c^{2} + a b^{4} c d - a b^{3} c^{2} d + b^{4} c^{2} d - 2 a b^{3} c + b^{4} c + a b^{2} c^{2} - b^{3} c^{2} \\{}+ 3 a b^{3} d - 2 b^{4} d - 2 a b^{2} c d + 2 b^{3} c d - a b c^{2} d + b^{2} c^{2} d + a b^{2} - b^{3} + 2 a b c \\{}- 2 b^{2} c - 2 a c^{2} + 3 b c^{2} - a b d + b^{2} d + a c d - 2 b c d + a - b + c - d \\[2ex] 2 a b^{3} c^{2} - 2 b^{4} c d + 6 a b^{2} c - 4 b^{3} c - 2 a b c^{2} \\{}+ 4 b^{2} c^{2} - 4 a b^{2} d + 2 b^{3} d + 4 a b c d - 6 b^{2} c d + 2 a c - 2 b d \\[2ex] a b^{4} c^{2} - a b^{4} c d + a b^{3} c^{2} d - b^{4} c^{2} d + 2 a b^{3} c - b^{4} c + a b^{2} c^{2} \\{}+ b^{3} c^{2} - a b^{3} d - 2 b^{3} c d + a b c^{2} d - b^{2} c^{2} d + a b^{2} - b^{3} \\{}+ 2 a b c + b c^{2} - a b d - b^{2} d + a c d - 2 b c d + a - b + c - d \end{pmatrix} $$ Let $G$ be the midpoint of $BC$. def midpoint(a, b): v = b[-1]a + a[-1]*b G = midpoint(B, C) $$ G=\begin{pmatrix} - b^{2} c^{2} + 1 \\ b^{2} c + b c^{2} + b + c \\ b^{2} c^{2} + b^{2} + c^{2} + 1 \end{pmatrix} $$ Then you can construct the line parallel to $EF$ through $B$ and intersect this with the radius $OG$ to obtain $K$. The parallel line ensures $\angle KBC=\angle EFC$. linf = vector([0, 0, 1]) O = vector([0, 0, 1]) K = meet(join(meet(join(E, F), linf), B), join(O, G)) $$ K=\begin{pmatrix} - a b c + b c d + a - d \\ a b + a c - b d - c d \\ a b c - a b d + a c d - b c d + a - b + c - d \end{pmatrix} $$ Now verify that the lines $KM$ and $AD$ are indeed orthogonal join(K, M)[:2].dot_product(join(A, D)[:2]).is_zero() This will print the desired answer, True. MvGMvG $\begingroup$ Quick proof that $K$ is the center of $\bigcirc BMC$: By the Inscribed Angle Theorem, all (and only) points $P$ separated from $K$ by $\overline{BC}$, such that $$\angle BPC = \frac{1}{2}(360^\circ - \angle BKC ) = \frac{1}{2}(360^\circ - (180^\circ - 2 (90^\circ - \angle AMB)) = 180^\circ - \angle AMB$$ lie on the (minor) arc $\stackrel{\frown}{BC}$ of the circle about $K$ through $B$ and $C$. $M$ is such a point. $\endgroup$ – Blue Apr 2 '14 at 20:34 $\begingroup$ Thank you! This problem has been haunting me for days, yet it can be solved so easily once you notice that little fact. $\endgroup$ – Deathkamp Drone Apr 3 '14 at 1:38 Not the answer you're looking for? Browse other questions tagged geometry contest-math or ask your own question. Perpendicular lines inside and outside a circle Properties of cyclic quadrilateral Geometry Problem about Cyclic Quadrilateral BMO1 2003/04 Question 2 - Geometry Prolem Prove that the midpoints of the sides of a quadrilateral lie on a circle if and only if the quadrilateral is orthodiagonal. A geometry problem - proving that points are concyclic Proof With Cyclic Quadrilateral and Circle Miquel Point of Cyclic Quad Need help on this geometry problem about angles Cyclic quadrilateral, and isogonal conjugates? Cyclic quadrilateral and trapezoid	CommonCrawl
Transactions of the American Mathematical Society Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics. The 2020 MCQ for Transactions of the American Mathematical Society is 1.43. Journals Home eContent Search About TRAN Editorial Board Author and Submission Information Journal Policies Subscription Information Collapsing of products along the Kähler-Ricci flow by Matthew Gill PDF Trans. Amer. Math. Soc. 366 (2014), 3907-3924 Request permission Let $X = M \times E$, where $M$ is an $m$-dimensional Kähler manifold with negative first Chern class and $E$ is an $n$-dimensional complex torus. We obtain $C^\infty$ convergence of the normalized Kähler-Ricci flow on $X$ to a Kähler-Einstein metric on $M$. This strengthens a convergence result of Song-Weinkove and confirms their conjecture. 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I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350, DOI 10.1002/cpa.3160310304 Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44, 53C55 Retrieve articles in all journals with MSC (2010): 53C44, 53C55 Matthew Gill Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093 Address at time of publication: Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, California 94720-3840 MR Author ID: 951451 Received by editor(s): June 14, 2012 Received by editor(s) in revised form: December 17, 2012 Published electronically: November 14, 2013 The copyright for this article reverts to public domain 28 years after publication. Journal: Trans. Amer. Math. Soc. 366 (2014), 3907-3924 MSC (2010): Primary 53C44; Secondary 53C55 DOI: https://doi.org/10.1090/S0002-9947-2013-06073-4	CommonCrawl
Rahul Khatri1, Michael Schmidt1 & Industrial enterprises represent a significant portion of electricity consumers with the potential of providing demand-side energy flexibility from their production processes and on-site energy assets. Methods are needed for the active and profitable participation of such enterprises in the electricity markets especially with variable prices, where the energy flexibility available in their manufacturing, utility and energy systems can be assessed and quantified. This paper presents a generic model library equipped with optimal control for energy flexibility purposes. The components in the model library represent the different technical units of an industrial enterprise on material, media, and energy flow levels with their process constraints. The paper also presents a case study simulation of a steel-powder manufacturing plant using the model library. Its energy flexibility was assessed when the plant procured its electrical energy at fixed and variable electricity prices. In the simulated case study, flexibility use at dynamic prices resulted in a 6% cost reduction compared to a fixed-price scenario, with battery storage and the manufacturing system making the largest contributions to flexibility. Ever increasing energy demands, integration of variable renewable energy resources and less reliance on building new generation capacity from conventional power plants is resulting in an increased challenge for power systems to match the supply and demand at all times during their operation. This gap between supply and demand is referred to as flexibility gap by authors in Papaefthymiou et al. (2018). Among the various options to address this gap, demand-side energy flexibility (DSEF) is considered to be one viable solution, in which electricity consumers adapt (increase, decrease or shift) their energy consumption, which is also referred to as demand-response (DR) (Roesch et al. 2019). The focus of this work lies in developing models and methods that allow industrial consumers to evaluate their DSEF potential and to participate in DR in a semi-automated or fully automated manner. The energy flexibility of industrial enterprises could play a significant role in closing the flexibility gap. In Germany, the industrial sector consumes 44% of the total electricity (Lund et al. 2015), thus a significant portion. According to a Umweltbundesamt report from 2015 (Langrock et al. 2015), there is a technical DR potential of 6.5 GW from the industrial sector in Germany for at least one hour; however, authors in Stede (2016) argue that under current regulatory and technical barriers, only 3.5 GW of the DR potential is estimated to be viable. Two categories of DSEF can be distinguished according to SEDC (2016): One is explicit DSEF, where consumers receive control signals from system operators or committed schedules from external agents and directly adjust their power consumption accordingly. In return, they receive a contracted remuneration. The other is implicit DSEF, in which consumers react to price signals. Various price based programs are being introduced to leverage implicit DSEF based on price signals, such as time of use (ToU), critical peak pricing (CPP) and real-time pricing (RTP) (Albadi and El-Saadany 2008). To facilitate DSEF, new market actors such as "aggregators" are also evolving, which act as intermediaries between end-consumers and electricity market. Their main role is to pool distributed units and market their generation capacity or DSEF on the several markets such as spot market with variable prices, balancing market and others (Stede et al. 2020). According to a white paper on the Electricity Market 2.0 by the Federal Ministry of Economics Affairs and Energy(BMWi), Germany (German Federal Ministry for Economic Affairs and Energy (BMWi) 2015), "aggregators can also open up the potential for flexibilization in medium-sized and small electricity consumers to the extent that they have direct access to energy markets". It is expected that industrial enterprises can also procure their electricity at variable prices in the future. This might apply not only to large and energy-intensive industries but also small and medium-sized enterprises (SMEs). In addition to easy access to electricity markets, another challenge for industrial enterprises is the effort involved in implementing and operating a local process control system that responds to dynamic price signals. Increased use of "digitalization" could be a key driver to tackle this challenge and to leverage energy flexibility. According to the report "SMEs Digital" from (BMWi) (German Federal Ministry for Economic Affairs and Energy (BMWi) 2018), 88% of all SMEs see a connection between digitalization and corporate success, but for 51% of those companies surveyed, digitalization is not the core part of their business strategy. The increased use of digitalization could therefore also prove to be a door opener for industrial enterprises, especially SMEs, to flexibly consume their energy and profitably participate in the electricity markets of today and tomorrow. Against this background, key research questions are whether it is really worthwhile for industrial enterprises to exploit their energy flexibility on the basis of dynamic prices, and how the technical entry barriers can be lowered. Methods and models are therefore needed which, on the one hand, can characterize the energy flexibility, the achievable profits and the associated effort in a company-specific and operation-focused manner and, on the other hand, enable a partially or fully automated implementation. The characterization and modeling of industrial enterprises' flexibility has been intensively investigated over the last years. Some research works (Tristán et al. 2020; Schott et al. 2019; Pierri et al. 2020; Weeber et al. 2017) present data models and characterization methodologies for available energy flexibility measures (EFMs) on-site, while some works (Seitz et al. 2019; Roesch et al. 2019; Schott et al. 2018) also discuss the overall conceptual layout for using EFMs by also presenting IT-based synchronization platforms between flexible companies and external demand response agents. When comprehensive automated data acquisition and process control are introduced as part of digitalization, the basis is also created for the use of advanced methods of data analysis, optimization and control (Scheidt et al. 2020). To also use these methods to provide automated use of energy flexibility, models are needed that can represent the energy flows themselves as well as the flexibility in the energy flows. This in turn requires modeling of the energy-relevant devices, plants, processes as well as all interdependencies. On the one hand, these models can be used in off-line simulations to assess the flexibility potential, or to assist in decisions on how to best use or increase flexibility. On the other hand, as part of model-based controls, they can help to realize semi-automated or fully automated operations with flexibility utilization, e.g. in response to external price signals. The paper (Roesch et al. 2019) argues that industrial enterprises face particular challenges in the optimal use of flexibility due to the complex nature of their production processes and interdependencies. In addition, each industrial enterprise is different from another based on their type of manufacturing, final products, and associated processes. This complexity challenge also applies in particular to the modeling of energy flows and their flexibilities, as well as the use of model-based control and optimization approaches for automated use of flexibility. The contribution of this work to the solution of the above-mentioned research questions and challenges is the development of a model library that is as generic as possible in order to be able to map the energy flows and energy flexibilities of various industrial companies with reasonable effort and to use them for model-based control and optimization. For this purpose, the models presented capture the energy, material and media flows within industrial companies, take into account so-called process dependencies and boundary conditions with sufficient accuracy and at the same time ensure a minimum complexity for mathematical optimization. To be applicable to as many companies as possible, the models are kept as generic and modular as possible. The developed model library is published as an open-source tool under GNU General Public License v3.0) (Offenburg University 2021). In the remainder the concept of the generic industrial enterprise model library is presented in detail, including the generic modeling of the manufacturing systems (MFS), the technical building services (TBS) and the energy systems (ES). Then, in a case study, the model library is used to model and analyze the flexibilities of a steel powder manufacturing plant, before the paper ends with a conclusion and outlook. A generic industrial enterprise model library A general industrial enterprise can be considered as composed of various industrial systems that work together to execute the intended production through core and auxiliary processes. The systems are interconnected by material, energy, media and data flows (Beier 2017). Focusing on modeling of energy flexibility, the proposed generic model library considers a industrial enterprise to be composed of three major technical units, namely MFS, TBS and ES, see Fig. 1. The research approach is taken from existing research works of (Beier et al. 2015; Tristán et al. 2020), where they represent the industrial facility in these sub-units for their methodology for energy flexible industrial systems. A generic representation of industrial enterprise for energy flexibility modeling purposes Each of these three units can contain different types of sub-units representing different generic key functionalities in industrial enterprise, for e.g cooling or heating systems as part of TBS. These sub-units can in turn be broken down to the lowest level, where concrete individual machines, devices and plants are modeled as the ultimate consumers and producers of electricity that directly affect the electric load profiles and flexibility. To model the enterprise's energy flexibility and DR potential, the control of the technical units has to be modeled in addition to the technical units themselves. In the generic model library, these control and management functionalities are combined and modeled in an additional fourth technical unit, the energy and manufacturing control (EMC) unit. It contains the optimal control algorithms and coordinates the processes and energy consumption of defined technical units, alongside with having interfaces to external signals based on the participation mechanisms. The sections below describe each of the technical units and relevant modeling parameters and constraints that are also chosen as inputs for the modules of the library. Manufacturing systems - MFS The MFS is the central and value-adding part of every industrial enterprise, determining the main electricity consumption activities of the production process. This is the system which transforms the energy, media, raw material and information into final products for the company (Beier 2017). For modeling this system as generically as possible for the purpose of optimal load control for DSEF, the MFS was modeled in form of a resource-task network (RTN) (Castro et al. 2009). Table 1 shows the corresponding modeling parameters and constraints. Table 1 Modeling parameters and constraints for MFS in the model library. The resources include raw materials that serve as inputs for the production systems, products that are finished goods having some required quantity at the end of the planning horizon, storage facilities that serve as buffers for materials in production systems and production machines which consume electricity and transform materials. Tasks represent the manufacturing steps or jobs that are carried out to produce an intermediate or final product depending on the production flow. When the task is executed, it uses a machine or combination of machines. The operation of machines in turn increases or decreases the quantity of raw materials and products inside storage facilities. Production machines alongside with electrical power might also require other media and energy such as compressed air, heating and cooling that is supplied by the TBS. When the production is being carried out, it might require special operation conditions, such as temperature or air quality, which can be ensured by heating and ventilation systems available in the TBS. The energy flexibility available in the MFS comes from degrees of freedom in production planning, where the execution of flexible production tasks can be interrupted or shifted according to external price signals. The relevant constraints for the MFS described in Table 1 are provided to the optimizer in the EMC module to make optimal use of the available flexibility. Technical building services - TBS The MFS needs certain media for its manufacturing processes, which must be provided by the TBS in requested quantities and at the right time. For this purpose, the TBS system contains corresponding units for the generation and intermediate storage of these media. For simplicity it is assumed that media cannot flow back to the TBS system as they might in a real enterprise. The sub-units of the TBS that are currently modeled in the library include air system (CAS), process cooling system (PCS), process heating system (PHS) and a simplified version of heating, ventilation and air conditioning (HVAC). Other building utilities such as lighting or humidity control can be included in the future. The main source of energy flexibility in the TBS is in the intermediate storage facilities for media, as they allow to alter and shift the load of media generation units. In the following, we discuss the sub-units and their components of TBS that are available in the library as modeling blocks. Compressed air system: A CAS is widely used in most of the industrial enterprises to supply the compressed air supporting the manufacturing process and equipment, including machine tools, clamping, spraying, material separation and material handling and pneumatic utility (Javied et al. 2018). The flexibility in these systems lies in the storage of compressed air in pressurized storage tanks, which can offer DSEF by making the energy consumption of compressors flexible. However, such systems have process constraints as, for example, that the air supply pressure in air distribution circuits or storage tanks must not fall out of the specified required operation levels. Table 2 shows the elements inside the CAS and considered parameters and optimization constraints. Table 2 Description of CAS and constraints in the model library Process heating and cooling system: PCS and PHS are important systems that provide heating and cooling as utility to MFS. For example, the operation temperatures of machines are required to be under certain limits, or materials need heating and cooling treatment before and after the production. The energy flexibility in these systems mainly comes from available thermal storage capabilities. Further, these systems can also supply the thermal energy to HVAC systems to maintain the desired operation temperature range and personal comfort depending on the climatic conditions. Table 3 shows the relevant parameters and constraints for energy flexibility modeling. Table 3 Description of PCS and PHS and constraints in the model library Heating, ventilation and air conditioning: HVAC systems are important for maintaining indoor comfort for personnel and favorable operating conditions for production machinery. For the purpose of energy flexibility, only thermal comfort has been considered in the developed model library so far, where the goal is to maintain the desired indoor temperature against the outdoor temperature using the dynamic indoor temperature prediction as in works of Harder et al. (2020); Hietaharju et al. (2018). Based on the desired thermal comfort, a heating or cooling demand is generated, which is provided by dedicated heating or cooling devices or also by thermal storage via PCS and PHS. Table 4 shows the parameters required for the modeling of the building HVAC system. Table 4 Description of HVAC system and constraint Energy systems - ES The ES of the industrial enterprise in the developed model library comprises on-site electricity generation and storage. This includes electricity supply from locally installed photovoltaics (PV), combined heat and power (CHP) units which can also supply heat energy to the thermal storage units inside the PHS, and battery energy storage (BES). Further systems such as wind turbines or diesel generators can also be added. For the purpose of modeling energy flexibility, the mentioned systems are not modeled in detail and a consideration on the power level is sufficient. BES play a particularly important role in terms of flexibility, because its operational constraints are not directly linked with the constraints of the MFS and TBS systems, and hence these system provide greater flexibility to the industrial enterprise. The optimizer inside the EMC module that also ensures the power balance with the external grid, optimally decides the power flows of all controllable ES components. Table 5 shows the generic parameters and constraints for the components of the ES. Table 5 Description of ES and constraints in the model library Energy and manufacturing control - EMC The EMC module in the model library serves as a central control entity equipped with model based optimization. The main objective is to adapt the overall load demand of the industrial enterprise at the connection point of the grid and hence provide available flexibility. It contains generic mixed-integer linear programming (MILP) based optimization formulations for the technical units, which automatically consider the defined input parameters and constraints (Tables 1, 2, 3, 4 and 5). Figure 2 shows the structural layout of the EMC. The generic formulation of technical units results in the load demand variables, and the power balance of the industrial enterprise reads for each time step t in optimization horizon T, $$ \begin{aligned} P_{Grid,t} = P_{MFS,t}+P_{PCS,t}+P_{PHS,t}-P_{PV,t}-& P_{CHP,t} - P_{BES,t} \\ & \forall t\in \mathbf{T} = [0,1,...,T_{f}] \end{aligned} $$ Optimization and control structure inside EMC where each of the given variables in Eq. 1 are optimally determined based on the provided process constraints. A positive grid power PGrid,t indicates consumption from the grid, whereas a positive electric storage power PBES,t indicates discharge. Negative values indicate feed-in to the grid and storage charging, respectively. The objective function of the optimization algorithm is to minimize the daily electricity procurement costs and CHP fuel costs, which is written as, $$ J = \sum_{t=0}^{t=T_{f}} (P_{Grid,t} \cdot \lambda^{electricity}_{t} \cdot \Delta_{t}) +\sigma^{CHP,fuel}_{t} \quad\quad\quad \forall t\in \mathbf{T} = [0,1,...,T_{f}] $$ where $\lambda ^{electricity}_{t}$ in €/kWH represents the electricity price at time step "t", which can be either fixed if the industrial enterprise procures its electricity at flat rate, or be variable based on the prices on the electricity market. Δt represents the sampling time step of the optimization horizon, which can be chosen generically as well. The input data for price signals is retrieved by the EMC from the Market block in the model library as shown in Fig. 3. Δt is the sampling time step of the optimization, which can also be adapted in the model. The term $\sigma ^{CHP,fuel}_{t}$ shows the fuel costs of CHP for time step "t". In the future, for example when other on-site energy generation assets such as diesel generator are added, their respective costs must also be accounted for in the objective function. Modeling diagram of generic industrial enterprise model library - representation of modules and sub-modules in objected oriented manner The developed optimization formulations contains the boundary conditions such as production planning from MFS, process and operational constraints from TBS and ES. Based on these degrees of freedom, the optimizer decides the values of load demand of individual units, which affects Pgrid,t. Further, for any specific technical units, their direct load forecast can also be added as external parameters, and the EMC ignores the optimization formulation for those units. Currently, the nature of the optimization is that it solves for all the time steps in optimization horizon T and finds a global solution. However, as a later work it is intended to develop a cascaded optimization with a moving horizon approach. This will help to tackle and simulate the short-term disturbances in process parameters and fluctuation in price signals. The software implementation of the generic model library is done in object-oriented Python. The development style is taken as motivation from the structure of open-source power system modeling tool pandapower (Thurner et al. 2018), where different elements of power grids can be defined generically and added in a data-based structure. Likewise, in this industrial enterprise model library, each of the technical units and their sub-units are represented by separate modules as classes and sub-classes with generic inputs. Figure 3 shows the corresponding modeling diagram. Once all units are created, the control-relevant parameters of the defined technical units become available in the EMC block, and are used there in generic optimization formulations. Case study - simulation To apply and validate the proposed methodology, a real-world case of a steel-powder manufacturing plant layout was taken from Yu et al. (2016). As we want to showcase our methodology, we model the manufacturing process using MFS based on tasks and resources. Further, the case study has a nitrogen plant, which we ignore in our analysis, and add CAS, PHS and CHP as additional units. For this, we also assume our parameters concerning the developed methodology. The basic purpose is to test the models developed in this exemplary industrial company. Figure 4 shows the technical units, their sub-units and interdependencies via material and energy flow for the chosen case study. Overall representation of the exemplary industrial process as divided by MFS; TBS; ES. Process layout and pictures for MFS taken from Yu et al. (2016) The MFS is composed of manufacturing tasks starting from melting of iron and coke to a final product coming out from the blending machine. For simplification of modeling, the manufacturing task related to blast furnace is not taken into account as it is a complex process in itself as per (Yu et al. 2016). The specifications of the modeled tasks in MFS is provided in Table 6. The tasks of type "INT" can be interrupted and shifted to feasible times following its run time constraints, while tasks of type "UN-INT" cannot be interrupted once they start; however, their start times can be adjusted. In the listed tasks, FRF is of type "CONST", which means that it is executed at all times in the planning horizon and has a uniform electricity demand. Table 6 Parameters for the tasks in MFS The parameters for the machines used by the manufacturing tasks are provided in Table 7. Table 7 Parameters for the modeled machines in MFS The other resources in the modeled MFS include a raw material RM with initial quantity of 2000 tons which is fed into the first machine, a sprayer SPR. For each task a dedicated storage facility is considered with maximum storage facility capacity of 300 tons. As the represented MFS system is of continuous manufacturing type, for each task an output material product is defined which becomes the input material for the next manufacturing task. In the production line, the last task is blending BL. Hence, the material coming out of this task represents the final product, for which a minimum required value of 150 tons was given. This represents the quantity that will be produced at the end of the planning horizon by the production process, which is ensured by the optimizer. From Fig. 4, it can be seen that the chosen case study also contains PCS, PHS, CAS and HVAC as part of the TBS which serve as utilities to manufacturing operation in the MFS. The details of the components and their parameters are shown in Table 8. Table 9 lists the parameters of sub-units of ES system, which include a BES, CHP and PV. Table 8 Details of the technical units inside TBS in MFS Table 9 Details for the ES of the chosen case study The maximum amount of power to be exchanged with the grid was limited to 1000 kW. A simulation horizon of 24 hours with a sampling time step of 15 minutes was chosen as optimization parameters inside the EMC, which formulates the time window of the optimization formulation accordingly. For solving the optimization problem inside the EMC, the commercially available solver GUROBI (Gurobi Optimization LLC 2021) was used with academic license. The hardware on which simulation was performed included a Windows PC with four 3.2 GHz processors with 16 GB of RAM. The relative MIP optimality criterion was chosen as 1% in the solver, and it took under 40 seconds to solve the simulated optimization problems. For the analysis, the following pricing mechanisms were assumed: Fixed price where the value of λbuy is 0.256 €/kWh including all the taxes and levies that are applied to industrial consumers. Variable prices in range of 0.2 - 0.32 €/kWh based on the variability in given wholesale electricity market prices plus the applied taxes and levies. Results - optimal scheduling of MFS tasks and load profiles Figures 5 and 6 show the simulation results for the load profiles of the machines in the MFS with fixed and variable price mechanism respectively. The resulting load profiles are the outcome of an optimal scheduling of the manufacturing tasks, while also following the given constraints for the MFS. In the case of fixed prices, the optimizer decides to schedule the tasks in such a way that the self-consumption of PV is increased, which reduces the overall electricity purchase costs. With variable price mechanism, the optimizer decides to execute and shift the operations of the tasks to time intervals when prices are lower. Load profile of MFS with fixed price Load profile of MFS with variable prices Results - load profiles and states of TBS systems Figures 7 and 8 show the effect of volatile prices on the load profiles of the components and corresponding state variables inside TBS respectively. The optimal operation with variable prices results in a shift of the load demands of the devices to intervals where prices are lower as compared to that with fixed price scenarios. The flexibility in shifting of the loads is dependent on boundary conditions of both input and state variables. For instance, it can be observed from Fig. 7 that the operation of binary controlled devices i-e compressor K2, chiller C1, and heat-pump HP-1 follow the respective run-time constraints. Load profiles for the components inside the TBS systems for both fixed and variable prices State variables for the components inside TBS systems for both fixed and variable price mechanisms Further, Fig. 8 shows that state variables such as the pressure of CAS storage, the temperature of thermal storages in PCS and PHS, and indoor thermal temperature follow the dynamic prices without violating their provided boundary limits. Results - energy system with BES, CHP and PV Figure 9 shows the results of the optimization of the ES. The energy content of the battery almost remains constant for fixed prices, except that it uses the available PV power after 16:00 HRS. However, for variable prices the optimizer decides to charge and discharge actions in direct relation to volatile prices, and the energy content of the battery varies accordingly between the given capacity limits. Figure 9 shows that also the CHP unit operates flexibility with respect to variable prices. Energy content, charging and discharging profiles for BES and CHP inputs Results - overall load profile and flexibility assessment Figure 10 shows the industrial enterprise's total load demand and its power exchange with the distribution grid for both fixed and variable prices. Figure 10 also shows the flexibility provided as a comparison between fixed and variable prices. When the prices are low, the industrial enterprise offers negative flexibility (red bars) by increasing its power from grid, and when prices are high, it offers positive flexibility (blue bars) by lowering its power intake from grid. Load profile, power exchange with the grid and flexibility in both fixed and variable prices mechanism The contribution of each system to the overall flexibility is shown in Fig. 11. For the modeled case, the greatest flexibility comes from the adjustment of production related tasks in the MFS and the energy storage capability of the battery. The flexibility of the CAS, the PCS and the PHS is limited because of the low rated power of the devices due to the given process constraints. Breakdown of the flexibility provided by each system in response to the variable electricity prices Results - daily costs Figure 12 shows the daily cost comparison between four case scenarios for which the simulations were performed for the modeled case study. The daily costs for this case are the sum of the total daily electricity purchase cost and fuel costs for CHP operation. These scenarios chosen for cost comparison are, Case 1: Fixed prices with no BES and no CHP Daily cost comparison in € Case 2: Fixed prices with BES and CHP Case 3: Variable prices with no BES and no CHP Case 4: Variable prices with BES and CHP The obtained results for daily costs for Case 1, Case 2, Case 3 and Case 4 were 971 €, 945 €, 968€ and 912€, respectively. From the figure, it becomes clear that for the modeled case, cost reduction is more pronounced when the industrial enterprise uses its flexible assets (BES and CHP) with variable costs, where cost reduction between Case 1 and Case 4 was around 6%. The cost reduction depends on several factors such as variability in dynamic prices, electrical ratings of loads inside the defined technical units, the capacity of storage, and fuel prices for CHP (or other flexible assets such as diesel generators). Further, operational constraints and their limits can also affect the load flexibility and hence the costs as well. The goal of this work was to show the methodology and use of the developed model. A detailed cost-benefit analysis is to be carried out as part of the future work. Conclusion and outlook This paper presents a generic modeling framework with optimal energy management and control for industrial enterprises to define their manufacturing systems, utility processes, and energy systems at a modular level. The developed framework also includes generic optimization formulations that take care of production constraints in the MFS, process constraints in the TBS and physical constraints in the ES for the defined units while providing the flexibility. Electricity prices are the external input to the optimization algorithm's cost function and serve as the industrial enterprise's interface to electricity markets. The developed model framework was applied to a case study of a steel-powder manufacturing plant for validation and for assessing the energy flexibility in terms of variable price signals. The simulation results showed that the optimizer decides to shift the load profiles of defined technical units towards the lower price intervals using the available flexibility potential. For the modelled case, the MFS and the BES provided most of the flexibility. In the case study, the proposed modeling framework fulfilled the requirements of the research task posed at the beginning: The framework enables the modeling of an exemplary industrial company with regard to its energy flexibility with little effort on the basis of a manageable number of company-specific parameters. The flexibility potential to be achieved by dynamic prices and the profit to be expected for the company could be estimated. The authors are convinced that this framework can be applied in the same way to a large number of other industrial companies and in this sense fulfills the claim of being "generic". In terms of an outlook, it can be assumed that the digitalization of industrial enterprises will progress rapidly in the next few years and that this will also create new prerequisites for advanced energy management in these companies. Model libraries such as the one proposed here could then form the basis for "digital twins" of industrial enterprises energy flows and flexibility potentials that in many ways facilitate semi-automated and fully automated flexibility provision and marketing. In order to prepare and research this development, an experimental twin of industrial enterprise is being set up at Offenburg University of Applied Sciences in parallel to the model library, which provides selected subsystems of the MFS, TBS and ES in real hardware. This will then exchange data with the digital twin and its EMC software module via standard industrial communication systems and be controlled automatically by it. This will require the development of further model library features such as data transfer, communication and control for real hardware. From the modeling perspective, the work described is still ongoing as part of a research project. A non-exhaustive list of planned future developments is: Adding granularity to the developed models which includes the representation of additional process constraints and characterization parameters; Extending the developed optimization formulations towards rolling horizon or Model Predictive Control (MPC) for energy management such as in works of Habib et al. (2018); Dongol et al. (2018); Sawant et al. (2020). This will provide the model library an ability to deal with short term process disturbances and variations in electricity market signals for provision of flexibility; Extending the market block of the model library to represent new and innovative flexibility products for industrial enterprises and modeling of those in cost functions and performing cost-benefit analyses for industrial enterprises. For the developed case study production layout information has been taken from the example of Yu et al. (2016) with adaptations and assumption to represent the developed methodology. The weather data as outdoor ambient temperature has been taken from weather station located at Offenburg University of Applied Sciences. The PV generation profile has been taken from a real roof-top generation installed at one industrial enterprise participating in the research project. 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This work was funded by Germany's Federal Ministry of Economics and Energy (BMWi) for the ongoing project "GaIN - Gewinnbringende Partizipation der mittelständischen Industrie am Energiemarkt der Zukunft" (Fkz: 03EI6019E). Publication funding was provided by the German Federal Ministry for Economic Affairs and Energy. Institute of Energy Systems Technology (INES), Offenburg University of Applied Sciences, Badstraße 24, Offenburg, 77652, Germany Rahul Khatri, Michael Schmidt & Rainer Gasper Rahul Khatri RK, MS, and RG conceived the idea presented. RK worked on software implementation, performed formal analysis and research, wrote the initial draft, wrote and revised drafts, and performed visualization. MS and RG arranged resources, wrote and edited review drafts, conducted analysis, and performed activities involving project management, funding, and acquisition. All authors read and approved the final manuscript. Correspondence to Rahul Khatri. The authors declare that they have no potential conflict of interest in relation to the study in this paper. Khatri, R., Schmidt, M. & Gasper, R. Active participation of industrial enterprises in electricity markets - a generic modeling approach. Energy Inform 4, 20 (2021). https://doi.org/10.1186/s42162-021-00173-5 Energy systems modeling Demand side flexibility Optimization and control	CommonCrawl
How could you tell if someone was messing around with your gravity? I'm working on a story involving first contact with an alien species that bases their space travel on directly manipulating gravity fields. My question is not about how that would work, but rather how would you be able to DETECT it working? More specifically: My intrepid human explorers are operating in at a technology level sufficiently advanced to allow interstellar travel, but not advanced enough to involve Faster-Than-Light technology of any kind. They are newly arrived in a previously unexplored system, and during the encounter with previously mentioned aliens, the aliens start moving the human's ship. So: If you're in interplanetary space (e.g. not close to a planet), and something creates an artificial gravity well which alters the orbital trajectory of your vehicle, how would you know what had happened? Obviously if you're paying close attention to your relative position with the planets and the star itself you'd notice that SOMETHING had altered your vector, but what other instrumentation would notice? The ideal answer would involve something that generates a "Well of COURSE any reasonably well-equipped scientific spacecraft would have one of those." reaction from the reader, rather than a "Wow, they're lucky they had one of those that they probably never thought they'd need or use." EDIT: You should be imagining the Endurance from the movie Interstellar, except mine isn't specifically exploring a black hole, so my ship would be even LESS likely to have specialized instrumentation to detect gravitational anomalies. space-travel hard-science gravity astrophysics Morris The Cat Morris The CatMorris The Cat This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information. $\begingroup$ Is this actually a "scientific spacecraft? $\endgroup$ – RonJohn Sep 1 '18 at 1:10 $\begingroup$ @RonJohn 100%. We're not talking Star Trek "heavily armed cruiser with a science lab" here. Imagine something much closer to the spaceship from Interstellar. $\endgroup$ – Morris The Cat Sep 1 '18 at 1:54 $\begingroup$ I didn't see that movie. $\endgroup$ – RonJohn Sep 1 '18 at 1:59 $\begingroup$ @RonJohn Buckaroo Banzai's RV? $\endgroup$ – Morris The Cat Sep 1 '18 at 2:10 $\begingroup$ Saw that, but a long while ago and don't remember much except that it was... different. (Good, but different.) $\endgroup$ – RonJohn Sep 1 '18 at 2:22 Measuring gravity to high precision is (relatively) easy, and doesn't need (much) high-tech equipment. An interstellar space ship -- even a warship -- will have enough equipment on board that this experiment could be performed. https://www.nature.com/articles/s41586-018-0431-5 The Newtonian gravitational constant, G, is one of the most fundamental constants of nature, but we still do not have an accurate value for it. Despite two centuries of experimental effort, the value of G remains the least precisely known of the fundamental constants. A discrepancy of up to 0.05 per cent in recent determinations of G suggests that there may be undiscovered systematic errors in the various existing methods. One way to resolve this issue is to measure G using a number of methods that are unlikely to involve the same systematic effects. Here we report two independent determinations of G using torsion pendulum experiments with the time-of-swing method and the angular-acceleration-feedback method. We obtain G values of 6.674184 × 10−11 and 6.674484 × 10−11 cubic metres per kilogram per second squared, with relative standard uncertainties of 11.64 and 11.61 parts per million, respectively. These values have the smallest uncertainties reported until now, and both agree with the latest recommended value within two standard deviations. If you think that They are fiddling with gravity, start taking measurements on a regular basis, and especially during a "gravitational anomaly event". Noticing any changes in G should tell you if They -- or Something -- are actually fiddling with gravity or you need to look somewhere else. RonJohnRonJohn $\begingroup$ Would anybody be likely to have an instrument like this operating if they didn't ALREADY think someone might be messing with gravity though? I'm specifically thinking of the scene that leads to the "Hey... they might be messing with gravity!" inspiration. $\endgroup$ – Morris The Cat Sep 1 '18 at 1:30 $\begingroup$ @MorrisTheCat funny thing, I was just thinking of a response to that. I think that you can use very similar apparatus to measure weight (it may even be the same thing, idr). So I would assume they would have a torsion scale and just be measuring hyper-fine differences in weight between whatever they were normally sciencing. Seeing it shift all of a sudden with no loads would definitely raise suspicion. Which you could then more accurately test I presume. $\endgroup$ – Black Sep 1 '18 at 1:38 $\begingroup$ Also (once again for today XD), Any local gravity worth noticing the effect of is going to have a higher gravitational gradient the smaller the source is. In other words, a scale is going to pick up the variation just fine I would assume (unless the whole point of a torsion scale is to negate that effect?.... research time). $\endgroup$ – Black Sep 1 '18 at 1:41 $\begingroup$ Can't remember the name of the other apparatus... and a torsion balance/pendulum is very specific (although why not add it to your kit? Aids in determining G of foreign planets which is relevant to just about all branches of science... even Chemistry)...That said I found this specifically the small size leading to an array of sensors for redundancy, which would (part of noise filtering) basically take a picture of local gravity shape; and the last paragraph which gives you a reason to have them. $\endgroup$ – Black Sep 1 '18 at 1:53 $\begingroup$ @MorrisTheCat maybe they'll have a G-measuring device as part of a suite of devices for testing whether or not the laws of physics that we've developed on Earth really are as universal as we think they are. Or they cobbled together the experiment from bits and bobs (small vacuum chamber, torsion bars, etc) in the ship's various equipment closets and laboratories. Remember: they're a long way from home and so must bring all sorts of stuff with them to meet many unusual contingencies. $\endgroup$ – RonJohn Sep 1 '18 at 1:58 What about... A human! Humans are great at detecting changes in acceleration, which is what a gravity change would feel like. If your ship has been traveling in a straight line on inertia alone, as long-distance ships are probably doing, running into a gravity field will feel like you've taken a sharp turn. Everything in the spacecraft not-tied down will likely crash into a nearby wall. If you've got a human on board, they'll probably notice. Even if the human is tied down or very distracted, they'll likely experience a sense of vertigo or confusion as the otoliths in their inner ear move about unexpectedly. DubukayDubukay $\begingroup$ This is useful, although it doesn't directly solve my problem. I'm envisioning a progression of "What the heck, we're moving?? Why are we moving?" leading to "hey, something MOVED us, how did they do that??" leading to "They must be messing around with our gravity...". This answer is very helpful for the first part of that, since I wasn't sure if a human inside a sealed spacecraft would even be able to TELL if the entire spacecraft started moving in a new direction due to gravitational pull. $\endgroup$ – Morris The Cat Sep 1 '18 at 1:33 $\begingroup$ @MorrisTheCat Everything would stay stuck to the wall after they felt the acceleration stop, including non metallic objects. Since the objects will continue to move towards the wall even after being moved away, gravity would be the first assumption. $\endgroup$ – Clay Deitas Sep 1 '18 at 2:32 $\begingroup$ @Clay Deitas that assumes the ship is applying its own thrust vector, doesn't it? If your ship isn't accelerating on its own, then the only acceleration ANYTHING would be feeling would be the gravitational field, and everything would react to it starting, stopping, or changing exactly the same way, wouldn't it? $\endgroup$ – Morris The Cat Sep 1 '18 at 4:05 $\begingroup$ @ClayDeitas I'm missing something here... why would anything hit the wall? The same force is being applied to the wall that's being applied to everything else. If the entire ship is (effectively) stationary, and our hypothetical aliens create spacetime curvature equal to the mass of Luna at a distance of ~1000km. The entire ship, humans, objects, etc, would start accelerating towards that point. If you turn it off, everything stops accelerating simultaneously. The only way the wall does anything different from the pen (or whatever) is if the wall is attached to something creating thrust. $\endgroup$ – Morris The Cat Sep 1 '18 at 4:21 $\begingroup$ There's your detection method. The crew knows their course changed. They would feel acceleration if the deviation was due to a conventional thrust method. Because they can't, it must be something that's accelerating the whole ship evenly, which is artificial gravity - or so near as to be indistinguishable. $\endgroup$ – Cadence Sep 1 '18 at 9:02 Equip your starship with sensors measuring the structural load at various points along its frame. In normal flight, this assures you that a) your engines are producing the thrust they're supposed to, and b) your spaceframe is still in one piece. It's especially valuable if your ship is supposed to perform any very-high-precision maneuvers or if you anticipate taking it into atmosphere at any point. More importantly, though, the signature of gravity accelerating your ship will be different than conventional means of acceleration: gravity will affect your whole ship more or less evenly, whereas conventional thrust will produce a pattern of stresses depending on the shape of your frame. Another way to look at it is that thrust originates from one point (the thruster) and is spread to the rest of the ship by the frame, whereas gravity acts on every point in the ship at once. CadenceCadence $\begingroup$ Ahh, so basically this would measure the tidal forces applied by different parts of the ship being close or further from the gravitational gradient... yeah... this might work. I'm not sure if I think something that a scientific probe would have though, unless they specifically had a reason to be looking for gravitational anomalies which I'm not sure they would. $\endgroup$ – Morris The Cat Sep 1 '18 at 1:37 $\begingroup$ That's why I would suggest pitching it to the reader as being a structural integrity or maneuvering/thrust sensor - something that they would be monitoring during normal flight just in case something went wrong with e.g. the engines. $\endgroup$ – Cadence Sep 1 '18 at 1:49 Well of course they use radar... If there is no FTL technology, then good old fashioned radar is still the best way to do range finding. Radar range finding off of multiple stars/planets should give you positional accuracy of less than one meter, easily, assuming enough computational power to handle the intricacies of Doppler effect and distance to target (minutes or more, in many cases). Any ship at sea will use radar to make sure it doesn't hit something. Any ship in space would want to use a navigational radar both to be on the look out for various small objects that you might run in to and to keep an accurate position relative to whatever planets/stars/celestial objects are nearby. I think that determination of position change is pretty trivial, and any navigational computer would detect an induced course change within a few minutes at most. For example, the navigation computer that I used 10 years ago in the US Navy would have told me about a ~1 degree course change within 5-10 minutes, as we started to deviate from our track towards a pre-set navigational waypoint. Also, I had a navigator on my bridge team whose job was specifically to tell me about such things. However, that was a military ship, a merchant ship would not have a full time navigation specialist on watch. An exception could be if the ship is doing something that causes it to transfer momentum; then unexpected distance changes might be harder to notice. Examples might be launching a shuttle, or transferring cargo to a nearby ship or something. ...unless you are in battle The only good reason to turn off your radar is if you are in some sort of wartime condition. Warships on Earth do this as well. There is some debate as to whether trying to hide is viable in space; I'm in the 'there is some stealth in space' camp so I think a military vessel would turn off its active sensors to try to be less obvious. That being said, there are alternatives. Directed beams like lidar would be nearly undetectable unless you are in just the right direction from the offending vessel, so you could still calculate your position from them. I don't know what the protocols would be for military warships in space, but there has to be some accommodation for safe navigation. kingledionkingledion $\begingroup$ This is useful, but a much more combative take than I was envisioning. The scenario is more like "A bunch of scientists in a mobile laboratory suddenly realize that someone is MOVING their lab and they don't know how.": $\endgroup$ – Morris The Cat Sep 1 '18 at 1:34 $\begingroup$ @MorrisTheCat Radar and a nav computer is the way that they would first discover that someone was moving their lab. 99% sure on that; as long as you are pre-FLT, radar is the best thing for rangefinding. $\endgroup$ – kingledion Sep 1 '18 at 1:40 $\begingroup$ How many years does one has to wait to complete a single radar measure? $\endgroup$ – L.Dutch♦ Sep 1 '18 at 3:40 $\begingroup$ @L.Dutch The OP says they are in a system, in some sort of orbit. So not that long. $\endgroup$ – kingledion Sep 1 '18 at 6:10 $\begingroup$ But your answer says "radar range finding off of multiple stars / planets". Which, of course, takes decades, requires the power of a star, and still won't work because radio waves don't bounce off seething balls of fusion. I'm also dubious as to whether they'd bounce off of gas giants at any detectable signal strength beyond it's own moons. $\endgroup$ – RonJohn Sep 1 '18 at 7:17 The scientists are in a large space station using lasers to more precisely measure gravity waves. One of the major functions of science is to reconcile all the forces which dictate the functions of the universe. Magnets and electricity were reconciled into electromagnetism. Your scientists will be in space helping to reconcile gravity and the other forces. Except out of no where large gravity waves appear, which is either the result of a major cosmic event, or a close by source of gravity. Your aliens who have already reconciled gravity with some of the other forces are able to use it in their technology. Everything gets wrapped up in a nice little package. Clay DeitasClay Deitas It's all in the creaking Generally speaking, gravity acts uniformly on all objects within its field, so existence within a gravity field feels exactly like freefall. So it almost seems like a perfect way to move a vessel without anyone detecting it. That being said, there may be an inescapable flaw in using an artificial gravity well, especially if it is too close, due to the inverse square law. The acceleration caused by gravity is proportional to the square of the distance from the center of the well. So in theory, on Earth, you feel a different amount of acceleration affecting your head versus affecting your feet, because your feet are closer to the center of the Earth. But Earth is so large, this difference is very very small. But with an artificial gravity well, which presumably is smaller than the Earth, the difference could easily be detectable. The acceleration due to gravity is computed as $acceleration=(gravitational \ constant) \times (mass \ of \ the \ body)/(distance)^2$ So plugging in some basic numbers, if you were to feel acceleration of 1 gravity at 200 meters, you would only feel about 0.98 g at 202 meters. So the height of a man yields an accelerational difference of 0.2 m/sec^2, possibly enough a person to detect, although possibly not. However, the ship itself is much longer than 6 feet (I hope). If the front end of the ship is closer to the gravity field and the tail is farther away, the ship would "stretch," i.e. the tip would be pulled harder than the tail. This may not cause any damage, but it may cause a certain amount of creaking or shuddering, and passengers may even be able to see the hull distort slightly, the same way you can detect the fuselage on an airplane changing shape if you pay careful attention. John WuJohn Wu $\begingroup$ You can see what do what on an airplane? $\endgroup$ – Clay Deitas Sep 1 '18 at 6:17 $\begingroup$ Have you never sat in the back seat and seen parts of the passenger compartment seem to sag? It is definitely detectable by the naked eye. $\endgroup$ – John Wu Sep 1 '18 at 6:27 $\begingroup$ No I haven't, and I hope I never do. Some kind of damn nightmare fuel going to make it impossible to see planes as non flying deathtraps. $\endgroup$ – Clay Deitas Sep 1 '18 at 7:05 $\begingroup$ How flexible is an aircraft fuselage? Sleep well. $\endgroup$ – John Wu Sep 1 '18 at 9:33 $\begingroup$ Please use Mathjax to format formulas $\endgroup$ – L.Dutch♦ Sep 1 '18 at 9:53 Not the answer you're looking for? Browse other questions tagged space-travel hard-science gravity astrophysics or ask your own question. Occultic FTL Drive When two planets are very close, what is the environment like? Could an advanced species, having evolved on a large planet with a deep gravity well, be helped out of it from above? Effect on our climate if gravity got lowered by alien tech Runaway Starship Ramps Using Jupiter as a Gravity Gun Zero Gravity Evolution in People How do you detect a rock in interstellar space? Ways to cushion an un-augmented human against high, prolonged g-forces?	CommonCrawl
