diff --git "a/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/load_file.txt" "b/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/HdE0T4oBgHgl3EQfhgHx/content/tmp_files/load_file.txt" @@ -0,0 +1,1306 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf,len=1305 +page_content='Asymptotic decay function of the stationary tail probabilities along an arbitrary direction in a two-dimensional discrete-time QBD process Toshihisa Ozawa Faculty of Business Administration, Komazawa University 1-23-1 Komazawa, Setagaya-ku, Tokyo 154-8525, Japan E-mail: toshi@komazawa-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='jp Abstract We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD pro- cess for short) on Z2 + ×S0, where S0 is a finite set, and consider a topic remaining unresolved in our previous paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In that paper, the asymptotic decay rate of the stationary tail probabilities along an arbitrary direction has been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' It has also been clarified that if the asymptotic decay rate ξc, where c is a direction vector in N2, is less than a certain value θmax c , the sequence of the stationary tail probabilities along the direction c geometrically decays without power terms, asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In this article, we give the function that the sequence asymptotically decays according to when ξc = θmax c , but it contains an unknown parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' To determine the value of the parameter is a next challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Keywards: quasi-birth-and-death process, Markov modulated reflecting random walk, Markov additive process, asymptotic decay rate, asymptotic decay function, stationary distribution, ma- trix analytic method Mathematics Subject Classification: 60J10, 60K25 1 Introduction We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) {Y n} = {(Xn, Jn)} on Z2 + × S0, where S0 is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This model is a Markov modulated reflecting random walk (MMRRW for short) whose transitions are skip free, and the MMRRW is a kind of reflecting random walk (RRW for short) with a background process, where the transition probabilities of the RRW vary depending on the state of the background process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' One-dimensional QBD processes have been introduced by Macel Neuts and studied in the literature as one of the essential stochastic models in the queueing theory (see, for example, [1, 5, 7, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The 2d-QBD process is a two-dimensional version of one-dimensional QBD process, and it enable us to analyze, for example, two-node queueing networks and two-node polling models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume the 2d-QBD process {Y n} is positive recurrent and denote by ν = (ν(x,j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (x, j) ∈ Z2 + × S0) the stationary distribution, where ν(x,j) is the stationary probability that the process is in the state (x, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Our interest is asymptotics of the stationary distribution ν, especially, tail asymptotics in an arbitrary direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let an integer vector c = (c1, c2) be nonzero and nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Two typical objects of our study are the asymptotic decay rate ξc and asymptotic decay function hc(k) defined as, for j ∈ S0, ξc = − lim k→∞ 1 k log ν(kc,j), lim k→∞ ν(kc,j) hc(k) = gj, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='02434v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='PR] 6 Jan 2023 where gj is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Under a certain condition, the asymptotic decay rate of the probability sequence {νx+kc,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ≥ 0} does not depend on x and j if it exists, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of Ozawa [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the case where c = (1, 0) or c = (0, 1), the asymptotic decay rate ξc has been obtained in Ozawa [10], see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [14], and the asymptotic decay function hc(k) in Ozawa and Kobayashi [11], see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The results in the case where c = (c, 0) or c = (0, c) for c ≥ 2 are automatically obtained from those in [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the case where c = (c1, c2) ≥ (1, 1), the asymptotic decay rate ξc has been obtained in Ozawa [14], see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' A condition ensuring the asymptotic decay function is given by hc(k) = e−ξck, an exponential function without a power term, has also been given in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In this article, we give the