Journal Home About Issues in Progress Current Issue All Issues Feature Issues Issue 24, pp. 37026-37039 •https://doi.org/10.1364/OE.403330 Localized photonic states and dynamic process in nonreciprocal coupled Su-Schrieffer-Heeger chain Wen-Xue Cui, Lu Qi, Yan Xing, Shutian Liu, Shou Zhang, and Hong-Fu Wang Wen-Xue Cui,1,2 Lu Qi,1 Yan Xing,1 Shutian Liu,1,3 Shou Zhang,2,4 and Hong-Fu Wang2,5 1School of Physics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China 2Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China [email protected] [email protected] [email protected] Shutian Liu https://orcid.org/0000-0001-9748-0175 Hong-Fu Wang https://orcid.org/0000-0002-6778-6330 W Cui L Qi Y Xing S Liu S Zhang H Wang Wen-Xue Cui, Lu Qi, Yan Xing, Shutian Liu, Shou Zhang, and Hong-Fu Wang, "Localized photonic states and dynamic process in nonreciprocal coupled Su-Schrieffer-Heeger chain," Opt. Express 28, 37026-37039 (2020) Transmission through a one-dimensional photonic lattice modulated by the side-coupled... Piao-Piao Huang, et al. J. Opt. Soc. Am. B 38(4) 1331-1340 (2021) Photonic two-particle quantum walks in Su–Schrieffer–Heeger lattices Friederike Klauck, et al. Photon. Res. 9(1) A1-A7 (2021) Engineering zero modes, Fano resonance, and Tamm surface states in the waveguide-array realization... Ying Yang, et al. Opt. Express 27(23) 32900-32911 (2019) Table of Contents Category Quantum Optics Nodal points Photons Quantum information processing Original Manuscript: July 22, 2020 Revised Manuscript: November 6, 2020 Manuscript Accepted: November 6, 2020 Energy eigenvalue spectrum and state distributions for 1D nonreciprocal coupled SSH chain Dynamic process in 1D nonreciprocal coupled SSH chain Realization of quantum simulation for 1D nonreciprocal coupled SSH chain Figures (11) We investigate the localized photonic states and dynamic process in one-dimensional nonreciprocal coupled Su-Schrieffer-Heeger chain. Through numerical calculation of energy eigenvalue spectrum and state distributions of the system, we find that different localized photonic states with special energy eigenvalues can be induced by the nonreciprocal coupling, such as zero-energy edge states, interface states and bound states with pure imaginary energy eigenvalues. Moreover, we analyze the dynamic process of photonic states in such non-Hermitian system. Interestingly, it is shown that the nonreciprocal coupling has an evident gathering effect on the photons, which also break the trapping effect of topologically protected edge states. In addition, we consider the impacts of on-site defect potentials on the dynamic process of photonic states for the system. It is indicated that the photons go around the defect lattice site and still present the gathering effect, and different forms of laser pulses can be induced with the on-site defect potentials in different lattice sites. Furthermore, we present the method for the quantum simulation of current model based on the circuit quantum electrodynamic lattice. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement After the concept of topology is introduced into the field of condensed matter physics, it rapidly promotes the development of fundamental study of physics. Many novel topological materials have also emerged. Su-Schrieffer-Heeger (SSH) model is one of the most simplest one-dimensional (1D) topological insulators, which is originally used to describe the polyacetylene [1]. The most important and interesting feature of the SSH model is the topologically nontrivial edge states protected by time-reversal symmetry. Based on this model, many efforts have been devoted to investigating topological phase [2–4] and topological states [5–9]. Recently, many attentions have been paid to two-coupled SSH model, which present more intriguing physical phenomena in the investigation of the topological nodal points [10], topological states [11], and topological phase transition [12,13]. Moreover, the investigation of topological states are also connected with periodically driven system [14,15]. Moreover, the non-Hermitian extension of topological system is becoming one of the research focuses. A series of novel phenomena have been found in non-Hermitian topological system, such as bulk-boundary correspondence breaking [16–19], non-Hermitian skin effect [20–22], and fractional topological number [23–25]. Most recently, Weidemann et al. demonstrated a highly efficient funnel for light by utilizing the non-Hermitian skin effect [26]. Moreover, many schemes have been proposed to investigate the SSH model with parity-time (${\mathcal {P}T}$) symmetry [27–30] and nonreciprocal coupling [16,31]. It benefits from advantage of topologically protected edge state with robustness against perturbations and defects, many schemes have been proposed to realize the topological lasers based on the theory and experiments [32–35]. On the other hand, with the rapid development of micro- and nanofabrication technologies, circuit QED system has become one of the most prospective platforms for the realization of quantum information processing [36–43], quantum computing [44,45], and quantum simulation [46,47]. Circuit QED system describes the interaction between the nonlinear superconducting circuit and the photon storing in the transmission line resonator. Similar to the energy level structure of the atom, superconducting qubit also has ground state and excited state. In the study of quantum information processing, just the lowest two energy levels can be considered, and the other higher energy levels are often negligible. When a superconducting qubit is coupled to a resonator, the quantum state of superconducting circuit can be manipulated and read out by the detector [48–50]. Moreover, circuit QED lattice system can be constructed when the highly coherent superconducting quantum qubits and microwave resonators are arranged in a periodic array, which can map to different one dimensional or even high dimensional topological systems [47,49,51–53]. Furthermore, resorting to the bosonic statistical properties of the circuit QED lattice, it is easy to realize the detection of the topological edge states and topological invariants [47,49]. Experimentally, the simulation of quantum spin has been realized in the circuit QED system, which provides a method for controlling and simulating spin-lattice dynamics [54]. Inspired by above, we investigate the different localized states in one-dimensional (1D) nonreciprocal coupled SSH chain, where the nonreciprocal coupling depends on the direction of photons tunneling. For the nonreciprocal coupling strength $\Delta =0$, the current system is decoupled into two independent chains with SSH model structure. To investigate the influence of the nonreciprocal coupling on the system, we numerically calculate the energy eigenvalue spectrum of the system. It is shown that the nonreciprocal coupling can induce different localized photonic states with special energy eigenvalue both in the topologically nontrivial regime and topologically trivial regime, such as the zero-energy edge states, the interface states and bound states with pure imaginary energy eigenvalues. We also analyze these localized photonic states from the perspective of state distributions. Moreover, we investigate the dynamic process of photonic states in current non-Hermitian system, which reveals that the nonreciprocal coupling has a gathering effect on the photons and break the trapping effect of the topologically protected edge states. In addition, we consider the influence of defect potential on the dynamic process of the system, which can induce different forms of laser pulses. Particularly, we find that the gathering effect caused by the nonreciprocal coupling is immune to the on-site defect potential. Further, we propose the method for the quantum simulation of current model based on the circuit QED lattice. The rest of this paper is structured as follows. In Sec. 2, we show the physical model and the corresponding Hamiltonian of the system and investigate the energy eigenvalue spectrum and the state distributions of the system. In Sec. 3, we investigate the dynamic process of the photonic states of the system. In Sec. 4, we present the quantum simulation of current model by using the circuit QED lattice. Finally, we give the conclusion in Sec. 5. 2. Energy eigenvalue spectrum and state distributions for 1D nonreciprocal coupled SSH chain As shown in Fig. 1, consider the nonreciprocal coupling model, which consists of two identical SSH chains. The corresponding Hamilton of the system can be written as (1)$$\begin{aligned} H=&\sum_{j=1}^{L-2}\left(t_{1}a^{\dagger}_{j}a_{j+1}+t_{2}a^{\dagger}_{j+1}a_{j+2}+b^{\dagger}_{j}b_{j+1}+t_{2}b^{\dagger}_{j+1}b_{j+2}+\rm{H.c.}\right)\cr &+\Delta a_{L}b^{\dagger}_{1}-\Delta b_{1}a^{\dagger}_{L}, \end{aligned}$$ where $t_{1}=t\left (1-\delta \cos \theta \right )$ and $t_{2}=t\left (1+\delta \cos \theta \right )$ are the intracell and intercell nearest neighbor hopping amplitudes, $\theta$ is a periodic parameter varying from $-\pi$ to $\pi$, and $\delta$ is the periodically modulated amplitude. The nonreciprocal coupling strengths $\pm \Delta$ between chain $\textrm {I}$ and chain $\textrm {II}$ depend on the direction of particles hopping. The hopping from the site $a_{j}$ to $b_{1}$ corresponds to the coupling strength $\Delta$. Instead, the hopping from the site $b_{1}$ to $a_{j}$ corresponds to the coupling strength $-\Delta$.Obviously, in the case of $\Delta =0$, the 1D non-Hermitian system is decoupled into two independent chains which possess SSH model structure with qubit-assisted on-site potential $\xi$. Under the open boundary condition, this model has topologically nontrivial phase in the regime of $-\pi /2< \theta < \pi /2$ characterized by the gapless zero-energy edge states with even-numbered sites. While, in the regimes $-\pi < \theta < -\pi /2$ and $\pi /2< \theta < \pi$, the system without zero-energy edge states belongs to the topologically trivial phase. For odd-numbered sites, this model holds a single zero-energy edge state in the whole parameter regions $\theta$. This is the typical even-odd effect of the SSH model. Fig. 1. Schematic diagram of 1D nonreciprocal coupled SSH chain with alternating coupling strength. Download Full Size \| PPT Slide \| PDF 2.1 Energy eigenvalue spectrum analysis In this section, we focus on the influence of nonreciprocal coupling strength on the system. We numerically calculate the energy eigenvalue spectrum of the system and present them as a function of $\theta$ with $L_\textrm {I}=50$ and $L_\textrm {II}=50$, as shown in Fig. 2. Here, $t$ is taken as the unit of energy, and the periodically modulated amplitude $\delta$ is set to be 0.5 throughout this paper. Fig. 2. The real and imaginary parts of the energy eigenvalue spectrum as a function of $\theta$ for 1D nonreciprocal coupled SSH chain, where we set $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the nonreciprocal coupling strengths (a) $\Delta =0.1$, (b) $\Delta =0.5$, (c) $\Delta =0.8$, (d) $\Delta =1$, (e) $\Delta =1.5$, (f) $\Delta =2$, (g) $\Delta =2.5$, and (h) $\Delta =3$, respectively. We first consider the topologically nontrivial regime $-\pi /2< \theta < \pi /2$. In the case of $\Delta =0.1$, the real part of the energy eigenvalue spectrum has a similar structure comparing with the standard SSH model, as shown in Fig. 2(a). The system possesses midgap modes in this regime. After checking the imaginary part of the energy eigenvalue spectrum, we find that the system holds two degenerate zero-energy edge states, and the energy eigenvalues of the rest ($L_\textrm {I}+L_\textrm {II}-2$) eigenstates are complex value. Notably, there also exists two interface states with pure imaginary energy eigenvalues (the real parts are degenerate at $E=0$), and the maximum and minimum absolute values of imaginary part occur at $\theta =0$ and $\theta =\pm \pi /2$, respectively. With the increase of $\Delta$, these two interface states gradually disappear and finally become two bound states with pure imaginary energy eigenvalues. While, the two degenerate zero-energy edge states can still exist. It is indicated that these zero-energy edge states are immune to the variation of the $\Delta$, as shown in Figs. 2(b)–2(h). Next, we investigate the energy eigenvalue spectrum of the system in the topologically trivial regimes, i.e., $-\pi < \theta < -\pi /2$ and $\pi /2< \theta < \pi$. For $\Delta ~\le ~0.5$, the energy eigenvalue spectrum of the system possesses ($L_\textrm {I}+L_\textrm {II}$) complex energy eigenvalues with the form of $\pm a \pm ib$, as shown in Figs. 2(a)–2(b). In Figs. 2(c) and 2(d), we depict the energy eigenvalue spectrum of the system with $\Delta =0.8$ and $\Delta =1$, respectively. There exists four bound states with complex energy eigenvalue. The maximum absolute values of imaginary part appear at $\theta =\pm \pi$. However, with the increase of $\Delta$, the minimum absolute values of imaginary part gradually spread to the phase boundary points $\theta =-\pi /2$ and $\theta =\pi /2$, and finally reach to them for $\Delta =1$. When $\Delta =1.5$, one can see that the imaginary parts of energy eigenvalue spectrum split into two branches nearby the phase boundary points $\theta =\pm \pi /2$, as shown in Fig. 2(e). To continue to increase $\Delta$, these branches with pure imaginary energy eigenvalue gradually spread to $\theta =-\pi$ and $\theta =\pi$ and finally form two new interface states and two bound states with pure imaginary energy eigenvalues (the real parts are degenerate at $E=0$), as shown in Figs. 2(f)–2(h). To see the change of energy eigenvalue spectrum of the system more clearly, we plot the real and imaginary parts of them as a function of the nonreciprocal coupling strength $\Delta$ for the system in different topological phases, as shown in Fig. 3. In Fig. 3(a), we show the energy eigenvalue spectrum of the system with $\theta =0$, and the system is in the topologically nontrivial phase. It is found that the system holds two degenerate zero-energy edge states in the whole parameter regions $\Delta$. Notably, one pair of imaginary part separates from the bulk states as long as $\Delta \not =0$, and the absolute value of energy eigenvalue presents an upward trend with the increase of $\Delta$, which corresponds to two interface states of the system (the real parts are degenerate at $E=0$). For $\theta =\pi /2$, this phenomenon begins to emerge at the phase boundary point when $\Delta > \Delta _{c,\pi /2}~=~1$, as shown in Fig. 3(b). For $\theta =\pi$, i.e., in the topologically trivial regime, one can see that two pairs of imaginary parts separate from the bulk states at $\Delta _{c_{1},\pi }=0.52$, which correspond to two pairs of degenerate bound states with complex energy eigenvalues. When the nonreciprocal coupling strength $\Delta _{c_{2},\pi }> 3$, these degenerate bound states with complex energy eigenvalues split into two branches and become two new interface states and two bound states with pure imaginary energy eigenvalues (the real parts are degenerate at $E=0$). Fig. 3. The real and imaginary parts of the energy eigenvalue spectrum as a function of $\Delta$ for 1D nonreciprocal coupled SSH chain. Here, we set $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the periodic parameters (a) $\theta =0$, (b) $\theta =\pi /2$, and (c) $\theta =\pi$, respectively. Furthermore, we also introduce the degree of complex energy levels, which is defined as the number of complex energy levels over the total number of energy levels. It can be used to indirectly determine the number of the zero energy existing in multiple degenerate levels. As shown in Fig. 4(a), we plot the degree of complex energy levels of the system as a function of nonreciprocal coupling strength $\Delta$ for choosing $\theta =0$. We note that the degree of complex energy levels is equal to 0.98 in the whole parameters regions $\Delta$. It is indicated that the system exists 98 complex energy levels, and the remaining two energy levels correspond to the two degenerate zero-energy edge states in the topologically nontrivial regime. Then, we show the degree of complex energy levels with $\theta =\pi /2$ and $\theta =\pi$, as shown in Figs. 4(b) and 4(c). We find that the degree of complex energy levels are both equal to 1. It is found that the corresponding energy eigenvalues both in the phase boundary and the topologically trivial phase are entirely complex energy eigenvalue. These results are consistent with previous analyses. Fig. 4. The degree of complex energy levels of the system, where $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$, and the periodic parameters (a) $\theta =0$, (b) $\theta =\pi /2$, and (c) $\theta =\pi$, respectively. The degrees of complex energy levels with blue line and pink line are equal 0.98 and 1. 2.2 State distributions analysis In above discussion, we have investigated the energy eigenvalue spectrum of the system. It is found that the nonreciprocal coupling can induce different localized photonic states with special energy eigenvalues, such as zero-energy edge states, interface states and bound states with pure imaginary energy eigenvalues. In the following, we analyze these localized photonic states of the system from the perspective of state distributions. In Fig. 5, we plot the state distributions of the system versus the lattice site $j$ and the nonreciprocal coupling strength $\Delta$ in different topological regimes. We first consider the state distributions with $\theta =0.35\pi$, as shown in Fig. 5(a). One can see that the interference fringes appear around the 1st and 100th lattice site and gradually disappear towards the center lattice site. Meanwhile, these interference fringes are not affected by the variation of nonreciprocal coupling strength. This phenomenon characterizes that the system holds two edge states steadily localized at two ends of system, which are immune to the effects of nonreciprocal coupling strength. On the other hand, for small $\Delta$, the same phenomena appear around the two middle lattice sites, which correspond to the left and right interface states of the system. With the increase of $\Delta$, we find that these interference fringes gradually disappear, and the maximum values of state distributions are in the 50th and 51st lattice sites. This process corresponds to the formation of two bound states. As shown in Fig. 5(b), we depict the state distributions of the system versus the lattice site $j$ and the nonreciprocal coupling strength $\Delta$ with $\theta =0.65\pi$. For $\Delta < 2$, the state distribution at each site are almost equal to 0, i.e., all the energy eigenstates are extended states. For $\Delta > 2$, we find that the interference fringes appear around the 49th and 51st lattice sites and gradually disappear towards the two boundaries of the system, which characterizes the formation of new left and right interface states of the system. In addition, the two bound states are always localized at the two middle lattice sites. Fig. 5. (a) The state distributions $\|\psi _{(50)}\|^{2}$, $\|\psi _{(51)}\|^{2}$ (two red zero-energy edge states in Fig. 2) and $\|\psi _{(49)}\|^{2}$, $\|\psi _{(52)}\|^{2}$ (two red interface states in Fig. 2) versus the nonreciprocal coupling strength $\Delta$ in topologically nontrivial regime $\theta =0.35\pi$. (b) The state distributions $\|\psi _{(50)}\|^{2}$, $\|\psi _{(51)}\|^{2}$ (two red bound states in Fig. 2) and $\|\psi _{(49)}\|^{2}$, $\|\psi _{(52)}\|^{2}$ (two new red interface states in Fig. 2) versus the nonreciprocal coupling strength $\Delta$ in topologically trivial regime $\theta =0.65\pi$. The other parameters are selected as $L_\textrm {I}=50$, $L_\textrm {II}=50$, and $\delta =0.5$, respectively. In the following, we present the state distributions of localized photonic states in nonreciprocal coupled SSH chain. As shown in Fig. 6(a), we find that two degenerate zero-energy edge states are highly localized at both ends of the system with $\theta =0$ and $\Delta =0.1$. Meanwhile, the system possesses two interface states with pure imaginary energy eigenvalues $+0.0889i$ and $-0.0889i$ at two middle lattice sites. In Fig. 6(b) we plot the state distributions of the localized states with $\theta =0$ and $\Delta =3$. It is noted that two degenerate zero-energy edge states are always highly localized at both ends of the system. However, the original two interface states become the two bound states with pure imaginary energy eigenvalues $+2.932i$ and $-2.932i$, which are localized at two middle lattice sites. Next, we consider the state distributions of the bound states with $\theta =\pi$ and $\Delta =0.8$, as shown in Fig. 6(c). Two pairs of degenerate bound states are localized at 49th, 50th, 51st, and 52th lattice sites with complex energy eigenvalues $1.3930+0.2437i$, $-1.3930-0.2437i$, $1.3930+0.2437i$, and $-1.3930-0.2437i$, respectively. In Fig. 6(d), we show the state distributions of the bound states $\theta =\pi$ and $\Delta =3$. In Fig. 6(d), the original bound states become two bound states with pure imaginary energy eigenvalues $1.102i$ and $-1.102i$, which are localized at two middle lattice sites. In addition, two new interface states appear at the 49th and 52th lattice sites with pure imaginary energy eigenvalues $1.184i$ and $-1.184i$. Fig. 6. The state distributions of localized photonic states for the system, where $L_\textrm {I}=50$, $L_\textrm {II}=50$, $\delta =0.5$. (a) $\theta =0$ and $\Delta =0.1$, (b) $\theta =0$ and $\Delta =3$, (c) $\theta =\pi$ and $\Delta =0.8$, (d) $\theta =\pi$ and $\Delta =3$. The red line, blue line, blue dotted line, green line, and green dotted line represent the state distributions of the zero-energy edge state, interface states with pure imaginary energy eigenvalues, new interface states with pure imaginary energy eigenvalues, bound states with complex energy eigenvalues, and bound states with pure imaginary energy eigenvalues, respectively. 3. Dynamic process in 1D nonreciprocal coupled SSH chain In this section, we investigate the dynamic process of the photonic states in nonreciprocal coupled SSH chain. The parameters are chosen as $\Delta =0.5$, $\theta =0$, $L_\textrm {I}=10$, and $L_\textrm {II}=10$, as shown in Fig. 7. In Fig. 7(a), when the 10th lattice site is excited, we find that the photons are localized on the two middle lattice sites with time evolution, which correspond to the site position of nonreciprocal coupling for the system. Then, when we excite the 5th and 15th lattice sites belonging to the bulk sites, the dynamic process of photonic states firstly presents the ballistic diffusion and finally shows the localization on the two middle lattice sites of the system, as shown in Figs. 7(b) and 7(c). In Figs. 7(d) and 7(e), for exciting the one and two lattice sites at the boundary, the photons are firstly trapped on the boundary with time evolution and finally localized on the two middle lattice sites of the system. When all the lattice sites are excited, one can see that all the photons are equally localized on the two middle lattice sites with time evolution. The above phenomena indicate that the nonreciprocal coupling of the system induces the gathering effect of the photons, which makes the dynamic process of the photons states towards the site position of nonreciprocal coupling. Fig. 7. The dynamic process of the photonic states in 1D nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, and $\Delta =0.5$, respectively. The excited positions are the 10th, 5th, 15th, 1st, 1st and 20th, and the whole lattice sites, corresponding to Figs. (a)-(f), respectively. Next, we show the dynamic process of photonic states with strong nonreciprocal coupling strength $\Delta =3$. The selected initial states and other parameters are same as that in Fig. 7. It is found that the gathering effect of the photons still appear in the system when we excite the lattice sites at the middle site, the bulk site, and the boundary site, respectively, as shown in Fig. 8. Apparently, compared with Fig. 7, the evolution time needed for the localization of dynamic process is reduced significantly with the increase of nonreciprocal coupling strength. In other words, the strong nonreciprocal coupling strength enhances the gathering effect of the photons. Fig. 8. The dynamic process of the photonic states in nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, and $\Delta =3$, respectively. The excited positions are the 10th, 5th, 15th, 1st, 1st and 20th, and the whole lattice sites, corresponding to Figs. (a)-(f), respectively. Furthermore, we show the dynamic process for exciting the lattice sites at the boundary in Hermitian SSH chain, correspondingly, the interchain coupling term can be described by $H_\textrm {I-II}=\Delta a_{L}b^{\dagger }_{1}+\Delta b_{1}a^{\dagger }_{L}$. For $\Delta =0.5$, one can see that the photons are always localized at the boundary sites with time evolution. Even in the strong coupling condition with $\Delta =3$, the dynamic process of photonic states also presents the localization on two boundary sites of the system, as shown in Fig. 9(b). This is because that, in such Hermitian system, the localization properties own to the topologically protected edge states, which have trapping effect on the photons. Obviously, compared with Figs. 7(d)–7(e) and 8(d)–8(e), we find that the results of dynamic process of Hermitian system are different from that in non-Hermitian system. It indicates that the nonreciprocal coupling breaks the trapping effect of the edge states. Fig. 9. The dynamic process of the photonic states in Hermitian SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $\theta =0$, (a) $\Delta =0.5$, and (b) $\Delta =3$. The excited positions are both the 1st and 20th lattice sites for Figs. (a)-(b). In the following, we investigate the influence of on-site defect potentials on the dynamic process of photonic states in nonreciprocal coupled SSH chain. The parameters are chosen as $\Delta =0.5$, $\theta =0$, $L_\textrm {I}=10$, $L_\textrm {II}=10$, and the initial state of the system is located at the 1st lattice site. In Fig. 10(a), consider that the defect is at the 1st lattice site, i.e., the left boundary site, one can see that the evolution time for the localization of dynamic process is longer than that without defect potential in Fig. 7(d). When the on-site defect potential is at the 2th lattice site, we find that the photons go around the defect lattice site, and they are finally localized on the two middle lattice sites corresponding to the site position of nonreciprocal coupling, as shown in Fig. 10(b). The same phenomena also occur for the defect potentials are in the other bulk sites ($j=3$ and $j=4$), as shown in Fig. 10(c)–10(d). It indicates that the gathering effect of the photons caused by the nonreciprocal coupling is not affected by the on-site defect potentials. Notably, we find that different forms of laser pulses can be induced around the left boundary site with the on-site defect potentials in different positions. However, the photons are finally localized on the two middle lattice sites with time evolution because of the nonreciprocal coupling. Fig. 10. The dynamic process of the photonic states with on-site defect potentials in nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$, $L_\textrm {II}=10$, $\delta =0.5$, $t=1$, $\theta =0$, and $\Delta =0.5$, respectively.The excited positions are the 1st, 2nd, 3rd, and 4th lattice sites, corresponding to Figs. (a)-(d), respectively. 4. Realization of quantum simulation for 1D nonreciprocal coupled SSH chain In this section, we briefly present the quantum simulation of the nonreciprocal SSH chain. We consider an array of transmission line resonators including chain I and chain II, in which these two chains have a nonreciprocal coupling with the strengths $\pm \Delta$. Moreover, we consider that each transmission line resonator is coupled to a two-level superconducting flux qubit represented by excited state $\|e\rangle$ and ground state $\|g\rangle$. The corresponding Hamiltonian of the system can be written as ($\hbar =1$) (2)$$H=H_\textrm{I}+H_\textrm{II}+H_\textrm{I-II},$$ (3)$$\begin{aligned} H_\textrm{I}=&\sum_{j=1}^{L}\left(\frac{\omega_{1}}{2}\sigma_{z, j}^{a}+\omega_{a}a_{j}^{\dagger}a_{j}+g_{a,j}\sigma_{a,j}^{+}a_{j}+g_{a,j}\sigma_{a,j}^{-}a_{j}^{\dagger}\right)\cr\cr &+\sum_{j=1}^{L-2}\left(t_{1}a_{j}^{\dagger}a_{j+1}+t_{2}a_{j+1}^{\dagger}a_{j+2}+\rm{H.c.}\right),\cr\cr H_\textrm{II}=&\sum_{j=1}^{L}\left(\frac{\omega_{2}}{2}\sigma_{z, j}^{b}+\omega_{b}b_{j}^{\dagger}b_{j}+g_{b,j}\sigma_{b,j}^{+}b_{j}+g_{b,j}\sigma_{b,j}^{-}b_{j}^{\dagger}\right)\cr\cr &+\sum_{j=1}^{L-2}\left(t_{1}b_{j}^{\dagger}b_{j+1}+t_{2}b_{j+1}^{\dagger}b_{j+2}+\rm{H.c.}\right),\cr\cr H_\textrm{I-II}=&\Delta a_{L}b_{1}^{\dagger}-\Delta b_{1}a_{L}^{\dagger}, \end{aligned}$$ where $a_{j}$ ($a_{j}^{\dagger }$) and $b_{j}$ ($b_{j}^{\dagger }$) are the annihilation (creation) operators, $\sigma _{a,j}^{+}$ ($\sigma _{a,j}^{-}$) and $\sigma _{b,j}^{+}$ ($\sigma _{b,j}^{-}$) are the raising (lowering) Pauli operators, $\sigma _{z, j}^{a}$ and $\sigma _{z, j}^{b}$ are the Pauli $z$ operators, $\omega _{1}$ and $\omega _{2}$ represent the transition frequencies between the excited state $\|e\rangle$ and the ground state $\|g\rangle$ of the resonators $Q_{j}^{a}$ and $Q_{j}^{b}$, $\omega _{a}$ and $\omega _{b}$ are the frequencies of the resonator $a_{j}$ and $b_{j}$, $g_{a,j}$ and $g_{b,j}$ denote the qubit-resonator coupling strengths, $t_{1}$ and $t_{2}$ are the nearest neighbor hopping amplitudes between the resonators. The nonreciprocal coupling device in Fig. 11 can be implemented by using a long Josephson junction operates in the flux-flow regime [55]. More specifically, the preferred direction of the electromagnetic wave can be created by combining the polarities of the bias current with magnetic field. The propagation is facilitated when the external microwave propagates along the direction of flux flow. On the contrary, the propagation is damped in opposite direction in a long Josephson junction. Fig. 11. Schematic diagram of 1D non-Hermitian circuit QED lattice system consisting of an array of transmission line resonators denoted by $a_{j}$ and $b_{j}$, each of them coupled to a two-level superconducting flux qubits $Q_{j}^{a}$ and $Q_{j}^{b}$. The nonreciprocal coupling between chain $\textrm {I}$ and chain $\textrm {II}$ can be implemented by using a long Josephson junction operates in the flux-flow regime. In the rotating frame with respect to the driving field frequency $\omega _\textrm {d}$, and all the superconducting flux qubits are prepared in their ground states, the effective Hamiltonian of the system is given by (4)$$\begin{aligned} H^{'}=&\sum_{j=1}^{L-2}\left(t_{1}a_{j}^{\dagger}a_{j+1}+t_{2}a_{j+1}^{\dagger}a_{j+2}+t_{1}b_{j}^{\dagger}b_{j+1}+t_{2}b_{j+1}^{\dagger}b_{j+2}+\rm{H.c.}\right)\cr\cr &+\sum_{j=1}^{L}\left(\Delta_{a}-\frac{g_{a,j}^{2}}{\Delta_{1}}\right)a_{j}^{\dagger}a_{j}+\sum_{j=1}^{L}\left(\Delta_{b}-\frac{g_{b,j}^{2}}{\Delta_{2}}\right)b_{j}^{\dagger}b_{j}\cr\cr &+\Delta a_{L}b_{1}^{\dagger}-\Delta b_{1}a_{L}^{\dagger}, \end{aligned}$$ where $\Delta _{a}=\omega _{a}-\omega _{d}$, $\Delta _{b}=\omega _{b}-\omega _{d}$, $\Delta _{1}=\omega _{1}-\omega _{d}$, and $\Delta _{2}=\omega _{2}-\omega _{d}$. Setting the parameters as follows: $\Delta _{a}-\frac {g_{a,j}^{2}}{\Delta _{1}}=\Delta _{b}-\frac {g_{b,j}^{2}}{\Delta _{2}}=\xi$, $t_{1}=t\left (1-\delta \cos \theta \right )$, and $t_{2}=t\left (1+\delta \cos \theta \right )$. In this parameter regime, the Hamilton in Eq. (4) becomes (5)$$\begin{aligned} H^{\prime\prime}=&\sum_{j=1}^{L-2}\Big[\left(1-\delta\cos\theta\right)a_{j}^{\dagger}a_{j+1}+\left(1+\delta\cos\theta\right)a_{j+1}^{\dagger}a_{j+2}\cr\cr &+\left(1-\delta\cos\theta\right)b_{j}^{\dagger}b_{j+1}+\left(1+\delta\cos\theta\right)b_{j+1}^{\dagger}b_{j+2}+\rm{H.c.}\Big]\cr\cr &+\sum_{j=1}^{L}\xi a_{j}^{\dagger}a_{j}+\sum_{j=1}^{L}\xi b_{j}^{\dagger}b_{j}+\Delta a_{L}b_{1}^{\dagger}-\Delta b_{1}a_{L}^{\dagger}. \end{aligned}$$ Obviously, the above Hamiltonian corresponds the implementation of nonreciprocal coupled SSH chain based on the circuit QED lattice. In conclusion, we have investigated the localized photonic states induced by the nonreciprocal coupling and dynamic process of photonic states in 1D nonreciprocal coupled SSH chain. Through analyzing the energy eigenvalue spectrum of the system, we find that different localized photonic states can be induced by the nonreciprocal coupling. In the topologically nontrivial regime, the interface states with pure imaginary energy eigenvalues gradually disappear and finally become the bound states with pure imaginary energy eigenvalues with the increase of the coupling strength. However, the zero-energy edge states are robust to the variation of nonreciprocal coupling, which are highly localized on two ends of the system. Meanwhile, two pairs of degenerate bound states with complex energy eigenvalues become two bound states and two new interface states with pure imaginary energy eigenvalues, respectively, in the topologically trivial regime. Further, we analyze the dynamic process of photonic states in such non-Hermitian system. Interestingly, it is shown that the nonreciprocal coupling has an evident gathering effect on the photons, which breaks the trapping effect of topologically protected edge states. In addition, we consider the impacts of on-site defect potentials on the dynamic process of the system. It is shown that the photons go around the on-site defect lattice site and still present the gathering effect. Particularly, different forms of laser pulses can be induced with the on-site defect potentials in different lattice sites. Moreover, we have presented a simple method to realize the quantum simulation of nonreciprocal coupled SSH chain. Our scheme provide a new approach to investigate the localized photonic states and the dynamic process in non-Hermitian system. National Natural Science Foundation of China (11874132, 61575055, 61822114). 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Wunner, G. Xia, Y. Xiao, L. Xing, Y. Xu, Z. Xue, P. Yan, L. L. Yang, C. Yang, C. P. Yao, S. Ye, J. Y. Yi, W. Yin, C. Yoshida, T. You, J. B. Yu, C. S. Yu, D. M. Yu, L. Yu, X. L. Yuce, C. Zhang, C. Zhang, G. Zhang, G. Q. Zhang, L. L. Zhang, S. Zhang, W. N. Zhang, Y. Zhao, X. Zheng, Z. F. Zhou, K. Zhou, L. W. Zhou, X. Zhu, A. D. Zhu, G. Zhu, S. L. Zhu, X. Y. Zilberberg, O. Zou, L. J. J. Phys. B (1) J. Phys.: Condens. Matter (1) Nat. Commun. (1) Nat. Phys. (2) New J. Phys. (3) Opt. Express (3) Phys. Rev. A (11) Phys. Rev. B (11) Phys. Rev. Lett. (9) Phys. Rev. X (1) Quantum Inf. Process. (2) Rev. Mod. Phys. (1) Sci. China Phys. Mech. Astron. (1) Sci. Rep. (4) Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here. Alert me when this article is cited. Click here to see a list of articles that cite this paper View in Article \| Download Full Size \| PPT Slide \| PDF Fig. 2. The real and imaginary parts of the energy eigenvalue spectrum as a function of $\theta$ for 1D nonreciprocal coupled SSH chain, where we set $L_\textrm {I}=50$ , $L_\textrm {II}=50$ , $\delta =0.5$ , and the nonreciprocal coupling strengths (a) $\Delta =0.1$ , (b) $\Delta =0.5$ , (c) $\Delta =0.8$ , (d) $\Delta =1$ , (e) $\Delta =1.5$ , (f) $\Delta =2$ , (g) $\Delta =2.5$ , and (h) $\Delta =3$ , respectively. Fig. 3. The real and imaginary parts of the energy eigenvalue spectrum as a function of $\Delta$ for 1D nonreciprocal coupled SSH chain. Here, we set $L_\textrm {I}=50$ , $L_\textrm {II}=50$ , $\delta =0.5$ , and the periodic parameters (a) $\theta =0$ , (b) $\theta =\pi /2$ , and (c) $\theta =\pi$ , respectively. Fig. 4. The degree of complex energy levels of the system, where $L_\textrm {I}=50$ , $L_\textrm {II}=50$ , $\delta =0.5$ , and the periodic parameters (a) $\theta =0$ , (b) $\theta =\pi /2$ , and (c) $\theta =\pi$ , respectively. The degrees of complex energy levels with blue line and pink line are equal 0.98 and 1. Fig. 5. (a) The state distributions $\|\psi _{(50)}\|^{2}$ , $\|\psi _{(51)}\|^{2}$ (two red zero-energy edge states in Fig. 2) and $\|\psi _{(49)}\|^{2}$ , $\|\psi _{(52)}\|^{2}$ (two red interface states in Fig. 2) versus the nonreciprocal coupling strength $\Delta$ in topologically nontrivial regime $\theta =0.35\pi$ . (b) The state distributions $\|\psi _{(50)}\|^{2}$ , $\|\psi _{(51)}\|^{2}$ (two red bound states in Fig. 2) and $\|\psi _{(49)}\|^{2}$ , $\|\psi _{(52)}\|^{2}$ (two new red interface states in Fig. 2) versus the nonreciprocal coupling strength $\Delta$ in topologically trivial regime $\theta =0.65\pi$ . The other parameters are selected as $L_\textrm {I}=50$ , $L_\textrm {II}=50$ , and $\delta =0.5$ , respectively. Fig. 6. The state distributions of localized photonic states for the system, where $L_\textrm {I}=50$ , $L_\textrm {II}=50$ , $\delta =0.5$ . (a) $\theta =0$ and $\Delta =0.1$ , (b) $\theta =0$ and $\Delta =3$ , (c) $\theta =\pi$ and $\Delta =0.8$ , (d) $\theta =\pi$ and $\Delta =3$ . The red line, blue line, blue dotted line, green line, and green dotted line represent the state distributions of the zero-energy edge state, interface states with pure imaginary energy eigenvalues, new interface states with pure imaginary energy eigenvalues, bound states with complex energy eigenvalues, and bound states with pure imaginary energy eigenvalues, respectively. Fig. 7. The dynamic process of the photonic states in 1D nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$ , $L_\textrm {II}=10$ , $\delta =0.5$ , $\theta =0$ , and $\Delta =0.5$ , respectively. The excited positions are the 10th, 5th, 15th, 1st, 1st and 20th, and the whole lattice sites, corresponding to Figs. (a)-(f), respectively. Fig. 8. The dynamic process of the photonic states in nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$ , $L_\textrm {II}=10$ , $\delta =0.5$ , $\theta =0$ , and $\Delta =3$ , respectively. The excited positions are the 10th, 5th, 15th, 1st, 1st and 20th, and the whole lattice sites, corresponding to Figs. (a)-(f), respectively. Fig. 9. The dynamic process of the photonic states in Hermitian SSH chain, where $L_\textrm {I}=10$ , $L_\textrm {II}=10$ , $\delta =0.5$ , $\theta =0$ , (a) $\Delta =0.5$ , and (b) $\Delta =3$ . The excited positions are both the 1st and 20th lattice sites for Figs. (a)-(b). Fig. 10. The dynamic process of the photonic states with on-site defect potentials in nonreciprocal coupled SSH chain, where $L_\textrm {I}=10$ , $L_\textrm {II}=10$ , $\delta =0.5$ , $t=1$ , $\theta =0$ , and $\Delta =0.5$ , respectively.The excited positions are the 1st, 2nd, 3rd, and 4th lattice sites, corresponding to Figs. (a)-(d), respectively. Fig. 11. Schematic diagram of 1D non-Hermitian circuit QED lattice system consisting of an array of transmission line resonators denoted by $a_{j}$ and $b_{j}$ , each of them coupled to a two-level superconducting flux qubits $Q_{j}^{a}$ and $Q_{j}^{b}$ . The nonreciprocal coupling between chain $\textrm {I}$ and chain $\textrm {II}$ can be implemented by using a long Josephson junction operates in the flux-flow regime. Equations on this page are rendered with MathJax. Learn more. (1) H = ∑ j = 1 L − 2 ( t 1 a j † a j + 1 + t 2 a j + 1 † a j + 2 + b j † b j + 1 + t 2 b j + 1 † b j + 2 + H . c . ) + Δ a L b 1 † − Δ b 1 a L † , (2) H = H I + H II + H I-II , (3) H I = ∑ j = 1 L ( ω 1 2 σ z , j a + ω a a j † a j + g a , j σ a , j + a j + g a , j σ a , j − a j † ) + ∑ j = 1 L − 2 ( t 1 a j † a j + 1 + t 2 a j + 1 † a j + 2 + H . c . ) , H II = ∑ j = 1 L ( ω 2 2 σ z , j b + ω b b j † b j + g b , j σ b , j + b j + g b , j σ b , j − b j † ) + ∑ j = 1 L − 2 ( t 1 b j † b j + 1 + t 2 b j + 1 † b j + 2 + H . c . ) , H I-II = Δ a L b 1 † − Δ b 1 a L † , (4) H ′ = ∑ j = 1 L − 2 ( t 1 a j † a j + 1 + t 2 a j + 1 † a j + 2 + t 1 b j † b j + 1 + t 2 b j + 1 † b j + 2 + H . c . ) + ∑ j = 1 L ( Δ a − g a , j 2 Δ 1 ) a j † a j + ∑ j = 1 L ( Δ b − g b , j 2 Δ 2 ) b j † b j + Δ a L b 1 † − Δ b 1 a L † , (5) H ′ ′ = ∑ j = 1 L − 2 [ ( 1 − δ cos ⁡ θ ) a j † a j + 1 + ( 1 + δ cos ⁡ θ ) a j + 1 † a j + 2 + ( 1 − δ cos ⁡ θ ) b j † b j + 1 + ( 1 + δ cos ⁡ θ ) b j + 1 † b j + 2 + H . c . ] + ∑ j = 1 L ξ a j † a j + ∑ j = 1 L ξ b j † b j + Δ a L b 1 † − Δ b 1 a L † . James Leger, Editor-in-Chief Feature Issues	CommonCrawl
\begin{definition}[Definition:Oblique] Two (straight) lines are '''oblique''' {{iff}} they are not perpendicular. \end{definition}	ProofWiki
# Understanding the ARIMA model The ARIMA model is a popular statistical model used in time series analysis to forecast future values based on past observations. It stands for Autoregressive Integrated Moving Average, and it's composed of three main components: - Autoregressive (AR): This component captures the relationship between the current value of the time series and its own past values. It's represented by the equation: $$y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \cdots + \phi_p y_{t-p} + \epsilon_t$$ where $y_t$ is the current value, $c$ is the constant term, $\phi_i$ are the autoregressive coefficients, and $\epsilon_t$ is the error term. - Integrated (I): This component captures the relationship between the current value and the difference between the current value and the previous value. It's represented by the equation: $$y_t - y_{t-1} = c + \phi_1 (y_{t-1} - y_{t-2}) + \phi_2 (y_{t-2} - y_{t-3}) + \cdots + \phi_p (y_{t-p} - y_{t-p-1}) + \epsilon_t$$ where $y_t$ is the current value, $c$ is the constant term, $\phi_i$ are the autoregressive coefficients, and $\epsilon_t$ is the error term. - Moving Average (MA): This component captures the relationship between the current value and the error terms from previous values. It's represented by the equation: $$y_t = c + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2} + \cdots + \theta_q \epsilon_{t-q} + \epsilon_t$$ where $y_t$ is the current value, $c$ is the constant term, $\theta_i$ are the moving average coefficients, and $\epsilon_t$ is the error term. ## Exercise Consider the following ARIMA model equation: $$y_t = 0.5y_{t-1} + 0.2y_{t-2} - 0.3\epsilon_{t-1} + 0.4\epsilon_{t-2}$$ 1. Identify the AR, I, and MA components of this model. 2. What is the order of the ARIMA model? The ARIMA model is widely used in forecasting because it captures the dependencies between past values, differences between values, and error terms. By understanding the ARIMA model and its components, you can effectively analyze and forecast time series data. # The role of autocorrelation in the ARIMA model Autocorrelation is a measure of the similarity between a time series and its own lagged (shifted) versions. In the context of the ARIMA model, autocorrelation plays a crucial role in understanding the dependencies between the time series values. The autocorrelation function (ACF) is a statistical tool used to measure the autocorrelation of a time series at different lags. The ACF can help us identify the order of the AR and MA components in the ARIMA model. The ACF is calculated as follows: $$ACF(h) = \frac{\sum_{t=1}^{n-h} (y_t - \bar{y})(y_{t+h} - \bar{y})}{\sum_{t=1}^{n} (y_t - \bar{y})^2}$$ where $ACF(h)$ is the autocorrelation at lag $h$, $y_t$ is the time series value at time $t$, and $\bar{y}$ is the mean of the time series. By analyzing the ACF, we can determine the order of the AR and MA components by looking for significant autocorrelation at different lags. For example, if the ACF has significant autocorrelation at lag 1, it indicates that the AR component has an order of 1. Similarly, if the ACF has significant autocorrelation at lag 2, it indicates that the MA component has an order of 2. ## Exercise Instructions: Consider the following ACF plot of a time series: ``` Lag 0 Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 0.5 0.3 0.1 0.05 0.02 0.01 ``` 1. What is the order of the AR component in the ARIMA model? 2. What is the order of the MA component in the ARIMA model? ### Solution None Autocorrelation is an essential concept in understanding the ARIMA model and its components. By analyzing the ACF, you can effectively determine the order of the AR and MA components in the ARIMA model, which is crucial for building and forecasting with the model. # Model selection and identification Selecting the appropriate ARIMA model is crucial for accurate forecasting. The process of model selection involves identifying the order of the AR, I, and MA components based on the ACF and PACF plots. The PACF (Partial Autocorrelation Function) is a statistical tool used to measure the partial autocorrelation of a time series at different lags. The PACF can help us identify the order of the AR component in the ARIMA model. The PACF is calculated as follows: $$PACF(h) = \frac{\sum_{t=1}^{n-h} (y_t - \bar{y})(y_{t+h} - \bar{y})}{\sum_{t=1}^{n-h} (y_t - \bar{y})^2}$$ where $PACF(h)$ is the partial autocorrelation at lag $h$, $y_t$ is the time series value at time $t$, and $\bar{y}$ is the mean of the time series. By analyzing the PACF, we can determine the order of the AR component by looking for significant partial autocorrelation at different lags. For example, if the PACF has significant partial autocorrelation at lag 1, it indicates that the AR component has an order of 1. ## Exercise Instructions: Consider the following PACF plot of a time series: ``` Lag 0 Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 0.5 0.3 0.1 0.05 0.02 0.01 ``` 1. What is the order of the AR component in the ARIMA model? ### Solution None Model selection and identification are essential steps in building an effective ARIMA model. By analyzing the ACF and PACF plots, you can determine the order of the AR and MA components, which is crucial for forecasting with the model. # Estimating the ARIMA parameters Once the AR and MA components of the ARIMA model are identified, the next step is to estimate the coefficients of these components. This can be done using various estimation techniques, such as maximum likelihood estimation (MLE) or method of moments (MoM). The MLE approach involves minimizing the negative log-likelihood of the model, which measures the goodness of fit between the model and the observed data. The MoM approach involves solving a system of equations that equate the sample mean and variance of the residuals to the theoretical values. In practice, many statistical software packages, such as R and Python, provide built-in functions to estimate the ARIMA parameters using MLE or MoM. ## Exercise Instructions: Consider the following ARIMA model equation: $$y_t = 0.5y_{t-1} + 0.2y_{t-2} - 0.3\epsilon_{t-1} + 0.4\epsilon_{t-2}$$ 1. What are the AR coefficients in this model? 2. What are the MA coefficients in this model? ### Solution None Estimating the ARIMA parameters is a crucial step in building an effective model. By using MLE or MoM, you can obtain the coefficients of the AR and MA components, which are essential for forecasting with the model. # Diagnostic checking for stationarity Before using the ARIMA model for forecasting, it's important to ensure that the time series is stationary. A stationary time series has constant mean, variance, and autocorrelation structure over time. To check for stationarity, you can use various diagnostic tests, such as the Augmented Dickey-Fuller (ADF) test, the Phillips-Perron (PP) test, and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. These tests help you determine if the time series is stationary and if the ARIMA model is appropriate for the data. In practice, many statistical software packages, such as R and Python, provide built-in functions to perform these diagnostic tests. ## Exercise Instructions: Consider the following time series data: ``` Time Value 1 100 2 110 3 120 4 130 5 140 ``` 1. Perform a stationarity test on this time series data using the ADF test. 2. If the time series is non-stationary, what transformation can be applied to make it stationary? ### Solution None Diagnostic checking for stationarity is crucial before using the ARIMA model for forecasting. By performing tests like the ADF test, you can determine if the time series is stationary and if the ARIMA model is appropriate for the data. # Forecasting using the ARIMA model Once the ARIMA model is selected, identified, and estimated, you can use it to forecast future values of the time series. The forecasting process involves using the estimated AR and MA coefficients to predict the future values. For example, if the ARIMA model is given by: $$y_t = 0.5y_{t-1} + 0.2y_{t-2} - 0.3\epsilon_{t-1} + 0.4\epsilon_{t-2}$$ To forecast the value at time $t+h$, you can substitute the estimated coefficients and the forecasted values of the error terms: $$\hat{y}_{t+h} = \hat{c} + \hat{\phi}_1 \hat{y}_{t+h-1} + \hat{\phi}_2 \hat{y}_{t+h-2} + \hat{\theta}_1 \hat{\epsilon}_{t+h-1} + \hat{\theta}_2 \hat{\epsilon}_{t+h-2}$$ where $\hat{y}_{t+h}$ is the forecasted value at time $t+h$, $\hat{c}$ is the estimated constant term, $\hat{\phi}_i$ are the estimated autoregressive coefficients, $\hat{\theta}_i$ are the estimated moving average coefficients, and $\hat{\epsilon}_{t+h-i}$ are the forecasted error terms. In practice, many statistical software packages, such as R and Python, provide built-in functions to forecast using the ARIMA model. ## Exercise Instructions: Consider the following ARIMA model equation: $$y_t = 0.5y_{t-1} + 0.2y_{t-2} - 0.3\epsilon_{t-1} + 0.4\epsilon_{t-2}$$ 1. What is the forecasted value of the time series at time 6? ### Solution None Forecasting using the ARIMA model is a powerful technique for predicting future values of a time series. By using the estimated AR and MA coefficients, you can accurately predict the future values of the time series. # Applications and extensions of the ARIMA model The ARIMA model has numerous applications and extensions in various fields, such as finance, economics, and engineering. Some of the applications and extensions of the ARIMA model include: - Seasonal ARIMA (SARIMA): This extension accounts for the seasonal dependencies in the time series data by introducing seasonal lags into the model. - Exponential smoothing state space models (ETS): These models combine the ARIMA model with exponential smoothing to capture both the autoregressive and exponential smoothing components. - State space models: These models extend the ARIMA model by introducing latent variables and state equations to capture more complex dependencies in the time series data. - Bayesian ARIMA: This extension incorporates Bayesian inference to estimate the ARIMA parameters, providing a more robust and flexible approach to modeling. - Machine learning techniques: These techniques, such as artificial neural networks and support vector machines, can be used to improve the forecasting performance of the ARIMA model. By understanding the applications and extensions of the ARIMA model, you can effectively apply this model to a wide range of time series data and forecasting problems. ## Exercise Instructions: Consider the following ARIMA model equation: $$y_t = 0.5y_{t-1} + 0.2y_{t-2} - 0.3\epsilon_{t-1} + 0.4\epsilon_{t-2}$$ 1. What are the potential applications of this model in finance? 2. What are the potential extensions of this model for improved forecasting performance? ### Solution None Applications and extensions of the ARIMA model are essential for understanding its practical applications and improving its forecasting performance. By exploring these applications and extensions, you can effectively use the ARIMA model in various domains and improve its accuracy. # Case studies and real-world examples To further understand the ARIMA model and its applications, it's important to analyze case studies and real-world examples. These examples can help you develop a deeper understanding of the model's strengths and limitations, as well as its practical use in various fields. Some case studies and real-world examples of the ARIMA model include: - Forecasting stock prices: The ARIMA model can be used to forecast the future values of stock prices based on historical data. - Economic forecasting: The ARIMA model can be used to forecast economic indicators, such as GDP growth or inflation, based on historical data. - Sales forecasting: The ARIMA model can be used to forecast sales data for businesses based on historical data. - Weather forecasting: The ARIMA model can be used to forecast weather patterns, such as temperature or precipitation, based on historical data. By analyzing case studies and real-world examples, you can effectively understand the practical applications of the ARIMA model and improve your ability to use it in various domains. ## Exercise Instructions: Consider the following ARIMA model equation: $$y_t = 0.5y_{t-1} + 0.2y_{t-2} - 0.3\epsilon_{t-1} + 0.4\epsilon_{t-2}$$ 1. What are the potential case studies and real-world examples for this model? ### Solution None Case studies and real-world examples are essential for understanding the practical applications of the ARIMA model and its extensions. By analyzing these examples, you can effectively use the model in various domains and improve its accuracy. Course Table Of Contents 1. Understanding the ARIMA model 2. The role of autocorrelation in the ARIMA model 3. Model selection and identification 4. Estimating the ARIMA parameters 5. Diagnostic checking for stationarity 6. Forecasting using the ARIMA model 7. Applications and extensions of the ARIMA model 8. Case studies and real-world examples Course	Textbooks
Temperature of an atom I read somewhere that the temperature of an atom is not defined. The definition of temperature is only for larger systems. Why is this so? thermodynamics quantum-chemistry atoms temperature LexiconLexicon Thermodynamic functions are strictly defined only for macroscopic systems (systems that have an essentially infinite number of atoms). You can't apply them to individual atoms because that would be confusing large-scale averages with individual microscopic values. Here's an analogy: the average speed of cars on a stretch of highway might be 55 mph, but it's possible that no individual car is traveling at that exact speed. Another analogy: the average global temperature is rising over long periods of time, but if I look at the thermometer outside my window, the local temperature is falling over a short period of time... I can't say that my local temperature drop disproves global warming, because weather is a different thing from climate. Suppose that you knew the speed of a single atom. Now, for a macroscopic ideal gas, the average speed of molecules $\overline{v}$ is $$\overline{v} = \sqrt{\frac{8 R T}{\pi M}}$$ It would be wrong for you to plug in the speed of your single atom and solve for its temperature, because that equation was derived by assuming that you had a macroscopic system (with a certain speed distribution over many molecules); it only applies to average velocities, not individual velocities. This applies to other thermodynamic functions, too. Let's look at a simple chemical example. For a macroscopic crystal, the Gibbs free energy might be written as $$G = Ng(P,T)$$ where $N$ is the number of atoms and $g$ is the Gibbs free energy per atom. Little $g$ is a function of pressure and temperature. So can we say that little $g$ is the Gibbs free energy of a single atom? No. It's the average Gibbs free energy per atom for a huge number of atoms. To see the difference, consider a small cluster of atoms from that same crystal, the previous equation will need some correction terms: $$G = Ng(P,T) + a(P,T)N^{2/3} + b(T)\ln{N} + c(P,T)$$ where the "a" term is a surface free energy and the last couple of terms might come from free energy contributed by things like rotation of the cluster (Source: T. L. Hill, Thermodynamics of Small Systems, Dover, 1962.) You can't say that $g(P,T)$ is the Gibbs free energy of a single atom in the cluster because of those correction terms. TL;DR: Thermodynamic functions are averages over very large numbers of particles; random fluctuations are tiny compared to the mean values. But with small systems, those fluctuations become quite important, and there are additional effects that cause macroscopic averages to poorly describe microscopic systems. Fred SeneseFred Senese $\begingroup$ Very nice answer. But wouldn't the ergodic hypothesis (which in short states that the ensemble-average is equal to the time-average) imply that thermodynamic functions would also be valid for a microscopic system (e.g. a single particle) if I consider the behavior of the system averaged over a very(!) long period of time? $\endgroup$ – Philipp Jan 26 '15 at 18:44 $\begingroup$ @Phillip - I agree with you - I think maybe it would be more correct to say that the instantaneous temperature of a single atom is undefined, even though the time-averaged temperature may be defined. This is also assuming that there is a suitable reference frame for determining the velocity of a single atom in the first place. Also, "very long" for atoms might not be that long for humans. As a rough guess, I think that at room temp a couple of nanoseconds would probably give a good average. $\endgroup$ – thomij Jan 26 '15 at 19:25 $\begingroup$ @Philipp and thomij, yes, you'll be comparing two averages, not an average and a single value. $\endgroup$ – Fred Senese Jan 26 '15 at 19:47 $\begingroup$ Aah i see. I remember reading the derivation of the average kinetic energy of the atoms of an ideal gas, and it involved quite a few assumptions. Thanks for the great answer sir! $\endgroup$ – Lexicon Jan 27 '15 at 15:17 While generally temperature in textbooks is connected with the speed of atoms/particles, strictly speaking not the velocity, but the distribution of velocity what counts. If you microscopically observe a bunch of particles, their velocity can be whatever as your frame of observation can be any inertia system. When you want to calculate temperature, you choose reference system which has the same velocity as the center of mass of those particles, and then you calculate the average velocities form that reference. The average will be small, if the distribution of velocities is sharp, and will be large if the distribution is broad. The calculated temperature can be low even for fast particles e.g. in a gas flow, if all the particles moving in the same directions, same velocity! Let's apply this to one atom: you don't have a distribution of different velocities! Once you adjusted your reference point to the center of mass, ie the particle, it has zero velocity and meaningless to look for a distribution of different velocities. $\begingroup$ There are electrons in the atom... $\endgroup$ – jinawee Oct 28 '18 at 17:51 $\begingroup$ @jinawee Electronic temperature can only be relevant is there are electronically excited states below kT, and a trivial coupling to the kinetics of the atom (generally no). Even so, their behavior is not classical, per def, so we don't discuss them in classical theories. $\endgroup$ – Greg Oct 29 '18 at 4:45 I'll just rephrase @Greg's answer: Thermodynamics distinguishes heat and mechanical work. From a microscopic point of view, heat is the unordered movement of atoms/molecules, while mechanical work is the ordered movement (all the atoms of the piston move at the same velocity = same speed & direction; free energy). As Greg says, the actual movement is decomposed into those two components: a macroscopic translation and an unordered part where the sum of all velocities is zero. This second part constitues heat (in terms of total energy of all those microscopic movements), and temperature (distribution of speed = \|velocity\|). A single atom has only one velocity, so we cannot decide which part of it is ordered and which is unordered. We can thus speak of the kinetic energy and velocity of the atom, but not of distinguishable free energy and heat, nor of temperature. Not the answer you're looking for? Browse other questions tagged thermodynamics quantum-chemistry atoms temperature or ask your own question. How is Negative Temperature Hotter than Infinite Temperature? Why is there no standard temperature? Why there is no temperature scale tied to normal human body temperature? Negative temperature as limiting case for two-level system Is heat just a change in temperature? Can somebody explain the low temperature Schottky anomaly for two level systems?	CommonCrawl
\begin{definition}[Definition:Piecewise Continuously Differentiable Function/Definition 2] Let $f$ be a real function defined on a closed interval $\closedint a b$. $f$ is '''piecewise continuously differentiable''' {{iff}}: :there exists a finite subdivision $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that: ::$f$ is continuously differentiable on $\closedint {x_{i − 1} } {x_i}$, where the derivative at $x_{i − 1}$ understood as right-handed and the derivative at $x_i$ understood as left-handed, for every $i \in \set {1, \ldots, n}$. \end{definition}	ProofWiki
# Euclidean distance and its role in hashing In geometric hashing, Euclidean distance plays a crucial role in determining the similarity between objects. It is the foundation for many geometric hashing algorithms and is used to compute the distance between objects in the feature space. For example, consider two objects A and B in a feature space. The Euclidean distance between A and B can be calculated as: $$ d(A, B) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2} $$ where $x_i$ and $y_i$ are the corresponding coordinates of the $i$-th feature in objects A and B, respectively. The Euclidean distance is used to measure the similarity between objects in geometric hashing. Two objects are considered similar if their Euclidean distance is small. Consider two objects A and B with the following coordinates in a 2-dimensional feature space: - Object A: (1, 2) - Object B: (3, 4) The Euclidean distance between A and B is: $$ d(A, B) = \sqrt{(1-3)^2 + (2-4)^2} = \sqrt{10} $$ ## Exercise Calculate the Euclidean distance between the following two objects in a 3-dimensional feature space: - Object A: (1, 2, 3) - Object B: (4, 5, 6) # Geometric transformations and their impact on hashing Geometric transformations such as rotations, translations, and scaling can affect the Euclidean distance between objects. Therefore, it is important to understand how these transformations affect the performance of geometric hashing algorithms. For example, consider two objects A and B in a feature space. After applying a rotation transformation to both objects, the Euclidean distance between A and B may change. In geometric hashing, it is often desirable to find a hashing function that is invariant to geometric transformations. This means that the hash values of two objects should remain the same even after applying a geometric transformation to both objects. Consider two objects A and B in a feature space. After applying a rotation transformation to both objects, the Euclidean distance between A and B changes. However, if the hashing function is invariant to the rotation transformation, the hash values of A and B remain the same. ## Exercise Determine if the following hashing function is invariant to scaling: $$ h(x) = \sum_{i=1}^{n} x_i^2 $$ # Grid-based indexing and its application in hashing Grid-based indexing is a technique used in geometric hashing to partition the feature space into a grid. Each cell in the grid corresponds to a hash value. For example, consider a feature space with dimensions 10x10. We can divide this space into a 2x2 grid, where each cell has dimensions 5x5. Grid-based indexing is useful in geometric hashing because it allows us to quickly search for similar objects by comparing their hash values. Consider a feature space with dimensions 10x10, divided into a 2x2 grid. If two objects have the same hash value in the same cell, they are considered similar. ## Exercise Design a grid-based indexing scheme for a feature space with dimensions 10x10. # Hashing algorithms and their performance There are several hashing algorithms used in geometric hashing, including locality-sensitive hashing (LSH), min-wise independent hashing (MWH), and product quantization (PQ). The performance of a hashing algorithm depends on factors such as the choice of hash function, the size of the hash table, and the distribution of objects in the feature space. Consider a hashing algorithm that uses a hash function $h(x) = x \mod m$, where $m$ is the size of the hash table. The performance of this algorithm depends on the choice of $m$ and the distribution of objects in the feature space. ## Exercise Evaluate the performance of the following hashing algorithm: - Hash function: $h(x) = x \mod m$ - Size of the hash table: $m = 100$ - Distribution of objects in the feature space: Uniformly distributed # Nearest neighbor search and its importance in geometric hashing Nearest neighbor search is a fundamental operation in geometric hashing. It is used to find the nearest neighbor of a query object in the feature space. In geometric hashing, the performance of the nearest neighbor search operation depends on the choice of hashing algorithm and the distribution of objects in the feature space. Consider a feature space with dimensions 10x10, divided into a 2x2 grid. To find the nearest neighbor of a query object, we can search the cells in the grid that are close to the query object's cell. ## Exercise Design a nearest neighbor search algorithm for a feature space with dimensions 10x10, divided into a 2x2 grid. # Application of geometric hashing in computer vision and robotics Geometric hashing has applications in computer vision and robotics, such as image retrieval, object recognition, and robot localization. For example, geometric hashing can be used to efficiently search for similar images in a large database. Consider a database of images, where each image is represented as a feature vector in a high-dimensional feature space. Geometric hashing can be used to compute the hash values of the images and store them in a hash table. When searching for similar images, we can compare the query image's hash value with the hash values in the hash table. ## Exercise Design a system that uses geometric hashing for image retrieval. # Challenges and future developments in geometric hashing There are several challenges and future developments in geometric hashing, including handling high-dimensional feature spaces, dealing with noise and outliers, and improving the performance of hashing algorithms. For example, in high-dimensional feature spaces, the curse of dimensionality can make it difficult to compute the Euclidean distance between objects. Consider a feature space with dimensions 1000x1000. Computing the Euclidean distance between two objects in this space can be computationally expensive. ## Exercise Discuss strategies for addressing the challenge of high-dimensional feature spaces in geometric hashing. In conclusion, geometric hashing is a powerful technique for searching similar objects in high-dimensional feature spaces. Future developments in geometric hashing will focus on addressing the challenges of high-dimensional feature spaces and improving the performance of hashing algorithms.	Textbooks
\begin{document} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newcommand {\afi}{\textbf{Claim: }} \newcommand {\prafi}{\textbf{Proof of claim: } } \newcommand {\D}{\displaystyle} \newcommand {\eps}{\varepsilon} \title[Universal C$^$-algebras and partial actions]{C$^$-algebras of endomorphisms of groups with finite cokernel and partial actions} \author{Felipe Vieira}\thanks{Supported by CAPES - Coordena\c{c}\~{a}o de Aperfei\c{c}oamento de Pessoal de N\'{i}vel Superior} \begin{abstract} In this paper we extend the constructions of Boava and Exel to present the C$^$-algebra associated with an injective endomorphism of a group with finite cokernel as a partial group algebra and consequently as a partial crossed product. With this representation we present another way to study such C$^$-algebras, only using tools from partial crossed products. \end{abstract} \keywords{Group, endomorphism, partial group algebra, partial crossed product.} \maketitle \section{Introduction} Consider an injective endomorphism $\varphi$ of a discrete countable group $G$ with unit $\{e\}$ with finite cokernel i.e, \begin{equation}\label{eqintro1} \left\|\dfrac{G}{\varphi(G)} \right\|<\infty, \end{equation} as above, for $H$ subgroup of $G$, we use $\frac{G}{H}$ to denote the set of left cosets of $H$ in $G$. Analyzing the natural representation of $G$ and $\varphi$ inside $\mathcal{L}(l^2(G))$ we construct a concrete C$^$-algebra $C_r^[\varphi]\subseteq \mathcal{L}(l^2(G))$ and a universal one denoted by $\mathds{U}[\varphi]$. Such constructions were presented by Hirshberg in \cite{Hirsh}, and were later generalized by Cuntz and Vershik in \cite{CunVer} and also in \cite{Viei}. Using a semigroup crossed product description of $\mathds{U}[\varphi]$ implies the existence of a (full corner) group crossed product description of it (\cite{Cuntz2}, \cite{CuntzTopMarkovII} and \cite{Laca1}), but it is not the only way to represent it as a crossed product: analogously to the work of G. Boava and R. Exel in \cite{BoEx} one can show that $\mathds{U}[\varphi]$ has a partial group crossed product description, which can also be related to an inverse semigroup crossed product by \cite{ExVi}. We present in this paper the latter construction cited above and show the simplicity of $\mathds{U}[\varphi]$, which is part of the conclusions in \cite{Hirsh}, using only the partial group crossed product description of that C$^$-algebra. \section{Definition} We repeat the constructions of \cite{Hirsh}. Let $G$ be a discrete countable group with unit $e$ and $\varphi$ an injective endomorphism (monomorphism) of $G$ with finite cokernel (\ref{eqintro1}). Consider the Hilbert space $l^2(G)$ with orthonormal basis $\xi_h$, taking every element of $G$ to 0 apart from the element $h$, which goes to 1. Define the following bounded operators on $l^2(G)$: $$ U_g(\xi_h)=\xi_{gh} $$ and $$ S(\xi_g)=\xi_{\varphi(g)}. $$ The invertibility property of groups and the injectivity of the endomorphism $\varphi$ imply that the $U_g$'s are unitary operators and $S$ is an isometry respectively. Therefore we define the following C$^$-algebra. \begin{definition}\label{defi1red}We denote $C_r^[\varphi]$ the reduced C$^$-algebra of $\varphi$, to be the C$^$-subalgebra of $\mathcal{L}(l^2(G))$ generated by the above defined unitaries $\{U_g:\;g\in G\}$ and isometry $S$. \end{definition} Inspired by the properties of the operators above: \begin{definition}\label{defi1}We call $\mathds{U}[\varphi]$ the universal C$^$-algebra generated by the unitaries $\{u_g:\;g\in G\}$ and one isometry $s$ such that: \begin{enumerate} \item[(i)] $u_gu_h=u_{gh}$; \item[(ii)] $su_g=u_{\varphi(g)}s$; \item[(iii)] $\D\sum_{g\in G/\varphi(G)}u_gss^u_{g^{-1}}=1$; \end{enumerate} for all $g$, $h\in G$. \end{definition} As the universal C$^$-algebra above is defined using relations satisfied by the generators of the reduced one, obviously there is a canonical surjective $$-homomorphism from $\mathds{U}[\varphi]$ onto $C_r^[\varphi]$. Note that the conditions (i) and (ii) above can be merged into the relation $$ u_gs^nu_hs^m=u_{g\varphi^n(h)}s^{n+m}. $$ By (ii) we have, for $g\in G$, the obvious relations $$ u_{g}s^=s^u_{\varphi(g)} $$ and $$ u_{\varphi(g)}ss^=ss^u_{\varphi(g)}. $$ Also note that in (iii) there is no ambiguity if we choose different representatives of the cosets: \begin{equation} \begin{split} u_{g\varphi(h)}ss^u_{(g\varphi(h))^{-1}}&=u_gu_{\varphi(h)}ss^u_{\varphi(h^{-1})}u_{g^{-1}}=u_gss^u_{\varphi(h)}u_{\varphi(h^{-1})}u_{g^{-1}}\\ &=u_gss^u_{g^{-1}}. \end{split} \end{equation} Condition (iii) implies that $u_gss^u_{g^{-1}}$ and $u_hss^u_{h^{-1}}$ are orthogonal projections if $g^{-1}h\notin\varphi(G)$, so the multiplication can be described as: \begin{equation} u_gss^u_{g^{-1}}u_hss^u_{h^{-1}}=\left\{ \begin{array}{cl} u_gss^u_{g^{-1}}, & \hbox{if }h\in g\varphi(G); \\ 0, & \hbox{otherwise.} \end{array} \right. \end{equation} This extends to the family of elements of type $u_gs^n{s^}^nu_{g^{-1}}$ for any $n\in\mathds{N}$ \begin{equation} u_gs^n{s^}^nu_{g^{-1}}u_hs^n{s^}^nu_{h^{-1}}=\left\{ \begin{array}{cl} u_gs^n{s^}^nu_{g^{-1}}, & \hbox{if }h\in g\varphi^n(G); \\ 0, & \hbox{otherwise.} \end{array} \right. \end{equation} And note that, for $g$, $h\in G$ and $n\geq m\in\mathds{N}$: \begin{equation} \begin{split} &u_gs^n{s^}^nu_{g^{-1}}u_hs^m{s^}^mu_{h^{-1}}\\ &=u_gs^n{s^}^nu_{g^{-1}}u_hs^m\left(\D\sum_{k\in\frac{G}{\varphi^{n-m}(G)}}u_ks^{n-m}{s^}^{n-m}u_{k^{-1}}\right){s^}^mu_{h^{-1}}\\ &=u_gs^n{s^}^nu_{g^{-1}}\left(\D\sum_{k\in\frac{G}{\varphi^{n-m}(G)}}u_{h\varphi^m(k)}s^n{s^}^nu_{(h\varphi^m(k))^{-1}}\right)\\ &=\left\{ \begin{array}{ll} u_gs^n{s^}^nu_{g^{-1}},&\hbox{ if }h\varphi^m(k)\in g\varphi^n(G)\hbox{ for some }k\in\frac{G}{\varphi^{n-m}(G)}; \\ 0,\hbox{ otherwise}. \end{array} \right. \end{split} \end{equation} \section{Crossed product description of $\mathds{U}[\varphi]$}\label{sectioncpdescr} In this section we present a semigroup crossed product description of $\mathds{U}[\varphi]$. The semigroup crossed product definition which we will use is the same as presented in Appendix A of \cite{Li1}, via covariant representations. In our case the semigroup implementing the action will be the semidirect product $$ S:=G\rtimes_\varphi\mathds{N}=\{(g,n)\;:\;g\in G, n\in\mathds{N}\} $$ with product $$ (g,n)(h,m)=(g\varphi^n(h),n+m). $$ We will also show that the action implemented by $S$ can be split i.e, the semigroup crossed product by $S$ can be seen as a semigroup crossed product by $\mathds{N}$. This crossed product description is a great tool to prove some properties of $\mathds{U}[\varphi]$: we will show that when $G$ is amenable this C$^$-algebra is nuclear and satisfies UCT. Secondly, that description allows one to use the six-term exact sequence introduced by M. Khoshkam and G. Skandalis in \cite{Khoska} on $\mathds{U}[\varphi]$. Moreover, due to M. Laca \cite{Laca1}, sometimes it is possible to see semigroup crossed products as full corners of group ones, which implies that both are Morita equivalent and therefore have the same K-groups. And in case the semigroup action is implemented by $\mathds{N}$, Laca's dilation turns this $\mathds{N}$-action into a $\mathds{Z}$-action, which fits the requirements to use the classical Pimsner-Voiculescu exact sequence \cite{Pivo1}. Set $$ \overline{G}:=\D\lim_{\leftarrow}\left\{\frac{G}{\varphi^m(G)}: p_{m,l+m}\right\} $$ where $$ p_{m,l+m}:\dfrac{G}{\varphi^{l+m}(G)}\rightarrow\dfrac {G}{\varphi^m(G)} $$ is the canonical projection. We can see $\overline{G}$ as $$ \overline{G}=\left\{(g_m)_m\in\D\prod_{m\in\mathds{N}}\frac{G}{\varphi^m(G)}:\;p_{m,l+m}(g_{l+m})=g_m\right\}, $$ with the induced topology on the product $\D\prod_{m\in\mathds{N}}\frac{G}{\varphi^m(G)}$, where each finite set $\dfrac{G}{\varphi^m(G)}$ carries the discrete topology, implying that $\overline{G}$ is a compact space.\\[2\baselineskip] Furthermore, we have the map \begin{equation} \begin{split} G&\rightarrow\overline{G}\\ g&\mapsto (g)_m, \end{split} \end{equation} which is an embedding when $\varphi$ is pure. Also set $$ \mathcal{G}:=\lim_{\rightarrow}\{\mathcal{G}_m:\phi_{l+m,m}\} $$ where $\mathcal{G}_m=\overline{G}$ for all $m\in\mathds{N}$ and $\phi_{l+m,m}=\varphi^l$. We can see $\mathcal{G}$ as $$ \mathcal{G}=\D\bigcup^._{m\in\mathds{N}}\mathcal{G}_m\diagup\thicksim $$ with $x_l\sim y_m$ if and only if $\varphi^m(x_l)=\varphi^l(y_m)$, $x_l\in\mathcal{G}_l$ and $y_m\in\mathcal{G}_m$. Note that $\mathcal{G}$ is a locally compact set. Denote by $q$ the canonical projection $$ q:\D\bigcup_{m\in\mathds{N}}^.\mathcal{G}_m\rightarrow\mathcal{G}, $$ and $i_m$ the embedding $$ \begin{array}{cccccc} i_m: & \overline{G} & = & \mathcal{G}_m & \hookrightarrow & \mathcal{G} \\ & x & = & x & \mapsto & q(x). \end{array} $$ Again we have the identification \begin{equation} \begin{split} \overline{G}&\hookrightarrow\mathcal{G}\\ x&\mapsto i_0(x). \end{split} \end{equation} \begin{remark}\em\label{obs12} Note that if we suppose that our endomorphism $\varphi$ is \emph{totally normal}, i.e. all the $\varphi^m(G)$ are normal subgroups of $G$, then $\overline{G}$ and $\mathcal{G}$ will be groups; one just has to consider the componentwise multiplication in $\overline{G}$ and $$ i_m(x)i_l(y)=i_{l+m}(xy),\; \forall\; x, y\in\overline{G} $$ on $\mathcal{G}$. \end{remark} \begin{proposition}\label{prop3}The map \begin{equation} \begin{split} \alpha: C^(P)&\rightarrow C(\overline{G})\\ u_gs^n{s^}^nu_{g^{-1}}&\mapsto p_{g\varphi^n(\overline{G})}, \end{split} \end{equation} where the latter denotes the characteristic function on the subset $g\varphi^n(\overline{G})\subseteq\overline{G}$, is an isomorphism. \end{proposition} \begin{proof} It is clear that $C^(P)$ is the inductive limit of $$ D_m:=C^\left(\left\{u_gs^m{s^}^mu_{g^{-1}}:g\in\frac{G}{\varphi^m(G)}\right\}\right) $$ with the inclusions (using (iii) of Definition \ref{defi1}) \begin{equation} \begin{split} D_m &\hookrightarrow D_{l+m}\\ u_gs^m{s^}^mu_{g^{-1}}&\mapsto\D\sum_{h\in\frac{G}{\varphi^l(G)}}u_{g\varphi^m(h)}s^{l+m}{s^}^{l+m}u_{\varphi^m(h^{-1})g^{-1}}. \end{split} \end{equation} Furthermore the pairwise orthogonality of the projections $u_gs^m{s^}^mu_{g^{-1}}$ for fixed $m\in\mathds{N}$ implies that $$ spec(D_m)\cong\dfrac{G}{\varphi^m(G)} $$ with \begin{equation} \begin{split} spec(D_{l+m})&\rightarrow spec(D_m)\\ \chi &\mapsto\chi\|_{D_m} \end{split} \end{equation} corresponding to \begin{equation} \begin{split} p_{l+m,m}:\frac{G}{\varphi^{l+m}(G)}&\rightarrow\frac{G}{\varphi^m(G)}\\ g\varphi^{l+m}(G) &\mapsto g\varphi^m(G). \end{split} \end{equation} Therefore $$ spec(C^(P))\cong\D\lim_{\leftarrow}\left\{\frac{G}{\varphi^m(G)}:p_{m,l+m}\right\}=\overline{G}. $$ Thus we get the isomorphism \begin{equation} \begin{split} \alpha:C^(P) &\rightarrow C(\overline{G})\\ u_gs^m{s^}^mu_{g^{-1}}&\mapsto p_{g\varphi^m(\overline{G})}. \end{split} \end{equation} \end{proof} \begin{definition}\label{defi2}The stabilization of \;$\mathds{U}[\varphi]$, denoted by $\mathds{U}^s[\varphi]$, is the inductive limit of the system $\{\mathds{U}_m^s[\varphi]:\psi_{m,l+m}\}$ where, $\forall\; m\in\mathds{N}$, $\mathds{U}_m^s[\varphi]=\mathds{U}[\varphi]$ and \begin{equation} \begin{split} \psi_{m,l+m}:\mathds{U}[\varphi]&\rightarrow\mathds{U}[\varphi]\\ x&\mapsto s^lx{s^}^l. \end{split} \end{equation} Furthermore define $C^(P)^s=\D\lim_{\rightarrow}\{C^(P)_m^s:\psi_{m,l+m}\}$ with $C^(P)_m^s=C^(P)$ and $\psi_{m,l+m}$ as above. \end{definition} \begin{proposition}\label{prop4}We have $C^(P)^s\cong C_0(\mathcal{G})$. \end{proposition} \begin{proof} The maps $\psi_{m,l+m}$, conjugated by $\alpha$, give maps $$ \widetilde{\psi}_{m,l+m}:=\alpha\circ\psi_{m,l+m}\circ\alpha^{-1}: C(\overline{G})\rightarrow C(\overline{G}), $$ where $\widetilde{\psi}_{m,l+m}(f)(x)=f(\varphi^{-l}(x))p_{\varphi^l(\overline{G})}(x):$ \begin{equation} \begin{split} \widetilde{\psi}_{m,l+m}(p_{g\varphi^m(\overline{G})})(x)&=\widetilde{\psi}_{m,l+m}\circ\alpha(u_gs^m{s^}^mu_{g^{-1}})(x)\\ &=\alpha\circ\psi_{m,l+m}(u_gs^m{s^}^mu_{g^{-1}})(x)\\ &=\alpha(u_{\varphi^l(g)}s^{l+m}{s^}^{l+m}u_{\varphi^l(g^{-1})})(x)\\ &=p_{\varphi^l(g)\varphi^{l+m}(\overline{G})}(x)\\ &=p_{g\varphi^m(\overline{G})}(\varphi^{-l}(x))p_{\varphi^l(\overline{G})}(x). \end{split} \end{equation} By the properties of inductive limits, we have an isomorphism $$ \overline{\alpha}: C^(P)^s\rightarrow\D\lim_{\rightarrow}\{C(\overline{G}):\widetilde{\psi}_{m,l+m}\}. $$ Additionally we consider the $$-homomorphisms \begin{equation} \begin{split} \kappa_k:C(\overline{G})&\rightarrow C_0(\mathcal{G})\\ f &\mapsto f\circ i_k^{-1}.p_{i_k(\overline{G})} \end{split} \end{equation} (where the $i$'s are as defined before Remark \ref{obs12}). These $$-homomorphisms satisfy $\kappa_{l+m}\circ\widetilde{\psi}_{m,l+m}=\kappa_m$, since \begin{equation} \begin{split} \kappa_{l+m}\circ\widetilde{\psi}_{m,l+m}(f)(x)&=\widetilde{\psi}_{m,l+m}(f)\circ i_{l+m}^{-1}(x)p_{i_{l+m}(\overline{G})}(x)\\ &=f(i_{l+m}^{-1}(\varphi^{-l}(x)))p_{\varphi^l(\overline{G})}(x)p_{i_{l+m}(\overline{G})}(x)\\ &=f(i_{m}^{-1}(x))p_{i_m(\overline{G})}(x)\\ &=\kappa_m(f)(x). \end{split} \end{equation} Hence we have a $$-homomorphism $$ \D\lim_{\rightarrow}\{C(\overline{G}):\widetilde{\psi}_{m,l+m}\}\rightarrow C_0(\mathcal{G}). $$ This is injective as each $\kappa_k$ is, because of $\kappa_k(f)\circ i_k=f$. It is also surjective as $\mathcal{G}=\overline{\D\cup_{m\in\mathds{N}^}i_m(\overline{G})}$ and using the Stone-Weierstrass Theorem. So we have $$ C^(P)^s\cong C_0(\mathcal{G}). $$ \end{proof} Now we have all the tools to describe our C$^$-algebra as a semigroup crossed product using $S=G\rtimes_\varphi\mathds{N}$. Consider the action \begin{equation} \begin{split} \alpha: S &\rightarrow \hbox{End}(C^(P))\\ (g,n)&\mapsto u_gs^n (.) {s^}^nu_{g^{-1}}. \end{split} \end{equation} \begin{theorem}\label{teo2}$\mathds{U}[\varphi]$ is isomorphic to $C^(P)\rtimes_{\alpha} S$. \end{theorem} \begin{proof} By definition, $C^(P)\rtimes_{\alpha} S$ together with \begin{equation} \begin{split} \iota_P: C^(P)&\rightarrow C^(P)\rtimes_{\alpha} S\\ x &\mapsto \iota_P(x) \end{split} \end{equation} and \begin{equation} \begin{split} \iota_S: S&\rightarrow \hbox{Isom}(C^(P)\rtimes_{\alpha} S)\\ (g,n) &\mapsto \iota_S(g,n) \end{split} \end{equation} satisfying $$ \iota_P(u_gs^nx{s^}^nu_{g^{-1}})=\iota_S(g,n)\iota_P(x)\iota_S(g,n)^ $$ is the crossed product of $(C^(P),S,\alpha)$. But note that the triple $\mathds{U}[\varphi]$, \begin{equation} \begin{split} \pi: C^(P)&\rightarrow \mathds{U}[\varphi]\\ x &\mapsto x \end{split} \end{equation} and \begin{equation} \begin{split} \rho: S&\rightarrow \hbox{Isom}(\mathds{U}[\varphi])\\ (g,n)&\mapsto u_gs^n \end{split} \end{equation} is a covariant representation of $(C^(P),S,\alpha)$ because: $$ \rho(g,n)\pi(x)\rho(g,n)^=u_gs^nx{s^}^nu_{g^{-1}}=\pi(\alpha_{(g,n)}(x)). $$ Therefore there exists a $$-homomorphism \begin{equation}\label{teoiso1} \Phi: C^(P)\rtimes_{\alpha} S \rightarrow \mathds{U}[\varphi] \end{equation} such that $\Phi\circ\iota_P=\pi$ and $\Phi\circ\iota_S=\rho$. In the other hand it is well known \cite{Laca2} that the crossed product $C^(P)\rtimes_{\alpha} S$ is generated as a C$^$-algebra by elements of the form $\iota_S(g,n)$ because we have$$\iota_P(u_gs^n{s^}^nu_{g^{-1}})=\iota_S(g,n)\iota_S(g,n)^.$$ But note that $\mathds{U}[\varphi]$ can be viewed as the universal C$^$-algebra generated by the unitaries $\{u_g:\; g\in G\}$ and the isometry $s$ changing conditions (i) and (ii) in Definition \ref{defi1} to the equivalent one $u_gs^nu_hs^m=u_{g\varphi^n(h)}s^{n+m}$. Therefore we identify $\iota_S(g,n)$ with $u_gs^n$ because the first ones satisfy the condition above, which generate $\mathds{U}[\varphi]$: $$ \iota_S(g,n)\iota_S(h,m)=\iota_S(g\varphi^n(h),n+m) $$ and \begin{equation} \begin{split} \D\sum_{g\in G/\varphi(G)}\iota_S(g,n)\iota_S(g,n)^=&\D\sum_{g\in G/\varphi(G)}\iota_P(u_gs^n{s^}^nu_{g^{-1}})\\ =&\iota_P\left(\D\sum_{g\in G/\varphi(G)}u_gs^n{s^}^nu_{g^{-1}}\right)\\ =&\iota_P(1)=1. \end{split} \end{equation} Thus we get another $$-homomorphism \begin{equation}\label{teoiso2} \begin{split} \Delta: \mathds{U}[\varphi] &\rightarrow C^(P)\rtimes_{\alpha} S\\ u_gs^n&\mapsto \iota_S(g,n). \end{split} \end{equation} As (\ref{teoiso1}) and (\ref{teoiso2}) are inverses of each other we can conclude that $\mathds{U}[\varphi]$ and $C^(P)\rtimes_{\alpha} S$ are isomorphic. \end{proof} In order to be able to apply the exact sequence presented in \cite{Khoska} we split the action of $S$ presented above: we show that its semigroup crossed product is isomorphic to a semigroup crossed product implemented by $\mathds{N}$, where $\mathds{N}$ acts on a group crossed product by $G$. \begin{proposition}\label{Prop1GN}The C$^$-algebra $\mathds{U}[\varphi]$ is also isomorphic to the semigroup crossed product $(C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$, where: \begin{equation} \begin{split} \omega: G &\rightarrow \hbox{Aut}(C^(P))\\ g &\mapsto u_g(\cdot)u_{g^{-1}}\\[2\baselineskip] \tau: \mathds{N} &\rightarrow \hbox{End}(C^(P)\rtimes_{\omega}G)\\ n &\mapsto s^n(\cdot){s^}^n \end{split} \end{equation} such that for $a_g\delta_g\in C^(P)\rtimes_{\omega}G$, $\tau_n(a_g\delta_g)=s^na_g{s^}^n\delta_{\varphi^n(g)}$. \end{proposition} \begin{proof} We will show that $C^(P)\rtimes_{\alpha} S$ and $(C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$ are isomorphic, by exploiting the universality of the semigroup crossed products, using two steps analogous to the first part of the proof of theorem above. Consider $C^(P)\rtimes_{\alpha} S$ together with \begin{equation} \begin{split} \iota_P: C^(P)&\rightarrow C^(P)\rtimes_{\alpha} S\\ x&\mapsto \iota_P(x) \end{split} \end{equation} and \begin{equation} \begin{split} \iota_S: S&\rightarrow \hbox{Isom}(C^(P)\rtimes_{\alpha} S)\\ (g,n)&\mapsto \iota_S(g,n) \end{split} \end{equation} satisfying $$ \iota_P(u_gs^nx{s^}^nu_{g^{-1}})=\iota_S(g,n)\iota_P(x)\iota_S(g,n)^* $$ being the crossed product of $(C^(P),S,\alpha)$. Analogously take $(C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$ with \begin{equation} \begin{split} \iota_G: C^(P)\rtimes_{\omega}G&\rightarrow (C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}\\ a\delta_g&\mapsto \iota_G(a\delta_g) \end{split} \end{equation} and \begin{equation} \begin{split} \iota_N: \mathds{N}&\rightarrow \hbox{Isom}((C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N})\\ n&\mapsto \iota_N(n) \end{split} \end{equation} satisfying $$ \iota_B(s^na\delta_g{s^}^n\delta_{\varphi^n(g)})=\iota_N(n)\iota_B(a\delta_g)\iota_N(n)^* $$ as the crossed product of $(C^(P)\rtimes_{\omega}G,\mathds{N},\tau)$, where $a\delta_g$ represents the generating elements of $C^(P)\rtimes_{\omega}G$, $g\in G$. Note that the triple $(C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$, \begin{equation} \begin{split} \varrho: C^(P)&\rightarrow (C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}\\ a&\mapsto \iota_G(a\delta_e) \end{split} \end{equation} and \begin{equation} \begin{split} \sigma: S&\rightarrow \hbox{Isom}((C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N})\\ (g,n)&\mapsto \iota_G(1\delta_g)\iota_N(n) \end{split} \end{equation} is a covariant representation of $(C^(P),S,\alpha)$: \begin{equation} \begin{split} \sigma(g,n)\varrho(a)\sigma(g,n)^&=\iota_G(1\delta_g)\iota_N(n)\iota_G(a\delta_e)\iota_N(n)^\iota_G(1\delta_g)^\\ &=\iota_G(1\delta_g)\iota_G(s^na{s^}^n\delta_e)\iota_G(1\delta_g)^\\ &=\iota_G(u_gs^na{s^}^nu_{g^{-1}}\delta_g)\iota_G(1\delta_{g^{-1}})\\ &=\iota_G(u_gs^na{s^}^nu_{g^{-1}}\delta_e)\\ &=\varrho(u_gs^na{s^}^nu_{g^{-1}}). \end{split} \end{equation} Therefore we get a $$-homomorphism \begin{equation}\label{teo12iso1} \Phi: C^(P)\rtimes_{\alpha}S\rightarrow (C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N} \end{equation} such that $\Phi\circ\iota_P=\varrho$ and $\Phi\circ\iota_S=\sigma$. Let us find an inverse for $\Phi$ using the fact that the triple $C^(P)\rtimes_{\alpha} S$, \begin{equation} \begin{split} \varpi: C^(P)\rtimes_{\omega}G&\rightarrow C^(P)\rtimes_{\alpha} S\\ a\delta_g&\mapsto \iota_P(a)\iota_S(g,0) \end{split} \end{equation} and \begin{equation} \begin{split} \vartheta: \mathds{N}&\rightarrow \hbox{Isom}(C^(P)\rtimes_{\alpha} S)\\ n&\mapsto \iota_S(e,n) \end{split} \end{equation} is a covariant representation of $(C^(P)\rtimes_{\omega}G,\mathds{N},\tau)$: \begin{equation} \begin{split} \vartheta(n)\varpi(a\delta_g)\vartheta(n)^&=\iota_S(e,n)\iota_P(a)\iota_S(g,0)\iota_S(e,n)^\\ &=\iota_S(e,n)\iota_S(g,0)\iota_P(u_{g^{-1}}au_g)\iota_S(e,n)^\\ &=\iota_S(\varphi^n(g),0)\iota_S(e,n)\iota_P(u_{g^{-1}}au_g)\iota_S(e,n)^\\ &=\iota_S(\varphi^n(g),0)\iota_P(s^nu_{g^{-1}a{s^}^n}u_g)\\ &=\iota_S(\varphi^n(g),0)\iota_P(u_{\varphi^n(g^{-1})}s^na{s^}^nu_{\varphi^n(g)})\\ &=\iota_P(s^na{s^}^n)\iota_S(\varphi^n(g),0)=\varpi(s^na{s^}^n\delta_{\varphi^n(g)}). \end{split} \end{equation} This implies the existence of a $$-homomorphism \begin{equation}\label{teo12iso2} \Delta: (C^(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N} \rightarrow C^(P)\rtimes_{\alpha} S \end{equation} satisfying $\Delta\circ\iota_G=\varpi$ and $\Delta\circ\iota_N=\vartheta$. Straightforward calculations show that the $$-homomorphisms (\ref{teo12iso1}) and (\ref{teo12iso2}) are inverses of each other. \end{proof} \begin{example}\em\label{ex2} For any finite group $G$, an injective endomorphism will be surjective and therefore the isometry $s$ defining $\mathds{U}[\varphi]$ will be a unitary (by item (iii) of Definition \ref{defi1}). Then as $C^(P)=\mathds{C}$, $$ \mathds{U}[\varphi]\cong C^(G)\rtimes_\tau\mathds{N} $$ where $$ \tau: \mathds{N}\rightarrow \hbox{End}(C^(G)) $$ with $$ \tau_n(\lambda u_g)=\lambda u_{\varphi^n(g)}. $$ If one has the description of the K-theory of $C^(G)$ it is easy to calculate the K-groups of $\mathds{U}[\varphi]$ by applying the Khoshkam-Skandalis sequence (\cite{Khoska}). \end{example} \begin{flushright} $\square$ \end{flushright} Since more results are known for group crossed products than for semigroup ones it is useful to find such a description of our C$^$-algebra. We can do this using the minimal automorphic dilation of the semigroup crossed product system above (for more details, see Section 2 in \cite{Laca1}). One important requirement to use this dilation is that the semigroup must be an Ore semigroup: an Ore semigroup is a cancellative semigroup which is right-reversible i.e, it satisfies $Ss\cap Sr\neq \emptyset$ for all $s,r\in S$. \begin{proposition}\label{propOre} The semidirect product $S=G\rtimes_\varphi\mathds{N}$ is an Ore semigroup. \end{proposition} \begin{proof} Consider $(g_i,n_i)\in S$ for $i\in\{1,2,3\}$. $S$ is cancellative: \begin{equation} \begin{split} &(g_1,n_1)(g_3,n_3)=(g_2,n_2)(g_3,n_3)\\ \Rightarrow\;&(g_1\varphi^{n_1}(g_3),n_1+n_3)=(g_2\varphi^{n_2}(g_3),n_2+n_3)\\ \Rightarrow\; &n_1=n_2 \hbox{ and }g_1\varphi^{n_1}(g_3)=g_2\varphi^{n_1}(g_3)\\ \Rightarrow\; &g_1=g_2 \end{split} \end{equation} \begin{equation} \begin{split} &(g_1,n_1)(g_2,n_2)=(g_1,n_1)(g_3,n_3)\\ \Rightarrow\; &(g_1\varphi^{n_1}(g_2),n_1+n_2)=(g_1\varphi^{n_1}(g_3),n_1+n_3)\\ \Rightarrow\; &n_2=n_3 \hbox{ and }\varphi^{n_1}(g_2)=\varphi^{n_1}(g_3)\\ \Rightarrow\; &g_2=g_3\hbox{ as }\varphi\hbox{ is injective}. \end{split} \end{equation} Also any two principal left ideals of $S$ intersect: \begin{equation} \begin{split} (\varphi^{n_2}(g_1^{-1}),n_2)(g_1,n_1)&=(e,n_2+n_1)\\ &=(\varphi^{n_1}(g_2^{-1}),n_1)(g_2,n_2)\in S(g_1,n_1)\cap S(g_2,n_2). \end{split} \end{equation} \end{proof} It follows that the semigroup $S$ can be embedded in a group, called the enveloping group of $S$, which we will denote as $env(S)$, such that $S^{-1}S=env(S)$ (Theorem 1.1.2 \cite{Laca1}). It also implies that $S$ is a directed set by the relation defined by $(g,n)< (h,m)$ if $(h,m)\in S(g,n)$. Let us define a candidate for $env(S)$. Consider $$ \mathds{G}:=\D\lim_{\rightarrow}\{G_n:\varphi^n\} $$ (with $G_n=G$ for all $n\in\mathds{N}$) and with the extended endomorphism $\overline{\varphi}$ construct the group $$ \overline{S}:=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}. $$ Then we can define an extended action $\overline{\alpha}$ of $\overline{S}$ over $C^(P)^s$: \begin{equation} \begin{split} \overline{\alpha}:\overline{S}&\rightarrow \hbox{Aut}(C^(P)^s)\\ (g_j,n)&\mapsto {s^}^{j}u_gs^{n+j}(\cdot){s^}^{n+j}u_{g^{-1}}s^{j} \end{split} \end{equation} (note that we can also find $g_j$ such that $j\geq \|\,n\|$). Moreover, consider $i: C^(P)\rightarrow C^(P)^s$ the canonical inclusion. \begin{proposition}\label{proporeG}The C$^$-dynamical system $(C^(P)^s,\overline{S},\overline{\alpha})$ is the minimal automorphic dilation of $(C^(P),S,\alpha)$. \end{proposition} \begin{proof} Since the subset of $S$ containing all elements of the type $(e,n)$ is cofinal in $S$, we need only prove that $\overline{S}=env(S)$ (to use Theorem 2.1.1 in \cite{Laca1}). For this we need to show that $S$ is a subsemigroup of $\overline{S}$ and $\overline{S}\subset S^{-1}S$ \cite{CliPre}. First it is obvious that $S$ is a subsemigroup of the group $\overline{S}$ via the inclusion $(g,n)\mapsto (g_0,n)$, where $g_0=g\in G=G_0\hookrightarrow\overline{G}$. Without loss of generality take $(g_i,j)\in\overline{S}$ with $i>\|j\|$. Then $$ (g_i,j)=(g_i,-i)(e,j+i)=(g_0,i)^{-1}(e,j+i)\in S^{-1}S. $$ \end{proof} We may conclude that the following theorem holds (Theorem 2.2.1 of \cite{Laca1}). \begin{theorem}\label{teo22}The C$^$-algebra $\mathds{U}[\varphi]$ is also isomorphic to the full corner $$\iota(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)\footnote{The isomorphism C$^(P)\cong C(\overline{G})$ implemented in Proposition \ref{prop3} implies that the projection $\iota(1)\in C^(P)^s$ corresponds to $p_{\overline{G}}\in C(\overline{G})$ viewed inside $C_0(\mathcal{G})$ via $i_0$ (defined before Remark \ref{obs12}).}.$$ \end{theorem} \begin{flushright} $\square$ \end{flushright} Let us denote the isomorphism given by the last theorem by $$ \beta: \mathds{U}[\varphi]\rightarrow \iota(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1), $$ and by Theorem 2.2.1 in \cite{Laca1} we know that $$ \beta({s^}^nu_{h^{-1}}fu_{h'}s^m)=i(1)U^_{(h,n)}i(f)U_{(h',m)}i(1). $$ Note that the isomorphism above implies that $\mathds{U}[\varphi]$ and $C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}$ are Morita equivalent and so they have the same K-groups. To finish our identifications: \begin{theorem}\label{teo3} The stabilization (Definition \ref{defi2}) $\mathds{U}[\varphi]^s$ is isomorphic to the group crossed product $C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}$. \end{theorem} \begin{proof} As in Theorem 2.4 in \cite{Laca1} we know that $\beta(u_g)=V(g_0,0)\iota(1)$ and $\beta(s^n)=V(e_0,n)\iota(1)$, where $V$ represents $\overline{S}$ in the crossed product $C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}$. Define \begin{equation} \begin{split} \widetilde{\gamma}_{m,l+m}:\iota(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)&\rightarrow \iota(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)\\ x &\mapsto V(e,l)xV(e,l)^. \end{split} \end{equation} Remembering from Definition \ref{defi2} that \begin{equation} \begin{split} \psi_{m,l+m}:\mathds{U}[\varphi]&\rightarrow\mathds{U}[\varphi]\\ x&\mapsto s^lx{s^}^l, \end{split} \end{equation} we can conclude that $$ \beta\circ\psi_{m,l+m}\circ\beta^{-1}= \widetilde{\gamma}_{m,l+m} $$ which implies the existence of an isomorphism $$ \overline{\beta}:\mathds{U}[\varphi]^s\rightarrow\D\lim_{\rightarrow}\{\iota(1)(C^(P)^s \rtimes_{\overline{\alpha}}\overline{S})\iota(1),\;\widetilde{\gamma}_{m,l+m}\}. $$ Moreover for $k\geq 0$ set \begin{equation} \begin{split} \lambda_k:\iota(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)&\rightarrow C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}\\ z &\mapsto V^(e,k)zV(e,k). \end{split} \end{equation} As \begin{equation} \begin{split} \lambda_{l+m}\circ\widetilde{\gamma}_{m,l+m}(z)&=V^(e,l+m)V(e,l)zV(e,l)^V(e,l+m)\\ &=V^(e,m)zV(e,m)=\lambda_m(z),\\ \end{split} \end{equation} we have a $$-homomorphism $$ \lambda:\D\lim_{\rightarrow}\{\iota(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1):\widetilde{\gamma}_{m,l+m}\} \rightarrow C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}. $$ It is injective because each $\lambda_k$ is. Moreover as $$ \lambda_k(\iota(1))=\overline{\alpha}_{(e,-k)}(\iota(1))V_{(e,0)}={s^}^k\iota(1)s^kV_{(e,0)} $$ is an approximate unit for $C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}$, for $z\in C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}$ we have \begin{equation} \begin{split} &\D\lim_k\lambda_k(\iota(1)V(e,k)zV^(e,k)\iota(1)\iota(1))\\ =&\D\lim_k[\lambda_k(\iota(1)V(e,k)zV^(e,k)\iota(1))\lambda_k(\iota(1))]\\ =&\D\lim_k[V^(e,k)\iota(1)V(e,k)zV^(e,k)\iota(1)V(e,k)][{s^}^k\iota(1)s^kV_{(e,0)}]\\ =&\D\lim_k{s^}^k\iota(1)s^kV_{(e,0)}z{s^}^k\iota(1)s^kV_{(e,0)}{s^}^k\iota(1)s^kV_{(e,0)}\\ =&z, \end{split} \end{equation} and so $\lambda$ is surjective. Consequently $ \mathds{U}[\varphi]^s\cong C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}. $ \end{proof} \begin{example}\em Consider a surjective endomorphism $\varphi$ of a group $G$. The surjectivity of $\varphi$ implies that $s$ is an isometry (by item (iii) of Definition \ref{defi1}). Moreover Proposition \ref{Prop1GN} together with the fact that $C^(P)=\mathds{C}$\, implies that $$ \mathds{U}[\varphi]\cong C^(G)\rtimes_\tau\mathds{N}, $$ where $$ \tau: \mathds{N} \rightarrow \hbox{End}(C^(G)) $$ is defined by $$ \tau_n(u_g)=u_{\varphi^n(g)}. $$ Using the six-term exact sequence introduced by Khoshkam and Skandalis in \cite{Khoska}, one can build the sequence $$ \begin{array}{ccccc} K_0(C^(G)) &\xrightarrow{1-K_0(\tau_1)} &K_0(C^(G)) &\rightarrow & K_0(\mathds{U}[\varphi]) \\ \uparrow & & & & \downarrow \\ K_1(\mathds{U}[\varphi]) &\leftarrow &K_1(C^(G)) &\xleftarrow{1-K_1(\tau_1)} & K_1(C^(G)) \\ \end{array} $$ (note that this example is very similar to Example \ref{ex2}). \end{example} \begin{flushright} $\square$ \end{flushright} \section{Properties} The crossed product description in last section implies two nice properties of $\mathds{U}[\varphi]$. \begin{proposition}\label{propl14}If $G$ is amenable then $\mathds{U}[\varphi]$ is nuclear. \end{proposition} \begin{proof} $G$ being amenable implies that $\overline{S}$ is amenable as well (amenability is closed under direct limits by \cite{vNeu} and also closed under semidirect products). But we know that $C^(P)^s$ is nuclear because it is commutative, therefore $C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}$ is nuclear by Proposition 2.1.2 in \cite{Ror}. Since hereditary C-subalgebras of nuclear C-algebras are nuclear by Corollary 3.3 (4) in \cite{ChoiEffros}, we conclude that $$ \mathds{U}[\varphi]\cong C^(P)\rtimes_{\alpha}S\cong i(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})i(1) $$ is nuclear. \end{proof} \begin{proposition}\label{propl15}If $G$ is amenable then $\mathds{U}[\varphi]$ satisfies the UCT property. \end{proposition} \begin{proof} Since $C^(P)^s$ is commutative, $C^(P)^s\rtimes_{\overline{\alpha}}\overline{S}$ is isomorphic to a groupoid C$^$-algebra. When the group $G$ is amenable then $\overline{S}$ also is, and the respective groupoid is also amenable. Therefore using a result by Tu (\cite{tutu} Proposition 10.7), the crossed product satisfies UCT. By Morita equivalence, $\mathds{U}[\varphi]$ also satisfies it. \end{proof} We will now prove that our algebra $\mathds{U}[\varphi]$ is purely infinite and simple. We will proceed in the same way as in \cite{Cuntz2} and in many other papers: we present a particular faithful conditional expectation and a dense $$-subalgebra of $\mathds{U}[\varphi]$ such that the conditional expectation of any positive element of this $$-subalgebra can be described using a finite number of pairwise orthogonal projections. For this purpose we will use the description in Theorem \ref{teo22} of $\mathds{U}[\varphi]$ as a corner of a group crossed product. To define the conditional expectation, we require the amenability of the group $G$: therefore the group $\overline{S}=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}$ (defined after Proposition \ref{propOre}) is also amenable (as mentioned in the proof of Proposition \ref{propl14}). This condition is necessary because we want to use the well-known result which says that there exists a canonical faithful conditional expectation on the reduced group crossed product, and the amenability of $\overline{S}$ implies that both the full and the reduced group crossed products (implemented by $\overline{S}$-actions) are isomorphic. The main tool of this section is the following (proven in Proposition 5.2 of \cite{Li1}). \begin{proposition}\label{prop13}Let $\widetilde{A}$ be a dense $$-subalgebra of a unital C$^$-algebra $A$. Assume that $\epsilon$ is a faithful conditional expectation on $A$ such that for every $0\neq x\in\widetilde{A}_{+}$ there exist finitely many projections $f_i\in A$ with \begin{itemize} \item[(i)] $f_i\bot f_j$,$\forall\; i\neq j$; \item[(ii)] $f_i\sim_{s_i} 1$, via\footnote{I.e.: $\exists\; s_i$ isometries such that $s_is_i^=f_i$,$\forall\; i$} isometries $s_i\in A$, $\forall\; i$; \item[(iii)] $\left\\|\D\sum_if_i\epsilon(x)f_i\right\\|=\\|\epsilon(x)\\|$; \item[(iv)] $f_ixf_i=f_i\epsilon(x)f_i\in\mathds{C}f_i$,$\forall\; i$. \end{itemize} Then $A$ is purely infinite and simple. \end{proposition} \begin{flushright} $\square$ \end{flushright} Moreover in order to find these projections it is also necessary to require that the injective endomorphism $\varphi$ is pure, i.e: $$ \D\bigcap_{n\in\mathds{N}}\varphi^n(G)=\{e\}. $$ In order to apply the proposition above the first step is to define a conditional expectation. As mentioned before, we require that the group $G$ is amenable. Remember the isomorphism from Theorem \ref{teo22}: $$ \beta: \mathds{U}[\varphi]\rightarrow \iota(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1). $$ \begin{proposition}\label{prop1}There exists a faithful conditional expectation \begin{equation} \begin{split} \epsilon: \mathds{U}[\varphi]&\rightarrow \beta^{-1}(\iota(1)C^(P)^s\iota(1))\\ {s^n}^u_{h^{-1}}fu_{h'}s^m&\mapsto\left\{ \begin{array}{ll} {s^n}^u_{h^{-1}}fu_hs^n, & \hbox{if }n=m\hbox{ and }h=h'; \\ 0, & \hbox{otherwise.} \end{array} \right. \end{split} \end{equation} for all $h$, $h'\in G$ and $n$, $m\in\mathds{N}$. \end{proposition} \begin{proof} As $\overline{S}$ is amenable, the isomorphism $\beta$ of Theorem \ref{teo22} can be expanded to include also the reduced group crossed product $$ \mathds{U}[\varphi]\cong \iota(1)(C^(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)\cong \iota(1)(C^(P)^s\rtimes_{r,\overline{\alpha}}\overline{S})\iota(1). $$ Let us denote the elements of $\overline{S}$ by $s$ and its identity by $e$. We will also use $\delta_s$ to denote the unitary elements implementing the action of $\overline{S}$ in the crossed product. Consider the well-known faithful conditional expectation on the reduced group crossed product: \begin{equation} \begin{split} E: C^(P)^s\rtimes_{r,\overline{\alpha}}\overline{S}&\rightarrow C^(P)^s\\ x\delta_s&\mapsto \left\{ \begin{array}{ll} x, & \hbox{if }s=e; \\ 0, & \hbox{otherwise.} \end{array} \right. \end{split} \end{equation} Straightforward calculations show that the following is also a faithful conditional expectation: \begin{equation} \begin{split} \overline{E}: \iota(1)(C^(P)^s\rtimes_{r,\overline{\alpha}}\overline{S})\iota(1)&\rightarrow \iota(1)C^(P)^s\iota(1)\\ \iota(1)x\delta_s\iota(1)&\mapsto \left\{ \begin{array}{ll} \iota(1)x\iota(1), & \hbox{if }s=e; \\ 0, & \hbox{otherwise.} \end{array} \right. \end{split} \end{equation} Using the isomorphism $\beta$ we can rewrite $\overline{E}$ to conclude that we have the faithful conditional expectation \begin{equation} \begin{split} \epsilon: \mathds{U}[\varphi]&\rightarrow \beta^{-1}(\iota(1)C^(P)^s\iota(1))\\ {s^n}^u_{h^{-1}}fu_{h'}s^m&\mapsto\left\{ \begin{array}{ll} {s^n}^u_{h^{-1}}fu_hs^n, & \hbox{if }n=m\hbox{ and }h=h'; \\ 0, & \hbox{otherwise.} \end{array} \right. \end{split} \end{equation} \end{proof} Now, to find projections to describe the image of $y\in span(Q)_{+}$ under the conditional expectation $\epsilon$ presented above, remember that $y$ has the form $$ y=\D\sum_{m,n,h,h',f}a_{(m,n,h,h',f)}{s^}^nu_{h^{-1}}fu_{h'}s^m $$ for $m$, $n\in\mathds{N}$, $h$, $h'\in G$, $f\in P$ and $a_{(\ldots)}\neq 0$. As we have finitely many projections of $C^(P)$ in the description of $y$, write them all as sums of (altogether $N$) mutually orthogonal projections $u_{g_i}s^M{s^}^Mu_{g_i^{-1}}$, with $g_i\in G/\varphi^M(G)$, for all $1\leq i\leq N$ and $M\in\mathds{N}$ big enough. \begin{proposition}\label{prop2}There are $N$ pairwise orthogonal projections $f_1,\ldots f_N\in P$ such that \begin{enumerate} \item[(i)] $\Phi$ defined by \begin{equation} \begin{split} C^(\{u_{g_1}s^M{s^}^Mu_{g_1^{-1}},\ldots,u_{g_N}s^M{s^}^Mu_{g_N^{-1}}\})&\rightarrow C^(\{f_1,\ldots,f_N\})\\ z&\mapsto\D\sum_{i=1}^Nf_izf_i \end{split} \end{equation} is an isomorphism, and \item[(ii)] $\Phi(\epsilon(y))=\D\sum_{i=1}^Nf_iyf_i$, $\forall\; y\in\mathds{U}[\varphi]$. \end{enumerate} \end{proposition} \begin{proof} Define $$ f_i:=u_{h_i}s^p{s^}^pu_{h_i^{-1}} $$ for some $p\in\mathds{N}$ bigger than $M$ (in fact, we may choose $p$ as big as we want), where $g_i^{-1}h_i\in\varphi^M(G)$. This implies that the set of the $f_i$'s is orthogonal and that (i) holds. For (ii), first note that when $\delta_{m,n}\delta_{h,h'}=1$ it is true that $\epsilon=Id$, and so (ii) is satisfied. So let us take a look on those summands in $y$ with $\delta_{m,n}\delta_{h,h'}=0$ (we will say that such an element has \emph{critical index} $(m,n,h,h',f)$). The conditional expectation $\epsilon$ maps these summands to 0 and in order for (ii) to be satisfied we need that, for all $1\leq i\leq N$, $$ f_i{s^}^nu_{h^{-1}}fu_{h'}s^mf_i=0. $$ We calculate \begin{equation} \begin{split} &f_i{s^}^nu_{h^{-1}}fu_{h'}s^mf_i\\ &={s^}^nu_{h^{-1}}(u_hs^nf_i{s^}^nu_{h^{-1}})f(u_{h'}s^mf_i{s^}^mu_{h'^{-1}})u_{h'}s^m\\ &={s^}^nu_{h^{-1}}[u_{h\varphi^n(h_i)}s^{n+p}{s^}^{n+p}u_{\varphi^n(h_i^{-1})h^{-1}}\\ &u_{h'\varphi^m(h_i)}s^{m+p}{s^}^{n+p} u_{\varphi^m(h_i^{-1})h'^{-1}}]fu_{h'}s^m. \end{split} \end{equation} Now, analysing only the expression between the brackets, \begin{equation} \begin{split} &[u_{h\varphi^n(h_i)}s^{n+p}{s^}^{n+p}u_{\varphi^n(h_i^{-1})h^{-1}}u_{h'\varphi^m(h_i)}s^{m+p}{s^}^{m+p} u_{\varphi^m(h_i^{-1})h'^{-1}}]\\ &=\left(\D\sum_{g\in G/\varphi^m(G)}u_{h\varphi^n(h_i)\varphi^{n+p}(g)}s^{m+n+p}{s^}^{m+n+p}u_{\varphi^{n+p}(g^{-1})\varphi^n(h_i^{-1})h^{-1}}\right)\\ &\times\left(\D\sum_{k\in G/\varphi^n(G)}u_{h'\varphi^m(h_i)\varphi^{m+p}(k)}s^{m+n+p}{s^}^{m+n+p}u_{\varphi^{m+p}(k^{-1})\varphi^m(h_i^{-1})h^{-1}}\right). \end{split} \end{equation} This product will be zero if the two sums are mutually orthogonal, which happens if for all $g\in G/\varphi^m(G)$ and $k\in G/\varphi^n(G)$, \begin{equation} h\varphi^n(h_i)\varphi^{n+p}(g)\varphi^{m+n+p}(x)\neq h'\varphi^m(h_i)\varphi^{m+p}(k)\varphi^{m+n+p}(y),\;\forall\; x,y\in G \end{equation} which is equivalent to \begin{equation} \varphi^{m+p}(k^{-1})\varphi^m(h_i^{-1})h'^{-1}h\varphi^n(h_i)\varphi^{n+p}(g)\neq\varphi^{m+n+p}(z),\;\forall\; z\in G. \end{equation} A sufficient condition for this to hold is that $\varphi^m(h_i^{-1})h'^{-1}h\varphi^n(h_i)\neq\varphi^p(z)$ $\forall\; z\in G$, for each critical index $(m,n,h,h',f)$. Using the fact that $\varphi$ is pure we may choose some $p_{(m,n,h,h',f)}\in\mathds{N}$ such that $$ \varphi^m(h_i^{-1})h'^{-1}h\varphi^n(h_i)\notin\varphi^{p_{(m,n,h,h',f)}}(G). $$ As we have a finite number of critical indices, it is sufficient to take the biggest $p_{(m,n,h,h',f)}$ and call it $p$. \end{proof} To understand this choice of $p$, consider the following example. \begin{example}\em\label{ex1} Let $G=\mathds{Z}$ and \begin{equation} \begin{split} \varphi:\mathds{Z}&\rightarrow\mathds{Z}\\ n &\mapsto 3n. \end{split} \end{equation} Then we have $\frac{G}{\varphi(G)}=\{\overline{0},\overline{1},\overline{2}\}$, $\frac{G}{\varphi^2(G)}=\{\overline{0},\overline{1},\ldots,\overline{8}\}$ and in general $$ \frac{G}{\varphi^n(G)}=\{\overline{0},\ldots,\overline{3^n-1}\}=\mathds{Z}_{3^n}. $$ Take the following $y\in span(Q)$ \begin{equation} \begin{split} y=&2{s^}^2u_{30}(u_5s^{4}{s^}^4u_{-5})u_{2187}s^{1}-4{s^}^7u_0(u_{10}s^{4}{s^}^4u_{-10})u_{-5}s^9\\ &+{s^}^8u_{20}s^{4}{s^}^4u_{-20}s^8 \end{split} \end{equation} and note that in $y$ we have two terms with critical indices (the first ones). Using the notation of the above proposition, $M=4$ and, for the first term of $y$: $n=2$, $m=1$, $h=-30$, $h'=2187$ and $g_1=5$. Choosing $h_1=86$, it is true that $-g_1+h_1=-5+86=81\in\varphi(G)$. Then: $$ \varphi^{1}(-86)-2187-30+\varphi^{2}(86)=-1701=\varphi^5(7)\notin\varphi^6(\mathds{Z}). $$ So $p_1:=p_{(1,2,-30,2187,f)}=6$ (or bigger). For the second term it is not hard to see that $p_2=1$: $$ \varphi^{9}(-91)+5-0+\varphi^{7}(91)=-1592131\notin\varphi^1(\mathds{Z}). $$ So one can choose any $p\geq 6$. \end{example} \begin{flushright} $\square$ \end{flushright} Using the description above of the faithful conditional expectation $$ \epsilon: \mathds{U}[\varphi]\rightarrow \beta^{-1}(\iota(1)C^(P)^s\iota(1)) $$ where $P=\{u_gs^n{s^}^nu_{g^{-1}}:\;g\in G,\; n\in\mathds{N}\}$, together with the dense $$-subalgebra $$ \hbox{span}(Q)=\hbox{span}(\{{s^}^nu_{h^{-1}}fu_{h'}s^m:\;f\in P, h, h'\in G, n,m\in\mathds{N}\}), $$ we can prove the main result of this section by applying Propositions \ref{prop2} and \ref{prop13} (the definition of pure infiniteness comes from \cite{Cuntz2}). \begin{theorem}\label{teo1}Let $G$ be a discrete countable amenable group and $\varphi$ a pure injective endomorphism of $G$ with finite cokernel. Then the C$^$-algebra $\mathds{U}[\varphi]$ is simple and purely infinite, i.e. for all non zero $x\in\mathds{U}[\varphi]$ there are $a$, $b\in\mathds{U}[\varphi]$ with $axb=1$. \end{theorem} \begin{flushright} $\square$ \end{flushright} \begin{corollary} When satisfied the conditions of the theorem above, the universal C$^$-algebra $\mathds{U}[\varphi]$ is isomorphic to $C_r^[\varphi]$, as defined in Definitions \ref{defi1} and \ref{defi1red} respectively. \end{corollary} \begin{flushright} $\square$ \end{flushright} \begin{theorem} If the conditions of the theorem above are satisfied, the universal C$^$-algebra $\mathds{U}[\varphi]$ is a Kirchberg algebra satisfying the UCT property. \end{theorem} \begin{flushright} $\square$ \end{flushright} It would be interesting to know if the conditions of the theorem above are also necessary: if we construct the C$^$-algebra associated with some injective endomorphism of an amenable group, is it simple and purely infinite only if $\varphi$ is pure? Unfortunately we don't answer this question here, but the next trivial example gives some idea about this direction. \begin{example}\em\label{ex0} For some commutative group $G$ (thus amenable), consider $\varphi = id_G$. As $u_gs=su_g$ for all $g\in G$ ($\varphi$ is trivial), our C$^$-algebra will be commutative. Now, as $\frac{G}{\varphi(G)}$ has only the element $\{e\}$, condition (iii) of Definition \ref{defi1} implies that the isometry $s$ is a unitary. Then $\mathds{U}[\varphi]$ is the commutative C$^$-algebra generated by the unitaries $\{u_g, s:g\in G\}$, and this one is the non-simple tensor product $C^(G)\otimes C^(\mathds{Z})=C^(G)\otimes C(\mathcal{S}^1)$. Moreover using the K\"{u}nneth Formula \cite{Schoc} we conclude that \begin{equation} K_0(\mathds{U}[\varphi])=K_1(\mathds{U}[\varphi])=K_0(C^(G))\oplus K_1(C^(G)). \end{equation} \end{example} \begin{flushright} $\square$ \end{flushright} \section{Description of $\mathds{U}[\varphi]$ via group partial crossed products}\label{partialcrpr} In \cite{BoEx} Boava and Exel constructed a partial group algebra isomorphic to the C$^$-algebra $\mathds{U}[R]$ associated with a integral domain $R$ \cite{Culi1}. Consequently due to Theorem 4.4 of \cite{ExLaQu} one can define a certain partial crossed product which is isomorphic to $\mathds{U}[R]$. With the latter description it is proven in \cite{BoEx}, using only tools from partial crossed products, that if $R$ is not a field then $\mathds{U}[R]$ is simple (which is part of the conclusion of Li \cite{Li1}, namely, Corollary 5.14). In this section we will present analogous results adapted to our case, i.e., given a C$^$-algebra $\mathds{U}[\varphi]$ associated with some injective endomorphism $\varphi$ of a group $G$ with unit $e$, we will show that $\mathds{U}[\varphi]$ can also be viewed as a partial group algebra and, consequently, as a partial crossed product. The ideas follow the ones presented in \cite{BoEx}. With this description we show that when $G$ is amenable we can rewrite the faithful conditional expectation $\epsilon$ presented in Proposition \ref{prop1} in terms of the partial group crossed product. To finish we use a well known result from the theory of group partial crossed products to prove a weaker result than Theorem \ref{teo1}: if $G$ is commutative and $\varphi$ is pure then $\mathds{U}[\varphi]$ is simple. We start with an introduction to partial actions, partial crossed products and partial group algebras, before presenting the right isomorphisms and descriptions of $\mathds{U}[\varphi]$. \begin{definition}\label{pcpdefi1}A partial action $\alpha$ of a group $G$ on a C$^$-algebra $A$ is a collection of closed two-sided ideals $\{D_g\}_{g\in G}$ of $A$ and $$-isomorphisms $\alpha_g: D_{g^{-1}}\rightarrow D_g$ satisfying \begin{itemize} \item[(PA1)] $D_e=A$; \item[(PA2)] $\alpha_h^{-1}(D_h\cap D_{g^{-1}})\subseteq D_{(gh)^{-1}}$; \item[(PA3)] $\alpha_g\circ\alpha_h(x)=\alpha_{gh}(x)$, $\forall\; x\in \alpha_h^{-1}(D_h\cap D_{g^{-1}})$. \end{itemize} \end{definition} Using (PA1) - (PA3) one can show that $\alpha_e=$ id$_A$, $\alpha_{g^{-1}}=\alpha_g^{-1}$ and that\nl $\alpha_h^{-1}(D_h\cap D_{g^-1})=D_{(gh)^{-1}}\cap D_{h^{-1}}$. Analogously, one can define a partial action of $G$ acting on a locally compact space $X$: just replace the ideals $D_g$ by open sets $X_g\subseteq X$ and the $$-isomorphisms $\alpha_g$ by homeomorphisms $\theta_g: X_{g^{-1}}\rightarrow X_g$. We call the triples $(\alpha,G,A)$ or $(\theta,G,X)$ partial dynamical systems, or partial actions when there is no possibility of misunderstanding. \begin{example}\em\label{Expa1} If $\theta$ is a partial action of $G$ on the locally compact space $X$ with $\theta_g: X_{g^{-1}}\rightarrow X_g$, one can easily construct a partial action of $G$ on the C$^$-algebra $C_0(X)$ considering $D_g=C_0(X_g)$ and \begin{equation} \begin{split} \alpha_g:C_0(X_{g^{-1}})&\rightarrow C_0(X_g)\\ f&\mapsto f\circ\theta_{g^{-1}}. \end{split} \end{equation} \end{example} \begin{flushright} $\square$ \end{flushright} Now we want to define partial crossed products. There are three ways to realize them: one using Fell bundles (and we recommend \cite{ExelFell}), another using enveloping C$^$-algebras (for details and some interesting examples look at Section 2 of \cite{Mc}) and the last one as a universal object with respect to covariant pairs (see Section 3 of \cite{QuRa}). We use the last way in our proofs and therefore we present it. Let us define first a particular set of representations called partial representations. \begin{definition}\label{pcpdefi2}A partial representation $\pi$ of a group $G$ into a unital C$^$-algebra $B$ is a map $\pi: G\rightarrow B$ satisfying \begin{itemize} \item[(PR1)] $\pi(e)=1$; \item[(PR2)] $\pi(g^{-1})=\pi(g)^$; \item[(PR3)] $\pi(g)\pi(h)\pi(h^{-1})=\pi(gh)\pi(h^{-1})$. \end{itemize} \end{definition} Then the partial group crossed product $A\rtimes_{\alpha}G$ is defined as the universal object with respect to a covariant pair $(\upsilon,\pi)$, which means a $$-homomorphism ($B$ being a unital C$^$-algebra) $$ \upsilon: A\rightarrow B $$ and a partial representation of $G$ $$ \pi: G\rightarrow B $$ satisfying \begin{equation} \begin{split} \upsilon(\alpha_g(x))&=\pi(g)\upsilon(x)\pi(g^{-1})\hbox{ for }x\in D_{g^{-1}},\\ \upsilon(x)\pi(g)\pi(g^{-1})&=\pi(g)\pi(g^{-1})\upsilon(x)\hbox{ for }x\in A. \end{split} \end{equation} To define a partial group algebra, consider the set $[G]:=\{[g]:\;g\in G\}$ (without any operations). \begin{definition}\label{defipagr}The partial group algebra of $G$, denoted $C^_p(G)$, is the universal C$^$-algebra generated by $[G]$ with respect to the relations \begin{itemize} \item[(R$_p$1)] $[e]=1$; \item[(R$_p$2)] $[g^{-1}]=[g]^$; \item[(R$_p$3)] $[g][h][h^{-1}]=[gh][h^{-1}]$. \end{itemize} \end{definition} The C$^$-algebra $C_p^(G)$ is universal with respect to partial representations of $G$ (note the equivalence between relations (R$_p$) and (PR) of Definition \ref{pcpdefi2}). In fact, one can define partial group algebras for more restricted situations, i.e., requiring that $[G]$ satisfies additional relations than the 3 relations above. Let us set $e_g:=[g][g^{-1}]$ and for our constructions consider $\mathcal{R}$ a set of (extra) relations on $[G]$ such that every relation is of the form \begin{equation}\label{relationspga} \sum_i\prod_je_{g_{ij}}=0. \end{equation} \begin{definition}The partial group algebra of $G$ with relations $\mathcal{R}$, denoted $C_p^(G,\mathcal{R})$, is defined to be the universal C$^$-algebra generated by $[G]$ with relations $R_p\cup\,\mathcal{R}$. This C$^$-algebra is universal with respect to partial representations which satisfy $\mathcal{R}$. \end{definition} An interesting fact is that the class of partial group algebras without restrictions and of the ones with extra relations of the type (\ref{relationspga}) is contained in the class of partial crossed products (Definition 6.4 of \cite{Exel1} and Theorem 4.4 of \cite{ExLaQu} respectively). In our case the C$^$-algebra $\mathds{U}[\varphi]$ will be isomorphic to a partial group algebra with additional relations of the form (\ref{relationspga}) above, and we will show how these can be viewed as partial crossed products. Consider the power set $\mathcal{P}(G)$ (of $G$) with the topology given by identifying it with the compact set $\{0,1\}^G$, and denote $X_G$ the subset of $\mathcal{P}(G)$ of the subsets $\xi$ of $G$ which contain $e\in G$. Note that using the product topology of $\{0,1\}^G$ implies that $X_G$ is compact and Hausdorff. Denote by $1_g$ the following function in $C(X_G)$: $$ 1_g(\xi)=\left\{ \begin{array}{ll} 1, & \hbox{if }g\in\xi; \\ 0, & \hbox{otherwise.