expression of the asymptotic decay function hc(k) when c = (c1, c2) ≥ (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' To this end, we clarify the analytic properties of the vector generating function of the stationary probabilities along the direction c, ϕc(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The point z = eξc is a singular point of the vector function ϕc(z), and if ξc is equal to a certain value θmax c , z = eθmax c is a branch point of ϕc(z) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From this result, we obtain the expression of hc(k), but it contains an unknown parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' To determine the value of the parameter, it suffices to prove that ϕc(z) diverges elementwise at z = eθmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' It seems to be a hard work and we leave it as a next challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We also generalize a part of existing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' One crucial point in analyzing the asymptotic decay function is how to analytically extend the G-matrix function appeared in the vector generating function of the stationary probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In [11], it has been done under the assumption that all the eigenvalues of the G-matrix function are distinct, see Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This assumption is not easy to verify in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We, therefore, remove the assumption and give a general formula of the Jordan decomposition of the G-matrix function, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In Section 2, we describe the 2d-QBD process in detail and state assumptions and main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In Section 3, an analytic extension of the G-matrix function is given in a general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The definition of G-matrix in the reverse direction and its properties are also given in the same section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' They are used in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The proof of the main results is given in Sections 4, where we demonstrate that the vector function ϕc(z) is elementwise analytic in the open disk with radius eξc + ε for some ε > 0, except for the point z = eξc, and clarify its singularity at the point z = eξc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The asymptotic decay function is obtained from those results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The paper concludes with some remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 2 Model description and main results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 Model description We consider the same model as that described in [14] and use the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by I2 the set of all the subsets of {1, 2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', I2 = {∅, {1}, {2}, {1, 2}}, and we use it as an index set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Divide Z2 + into 22 = 4 exclusive subsets defined as Bα = {x = (x1, x2) ∈ Z2 +;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' xi > 0 for i ∈ α, xi = 0 for i ∈ {1, 2} \\ α}, α ∈ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let {Y n} = {(Xn, Jn)} be a 2d-QBD process on S = Z2 + × S0, where S0 = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let P be the transition probability matrix of {Y n} and represent it in block form as P = � Px,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2 + � , where Px,x′ = (p(x,j),(x′,j′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' j, j′ ∈ S0) and p(x,j),(x′,j′) = P(Y 1 = (x′, j′) | Y 0 = (x, j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For α ∈ I2 and i1, i2 ∈ {−1, 0, 1}, let Aα i1,i2 be a one-step transition probability block from a state in Bα, where we assume the blocks corresponding to impossible transitions are zero (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the level process is skip free, for every x, x′ ∈ Z2 +, Px,x′ is given by Px,x′ = � Aα x′−x, if x ∈ Bα for some α ∈ I2 and x′ − x ∈ {−1, 0, 1}2, O, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) We assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 2 Figure 1: Transition probability blocks Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The 2d-QBD process {Y n} is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Next, we define several Markov chains derived from the 2d-QBD process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For a nonempty set α ∈ I2, let {Y α n} = {(Xα n, Jα n )} be a process derived from the 2d-QBD process {Y n} by removing the boundaries that are orthogonal to the xi-axis for each i ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y {1} n } is a Markov chain on Z × Z+ × S0 whose transition probability matrix P {1} = (P {1} x,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z × Z+) is given as P {1} x,x′ = � � � � � A{1} x′−x, if x ∈ Z × {0} and x′ − x ∈ {−1, 0, 1} × {0, 1}, A{1,2} x′−x, if x ∈ Z × N and x′ − x ∈ {−1, 0, 1}2, O, otherwise, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) where N is the set of all positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y {2} n } on Z+ × Z × S0 and its transition probability matrix P {2} = (P {2} x,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z+ × Z) are analogously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y {1,2} n } is a Markov chain on Z2 × S0, whose transition probability matrix P {1,2} = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) is given as P {1,2} x,x′ = � A{1,2} x′−x, if x′ − x ∈ {−1, 0, 1}2, O, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) Regarding X{1} 1,n as the additive part, we see that the process {Y {1} n } = {(X{1} 1,n , (X{1} 2,n , J{1} n ))} is a Markov additive process (MA-process for short) with the background state (X{1} 2,n , J{1} n ) (with respect to MA-processes, see, for example, Ney and Nummelin [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y {2} n } = {(X{2} 2,n , (X{2} 1,n , J{2} n ))} is also an MA-process, where X{2} 2,n is the additive part and (X{2} 1,n , J{2} n ) the background state, and {Y {1,2} n } = {(X{1,2} 1,n , X{1,2} 2,n ), J{1,2} n )} an MA-process, where (X{1,2} 1,n , X{1,2} 2,n ) the additive part and J{1,2} n the background state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We call them the induced MA-processes de- rived from the original 2d-QBD process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let { ¯A{1} i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i ∈ {−1, 0, 1}} be the Markov additive kernel (MA-kernel for short) of the induced MA-process {Y {1} n }, which is the set of transition probability blocks and defined as, for i ∈ {−1, 0, 1}, ¯A{1} i = � ¯A{1} i,(x2,x′ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x2, x′ 2 ∈ Z+ � , ¯A{1} i,(x2,x′ 2) = � � � � � A{1} i,x′ 2−x2, if x2 = 0 and x′ 2 − x2 ∈ {0, 1}, A{1,2} i,x′ 2−x2, if x2 ≥ 1 and x′ 2 − x2 ∈ {−1, 0, 1}, O, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 3 X2 ^ B(2] B(1,2] [2] [1,2] 12 i1,i2 17 ,12 B(1) Bo 0 x1Let { ¯A{2} i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i ∈ {−1, 0, 1}} be the MA-kernel of {Y {2} n }, defined in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' With respect to {Y {1,2} n }, the MA-kernel is given by {A{1,2} i1,i2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i1, i2 ∈ {−1, 0, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The induced MA-processes {Y {1} n }, {Y {2} n } and {Y {1,2} n } are irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' According to [14], we assume several other technical conditions for the induced MA-process {Y {1,2} n }, concerning irreducibility and aperiodicity on subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let {Y + n } = {(X+ n , J+ n )} be a lossy Markov chain derived from the induced MA-process {Y {1,2} n } by restricting the state space of the additive part to N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y + n } is a Markov chain on N2 × S0 whose transition probability matrix P + is given as P + = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ N2), where P + is strictly substochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The process {Y + n } is also a lossy Markov chain derived from the original 2d-QBD process {Y n} by restricting the state space of the level to N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' {Y + n } is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ Z, let Z≤k and Z≥k be the set of integers less than or equal to k and that of integers greater than or equal to k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We also assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For what this assumption implies, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (i) The lossy Markov chain derived from the induced MA-process {Y {1,2} n } by restricting the state space to Z≤0 × Z≥0 × S0 is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (ii) The lossy Markov chain derived from {Y {1,2} n } by restricting the state space to Z≥0×Z≤0×S0 is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The stability condition of the 2d-QBD process has already been obtained in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let a{1}, a{2} and a{1,2} = (a{1,2} 1 , a{1,2} 2 ) be the mean drifts of the additive part in the induced MA-processes {Y {1} n }, {Y {2} n } and {Y {1,2} n }, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [12], the stability condition of the 2d-QBD process {Y n} is given as follows: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (i) In the case where a{1,2} 1 < 0 and a{1,2} 2 < 0, the 2d-QBD process {Y n} is positive recurrent if a{1} < 0 and a{2} < 0, and it is transient if either a{1} > 0 or a{2} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (ii) In