} \end{array} \right. $$ Denote $\widehat{\mathcal{R}}$ the subset of $C(X_G)$ given by the functions $\sum_i\prod_j1_{g_{ij}}$ where the relation $\sum_i\prod_je_{g_{ij}}=0$ is in $\mathcal{R}$. The \emph{spectrum} of the relations $\mathcal{R}$ is defined to be the compact (Proposition 4.1 \cite{ExLaQu}) space $$ \Omega_\mathcal{R}:=\{\xi\in X_G:\; f(g^{-1}\xi)=0,\;\forall\; f\in\widehat{\mathcal{R}},\;\forall\; g\in\xi\}. $$ Now for $g\in G$, consider $$ \Omega_g:=\{\xi\in\Omega_\mathcal{R}:\; g\in\xi\} $$ and let us define \begin{equation} \begin{split} \theta_g: \Omega_{g^{-1}}&\rightarrow\Omega_g\\ \xi&\mapsto g\xi. \end{split} \end{equation} Then we have defined a partial action $\theta$ of $G$ on $\Omega_\mathcal{R}$. Turning this partial action (as in Example \ref{Expa1}) into a partial action $\alpha$ of $G$ on $C(\Omega_\mathcal{R})$, it is well known (by Theorem 4.4 (iii) in \cite{ExLaQu}) that \begin{equation}\label{parcpc14} C_p^(G,R)\cong C(\Omega_\mathcal{R})\rtimes_{\alpha}G. \end{equation} Now let us find a partial group C$^$-algebra description of $\mathds{U}[\varphi]$. Therefore recall the set $\overline{S}=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}$ whose elements will be denoted by $(g_i,n)$ with $g_i\in G_i\subseteq\mathds{G}$. In case $g\in G=G_0\subseteq\mathds{G}$ we will use the notation $(g,n)$. Consider the following relations $\mathcal{R}$: \begin{itemize} \item[($\mathcal{R}_1$)] $[(g,0)][(g,0)^{-1}]=1,\;\forall\; g\in G$; \item[($\mathcal{R}_2$)] $[(e,-n)][(e,-n)^{-1}]=1\;\forall\; n\in\mathds{N}$; \item[($\mathcal{R}_3$)] $\D\sum_{g\in\frac{G}{\varphi^n(G)}}[(g,n)][(g,n)^{-1}]=1,\;\forall\; n\in\mathds{N}$. \end{itemize} Consider also the partial group algebra relations in this case i.e, on the group $\overline{S}$: \begin{itemize} \item[(R$_p$1)] $[(e,0)]=1$; \item[(R$_p$2)] $[(g_i,n)^{-1}]=[(g_i,n)]^,\;\forall\; n\in\mathds{Z},\;\forall\; g_i\in\mathds{G}$; \item[(R$_p$3)] $[(g_i,n)][(h_j,m)][(h_j,m)^{-1}]=[(g_i\varphi^n(h_j),n+m)][(h_j,m)^{-1}],\nl\;\forall\; m,n\in\mathds{Z},\;\forall\; g_i,h_j\in\mathds{G}$. \end{itemize} Define \begin{equation} \begin{split} \pi: \overline{S}&\rightarrow\mathds{U}[\varphi]\\ (g_i,n)&\mapsto {s^}^iu_gs^{n+i}, \end{split} \end{equation} remembering that we can always suppose $i\geq \|\,n\|$. Note that when $g\in G$, $\pi(g,n)=u_gs^n$. \begin{proposition}The map $\pi$ is a partial representation of $\overline{S}$ which satisfies the relations $\mathcal{R}$. \end{proposition} \begin{proof} First we prove that $\pi$ is a partial representation of $\overline{S}$. (R$_p$1): $\pi((e,0))=u_e=1$; (R$_p$2): \begin{equation} \begin{split} \pi((g_i,n)^{-1})&=\pi((g^{-1}_{i+n},-n))={s^}^{i+n}u_{g^{-1}}s^i=({s^}^iu_gs^{i+n})^\\ &=(\pi((g_i,n)))^; \end{split} \end{equation} (R$_p$3): \begin{equation} \begin{split} &\pi((\varphi^j(g)\overline{\varphi}^{i+n}(h)_{i+j},n+m))\pi((h_j,m)^{-1})\\ &={s^}^{i+j}u_{\varphi^j(g)\varphi^{i+n}(h)}s^{i+j+n+m}{s^}^{j+m}u_{h^{-1}}s^j\\ &={s^}^iu_g{s^}^js^{i+n}\underbrace{u_hs^{j+m}{s^}^{j+m}u_{h^{-1}}}\underbrace{s^j{s^}^j}s^j\\ &={s^}^iu_g{s^}^js^{i+n}s^j{s^}^ju_hs^{j+m}{s^}^{j+m}u_{h^{-1}}s^j\\ &={s^}^iu_gs^{i+n}{s^}^ju_hs^{j+m}{s^}^{j+m}u_{h^{-1}}s^j\\ &=\pi((g_i,n))\pi((h_j,m))\pi((h_j,m)^{-1}). \end{split} \end{equation} Now we show that $\pi$ satisfies the extra relations $\mathcal{R}$. ($\mathcal{R}_1$): $\pi((g,0))\pi((g,0)^{-1})=u_e=1$; ($\mathcal{R}_2$): $\pi((e,-n))\pi((e,-n)^{-1})=\pi((e_n,-n))\pi((e_n,-n)^{-1})={s^}^ns^n=1$; ($\mathcal{R}_3$): $\D\sum_{g\in\frac{G}{\varphi^n(G)}}\pi((g,n))\pi((g,n)^{-1})=\D\sum_{g\in\frac{G}{\varphi^n(G)}}u_gs^n{s^}^{-n}u_{g^{-1}}=1$. \end{proof} It follows from the universality of the partial group algebra $C_p^(\overline{S},\mathcal{R})$ that there exists a $$-homomorphism \begin{equation}\label{eqpga1} \begin{split} \Phi: C^_p(\overline{S},\mathcal{R})&\rightarrow\mathds{U}[\varphi]\\ [(g_i,n)]&\mapsto {s^}^iu_gs^{n+i}. \end{split} \end{equation} Let us find an inverse for $\Phi$ by using the relations which define $\mathds{U}[\varphi]$. \begin{proposition}The (obviously) unitary elements $[(g,0)]$ and isometries $[(e,n)]$ of $C_p^(\overline{S},\mathcal{R})$ satisfy the relations which define $\mathds{U}[\varphi]$. \end{proposition} \begin{proof} Let us show that the elements above satisfy the relations (i) - (iii) of Definition \ref{defi1}. (i):\begin{equation} \begin{split} [(g,0)][(h,0)]&=[(g,0)][(h,0)][(h^{-1},0)][(h,0)]=[(gh,0)][(h^{-1},0)][(h,0)]\\ &=[(gh,0)]; \end{split} \end{equation} (ii):\begin{equation} \begin{split} [(e,1)][(g,0)]&=[(e,1)][(g,0)][(g^{-1},0)][(g,0)]=[(\varphi(g),1)][(g^{-1},0)][(g,0)]\\ &=[(\varphi(g),1)]=[(\varphi(g),1)][(e,-1)][(e,1)]\\ &=[(\varphi(g),0)][(e,1)][(e,-1)][(e,1)]\\ &=[(\varphi(g),0)][(e,1)]; \end{split} \end{equation} (iii): \begin{equation} \begin{split} [(g,0)][(e,1)][(e,-1)][(g^{-1},0)]&=[(g,1)][(e,-1)][(g^{-1},0)]\\ &=[(g,1)][(e,-1)][(g^{-1},0)][(g,0)][(g^{-1},0)]\\ &=[(g,1)][(g^{-1}_1,-1)][(g,0)][(g^{-1},0)]\\ &=[(g,1)][(g^{-1}_1,-1)]=[(g,1)][(g,1)]^, \end{split} \end{equation} and using $\mathcal{R}_3$ we see that it satisfies condition (iii). \end{proof} Consequently we have a $$-homomorphism \begin{equation}\label{eqpga2} \begin{split} \Psi: \mathds{U}[\varphi]&\rightarrow C^_p(\overline{S},\mathcal{R})\\ u_g&\mapsto [(g,0)]\\ s^n&\mapsto [(e,n)]. \end{split} \end{equation} \begin{theorem}\label{teouppga} The C$^$-algebra $\mathds{U}[\varphi]$ is isomorphic to $C^_p(\overline{S},\mathcal{R})$. \end{theorem} \begin{proof} We just have to show that the $$-homomorphisms (\ref{eqpga1}) and (\ref{eqpga2}) are inverses of each other on the generators of the respective C$^$-algebras. $\,\;\;\;\;\;\;\bullet\;\;\Phi\circ\Psi(u_g)=\Phi([(g,0)])=u_g$; $\,\;\;\;\;\;\;\bullet\;\;\Phi\circ\Psi(s^n)=\Phi([(e,n)])=s^n$; \begin{equation} \begin{split} \bullet\;\;\Psi\circ\Phi([(g_i,n)])&=\Psi({s^}^iu_gs^{n+i})=[(e,-i)][(g,0)][(e,n+i)]\\ &=[(e,-i)][(g,0)][(e,n+i)][(e,-n-i)][(e,n+i)]\\ &=[(e,-i)][(g,n+i)][(e,-n-i)][(e,n+i)]\\ &=[(e,-i)][(e,i)][(e,-i)][(g,n+i)]\\ &=[(e,-i)][(e,i)][(\overline{\varphi}^{-i}(g),n)]\\ &=[(g_i,n)]. \end{split} \end{equation} \end{proof} In order to define a partial crossed product isomorphic to $C_p^(\overline{S},\mathcal{R})$ which by the theorem above is isomorphic to $\mathds{U}[\varphi]$, consider $X_{\overline{S}}$ the subset of $\mathcal{P}(\overline{S})$ of the subsets $\xi$ of $\overline{S}$ which contain $(e,0)\in \overline{S}$. Also $1_s\in C(X_{\overline{S}})$ is given by $$ 1_s(\xi)=\left\{ \begin{array}{ll} 1, & s\in\xi; \\ 0, & \hbox{otherwise.} \end{array} \right. $$ and the partial group algebra relations $\mathcal{R}$ are \begin{itemize} \item[($\mathcal{R}_1$)] $e_{(g,\,0)}-1=0,\;\forall\; g\in G$; \item[($\mathcal{R}_2$)] $e_{(e,-n)}-1=0\;\forall\; n\in\mathds{N}$; \item[($\mathcal{R}_3$)] $\D\sum_{g\in\frac{G}{\varphi^n(G)}}e_{(g,n)}-1=0,\;\forall\; n\in\mathds{N}$. \end{itemize} This implies that $\widehat{\mathcal{R}}$ is the subset of $C(X_{\overline{S}})$ consisting of the functions \begin{itemize} \item[($\widehat{\mathcal{R}}_1$)] $1_{(g,\,0)}-1_{(e,\,0)},\;\forall\; g\in G$; \item[($\widehat{\mathcal{R}}_2$)] $1_{(e,-n)}-1_{(e,\,0)}\;\forall\; n\in\mathds{N}$; \item[($\widehat{\mathcal{R}}_3$)] $\D\sum_{g\in\frac{G}{\varphi^n(G)}}1_{(g,n)}-1_{(e,\,0)},\;\forall\; n\in\mathds{N}$. \end{itemize} The spectrum of the relations $\mathcal{R}$ is defined to be $$ \Omega_\mathcal{R}=\{\xi\in X_{\overline{S}}:\; f(g^{-1}\xi)=0,\;\forall\; f\in\widehat{\mathcal{R}},\;\forall\; g\in\xi\}. $$ Consider $$ \Omega_s=\{\xi\in\Omega_\mathcal{R}:\; s\in\xi\} $$ and define the partial action $\varpi$ of $\overline{S}$ on $\Omega_\mathcal{R}$ by \begin{equation}\label{isouapcp} \begin{split} \varpi_s: \Omega_{s^{-1}}&\rightarrow\Omega_s\\ \xi&\mapsto s\xi. \end{split} \end{equation} Then it is well known by Theorem \ref{teouppga} and (\ref{parcpc14}) respectively that \begin{equation}\label{isouapcp15} \mathds{U}[\varphi]\cong C^_p(\overline{S},\mathcal{R})\cong C(\Omega_\mathcal{R})\rtimes_{\alpha}\overline{S}, \end{equation} where \begin{equation}\label{isouapcp16} \begin{split} \alpha_s: C(\Omega_{s^{-1}})&\rightarrow C(\Omega_s)\\ f&\mapsto f\circ\varpi_{s^{-1}}. \end{split} \end{equation} The partial crossed product description of $\mathds{U}[\varphi]$ presented above together with the requirement that $G$ is amenable (which implies that $\overline{S}$ is as well) makes it possible to define a certain conditional expectation as done in \cite{ExelFell} Proposition 2.9 (as in the classical group crossed product construction the amenability of the group implies the isomorphism of both reduced and full constructions by \cite{Mc}, and a faithful conditional expectation exists for the reduced one). We will show that this conditional expectation is the same - modulo the isomorphism already established - as $\epsilon$ as given by Proposition \ref{prop1}. The conditional expectation of $C(\Omega_\mathcal{R})\rtimes_{\alpha}\overline{S}$ is given by \begin{equation} \begin{split} \overline{E}: C(\Omega_\mathcal{R})\rtimes_{\alpha}\overline{S}&\rightarrow C(\Omega_\mathcal{R})\\ f\delta_s &\mapsto \left\{ \begin{array}{ll} f, & \hbox{if }s=(e,0); \\ 0, & \hbox{otherwise.} \end{array} \right. \end{split} \end{equation} Identifying $C^_p(\overline{S},\mathcal{R})$ with $C(\Omega_\mathcal{R})\rtimes_{\alpha}\overline{S}$, $\overline{E}$ becomes \begin{equation} \begin{split} E: C^_p(\overline{S},\mathcal{R})&\rightarrow C^(e_{(g_i,n)})\footnotemark\\ \D\prod_{(g_i,n)\in\overline{S}}^{\scriptscriptstyle{\hbox{finite}}}[(g_i,n)]&\mapsto \left\{ \begin{array}{ll} \D\prod_{(g_i,n)\in\overline{S}}^{\scriptscriptstyle{\hbox{finite}}}[(g_i,n)], & \hbox{if }\D\prod_{(g_i,n)\in\overline{S}}^{\scriptscriptstyle{\hbox{finite}}}(g_i,n)=(e,0); \\ 0, & \hbox{otherwise.} \end{array} \right. \end{split} \end{equation}\footnotetext{$e_{(g_i,n)}:=[(g_i,n)][(g_i,n)^{-1}]$ with $(g_i,n)\in\overline{S}=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}$} Using the isomorphism $\Psi$ (from (\ref{eqpga2})) and $\epsilon$ (from Proposition \ref{prop1}), we shall prove the following. \begin{proposition} $E\circ\Psi=\Psi\circ\epsilon$. \end{proposition} \begin{proof} Let us prove the equality on the dense $$-subalgebra of $\mathds{U}[\varphi]$ given by $$ \hbox{span}(Q)=\hbox{span}(\{{s^}^nu_{h^{-1}}fu_{h'}s^m:\;f\in P,\,h, h'\in G,\,n,m\in\mathds{N}\}). $$ Consider $f=u_gs^k{s^}^ku_{g^{-1}}\in P$, $h, h'\in G$, and $n,m\in\mathds{N}$. \begin{equation} \begin{split} &E\circ\Psi({s^}^nu_{h^{-1}}fu_{h'}s^m)=E\circ\Psi({s^}^nu_{h^{-1}}u_gs^k{s^}^ku_{g^{-1}}u_{h'}s^m)\\ &= E([(e,-n)][(h^{-1},0)][(g,0)][(e,k)][(e,-k)][(g^{-1},0)][(h',0)][(e,m)])\\ &=\delta_{n,m}\delta_{h,h'}[(e,-n)][(h^{-1},0)][(g,0)][(e,k)][(e,-k)][(g^{-1},0)][(h,0)][(e,n)]\\ &=\delta_{n,m}\delta_{h,h'}[(e,-n)][(h^{-1},0)]\Psi(f)[(h,0)][(e,n)], \end{split} \end{equation} while \begin{equation} \begin{split} \Psi\circ\epsilon({s^}^nu_{h^{-1}}fu_{h'}s^m)&=\Psi(\delta_{n,m}\delta_{h,h'}{s^n}^u_{h^{-1}}fu_hs^n)\\ &=\delta_{n,m}\delta_{h,h'}[(e,-n)][(h^{-1},0)]\Psi(f)[(h,0)][(e,n)]. \end{split} \end{equation} This shows that both conditional expectations $E$ and $\epsilon$ are the same, up to the isomorphism $\Psi$. \end{proof} \subsection{Simplicity of $\mathds{U}[\varphi]$} To prove that $\mathds{U}[\varphi]$ is simple using partial crossed product theory, we suppose that $G$ is commutative. Therefore our group is amenable and the endomorphism $\varphi$ is totally normal i.e, the images of $\varphi$ are normal subgroups of $G$. This implies that the set $\overline{G}$, defined in the beginning of Section \ref{sectioncpdescr}, is a group. We need some definitions (from \cite{ExLaQu}) concerning partial actions, as they play a role in the proof that $\mathds{U}[\varphi]$ is simple. Consider $(\theta,H,X)$ a partial dynamical system where $X$ is a locally compact space with $X_h$ being the open sets (Definition \ref{pcpdefi1}). \begin{definition}We say that a partial action $\theta$ is topologically free if for every $h\in H\backslash\{e\}$ the set $F_h:=\{x\in X_{h^{-1}}:\;\theta_h(x)=x\}$ has empty interior. \end{definition} In order to define the minimality of $\theta$, we adjust the classical definition of invariance: a subset $V$ of $X$ is invariant under the partial action $(\theta,H,X)$ if $\theta_h(V\cap X_{h^{-1}})\subseteq V$ $\forall\; h\in H$. \begin{definition}The partial action $\theta$ is minimal if there are no invariant open subsets of $X$ other than $\emptyset$ and $X$. \end{definition} Suited to our setting, there is a result due to Exel, Laca and Quigg (Corollary 2.9 of \cite{ExLaQu}) which says that the partial action $\varpi$ defined in (\ref{isouapcp}) is topologically free and minimal if and only if $C(\Omega_\mathcal{R})\rtimes_{\alpha}(\overline{S})$, as defined in (\ref{isouapcp15}) and (\ref{isouapcp16}), is simple (in fact their result applies to the reduced crossed product, but as we are assuming $G$ is commutative and thus amenable, we know that $\overline{S}$ is amenable and this implies that both the full and reduced partial crossed products are isomorphic by \cite{Mc} Proposition 4.2), so it is clear that we have to understand the topology of $\Omega_\mathcal{R}$, which unfortunately is not an easy task. To avoid difficulties we present a new set which is homeomorphic to $\Omega_\mathcal{R}$, and for which we can easily understand the topology. Consider $\frac{G}{\varphi^k(G)}=\{e\}$ for negative integers $k$ and for $m\leq n$ both integers the canonical projection $$ p_{m,n}: \dfrac{G}{\varphi^n(G)}\rightarrow\dfrac{G}{\varphi^m(G)}. $$ Using these, define \begin{equation} \begin{split} \widetilde{G}:&=\lim_{\leftarrow \atop n}\left\{\dfrac{G}{\varphi^n(G)}:\;p_{m,n}\right\}\\ &=\left\{(g_n\overline{\varphi}^n(G))_{n\in\mathds{Z}}\in\prod_{n\in\mathds{Z}}\dfrac{G}{\varphi^n(G)}:\;p_{m,n}(g_n)=g_m,\hbox{ if }m\leq n\right\}, \end{split} \end{equation} where $\overline{\varphi}$ is the extension of $\varphi$ defined after Proposition \ref{propOre}. Note that when $n\leq0$, $\frac{G}{\varphi^n(G)}=\{e\}$ and therefore for any element in $\widetilde{G}$, the entries indexed by negative integers are $e$. Moreover, when $n>0$, $\overline{\varphi}^n=\varphi^n$. Particularly it makes not necessary to carry the bar over $\varphi$ when denoting the elements of $\widetilde{G}$, and we will also use the notation $(g_m)_{n\in\mathds{Z}}\in\widetilde{G}$. One can see $G$ inside $\widetilde{G}$ through the map $g\mapsto (g\varphi^n(G))_n$, which is injective if $\varphi$ is pure. Another fact is that the set defined above is isomorphic as a topological group to our previous defined $\overline{G}$ (beginning of Section \ref{sectioncpdescr}), because that set is exactly this one except for the negative entries of the vectors in $\widetilde{G}$, which are always $e$. Therefore $\widetilde{G}$ is compact. Consider \begin{equation} \begin{split} \rho:\widetilde{G}&\rightarrow \hbox{P}(\overline{S})\\ (g_n\overline{\varphi}^n(G))_{n\in\mathds{Z}}&\mapsto\{(g_n\overline{\varphi}^n(h),n):\;n\in\mathds{Z},\;h\in G\}. \end{split} \end{equation} \begin{lemma} The set $\rho(\widetilde{G})$ is contained in $\Omega_\mathcal{R}$. \end{lemma} \begin{proof} Take $(g_m)_m\in\widetilde{G}$ and it is clear from the definition of $\widetilde{G}$ that $$g_m=g_{m-n}\overline{\varphi}^{m-n}(\overline{k}_1)$$ and $$g_{m+n}=g_m\overline{\varphi}^m(\overline{k}_2)$$ for $n\in\mathds{N}$ and $\overline{k}_1,\overline{k}_2\in G$. Denote $\xi:=\rho((g_m)_m)$. We have to show that $f(g^{-1}\xi)=0$ for all $g\in\xi$ and all $f\in\widehat{\mathcal{R}}=\widehat{\mathcal{R}}_1\cup\widehat{\mathcal{R}}_2\cup\widehat{\mathcal{R}}_3$. Therefore fix $g=(g_m\overline{\varphi}^m(k),m)\in\xi$ for $m\in\mathds{Z}$ and $k\in G$. $\bullet\; f=1_{(h,0)}-1\in\widehat{\mathcal{R}}_1$: Then $f(g^{-1}\xi)=0\Leftrightarrow g(h,0)\in\xi$, which is true because $g(h,0)=(g_m\overline{\varphi}^m(kh),m)\in\xi$. $\bullet\; f=1_{(e,-n)}-1\in\widehat{\mathcal{R}}_2$: Similarly $f(g^{-1}\xi)=0\Leftrightarrow g(e,-n)\in\xi$ and the latter holds as $g(e,-n)=(g_m\overline{\varphi}^m(k),m-n)=(g_{m-n}\overline{\varphi}^{m-n}(\overline{k}_1\overline{\varphi}^n(k)),m-n)\in\xi$. $\bullet\; f=\D\sum_{h\in\frac{G}{\varphi^n(G)}}1_{(h,n)}-1\in\widehat{\mathcal{R}}_3$: Here $f(g^{-1}\xi)=0\Leftrightarrow$ there exists only one class $h\varphi^n(G)$ such that $g(h,n)\in\xi$. But $$ g(h,n)=(g_m\overline{\varphi}^m(kh),m+n)=(g_{m+n}\overline{\varphi}^m(\overline{k}_2^{-1}kh),m+n) $$ belongs to $\xi$ if and only if $\overline{k}_2^{-1}kh\in\overline{\varphi}^n(G)=\varphi^n(G)$ (as $n\in\mathds{N}$), which is the same as requiring $h\in k^{-1}\overline{k}_2\varphi^n(G)$, and this can be true only for one class in $\frac{G}{\varphi^n{G}}$. \end{proof} \begin{proposition}\label{homeorho} $\rho:\widetilde{G}\rightarrow \Omega_\mathcal{R}$ is a homeomorphism. \end{proposition} \begin{proof} If $\rho((g_m)_m)=\rho((h_m)_m)$ then $h_m=g_m\varphi^m(k_m)$ for all $m\in\mathds{N}$, with $k_m\in G$. Then $g_m=h_m$ in $\frac{G}{\varphi^m(G)}$ for all $m\in\mathds{N}$ and $(g_m)_m=(h_m)_m$ (note that for $m<0$, $g_m=h_m=e$). Now let us prove that $\rho$ is surjective. Take $\xi\in\Omega_\mathcal{R}$ and remember that $(e,0)\in\xi$ which, using $f_1^h:=1_{(h,0)}-1\in\widehat{\mathcal{R}}_1$, implies that $(h,0)\in\xi$ $\forall\; h\in G$. Also for each $j\in\mathds{N}$, set $f_3^j:=\D\sum_{h\in\frac{G}{\varphi^j(G)}}1_{(h,j)}-1\in\widehat{\mathcal{R}}_3$. As $f_3^j((e,0)\xi)=0$, for each $j$ there exists only one class $u_j\varphi^j(G)\in\frac{G}{\varphi^j(G)}$ such that $(u_j,j)\in\xi$. Using functions of the type $f_2^n:=1_{(0,-n)}-1\in\widehat{R}_2$, for $n\in\mathds{N}$, one sees that $(u_j\varphi^j(G))_{j\in\mathds{Z}}\in\widetilde{G}$. Now we prove that $\rho((u_j\varphi^j(G))_j)=\xi$. By construction $(u_j,j)\in\xi$, which implies (using $f_1^h\in\widehat{\mathcal{R}}_1$ defined above) that $(u_j,j)(h,0)=(u_j\varphi^j(h),j)\in\xi$ for all $h\in G$. Doing the same for every $j$ it follows that $\rho((u_j\varphi^j(G))_j)\subseteq\xi$. Suppose that $h=(k,i)\in\xi\backslash\rho((u_j\varphi^j(G))_j)$ and note that $$ (k,i)\notin\rho((u_j\varphi^j(G))_j)\Leftrightarrow(k,i)\notin(u_i\varphi^i(G),i)\Leftrightarrow u^{-1}_ik\notin\varphi^i(G). $$ Now consider the elements $g=(u_i,0)$ and $h'=(u_i,i)$ of $\rho((u_j\varphi^j(G))_j)\subseteq\xi$. Since $u^{-1}_ik\notin\varphi^i(G)$, we have that $g^{-1}h=(u^{-1}_ik,i)$ and $g^{-1}h'=(e,i)$ are different, which implies that $f_3^i(g^{-1}\xi)\neq 0$, and this contradicts the fact that $\xi\in\Omega_\mathcal{R}$. Last, let us prove that $\rho$ preserves the topology. As the sets are compact and Hausdorff, it is enough to prove that $\rho^{-1}$ is continuous, which we will prove by showing that $\pi_m\circ\rho^{-1}$ is continuous for all $m\in\mathds{Z}$ where $\pi_m:\widetilde{G}\rightarrow\frac{G}{\varphi^m(G)}$ is the canonical projection. As $\frac{G}{\varphi^m(G)}$ is discrete we just have to show that $\rho\circ\pi_m^{-1}(\{u_m\varphi^m(G)\})$ is open in $\Omega_R$ for all $u_m\varphi^m(G)\in\frac{G}{\varphi^m(G)}$. But note that (by the proof of surjectivity above) $$ \rho\circ\pi_m^{-1}(\{u_m\varphi^m(G)\})=\{\xi\in\Omega_R:\;(u_m,m)\in\xi\}, $$ which is open in $\Omega_R$ (induced by the product topology in $\{0,1\}^{\overline{S}}$). Then $\rho:\widetilde{G}\rightarrow \hbox{P}(\overline{S})$ is a homeomorphism. \end{proof} Using the proposition above, we identify $\Omega_\mathcal{R}$ with $\widetilde{G}$, and thus view $\varpi$ as a partial action of $G$ on $\widetilde{G}$. Remember that $$ \Omega_s=\{\xi\in\Omega_\mathcal{R}:\; s\in\xi\}. $$ Set $$ \widetilde{G}_s:=\rho^{-1}(\Omega_s) $$ and define $$ \varpi_s:\widetilde{G}_{s^{-1}}\rightarrow\widetilde{G}_s. $$ Using $\rho$ we can conclude that for $(g_i,n)\in\overline{S}=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}$ ($g_i\in G_i\hookrightarrow\mathds{G}$) $$ \widetilde{G}_{(g_i,n)}=\{(h_m\varphi^m(G))_{m\in\mathds{Z}}\in\widetilde{G}:\;h_n\varphi^n(G)=g_i\varphi^n(G)\} $$ (where $h_n$ is viewed inside $G=G_0\subseteq\mathds{G}$) and $$ \varpi_{(g_i,n)}((h_m\varphi^m(G))_m)=(g_i\varphi^n(h_m)\varphi^{n+m}(G))_{n+m}=(g_i\varphi^n(h_{m-n})\varphi^m(G))_m. $$ An easily proven and useful result follows. \begin{lemma} \label{lema111}For $(g_i,n)\in\overline{S}$ the following holds: \begin{enumerate} \item[(i)] $\widetilde{G}_{(g_i,n)}=\emptyset\Leftrightarrow g_i\notin G\varphi^n(G)$; \item[(ii)] $\widetilde{G}_{(g_i,n)}=\widetilde{G}\Leftrightarrow G\subseteq g_i\varphi^n(G)$. \end{enumerate} \end{lemma} \begin{flushright} $\square$ \end{flushright} For $m\in\mathds{Z}$ and a subset $C_m\subseteq\frac{G}{\varphi^m(G)}$ (containing whole cosets) define the open set (it is open because it is the inverse image of a point via a projection) $$ V_m^{C_m}=\{(u_n\varphi^n(G))_n\in\widetilde{G}:\;u_m\varphi^m(G)\in C_m\}. $$ Clearly when $m\leq n$ then $V_m^{C_m}=V_n^{C_n}$ where $$ C_n=\left\{u\varphi^n(G)\in\dfrac{G}{\varphi^n(G)}:\;u\varphi^m(G)\in C_m\right\}. $$ From the definition of the product topology, we know that finite intersections of open sets $V_m^{C_m}$ form the base for the topology in $\widetilde{G}$. Since $V_{m_1}^{C_{m_1}}\cap V_{m_2}^{C_{m_2}}=V_{m}^{C_{n_1}}\cap V_{m}^{C_{n_2}}=V_m^{C_{n_1}\cap C_{n_2}}$ for $m\geq m_1,m_2$, $\left\{V_m^{C_m}\right\}$ is already a base for the topology. Also note that if $C_m\neq\emptyset$ then for $k>0$, $C_{m+k}$ has at least 2 elements and therefore we can assume that if $V_m^{C_m}$ is not empty then $C_m$ has at least 2 elements (replacing $V_m^{C_m}$ by $V_n^{C_n}$ for $n>m$ if necessary). \begin{proposition}\label{prop111}When $\varphi$ is a pure injective endomorphism of a commutative group $G$, the partial action $\varpi$ from $\widetilde{G}$ defined above is topologically free. \end{proposition} \begin{proof} Let us show that $$ F_{(g_i,n)}=\{x\in\widetilde{G}_{(g_i,n)^{-1}}:\;\varpi_{(g_i,n)}(x)=x\} $$ has empty interior, for $(g_i,n)\neq (e,0)$. $\bullet$ Case 1: $n=0$. If $g_i\notin G$ then Lemma \ref{lema111} (i) assures that $F_{(g_i,0)}=\emptyset$. Therefore suppose that $g_i\in G$. If $F_{(g_i,0)}\neq\emptyset$ the equation $\varpi_{(g_i,0)}(x)=x$ implies $g_i\in\varphi^m(G)$ for all $m\in\mathds{Z}$ (using the commutativity of $G$). As $\varphi$ is pure we conclude that $g_i=e$, and then $F_{(g_i,0)}=\emptyset$ for $g_i\neq e$. $\bullet$ Case 2: Let $(g_i,n)$ with $n\neq 0$. Using again Lemma \ref{lema111} (i) we can assume that $g_i\in G\varphi^n(G)$. Take $V$ a non-empty open set of $\widetilde{G}_{(g_i,n)^{-1}}$ and, if needed, shrink $V$ so that $V=V_m^{C_m}$ (and we can assume that $m=ln>0$ for some big $l>0$). Note that we can assume that $C_m$ has at least 2 distinct elements, say $u_1\varphi^m(G)\neq u_2\varphi^m(G)$, which implies that $u_2^{-1}u_1\notin\varphi^m(G)$. Suppose for a contradiction that $\varpi_{(g_i,n)}(x)=x$, $\forall\; x\in V$. Then, since $(u_j\varphi^k(G))_k\in V$ for $j=1,2$, we have \begin{equation} \begin{split} \varpi_{(g_i,n)}((u_j\varphi^k(G))_k)=(u_j\varphi^k(G))_k&\Rightarrow (g_i\varphi^n(u_j)\varphi^k(G))_k=(u_j\varphi^k(G))_k\\ &\Rightarrow u_j^{-1}g_i\varphi^n(u_j)\in\varphi^k(G)\hbox{ for }j=1,2\\ &\Rightarrow\varphi^n(u_2^{-1})u_2u_1^{-1}\varphi^n(u_1)\in\varphi^k(G), \forall\; k\in\mathds{Z} \end{split} \end{equation} (again we used the commutativity of $G$ to cancel the $g_i$'s). But as $\varphi$ is pure, \begin{equation} \begin{split} \varphi^n(u_2^{-1}u_1)=u_2^{-1}u_1&\Rightarrow\varphi^{ln}(u_2^{-1}u_1)=u_2^{-1}u_1\Rightarrow u_2^{-1}u_1\in\varphi^m(G) \end{split} \end{equation} which contradicts our hypothesis. So no open set can be contained in $F_{(g_i,n)}$, which implies that it has empty interior. \end{proof} \begin{proposition}\label{prop112}The partial action $\varpi$ is minimal. \end{proposition} \begin{proof} We will show that all $x\in\widetilde{G}$ has dense orbit by showing the following: if $V$ is a non-empty open set then there exists $(g,n)\in\overline{S}$ such that $x\in\widetilde{G}_{(g,n)^{-1}}$ and $\varpi_{(g,n)}(x)\in V$. Take $x=(u_m\varphi^m(G))_{m\in\mathds{Z}}\in\widetilde{G}$ and $V=V_k^{C_k}\neq\emptyset$. Consider $u\varphi^k(G)\in C_k$ and define $(uu_k^{-1},0)$. By Lemma \ref{lema111} (ii), since $uu_k^{-1}G=G$, it follows that $\widetilde{G}_{(uu_k^{-1},0)^{-1}}=\widetilde{G}$ and therefore $x\in\widetilde{G}_{(uu_k^{-1},0)^{-1}}$. To finish, note that $$ \varpi_{(uu_k^{-1},0)}(x)=\varpi_{(uu_k^{-1},0)}((u_m\varphi^m(G))_m)=(uu_k^{-1}u_m\varphi^m(G))_m\in V. $$ \end{proof} We can now conclude (and this result agrees with the previous obtained Theorem \ref{teo1}): \begin{theorem}If $\varphi$ is a pure injective endomorphism with finite cokernel of some commutative discrete countable group $G$ then the C$^$-algebra $\mathds{U}[\varphi]$ is simple. \end{theorem} \begin{flushright} $\square$ \end{flushright} \begin{corollary}In the conditions of theorem above, we have$$C_r^[\varphi]\cong\mathds{U}[\varphi].$$ \end{corollary} \begin{flushright} $\square$ \end{flushright} \footnotesize DEPARTAMENTO DE MATEM\'{A}TICA - UFSC BLUMENAU - BRAZIL ([email protected]) \end{document}	arXiv
Conway circle theorem In plane geometry, the Conway circle theorem states that when the sides meeting at each vertex of a triangle are extended by the length of the opposite side, the six endpoints of the three resulting line segments lie on a circle whose centre is the incentre of the triangle. The circle on which these six points lie is called the Conway circle of the triangle.[1][2][3] The theorem and circle are named after mathematician John Horton Conway. The radius of the Conway circle is ${\sqrt {r^{2}+s^{2}}}$ where $r$ and $s$ are the inradius and semiperimeter of the triangle.[3] Conway's circle is a special case of a more general circle for a triangle that can be obtained as follows: Given any △ABC with an arbitrary point P on line AB. Construct BQ = BP, CR = CQ, AS = AR, BT = BS, CU = CT. Then AU = AP, and PQRSTU is cyclic.[4] See also • List of things named after John Horton Conway References 1. "John Horton Conway". www.cardcolm.org. Archived from the original on 20 May 2020. Retrieved 29 May 2020. 2. Weisstein, Eric W. "Conway Circle". MathWorld. Retrieved 29 May 2020. 3. Francisco Javier García Capitán (2013). "A Generalization of the Conway Circle" (PDF). Forum Geometricorum. 13: 191–195. 4. Michael de Villiers (2023). "Conway's Circle Theorem as a Special Case of a More General Side Divider Theorem". Learning and Teaching Mathematics (34): 37–42. External links • Kimberling, Clark. "Encyclopedia of Triangle Centers".{{cite web}}: CS1 maint: url-status (link) • Conway’s Circle Theorem as special case of Side Divider Theorem at Dynamic Geometry Sketches, interactive geometry sketches.	Wikipedia
QSAR, molecular docking, and design of novel 4-(N,N-diarylmethyl amines) Furan-2(5H)-one derivatives as insecticides against Aphis craccivora Yusuf Isyaku1, Adamu Uzairu1, Sani Uba1, Muhammad Tukur Ibrahim1 & Abdullahi Bello Umar1 Aphis craccivora has many plant hosts, though it seemingly forechoice to groups of bean family. Other plants it hosts are families of Solanaceae, Rosaceae, Malvaceae, Chenopodiaceae, Caryophyllaceae, Ranunculaceae, Cucurbitaceae, Brassicaceae, and Asteraceae. A computational study was carried out on a series of twenty compounds of novel 4-(N,N-diarylmethylamines) furan-2(5H)-one derivatives against Aphis craccivora insect. Optimization of the compounds was performed with the aid of Spartan 14 software using DFT/B3LYP/6-31G quantum mechanical method. Using PaDel descriptor software to calculate the descriptors, Generic Function Approximation (GFA) was employed to generate the model. Model 1 found to be the optimal out of four models generated which has the following statistical parameters; R2 = 0.871489, R2adj = 0.83644, cross-validated R2 = 0.790821, and external R2 = 0.550768. Molecular docking study occurred between the compounds and the complex crystal structure of the acetylcholine (protein AChBP) (PDB CODE 2zju) in which compound 13 was identified to have the highest binding energy of − 8.4 kcalmol−1. Statistical analyses, such as variance inflation factor, mean effect, and the applicability domain, were conducted on the model. This compound has a strong affinity with the macromolecular target point of the A. craccivora (2zju) producing H-bond and as well the hydrophobic interaction at the target point of amino acid residue. Molecular docking gave an insight into the structure-based design of the new compounds with better activity against A. craccivora in which three compounds A, B, and C were designed and discovered to be of high quality and have greater binding affinity compared to the one obtained from the literature. The QSAR model was generated by the employment of Genetic Function Approximation (GFA). The model was found to be robust and possessed a good statistical parameter. Furthermore, a molecular docking study was performed to get an idea for structure-based design in which three (3) compounds A, B, and C were designed and were found to be more active than the template (compound 13, i.e., the one with highest docking score). QSAR model was developed to give an insight into the ligand/template-based design of computer-aided drug design. Aphis craccivora, known as cowpea aphid, peanut (groundnut) aphid, or black legume aphid, is one of the most dangerous agricultural pests which directly causes harm to plants by delaying and deforming the growth of the plants through malnutrition. The molasses manufactured by the vector are placed on plants and stimulate the growth of molds with soot that limits photosynthesis. Aphis craccivora has many plant hosts, though it seemingly forechoice to groups of bean family. Other plants it hosts are families of Solanaceae, Rosaceae, Malvaceae, Chenopodiaceae, Caryophyllaceae, Ranunculaceae, Cucurbitaceae, Brassicaceae, and Asteraceae. Aphid is a vector of a series of plant viruses that include peanut rosette virus, groundnut mottle virus, mosaic virus common bean, alfalfa mosaic virus, and cucumber mosaic virus (Wikipedia, 2018). Furan- and amide-containing compounds are among the molecular structures found to have an extensively wide range of applications in the field of medicine and agrochemical due to their extensive range of biological activity like antimicrobial/anti-inflammatory activity (Huczyński et al. 2012; Ravindra et al. 2006; Özden et al. 2005), as antibiotic activity (Arjona et al. 1999), as pesticides such antifungal (Yao et al. 2017), and insecticides (Teixeira et al. 2015; Wang et al. 2013) among others. The furan-2(5H)-one was examined to be a potential inhibitor of nicotinic acetylcholine receptor (Tian et al., 2019), and this was the reason for the docking study on the crystal structure of this protein. The quantitative analysis of the structure-activity relationship (QSAR) is among the most efficient ways to optimize the main compounds and design new compounds. QSAR can be used to predict bioactivities, like toxicity, carcinogenicity, and mutagenicity, depending on the structural characteristics of the molecules and the actual mathematical models. Nowadays, one can easily and accurately calculate quantum chemical parameters of the compounds due to fast development in computer technology as well as theoretical quantum chemical study which helps in predicting the new compounds with better activity than the existing ones. This quantum chemical calculation is extensively applied while forming the QSAR models (Gagic et al. 2016). Molecular docking helps to investigate the capacity of the prepared compounds toward the interaction with the protein residue of the target organism and to also predict the preferred orientation of the molecules. The objective of the research is to discover a new model that predicts the activity of chemical products with better activity capable of destroying Aphis craccivora using Genetic Function Approximation (GFA) or molecular docking techniques. In this work, we used a dataset of 20 compounds to design a relation between the chemical traces of compounds and their insecticidal activity. These 20 compounds of novel4-(N,N-diarylmethylamines) furan-2(5H)-one derivatives were obtained from the literature (Tian et al., 2019). The logarithm of the measured LC50 (μg mL−1) against insecticidal activity given by p LC50 (p LC50 = −log 1/LC50) was taken as a dependent parameter; therefore, the data was linearly correlated with the independent parameter/descriptors (Edache et al. 2017). Optimization/molecular descriptor calculation The database (see Fig. 1 and Table 1) was optimized at a density function theory level using the "Becke's three-parameter read-Yang-Parr hybrid" (B3LYP) function together with "6-31G" basis set of Spartan14 software (Arthur et al. 2016a). Graphical-user-interface of Spartan14 was utilized in drawing the 2D molecular structures of the dataset which were later exported in the form of 3D. The optimized structures were then taken to PaDel descriptor software to calculate the quantum molecular descriptors (Yap 2011). The parent compound of the dataset Table 1 Dataset compounds Data division To get a validated model, the dataset was split into (3:1) train test sets. Accordingly, the split was done in such a way that the compounds forming the train (70% of the data) and the test sets (30% of the data) are shared within an entire descriptive space filled by the complete dataset as described by Kennard-Stone Algorithm method (Arthur et al. 2016b). The generated molecular descriptors were taken for regression analysis, with experimental activities as dependent parameters where the molecular descriptors served as independent parameters. With the Genetic Function Approximation method (GFA) incorporated in "Material Studio 2017" software, the compounds of train sets were utilized to develop the QSAR model. Four QSAR models were built where the best model was chosen according to the one with the lowest score of lack of fit (LOF) given as follows: $$ \mathrm{LOF}=\mathrm{SSE}{\left(1-\frac{c+ dp}{M}\right)}^2 $$ where SSE represents the sum of squares of errors, d is a smoothing parameter defined by the user, c = number of terms a model possessed in addition to the constant term, M is equal to the number of samples present in the training set, and p = overall number of descriptors present in all terms of the model excluding the constant term (Edache et al. 2015). Internal validation The generated model was validated internally by the following parameters: (a) The correlation coefficient (R2): explain the division of overall variation ascribed to the built model. The accepted value of R2 ranges from 0.5 to <1 and more the value of R2 and the model considered to be a better model as R2 approaches 1.0, though there are other analyses that the model passed to be a better one. Being the most common internal validation pointer, R2 is expressed as follows: $$ R2=1-\frac{\sum {\left(Y\mathrm{expt}-Y\mathrm{perdt}\right)}^2}{\sum {\left(Y\mathrm{expt}-\overline{Y}\mathrm{train}\right)}^2} $$ where Yexpt, Ypredt, and $ \overline{Y} $train represent the experimental, predictive, and average activities of the training set (Adeniji et al. 2018). (b) Adjusted R2: The value of R2 is inconsistent to evaluate the power of the built model. Thus, R2 is adjusted to restore and stabilize the model. This adjusted R2 is defined in equation iii as: $$ R2\mathrm{adj}=\left(1-R2\right)\frac{\left(n-1\right)}{n-P-1}=\frac{\left(n-1\right)\left({R}^2-P\right)}{n-P+1} $$ where p presents "the number of descriptors constituted the model," while n = number of training set compounds (Ibrahim et al. 2018a). (c) Cross-validated R2: The validity of the models was identified by a cross-validation test measured by predictive Q2cv. For a "leave one out (LOO) cross-validation," a data point is eliminated (left-out) in the set and the model is readjusted; the predicted value of the eliminated data point is compared to its real value. This is repeated until each data removed. We can then calculate the value of Q2 using the sum of the squares of these elimination residues as in the equation below: $$ Q2\mathrm{cv}=1-\frac{\sum {\left(Y\mathrm{predt}-Y\mathrm{expt}\right)}^2}{\sum {\left(Y\mathrm{expt}-\overline{Y}\mathrm{train}\right)}^2} $$ where Yexpt, Ypredt, and $ \overline{Y} $train represent the experimental, predictive, and average activities of the training set (Adedirin et al. 2018). External validation The prediction ability of the model was examined by an external validation through the ability of the model to predict the activity values of the test set compounds as well as its application in the calculating the predicted value of R2pred according to the equation below: $$ R2=1-\frac{\sum {\left(Y\mathrm{predt}-Y\mathrm{expt}\right)}^2}{\sum {\left(Y\mathrm{expt}-\overline{Y}\mathrm{train}\right)}^2} $$ where Ypredt and Yexpt are the test set's experimental and predicted activities while Ytrain indicates the average activities of the training set (Edache et al., 2017). Statistical analysis of the descriptors Variance inflation factor (VIF) VIF is defined as the measure of multicollinearity amongst the independent variables (i.e., descriptors). It quantifies the extent of correlation between one predictor and the other predictors in a model. $$ \mathrm{VIF}=\frac{1}{\left(1-{R}^2\right)} $$ where R2 gives multiple correlation coefficient between the variables within the model. If the VIF is equal to 1, it means there is no intercorrelation in each variable, and if it ranges from 1 to 5, then it is said to be suitable and acceptable. But if the VIF turns out to be greater than 10, this indicates the instability of the model and need to be reexamined (Pourbasheer et al. 2015; Karthikeyan et al. 2009). Mean effect (ME) The average effect (mean effect) correlates the effect or influence of given molecular descriptors to the activity of the compounds that made up the model. The sign of descriptors shows the direction of their deviation toward the activity of compounds. That is an increase or decrease in the value of the descriptors will improve the activity of the compounds. The mean effect is defined by the following: $$ \mathrm{Mean}\ \mathrm{effect}=\frac{B_j{\sum}_i^n{D}_j}{\sum_j^m\left({B}_j{\sum}_i^n{D}_j\right)} $$ where Bj and Dj are the j-descriptor coefficient in a model and the values of each descriptor in training set, while m and n stand for the number of molecular descriptors as well as the number of compounds in the training set. To evaluate the significance of the model, the ME of all the descriptors was calculated (Edache et al. 2015). Applicability domain To confirm the reliability of the model and to examine the outliers as well as the influential compounds, it is very important to evaluate its domain of applicability. It aimed at predicting the uncertainty of a compound depends on its similarities to the compounds used in building the model and also the distance between the train and test set of the compounds. This can be achieved by employing William's plot which was plotted using standardized residuals versus the leverages. The leverages for a particular chemical compound are given as follows: $$ {h}_{\mathrm{i}}={Z}_{\mathrm{i}}{\left({Z}^{\mathrm{T}}\ Z\right)}^{-1}\ {Z_{\mathrm{i}}}^{\mathrm{T}} $$ (viii) where hi = leverage for a particular compound, Zi = matrix i of the training set. Z = nxk descriptor-matrix for a training set compounds. ZT = transpose of Z matrix. The warning leverage (h) that is the boundary for usual values of Z outliers is given as follows: $$ {h}^{\ast }=3\frac{\left(p+1\right)}{n} $$ where n = number of compounds in the training set whereas p gives the number of descriptors present in the model (Ibrahim et al. 2018a). Ligand and receptor preparation From the RCSBPDB (www.rcsb.org), the PDB format of the receptor was successfully downloaded. This was then taken to the discovery studio for an appropriate preparation where all the residues associated with it such as a ligand, water molecules, and other traces associated with the receptor were removed. The ligands (the optimized compounds) which were in the SDF file were transformed into the PDB file format. Figures 2 and 3 showed the prepared receptor and ligand (Ibrahim et al. 2018b). Prepared receptor Prepared ligand QSAR model Genetic function algorithm (GFA) was used to generate the three QSAR models which predicted the activity of the compounds. The first model was chosen as the optimal model due to its statistical significance was presented in equation (x) below: $$ Y=0.262046205\ast \mathbf{nCl}-5.412377232\ast \mathbf{ATSc4}-5.657053582\ast \mathbf{weta}\mathbf{3}.\mathbf{polar}+1.799880140 $$ All the validation/statistical parameters that signified the stability, robustness, and the prediction capability of the model were presented in Table 2. Table 2 Validation parameter of the model The name, symbol, and class of the three selected descriptors that made up the model are presented in Table 3. Table 3 Descriptors and their description Since the selected model is internally valid, then an external validation is the next step. Tables 4 and 5 represent the external validation as well as the calculation of predicted R2 of the best-chosen model. Table 4 External validation Table 5 Continuation of external validation The experimental, predictive, and residual activity for both training sets and test sets are shown in Table 6. This table is represented to show the robustness of the model considering the lower residual values. Table 6 The experimental, predictive, and residual activity Statistical analyses on the model's descriptors are very necessary in order to know how related they are. For that, Pearson's correlation, variance inflation factor, mean effect (which contains regression analysis), and applicability domain were carried out. Pearson's correlation (Tables 7 and 8) Figures 4 and 5 are the plot of predicted activity against the experimental activity (pLC50) and plot of standardized residual against the experimental activity (pLC50). Table 7 Pearson's correlation Table 8 Standard regression coefficients "bj", the values of mean effect (ME) and confidence interval (p values) A plot of predicted activity against experimental activity (pLC50) A plot of standardized residual against experimental activity (pLC50) A graph of leverages of each compound of dataset versus their standardized residuals terms William's plot was presented in the Fig. 6 below. William's plot Docking studies Molecular docking analysis was carried out between the ligands (compounds) and the receptor to evaluate the binding affinity at the ligand-receptor interface. Result of the design The structure of the three (3) compounds which were designed using an optimization method of structure-based design. The structure of the chosen scaffold (compound 13) was presented in Fig. 9 below. The QSAR examination was carried out to relate the structure-activity relationship of novel 4-(N,N-diarylmethylamines) furan-2(5H)-one derivatives as a potent inhibitor of Aphis craccivora. Three descriptors were utilized in constructing the QSAR model which predicted the activity of the compounds based on the genetic function algorithm (GFA). The first model was chosen as the optimal model due to its statistical significance. The best-chosen model constructed was presented in equation. All the validation/statistical parameters that signified the stability, robustness, and the prediction capability of the model were presented in Table 2. From the table, the highly calculated R2 value (0.8715) for the predicted activity indicated the robustness of the model. The descriptors, definitions, and their classes are represented in Table 3 below. The fact that 2D and 3D descriptors are present in the model implies that the descriptors used in the model can determine a better insecticidal activity of the compounds. The individual capability and inducing power of the selected descriptors toward the activity of the compounds depend on their values, signs, and as well their mean effects. Tables 4 and 5 represent the external validation as well as the calculation of predicted R2 of the best-chosen model. The experimental, predictive, and residual activity for both training sets and test sets are shown in Table 6. The residual value is the difference between the predicted and actual activity. To evaluate the relationships between each descriptor used in the built model, Pearson's correlation was carried out on the values of the model's descriptors and the results were presented in Table 7. The results show that the descriptors are not significantly inter-correlated for the fact that none of their correlation coefficients are up to 0.5, and this indicates the robustness as well as the stability of the built model. The variance inflation factor (VIF) values for each of the three descriptors were not up to 2, which indicates that the descriptors and the model are stable and accepted. Table 8 showed the standard regression coefficients "bj," the values of mean effect (ME), and confidence interval (p values). These give vital information on the impact and contribution of the descriptors toward the built model. The individual capability and inducing power of the selected descriptors toward the activity of the compounds depend on their values, signs, and as well their mean effects. The p values of the three descriptors (at 95% c.l.) that made up the model are all < 0.05; this implies that there is a significant relationship among these descriptors (as contrary to the null hypothesis) and the inhibitory concentration of the compounds. Figure 5 which presented a graph of observed activity versus standardized residual shows a random dispersion at the baseline where the standardized residual is zero. Therefore, no systematic error occurred in the built model. A graph of leverages of each compound of dataset versus their standardized residual terms William's plot was plotted to discover the outliers as well as the chemical influential values of the model. The domain of applicability was established within a box at ± 3.0 limit for the residuals and leverage threshold h (where h* calculated to be 0.8). The result indicates that except three compounds from the test set, all the molecules in the dataset are within the box of the applicability domain of the model. This may be characterized by their clear differences in chemical structures by considering the rest of the compounds highlighted in the dataset. Molecular docking study Due to unavailability of the crystal structure of Aphis craccivora, the complex crystal structure of the acetylcholine (protein AChBP) from Lymnaea stagnalis and imidacloprid with PDB code 2ZJU was utilized for the docking analysis since it possess "high homology extracellular domain" of the A. craccivora protein (nAChR) which has been used for many docking studies involving A. craccivora. The docking studies were performed between 2ZJU and the ligands (compounds) of novel4-(N,N-diarylmethylamines) furan-2(5H)-one derivatives to investigate the binding energy of the compounds to the target site of the insect. The ligands show a good interaction with the active site of the Aphis craccivora that is to say they inhibit the activity of the insect. Some ligands show high binding energy that varies from − 7.9 to − 8.4 kcalmol−1 as presented in Table 9. However, compound 13 with the highest binding score (− 8.4 kcal/mol) possessed an interaction mode with H-bond of ARG137 and 2.60716 bond length and hydrophobic interaction of TYR89, TYR89, ASN90, VAL183, TRP53, TYR89, and TYR185. The interaction between the compound with highest binding energy and the binding pocket of the receptor is shown in Fig. 7 while Figs. 8 and 9 is the 2D hydrogen bond interaction of compound 13 with the receptor. Table 9 Ligands, binding affinity, H-bond, and hydrophobic interaction between high binding score compounds and receptor The interaction between the compound with the highest docking score and receptor 2D interaction of compound 13 2D structure of the template In our research, we utilized the method of structure-based design to design a new (novel) insecticidal compound with a better activity by taking the compound with the highest docking score which is compound 13 (with binding energy of − 8.4 kcal/mol) as our template compound and thus provides suitable insecticidal activity and appeared very inspiring as a noteful scaffold. Compound 13 was selected as a synthetically best structure in which some structural advancement was performed on it. The newly designed compounds (A) 4-(((2-chloro-4-(trichloromethyl)pyridine-1(2H)-yl)methyl)(2-chloro-4-(trifluoromethyl)benzyl)amino)furan-2(5H)-one, (B) 4-(((2-chloro-4-(trichloromethyl)pyridine-1(2H)-yl)methyl)(3-chloro-4-(trifluoromethyl)benzyl)amino)furan-2-(5H)-one, and (C) 4-(((2-chloro-4-(2,2,2-trichloroethyl)pyridin-1(2H)-yl)methyl)(4-(trifluoromethyl)benzyl)amino)furan-2(5H)-one with their binding energy of – 8.9, – 9.1, and – 9.0 kcal/mol (as shown in Figs. 10, 11, and 12) were discovered to be of high quality and have greater binding affinity compared to the one obtained from the literature. 2D structure of the designed compound (A) 2D structure of the designed compound (B) 2D structure of the designed compound (C) This research involves a QSAR and molecular docking studies on 20 compounds of novel 4-(N,N-diarylmethylamines) furan-2(5H)-one derivatives against Aphis craccivora. Using DFT for molecule optimization, Genetic Function Approximation (GFA) was employed in generating the built model. Out of three models built, the first model was identified to be the optimal constituted with good statistical parameters such as R2 = 0.871489, R2adj = 0.83644, cross-validated R2 = 0.790821, and external R2 = 0.550768. A decrease in negative coefficient descriptors (like ATSc4 and Weta3.polar) and an increase in positive coefficients descriptors (like nCl) will improve the activities of the compounds against A. craccivora. According to the docking scores, most of the ligands (compounds) show good inhibitory activity against A. craccivora protein. However, ligands 13 showed a higher binding affinity of – 8.4 kcal/mol. This compound has a strong affinity with the macromolecular target point of the A. craccivora (2zju) producing H-bond and as well the hydrophobic interaction at the target point of amino acid residue. Molecular docking gave an insight into the structure-based design of the new compounds with better activity against A. craccivora in which three compounds A, B, and C were designed and discovered to be of high quality and have greater binding affinity compared to the one obtained from the literature. The section is not applicable to this research work B3LYP: Becke's three-parameter read-Yang-Parr hybrid DFT: Density function theory GFA: Genetic Function Approximation PDB: QSAR: Quantitative structure-activity relationship Adedirin O, Uzairu A, Shallangwa GA, Abechi SE (2018) QSAR and molecular docking based design of some n-benzylacetamide as γ-aminobutyrate-aminotransferase inhibitors. J Eng Exact Sci 4(1):0065–0084 Adeniji SE, Uba S, Uzairu A (2018) QSAR modeling and molecular docking analysis of some active compounds against mycobacterium tuberculosis receptor (Mtb CYP121). 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J Comput Methods Mol Des. 5(3):135–149 Gagic Z, Nikolic K, Ivkovic B, Filipic S, Agbaba D (2016) QSAR studies and design of new analogs of vitamin E with enhanced antiproliferative activity on MCF-7 breast cancer cells. J Taiwan Instit Chem Eng 59:33–44 Huczyński A, Janczak J, Stefańska J, Antoszczak M, Brzezinski B (2012) Synthesis and antimicrobial activity of amide derivatives of polyether antibiotic—salinomycin. Bioorg Med Chem Lett 22(14):4697–4702 Ibrahim MT, Uzairu A, Shallangwa GA, Ibrahim A (2018a) Computational studies of some biscoumarin and biscoumarin thiourea derivatives AS⍺-glucosidase inhibitors. J Eng Exact Sci 4(2):0276–0285 Ibrahim MT, Uzairu A, Shallangwa GA, Ibrahim A (2018b) In-silico studies of some oxadiazoles derivatives as anti-diabetic compounds. Journal of King Saud University-Science Karthikeyan C, Moorthy NHN, Trivedi P (2009) QSAR study of substituted 2-pyridinyl guanidines as selective urokinase-type plasminogen activator (uPA) inhibitors. 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J Braz Chem Soc 26(11):2279–2289 Tian P, Liu D, Liu Z, Shi J, He W, Qi P, Song B (2019) Design, synthesis, and insecticidal activity evaluation of novel 4-(N,N-diarylmethylamines) furan-2 (5H)-one derivatives as potential acetylcholine receptor insecticides. Pest Manag Sci 75(2):427–437 Wang BL, Zhu HW, Ma Y, Xiong LX, Li YQ, Zhao Y, Zhang JF, Chen YW, Zhou S, Li ZM (2013) Synthesis, insecticidal activities, and SAR studies of novel pyridylpyrazole acid derivatives based on amide bridge modification of anthranilic diamide insecticides. J Agric Food Chem 61(23):5483–5493 Wikipedia contributors (2018). Retrieved August 6, 2019, from https://en.wikipedia.org/wiki/Aphis_craccivora#cite_note-ITIS-2 Yao TT, Xiao DX, Li ZS, Cheng JL, Fang SW, Du YJ, Zhao JH, Dong XW, Zhu GN (2017) Design, synthesis, and fungicidal evaluation of novel pyrazole-furan and pyrazole-pyrrole carboxamide as succinate dehydrogenase inhibitors. J Agric Food Chem 65(26):5397–5403 Yap CW (2011) PaDEL-descriptor: an open-source software to calculate molecular descriptors and fingerprints. J Comput Chem 32(7):1466–1474 Article CAS MathSciNet Google Scholar No fund was collected on this research work. Department of Chemistry, Ahamadu Bello University, P.M.B, Zaria, 1044, Nigeria Yusuf Isyaku, Adamu Uzairu, Sani Uba, Muhammad Tukur Ibrahim & Abdullahi Bello Umar Yusuf Isyaku Adamu Uzairu Sani Uba Muhammad Tukur Ibrahim Abdullahi Bello Umar YI: contributed throughout the research work. AU: gives directives and technical advices. SU: partake in technical activities. MTI: partake in technical activities. ABU: partake in technical activities. All authors have read and approved the final manuscript. Correspondence to Yusuf Isyaku. The authors declared no competing interest. Isyaku, Y., Uzairu, A., Uba, S. et al. QSAR, molecular docking, and design of novel 4-(N,N-diarylmethyl amines) Furan-2(5H)-one derivatives as insecticides against Aphis craccivora. Bull Natl Res Cent 44, 44 (2020). https://doi.org/10.1186/s42269-020-00297-w QSAR Molecular docking	CommonCrawl
An automated framework for NMR chemical shift calculations of small organic molecules Yasemin Yesiltepe1,2, Jamie R. Nuñez2, Sean M. Colby2, Dennis G. Thomas2, Mark I. Borkum2, Patrick N. Reardon3, Nancy M. Washton2, Thomas O. Metz2, Justin G. Teeguarden2, Niranjan Govind2 & Ryan S. Renslow ORCID: orcid.org/0000-0002-3969-55701,2 When using nuclear magnetic resonance (NMR) to assist in chemical identification in complex samples, researchers commonly rely on databases for chemical shift spectra. However, authentic standards are typically depended upon to build libraries experimentally. Considering complex biological samples, such as blood and soil, the entirety of NMR spectra required for all possible compounds would be infeasible to ascertain due to limitations of available standards and experimental processing time. As an alternative, we introduce the in silico Chemical Library Engine (ISiCLE) NMR chemical shift module to accurately and automatically calculate NMR chemical shifts of small organic molecules through use of quantum chemical calculations. ISiCLE performs density functional theory (DFT)-based calculations for predicting chemical properties—specifically NMR chemical shifts in this manuscript—via the open source, high-performance computational chemistry software, NWChem. ISiCLE calculates the NMR chemical shifts of sets of molecules using any available combination of DFT method, solvent, and NMR-active nuclei, using both user-selected reference compounds and/or linear regression methods. Calculated NMR chemical shifts are provided to the user for each molecule, along with comparisons with respect to a number of metrics commonly used in the literature. Here, we demonstrate ISiCLE using a set of 312 molecules, ranging in size up to 90 carbon atoms. For each, calculation of NMR chemical shifts have been performed with 8 different levels of DFT theory, and with solvation effects using the implicit solvent Conductor-like Screening Model. The DFT method dependence of the calculated chemical shifts have been systematically investigated through benchmarking and subsequently compared to experimental data available in the literature. Furthermore, ISiCLE has been applied to a set of 80 methylcyclohexane conformers, combined via Boltzmann weighting and compared to experimental values. We demonstrate that our protocol shows promise in the automation of chemical shift calculations and, ultimately, the expansion of chemical shift libraries. Metabolomics is being increasingly applied in biomedical and environmental studies, despite the technical challenges facing comprehensive and unambiguous identification of detected metabolites [1,2,3]. The capability to routinely measure and identify even a modicum of biologically important molecules within all of chemical space—greater than 1060 compounds [4]—remains a grand challenge in biology. The prevention and treatment of metabolic diseases, determining the interactions between plant and soil microbial communities, and uncovering the building blocks that led to abiogenesis will all strongly depend on confidently identifying small molecules, and thus understanding the mechanisms involved in the complex processes of metabolic networks [5,6,7]. The current gold standard for chemical identification requires matching chemical features to those measured from an authentic chemical standard. However, this is not the case with the vast majority of molecules. For example, only 17% of compounds found in the Human Metabolome Database (HMDB) and less than 1% of compounds found in exposure chemical databases like the U.S. Environmental Protection Agency (EPA) Distributed Structure-Searchable Toxicity (DSSTox) Database [8] can be purchased in pure form [9, 10]. Although analytical techniques like nuclear magnetic resonance (NMR) spectroscopy [11,12,13] and mass spectrometry (MS) [14,15,16] have been applied for the identification of metabolites and to build libraries [17,18,19,20,21], determining the complete composition of entire metabolomes is still non-trivial for both technical and economic reasons. In this regard, libraries constructed of experimentally obtained data are too limited, expensive, and slow to build, even for libraries with thousands of metabolites [22,23,24,25]. The most practical approach expand reference libraries for comprehensive identification of compounds detected in metabolomics studies is through in silico calculation of molecular attributes. Molecular properties that can be both accurately predicted computationally and consistently measured experimentally may be used in "standards free" metabolomics identification approaches. The metabolomics community has made many advances in calculations of measurable chemical attributes, such as chromatographic retention time [26, 27], tandem mass spectra [28,29,30], ion mobility collision cross section [31, 32], and NMR chemical shifts [33]. Recently, high throughput computation of chemical properties has been demonstrated using machine learning approaches [34,35,36,37]. These tools are a good resource for the metabolomics community, however, machine learning methods are limited by the size and scope of the initial training set, and thus ultimately limited by the number of authentic chemical standards available for purchase. In contrast, structure-based approaches, utilizing first principles of quantum chemical calculations, leverage our understanding of the underlying chemistry and physics to directly predict chemical properties of any chemically valid molecule. Thus, quantum chemical calculations enable us to overcome the reliance on authentic chemical standards in metabolomics. In this study, we focus on expanding the utility of density functional theory (DFT), a widely used electronic structure approach, which has been applied to predict NMR chemical shifts [38,39,40,41]. DFT enables examination of molecular conformers [42,43,44,45] and allows custom solvent conditions [46,47,48]. Ultimately, computational modeling can be used in the rapid identification and study of thousands of metabolites, culminating in in silico metabolome libraries of multiple chemical properties. Furthermore, the same tools that can be used to aid identification of small molecules in complex samples can also be used for structure confirmation and correction. For example, we recently used the tool described in this manuscript to help correct the misidentification of the isoflavonoid wrightiadione to the actual structure as an isobaric isostere, the alkaloid tryptanthrin [49]. Metabolomics researchers unfamiliar with DFT or similar calculations may find the application of quantum chemical calculations complicated or challenging to apply quickly, and thus avoid these techniques. To this end, and to help bring DFT calculations to large sets of small organic molecules relevant to the mainstream metabolomics community, we have developed a Python-based workflow and analysis package, the ISiCLE (in silico Chemical Library Engine) NMR chemical shift module employs DFT methods through use of NWChem [50], a high-performance quantum chemistry software package developed at Pacific Northwest National Laboratory (PNNL). The module automates calculations of NMR chemical shifts, including solvent effects, via the COnductor-like Screening Model (COSMO) [51] of user-specified NMR-active nuclei for a given set of molecules for multiple DFT methods. ISiCLE also calculates the corresponding errors if experimental values are available. In this paper, we describe ISiCLE's NMR module, provide a working tutorial example, demonstrate its use through the calculation of chemical shifts for a large set of small molecules, and, finally, show how ISiCLE can be applied to rapidly calculate chemical shifts of arrays of Boltzmann-weighted conformers to yield high accuracy chemical shift calculations. In silico Chemical Library Engine (ISiCLE)—NMR module ISiCLE is a Python module that provides straightforward automation of DFT using NWChem, an open source, high-performance computational quantum chemistry package, developed at Pacific Northwest National Laboratory (PNNL), for geometry optimization and chemical shift and solvent effect calculations. Figure 1 shows a schematic representation of ISiCLE. For typical use, ISiCLE requires only a list of molecules and a list of desired levels of DFT theory from the user. For more advanced use cases, users may adjust NWChem parameters by modifying the provided .nw template file. Schematic representation of inputs and outputs of the ISiCLE NMR module Here, we describe each step of a typical ISiCLE run (see Fig. 2 for a general workflow for using the ISiCLE NMR module). The step-by-step conceptual workflow for the ISiCLE NMR module. Conformer generation with Boltzmann weighting is optional and will be automated in subsequent versions. Please see github.com/pnnl/isicle for the latest versions To start, users must prepare File A, containing a list of molecules, and File B, containing a list of DFT combinations, which both are required to be in Excel format (.xls or .xlsx). File A must contain all input molecules either as (i) International Union of Pure and Applied Chemistry (IUPAC) International Chemical Identifier (InChI) strings [52, 53] or (ii) XYZ files, a free-format text file having XYZ coordinates of atoms. In subsequent versions, alternative file formats will be supported, such as TSV for inputs and outputs. Once prepared, the user runs ISiCLE. First, ISiCLE opens File A for the input molecules. OpenBabel, an open-source chemical informatics toolbox available with Python wrappers [54, 55], is called to generate geometry files. For InChI inputs, OpenBabel generates .xyz files for each molecule, unless .xyz files are provided, and converts InChI to InChIKey for naming files (otherwise, the base names of XYZ files are used for naming subsequent files). Next, OpenBabel applies the Merck molecular force field (MMFF94) [56] to generate a rough three-dimensional (3D) structure for each molecule, resulting in associated .mol files. ISiCLE then prepares NWChem input files based on the specified DFT methods, solvents, shielding parameters and regarding task directives given by the user-prepared File B. Finally, ISiCLE submits the appropriate files to, if relevant, a remote NWChem installation (typically on a non-local, networked, high-performance computer), and then retrieves the output files once the calculations are complete. Additional information and further details about ISiCLE is provided in Additional file 1 (S1). Note that future versions of ISiCLE will automatically generate conformers of a given molecule, as part of the seamless pipeline. For each molecule, ISiCLE generates MDL Molfiles (.mol) [57] that contain isotropic shieldings and NMR chemical shifts. ISiCLE exports isotropic shieldings for each molecule and appends them to a MDL Molfile in the same atomic order of the original XYZ files. Then, ISiCLE converts isotropic shieldings to NMR chemical shifts by subtracting the isotropic shielding constants for the specified nuclei of the molecule of interest from those of a reference compound computed at the same level of theory (Eq. 1). For this manuscript, tetramethylsilane (TMS) is used as a reference compound. The experimental chemical shifts of TMS are assigned a value of zero, thus the calculation of NMR chemical shifts needs only isotropic shieldings of TMS [58,59,60,61]. Any molecule can be used as reference in ISiCLE as long as it has the specified nuclei and its experimental (or calculated) chemical shifts are supplied. It is explained in detail in a supplemental tutorial how a user inputs experimental data. The equation for calculating chemical shifts from isotropic shieldings is: $$ \delta_{i} = \sigma_{ref} - \sigma_{i} + \delta_{ref} $$ where $ \delta_{i} $ and $ \delta_{ref} $ are the chemical shifts of atom i (of the molecule of interest) and the reference molecule, respectively. $ \sigma_{i} $ and $ \sigma_{ref} $ are the isotropic shielding constants of atom i and the reference molecule, respectively. ISiCLE also calculates errors in NMR chemical shifts if experimental data is provided in the MDL Molfiles in a required way as explained in the tutorial. The errors are quantified in terms of mean absolute error (MAE) (Eq. 2), corrected mean absolute error (CMAE) (Eq. 3), root mean square error (RMSE) (Eq. 4), and maximum absolute error (Eq. 5). $$ MAE = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left\| {\delta_{exp} - \delta_{calc} } \right\|}}{N} $$ $$ CMAE = \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left\| {\delta_{exp} - (\delta_{calc} - intercept)/slope} \right\|}}{N} $$ $$ RMSE = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {\delta_{exp} - \delta_{calc} } \right)^{2} }}{N}} $$ $$ \mathop {\hbox{max} }\limits_{i = 1,2, \ldots ,N} \left\| {\delta_{exp} - \delta_{calc} } \right\| $$ where N is the total number of chemical shifts, and $ \delta_{calc} $ and $ \delta_{exp} $ are the lists of calculated and experimental chemical shits, respectively. Empirical scaling of isotropic shieldings or NMR chemical shifts is the most common approach to remove systematic errors. If experimental data is provided, ISiCLE uses two optional approaches for its linear regression method, where slope and intercept values are derived from (i) regression of computed NMR chemical shifts versus experimental NMR chemical shifts using (Eq. 6), and/or (ii) regression of computed isotropic shieldings versus experimental NMR chemical shifts using (Eq. 7). $$ \delta_{exp} = \frac{{intercept - \delta_{calc} }}{ - slope} $$ $$ \delta_{exp} = \frac{{intercept - \sigma_{calc} }}{ - slope} $$ where $ \sigma_{calc} $ is the list of isotopic shielding constants of molecules. Alternatively, if the user does not provide experimental NMR chemical shifts, ISiCLE can scale NMR chemical shifts using provided intercept and slope values. The scaled NMR chemical shifts are appended to MDL Molfiles. A detailed description of InChIs and InChIKeys, and why they were chosen, can be found in the Additional file 1 (S2). Similarly, justification for the use of MDL Molfiles is explained in Additional file 1 (S2). In the next version, ISiCLE will be compatible with other file formats, such as the NMReDATA [62] format that has been recently designed for NMR data use. To help ease the use of our data, we provide NMReDATA files for the demonstration set in the Additional file 2. Furthermore, installation details for OpenBabel and other required Python packages are provided in the tutorial (see Additional file 2). The Windows-based tutorial provides step-by-step instructions for running ISiCLE for the first time, including information for installation of packages, properly preparing input files, running a calculation, and obtaining output files. The tutorial includes example molecules with anticipated output files for use as a practice set and for benchmarking purposes. It is designed to guide users of ISiCLE and NWChem in the use of the input files and scripts, demonstrated using three small molecules: methanol, methyl-isothiocyanate, and nitromethane. Calculation time may vary (depending on network speed, local computational power, etc.), but it is expected to take less than 10 min. Demonstration set For an initial demonstration of ISiCLE, we have compiled a molecule set of 312 compounds from previous studies: Alver [63], Asiri et al. [64], Bally and Rablen [65], Bagno et al. [66], Borkowski et al. [67], Coruh et al. [68], Fulmer et al. [69], Hill et al. [70], Izgi et al. [71], Karabacak et al. [72], Krishnakumar et al. [73,74,75], Kwan and Liu [45], Li et al. [76], Lomas [77], Osmialowski et al. [78], Parlak et al. [79], Perez et al. [80], Rablen et al. [81], Sarotti and Pellegrinet [82, 83], Sebastian et al. [84], Seca et al. [52], Senyel et al. [85, 86], Sridevi et al. [87], Tormena and da Silva [88], Vijaya and Sankaran [89], Watts et al. [53], Wiitala et al. [90, 91], Willoughby et al. [92], and Yang et al. [93]. We aimed to cover a broad chemical space and distribution of sizes. Our criteria also included the existence of all 1H and/or 13C NMR experimental data in chloroform solvent, referenced to TMS at room temperature, for comparisons. Note that the NMR spectra of each molecule set were not recorded at the same magnetic field strengths. A summary of the demonstration set compounds are given in Table 1. Detailed information about the individual sets is given in Additional file 1 (S3). Table 1 Demonstration set sources and details As a first demonstration of ISiCLE, a benchmark study was performed with 8 different DFT methods to predict 1H and 13C NMR chemical factors for the calculations of chemical shifts in chloroform. Each compound was optimized with the Becke three-parameter Lee–Yang–Parr (B3LYP) hybrid functional [94,95,96] and the 6-31G(d) split-valence basis set [97]. This level of theory in geometry optimization was chosen because of its broad application in the literature for organic molecules [98, 99]. Isotropic magnetic shielding constants were calculated with the 4 different functionals, BLYP [94, 95], B3LYP [97,98,99], B35LYP, and BHLYP [100]. DFT methods were selected with different Hartree–Fock (HF) ratios: BLYP (0% HF), B3LYP (20% HF), B35LYP (35% HF), BHLYP (50% HF). Each method was tested with 2 different correlation-consistent Dunning basis sets (double-zeta cc-pVDZ [101] or triple-zeta cc-pVTZ [101]). All basis sets were obtained from the Environmental Molecular Sciences Laboratory (EMSL) Basis Set Exchange [102,103,104]. For each optimized geometry, 1H and 13C NMR chemical shifts were computed relative to TMS using the Gauge Including Atomic Orbitals (GIAO) formalism [105]. Chloroform solvation effects were simulated using COSMO. For a second demonstration of ISiCLE, the NMR chemical shifts, along with frequency calculations (and subsequent Boltzmann weighting), two sets of axial and equatorial conformers (40 conformers each) of methylcyclohexane were processed. We performed in vacuo molecular dynamics (MD) simulations, using the sander MD software program from AmberTools (version 14) [106], to generate 80 conformers of the methylcyclohexane compound. These conformers were generated in four stages. First, the initial geometries of axial and equatorial conformers were taken from the study of Willoughby et al. [92]. Second, a short energy minimization run was performed to relax the initial structure and to remove any non-physical atom contacts. Third, a short 50 ps MD run was performed (in 0.5 fs time steps) to heat the structure from 0 to 300 K, without non-bonded cutoffs. In the fourth step, we performed 8 simulated annealing cycles, where each cycle was run for 1600 ps in 1 fs MD steps with the following temperature profile: heating from 300 to 600 K (0–300 ps), equilibration at 600 K (300–800 ps), cooling from 600 to 300 K (800–1100 ps), and equilibration at 300 K (1100–1600 ps). Ten conformers from the equilibration stage at 300 K, of each simulated annealing cycle, were randomly selected to obtain the 80 conformers. After the conformers were obtained, M06-2X was used with the basis set of 6-31 + G(d,p) for the geometry optimization and frequency calculations and B3LYP with 6-311 + G(2d,p) method for the calculations of NMR chemical shifts. Relative free energies of the conformations and Boltzmann weighted NMR chemical shifts were compared to those found in the literature [92, 107,108,109]. All results shown in this manuscript were generated using the Cascade high-performance computer (1440 compute nodes, 23,040 Intel Xeon E5-2670 processor cores, 195,840 Intel Xeon Phi 5110P coprocessor cores, and 128 GB memory per compute node [110]), in EMSL (a U.S. national scientific user facility) located at PNNL. Cascade is available for external users through a free, competitive proposal process. ISiCLE can utilize local clusters or high-performance computing resources available to the user. NWChem is freely available and can be downloaded from the website [111, 112]. NMR chemical shift calculations have been used successfully to identify new molecules, determine metabolite identifications, and eliminate structural misassignments [59, 113]. In the last two decades, many research groups have performed benchmark DFT studies on the accuracy of optimized molecular geometry [92, 114,115,116], functionals [117, 118], basis sets [88, 119], and solvation models [90, 120, 121] for NMR chemical shifts [60, 122,123,124]. Each group uses a molecule set focusing on a unique chemical class [78, 125,126,127,128,129] and several groups have recommended different exchange–correlation (XC) energy functionals with a different basis set for a particular condition or suitable to specific chemical functionalities and properties [70, 77, 130,131,132,133]. The prevailing opinion is that reliable isotropic NMR chemical shifts strongly depend on accurate calculations of molecular geometries and inclusion of HF exchange in selected DFT methods, to an extent [134, 135]. On the other hand, the size of the basis set does not increase the accuracy after a point [136, 137]. The ISiCLE software can be installed locally. As seen in Fig. 1, it requires only two input files, prepared in Excel: a sequence of InChI or XYZ molecule geometry files, and a sequence of DFT methods of the user's choice. Preparation of NWChem "run files," 3D molecule geometry files, and/or Linux/Unix shell script "drivers" are not required. As output, ISiCLE prints isotropic shielding, calculated by NWChem, and calculated chemical shifts with respect to a reference molecule and/or application of a user-specified linear regression technique. ISiCLE is a promising tool contributing to standards-free metabolomics, which depends on the ability to calculate properties for thousands of molecules and their associated conformers. Application 1: chemical shift calculations for a demonstration set of molecules To test ISiCLE, we generated a set of 312 molecules. This set is large relative to other metabolomic molecule sets found in the literature, which in our literature survey averaged 34 molecules (Table 1). Our molecule set ranges from small- to large-sized molecules (number of carbon atoms ranging from 1 to 90), and experimental 13C and 1H NMR data in chloroform were available for each of them. Our set also spans a wide array of chemical classes, including acetylides, alkaloids, benzenoids, hydrocarbons, lipids, organohalogens, and organic nitrogen and oxygen compounds. ISiCLE was used to successfully perform DFT calculations for this set under chloroform solvation using eight different levels of DFT theory (4 different functionals and 2 basis sets for 13C and 1H). A total of 2494 carbon nuclei and of 3127 hydrogen nuclei were calculated for all 312 molecules of the demonstration set and compared with experimental data. Deviation bars indicating MAE and MAXAE are plotted for each method in Fig. 3. For both 13C and 1H NMR chemical shifts, the MAE of each method with cc-pVTZ is higher than those with cc-pVDZ. For 13C, the MAE of each method with cc-pVTZ (7–10 ppm) is higher than those with cc-pVDZ (5–6 ppm). MAE of methods with a larger basis set deviate more compared to those with a smaller basis set. The smallest deviations are observed for B3LYP and B35LYP, both in MAE and MAXAE results. The same situation is observed for 1H NMR chemical shifts as well: MAE of each method with cc-pVTZ (~ 0.35 ppm) is higher than those with cc-pVDZ (~ 0.30 ppm). In contrast to 13C NMR chemical shifts, 1H NMR chemical shifts are better predicted with methods using larger basis sets (cc-pVTZ). Although the error differences among each method may be too low to confidently identify the outperforming method, B3LYP/cc-pVDZ is the most successful combination in the calculation of 13C and 1H NMR chemical shifts for our application shown here. Mean absolute errors (MAE) and maximum absolute errors (MAXAE) of chemical shifts for the demonstration set. The grey bars represent MAE, the black bars represent MAXAE. For all methods, geometries are optimized at B3LYP/6-31G(d) in chloroform Figure 4 shows computational costs of DFT combinations for the demonstration set. We found that the smaller basis set (cc-pVDZ) in the calculation of both 13C and 1H NMR chemical shifts was an acceptable compromise between accuracy and computational performance, compared with the larger cc-pVTZ basis. This finding is similar to a recent benchmark study [138] that showed B3LYP/cc-pVDZ is a reliable combination, balancing accuracy with computational cost in 13C chemical shifts calculation. The larger basis set (cc-pVTZ) took 2–3 times longer to complete than cc-pVDZ (in terms of total CPU time). The computational times of the isotropic shielding and chemical shift calculations for this demonstration set are given in the file of DemonstrationSet_CPUtimes.xlsx in the Additional file 2. Computational costs of DFT methods performed for the demonstration set. Each bar is for two DFT methods with basis sets of cc-pVDZ and cc-pVTZ. The grey bars represent CPU times for the methods with cc-pVDZ and the black bars represent those with cc-pVDZ and the black bars represent those with cc-pVTZ Effect of scaling by linear regression We performed the most general approach to error reduction, empirical scaling. Our molecule set has 1554 and 1830 experimental 13C and 1H NMR chemical shifts, respectively. It provides confidence for applying linear regression effectively as it reduces the possibility of overfitting. Empirical scaling was applied to the data obtained with the best combination, B3LYP/cc-pVDZ, using two different relationships: computed shifts versus experimental chemical shifts (Eq. 6), and computed isotropic shieldings versus experimental shifts (Eq. 7). Once the empirical scaling was applied, the accuracy for 13C chemical shifts and 1H chemical shifts improved by 0.7 and 0.11 ppm, respectively. Our computed NMR chemical shifts and shieldings deviate from unity (desired slope = 1) by 0.02 for both 13C and 1H NMR chemical shifts. Linear fits with correlation coefficients of 0.99 (Fig. 5a, b) and 0.93 (Fig. 5c, d) for 13C and 1H NMR chemical shifts, respectively, were observed, which also shows that B3LYP/cc-pVDZ is able to produce data free from random error. Results of linear regression to the 13C and 1H NMR chemical shifts obtained by other DFT methods are given in the Additional file 1 (S5). Linear correlation plots of a 13C and c 1H isotropic shielding values, and b 13C and d 1H NMR chemical shifts versus experimental NMR chemical shifts. Chemical shifts are calculated using the GIAO/B3LYP/cc-pVDZ//B3LYP/6-31G(d) level of theory for the demonstration set in CDCl3 (312 molecules (1554 carbons and 1830 hydrogens)). R2 indicates the correlation coefficient Detailed look at 13C NMR chemical shifts The carbon (13C) magnetic shieldings and chemical shifts derived from the various DFT methods are highly correlated, as shown by a correlation coefficient of 0.99 (Fig. 5). The inclusion of a scaling factor enhances the performance of theoretical calculations with B3LYP/cc-pVDZ//B3LYP/6-31G(d) and decreases the MAE in 13C NMR chemical shifts for this set by approximately 13%. There has been a trend toward using multiple references, such that each molecule should be referenced to a molecule with similar properties to improve accuracy of NMR chemical shifts [123, 134, 139]. Sarotti et al. examined the influence of the reference compound used in the 13C [83] and 1H [82] NMR chemical shift calculations over a set of organic compounds, all of which were included in our calculations. They recommended the use of benzene and methanol as a reference standard in the calculations of chemical shifts of sp-sp2- and sp3- hybridized carbon atoms, respectively, instead of TMS for all type of carbon atoms [140]. Propelled by the discussion in the study of Grimblat et al. [141] about the distribution of the errors observed in sp2- and sp3- carbons, we determined the distribution of the data of chemical shifts of sp2- (933 carbons) and sp3- (745 carbons) hybridized carbons (Fig. 6, the sp2- and sp3- derived series of carbons show two separate chemical shift distributions and two separate error distributions over a much larger variety of compounds than Grimblat et al. For our demonstration set, both errors between calculated and experimental sp2- and sp3- chemical shifts more closely resemble a Student's t-distribution [58, 142], rather than a normal distribution [138, 143]. The correlation coefficients of the errors of sp2- and sp3- carbons are 0.93 and 0.78, and 0.98 and 0.95 for Student's t-distribution and normal distribution, respectively. Chemical shifts of sp2- and sp3- hybridized carbon atoms. a Chemical shifts, b associated errors. Chemical shifts were calculated using the B3LYP/cc-pVDZ//B3LYP/6-31G(d) level of theory in CDCl3 Furthermore, we looked for the bonded neighbors of each carbon and hydrogen extensively in Fig. 7. For carbon shifts, error was measured for carbon (n = 1709), chlorine (n = 149), fluorine (n = 8), hydrogen (n = 1161), nitrogen (n = 199), oxygen (n = 251) and sulfur (n = 20) attachments. The largest deviations occur in carbon–chlorine and carbon–sulfur attachments with MAEs of 11.2 and 5.8 ppm and MAXAE 39.7 and 16.5 ppm, respectively. The study by Li et al. [76], which used a set of chlorinated carbons, reports the same conclusion: calculation accuracy decreases as the size of the basis set used increases, but improvement was obtained after linear regression corrections for B3LYP/6-31 + G(d,p) with slope of 0.98. Other than chlorine and sulfur, carbon-hydrogen attachments also make the 13C NMR chemical shift DFT calculations deviate significantly from experimental values, with MAE of 4.7 and 3.9 ppm and with MAXAE of 51.6 and 34.9 ppm. Carbon was found in rings in 70% of the cases, and these carbons show a MAE of 4.6 ppm. Also, the MAE of 13C NMR chemical shift is 3.9 ppm for carbons bonded to a hydrogen atom but reaches 9.7 ppm in all other cases. Chemical shift prediction errors for different functional groups. a 13C NMR chemical shifts, b 1H NMR chemical shifts. All molecules are from the demonstration set and are calculated using the GIAO/B3LYP/cc-pVDZ//B3LYP/6-31G(d) level of theory in chloroform Oxygen and nitrogen attachments to carbon led to 13C NMR chemical shifts with MAE up to 3.1 and 4.2 ppm and MAXAE of 32.5 and 23.6 ppm, respectively. Interestingly, half the C–O attachments found in ring-form had chemical shifts with a MAE of 2.8 ppm, compared to the chemical shifts of C–O attachments not found in a ring, which had a relatively higher MAE of 3.2 ppm, leading to a percent difference of 14.1%. NMR chemical shifts of C–N attachments, present in a ring or not, show close MAE of 4.19 and 4.26 ppm, respectively, a percent difference of 1.6%. This is to be expected, since C–O attachments are expected to show some deviation in chemical shift due to the polarization of the electron distribution caused by the high electronegativity of oxygen, while nitrogen atoms have a lower electronegativity, leading to a lower deviation in C–N chemical shifts. The correlation plot (Fig. 5a, b) shows a linear pattern with only minor deviations of the predicted 13C shieldings or 13C chemical shifts from the fitted line. It is verified by the correlation coefficient of 0.99, as observed in previous studies [130, 144], that the deviation of the slope from unity within the range of 0.95 and 1.05 is an indicator of a reliable method. However, when placed in subgroups of different attachment types, distant outliers are observed, with some more than 15 ppm away (Fig. 7a). Most outliers are observed in the C–C and C–H attachments, respectively. C–Cl and C–N chemical shifts have a high occurrence of outliers, which may be due to the chemical properties of chlorine and nitrogen such as being remarkably close to first ionization energies. We suspected that some cases of high calculation errors could be due to the consideration of only a single conformer. For the highest accuracy, proper conformational sampling must be considered, as demonstrated below in "Application 2: Boltzmann-weighted NMR chemical shifts of methylcyclohexane" section Detailed look at 1H NMR chemical shifts Proton (1H) chemical shifts are significantly affected by intermolecular interactions, particularly in aqueous states, especially compared to 13C chemical shifts. Agreement with experimental values improves as empirical linear scaling is performed for 1H chemical shifts. GIAO/B3LYP/cc-pVDZ//B3LYP/6-31G(d) yields scaled 1H chemical shifts in chloroform solution having a MAE of 0.30 ppm in comparison with solution experimental values. The 1H chemical shifts in the range of 10–17 ppm show the largest deviation, occurring higher than 5 ppm. In Fig. 7b, error bars are shown for 1H chemical shifts when the hydrogen attaches to carbon (n = 1793), nitrogen (n = 17), and oxygen (n = 41). Oxygen-bound hydrogen nuclei have the largest errors (up to 10 ppm), which is to be expected due to the electronegative property of oxygen atoms, as discussed in the previous section. It is followed by less electronegative nitrogen-bound hydrogen atoms, with an MAE of 0.71 ppm and a MAXAE of 2.25 ppm. About 95% of the 1H NMR chemical shifts calculated for this set are from H–C attachments. These chemical shifts had a MAE of 0.27 ppm and a MAXAE of 4.41 ppm. The high occurrence of outliers could be evidence of how 1H NMR chemical shifts are sensitive to intermolecular interactions. H–O attachments are highly sensitive (to concentration, solvent, temperature, etc.), and it is non-trivial to determine the NMR chemical shift value of arbitrary protons experimentally as well as predict them by using a single, "catch all" DFT method, which explains the relatively low correlation coefficient of 0.93 (Fig. 5c–d). For future studies, we may need to consider the use of different DFT methods, including the use of explicit solvation, particularly in the calculation of 1H NMR chemical shifts in the presence of H–O attachments. Application of empirical scaling to functional groups are given in detail in the Additional file 1 (see S7 and S8). As a final assessment for the data collected with the demonstration set, we assessed the stability and accuracy of the linear regression approach using cross-validation [145]. Cross-validation is a technique mostly used in prediction problems to evaluate how much a given model generalizes to an independent set of data. Specifically, we performed Monte Carlo cross-validation [146, 147]. The procedure of application of Monte Carlo cross-validation method is explained in Additional file 1 (S9). We observed that the estimated linear model parameters (i.e. slope and intercept) from the training set do not differ from that of the entire set. Therefore, the predictive linear model is stable to be accurately estimated and the subsets of 13C and 1H NMR chemical shifts generalize well to the groups that are not represented in the training fold. Application 2: Boltzmann-weighted NMR chemical shifts of methylcyclohexane Metabolites were experimentally interrogated using solution-state NMR, where the observed signal arises from the combined signals of present conformers. It is routine that NMR chemical shifts calculations are carried out on a single dominant conformer. However, it is well known that metabolites do not comprise a single conformer in solution and are instead found in a collection of various conformers [148], and the accuracy of NMR chemical shifts heavily depends on molecular geometries and conformation consideration [46]. It has been shown that for the highest accuracy NMR chemical shift calculations, consideration of conformers is critical, even for relatively small molecules [149]. As a second demonstration of ISiCLE, a conformational analysis based on DFT was performed on a set of 80 conformers of methylcyclohexane using a Boltzmann distribution technique. Boltzmann weighting determines the fractional population of each conformer based on its energy level [92]. High-throughput and straightforward DFT-based NMR chemical shift calculations of all 80 methylcyclohexane conformers was performed by ISiCLE then compared to experimental values. It has been shown by Willoughby et al. [92] that the effects of molecular flexibility on NMR chemical shifts can be captured by Boltzmann weighting analysis, as demonstrated with methylcyclohexane (Fig. 8). Methylcyclohexane is a well-studied small molecule [150,151,152,153,154]. It is flexible, composed of a single methyl group attached to a six-membered ring, and known to exist as an assembly of two chair conformers. There are two distinct conformations of which chair–chair interconversion is rapid and dominated by equatorial to axial conformation. We weighted 40 axial and 40 equatorial conformers in chloroform and obtained a relative free energy of 1.99 kcal/mol with NWChem, similar to calculations using Gaussian [155] by Willoughby et al. [92], and similar to experimental findings (1.73 kcal/mol [156], 1.93 kcal/mol] [108]) and computed (2.15–2.31 kcal/mol [107] and 1.68–2.48 kcal/mol [109]) values. We compared the Boltzmann-weighted 1H and 13C chemical shifts (a ratio of 3% axial to 97% equatorial) to experimental values reported by Willoughby et al. [92]. The Boltzmann-weighted, scaled MAE was 0.017 ppm for 1H chemical shifts ($ \delta_{exp} = 1.00 \times \delta_{comp} + \sim0.00 $), similar to the experimental value of 0.018 ppm in the study of Willoughby et al. Also, the MAE for 13C chemical shifts was 4.4 ppm and decreases to 0.8 ppm when the chemical shifts were scaled ($ \delta_{exp} = 0.99 \times \delta_{comp} + 0.13 $) (Fig. 9). Further details can be found in the Additional file 1 (S10). The a equatorial and b axial structures of methylcyclohexane Experimental and scaled chemical shifts (ppm) of methylcyclohexane We introduce the first release of ISiCLE, which predicts NMR chemical shifts of any given set of molecules relevant to metabolomics for a given set of DFT techniques. ISiCLE calculates the unscaled or scaled NMR chemical shifts (depending on the user's choice of DFT method) and writes the data to appended MDL Molfiles. It also quantifies the error in calculated NMR chemical shifts if the user provides experimental values. The functionality of ISiCLE is demonstrated on a molecule set consisting of 312 molecules, with experimental chemical shifts reported in chloroform solvent. 1H and 13C NMR chemical shifts were calculated using 8 different levels of DFT (BLYP, B3LYP, B35LYP, BHLYP and, cc-pVDZ, and ccpVTZ), referenced to TMS in chloroform by carrying initial geometry optimizations out at B3LYP/6-31G(d) for all molecules. The optimal combination for this set was found to be B3LYP/cc-pVDZ//B3LYP/6-31G(d) with mean absolute error of 0.33 and 3.93 ppm for proton and carbon chemical shifts, respectively. We show that DFT calculations followed by linear scaling do in fact provide an analytically useful degree of accuracy and reliability. Finally, we used ISiCLE for the calculation of NMR chemical shifts of 80 Boltzmann-weighted conformers of methylcyclohexane and compared our results with earlier studies in the literature. ISiCLE is a promising automated framework for accurate NMR chemical shift calculations of small organic molecules. Through this tool, we hope to expand chemical shift libraries, without the need for chemical standards run in the laboratory, which could lead to significantly more identifiable metabolites. Future work includes wrapping individual steps of the ISiCLE NMR module into a formal workflow management system such as Snakemake, to include better fault tolerance, modularization, and improved data provenance. Furthermore, additional chemical properties will be included, such as ion mobility collision cross section and infrared spectra. Finally, ISiCLE will be adapted to run seamlessly on cloud computer resources such as Amazon AWS, Microsoft Azure, and Google Cloud. 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NG provided technical support. RR conceived and supervised the project. All authors participated in the manuscript preparation. All authors read and approved the final manuscript. This research was supported by PNNL Laboratory Directed Research and Development program, the Microbiomes in Transition (MinT) Initiative. This work was performed in the W. R. Wiley Environmental Molecular Sciences Laboratory (EMSL), a DOE national scientific user facility at the PNNL. The NWChem calculations were performed using the Cascade supercomputer at the EMSL. PNNL is operated by Battelle for the DOE under contract DE-AC05-76RL0 1830. Project name: ISiCLE. Project home page: github.com/pnnl/isicle. Programming language: Python. Operating system(s): Platform independent. License: GNU GPL license. Free academic and non-profit research use only. All data files, source code, and tutorial are provided in the additional files. The Gene and Linda Voiland School of Chemical Engineering and Bioengineering, Washington State University, Pullman, WA, USA Yasemin Yesiltepe & Ryan S. Renslow Earth and Biological Sciences Division, Pacific Northwest National Laboratory, Richland, WA, USA Yasemin Yesiltepe, Jamie R. Nuñez, Sean M. Colby, Dennis G. Thomas, Mark I. Borkum, Nancy M. Washton, Thomas O. Metz, Justin G. Teeguarden, Niranjan Govind & Ryan S. Renslow Nuclear Magnetic Resonance Facility, Oregon State University, Corvallis, OR, 97331, USA Patrick N. Reardon Yasemin Yesiltepe Jamie R. Nuñez Sean M. Colby Dennis G. Thomas Mark I. Borkum Nancy M. Washton Thomas O. Metz Justin G. Teeguarden Niranjan Govind Ryan S. Renslow Correspondence to Ryan S. Renslow. Supporting information document. Supporting information files. Including tutorial, code, and all other files. Yesiltepe, Y., Nuñez, J.R., Colby, S.M. et al. An automated framework for NMR chemical shift calculations of small organic molecules. J Cheminform 10, 52 (2018). https://doi.org/10.1186/s13321-018-0305-8 NWchem Quantum chemistry	CommonCrawl
\begin{document} \draft \title{Experimental Study of A Photon as A Subsystem of An Entangled Two-Photon State} \author{Dmitry V. Strekalov\thanks{Present address: Department of Physics, New York University, NY 10003},Yoon-Ho Kim, and Yanhua Shih} \address{Department of Physics, University of Maryland, Baltimore County,\\ Baltimore, MD 21250} \maketitle \begin{abstract} The state of the signal-idler photon pair of spontaneous parametric down conversion is a typical nonlocal entangled pure state with zero entropy. The precise correlation of the subsystems is completely described by the state. However, it is an experimental choice to study only one subsystem and to ignore the other. What can we learn about the measured subsystem and the remaining parts? Results of this kind of measurements look peculiar. The experiment confirms that the two subsystems are both in mixed states with entropy greater than zero. One can only obtain statistical knowledge of the subsystems in this kind of measurement. \end{abstract} \pacs{PACS Number: 03.65.Bz, 42.50.Dv} One of the most surprising consequences of quantum mechanics is the entanglement of two or more distant particles. The first example of a two-particle entangled state was suggested by Einstein, Podolsky, and Rosen in their famous {\em gedankenexperiment} in 1935 \cite{epr}. The EPR state is a pure state of two spatially separated particles which can be written as, \begin{equation} \left\| \Psi \right\rangle =\sum_{a,b}\delta \left( a+b-c_{0}\right) \left\| a\right\rangle \left\| b\right\rangle \label{eprst} \end{equation} where $a$ and $b$ are the momentum or the position of particle 1 and 2 respectively and $c_{0}$ is a constant. It is clear that state (\ref{eprst}) is a two-particle state; however, it cannot be factored into a product of the state of particle 1 and the state of particle 2. This type of states was defined by Schr$\ddot{o}$dinger as {\em entangled states} \cite{schro}. One, perhaps the most easily accessible, example of an entangled state is the state of a photon pair emitted in Spontaneous Parametric Down Conversion (SPDC). SPDC is a nonlinear optical process from which a pair of signal-idler photon is generated when a pump laser beam is incident onto an nonlinear optical crystal. The signal-idler two-photon state can be calculated by first order perturbation from the SPDC nonlinear interaction Harmiltonion \cite{Klyshkobook}, \begin{equation} \left\| \Psi \right\rangle =\sum_{s,i}\delta \left( \omega _{s}+\omega _{i}-\omega _{p}\right) \delta \left( {\bf k}_{s}+{\bf k}_{i}-{\bf k} _{p}\right) a_{s}^{\dagger }(\omega ({\bf k}_{s}))\ a_{i}^{\dagger }(\omega ( {\bf k}_{i}))\mid 0\rangle \label{state} \end{equation} where $\omega _{j}$, {\bf k$_{j}$ (}j = s, i, p) are the frequency and wavevectors of the signal (s), idler (i), and pump (p) respectively, $\omega _{p}$ and {\bf k}$_{p}$ can be considered as constants, usually a single mode laser is used for pump, $a_{s}^{\dagger }$ and $a_{i}^{\dagger }$ are the respective creation operators for signal and idler photon. The delta functions of the state ensure energy and momentum conservation. It is indeed the conservation laws that determine the values of an observable for the pair. Quantum mechanically, state (\ref{state}) only provides precise momentum (energy) {\em correlation} of the {\em pair} but no precise momentum (energy) determination for the signal photon and the idler photon. In EPR's language: the momentum (energy) of neither the signal nor the idler is determined by the state; however, if one is known to be at a certain value the other one is determined with certainty. Notice also that state ( \ref{state}) is a pure state. It provides a complete description of the entangled two-photon system. Following the creation of the pair, the signal and idler may propagate to different directions and be separated by a considerably large distance. If it is a free propagation, the state will remain unchanged except for the gain of a phase, so that the precise momentum (energy) {\em correlation} of the {\em pair} still holds. The conservation laws guarantee the precise value of an observable with respect to the pair (not to the individual subsystems). It is in this sense, we say that the entangled two-photon state of SPDC is {\em nonlocal}. Quantum theory does allow a complete description of the precise {\em correlation} for the spatially separated subsystems, but no complete description for the physical reality of the subsystems defined by EPR. It is in this sense, we say that quantum mechanical description (theory) of the entangled system is{\em \ nonlocal}. So far, our discussion involves no measurement. In a type of measurements when ``joint detections'' are involved, for example a coincidence detection for the SPDC pair, it corresponds to the intensity correlation, $\left\langle \Psi \right\| \hat{I}_{1}\otimes \hat{I} _{2}\left\| \Psi \right\rangle $ or the fourth order correlation of the fields, $\left\langle \Psi \right\| \hat{E}_{1}^{(-)}\hat{E}_{2}^{(-)}\hat{E} _{2}^{(+)}\hat{E}_{1}^{(+)}\left\| \Psi \right\rangle $. One may cooperate spin (polarization for photon) correlations into the coincidence joint detection too, as in the measurements for the EPR-Bohm state \cite{Bohm}. If the correlations of the pair have been built up in the entangled two-particle state from the beginning, it comes as no surprise that the intensity correlation reflects {\em perfect} correlation (EPR, EPR-Bohm or EPR-Bell type correlation) of the pair. The distance between the detectors would not matter. There is no{\em \ action-at-a-distance} involved, if we are willing to give up the classical EPR reality. The entangled state indeed indicates and represents a very different physical reality: Does the signal photon or idler photon have a defined momentum (energy) in state (\ref{state} )? No! Does the pair have a defined {\em total momentum} ({\em energy}) in state (\ref{state})? Yes! In state (\ref{state}) the precise value of an observable is determined in the form of {\em total} {\em value} by conservation laws. In addition, one cannot ``assume'' or ``imagine'' two individual wavepackets each associated with the signal photon and the idler photon. It is a non-factorizable two-dimensional ``wavepacket'' associated with the entangled two-particle system \cite{Rubin}. For this very reason we have named the signal-idler pair the ``{\em biphoton''}. Many interesting phenomena involving biphoton have been demonstrated in two-photon interferometry and in two-photon correlation type experiments \cite{sample}. Several recent experiments have clearly shown that the two-photon interference is not the interference between two photons. It is not the signal and idler photon wavepackets but the two-dimensional biphoton wavepacket that plays the role \cite{2and1}. Recently, there has been a lot of interest in a type of measurements in which only one subsystem of an entangled multi-particle state is measured and the remaining parts are left undisturbed. One of the popular misconceptions is to believe that the ``state'' of the undisturbed remaining parts is completely determined by this kind of ``distance measurement'': if the measurement of the subsystem either yields the result of a value for an observable or indicates the ``state'' of that subsystem, the undisturbed remaining parts is then ``forced'' into a ``pure state''. Do we have to accept {\em action-at-a-distance} in quantum theory? It is a fact that the experimentalist can choose to look at one part of the entangled system and to ignore the other. The subsystems may well be separated spatially. For instance, one can use a photon counting detector to register a ``click'' event of the signal photon of the two-photon state of SPDC (not a ``click-click'' event!) to study only the properties of the signal and leave the idler undisturbed or to be predicted. What can we predict for the idler photon in this kind of measurement? Before answering this question, it may be better to ask first: ``what can we learn about the signal photon in this kind of measurement?'' An interesting situation arises: while the two-photon state of SPDC is a pure state, the respective states of the signal and idler photon are not. The states of the individual signal and idler are both in thermal (mixed) states. This has been pointed out by several researchers, e.g. \cite {takahashi75,yurke87,klyshko96,cerf97}, from different perspectives. Significance of two parts in a mixed state constitute a quantum mechanical system in a pure state was emphasized by B. Yurke and M. Potasek \cite {yurke87} as an example of purely quantum thermalization, that is obtaining mixed states out of pure states in a Hamiltonian system. N.J. Cerf and C. Adami \cite{cerf97} introduced the {\em mutual} ($S_{A:B}=S_{B:A}$) and {\em conditional} ($S_{A\|B}$, $S_{B\|A}$) entropy (or information) for a two-particle system similar to the mutual and conditional entropies as defined in classical probability theory: \begin{equation} S=S_{A\|B}+S_{B\|A}+S_{A:B},\quad S_{A}=S_{A\|B}+S_{A:B},\quad S_{B}=S_{B\|A}+S_{A:B}. \label{smisc} \end{equation} For an entangled two-particle system in a pure state (so that $S=0$), the relations in (\ref{smisc}) give, \begin{equation} S_{A}+S_{B\|A}=0,\qquad S_{B}+S_{A\|B}=0. \label{s0} \end{equation} The paradox of the whole system entropy $S$ being zero while an entropy of its either part $S_{A}$ and $S_{B}$ are both positive (which is a formal expression of the statement that the information contained in the whole system is less than the information contained in its parts) is suggested resolvable by letting the conditional entropy to take on negative values. In this paper we report an experimental work along the lines of this discussion. The reported experiment hinges on a typical Fourier spectroscopy measurement. The schematic setup is shown in Fig. \ref{fig:setup}. The measurement is based on a ``click'' type single photon detection; however, the photon source is an entangled two-photon source of SPDC: a $3mm$ BBO ($ \beta -BaB_{2}O_{4}$) crystal pumped by a $351.1nm$ CW Argon ion laser line. The orthogonally polarized signal-idler photon pairs are generated by satisfying the collinear degenerate (centered at wavelength $702.2nm$) type-II phase matching condition \cite{Yariv}. The idler (extraordinary ray of BBO) is removed by a polarizing beamspliter PBS. The signal (ordinary ray of BBO) is then sent to a Michelson interferometer. A photon counting detector is coupled to the output port of the interferometer through a $25mm$ focal lens. A $702.2nm$ spectral filter with Gaussian transmittance function (bandwidth $83nm$ FWHM) is placed in front of the detector. The counting rate of the detector is recorded as a function of the optical arm length difference, $\Delta L$, of the Michelson interferometer. The SPDC spectrum closely resembles a rainbow ranging from red to blue. The spectrum collected by the $25mm$ focal lens is much wider then $83nm$. It would be reasonable to expect a Gaussian spectrum with $83nm$ FWHM (determined by the spectrum filter used for the detector) from the above measurement. (This conjecture is different from the wrong belief or imagination that each individual of the signal and idler photons is associated with a Gaussian wavepacket.) On the contrary, we instead observed an ``unexpected'' result, the observed spectrum is not Gaussian and its width is only $2.2nm$ (far from $83nm$). The experimental data is reported in Fig. \ref{fig:wpacket}. The envelope of the sinusoidal modulations (in segments) is fitted very well by two ``notch'' functions (upper and lower part of the envelope). The width of the triangular base is about $225\mu m$ which corresponds to roughly a spectral band width of $2.2nm$. To seek an explaination of this result, we must first examine the two-photon state of SPDC. We cannot assume a state for either the signal photon or idler photon. The single photon state is obtained by taking a partial trace of the two-photon state density operator, integrating over the spectrum of the idler and vice versa: \begin{equation} \hat{\rho}_{s}=tr_{i}\ \hat{\rho},\quad \hat{\rho}_{i}=tr_{s}\ \hat{\rho} \label{ro} \end{equation} with \begin{equation} \hat{\rho}\equiv \left\| \Psi \right\rangle \left\langle \Psi \right\| , \label{stateII} \end{equation} where $\hat{\rho}$ the density matrix operator and $\left\| \Psi \right\rangle $ the two-photon state (\ref{state}). First, it is very interesting to find that even though the two-photon EPR state of SPDC is a pure state, i.e., $\hat{\rho}^{2}=\hat{\rho},$ the corresponding single photon state of the signal and idler are not, i.e., $ \hat{\rho}_{s,i}^{2}\neq \hat{\rho}_{s,i}$. This accords with the earlier mentioned fact that the entropy of the system is zero (pure state) while each subsystem has an entropy greater than zero (mixed state). The zero entropy condition for a system in a pure state reflects the fact that the quantum state $\|\Psi \rangle $ provides a {\em complete} description of the system . On the other hand, the mixed state of each subsystem only reveals their statistical nature. In the experiment, we realize a collinear degenerate type-II phase matching \cite{Yariv}. This means that the SPDC crystal orientation is such that the orthogonally polarized signal-idler pair with degenerate frequency $\omega =\omega _{p}/2$, are emitted collinearlly. We select this direction by a set of pinholes during the experimental alignment process. Then the integral in Eq.(\ref{state}) can be simplified to an integral over a frequency detuning parameter $\nu $, (the detailed calculation can be find in ref.\cite{Rubin} ): \begin{equation} \|\Psi \rangle =A_{0}\int d\nu \ \Phi (DL\nu )\ a_{s}^{\dagger }(\omega +\nu )a_{i}^{\dagger }(\omega -\nu )\ \|0\rangle . \label{psinu2} \end{equation} where the $sinc$-like function $\Phi (LD\nu )$ followed from Eq.