the case where a{1,2} 1 ≥ 0 and a{1,2} 2 < 0, {Y n} is positive recurrent if a{1} < 0, and it is transient if a{1} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (iii) In the case where a{1,2} 1 < 0 and a{1,2} 2 ≥ 0, {Y n} is positive recurrent if a{2} < 0, and it is transient if a{2} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (iv) If one of a{1,2} 1 and a{1,2} 2 is positive and the other is non-negative, then {Y n} is transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For the explicit expression of the mean drifts, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [12] and its related parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The condition in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 that ensures the 2d-QBD process {Y n} is positive recurrent holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ν the stationary distribution of {Y n}, where ν = (νx, x ∈ Z2 +), νx = (ν(x,j), j ∈ S0) and ν(x,j) is the stationary probability that the 2d-QBD process is in the state (x, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 4 Figure 2: Domains Γ{1,2}, Γ{1} and Γ{2} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 Main results Let ¯A{1} ∗ (z) and ¯A{2} ∗ (z) be the matrix generating functions of the MA-kernels of {Y {1} n } and {Y {2} n }, respectively, defined as ¯A{1} ∗ (z) = � i∈{−1,0,1} zi ¯A{1} i , ¯A{2} ∗ (z) = � i∈{−1,0,1} zi ¯A{2} i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The matrix generating function of the MA-kernel of {Y {1,2} n } is given by A{1,2} ∗,∗ (z1, z2), defined as A{1,2} ∗,∗ (z1, z2) = � i1,i2∈{−1,0,1} zi1 1 zi2 2 A{1,2} i1,i2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let Γ{1}, Γ{2} and Γ{1,2} be regions in which the convergence parameters of ¯A{1} ∗ (eθ1), ¯A{2} ∗ (eθ2) and A{1,2} ∗,∗ (eθ1, eθ2) are greater than 1, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Γ{1} = {(θ1, θ2) ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' cp( ¯A{1} ∗ (eθ1)) > 1}, Γ{2} = {(θ1, θ2) ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' cp( ¯A{2} ∗ (eθ2)) > 1}, Γ{1,2} = {(θ1, θ2) ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' cp(A{1,2} ∗,∗ (eθ1, eθ2)) > 1}, where, for a nonnegative square matrix A with a finite or countable dimension, cp(A) denote the convergence parameter of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', cp(A) = sup{r ∈ R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' �∞ n=0 rnAn < ∞, entry-wise}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We have cp(A{1,2} ∗,∗ (eθ1, eθ2)) = spr(A{1,2} ∗,∗ (eθ1, eθ2))−1, where for a square complex matrix A, spr(A) is the spectral radius of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of Ozawa [13], cp( ¯A{1} ∗ (eθ))−1 and cp( ¯A{2} ∗ (eθ))−1 are log- convex in θ, and the closures of Γ{1} and Γ{2} are convex sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' spr( ¯A{1,2} ∗ (eθ1, eθ2)) is also log-convex in (θ1, θ2), and the closure of Γ{1,2} is a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of Ozawa [13], Γ{1,2} is bounded under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We depict an example of the domains Γ{1,2}, Γ{1} and Γ{2} in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We define several extreme values and several functions with respect to the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i ∈ {1, 2}, define θmin i and θmax i as θmin i = inf{θi ∈ R : (θ1, θ2) ∈ Γ{1,2}}, θmax i = sup{θi ∈ R : (θ1, θ2) ∈ Γ{1,2}}, and for a direction vector c = (c1, c2) ∈ N2, θmax c as θmax c = sup{c1θ1 + c2θ2 : (θ1, θ2) ∈ Γ{1,2}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For θ1 ∈ [θmin 1 , θmax 1 ], there exist two real solutions to equation spr(A{1,2} ∗,∗ (eθ1, eθ2)) = 1, counting multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote them by θ2 = η2(θ1) and θ2 = ¯η2(θ1), respectively, where η2(θ1) ≤ ¯η2(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For 5 C101 + C202 = 0max 02 个 01 = 0 02 = 02 r(1) spr amax n2 (0) r(2] r(1,2] n2(0) > 0 0 1 0 01 0Figure 3: Classification θ2 ∈ [θmin 2 , θmax 2 ], also denote by θ1 = η1(θ2) and θ1 = ¯η1(θ2) the two real solutions to the equation spr(A{1,2} ∗,∗ (eθ1, eθ2)) = 1, where η1(θ2) ≤ ¯η1(θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i ∈ {1, 2}, define θ∗ i as θ∗ i = sup{θi ∈ R : (θ1, θ2) ∈ Γ{i}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For another characterization of θ∗ i , see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7 of Ozawa [10], where θ∗ i is denoted by z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In terms of these points and functions, we geometrically classify the model into four types according to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define two points Q1 and Q2 as Q1 = (θ∗ 1, ¯η2(θ∗ 1)) and Q2 = (¯η1(θ∗ 2), θ∗ 2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Using these points, we define the following classification (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Type 1: θ∗ 1 ≥ ¯η1(θ∗ 2) and ¯η2(θ∗ 1) ≤ θ∗ 2, Type 2: θ∗ 1 < ¯η1(θ∗ 2) and ¯η2(θ∗ 1) > θ∗ 2, Type 3: θ∗ 1 ≥ ¯η1(θ∗ 2) and ¯η2(θ∗ 1) > θ∗ 2, Type 4: θ∗ 1 < ¯η1(θ∗ 2) and ¯η2(θ∗ 1) ≤ θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [14], for any direction vector c = (c1, c2) ∈ N2, the asymptotic decay rate in the direction c is space homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, we denote it by ξc, which satisfies, for any (x, j) ∈ Z2 + × S0, ξc = − lim k→∞ 1 n log ν(x+kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4) The asymptotic decay rate ξc has already been obtained in [14], and as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [14], it is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let c = (c1, c2) be an arbitrary direction vector in N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Type 1: ξc = � � � c1θ∗ 1 + c2¯η2(θ∗ 1) if − c1 c2 < ¯η′ 2(θ∗ 1), θmax c if ¯η′ 2(θ∗ 1) ≤ − c1 c2 ≤ ¯η′ 1(θ∗ 2)−1, c1¯η1(θ∗ 2) + c2θ∗ 2 if − c1 c2 > ¯η′ 1(θ∗ 2)−1, where ¯η′ 2(x) = d dx ¯η2(x) and ¯η′ 1(x) = d dx ¯η1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Type 2: ξc = � � � c1θ∗ 1 + c2¯η2(θ∗ 1) if − c1 c2 ≤ θ∗ 2−¯η2(θ∗ 1) ¯η1(θ∗ 2)−θ∗ 1 , c1¯η1(θ∗ 2) + c2θ∗ 2 if − c1 c2 > θ∗ 2−¯η2(θ∗ 1) ¯η1(θ∗ 2)−θ∗ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 0* 02 r(1,2] > 01 0* 1 1 Type 1 Type 2 Type 3 Type 4Type 3: ξc = c1¯η1(θ∗ 2) + c2θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Type 4: ξc = c1θ∗ 1 + c2¯η2(θ∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The asymptotic decay function hc(k) in the direction c is defined as the function that satisfies, for some positive vector gc, lim k→∞ νkc hc(k) = gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) It is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let c be an arbitrary direction vector in N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' hc(k) = � k− 1 2 (2l−1)e−ξck if ¯η′ 2(θ∗ 1) < − c1 c2 < ¯η′ 1(θ∗ 2)−1 in Type 1, e−ξck otherwise, as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) where l is some positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Except for the case where ¯η′ 2(θ∗ 1) ≤ − c1 c2 ≤ ¯η′ 1(θ∗ 2)−1 in Type 1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 has already been proved in [14], see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, to this end, it suffices to prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1 and set c = (c1, c2) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, the asymptotic decay function hc(k) is given as hc(k) = � k− 1 2 (2l−1)e−θmax c k if ¯η′ 2(θ∗ 1) < − c1 c2 = −1 < ¯η′ 1(θ∗ 2)−1, e−θmax c k if ¯η′ 2(θ∗ 1) = −1 or ¯η′ 1(θ∗ 2) = −1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7) where l is some positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From this proposition, we can obtain the same result for a general direction vector c ∈ N2, by using the block state process derived from the original 2d-QBD process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We, therefore, prove the proposition in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From the corresponding results for a 2d-RRW without a background process obtained in Malyshev [6], it is expected that the value of l in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 is one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', hc(k) = k− 1 2 e−ξck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 3 Preliminaries Let z and w be complex valuables unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For a positive number r, denote by ∆r the open disk of center 0 and radius r on the complex plain, and ∂∆r the circle of the same center and radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We denote by ¯∆r the closure of ∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For a, b ∈ R+ such that a < b, let ∆a,b be an open annular domain on C defined as ∆a,b = {z ∈ C : a < |z| < b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We denote by ¯∆a,b the closure of ∆a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For r > 0, ε > 0 and θ ∈ [0, π/2), define ˜∆r(ε, θ) = {z ∈ C : |z| < r + ε, z ̸= r, | arg(z − r)| > θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For r > 0, we denote by “ ˜∆r ∋ z → r” that ˜∆r(ε, θ) ∋ z → r for some ε > 0 and some θ ∈ [0, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the rest of the paper, instead of proving that a function f(z) is analytic in ˜∆r(ε, θ) for some ε > 0 and θ ∈ [0, π/2), we often demonstrate that the function f(z) is analytic in ∆r and on ∂∆r \\ {r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In order to give general results, this section is described independently from other parts of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 Analytic extension of a G-matrix function First, we define a G-matrix function according to Ozawa and Kobayashi [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i, j ∈ {−1, 0, 1}, let Ai,j be a substochastic matrix with a finite dimension