(\ref{state} ) considering a finite length of the SPDC crystal \cite{Klyshkobook}. It represents a spectral width of the two-photon state, \begin{equation} \Phi (DL\nu )=\frac{1-e^{-iDL\nu }}{iDL\nu }, \label{sinc} \end{equation} which is determined by the finite crystal length $L$ and, specifically for the collinear degenerate type-II SPDC, by the difference of inverse group velocities for the signal (ordinary ray) and the idler (extraordinary ray): $ D\equiv 1/u_{o}-1/u_{e}$. The constant $A_{0}$ is found from the normalization condition $tr\rho =\langle \Psi \|\Psi \rangle =1$ (dimensionless): \[ A_{0}=\sqrt{\frac{DL}{4\pi }}. \] Substituting $\|\Psi \rangle $ in the form of Eq.(\ref{psinu2}) into Eq.(\ref {ro}) the density matrix of signal is calculated to be, \begin{equation} \hat{\rho}_{s}=A_{0}^{2}\int d\nu \ \left\| \Phi (\nu )\right\| ^{2}\ a_{s}^{\dagger }(\omega +\nu )\left\| 0\right\rangle \left\langle 0\right\| \ a_{s}(\omega +\nu ) \label{densityIs} \end{equation} where \begin{equation} \left\| \Phi (\nu )\right\| ^{2}={\rm sinc}^{2}\frac{DL\nu }{2} \label{sincsignal} \end{equation} In (\ref{densityIs}) we consider a multimode (a continues frequency spectrum) entangled system with a single quantum, $n=1$. The operator (\ref {densityIs}) describes the statistical distribution of this quantum. This is a good approximation since the coupling in SPDC is week and greater number states $n>1$ that correspond to higher perturbation orders are extremely unlikely. On the other hand, $n=0$ represents vacuum fields that do not result in detections \cite{Squeezed}. By now, we can understand very well the experimental results. (1) For a spectrum of $sinc$-square function we do expect a double ``notch'' envelope in the measurement and the base of the triangle, which is determined by $DL$, is calculated to be $225\mu m$ (we have considered the optical path difference is twice of the arm difference in Michelson interferometer), corresponding to a $2.2nm$ bandwidth. The experimental result, from fitting, is about $225nm$, which agrees well with the prediction \cite{Double}. (2) We see that the spectrum of the {\em signal} photon is dependent on the group velocity of the {\em idler} photon which is not measured at all in our experiment. However, this comes as no surprise, because the state of the signal photon is calculated from the two-photon state by integrating over the idler modes. (3) We also see immediately that $\hat{\rho}_{s}^{2}\neq \hat{\rho}_{s}$, so the signal and idler single-photon states are both mixed states. It is then straightforward to evaluate numerically the Von Neuman entropy $S$ \cite{inf} of the signal (or idler) subsystem, \begin{equation} S_{s}=-tr[\hat{\rho}_{s}\log \hat{\rho}_{s}], \label{nentropy} \end{equation} based on the ``double notch'' fitting function. Note that operator (\ref {densityIs}) is diagonal. Taking its trace is simply to perform an integration over the frequency spectrum with the spectral density of Eq.(\ref {sincsignal}). To compute the integral of Eq.(\ref{nentropy}) for the density matrix $\hat{\rho}_{s}$ of Eq.(\ref{ro}), we replace variable $\nu $ by a dimensionless variable $DL\nu /2$ and evaluate the integral numerically. The calculation yields, \[ S_{s}\approx 6.4>0. \] This again indicates the statistical mixture nature of the state of a photon (subsystem) in an entangled two-photon system. Based on the experimental data, we conclude that the entropy of signal and idler are both greater then zero (mixed state); while the entropy of the signal-idler two-photon system is zero (pure state). This may mean that negative entropy is present somewhere in the system, perhaps in the form of the conditional entropy \cite{cerf97}. By definition of the conditional entropy, one is tempted to say that {\em given the result of a measurement over one particle, the result of measurement over the other must yield negative information.} This paradoxical statement is similar to and in fact closely related to the EPR ``paradox''. We suggest that the paradox comes from the same philosophy. Conclusion: In these kind of measurements, in which the experiment only measures a subsystem of an entangled multi-particle system and leave the remaining parts undisturbed, one can only obtain statistical knowledge of the subsystems. Neither the measured subsystem nor the remaining parts is in pure state. The individual subsystems are described statistically by the quantum theory before the measurement and after the measurement. The measurements can never ``force'' the undisturbed subsystems into a pure state. Again, we emphasize that no {\em action-at-a-distance} in any format. We gratefully acknowledge the many useful discussions with M.H. Rubin. This work was supported, in part, by the U.S. Office of Naval Research and National Security Agency. \begin{references} \bibitem{epr} A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {\bf 47} , 777 (1935). \bibitem{schro} E.Schr$\ddot{o}$dinger, Naturwissenschaften {\bf 23}, 807, 823, 844 (1935); the English translation appears in {\em Quantum Theory and Measurement}, ed. J.A. Wheeler and W.H. Zurek, Princeton University Press, New York, (1983). \bibitem{Klyshkobook} D.N. Klyshko, {\em Photon and Nonlinear Optics}, Gordon and Breach Science, New York, (1988). \bibitem{Bohm} D. Bohm, {\em Quantum Theory}, Prentice Hall Inc., New York, (1951). \bibitem{Rubin} M.H. Rubin, D.N. Klyshko, Y.H. Shih, and A.V. Sergienko, Phys. Rev. A {\bf 50}, 5122 (1994). \bibitem{sample} For example the ``ghost image'' and ``ghost interference'': T.B. Pittman, Y.H. Shih, D.V. Strekalov, and A.V. Sergienko, Phys. Rev. A {\bf 52}, R3429 (1995); D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, Phys. Rev. Lett., {\bf 74}, 3600 (1995). The ``superluminal tunneling'': A.M. Steinberg, P.G. Kwiat, and R.Y. Chiao, Phys. Rev. Lett., {\bf 71}, 708 (1993). The ``nonlocal Franson interferometer'': J.D. Franson, Phys. Rev. Lett., {\bf 62}, 2205 (1989); D.V. Strekalov, T.B. Pittman, A.V. Sergienko, Y.H. Shih, and P.G. Kwiat, Phys. Rev. A {\bf 54}, R1 (1996). The ``quantum eraser'': M.O. Scully and K. Dr\"{u}hl, Phys. Rev. A {\bf 25}, 2208 (1982); P.G. Kwiat, A.M. Steinberg, and R.Y. Chiao, Phys. Rev. A {\bf 45}, 7729 (1992); T.J. Herzog, P.G. Kwiat, H. Weinfurter, and Zeilinger, Phys. Rev. Lett., {\bf 75}, 3034 (1995);Y.H. Kim, S.P. Kulik, M.O. Scully, and Y.H. Shih, (1998). The ``induced coherence'': X.Y. Zou, L.J. Wang, and L.Mandel, Phys. Rev. Lett., {\bf 67}, 318 (1991). The ``frustrated two-photon interference'': T.J. Herzog, J.G. Rarity, H. Weinfurter, and Zeilinger, Phys. Rev. Lett., {\bf 72}, 629 (1994). As well as many Bell's inequality violation experiments. \bibitem{2and1} T.B. Pittman, D.V.Strekalov, A. Migdall, M.H. Rubin, A.V. Sergienko, and Y.H. Shih, Phys. Rev. Lett., {\bf 77}, 1917 (1996); D.V.Strekalov,T.B. Pittman, and Y.H. Shih, Phys. Rev. A {\bf 57}, 567 (1998). \bibitem{takahashi75} Y. Takahashi and H. Umezawa, Collect. Phenom., {\bf 2} , 55, (1975). \bibitem{yurke87} B. Yurke and M. Potasek, Phys. Rev. A, {\bf 36}, 3664, (1987). \bibitem{klyshko96} D. N. Klyshko, Uspekhi Fizicheskikh Nauk, {\bf 166}(6), 613-638 (1996) [Physics -- Uspekhi {\bf 39}(6), 573-596 (1996)]. \bibitem{cerf97} N.J. Cerf and C. Adami, Phys. Rev. Lett., {\bf 79}, 5194-7, (1997). \bibitem{Yariv} A. Yariv, {\em Quantum Electronics}, John Wiley and Sons, New York, (1989). \bibitem{Squeezed} Our approach therefore is quite different from the early ``squeezed state'' studies, e.g. \cite{takahashi75,yurke87} where the signal and idler radiation of SPDC is shown to be in the thermal (and hence mixed) state based on a single-mode model which allow all number-states but omits entanglement. \bibitem{Double} It is interesting to notice that apparently the same result, a triangular correlation function was also obtained in a two-photon interference experiment \cite{sergienko95} based on {\em joint coincidences detections}. The two experiments are fundamentally different: the earlier one \cite{sergienko95} demonstrates the overlap of two biphoton amplitudes while the present reveals the shape of a single-photon wave packet. Even though we understand, mathematicaly, why the results are identical, the physical meaning of this fact remains puzzling. \bibitem{sergienko95} A.V. Sergienko, Y.H. Shih and M.H. Rubin, J. Opt. Soc. B, {\bf 12}, 859-862, (1995). \bibitem{inf} See e.g. C.E. Shannon and W. Weaver, {\em The mathematical theory of communication}, University of Illinois Press, 1949; C.H. Bennett, Physics Today, {\bf 48(10)}, 24, (1995). \end{references} \begin{figure} \caption{ Schematic of the experimental set up. A Michelson interferometer is used to study the spectrum of the signal of SPDC. The SPDC spectrum closely resembles a rainbow ranging from red to blue. A band pass spectral filter centered at $702.2nm$ with $83nm$ FWHM of a Gaussian transmittance function, is placed in front of the photon counting detector.} \label{fig:setup} \end{figure} \begin{figure} \caption{Experimental data indicated a ``double notch'' envelope of the interference pattern. The X-axis, $\Delta L$ in $\mu m$, is the optical arm difference of the Michelson interferometer. Each of the doted single vertical segment contains many cycles of sinusoidal modulations. The spike at $\Delta L=0$, usually called ``white light condition'' for observing``white light'' interference, is a broad band interference pattern which is determined by the spectral filter. Two vertical lines show $ +122.5\mu m$ and $-122.5 \mu m$.} \label{fig:wpacket} \end{figure} \centerline{\epsffile{setup1.eps}} Figure 1. Dmitry V. Strekalov, Yoon-Ho Kim, and Yanhua Shih \centerline{\epsffile{wpacket.EPS}} Figure 2. Dmitry V. Strekalov, Yoon-Ho Kim, and Yanhua Shih \end{document}	arXiv
Development of pathogenicity predictors specific for variants that do not comply with clinical guidelines for the use of computational evidence Elena Álvarez de la Campa1,2, Natàlia Padilla1 & Xavier de la Cruz1,3 Strict guidelines delimit the use of computational information in the clinical setting, due to the still moderate accuracy of in silico tools. These guidelines indicate that several tools should always be used and that full coincidence between them is required if we want to consider their results as supporting evidence in medical decision processes. Application of this simple rule certainly decreases the error rate of in silico pathogenicity assignments. However, when predictors disagree this rule results in the rejection of potentially valuable information for a number of variants. In this work, we focus on these variants of the protein sequence and develop specific predictors to help improve the success rate of their annotation. We have used a set of 59,442 protein sequence variants (15,723 pathological and 43,719 neutral) from 228 proteins to identify those cases for which pathogenicity predictors disagree. We have repeated this process for all the possible combinations of five known methods (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2). For each resulting subset we have trained a specific pathogenicity predictor. We find that these specific predictors are able to discriminate between neutral and pathogenic variants, with a success rate different from random. They tend to outperform the constitutive methods but this trend decreases as the performance of the constitutive predictor improves (e.g. with PON-P2 and PolyPhen-2). We also find that specific methods outperform standard consensus methods (Condel and CAROL). Focusing development efforts on the case of variants for which known methods disagree we may obtain pathogenicity predictors with improved performances. Although we have not yet reached the success rate that allows the use of this computational evidence in a clinical setting, the simplicity of the approach indicates that more advanced methods may reach this goal in a close future. The application of NGS in the clinical setting is limited, among other things, by our inability to accurately pinpoint the causative variant of a patient's condition from the set of variants identified in sequencing experiments [1]. Frequently, this is due to a lack of information on the pathogenicity of these variants. In this situation, pathogenicity predictors, designed to estimate the damage caused by sequence variants [2, 3], can provide valuable information. For variants resulting in amino acid substitutions, these tools combine properties that measure different aspects of protein structure/function. For example, some of the properties (like hydrophobicity or volume differences) are related to changes in protein stability upon mutation, while others indicate whether the functional site of the protein has been damaged [2]. Using this information, in silico predictors produce a numerical score that is transformed into a binary prediction (pathogenic/neutral) through the use of a decision threshold. The accuracy of these predictions is around 80% [2, 3]. Although this value is not a fundamental threshold limiting the usage of in silico tools in the clinical, this kind of application was not initially advocated [3,4,5]. However, this situation is changing due to three facts. First, the drop in sequencing costs is leaving variant interpretation as one of the main bottlenecks in clinical applications of NGS [1] thus creating an important pressure for finding strategies that alleviate this problem. Second, and further in this direction, clinical users increasingly consider the possibility of using pathogenicity predictions as supporting evidence that can be combined with medical data to support diagnostic decisions [6,7,8]. This view has been facilitated by the clarification of the probabilistic nature of computational evidence [9]. And, third, the fact that the success rate of pathogenicity predictors remains around 80%, regardless of the technical differences between them [2, 3], indicates that these tools recognize a signal common to many pathogenic variants but absent in neutral ones [2, 3]. In this scenario, where pathogenicity predictions can be useful but are still imperfect, the idea of scoring variants with several predictors is gaining support in healthcare applications [3, 9, 10]. The underlying rationale is that because different methods implement (partially) complementary representations of the variant's impact, coincidence in their predictions would be reinforcing. This idea is included in the guidelines for variant interpretation of the American College of Medical Genetics and Genomics (ACMG) and the Association for Molecular Pathology (AMP) [11]. There, the application of more than one predictor is considered advantageous and, to combine the resulting evidence, it is proposed that "If all of the in silico programs tested agree on the prediction, then this evidence can be counted as supporting. If in silico predictions disagree, however, then this evidence should not be used in classifying a variant". The value of this type rule (to which we will refer to as the coincidence rule) has been observed in different works [12,13,14]. However, when pathogenicity predictions disagree this rule will result in the rejection of computational evidence and, consequently, in a reduction of the data available to make medical decisions. Not only this, this effect will affect an increasing number of variants if we combine more predictors in a quest for higher reliability. In this work, we address this problem and study whether we can develop specific, competitive pathogenicity predictors for those variants for which known methods give contradictory results. To this end, we have developed a series of neural network-based predictors using a dataset of pathogenic and neutral variants for which five known predictors (SIFT, PolyPhen-2, CADD, PON-P2 and MutationTaster2) disagreed in their results (Additional file 1: Figure S1; this set will be called PRDIS). To build our tools we have explored different options (Additional file 2: Figure S2), including the use of two neural networks (NN) -a model with no hidden layer and one with a single hidden layer and two nodes, and different combinations of input attributes (using prediction scores and molecular/evolutionary properties). Note: since in this work we will frequently compare different sets of predictors, to avoid confusion we will refer to SIFT, PolyPhen-2, CADD, PON-P2 and MutationTaster2 as reference tools/predictors/methods, and to Condel and CAROL as consensus tools/predictors/methods. The results obtained show that there is a high number of variants, between 10% and 45% of the cases studied, for which contradictory predictions are obtained. For these variants we find that we can build specific pathogenicity predictors with non-random success rates. In fact, the performance of these specifically trained tools generally improves on that of the reference tools used (SIFT, PolyPhen-2, CADD, PON-P2 and MutationTaster2) and on that of consensus pathogenicity predictors (Condel and CAROL). Finally, we provide a global view of what prediction performance can be reached when combining in a hybrid method the coincidence (or ACMG/AMP) rule and the predictors for PRDIS. Note. We will use the terms 'specific' or 'PRDIS specific' predictors for those predictors obtained using variants from PRDIS only. General protocol for building the PRDIS specific predictors The goal of this work is to study whether we can obtain better pathogenicity predictions by developing methods specific for subsets of the variant. More precisely, in this work we have used the coincidence rule to partition our set of variants (Additional file 1: Figure S1) and develop specific predictors for PRDIS (Additional file 2: Figure S2), the subset of variants for which the reference predictors disagree. We have studied this problem for all possible combinations of five reference predictors (Additional file 2: Figure S2): SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2. For each of these combinations, we will obtain a PRDIS and this set will be used for training a neural network predictor following a standard protocol that has been described in our previous work [2, 15, 16]. For each PRDIS, this protocol is divided into three steps: (i) characterization of variants with several properties, (ii) build a neural network model for variant prediction and (iii) estimate its performance. Below we describe these steps, although more information can be found in our previous work [2, 15, 16]. Variant datasets The development of the pathogenicity predictors PRDIS required, in a first step, to build an initial set of pathological and neutral variants; in a second step, this set of variants is processed to give the PRDIS sets that will be used to derive the predictors tested in this work. Below, we devote a specific section to each of these two steps. The initial variant dataset This dataset, constituted by pathological and neutral variants, was obtained following commonly used procedures [2, 15, 16]. Pathological variants were retrieved from UniProt [17] and corresponded to sequence variants labeled as "Disease" in Humsavar (version 06-JUL-2016). However, not all of them were included in our initial dataset; we removed those variants from proteins contributing less than 30 independent variants to Humsavar. For example, if for a protein there were only two known variants in Humsavar, none of them was included in our initial dataset. On the contrary, if for a protein there were 31 variants listed in Humsavar, all of them were included in our initial dataset. The reason for this filter is to avoid the large imbalances between the number of pathological and neutral variants in the dataset, caused by proteins contributing few pathological but many neutral variants. The threshold (thres) value of 30 pathological variants per protein was chosen after exploring the dependence of the ratio of neutral to pathological variants on different thres values: 12.5 (thres = 0), 6.3 (thres = 5), 4.8 (thres = 10), 2.8 (thres = 30) and 2.0 (thres = 50). On the basis of our previous work (Fig. 4 in [15]), where we found that for ratios above 5 the sample imbalance becomes increasingly difficult to correct, we chose a conservative threshold (thres = 30) for this work. Higher values were discarded because the number of proteins dropped substantially, e.g. for thres = 50, only 130 proteins contributed variants to the dataset, compared to 228 for thres = 30. At the end of the process, we obtained 15,723 pathogenic variants, distributed over a total of 228 proteins. For neutral variants, we used the homology-based model described in our previous work [2, 15, 16], where variants are obtained from a multiple sequence alignment (MSA) for each protein family. More precisely, they correspond to those sequence deviations from the human representative observed in close homologs (sequences from other species > = 95% identical to the human one) [18]. The technical steps are well described in Riera et al. [15]. Here, we briefly summarize them. First, for each of the 228 proteins we retrieved their sequence from UniProt and used it to query UniRef100 (06-JUL-2016) [19], running a PsiBlast [20] query (e-value 0.001, two iterations). From this output, we eliminated those sequences less than 40% identical to the human protein. Second, the remaining sequences were aligned with Muscle [21]. And third, we collected all the deviations from the human sequence found in homologs > = 95% sequence identity. These deviations constituted the set of neutral variants for this protein. Following this protocol for the 228 proteins, we obtained a total of 43,651 neutral variants. Together with the patological cases, we obtained a set of 59,442 variants spread over 228 proteins, that we called VS228. An annotated list of the variants in VS228, plus the pathogenicity predictions for the tools used in this work are provided as Additional file 3 (pathological variants) and Additional file 4 (neutral variants). To check the reach of the conclusions in this work for proteins not represented in VS228, we employed those variants discarded when building VS228 because their proteins did not have 30 or more cases. The new dataset, which was not utilized during the training of our predictors, was constituted by a total of 322,270 variants (29,259 pathological and 293,011 neutral) spread over 2168 proteins. This independent, validation dataset was called VS2168. Note that in this set pathogenic variants, apart from UniProt [17], were also obtained from HGMD Professional [22], to which we have recently bought a subscription. The PRDIS variant datasets As explained at the start of the Materials and Methods, we tried different versions of the coincidence rule, each corresponding to one of the combinations of five reference methods (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2). Application of this rule to VS228 (Additional file 1: Figure S1) was used to produce a given PRDIS. Repeated application of all possible versions of the rule results in all the PRDIS used in this work (Additional file 2: Figure S2). Equivalent PRDIS datasets were also obtained from VS2168. They were used to test if the conclusions reached with VS228 also hold for proteins (and their variants) not included in the development of our predictors. Characterization of variants in terms of discriminant features We tried two different sets of features to characterize variants for building our specific methods (Additional file 2: Figure S2). In one we only used the scores of the reference predictors employed to include the variant in the PRDIS (Additional files 1, 2: Figures S1, S2). For example, when the PRDIS was built using SIFT and PON-P2, we used the SIFT and PON-P2 scores as input for our predictor; when the PRDIS was built using PolyPhen-2, CADD and PON-P2, then our input was constituted by the scores of these three methods; etc. In the second set, we enriched the previous scores with three additional properties: the element of the Blosum62 matrix [23] corresponding to the amino acid replacement and two properties related to the sequence conservation pattern at the variant locus. The first was Shannon's entropy; it is equal to -Σipi.log(pi), where the index i runs over all the amino acids at the variant's MSA column. The second property was the value of the position-specific scoring matrix [15, 24] for the native amino acid, pssmnat, which is equal to log(fnat,i/fnat,MSA), where fnat,i and fnat,MSA are the frequencies of the native amino acid at the variant locus i and in the whole alignment, respectively. Both Shannon's entropy and the position-specific scoring matrix element were computed from the MSA of the protein family. Building the specific predictors All our predictors were built with WEKA (v3.6.8) [25]. We tried two neural network models. One was the simplest neural network possible: a single-layer perceptron (WEKA defaults: L = 0.3, M = 0.2, N = 500, V = 0, S = 0, E = 20), with no hidden layers [26]. This model was chosen because we have used it with good results in our previous work [15, 16]. The second model was a slightly more complex neural network with one hidden layer having two nodes (WEKA parameters: L = 0.3, M = 0.2, N = 500, V = 0, S = 0, E = 20, H = 2). We used SMOTE [27] to correct for the imbalance between pathological and neutral variants in the training sets (not in the test/validation sets). For each PRDIS, the whole procedure described in this section was applied to the two possible sets of features here described. Performance estimates are obtained following a standard 5-fold cross-validation procedure, such as that described in our previous work. The success rate of the predictors was measured using six parameters [15, 16, 28, 29]: sensitivity, specificity, accuracy, positive predictive and negative predictive values, and Matthew's correlation coefficient (MCC). They are computed as shown below. .- Sensitivity: $$ \frac{TP}{TP+FN} $$ .- Specificity: $$ \frac{TN}{TN+FP} $$ .- Accuracy: $$ \frac{TP+TN}{TP+FP+TN+FN} $$ .- Positive predictive value (PPV): $$ \frac{TP}{TP+FP} $$ .- Negative predictive value (NPV): $$ \frac{TN}{TN+FN} $$ .- Matthews Correlation Coefficient: $$ \frac{TP\cdotp TN-FP\cdotp FN}{\sqrt{\left(TP+FN\right)\cdotp \left(TN+FP\right)\cdotp \left(TP+FP\right)\cdotp \left(TN+FN\right)}} $$ In all the previous equations: TP and FN are the numbers of correctly and incorrectly identified pathological variants, respectively; TN and FP are the numbers of correctly and incorrectly identified neutral variants, respectively. The values of these parameters are provided in Additional files 5, 6: Tables S1, S2 (including also the corresponding TN, TP, FN, FP) for VS228; Additional file 7: Table S3, for VS2168. For simplicity, our analyses focus on the values of the MCC, but comparable results are obtained using accuracy (Additional files 8, 9, 10: Figures S3, S4, S5). External predictors Application of the coincidence rule requires a minimum of two pathogenicity predictors. In our case we tried all possible combinations of the following five tools: PolyPhen-2 [30], SIFT [31], PON-P2 [32], MutationTaster2 [33] and CADD [34]. We chose them because their results are provided by software suites broadly used in the annotation of sequencing results in the clinical setting: SIFT, PolyPhen-2, CADD and MutationTaster2 are in ANNOVAR [35]; SIFT, PolyPhen-2 (after submission) and MutationTaster are in Alamut (http://www.interactive-biosoftware.com/doc/alamut-visual/2.9/missense-pred.html), SIFT and PolyPhen are included in Illumina's Variant Studio software (http://support.illumina.com/downloads/variantstudio_userguide.html). PON-P2 is not included in none of them, but it was added because of its top-ranking performance relative to other predictors [15]. PolyPhen-2 (v2.2.2) was run locally with default parameters. SIFT and PON-P2 were run online (at http://sift.jcvi.org and http://structure.bmc.lu.se/PON-P2/, respectively). MutationTaster2 (http://www.mutationtaster.org) and CADD (http://cadd.gs.washington.edu) predictions were obtained using ANNOVAR [35]. The coverage of MutationTaster2, nor CADD tends to be lower than that of other methods because these two programs do not give predictions for amino acid substitutions resulting from more than one nucleotide change. We also compared the performance of our method with that of two well-established consensus methods Condel [36] and CAROL [37]. We chose them because they build their consensus utilizing a minimum number of tools: Condel combines FATHMM [38] and MutationAssessor [39]; CAROL combines PolyPhen and SIFT. This makes them a good baseline for the performance of our approaches, which in their simpler form also combine two reference predictors. In the case of CONDEL the predictions were retrieved from the file 'fannsdb.tsv.gz', available for download at the website http://bg.upf.edu/fannsdb/. For CAROL run locally the R version of the program, downloaded from its website at the Sanger Institute (http://www.sanger.ac.uk/science/tools/carol). Our goal is to test whether pathogenicity predictors with improved performance can be obtained for variants for which known methods do not agree in their predictions (these variants will be considered as pathogenic or neutral, depending on the method). The next two sections correspond to the two main steps followed to address this problem: (i) application of different versions of the coincidence rule for building the variant dataset; and (ii) development of the predictors. In a third and final section we describe what would be the overall state of the prediction problem, when considering together the cases that follow and the cases that break the coincidence rule. NOTE. The results of this work apply to any single amino acid replacement, irrespective of whether it is the result of a single nucleotide change or not. These results remain essentially the same, except from minor variations that do not affect our conclusions, when we restrict our analyses to those variants resulting from a single nucleotide replacement only (Additional files 11, 12, 13, 14: Figures S6, S7, S8, S9). Applying the coincidence rule to build the variant dataset To obtain the variant dataset for deriving our predictors we followed a simple protocol (Additional file 1: Figure S1) in which we first retrieved a total of 59,442 variants (15,723 pathogenic and 43,719 neutral variants, see Methods) distributed over a total of 228 proteins. Then, a set of known pathogenicity predictors was applied to these variants, keeping only those for which the predictors disagreed: these constituted our variant dataset, which we called PRDIS. Looking at this protocol, we see that each combination of pathogenicity predictors will give a different dataset. In this work, we have tried all possible combinations of five reference methods: SIFT, PolyPhen-2, CADD, PON-P2 and MutationTaster2. For example, for the case of two predictors, we produced a variant dataset for each of the following options: SIFT-PON-2, SIFT-PolyPhen-2, SIFT-CADD, SIFT-MutationTaster2, PON-P2-CADD, PON-P2-PolyPhen-2, etc. This resulted in a total of 26 PRDIS datasets. The first thing we observe during this process is that part of the 59,442 initial variants are discarded because predictions are not provided by all the methods for all variants (Fig. 1a; Additional file 5: Table S1). For example, there are only 57,349 (96% of the total of variants) instances for which the two predictors in the SIFT-PolyPhen (HDIV version) combination give an output; this number drops to 32,741 (55% of the total of variants) for the SIFT-MutationTaster2. These numbers reflect the original coverage of the reference methods. For example, SIFT, PolyPhen-2 (HDIV version) and MutationTaster2 generate results for 97%, 99% and 55% of the initial variants, respectively. It is then to be expected that the SIFT-PolyPhen-2 (HDIV version) combination gives more predictions than the SIFT-MutationTaster2 one. We also notice (Fig. 1b; Additional file 6: Table S2) that for those variants that pass the first step, there is an important percentage of cases for which predictors disagree, varying, for example, between 10% and 35% for the combinations of two predictors. For the remaining PRDIS datasets, the total number of mutations is large enough to suport the development of pathogenicity predictors; e.g. the combination of SIFT, PolyPhen (HDIV version), PON-P2, CADD and MutationTaster2 gives a PRDIS set with 5815 variants. As a reference, protein-specific predictors have been developed with variant datasets with 50/50 neutral/pathogenic instances [15]. Statistics for the variant datasets in this study. a Percentage of cases that entered the study. The X-axis corresponds to the number of reference methods combined; each point corresponds to a specific combination of reference predictors (a slight offset is used for clarity purposes). b Composition of the PRDIS sets built from the combination of two reference predictors only. Each of the lines (percentage of agreements and disagreements to the left and right, respectively) corresponds to a point in (B), at x = 2 We checked the success rate of the coincidence rule for the variants for which the combined predictors agreed (Additional file 1: Figure S1). We found that using this rule always gives better results than using the predictors alone (Fig. 2a): it has the ability to select the subset of predictions, from a given method, that are more accurately predicted. For example, in the case of PolyPhen-2 (HDIV version) the individual MCC is 0.57, while that of the SIFT-PolyPhen-2 (HDIV version) is 0.70. For PON-P2 the individual MCC is 0.70, while that of its combination with MutationTaster2 is 0.79. We also see that increasing the number of methods results in better success rates, although the trend is asymptotic (Fig. 2b). Success rate of predictions obtained following the coincidence rule. In the coincidence rule (see main text) computational information is accepted as supporting evidence in clinical settings only when the pathogenicity predictions of different methods agree. Here we describe how the success rate of this rule depends on the chosen in silico predictors. a Violin plots for the Matthews Correlation Coefficients (MCC) grouped by method. Each violin plot corresponds to all possible combinations of reference predictors that include the method shown at the bottom. For example, the first plot to the left represents all combinations of five reference predictors (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2) that include MutationTaster2. The thick lines at the bottom of each violin plot represent the individual performance of the reference predictors. b Dependence of MCC values on the number of predictors used to implement the coincidence rule Building specific tools for the variants with discordant predictions (PRDIS) This section is divided into four subsections. In the first one ("Obtaining predictors..."), we show that we can obtain non-random predictors for variants in PRDIS. The remaining three subsections ("Can specific predictors outperform reference..."; "Can specific predictors outperform simple..."; "Testing the reach...") are devoted to compare the performance of these specific predictors with that of (i) reference tools (PolyPhen-2, SIFT, PON-P2, CADD and MutationTaster2), (ii) with that of consensus tools (Condel and CAROL), and (iii) extending the main conclusion to proteins outside VS228. Obtaining specific predictors for PRDIS For each PRDIS dataset we derived a set of four specific predictors (Additional file 2: Figure S2). They correspond to the different combinations of the following options: two possible inputs and two models of different complexity. The two inputs were: (i) a simple one, having only the prediction scores from the reference methods; and (ii) and extended version of the simple input augmented with three additional properties (Blosum62 matrix elements, Shannon's entropy and the position-specific scoring matrix elements). The two complexity levels for the models were: a neural network with no hidden layers and one with one hidden layer and two nodes. The performance figures are the average of 10 replicas of the 5-fold cross validation process, to smooth out fluctuations. Our results show (Fig. 3) that the vast majority of the specific predictors have performances above those of a random method. That is, there is a signal in PRDIS allowing the discrimination between pathogenic and neutral mutations; this signal can be recognized with the variant features employed in this work. Performance of the PRDIS specific predictors. a and c. Frequency distribution of MCC values for all the specific predictors generated in this work: (a) data for simple neural networks; (c) data for neural networks with one hidden layer and two nodes. Shown with a dashed line is 0, the MCC value for a random predictor. We see that specific predictors are systematically better than the random predictor. b and d. Contribution of the three biochemical/biophysical properties (Blosum62 elements, Shannon's entropy and Position specific scoring matrix elements; see Materials and Methods) to improve the performance of the specific predictors. Points above the dotted line correspond to cases where use of these properties improves the performance of a specific predictor. We see that this is essentially always the case. b and d correspond to the simpler and to the one hidden layer neural networks, respectively We also observe bimodality in the MCC distributions (Fig. 3a and c); the peaks at high and low MCC values predominantly correspond to methods using the extended and the reduced inputs, respectively. This is in agreement with our previous experience where the use of biochemical/biophysical features allowed us to resolve a contradiction between SIFT and PolyPhen-2 predictions for variant F367 V in FOXP3 [40]. These results remain essentially unchanged whether we use the simple (Fig. 3a and b) or the complex neural network model (Fig. 3c and b). Can specific predictors outperform reference (SIFT, PolyPhen-2, PON-P2, CADD, MutationTaster2) methods? In Fig. 4, for each reference predictor (SIFT, PON-P2, etc) we plot both its performance (MCC) distribution (black boxplot) and that of the specific predictors that include its score among their input attributes (color violin plot). The first thing we notice is that here the performance of the reference predictors is lower than for the case of variants with concordant predictions (Fig. 2a). The same happens when we consider specific tools instead. For these, the upper-bounds of the MCC distributions are between 0.6 and 0.7 (Fig. 4b), while for the consistency rule MCCs values can reach 0.9 (Fig. 2b). With the lower-bounds we see a similar effect. For example, for specific predictors involving MutationTaster2 and CADD the lower-bounds are around 0.2; for applications of the consistency rule involving these two predictors, the values are above 0.6. Overall, this indicates that the problem of discriminating between neutral and pathogenic variants is harder for PRDIS than for non-PRDIS variants. The contribution of reference methods to PRDIS specific predictors. In (a) we compare the performance of PRDIS specific methods, represented with violin plots with that of the reference methods (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2), represented with black boxplots. We see that specific methods are frequently better than reference methods, but there is an increasing overlap between both approaches as the performance of the reference method grows (e.g. in the cases of PON-P2 or PolyPhen-2). b Performance depends on the number of reference predictors combined: the more we use, the more likely we are to obtain higher performances We also observe how the performances (MCC) of the reference and specific predictors are related. In particular, we see that when the success rate of the reference predictor is high, the same happens with that of the derived specific predictors. For PON-P2, the method with the highest success rates in this work, specific's MCCs concentrate near 0.65; for the next performer, PolyPhen-2 (HDIV version), specific's MCCs show a shift towards lower values; and so on. We also find that as the individual performances of the reference methods drop (when the black boxplots move towards 0 in Fig. 4a) the difference between specific and reference predictors grows. In summary, the better the performance of the reference method is, the more it resembles that of its related specific predictors. We have seen that prediction of PRDIS variants is a hard problem and that specific tools provide a promising approach to its solution. In this context, a natural question is: has the performance of specific tools in PRDIS variants (Additional file 6: Table S2) reached that of reference methods in average variants (Additional file 5: Table S1)? Our results indicate that, in general, this is not yet the case. The differences between reference methods do not generally alter this conclusion; there is, however, some variability that depends on the parameters considered. More precisely, for MCC we see that SIFT is above 39 out of 57 (68%) specific predictors, PolyPhen-2 (HDIV version) is above 75%, CADD is above 88%, and PON-P2 is above 100%. If we turn to variant-specific parameters, like sensitivity (pathogenic) and specificity (neutral) we find that for sensitivity, all reference methods are above all specific predictors, except for SIFT, which is above 72% of them. For specificity, the situation is somewhat reversed. The specificity of MutationTaster2 for the average variant, 0.47, is below that of all specific tools in PRDIS variants; in our dataset, this method shows a prediction bias towards pathogenicity. This bias is also present in the other reference methods, which show specificities below their sensitivities. However, the difference with specific methods becomes gradually smaller, from PolyPhen-2 (HDIV version), which is above 14%, to PON-P2, which is above 56%. The other variant-related parameters (PPV and NPV) are also of interest; however, they have a high dependency on the sample composition that makes difficult the comparison. Having said that, for PPV we see that reference methods, when applied to the average variant, outperform specific methods, when applied to PRDIS variants, in different degrees: MutationTaster2 is above 39%, PolyPhen-2 (HDIV version) is above 70%, SIFT is above 77%, CADD is above 86%, and PON-P2 is above 100%. Given that the sample effect is unclear in this case, we also give (Additional file 15: Figure S10) the comparison of PPV values when applying to PRDIS variants both reference and specific methods. We find that the latter clearly outperforms the former. On the basis of both results we believe that for PPV there is a complementary situation where both approaches mutually outperform each other; however, we cannot go any further, given the sample differences. In summary, the overall view is that the performance of specific methods in the hard problem of PRDIS variants has not yet reached that of reference methods in the problem of average variants. Consequently, the success rates of specific methods are still below the levels above which bioinformatics evidence is considered as supporting evidence in the clinical setting [11]. It must be noted that the size of the PRDIS sets varies gradually, increasing as we add more predictors (Fig. 5). Coverage of the specific predictors. The number of variants used to obtain specific predictors grows as we increase the number of standard methods used to build the PRDIS set. This is to be expected, since the more methods we use, the easier it is to find a discordance between predictions Can specific predictors outperform simple consensus (Condel, CAROL) methods? As we have seen before, our specific predictors are obtained using as input the score of reference predictors (enriched, in some cases, with other features). In this sense they are similar to consensus methods [12, 13, 37, 41], which also use the output of known predictors as their input. Here we compare PRDIS specific predictors with Condel and CAROL. These two methods constitute an interesting reference since, in spite of their good performance, they are technically simple: they utilize a minimum number of known predictors to build their consensus, (MutationAssessor, FATHMM) for Condel and (PolyPhen, SIFT) for CAROL. We see (Fig. 6) that PRDIS specific predictors outperform always Condel and almost always CAROL. This indicates that using PRDIS data for developing specific predictors is a good option relative to the technically simple (but powerful) predictors such as Condel and CAROL. Comparison between the performance of PRDIS specific and conventional consensus predictors. We represent the MCC of PRDIS specific predictors (Y-axis) against that of conventional consensus methods (Condel and CAROL; X-axis). Points above the diagonal indicate that the former tend to outperform the latter, for PRDIS variants. We see that this is generally the case, although with a trend in the performance of CAROL predictions to reach the level of specific methods Testing the reach of the specific approach The specific approach presented here is based on identifying the variants that do not follow the coincidence rule and train predictors specific for them. In Results section "Can specific predictors outperform reference (SIFT, PolyPhen-2, PON-P2, CADD, MutationTaster2) methods?" we have seen (Fig. 4), using a standard cross-validation scheme, that this approach generally outperforms reference predictors (PolyPhen-2, SIFT, etc.) for variants in VS228. To test if this conclusion also holds for proteins not represented in VS228, we applied our specific models to PRDIS sets obtained from VS2168. It is important to note that VS228 and VS2168 contain variants from different proteins. That is, proteins contribute variants either to one set or the other, but not to both. In Fig. 7, which is analogous to Fig. 4, for each reference predictor (SIFT, PON-P2, etc.) we plot both its performance distribution (black boxplot) and that of the specific predictors that include its score among their input attributes (color violin plot). We see that, apart from an overall trend towards lower success rates, the results are comparable to those obtained for VS228: specific predictors tend to outperform reference predictors and, as the performance of the latter improves, the difference between approaches decreases. The relationship between PRDIS specific predictors and reference methods for proteins in VS2168 dataset. In this figure we compare the performance of PRDIS specific methods when applied to the variants in VS2168. None of the proteins represented in this set contributes a variant to VS228, which is the dataset used to train the specific predictors and obtain a cross-validated estimate of their performance (Fig. 4). The MCCs of the specific methods are represented with violin plots and those of the reference methods (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2) are represented with black boxplots. We see that, in spite an overall decrease in performance for all tools displayed, specific methods are frequently better than reference methods, but there is an increasing overlap between both approaches as the performance of the reference method grows (e.g. in the cases of PON-P2 or PolyPhen-2). This result confirms the conclusion obtained with the VS228 set (Fig. 4) Given that the overall success rates for VS2168 have decreased, these results do not affect the previous observation according to which the performance of reference methods applied to the average variant is higher than that of specific predictors applied to PRDIS variants. How good is the combination of the coincidence rule and PRDIS specific predictors? Combined use of the coincidence rule and PRDIS specific predictors results in a hybrid method that can produce predictions for the major part of the variant dataset (Additional file 16: Figure S11). There is one hybrid method for each combination of reference predictors; for example, when our reference methods are SIFT and PolyPhen-2, we will have one associated coincidence rule and one PRDIS specific predictor. We see (Fig. 8) that, when applying the hybrid approach to the original dataset of variants, most of the hybrid methods have performances higher than those of the reference methods (estimated on the same dataset). For example, MutationTaster2 is outperformed by all hybrid methods while, at the other end of the scale, PON-P2 outperforms 50% of the hybrid methods. This is not related to coverage (percentage of variants predicted), since both MutationTaster2 and PON-P2 have very similar values, 55% and 51%, respectively. Detailed performance results are provided in Additional file 17: Table S4. Performance of the hybrid predictor. For each possible hybrid predictor in this work, we computed its MCC. In the figure we show the frequency histogram of these values. With dashed lines we show the prediction of the different reference methods, estimated on the original set of 59,442 variants. We see that hybrid methods tend to outperform reference methods, although this depends on the latter. For example, PON-P2 alone is better than many of the hybrids In the last years the use of computational evidence for the identification of pathogenic sequence variants in the clinical setting is being gradually reconsidered [6,7,8]. However, given their still limited accuracy, the unrestricted use of pathogenicity predictors is not advised [42]. This idea has taken a more precise shape in the ACMG/AMP guidelines for variant interpretation [11], where computational results are considered as supporting evidence only when the tools used to generate them agree (what we call consistency rule in this work). Otherwise, computational data are rejected. Seeking agreement between methods is a natural approach to enhance our prediction ability and is particularly valuable when several (partial) solutions to the same problem are available [43]. For the case of pathogenicity predictions this approach has also been tried. For example, using a small set proteins Chan et al. [44] find that taking the consensus of four prediction tools (naive use of Blosum62, SIFT, PolyPhen and A-GVGD) results in an increased predictive value, although at the price of a substantial reduction in the number of predictions. In general, it is accepted that this approach may produce detectable improvements over the use of single methods [10, 12, 13, 44], although combining tools may have its problems [12]. In our case, we observe that simple application of the consistency rule to our variant dataset (Additional file 1: Figure S1) also results in high success rates (Fig. 2b), better than those of the reference methods employed to implement the rule. However, there is a percentage of cases -considered to be hard to predict by Capriotti et al. [13], for which reference predictors disagree and consequently computational evidence should be discarded in a medical environment [11]. These cases represent about 10% to 45% of the total number of variants (Additional file 5: Table S1) and their prediction constitutes the main goal of our work. In particular, we have explored whether by focusing our efforts on these cases we can derive specific predictors outperforming known methods. We have tested this idea on VS228, a set of 59,442 variants spread over 228 proteins of medical interest. To this end we have trained a series of neural network predictors (Additional file 2: Figure S2), trying two different inputs and two different complexity levels, and estimated their performance using a 5-fold cross-validation procedure. Our results indicate that indeed using this specific approach gives tools with increased success rates, which are better than those of the reference (Fig. 4; SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2) and consensus (Fig. 6; Condel, CAROL) methods considered. We also observe that the overall performance of PRDIS specific tools (Fig. 4a) is still below that obtained for variants for which predictors agree in their predictions (Fig. 2a). This reflects the gap described by Capriotti et al., [13] between easier and harder cases. However, the simplicity of our models suggests that there is still room for the development of models that can close this gap. And, even at this early stage, specific tools can already be useful. For example, let us consider the following variants: Y482C, in ATP-binding cassette sub-family A member 1, which causes High-density lipoprotein deficiency; Y72C, in Hypoxanthine-guanine phosphoribosyltransferase, which causes hyperuricaemia and chronic tophaceous gout, and W453R, in Cytochrome b-245 heavy chain, which causes X-linked Chronic Granulomatous disease. The three variants are correctly predicted by SIFT, PON-P2, CADD and MutationTaster2, but are missed by PolyPhen-2 (HDIV version). Our specific method that uses the five scores as input features correctly identifies the variants as pathogenic (scores: 0.67, 0.62 and 0.57; all above 0.5). In addition, if our tool also includes the three biological features as part of the input, the reliability of the predictions is higher (scores: 0.82, 0.92 and 0.77; all above 0.5). Apart from showing the potential of the PRDIS specific predictors, this example can be used to understand why sometimes predictions by PolyPhen-2 are in contradiction with those from the other methods. A detailed analysis of PolyPhen-2's MSAs shows that, for the three variants considered, the pathogenic amino acid appears once in the column of the mutated amino acid, in a non-human species: for Y482C the cysteine is present in S. harrisii, for Y72C the cysteine is present in P. tricornutum, and in W453R the tryptophan appears in R. norvegicus. Since the score of PolyPhen-2 takes into account this fact, this could explain the deviating prediction. We reran PolyPhen-2 after eliminating the affected sequences from the MSAs and the three variants were now correctly predicted as pathological, in accordance with the other reference methods. We had previously found a similar situation in a FOXP3 variant, when integrating PolyPhen-2, SIFT and structural evidence [40]. We have extended the validity of our principal conclusion applying our trained predictors to the 322,270 variants in VS2168, which are distributed over 2168 proteins not represented in VS228. Our results (Fig. 7) indicate that, in spite an overall decrease in success rate, the main conclusion of this work holds: specific predictors tend to outperform reference methods. Partitioning the variant space and focusing on the hardest problems Methodologically, the approach presented is based on the idea of partitioning the dataset of variants according to a given criterion and then derive a specific predictor for some, or for all, of the resulting subsets. The underlying rationale is that the partitioning step may give improved prediction tools either because the resulting subsets are more homogeneous or because it allows us to put our efforts on tackling the more difficult parts of the prediction problem. The development of protein-specific predictors [15] corresponds to the first situation: every subset is constituted by variants from a single protein. The specific predictors show good performances relative to non-specific methods (e.g. PolyPhen-2, CADD, etc) although not always (in many cases PON-P2 outperforms the protein-specific methods). This may be due to different factors, for example the new prediction problem defined by the data in the protein-specific subset may require also an adaptation of the model, e.g. including specific terms for the protein. This is for example what has been recently done by [45] for KinMutRF, their pathogenicity predictor for kinases; in this tool the authors employ kinase-specific features in their input, such as specific Gene Ontology terms. Our work corresponds to the second case, in which partitioning through application of the coincidence rule separates variants "easy" to predict from those that are harder to predict, which are those for which known methods disagree (PRDIS in our case). This difficulty gap has been already mentioned by Capriotti et al. [13] who describe how their consensus predictor Meta-SNP performs much better for those cases for which their four constituting predictors PANTHER, PhD-SNP, SIFT and SNAP agreed in their verdict than for those where they disagreed. Here we have shown that developing specific predictors for this hard case benefits our performance for PRDIS and improves overall prediction performance (Fig. 8). It is worth noting, however, that improvement size varies depending on the performance of the reference predictors, a trend already observed in the case of protein-specific predictors. That is, when the performance of the reference predictor is high (e.g. like in the case of PON-P2), it is more difficult to obtain outperforming specific predictors (Fig. 4). In the clinical setting, the use of computational evidence on variant pathogenicity is restricted to those cases where there is a full coincidence between in silico tools (see ACMG/AMP guidelines [11]). This coincidence rule results in a loss of information for a percentage of variants that varies between 10% and 35%, when combining two predictors. In this work, we have focused on the development of specific tools for these variants and on testing whether we can obtain better success rates than known methods. We find that this is indeed the case, although some existing methods (PON-P2 and PolyPhen-2) already give a competitive performance (with varying coverages) that is more difficult to improve. Matthews Correlation Coefficient PPV and NPV: positive and negative predictive values. 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BMC Genomics. 2016;17 This work has been supported by the spanish Ministerio de Economía y Competitividad (BIO2012–40133; SAF2016–80255-R). It has also been supported, and the publication costs have been defrayed, by the European Regional Development Fund (ERDF), through the Interreg V-A Spain-France-Andorra programme (POCTEFA 2014–2020), research grant PIREPRED (EFA086/15). We provide as Additional files all the pathological [Additional file 3] and neutral variants [Additional file 4] of the VS228 dataset, which constitutes the core of this work (used in all figures, except Fig. 7). These files also contain, for dataset VS2168 (used only in Fig. 7), the neutral variants (obtained from our homology-based model) and the pathogenic variants not retrieved from HGMD professional (access to this repository is restricted to subscribers). We also provide as Additional files four tables (Additional files 5, 6, 7, 17: Tables S1, S2, S3, S4) with the summary performances for the predictors presented in this work. Research Unit in Translational Bioinformatics, Vall d'Hebron Institute of Research (VHIR), Universitat Autònoma de Barcelona, Barcelona, Spain Elena Álvarez de la Campa, Natàlia Padilla & Xavier de la Cruz Department of Molecular Genomics, Instituto de Biología Molecular de Barcelona (IBMB), Consejo Superior de Investigaciones Científicas (CSIC), Barcelona, Spain Elena Álvarez de la Campa ICREA, Barcelona, Spain Xavier de la Cruz Natàlia Padilla EC did most of the technical work (developing predictors, executing known predictors, etc) required to generate and present (in the form of Figures and Tables) the data provided in this article. NP generated part of the data and of the data analyses. XC conceived the study, analyzed the data and wrote the article. All authors read and approved the final manuscript. Correspondence to Xavier de la Cruz. The authors declare that they have no competing of interests. Additional file 1: Figure S1. Obtention of the variant datasets. The figure shows how we obtained the subsets of variants for which pathogenicity predictors disagreed (PRDIS, within the red contour) and agreed (within the blue contour), respectively. For a certain percentage of cases, some predictors would not give a prediction for the variables (indicated as "No output for predictor(s)"). The original set of protein sequence variants was obtained from (see Materials and Methods): (i) UniProt database, for pathogenic variants; (ii) a homology-based model, for neutral variants. (PNG 673 kb) Obtention of specific predictors for PRDIS variants. For each combination of the five reference methods used in this work (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2) we obtained PRDIS, the subset of those variants for which the reference predictors disagreed. Then, for each of these PRDIS sets, we produced four different predictors, which differed either in the neural network model or in the neural network input. For the neural network model we tried two options: (i) no hidden layers (NN: 0); and (ii) one hidden with two nodes (NN: 2). For the neural network inputs, we tried two options: (i) the scores of the reference predictors; and (ii) the scores of the reference predictors enriched with three biological features (Blosum62 matrix elements, Shannon's entropy, Position-specific scoring matrix elements; see Materials and Methods). Boxed in red is the case where PRDIS was obtained using SIFT and PolyPhen-2 as reference methods. (PNG 666 kb) Pathogenic variants. Each line corresponds to a variant, providing: the amino acid replacement and its location in the protein sequence, the UniProt code for the protein, the values of the contribution of the three biochemical/biophysical properties (Blosum62 elements, position specific scoring matrix elements and Shannon's entropy) followed by the output of the pathogenicity predictions for the reference methods used in this work (for PolyPhen-2 we give the output of its two versions –HDIV and HVAR- although in this work we only used HDIV predictions), and '?' is given when no output was provided by the method. The last column gives the dataset where the variant belongs, either VS228 or VS2168. (CSV 1616 kb) Neutral variants. Each line corresponds to a variant, providing: the amino acid replacement and its location in the protein sequence, the UniProt code for the protein, the values of the contribution of the three biochemical/biophysical properties (Blosum62 elements, position specific scoring matrix elements and Shannon's entropy) followed by the output of the pathogenicity predictions for the reference methods used in this work (for PolyPhen-2 we give the output of its two versions –HDIV and HVAR- although in this work we only used HDIV predictions), and '?' is given when no output was provided by the method. The last column gives the dataset where the variant belongs, either VS228 or VS2168. (ZIP 5531 kb) Success rate of the coincidence rule, for the all the different combinations of reference predictors (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2). The performance measures are the six standard measures (MCC, accuracy, sensitivity, specificity, PPV and NPV) described in the Materials and Methods section. We give: the raw TP, TN, FP and FN values; the coverage relative to the original dataset of 59,442 variants (VS228) and the number of cases where the predictors coincide. (PDF 28 kb) Prediction performance for the PRDIS specific predictors in this work for VS228; each corresponds to a different combination of the reference predictors (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2). The performance measures are the six standard measures (MCC, accuracy, sensitivity, specificity, PPV and NPV) described in the Materials and Methods section. We also give: the total number and the percentage of cases, and the raw TP, TN, FP and FN values. (PDF 26 kb) Prediction performance for the PRDIS specific predictors in this work for VS2168 dataset; each corresponds to a different combination of the reference predictors (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2). The performance measures are the six standard measures (MCC, accuracy, sensitivity, specificity, PPV and NPV) described in the Materials and Methods section. We also give: the total number and the percentage of cases, and the raw TP, TN, FP and FN values. (PDF 26 kb) In the coincidence rule (see main text) computational information is accepted as supporting evidence in clinical settings only when the pathogenicity predictions of different methods agree. Here we describe how the success rate of this rule depends on the chosen in silico predictors. (A) Violin plots for the Accuracy grouped by method. Each violin plot corresponds to all possible combinations of reference predictors that include the method shown at the bottom. For example, the first plot to the left represents all combinations of five reference predictors (SIFT, PolyPhen-2, PON-P2, CADD and Mutation Taster2) that include MutationTaster2. (B) Dependence of Accuracy values on the number of predictors used to implement the coincidence rule. (PNG 135 kb) (A) and (C). Frequency distribution of accuracy values for all the specific predictors generated in this work: (A) data for simple neural networks; (C) data for neural networks with one hidden layer and two nodes. Shown with a dashed line is 0.5, the accuracy value for a random predictor. We see that specific predictors are systematically better than the random predictor. (B) and (D). Contribution of the three biochemical/biophysical properties (Blosum62 elements, Shannon's entropy and Position specific scoring matrix elements; see Materials and Methods) to improve the performance of the specific predictors. Points above the dotted line correspond to cases where use of these properties improves the performance of a specific predictor. We see that this is essentially always the case. (B) and (D) correspond to the simpler and to the one hidden layer neural networks, respectively. (PNG 194 kb) Additional file 10: Figure S5. In (A) we compare the performance of PRDIS specific methods, represented with violin plots with that of the reference methods (SIFT, PolyPhen-2, PON-P2, CADD and Mutation Taster2), represented with black boxplots. We see that specific methods are frequently better than reference methods, but there is an increasing overlap between both approaches as the performance of the reference method grows (e.g. in the cases of PON-P2 or PolyPhen-2). (B) Performance depends on the number of reference predictors used: the more predictors are used, the more likely to obtain higher performances. (PNG 219 kb) The results in this figure are computed for the subset of amino acid variants resulting from single nucleotide replacements only. (A) Percentage of cases that entered the study. The X-axis corresponds to the number of reference methods combined; each point corresponds to a specific combination of reference predictors (a slight offset is used for clarity purposes). (B) Composition of the PRDIS sets built from the combination of two reference predictors only. Each of the lines (percentage of agreements and disagreements to the left and right, respectively) corresponds to a point in (B), at x = 2. (PNG 115 kb) The results in this figure are computed for the subset of amino acid variants resulting from single nucleotide replacements only. In the coincidence rule (see main text) computational information is accepted as supporting evidence in clinical settings only when the pathogenicity predictions of different methods agree. Here we describe how the success rate of this rule depends on the chosen in silico predictors. (A) Violin plots for the Matthews Correlation Coefficients (MCC) grouped by method. Each violin plot corresponds to all possible combinations of reference predictors that include the method shown at the bottom. For example, the first plot to the left represents all combinations of five reference predictors (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2) that include MutationTaster2. (B) Dependence of MCC values on the number of predictors used to implement the coincidence rule. (PNG 113 kb) The results in this figure are computed for the subset of amino acid variants resulting from single nucleotide replacements only. (A) and (C). Frequency distribution of MCC values for all the specific predictors generated in this work: (A) data for simple neural networks; (C) data for neural networks with one hidden layer and two nodes. Shown with a dashed line is 0, the MCC value for a random predictor. We see that specific predictors are systematically better than the random predictor. (B) and (D). Contribution of the three biochemical/biophysical properties (Blosum62 elements, Shannon's entropy and Position specific scoring matrix elements; see Materials and Methods) to improve the performance of the specific predictors. Points above the dotted line correspond to cases where use of these properties improves the performance of a specific predictor. We see that this is essentially always the case. (B) and (D) correspond to the simpler and to the one hidden layer neural networks, respectively. (PNG 172 kb) The results in this figure are computed for the subset of amino acid variants resulting from single nucleotide replacements only. In (A) we compare the performance of PRDIS specific methods, represented with violin plots with that of the reference methods (SIFT, PolyPhen-2, PON-P2, CADD and MutationTaster2), represented with black boxplots. We see that specific methods are frequently better than reference methods, but there is an increasing overlap between both approaches as the performance of the reference method grows (e.g. in the cases of PON-P2 or PolyPhen-2). (B) Performance depends on the number of reference predictors combined: the more we use, the more likely we are to obtain higher performances. (PNG 258 kb) Additional file 15: Figure S10. Comparison between PPV values for PRDIS specific and reference predictors. The figure shows that combination of reference methods (specific predictors) gives better PPV than reference methods alone: for only seven cases the reference approach outperformed the specific approach. (PNG 68 kb) A hybrid predictor. A hybrid method is implicitly defined if the coincidence rule is used as a pre-classification step. In this method, the variants for which standard methods agree will be assigned this coinciding prediction; for PRDIS variants, a prediction will be obtained from the PRDIS specific method. The final performance of this hybrid method is obtained by combining that of the two cases. (PNG 607 kb) Additional file 17: Table S4. Prediction performance for the hybrid predictor. We give the raw TP, TN, FP and FN values and the values of the six standard measures (MCC, accuracy, sensitivity, specificity, PPV and NPV) described in the Materials and Methods section. (PDF 24 kb) de la Campa, E.Á., Padilla, N. & de la Cruz, X. Development of pathogenicity predictors specific for variants that do not comply with clinical guidelines for the use of computational evidence. BMC Genomics 18, 569 (2017). https://doi.org/10.1186/s12864-017-3914-0 In silico pathogenicity predictors Protein sequence variants Missense variants	CommonCrawl
\begin{definition}[Definition:Cross-Ratio/Lines through Origin] Let $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$ be lines through the origin $O$ whose equations embedded in the Cartesian plane are as follows: {{begin-eqn}} {{eqn \| ll= \LL_1: \| l = y \| r = \lambda x }} {{eqn \| ll= \LL_2: \| l = y \| r = \mu x }} {{eqn \| ll= \LL_3: \| l = y \| r = \lambda' x }} {{eqn \| ll= \LL_4: \| l = y \| r = \mu' x }} {{end-eqn}} The '''cross-ratio''' of $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$, in that specific order, is defined and denoted: :$\tuple {\lambda \mu, \lambda', \mu'} := \dfrac {\paren {\lambda - \lambda'} \paren {\mu - \mu'} } {\paren {\lambda - \mu'} \paren {\mu - \lambda'} }$ \end{definition}	ProofWiki
Methodology article Patterns of cross-contamination in a multispecies population genomic project: detection, quantification, impact, and solutions Marion Ballenghien1,2, Nicolas Faivre1 & Nicolas Galtier1 Contamination is a well-known but often neglected problem in molecular biology. Here, we investigated the prevalence of cross-contamination among 446 samples from 116 distinct species of animals, which were processed in the same laboratory and subjected to subcontracted transcriptome sequencing. Using cytochrome oxidase 1 as a barcode, we identified a minimum of 782 events of between-species contamination, with approximately 80% of our samples being affected. An analysis of laboratory metadata revealed a strong effect of the sequencing center: nearly all the detected events of between-species contamination involved species that were sent the same day to the same company. We introduce new methods to address the amount of within-species, between-individual contamination, and to correct for this problem when calling genotypes from base read counts. We report evidence for pervasive within-species contamination in this data set, and show that classical population genomic statistics, such as synonymous diversity, the ratio of non-synonymous to synonymous diversity, inbreeding coefficient FIT, and Tajima's D, are sensitive to this problem to various extents. Control analyses suggest that our published results are probably robust to the problem of contamination. Recommendations on how to prevent or avoid contamination in large-scale population genomics/molecular ecology are provided based on this analysis. Contamination is a well-known and ancient problem in molecular biology research. Most people who have worked in a molecular biology laboratory for a while have at least once observed an extra band on a gel, or obtained DNA sequence data originating from an unexpected species. Projects involving a polymerase chain reaction (PCR) step are particularly sensitive to contamination because initially small amounts of foreign DNA can accidentally be amplified by PCR, and transferred from tube to tube. A number of published results obtained by Sanger sequencing were subsequently demonstrated to most likely result from laboratory contamination [1–4]. Massive sequencing projects and databases based on next-generation sequencing (NGS) technologies are far from immune from contamination issues [5–8]. The problem is perhaps even exacerbated, with contaminant sequence reads being lost in the myriad of reads from the target sample, and therefore difficult to detect and clean out. The concern is particularly serious when the target sample is from a non-model species lacking a reference genome, so that genuine sequence reads cannot be easily identified by similarity. It should be noted that many NGS library construction protocols involve one or multiple PCR amplification steps that generate elevated concentrations of DNA, thereby increasing the risk of contamination. Contamination is a well-identified problem in projects targeting very small amounts of DNA, such as ancient DNA projects [9, 10] and low-frequency variant analysis [11–13], in which small amounts of contamination can be sufficient to confound the results. A couple of recent studies, however, have revealed that both cross-contamination [14] and environmental contamination [15] can cause serious problems even in standard NGS projects, that is, when target DNA is a priori thought to be much more abundant than contaminant DNA [16]. These studies ring an alarm bell and call for a systematic examination of the prevalence of contamination in past, present and future NGS datasets. Recently, we conducted a multispecies population genomic project in which one to ten individuals from each of >100 non-model species of animals were subjected to RNA sequencing (RNAseq), leading to a number of scientific publications [17–30]. The co-occurrence in the same laboratory of many samples from many distinct species in a relatively short period of time provides an ideal situation for investigating the effect of cross-contamination in a molecular ecology project. Our goals in this study were multiple. First, we aimed to quantify the prevalence of cross-contamination, identifying at which steps of the experimental protocol it most often happens, and if possible delivering guidelines on how to avoid it. Second, we wanted to check the robustness of our published results to the problem of contamination, and if possible identify solutions to this problem. Two distinct, complementary approaches were taken. Regarding between-species contamination, cytochrome oxidase 1 (cox1) was used to detect the occurrence of foreign cDNA sequences in a sample and trace their likely sources. cox1 is a high-expressed gene, and is therefore expectedly prevalent in RNAseq data. It is the standard DNA barcoding tool in animals, so a huge database of cox1 reference sequences from many distinct species of animals is available. Regarding within-species contamination, patterns of read counts were analyzed to search for evidence of allele leakage across individuals. The inferred patterns of contamination across individuals and species were considered in the light of laboratory metadata – dates of entry and processing of samples in the laboratory, identity of technicians in charge of the samples, date of shipment to sequencing center, identity of sequencing center, flowcell number, and lane number. A modified single nucleotide polymorphism (SNP)-calling method that accounts for among-individual contamination was introduced and a re-analysis of our main published results was conducted. Project overview and protocols European Research Council project 232971 "PopPhyl" took place at the Institute of Evolutionary Sciences Montpellier, France, from June 2009 to December 2014. During this period, samples from >3800 distinct individuals of 180 species from eight phyla of animals entered the laboratory located in building 32 of University Montpellier, France. Samples were either collected by ourselves in the field or shipped by colleagues in RNAlater® (Qiagen, Dusseldorf, Germany) buffer. A fraction of the samples were barcoded after DNA extraction and cox1 amplification. Roughly 1200 samples were subjected to RNA isolation following standard or modified protocols [31]. The quality and quantity of extracted RNA were assessed using spectrophotometry and capillary electrophoresis. Total RNA from 446 of these samples was sent out for Illumina sequencing on either a Genome Analyzer II (2009–2010) or a HiSeq 2000 (2011–2014). Illumina library construction, DNA fragment tagging, pooling, and demultiplexing were achieved in the sequencing centers. Short-read data were returned to us as one or several FASTQ files per sample and we performed the downstream bioinformatic analyses [18]. No more than one sample per individual was sent out for sequencing. The individual samples that were sent out for sequencing belonged to 116 distinct species. Sixty-three additional species were subjected to RNA extraction but not sent out for Illumina sequencing. In our laboratory, eight distinct persons, referred to below as "technicians," processed the samples. Two technicians collectively processed ~80% of the samples. These two technicians, and the majority of the other technicians involved, were 100% dedicated to the project and did not (or very rarely) manipulate biological material coming from species not included in the project. In 141 species, the same technician processed all the samples, whereas in 39 species, two distinct technicians were involved. All samples were processed at the same laboratory bench, in a room almost entirely dedicated to the project, with specific materials shared by the involved technicians. Samples were sent to sequencing centers in dry ice at 15 distinct dates, from September 23, 2009, to February 13, 2014. Shipments typically involved several individuals from several distinct species. In each shipment, samples were contained in separate, labeled tubes that were gathered in a single box and accompanied by a form briefly describing the label and content of each tube. Tubes in boxes were organized by species and ordered consistently with the form. When more than one technician was involved in a shipment, tubes were ordered by technician, that is, samples processed by technician 1 first, then samples processed by technician 2. Tubes were not opened at shipment stage: they were simply taken out from freezers, packed, and transferred to the carrier. Samples were sent to three distinct sequencing centers, of which one (SC1) processed ~85% of the samples. The dates of first entry in the laboratory, first and last experiment in the laboratory, and shipment(s) were recorded per species. This information is available in Additional file 1: Table S1. Details on experimental dates per sample are available on request. We also retrieved flowcell identifiers and lane numbers from read headers in FASTQ files – this information was absent from 88 of the files we received, though. We have no information on the dates of library construction and the identity of the technicians involved in library construction and sequencing. Short-read data sets We analyzed Illumina short-read data sets from 446 individuals of 116 species (1–11 individuals per species; Additional file 1: Table S1). Read length was 100 in 358 individuals (92 species), 75 in 12 individuals (three species), and 50 in 76 individuals (21 species). Single-end sequencing was ordered in all cases. Occasionally, sequencing centers still returned paired-end reads, which were treated as single-end reads in our analyses for the sake of homogeneity – meaning that both reads were treated as independent events. The total number of reads varied from 2.44 to 76 millions among samples. Samples sequenced later in the project typically received more reads than early-sequenced samples. Most of the generated data sets have been submitted to the National Center for Biotechnology Information (NCBI) Sequence Reads Archive (SRA) under bioprojects PRJNA230239, PRJNA249058, PRJNA268920, PRJNA278516, PRJNA322119, and PRJNA326910. We here publish data from an additional 27 individuals from 19 species, which were submitted to NCBI-SRA under bioproject PRJNA374528. A list of sequenced individuals with associated identifiers is provided in Additional file 2: Table S2. cox1 reference database We created a reference database of cox1 sequences for subsequent sequence similarity searches. This database had three components: a target species component, a companion species component, and a model species component. The target species component corresponds to cox1 sequences from the species that have been subjected to DNA barcoding and/or RNA extraction in our laboratory between 2009 and 2014. We automatically downloaded target species cox1 sequences from the Barcode Of Life Database [32] (May 6, 2016), accounting for taxonomic synonymy – we equated Myodes with Clethrionomys, Cervus with Rucervus, Parus with Cyanistes, Mellicta with Melitaea, Physa with Physella, Galba with Lymnaea, Lineus with Ramphogordius, and Abatus agassizi with Abatus agassizii. Sequences in the Barcode of Life Data System (bold) database are binned based on similarity. We kept a single cox1 sequence per bin per target species, maximizing sequence length and number of annotations – geographic origin, collector, sampling date, lifestyle, tissue, and existence of voucher. Sequences not assigned to a bin were excluded. A similarity search was performed by BLAST to the NCBI non-redundant (NR) database. Thirteen sequences not hitting any cox1 sequences from NR were removed or manually replaced. Some of our target species were not represented in the bold database. For these we performed a manual search in GenBank and retrieved additional cox1 sequences. The companion species component of our reference cox1 database corresponds to species that never entered our laboratory, but that are phylogenetically related to target species. For each genus of our target species sample, we identified a companion genus from the same family or same order in which cox1 sequences were available in a roughly equivalent number of species (Additional file 1: Table S1). The same automatic and cleaning procedure as described above for target species was applied to companion species, with the exception that we did not manually search GenBank for companion species. The companion species component was added as a negative control, that is, a measure of the prevalence of seemingly foreign cox1 sequences in our samples due to experimental noise, in the absence of contamination. The model species component of our reference cox1 database corresponds to species of animals that are frequently subjected to NGS projects, with which our samples might have been in contact at some point during the experimental protocol – including Homo sapiens. We selected the 20 species of animals with the largest number of entries in the NCBI-SRA database (March 15, 2016), and retrieved the complete cox1 sequence of each of these. cox1 sequences from the target species, companion species, and model species were aligned using MACSe [33]. A single segment of the cox1 sequence was selected, from position 6189 to position 6539 (revised Cambridge reference sequence). Sequences for which the segment was not entirely determined were discarded. For each genus of the target species component, a phylogenetic tree was reconstructed using PHYML in SEAVIEW [34] and inspected by eye. Obvious anomalies were corrected by removing the misplaced sequences, in light of existing taxon-specific literature when available. Our reference cox1 sequence alignment is provided as Additional file 3. Detection of between-species contamination Short-read Illumina data sets were cleaned for low-quality reads or read portions as described previously [35]. Reads were mapped to the reference cox1 database using Burrows-Wheeler Aligner (BWA) software [36]. Mapping tolerance was one mismatch for data sets in which read length was 50 and two mismatches for data sets in which read length was 75 or 100. The best scoring hit of each read, if any, was recorded – note that in case of equal mapping scores, BWA will randomly output one of the highest scoring hits. For each individual, the number of hits for each sequence of the reference database was recorded and normalized by total number of reads for the considered individual. The results were then summed up by species in order to calculate, for every pair of species (sp1, sp2), the total number of cox1 reads from sp1 mapping to a reference sequence from sp2. Species from the reference database between which cox1 divergence was <5% were considered as non-diagnostic, meaning that a hit of a sp1 read to a sp2 read was only considered to reflect contamination if cox1 divergence between sp1 and sp2 was >5%. When more than one reference sequence per species was available, we required that the minimal divergence between any two cox1 sequences from the two species be >5%. Sixty such closely related species pairs were identified. These analyses were performed using homemade programs in C++ and R. Read counts, homo-quartets, and detection of within-species contamination For each species including at least four individuals, transcriptome assembly was performed using a combination of ABySS [37] and CAP3 [38], as described previously [35]. Reads were mapped to predicted cDNAs using BWA, following [18] and [21], and potential PCR duplicates were removed – meaning that identical reads in any particular individual were counted only once. Open reading frames were predicted as in [18] and coding sequences were retained. For each position of each coding sequence and each individual, the number of reads for the four possible states A, C, G, and T were recorded. Below we refer to these vectors of counts as "quartets." To characterize within-species, between-individual contamination, we focused on quartets in which (1) exactly two states were observed, (2) read count was high (above 40) for the most prevalent state, and (3) read count was exactly one for the other state. Such quartets were assumed to correspond to genotypes that are homozygous for the major state and in which an error had been introduced – the minor state. We called these quartets "homo-quartets." We defined two categories of homo-quartets, depending on the read counts in other individuals at the same position. The first category corresponds to homo-quartets occurring at positions dominated by the major state – specifically, when the sum across individuals of allele counts for the major state was >95% of total counts. Such positions were called monoallelic (Fig. 1, black). The second category corresponds to homo-quartets occurring at positions in which two alleles were found at a substantial frequency – specifically, when the sum across individuals of read counts for the second more frequent state was more than 10n, n being the number of genotyped individuals. Such positions were called biallelic (Fig. 1, red). Homo-quartets not falling in either of these two categories were disregarded. Detection of within-species contamination through homo-quartet analysis. Each multicolored square represents a quartet, that is, read counts for states A (green), C (yellow), G (blue), and T (orange) at a specific position in a specific individual, zeros being omitted. A fictive dataset of four individuals (Ind 1 to Ind 4 ) and five positions (Pos 1 to Pos 5 ) is shown. At all five positions, the quartet for individual Ind1 is a homo-quartet (thick borders): the major state has more than 40 reads, and the minor state has exactly one read. Positions Pos1 and Pos2 are monoallelic: the major state represents more than 95% of reads across the four individuals. These two positions inform on the contamination-free error pattern. Positions Pos3, Pos4, and Pos5 are biallelic: besides the major state, another allele segregates in the sample. At Pos3 the Ind1, the minor state (G) differs from the other segregating allele (C); this error cannot result from within-species contamination. At Pos4 and Pos5, the Ind1 minor state is identical to the other segregating allele (T), potentially reflecting allele leakage between individuals, as indicated by red arrows. The proportions of these different types of position inform on the prevalence of within-species contamination We reasoned that, in the absence of contamination, the identity of the minor state in any given homo-quartet should be independent of read counts for other individuals, so that a similar pattern of sequencing error would be expected for the two categories of homo-quartets. If, however, the distribution of the minor state in homo-quartets differed between monoallelic and biallelic positions, and was influenced at biallelic positions by the identity of alleles segregating in the sample, then such a pattern would demonstrate the existence of within-species contamination (Fig. 1). Formally, for each species, we first considered homo-quartets occurring at monoallelic positions and calculated P, the matrix of minor state prevalence given the major state: $$ P\left( a, b\right)={h}_{mono}\left( a, b\right)/{\displaystyle \sum }{h}_{mono}\left( a, k\right) $$ where a and b are two of the four A, C, G, and T states (b ≠ a), and h mono(a,b) is the number of homo-quartets occurring at monoallelic positions and having a as the major state and b as the minor state. P can be understood as an estimate of the error matrix at monoallellic positions. Then we considered homo-quartets occurring at biallelic positions and calculated q obs , the observed prevalence as a minor state of the other allele segregating at the considered position: $$ {q}_{obs}={\displaystyle \sum {r}_z}(k)/{h}_{bi} $$ where h bi is the number of homo-quartets occurring at biallelic positions, z is the allele different from the major state segregating at the position at which homo-quartet k occurs, and r z (k) is read count for state z at homo-quartet k. By definition of homo-quartets, r z (k) must be zero or one. This number was compared to q exp , the expected prevalence as a minor state in homo-quartets of the other segregating allele assuming no contamination, that is, assuming that sequencing errors at homo-quartets occurring at biallelic positions are well predicted by P: $$ {q}_{\exp }={\displaystyle \sum P}\left( a, z\right)/{\displaystyle \sum {\displaystyle \sum P}}\left( a, i\right) $$ where a is the major state of homo-quartet k and z is the allele different from a segregating at the position at which k occurs. We defined λ, the index of allele leakage among individuals within a species as: $$ \lambda =\left({q}_{obs}{\textstyle \hbox{-} }{q}_{exp}\right)/{q}_{exp} $$ λ is expected to equal zero in the absence of contamination, that is, when the identity of the minor state for a given homo-quartet is independent of the genotypes and read counts of other individuals. Contamination-aware genotype calling We modified our genotype-calling procedure [17, 18] to account for between-individual, within-species contamination. Following [17], we describe a quartet R by r 1 (number of A reads at a given position for a given diploid individual), r 2 (C reads), r 3 (G reads), and r 4 (T reads), and define r = r 1 + r 2 + r 3+ r 4. Let us call f 1, f 2, f 3, and f 4 the frequencies of alleles A, C, G, and T in the population at the considered position. Assuming Hardy-Weinberg equilibrium and a constant error rate ε, the probability of a quartet can be written as: $$ \Pr (R)={\displaystyle \sum_{a=1}^4{\displaystyle \sum_{b= a}^4{f}_a{f}_b\frac{r!}{r_1!{r}_2!{r}_3!{r}_4!}{\displaystyle \prod_{x=1}^4{q}_x{\left[ ab\right]}^{r_x}}}} $$ $$ {q}_a\left[ aa\right]=1-3\upvarepsilon $$ $$ {q}_b\left[ aa\right]=\varepsilon $$ $$ {q}_a\left[ ab\right]=1/2-\varepsilon $$ $$ {q}_c\left[ ab\right]=\varepsilon $$ where a ≠ b ≠ c are in {A,C,G,T}. q x [yz] is the probability of calling state x from an individual carrying genotype {y,z}. Equations 5, 6, 7, 8, and 9 assume that, for every read, the genuine state will be called with probability 1 − 3ε, whereas an erroneous state will be called with probability ε. Equation 5 sums the contributions of the 10 possible diploid genotypes to the likelihood of R. In our implementation, the allele frequencies f k were estimated from observed read counts, the error rate was estimated by maximizing the likelihood function, and distinct rates were assumed for transition-type vs. transversion-type errors [17]. Here, we introduce a generalization of Eqs. 6, 7, 8, and 9: $$ {q}_a\left[ aa\right]=\left(1-\gamma \right)\left(1-3\upvarepsilon \right)+\gamma \left({f}_a\hbox{'}\left(1-3\upvarepsilon \right)+\left(1-{f}_a\hbox{'}\right)\varepsilon \right) $$ $$ {q}_b\left[ aa\right]=\left(1-\gamma \right)\varepsilon +\gamma \left({f}_b\hbox{'}\left(1-3\upvarepsilon \right)+\left(1-{f}_b\hbox{'}\right)\varepsilon \right) $$ $$ {q}_a\left[ ab\right]=\left(1-\gamma \right)\left(1/2-\varepsilon \right)+\gamma \left({f}_a\hbox{'}\left(1-3\upvarepsilon \right)+\left(1-{f}_a\hbox{'}\right)\varepsilon \right) $$ $$ {q}_c\left[ ab\right]=\left(1-\gamma \right)\varepsilon +\gamma \left({f}_c\hbox{'}\left(1-3\upvarepsilon \right)+\left(1-{f}_c\hbox{'}\right)\varepsilon \right) $$ where f x' is the frequency of reads of state x at the considered position excluding the focal individual. Equations 10, 11, 12, and 13 assume that with probability (1 − γ) a read state is determined by genotype and error rate, as in the original method, whereas with probability γ a read state is obtained by randomly sampling one state at the considered position, excluding reads from the focal individual. Here, γ is the probability of contamination, and is assumed to be homogeneously distributed across individuals of the sample. Note that even a contaminant read can be affected by sequencing error, as expressed in the right-hand term of Eqs. 10, 11, 12, and 13. Genotypes and SNPs were called assuming four distinct values for γ, namely γ = 0 (no contamination), γ = 0.05, γ = 0.1, and γ = 0.2. We created a reference database of 624 aligned, partial, 351-bp-long cox1 sequences. The database included a mixture of sequences from our target species (378 sequences from 149 species), companion species (226 sequences from 139 species), and model species (20 sequences from 20 species). Target species were intended to trace cross-contamination among samples. Companion species were introduced as negative controls. Model species were introduced to search for contamination by standard laboratory organisms. In our reference databases, 31 of our target species were not represented at all, 98 were represented by a single cox1 sequence, and six were represented by more than ten cox1 sequences, implying that our ability to detect the occurrence of a given species in a given sample varied among species. Patterns of between-species contamination Short sequence reads from each of 446 samples (individuals) from 116 species were aligned to our reference cox1 database using BWA. The number of hits to each reference sequence was recorded and divided by the number of millions of reads of the considered sample. For each sample, we calculated the prevalence of cox1 hits to a reference sequence from the expected species, and the prevalence of cox1 hits to a reference sequence from an unexpected species – that is, a species differing from the expected species by >5% of cox1 divergence. Hits to a species different but <5% divergent from the expected one were not counted. Figure 2 shows an overview of the contamination pattern in this large-scale data set. Figure 2a shows the across-samples distribution of the prevalence of expected (gray) vs. unexpected (red) cox1 reads, while Fig. 2b plots these two variables. The across-samples median prevalence of expected cox1 reads was 674 cox1 reads per million. The prevalence of expected cox1 reads was sometimes low: it was <10 per million in 86 samples, and zero in 52 samples, of which 13 were from a species that was represented in our reference cox1 database. This is quite surprising, given that cox1 is considered a generally high-expressed gene. This result might be explained by insufficient/inappropriate species representation in the reference database for these particular samples. It might also be that in some taxa mitochondrial transcripts lack a polyA tail (or use it as a degradation signal, as in plants [39]) and were therefore excluded at the retrotranscription stage in our protocol. Overall pattern of between-species contamination. a Among-sample distribution of the prevalence of reads mapping to a cox1 reference from the expected (gray) or an unexpected (red) species. Prevalence is defined as the number of cox1 reads per million reads. b Relationship between the prevalence of cox1 reads mapping to the expected (x-axis) vs. an unexpected (y-axis) species, again per million reads. Each dot represents a sample. Plain line: ratio of unexpected to expected cox1 reads is one. Dotted lines: ratio of unexpected to expected cox1 reads is 0.1 (respectively, 0.01). Samples from species not represented in our cox1 reference database are not shown We found at least one hit to an unexpected species in 353 of the 446 samples. The prevalence of unexpected cox1 hits was >50 per million in 22 samples, and >500 per million in seven samples. One species, woodlouse Armadillidium vulgare, was particularly affected by unexpected hits – six individuals out of ten showed >50 per million unexpected hits. Twelve samples for which the prevalence of expected hits was >100 per million had a ratio of unexpected to expected hits >0.1, and two samples, GA24O (earthworm Allolobophora chlorotica L1) and GA17L (brine shrimp Artemia tibetiana), had a ratio >1.0. In summary, expected cox1 reads clearly dominated but contaminant reads were common and reached a high prevalence in a substantial number of samples. The vast majority (99.54%) of the 385,597 unexpected cox1 reads originated from target species. Only 0.11% of the unexpected hits were assigned to a companion species, and 0.35% to a model species. The low prevalence of companion species was expected and confirmed that unexpected cox1 hits result almost uniquely from contamination. Regarding model species, we detected human cox1 reads in ten samples from nine distinct species, but always at very low prevalence – the total number of reads hitting a human cox1 sequence was 92. Mus musculus and Bos taurus were more prevalent in terms of total reads (507 and 447, respectively), but concerned a smaller number of samples (five and three) and species (three and three, respectively). Among the 446 analyzed samples, 353 included at least one read mapping to an unexpected species – that is, showed evidence for between-species contamination. Of these, 205 were contaminated by at least two species, and we detected up to eight contaminant species in samples GA08R (Glanville fritillary Melitaea cinxia) and GA34L (mosquito Culex hortensis). Summing contaminant species across samples, we found that the data set had been affected by at least 782 distinct events of between-species contamination. This is an underestimate, due to the incompleteness of our reference database, our inability to detect contamination between closely related species, and the possibility of multiple events of contaminations of a given sample by a given species. The number of expected cox1 reads, unexpected cox1 reads, and contaminant species per sample are available in Additional file 2: Table S2. Reversely, 94 of the 180 species we processed in this project did contaminate at least one sample from another species. Among these, four species contaminated more than 15 distinct samples, and one, king penguin Aptenodytes patagonicus, contaminated samples from as many as 11 distinct species (Additional file 4: Figure S1). We found that the mean prevalence of expected cox1 reads of a species was significantly correlated with the number of individuals it contaminated (r = 0.35, p < 10−3) and with the total number of contaminant reads it contributed (r = 0.45, p < 10−4, log-transformed number of contaminant reads). Dubious samples Two samples resulted in unexpected patterns. Sample GA36K, assigned to species Mytilus trossulus (bay mussel), yielded a single cox1 read that mapped to a M. trossulus reference, but >18,000 cox1 reads that mapped to a sequence from either M. edulis or M. galloprovincialis, two interbreeding species of European mussels (Fig. 2b, top left dot). By contrast, 99% of cox1 reads from the other M. trossulus sample that we analyzed, GA36L, mapped to a M. trossulus reference. The GA36K sample was collected in Seattle, WA, USA, a state in which invasive populations of European mussels are documented [40, 41]. Sample GA36K therefore probably results from an identification error, or reflects M. galloprovincialis/edulis mtDNA introgression into M. trossulus. Similarly, sample GA08F, assigned to Glanville fritillary Melitaea cinxia (Lepidoptera), did not yield a single cox1 read that mapped to a M. cinxia reference, but >26,000 cox1 reads that mapped to a reference from the Spanish fritillary Euphydryas desfontainii. This species is quite divergent from M. cinxia, both morphologically and molecularly (cox1 divergence >25%), so mtDNA introgression and misidentification appear unlikely in this case. According to our records, the GA08F sample came from Aland, Finland, a place where E. desfontainii does not occur. We did, however, sample E. desfontainii, together with M. cinxia, in Morocco. The problem, therefore, probably resulted from sample mislabeling. The GA08F sample very likely belongs to E. desfontainii and was mistaken for an M. cinxia individual in our published analyses. We checked, however, that our main results are robust to these problems (see final paragraph of the "Results" section). Analysis of laboratory metadata We created a between-species contamination matrix M in which cell m ij contained zero in the absence of evidence for contamination of species j by species i, one in case of the detected contamination of species j by species i, and missing data if species i and j were <5% divergent cox1-wise, such that contamination detection was assumed to be unreliable. Here, a single read from any individual of species i hitting a reference sequence from species j was considered sufficient to attest for an event of contamination of i by j. Requiring at least ten unexpected reads, instead of just one, yielded qualitatively similar results. The 39 samples from species not represented in our reference cox1 database were here disregarded, so that sample size was 407 in this analysis. The total number of ones in M was 362, and the total number of pairs of species sufficiently divergent such that contamination detection was possible was 27,251, so that the proportion of species pairs for which an event of contamination was detected was p = 0.0133. We focused on five predictors of the probability for two species to be connected by contamination, namely lab_overlap, same_technician, same_shipment, same_flowcell, and same_lane. To calculate the lab_overlap variable, we first defined the processing period of any given species as the period from date of entry into our laboratory to date of last shipment to a sequencing center. For any given pair of species, lab_overlap was defined as the length, in days, of the intersection between the processing periods of the two species. The same_technician variable was a Boolean variable set to one if at least one sample of each of the two considered species was treated by the same person in our laboratory, and to zero otherwise. Similarly, the same_shipment, same_flowcell, and same_lane variables indicated whether at least one sample of each of the two considered species had been shipped on the same day to the same sequencing center, or sequenced on the same flowcell/same lane, respectively. We calculated the average value of these variables across all pairs of species for which an event of contamination was attested (Fig. 3, red vertical bars), and compared these to null distributions obtained by shuffling zeros and ones in the contamination matrix (Fig. 3, white histograms, 1000 replicates). More precisely, each cell of a randomized matrix was assigned one with probability p, or zero with probability (1 − p), with missing data being left unchanged, where p = 0.0133 was the overall probability of contamination (see above). We detected a strong and significant effect of each of the five variables: compared to the average species pair, species contaminating each other tended to have a longer period of overlap in our laboratory, to be handled by the same technician, and to be sent the same day and sequenced on the same flowcell. The effect of sequencing center-associated variables was particularly strong. For instance, the probability for two species that were shipped together to be connected by an event of contamination was 0.13, that is, more than ten times the unconditional probability. The same_lane pattern was very similar to same_flowcell and is not shown in Fig. 3. Effect of laboratory metadata on the probability of between-species contamination. Four statistics are shown: lab_overlap (top left), same_technician (top right), same_shipment (bottom left), same_flowcell (bottom right). x-axis: average value of each statistics. Vertical red line: actual data set. y-axis: number of randomized data sets (out of 1000). White histograms: expected distribution assuming random probability of contamination. Blue histograms: expected distribution assuming that contamination is dependent on same_shipment. Green histograms: expected distribution assuming that contamination is dependent on lab_overlap and same_technician The five analyzed variables were significantly correlated with each other. We tried to disentangle their effects, and particularly distinguish the influence of our laboratory from that of sequencing centers. To this aim, we compared the observed value of lab_overlap and same_technician to null distributions obtained by reshuffling M in a way that controls for the effects of same_shipment (Fig. 3, top, blue histograms). In this analysis, each (i, j) cell of a randomized matrix was assigned one with probability p ij , or zero with probability (1 − p ij ), again leaving missing data unchanged, where p ij was the probability of contamination knowing same_shipment(i, j). These were obtained by calculating the proportion of ones in M conditional on values 0 or 1 for same_shipment. Similarly, the null distributions of same_shipment and same_flowcell conditional on lab_overlap and same_technician were generated (Fig. 3, bottom, green histograms). The effects of the five variables were still significant in these control analyses: a laboratory effect was detected when controlling for sequencing center-associated variables and a sequencing center effect was detected when controlling for laboratory-associated variables. To analyze this effect more deeply, we created two synthetic variables summarizing the effect of laboratory (LAB) and sequencing center (CENTER), respectively. The LAB variable was positive when same_technician was true and lab_overlap was >200 days, but negative otherwise. The CENTER variable was negative for pairs of species shipped on distinct dates, but positive otherwise. Regarding species pairs that were sent together, we distinguished pairs sequenced on distinct flowcells (CENTER+), the same flowcell but distinct lanes (CENTER++), and the same lane (CENTER+++). In this analysis we focused on the 97 species for which information on shipment dates, flowcell, and lane numbers was available for all individuals. As far as species sent on distinct dates were concerned (CENTER-), the contamination probability was very low regardless of LAB (Table 1, first line). This seems to be incompatible with the hypothesis of a substantial level of contamination in our laboratory. In contrast, the probability that two species shipped on the same day were connected by an event of contamination was as high as 0.2, and further increased in case of shared flowcell and shared lane (Table 1, lines 2 to 4), reaching values >0.5. Table 1 Effect of laboratory and sequencing center variables on the probability of contamination Surprisingly, we detected a strong and significant interaction between the LAB and CENTER variables (Table 1). Two species being shipped the same day (CENTER+), overlapping in our laboratory, and being handled by the same technician (LAB+) substantially increased the probability of contamination. We suggest that this is an induced effect resulting from the fact that tubes in shipped boxes were ordered by technician, so that samples processed by the same technician in our laboratory were presumably more likely to be processed together by sequencing centers, and therefore to contaminate each other. To test this hypothesis, we subsampled species in such a way that a single species per technician per shipment was kept, so that no induced effect of same_shipment on same_technician was possible. We found eight events of contamination between the 24 species of the subsample. There was still a significant effect of same_shipment on contamination probability in this subsample, but no effect of lab_overlap or same_technician was detected (Additional file 5: Figure S2), suggesting that the LAB effect conditional on CENTER+ reported in Table 1 was an induced effect. These analyses therefore indicate that the vast majority of the events of between-species contamination we detected occurred in sequencing centers. The results were qualitatively unchanged when a 10% threshold was used, instead of 5%, for the minimal cox1 divergence between contaminant and contaminated species (Additional file 6: Table S3). Laboratory contamination: detailed analysis Eight events of contamination were detected between species that were not shipped on the same date. Of these, four involved Glanville fritillary M. cinxia. This is the one species in our data set that included samples for which data on shipment date are missing (GA08B to GA08F, Additional file 2: Table S2). The three species that contaminated or were contaminated by M. cinxia but lacked an attested shared shipment date with M. cinxia – Iberian hare Lepus granatensis, mountain hare L. timidus and ascidian Ciona intestinalis A – were shipped the same day, May 26, 2010. It seems therefore possible, not to say probable, that samples GA08B to GA08F were actually sent out for sequencing on May 26, 2010, and that contamination occurred in the sequencing center in this case, too. Besides these four cases, one detected event of contamination between species not shipped on the same date involved gorgonian Eunicella cavolini and European blue mussel M. galloprovincialis. E. cavolini, however, shares a shipment date (January 23, 2013) with M. edulis, the other species of European mussel, which hybridizes with M. galloprovincialis – the two species have very similar haplotypes in our reference cox1 database. A closer inspection of the data revealed that the single E. cavolini sample, GA31L, affected by contamination from M. galloprovincialis is the single E. cavolini sample that was shipped on January 23, 2013. Eight cox1 reads from this sample mapped to a M. edulis reference and two mapped to a M. galloprovincialis reference. In conclusion, only three events of between-species contamination out of 782 can be unambiguously assigned to our laboratory: contamination of European pond turtle Emys orbicularis by ascidian Ciona intestinalis A and of seahorses Hippocampus hippocampus and H. guttulatus by each other. Within-species contamination The above analyses suggest that there was substantial contamination in this project, and primarily involves samples that were shipped together. This is worrisome because samples from distinct individuals of the same species, between which contamination is most problematic and difficult to detect, were typically sent together. To quantify the amount of within-species contamination, we examined the prevalence as the minor state ("errors") at homozygous genotypes of alleles segregating in the sample. First focusing on homo-quartets (i.e., positions at which the read count for the major state was >40 and the read count for the minor state equaled 1) that occurred at monoallelic positions, we determined P, the error matrix in the absence of contamination. This was done separately for each of the 39 species of the sample in which at least four individuals were sequenced. Note that in this study we did not use strand information, so we could not distinguish between X → Y and X* → Y* errors, where X* is the complementary of base X. Error matrices revealed two main features. First, the A → C or T → G errors were often more frequent than the other three transversion-type errors, namely A → T or T → A, C → G or G → C, and C → A or G → T. The ratio of A → C or T → G to other transversion-type errors varied between 0.29 and 0.79 among species (correcting for base composition), when a ratio of 0.67 would be expected under random error. This is consistent with documented error biases of the Illumina technology [42, 43]. Second, transition-type errors, C → T or G → A and T → C or A → G, were typically more numerous than expected. The ratio of transition-type to transversion-type errors varied from 0.47 to 1.14 among species (correcting for base composition, median = 0.79), when the expected ratio would be 0.5 under random error, and <0.5 according to [43]. Knowing that DNA polymerases typically generate more transition-type than transversion-type errors, this result suggests that a fraction of the sequencing errors affecting our data was introduced prior to sequencing, presumably at the PCR step during library construction. We then considered homo-quartets occurring at biallelic positions, where two alleles segregate at substantial frequency. Here, we only considered the 33 species in which at least 50 such homo-quartets were found. We asked whether the minor state at such homo-quartets tended to correspond with the other segregating allele more often than expected based on P. We found that the relative prevalence of the other segregating allele was above its expected value in all 33 species. The index of allele leakage, λ, varied from 0.19 to 8.5, when λ = 0 would be expected in the absence of contamination. This analysis therefore indicates that within-species contamination is widespread in our dataset and probably affects all the sequenced species. We investigated the influence of laboratory metadata, and particularly the date of shipment to sequencing centers, on the prevalence of within-species contamination. To this end, we focused on the 12 species of our data set in which not all samples were shipped the same day – that is, most often at two distinct dates, and up to four dates in the blue tit Parus caeruleus. In these species, we measured λ', the index of allele leakage between samples sent on different dates. This was achieved by only considering homo-quartets occurring at positions that were biallelic across the whole sample of individuals, but monoallelic in the subsample of individuals shipped the same day as the focal individual (Additional file 7: Figure S3). This analysis could not be performed species by species due to the small number of relevant homo-quartets per species. We therefore pooled homo-quartets across the 12 species, still accounting for species-specific error matrices P, and obtained an index of allele leakage between samples sent on different dates of λ' = 0.59. This figure was twice as small as the index calculated as above, that is, irrespective of shipment date, which for these 12 pooled species was λ = 1.21, demonstrating an effect of same_shipment on the prevalence of within-species contamination. Contamination-aware SNP calling To assess the robustness of our published results to the problem of within-species contamination, we re-called SNPs and genotypes using a modified method accounting for allele leakage between individuals. Compared to our original SNP-calling method, a parameter γ was added, which represents the probability that a read originates from another individual of the sample. Three arbitrary values of γ were used: 0.05, 0.1, and 0.2. Contamination-aware SNP calling was applied to the 39 species of our sample in which at least four individuals were available. Classical population genomic statistics were calculated from this data set using the same pipeline as in [18]. To save computational time, SNP calling was applied to reduced data sets consisting of exactly one million positions per species, instead of the 1.8–27 million positions in full data sets. We found that the number of called SNPs and the estimate of πS, the genetic diversity at synonymous positions, decreased with increasing γ (Fig. 4a). This was expected: contamination spuriously increases heterozygosity by moving alleles around. The relative bias was substantial – the median ratio of corrected to uncorrected πS was 0.90 when γ was 0.1, and 0.81 when γ was 0.2. The relative bias, however, was fairly constant across species, and much smaller that the between-species differences in πS, suggesting that our published comparative analyses of πS across species [17, 19, 21, 22] are robust to within-species contamination. We checked that the correlation reported by Romiguier et al. [21] between πS and species life history traits were still valid after control for contamination. We found that the correlation coefficient between log-transformed πS and log-transformed longevity was very similar in all four analyses, that is, between −0.517 and −0.524, the most negative coefficient being obtained when γ = 0.1. Similarly, the relationship between log-transformed πS and log-transformed propagule size [21] was very robust to changes in γ (correlation coefficient between and −0.772 and −0.758, minimal value when γ = 0). Robustness of population genomic estimates to contamination-aware single-nucleotide polymorphism (SNP) calling. a Synonymous diversity πS; b ratio of non-synonymous to synonymous diversity, πN/πS; c FIT; d Tajima's D, synonymous SNPs only. Each dot represents a species. x-axis: estimates obtained assuming no contamination. y-axis: estimates obtained from contamination-aware SNP calling. Black dots: γ = 0.05; blue dots: γ = 0.1; red dots: γ = 0.2 synonymous diversity πS; top right: πN/πS ratio; bottom left: FIT; bottom right: Tajima's D, synonymous SNP's only The ratio of non-synonymous to synonymous diversity, πN/πS, was only slightly modified when we controlled for contamination (Fig. 4b), the median relative bias being close to 0.96 for all three positive values of γ. The synonymous (Fig. 4d) and non-synonymous Tajima's D, a statistic measuring the departure of the distribution of minor allele frequency from the standard coalescent, were also only moderately affected. These two results suggest that published inferences based on πN/πS and site-frequency spectra [18, 27] are presumably robust enough to within-species contamination. The FIT statistics measures the excess of individual homozygosity compared to Hardy-Weinberg expectations. A positive FIT is expected in cases of inbreeding and/or population substructure. Figure 4c shows that our FIT estimate is particularly sensitive to contamination issues. Controlling for contamination resulted in a substantial increase in FIT in all the analyzed species, reflecting the fact that within-species contamination tends to increase individual heterozygosity. In our uncorrected analysis (γ = 0), a negative estimate of the genome-average FIT was obtained in nine species [21]. This is an unexpected result, given that processes leading to heterozygote excess, such as balancing selection, are presumably limited to a small fraction of the genome [44]. In our contamination-aware analyses, a negative FIT was obtained in just four, two, and one species when γ was set to 0.05, 0.1, and 0.2, respectively, suggesting that within-species contamination might explain, at least partly, our previously unexpected report of negative estimates of FIT [21]. Harvest ant Messor barbarus was not included in this analysis because the genome-average FIT is very negative in this species as a consequence of its peculiar mating system, such that worker individuals are highly heterozygous [45]. We have not commented on FIT estimates in our published analyses, with the exception of [19], in which the lack of detectable population substructure (i.e., low FIT) in the giant Galapagos tortoise Chelonoidis nigra provided evidence against the definition of as many as 12 species in this taxon [46]. This result was here corroborated: C. nigra is one of the two species still showing a slightly negative FIT estimate after correction for contamination. We have, however, published a couple of analyses assessing the prevalence of hybridization and gene flow between diverged species or populations [20, 28, 30]. These results should be confirmed by reproducing the analyses using contamination-corrected data. We compared for each species the likelihoods of the four considered values of γ. The maximally likely γ, which we called γ, was 0 in ten species, 0.05 in 15 species, 0.1 in five species, and 0.2 in nine species. We detected a strong effect of species diversity on γ : the median πS was 0.034 among species for which γ* was 0, but 0.003 among species for which γ* was 0.2. This was unexpected and probably reflects the existence of factors that confound contamination detection (see section 3 of the Discussion "Modeling contamination"). Finally, we reproduced the analyses of Romiguier et al. (2014) [21], accounting for the dubious GA36K and GA08F samples. The published relationships between genetic diversity and species life history traits were robust to the exclusion of M. trossulus and M. cinxia: the correlation coefficient between πS and propagule size was almost unchanged compared to the uncorrected analysis (0.766 vs. 0.771), whereas the correlation coefficient between πS and longevity was slightly increased (0.594 vs. 0.569), as was the case for correlations between the πN/πS ratio and life history traits. We recalculated population genomics statistics in M. cinxia after excluding individual GA08F, that is, based on just nine individuals instead of ten. Excluding GA08F resulted in a substantial decrease in genome-average πS (0.025 vs. 0.034), πN (0.0027 vs. 0.0032), and FIT (0.38 vs. 0.52). Correlation coefficients with life history traits, however, were hardly affected by this correction. Here, we analyzed the prevalence and impact of between-species and within-species contamination in an RNAseq project involving 446 samples from 116 distinct species of animals. We focused on cross-contamination and contamination from model animals. We did not investigate contamination from, for example, microbes, which can be highly problematic, too [15]. This is in part because our experimental process targets polyA-containing RNAs, which filters out the bulk of bacterial mRNAs, and in part because a BLAST search that was performed in a previous study [21] indicated very low levels of microbial contamination in our final sets of contigs. Our analysis indicates that cross-contamination was widespread: approximately 80% of our samples showed evidence of contamination by a foreign species, and traces of within-species contamination were detected in all the species we analyzed. Contamination in this project was not an accident, it was a pattern. A single unexpected cox1 read was taken as sufficient to document an event of between-species contamination in our analysis. This might appear too liberal a criterion: contamination at such low levels is very unlikely to affect the analyses or the conclusions. The near-zero detection of contaminant reads from companion species, however, demonstrates that the unexpected cox1 reads we uncovered do not result from environmental contamination or experimental noise, but indeed trace transfers of genetic information from sample to sample. Even at very low prevalence, therefore, unexpected reads do provide relevant information on when and how events of contamination have happened. Our results were qualitatively unchanged when we only counted events of contamination supported by at least ten reads (Additional file 8: Figure S4). Contamination occurred in sequencing centers We uncovered indirect evidence that the vast majority of the events of cross-contamination occurred in sequencing centers. This was attested to by the very strong effect of same_shipment on the probability of between-species contamination, and confirmed by the reduced within-species allele leakage when only samples sent on different dates were considered. Roughly 15% of species pairs sharing a common shipment were connected by at least one event of contamination, and we detected only three events of contamination between species that were not shipped together. In this project, libraries were constructed in the sequencing centers. This step involves PCR amplification and might be more prone to contamination than RNA extraction, purification, and quantification, which were achieved in-house. It might also be that sequencing centers were simply less careful than our technicians about contamination. Sequencing center SC2 only handled two shipments, and SC3 only one, so we do not have sufficient power to compare centers in this analysis. In principle, contamination could occur during the library preparation stage through physical transfer of material, or during the sequencing stage through mis-tagging – That is, when the identifier assigning a read to its source sample is in error. In this project, we used simple indexing of samples, which can result in a non-negligible rate of sample misidentification [47]. We detected a strong effect of shipment, and on top of this a significant effect of flowcell and lane identity (Table 1), suggesting that contamination occurred during both stages. However, in the absence of data about which samples were handled together during library preparation, it is difficult to firmly conclude at which experimental steps contamination most often occurred – especially if libraries prepared together were more likely to be sequenced in the same flowcell/lane, as might well have been the case. It should be noted that our index of allele leakage λ was still significantly higher than zero in the control analysis when only samples sent at different dates were considered. This might indicate that a fraction of the events of within-species contamination did occur in our laboratory. Alternatively, λ might be inflated by processes different from contamination, such as hotspots of systematic errors [43], mosaicism [48], hidden paralogy, and variable expression level between alleles and individuals [17, 18]. Approaching and quantifying within-species contamination is actually a difficult problem, especially with RNAseq data, because a number of distinct processes can potentially generate asymmetric read counts. Families of recently duplicated genes are particularly tricky in this respect: they will yield an unpredictable number of contigs after de novo assembly, which will each attract a fraction of the reads of the distinct individuals at the mapping step. This might generate patterns similar to the ones shown in Fig. 1, confounding within-species contamination detection (see below). Modeling contamination We introduced a modified SNP-calling method that accounts for within-species contamination by assuming that a fixed fraction of the observed reads originates from other individuals of the sample. Estimates of the classical population genomic statistics were affected to various extents by this correction, depending on the assumed contamination rate γ. FIT was particularly sensitive to γ, calling for caution as far as studies of population substructure and gene flow are concerned. The effect of γ on population genomic estimates was essentially homogeneous among species, suggesting that our published comparative analyses are reasonably robust, as we explicitly checked in some cases. In these control analyses, γ was fixed to arbitrary values. When we tried to estimate the contamination rate in the maximum-likelihood framework, together with the sequencing error rate and transition/transversion ratio, we obtained estimates of γ that were negatively correlated to species genetic diversity. We suspect that this might reflect the confounding effect of hidden paralogy, when the reads corresponding to two paralogous genes map to a single reference cDNA due to erroneous assembly. Hidden paralogy tends to mimic contamination by resulting in sites at which all individuals carry similar read counts for two distinct "alleles" [18]. Such sites tend to inflate the estimate of γ, particularly in low-diversity species, where the ratio of spurious to correct SNPs is maximal. Read counts in NGS projects are typically over-dispersed compared to the multinomial distribution that is assumed by SNP-calling methods, and contamination is one out of several sources of over-dispersion (see above). To distinguish between these various effects is a methodological challenge that would require further developments. One consequence is that we do not know which values of γ in our analyses are closer to the true contamination rates – which perhaps differ between species and between pairs of individuals. It is therefore premature to draw conclusions on the quantitative impact of within-species contamination on population genomic statistics based on this analysis. We still believe that the newly introduced γ parameter, which likely captures a combination of undesired effects, tends to improve the accuracy of predicted SNPs and genotypes – as reflected by the positive values of genome-average FIT we obtained when assuming non-zero γ. For this reason, this approach and recently published related approaches [10, 49] deserve to be further developed. Are our results generalizable to other NGS-based population genomic/molecular ecology studies? We are not sure, mainly because sequencing centers were critical in the contamination patterns we detected, and our sampling of sequencing centers was poor. There is, however, no reason to a priori believe that the patterns we uncovered here will not apply to other studies – particularly those having relied on SC1 in 2009–2014. The three shipments addressed to centers SC2 and SC3 were not devoid of contamination, so the problem is probably not specific to SC1. Can guidelines for avoiding contamination be deduced from this analysis? Possibly not, again because sequencing centers were critical and we have no control over, or even knowledge of, their detailed experimental processes. We do, however, still make a number of recommendations. First, we suggest taking the cost of potential contamination into account when deciding to subcontract, or not, part of a research project in molecular biodiversity. Second, if samples have to be shipped for sequencing, we would suggest, whenever possible, sending together samples that are as genetically divergent as possible, such that contamination would be both easier to detect and less problematic. Third, when possible, we would suggest sending replicated samples, preferably on distinct dates, as controls for contamination. This can be expensive but is a direct way to identify and clean contaminant sequences, and measure their prevalence. 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Don't discard them, transmissible cancer could be an explanation. Evol Appl. 2016;10:140–5. Flickinger M, Jun G, Abecasis GR, Boehnke M, Kang HM. Correcting for sample contamination in genotype calling of DNA sequence data. Am J Hum Genet. 2015;97:284–90. We thank Mark Blaxter, Claire Daguin, Clémentine François, Philippe Gayral, Sylvain Glémin, Hervé Philippe, Jonathan Romiguier, Marjolaine Rousselle, Paul Simion and one anonymous reviewer for comments and discussions. This work was supported by European Research Council grant 232971, Swiss National Foundation grant CRSII3_160723, and Agence Nationale de la Recherche grants ANR-10-BINF-01-01 and ANR-15-CE12-0010. The data sets generated or analyzed during the current study are available from the NCBI-SRA repository under bioprojects PRJNA230239, PRJNA249058, PRJNA268920, PRJNA278516, PRJNA322119, PRJNA326910, and PRJNA374528 (Additional file 2: Table S2). MB and NF established the RNA extraction protocols and generated the data. NG performed the statistical analyses and drafted the manuscript. All authors read and approved the final manuscript. The authors declare they have no competing interests. UMR5554 – Institute of Evolutionary Sciences, University Montpellier, CNRS, IRD, EPHE, Place Eugène Bataillon, CC64, 34095, Montpellier, France Marion Ballenghien , Nicolas Faivre & Nicolas Galtier UMR7144 - Adaptation et Diversité en Milieu Marin - CNRS, Université Pierre et MarieCurie, Station Biologique de Roscoff, 29680, Roscoff, France Search for Marion Ballenghien in: Search for Nicolas Faivre in: Search for Nicolas Galtier in: Correspondence to Nicolas Galtier. Additional file 1: Table S1. List of species and laboratory metadata. (XLS 55 kb) Table S2. List of samples, shipment dates, and prevalence of expected and unexpected cox1 reads. (XLS 101 kb) Alignment of reference cox1 sequences used in this study. The fragment we used corresponds to positions 6189–6539, Cambridge reference sequence. (TXT 241 kb) Additional file 4: Figure S1. Contaminant species-associated statistics. Each dot is for a contaminant species, that is, a species for which at least one cox1 read was found in a sample from another species. x-axis: number of contaminated species (median = 2). y-axis, bottom: number of contaminated individuals (median = 4). y-axis, top: number of contaminant reads (median = 65). (PDF 40 kb) Effect of laboratory metadata, one species per technician per date. See legend to Fig. 3. Here, a single species per technician per shipment was kept, removing any possible induction by same_shipment of a same_technician effect on the probability of between-species contamination. No significant effect of laboratory-associated variables is detected in this control. (PDF 5 kb) Effect of laboratory and sequencing center variables on the probability of contamination, with the minimal cox1 divergence between contaminant and contaminated species being set to 10% instead of 5%. (XLS 6 kb) Homo-quartet analysis of contamination between shipments. See legend to Fig. 1. Here we assume that Ind1 and Ind2 have been shipped together on a date different from the shipment date of Ind3 and Ind4. We want to specifically assess the prevalence of contamination between individuals shipped at different dates. Here, Pos4 is not considered because the {Ind1, Ind2} group is not monoallelic, so that contamination could involve two individuals shipped together. Only Pos5 is identified as a candidate for between-shipment contamination. (PDF 27 kb) Effect of laboratory metadata, at least ten reads per contaminant. See legend to Fig. 3. Here, at least ten unexpected cox1 reads were required to call a contaminant, instead of one read in the main analysis. (PDF 5 kb) Ballenghien, M., Faivre, N. & Galtier, N. Patterns of cross-contamination in a multispecies population genomic project: detection, quantification, impact, and solutions. BMC Biol 15, 25 (2017) doi:10.1186/s12915-017-0366-6 Transcriptome SNP calling Within-species	CommonCrawl
Gmat Test Sample Math Home » Hire Someone to do GMAT Verbal » Gmat Test Sample Math Gmat Test Sample Math. About Math. From Greek matia, kammae, magnae – a simple geometric realization of the rational Godel property. Test Sample, Math. About Matrices. MATLAB is an electronic solver that finds and understands matrices and their even-odd rank. Matrices can be used as a database for different types of operations and may be described in any order. MATLAB operates with many different matrices, but MATLAB uses its own native library to handle specific types of computation, such as multiplication, division and stashing. Matrices are defined mathematically as the number of elements in each column of each row of matrices, which may take any number of elements. The entries in each column of the matrices can then be combined to form a matrix. Common Matrices are different types of matrices and may be read in from other forms of databases. 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The minimum value for _max_ and the maximum value for _log_, also called _max-min_, are _maxmax_ and _maxmax-min_. ## Chapter 8.11 Application to Image Analysis ### Example 7.2 Output dig this mean and standard deviation of image values Consider the following example with an output sample of Figure 8. This can be seen in Figure 8. > \| > \| 2 * 4 = 2 * 4 * 5 = 2 * 5 > \| 2* 5 = 2* 5 * 5 > \| (6) This operation will result in: > \| 2* 5 = 2* 5Gmat Test Sample Math Tracers}*** Hilleke Matig test sample $c_3^6 \circ c_2^4$ Matig Test Sample \#3 (2004);\ $MATCHU\#1a**$\ $MATCHU\#1b****$\n" A two-row Mat3 (2010) test sample 2CMAT [$d$]{}Test Sample Number 0 (c2)1. The test sample is 86834 [P]{}artition 1.\ \ Table \[S:mat3\] summarizes the results from a $7$-row Matz 5-6 CART test [MPT]{} [@Kossak]. In this sample, there are ten common sampling schemes, and only one of the non-specific sampling schemes is repeated in a single row. The distribution of the test sample's variance is linear across different rows. The box-and-whisk method returned the symmetric distribution, with the empirical error of 95%. On the flip side, for the same test, there are three non-specific sampling schemes – the three random permutations chosen by the ABI; the black box random permutation among the permutations chosen by BAC; and the mean-square error values of the permutation produced by the ADU algorithm taken from @Bauer96. All these three non-specific sampling modes yielded the following mean-square deviation from expected mean-square deviation for each test case: 1. [bac]{} [PMT]{} [MPT]{}, $d_{\# 3 } = 47$; 2. [BAI]{} [MKP]{} [MPT]{}, $d_{\# 3 } = 57$; 3. [MRT]{} [MIAT]{} [MPT]{}, $d_{\# 3 } = 19$; 4. [NANAL]{} [MFAT]{} [PMT]{}, $d_{\# 6 } = 27$; and 5. Do My Online Class For Me and [MIAT]{} [BANAL]{} [MKP]{}, $d_{\# 6 } = 41$ (DGBA) and $d_{\# 6 } = 30$ (BANAL) respectively. The common and non-specific methods yielded mean-square deviation values of 63 and 11 for the symmetric case, 66 and 12 for the non-symmetric, respectively. As shown in Fig. \[Meag2\], both error distribution and mean-square deviation are in a relatively straight line with the best fitting equations and the next page value measured from the log-normal distribution. Following @Lawson85 and @Miller17, the median error was $1.24\pm 0.14$ and the 95% confidence region was $-0.42\pm 0.14$, respectively. To achieve this sample size, we compared the sample complexity models for our test case relative to the various other matrices used in this work (see Table \[TableB\]). The average cost (calculation by base-band) for running our grid-based test is only $2.65\times 10^{-10}$ years for every matrix under consideration and it takes $13,000$ hours of computer time to estimate the required matrices, regardless of the number of users ($1,200$ matrices in this example). While the cross-validation performance is identical, the median time to run the test is $3580$ hours per user. Given the significant differences in the test case and other matrices, this figure suggests that the model performance for $MATCHU$ was substantially increased. Discussion {#Disc} ========== Our calculations allow us to demonstrate that using matrices with variable complexity leads to reasonable results: with the use of non-specific matrices gives a superior performance and can be used successfully in any model of interest. Although it poses the problem of the matrix overfitting, the high number of available matrices Do You Have To Report All Your Gmat Scores? Crack Gmat Verbal Gmat Tricks Verbal	CommonCrawl
For go back n arq protocol the value of n cannot be Go-Back-N Automatic Repeat Request (ARQ) Protocol To improve the efficiency of transmission (filling the pipe), multiple frames must be in transition while waiting for acknowledgment. The receiver keeps track of only one variable, and is no need to buffer out-of-order frames. Go-Back-N ARQ simplifies the process at the receiver site. e window will carry frames from 0 to 6 which are 7 in number. next : 7 8 9 X +N = 1 −PL 1 +PL(N −1) 3. I'll call the Go-Back-One G-1. It is needed to design a Go-Back-7 sliding window protocol for this network. The window. These programs are hardcoded to transfer to compile: gcc -o server Flow Control – set of procedures used to restrict the amount of data that Go-Back-N ARQ (Cont. Go-Back-N ARQ is a specific instance of the Automatic Repeat-reQuest (ARQ) Protocol, in which the sending process continues to send a number of frames specified by a window size even without Go-Back-N ARQ is a specific instance of the automatic repeat request (ARQ) protocol, in which the sending process continues to send a number of frames specified by a window size even without receiving an acknowledgement (ACK) packet from the receiver. Since in the two-way transmission, data frames and acknowledgement of frames are interleaving. • Node A sends a new Node B otherwise may not be able to tell if a frame contains a new packet or if a frame . It is a special case of the general Go-Back-N ARQ simplifies the process at the receiver site. The client transfers a single file to the server's local directory over UDP on a VERY unreliable network. c implements a reliable data transfer over UDP in C client. I bought a FireTV so I could Chromecast to it (DIAL protocol for nerds) and now I find out Google & Amazon are bickering? I spent $40 on an app + $69 for the FireTV 4K. But even worse, if a frame is lost and transmitter doesn't have a frame to send, the window will not be exhausted and recovery will not commence. e. Even with this delay, the system is able to provide good throughput. 7 Go-Back-N ARQ is a specific instance of the automatic repeat request in cases of any p(ARQ) protocol, in which the sending process continues to send a number of frames specified by a window size even without receiving an acknowledgement (ACK) packet from the receiver. The wildcard address is used by applications (typically servers) that intend to accept connections on any of the hosts's network addresses. 42 modem. However, this protocol is very inefficient for a noisy link. The station A has infinite supply of frames to send to B and B is acknowledging them according to the protocol. Automatic Repeat ReQuest (ARQ) When the receiver detects errors in a packet, how does it let Go-Back N ARQ: In this sliding window go-back-n ARQ method, if one frame is lost or damaged, all frames sent since the last frame acknowledged are retransmitted. If node is not NULL, then the AI_PASSIVE flag is ignored. where m is thte size of the sequence number field in bits. Dr Beyza Unal, a research fellow in nuclear policy at think tank Chatham House, argued there is currently no evidence that terrorist groups could build a nuclear weapon. There are a few things to keep in mind when choosing a value for N : The sender must not transmit too fast. Aug 23, 2019 · The sliding window method using cumulative ACK is known as the Go-Back-N ARQ protocol. Suppose that frames from station A to station B are one unit long and use a time-out value of 2. Therefore, a buffer is not needed at the receiver. Typical values of N are 2 and 4. In Go-Back-N Automatic Repeat Request, we can send several frames before receiving acknowledgments; we keep a copy of these frames until the acknowledgments arrive. ) 20 Sender Sliding Window • all frames are stored in a buffer, outstanding frames are enclosed in a window frames to the left of the window are already ACKed and can be purged frames to the right of the window cannot be sent until the window slides over them whenever a new ACK arrives, the window slides The figure above shows the efficiency of Go-Back-N and Selective Repeat ARQ as a function of frame size. /server. In Selective repeat-ARQ the windows size can at the most be half of the largest possible sequence number. Send again starting from certain packet (its number is the number on an NAK). In a noisy link a frame has a higher probability of damage, execute layer-n protocol Peer-to-Peer Protocols and Data Link Layerand Data Link Layer ARQ Protocols and Reliable Go-Back N ARQ Server/Client Go-Back-N. For example, sender may send frames 1,2,3,4 and get an NAK with a value of 2. Go-Back-N ARQ is a specific instance of the Automatic Repeat-reQuest (ARQ) Protocol, in which the sending process continues to send a number of frames specified by a window size even without Automatic Repeat ReQuest (ARQ), also called Automatic Repeat Query, is an error-control protocol that automatically initiates a call to retransmit any data packet or frame after receiving flawed or incorrect data. The propagation speed of media is 2 × 106 mps. Analysis of Go-Back-N ARQ in Block Fading Channels Kamtorn Ausa vapat tanakun and Aria Nosratini a, Senior Member , IEEE Abstract — This work analyze s the thro ughput of Go-Back-N The value of N is usually chosen such that the time taken to transmit the N words is less than the round trip delay from transmitter to receiver and back again. An implementation of the Go Back N ARQ in FTP protocol, written in Python. Receiver window size is 1. Feb 18, 2013 · Consider a bidirectional link that uses Go-Back-N with N = 3. so the above whole slot would be resend according to the procedure of go back n. The receiver process keeps track of the sequence number of the next frame it expects to receive, and sends that number with every ACK it sends. The rst type of generalized ARQ protocols is go back n ARQ in which the sender is allowed to transmit n frames before stopping to wait for acknowledgments. The solution is to use a timeout with each frame. • Use of piggybacking. from 0 - 7 ) and maximum window size = 2 3 - 1 = 7 i. Go back means sender has to go back N Assume that sequence space was four (sequence numbers 0,1,2,3). A method for minimizing feedback responses in an ARQ protocol is disclosed, whereby different mechanisms can be used to indicate erroneous D-PDUs and construct S-PDUs. RECEIVER SLIDING WINDOW The size of the window at the receiver site in this protocol is 1. Feb 18, 2017 · The timer of a system using the Stop-and-Wait ARQ Protocol has a time-out of 6 ms. The protocol uses a 3-bit go-back-n sliding window protocol. In this method, a station may send a series of frames sequentially numbered modulo some maximum value. Figure 2. now the counting of the 5th packet will start from the last packet after which the packet was lost i. Aug 03, 2019 · What is Go-Back-N Protocol? It is completely totally different from totally different codecs as a result of it does not require any authentication when the knowledge strikes between the system, the place the receiver will solely take the following price prepared in line and not all of the issues coming its method. Design the algorithm for Stop and Wait ARQ and Go Back N protocol. Cannot retrieve contributors at this time. Solution- Given-Bandwidth = 20 Kbps May 20, 2016 · Key Differences Between Go-Back-N and Selective Repeat. c - implementation of go-back-n ARQ in C Server. Go back N ARQ is an implementation of sliding window protocol like Selective Repeat Protocol. • Window size = N. Assignment 1 and implement the simplified version of the Go-Back-N ARQ protocol. Frame 8 must have be received by the receiver and the ACK has been sent back to the sender. Thus for simplicity, we used a separate out[7:0] register to output the data. It can transmit N frames to the peer before requiring an ACK. Find the optimum frame length that maxi mized transmission efficiency for a chan- Figure 2. Orin binary, 0100, 0101, 0110, 0111, 1000, 1001. The average packet size is 107 bits. ) 23. Therefore, it continues to makeprogress. In this method, if one frame is lost or damaged all frames sent, since the last frame acknowledged are retransmitted. An alternative strategy, the selective repeat protocol, is to allow the receiver to accept and buffer the frames following a damaged or lost one. Assuming that each frame is 100 bytes long, what is the maximum data rate possible? 5 Kbps; 10 Kbps; 15 Kbps; 20 Kbps . Therefore, frame 3 must not be included in sender's window. In a noisy link a frame has a higher probability of damage, which means the resending of multiple frames. The receiver keeps track of only one variable, and there is no need to buffer out-of-order frames; they are simply discarded. Go-Back-N ARQ In Go-Back-N ARQ, what is the maximum size of the send window if the protocol uses 5-digit sequence numbers? May 15, 2016 · Python Implementation of the Go-back-N protocol for file transfer using Client server communication over UDP. 1 would mean that one in ten packets (on average) are lost. Go-Back-N has very low efficiency (always below 10%) for all values of n. • GBN is a practical approach of sliding window protocol. Dec 06, 2006 · In this project, you will be implementing a go-back-n based reliable duplex data transfer protocol described in section 3. Here find details of Go-Back-N ARQ protocol. Hi, I have a large client server project to write in java demonstating the go-back-n protocol, but I cannot find an example of any go-back-n prog's in Go Back N ARQ (Java in General forum at Coderanch) Features of Go Back N. 9 show the effect of m and k values on throughput efficiency in the Rayleigh fading environment. 2 Correctness of go-back-N Safety: packets are receivedinthe order transmitted. you can't always send an acknowledgement every two packets: analysis. When the transmitter reaches the end of its window, or times out, it goes back and retransmits packets starting from SNmin. choosing a new protocol restarts the simulation 7 Oct 2019 An initial idea of Stop-and-Wait ARQ. – Sender cannot send packet i+N until it has received the ACK for packet i. If transmission and ack receipt do not complete within the expected time, Flow Control – set of procedures used to restrict the amount of data that Go-Back-N ARQ (Cont. Go – Back – N ARQ. 9. configuration. ] 2. protocol. Go Back N ARQ • The transmitter has a "window" of N packets that can be sent without acknowledgements • This window ranges from the last value of RN obtained from the receiver (denoted SN min) to SN min+N-1 • When the transmitter reaches the end of its window, or times out, it goes back and retransmits packet SN min Let SN the use of the simple Go-Back-N ARQ protocol, where up to N code blocks are sent for each retransmission. Figure 2: Example showing the window size in Go-back-N must be strictly less than 2m. Assume no data frame In Go-Back-N ARQ, the size of the sender window must be less than 2^m; the size of the receiver window is always 1. Go-Back-N ARQ is a specific instance of the Automatic Repeat-reQuest (ARQ) Protocol, in which the sending process continues to send a number of frames specified by a window size even without to switch between different Go-Back- ARQ protocols, and subsequently adapted to the time varying nature of the wire-less channels. the sink process receives a packet, an acknowledgment may or may not be sent 25 Apr 2007 most common are known as stop-and-wait ARQ, go-back-N ARQ and . The receiveronly releases the packets to the upper layer in the correct order. Go back N Protocol in computer networks is a Sliding Window Protocol. We are able to address erasure errors on both the forward and reverse link by using block-transition probabilities. When the transmitter goes back 3, the retransmitted frame 0 is not what the receiver is waiting for. The throughput analysis takes into account the effects of feedback errors and imperfect channel When the transmitter goes back 3, the retransmitted frame 0 is not what the receiver is waiting for. The receiver's algorithm under go back n is the same as in stop and wait ARQ. Go-Back-N ARQ will use the HDLC, a bit-oriented data link control protocol, as well as a V. Oct 17, 2017 · PROBLEM 3 (16 points) Consider a TCP session whose current values of EstimatedRTT and DevRTT are 50 ms and 10 ms, respectively. Eytan Modiano Sender cannot send packet i+N until it has received the ACK for packet i This window ranges from the last value of RN obtained from the goes back and retransmits packet SN min. . As frame 9 hasn't been received yet, 9 must be greater than or equal to Slast. Stop and wait ARQ is not efficient there is only one frame that is sent can we from IT 121 at Ambo University Nov 10, 2011 · (Remember Stop-and-Wait method have packets numbered 0 and 1 only) b- Go-back-n ARQ. I need to run it on Windows and using C++ only. To make ter has received an ACK, this deadlock cannot be resolved through resenting channel quality at time t, taking values from the set. receives an ACK with a given value Rnext, it can assume that all prior frames. Problems with Go-Back-N • in Go-Back-N, receiver does There are three basic ARQ protocols these are stop and wait ARQ, Go back N ARQ, and Selective repeat ARQ. When there is a communication error, the Go-Back-N. In G-n, the recipient sends a single ack for a burst of N packets. Remember that Go-back-N ARQ utilization for error-free Examining the operation of Go. This would make the code very complicated. The number of bits in a sequence number is m-2. Nov 21, 2012 · 2. That's why it is called Go Back N. the same as the value of R, the frame is accepted, if not it is rejected. !! Formal!Methods!of!Communication. Im pasting the code for Go Back N ARQ here, but theres a problem with this one too. Jan 06, 2017 · DEFINITION • Go-Back-N ARQ is a specific instance of the automatic repeat request (ARQ) protocol, in which the sending process continues to send a number of frames specified by a window size even without receiving an acknowledgement(ACK) packet from the receiver. Is it the one given in Go-Back-N or (2 N-1). Example of Goback 7 ARQ Note that packet RN–1 must be accepted at Node B before a frame containing request RN can start transmission at Node B. number of selective repeat ARQ (SR-ARQ) and FEC code rate. Go Back N Protocol \|\| Sliding window protocol the chi square test, the p value The Wikipedia page on ARQ already answers your question about TCP: "The Transmission Control Protocol uses a variant of Go-Back-N ARQ to ensure reliable transmission of data over the Internet Protocol, which does not provide guaranteed delivery of packets; with Selective Acknowledgement (SACK), now 5th packet is lost. Chapter 3 - Data Link Layer - Free download as Powerpoint Presentation (. 0 Time pkt How to implement an ARQ stop and Wait Protocol. RDT Technologies Details. the send state variable (V(S)) is assigned the value of the received sequence number (N(R))). One such criterion used is to minimize the size of the S-PDUs. 3 The choice of N depends on data rate and latency for acknowledgment. The go- back-N protocol allows transmission of new packets before earlier ones are be the latest value of RN obtained from the receiver. While in the "good" channel state, the transmitter follows the basic Go-Back- procedure. Why in TCP's Go-Back-N Algorithm window size(N) has to be smaller than the sequence number space(S): S>N? I tried figuring it out myself but don't quiet get it Nov 17, 2013 · In Go-Back-N ARQ, if m is the number of bits for the sequence number, then the size of the send window must be at most 2m−1; the size of the receiver window is always 1. You are asked to specify a packet loss probability. Problems with Go-Back-N • in Go-Back-N, receiver does Go-Back-N ARQ It is a special case of the general sliding window protocol with the transmit window size of N and receive window size of 1. A similar analysis is also carried out for Rayleigh fding channel. The sender does not worry about these frames and keeps no copies of them. Go- Back-N ARQ is a fairly straight-forward protocol and has been adopted in Since we assume that the network is relatively error free, a static timeout value is used. 20 Jul 2017 Maximum sequence number in GBN is same as window size. If network used as its full capacity, then the bandwidth of network is_____ Mbps Show that in Go-Back-N ARQ, the size of the send window must be less than2 m. The transmitter then retransmits the requested I-frame followed by all successive I-frames. What is the current TimeoutInterval value? Suppose the next measured value of SampleRTT is 70 ms. Index Terms— ARQ, block fading, Go-Back-N protocol, Markov model , cannot be used. View Notes - The Data Link Layer: ARQ Protocols notes from CSC 4999 at Georgia State University. Before discussing on our reliable transmission protocol, we have a brief . The adaptive automatic repeat request (ARQ) strategy for a Gilbert–Elliott channel is depicted in Fig. for all ARQ processes are piggybacked onto every frame. 1. In our approach, if the window is full, the request is just ignored and the network layer needs to try again. Go-Back-N ARQ is a specific instance of the automatic repeat request (ARQ) protocol, in which Choosing a window size (N)[edit]. Also, because the transmit-ter has received an ACK, this deadlock cannot be resolved through a timeout. Sender in Sliding Window (using cumulative ACK) When sender has a new frame to transmit and there is some unused sequence number in sender window, use the next sequence number and send new frame labeled with that number. Reference [4] proposed a novel adaptive hybrid ARQ for low earth orbit (LEO) systems, in which the side information of turbo decoder is utilized to choose different modes of link layer protocol—Go-Back-N ARQ for large-scale burst errors and FEC with interleaving for short-scale (random) errors. , it varies from 0 to 7. When local area networks (LAN) have noisy environments, Selective Repeat ARQ is employed with packet segmentation. txt) or view presentation slides online. Figure 5. number of sequence bits needed = ceil (log 2 (1+window size)) = 3 PRACTICE PROBLEMS BASED ON GO BACK N PROTOCOL- Problem-01: A 20 Kbps satellite link has a propagation delay of 400 ms. Stack!Processing!Algorithm!for!Go!Back!N!ARQ!Protocol. For example, if m = 4, the sequence numbers go from 0 to 15, but the size of the window is just 8 (it is 15 in the Go-Back-N Protocol). The receiver refuses to accept any packet but the next one in sequence. c implements a reliable data transfer client over UDP in C Both of these programs use the go-back-n ARQ, that is lost data is automatically resent. I have the code for Go Back N ARQ which is similar to this algo but a few minor differences such as the size of sender and reciever window in this should be same and each frame has its own timer. On the other hand, Selective Repeat protocol retransmits only that frame that is damaged or lost. Frames in the opposite directions are 2. The sequence number is 3-bit wide, i. The returned socket address will contain the "wildcard address" (INADDR_ANY for IPv4 addresses, IN6ADDR_ANY_INIT for IPv6 address). In the sender's algorithm for go back n ARQ, the sender A keeps a window of n packets, In this paper the efficiency of the well-known Stop-and-Wait (SW) mechanism and the enhanced Burst Acknowledgment (BurstAck) behavior, utilized as a Go-Back-N (GBN) Automatic Repeat Request (ARQ) scheme with Sliding Window is studied. The frames are sequentially numbered and a finite number of frames are sent. Now, for sender window = 5. ! EE6777!!! Group&Members:& & Vinti&Vinti&(vv2236)& Garvit&Singh&(gs2731)& Go-Back-N ARQ is the sliding window protocol with wt>1, but a fixed wr=1. Go Back N Sender Rules •SN min = 0; SN max = 0 • Repeat – If SN max < SN min + N (entire window not yet sent) Send packet SN max ; SN max = SN max + 1; – If packet arrives from receiver with RN > SN min SN min = RN; – If SN min < SN max (there are still some unacknowledged packets) and sender cannot send any new packets Choose some packet between SN Go-Back-N ARQ Protocol Throughput Question. It's discarded and ambiguity removed. The protocol that determines who can transmit on a broadcast channel . Go-Back-N ARQ. Selective Repeat / Go Back N. The go-back-n protocol works well if errors are less, but if the line is poor it wastes a lot of bandwidth on retransmitted frames. e 6. Dec 06, 2006 · The go-back-n protocol provides APIs for application layer programs to send and receive messages, just like TCP does in practice. Lets say window size was also 4. The smaller window size means less efficiency in filling the pipe, but the fact that there are fewer duplicate frames can compensate for this. They transfer data back on the SDA line which is read by the master. The input[7:0] data in the master is a used line. In a noisy link a frame has a higher probability of damage, which means the resending of multiple Dec 28, 2016 · Flow Control and ARQ 3: Go-back-N and Selective-reject ARQs - Duration: 12:53. c- Selective-reject ARQ. In G-1, the recipient acks each packet individually. Go-Back-N protocol is design to retransmit all the frames that are arrived after the damaged or a lost frame. Selec-tive Repeat achieves a maximum of about 95% efficiency at around n=2500 bits. This window ranges from the last value of RN obtained from the receiver (denoted SNmin) to SNmin+n–1. Jacob Schrum 66,231 views. 11 for four frames if the round trip delay is 4 ms. – Receive packets in order – Receiver cannot accept packet out of sequence – Send RN = i + 1 => ACK for all packets up to and including i. ppt), PDF File (. next : 5 6 7. timeout period is usually set to a larger value than this, since the reliable protocols . It uses the concept of sliding window, and so is also called sliding window protocol. Consider n = 3, sequence numbers will be 2 3 = 8 ( i. TCP uses a variant of Go-Back-N ARQ to ensure reliable data transmission over the Internet protocol. A value of 0. Note that the optimum value of n is dependent on the value of no. A pair of test applications for the go-back-n protocol are provided to you. In Selective- Repeat ARQ, the size of the sender and receiver window must be at most 2m−1. Designed for easy installation within an ultra-lightweight and low profile enclosure, the T301n is ideal for venue owners and enterprises looking to quickly and economically deploy Wi-Fi in Graphene liquid cell electron microscopy can be used to observe nanocrystal dynamics in a liquid environment with greater spatial Despite Obama's remarks in 2016 and these two incidents, experts and officials contest the viability of the nuclear terrorism threat. 7-3. Go – Back – N ARQ provides for sending multiple frames before receiving the acknowledgement for the first frame. If TCP uses Selective Acknowledgement (SACK), Selective Repeat ARQ is used. Go back n Consider a network connecting two systems, 'A' and 'B' located 6000 km apart. Two nodes are connected to one another via a transmission link. It won't work if data is The T301n, a dual-band 802. Problem 2. It is a special case of the general Go-Back-N ARQ is a specific instance of the automatic repeat request (ARQ) protocol, in which the sending process continues to send a number of frames specified by a window size even without receiving an acknowledgement (ACK) packet from the receiver. Application layer . 3. [Hint: for simplicity, you may look at a sequence of N consecutive frames passing through the system, and compare different behaviors under these two protocols. Figures 3. pdf), Text File (. Upon receipt of a REJ frame (by the local node), the transmitter winds-back the sequence of I-frames pending transmission to the indicated I-frame (i. The sliding window method using cumulative ACK is known as the Go-Back-N ARQ protocol. Liveness: Eventually a packet is received, and eventually the ack for that packet is returned. Draw the flow diagram similar to Figure 11. Stations A and B are using Go-Back-N ARQ protocol to communicate where window size is N. c and client. To prevent deadlock when a NACK is lost in transition, we use a time-out mechanism as in [7], [8]. 4 of our textbook (Computer Networks, Tanenbaum, 2002). The S-PDUs are constructed so as to optimize performance in accordance with certain criteria. ⇨ Cannot detect even numbers of bit errors . 17 Send window for Selective Repeat ARQ Repeat Question 1 when Go-back-N ARQ is used and we assume that from the network layer, the request for the first and the second data frame arrive at time instant 0 ms and 1 ms, respectively. Sender sends 4 packets with sequence 16 Mar 2014 LECTURE OVERVIEW Go-Back N-ARQ Selective Repeat ARQ of the window cannot be sent before the window slides over them. It is stated that the sequence number is represented by 7 bits, thus a total of 27 2 = 128 2 = 64 frames can be monitored. In particular, we assume that frames, while in transit, cannot pass previously . The second region, colored in Figure 2. numbers to the right of this window define the frames that cannot be received. protocol may have to retransmit up to N outstanding frames. Go-Back-N ARQ is a form of ARQ protocol in which the sender continuously sends a number of packets (determined by the duration of transmission window) without receiving an ACK signal from the receiver. However, it does not guarantee delivery of data packets. three standard ARQ schemes, stop-and-wait,go-back-N,and ideal selective repeat ARQ. Explain the reason for moving from the stop-and-wait ARQ Protocol to the Go-Back-N ARQ Protocol. When the transmitting device fails to receive an acknowledgement signal to confirm the data has been received, it usually protocol will go into deadlock. It will receive data back from the slave if the rw bit is meant to read data. Go-Back-N ARQ (cont. Layered Architecture For the purpose of this assignment, assume that each node in the network has three layers: physical layer, datalink layer (DLC) and application layer. Go back N Selective Repeat. If a packet is lost in transit, following packets are ignored until the missing packet is retransmitted, a minimum loss of one round trip time. The efficiency of a Go-Back-N protocol is $\eta = \frac{N}{1+ 2 \frac{Propogation Delay}{Transmission Delay}}$ In this equation what is the exact value of N. So for the frame length n = 5 11 bits, the N value of a Go-Back N protocol will be 1370. Qualitatively, compare the relative performance of this protocol with Go-Back-N ARQ and with Stop-and-Wait ARQ. Oct 17, 2017 · I'll call it G-n The comparison is with Go-Back-1 is interesting. 11ac outdoor access point (AP), is designed explicitly for high density public venues such as airports, conventions centers, plazas & malls, and other dense urban environments. 2. Once the sender has sent all of the frames in its window, it will detect that all of the frames since the first lost frame are outstanding, and will go back to the sequence number of the last ACK it received from the receiver process and fill its window starting with that frame and continue the process over again. Go back N ARQ is an implementation of sliding window protocol like are all possible values of the ACK field in the messages currently propagating back to the Please refer this as a prerequisite article: Sliding Window Protocol (sender side)\| set 1 window size (1+2a) in order to increase the efficiency of stop and wait arq . Unlike Stop-and-Wait ARQ, this protocol allows several requests from the network layer without the need for other events to occur; we just need to be sure that the window is not full (line 12). 15 shows how the resulting Go-Back-N ARQ protocol operates. Go-Back-N Automatic Repeat Request (ARQ) Protocol The size can be fixed and set to the maximum value. 12 a, defines the range of sequence numbers belonging to the frames that are sent and have an unknown status. Go Back N ARQ • The transmitter has a "window" of N packets that can be sent without acknowledgements • This window ranges from the last value of RN obtained from the receiver (denotedSN min) to SN min +N-1 • When the transmitter reaches the end of its window, or times out, it goes back and retransmits packet SN min Let SN considered. 12:53. DEFINITION • Go-Back-N ARQ is a specific instance of the automatic repeat request (ARQ) protocol, in which the sending process continues to send a number of frames specified by a window size even without receiving an acknowledgement (ACK) packet from the receiver. 12 Send window for Go-Back-N ARQ . next : 6 7 8. 5 units long and use a time-out value of 4. Let alpha = 1/8 and beta = 1/4 be the exponential smoothing weights. • Receiver operates just like in Stop and Wait. Jan 06, 2017 · Go back-n protocol. In the _____protocol, if an acknowledgment for a transmitted frame does not arrive until a time-out occurs, the transmitter resends all outstanding frames. - gms298/Go-back-N-ARQ-Protocol. The transmitter employs the "go back n ARQ" scheme with n set to 10. Jul 20, 2017 · For sequence bits = n, number of sequence numbers = 2 n and window size = 2 n - 1. Go-Back-N ARQ is a specific instance of the automatic repeat request in cases of any p(ARQ) protocol, in which the sending process continues to send a number of frames specified by a window size even without receiving an acknowledgement (ACK) packet from the receiver. Therefore, Slast can be any of these: 4 to 9 in decimal. It begins with reading chunks of data from a file and sending them as packets (packetsize < 1KB) to the server that acknowledges the packets received. For sequence bits = n, number of sequence numbers = 2n and window size = 2n - ARQ Protocols: Go Back N and SRP. [[Jun 2013]] Computer Networks - I Computer / Information Science & Engineering VisvesvarayaTechnological University Answer The receiver has a control variable, which we call Rn (receiver, next frame expected), that holds the number of the next frame expected. 2 would mean that one in five packets (on average) are corrupted. for go back n arq protocol the value of n cannot be	CommonCrawl

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