s0, and define the following matrix functions: A∗,j(z) = � i∈{−1,0,1} ziAi,j, j = −1, 0, 1, A∗,∗(z, w) = � i,j∈{−1,0,1} ziwjAi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' A∗,∗(1, 1) is stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let χ(z, w) be the spectral radius of A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='∗(z, w), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', χ(z, w) = spr(A∗,∗(z, w)), and Γ be a domain on R2 defined as Γ = {(θ1, θ2) ∈ R2 : χ(eθ1, eθ2) < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume the following condition throughout this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The Markov modulated random walk on Z2 × {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0} that is governed by {Ai,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i, j ∈ {−1, 0, 1}} is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Under this assumption, A∗,∗(1, 1) is also irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11], Γ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since χ(eθ1, eθ2) is convex in (θ1, θ2) ∈ R2, the closure of Γ is a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define extreme points θmin 1 and θmax 2 as follows: θmin 1 = inf (θ1,θ2)∈Γ θ1, θmax 1 = sup (θ1,θ2)∈Γ θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For θ1 ∈ [θmin 1 , θmax 1 ], let θ2(θ1) and ¯θ2(θ1) be the two real solutions to equation χ(eθ1, eθ2) = 1, counting multiplicity, where θ2(θ1) ≤ ¯θ2(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We set zmin 1 = eθmin 1 and zmax 1 = eθmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For n ≥ 1, define the following set of index sequences: In = � i(n) ∈ {−1, 0, 1}n : k � l=1 il ≥ 0 for k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', n − 1} and n � l=1 il = −1 � , where i(n) = (i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', in), and define the following matrix function: Dn(z) = � i(n)∈In A∗,i1(z)A∗,i2(z) · · · A∗,in(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function G(z) as G(z) = ∞ � n=1 Dn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11], this matrix series absolutely converges entry-wise in z ∈ ¯∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We call this G(z) the G-matrix function generated from {Ai,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i, j ∈ {−1, 0, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For z ∈ ¯∆zmin 1 ,zmax 1 , G(z) satisfies the inequality |G(z)| ≤ G(|z|) and the following matrix quadratic equation: A∗,−1(z) + A∗,0(z)G(z) + A∗,1(z)G(z)2 = G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) Furthermore, for z ∈ [zmin 1 , zmax 1 ], it is the minimum nonnegative solution to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, G(z) is an extension of a usual G-matrix in the queueing theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' see, for example, [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 of [11], we see that, for z ∈ [zmin 1 , zmax 1 ], the Perron-Frobenius eigenvalue of G(z) is given by eθ2(log z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', spr(G(z)) = eθ2(log z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11], G(z) satisfies I − A∗,∗(z, w) = w−1 (I − A∗,0(z) − wA∗,1(z) + A∗,1(z)G(z)) (wI − G(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11], the following property holds true for G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 8 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' G(z) is entry-wise analytic in the open annular domain ∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We give the eigenvalues of G(z) according to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that our final aim in this subsection is to give an analytic extension of G(z) through its Jordan canonical form without assuming all the eigenvalues of G(z) are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' On the other hand, in [11], the eigenvalues were assumed to be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function L(z, w) as L(z, w) = zw(I − A∗,∗(z, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Each entry of L(z, w) is a polynomial in z and w with at most degree 2 for each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We use a notation Ξ, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let f(z, w) be an irreducible polynomial in z and w and assume its degree with respect to w is m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let a(z) be the coefficient of wm in f(z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set Ξ(f) as Ξ(f) = {z ∈ C : a(z) = 0 or (f(z, w) = 0 and fw(z, w) = 0 for some w ∈ C)}, where fw(z, w) = (∂/∂w)f(z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Each point in Ξ(f) is an algebraic singularity of the algebraic function w = α(z) defined by polynomial equation f(z, w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each point z ∈ C \\ Ξ(f), f(z, w) = 0 has just m distinct solutions, which correspond to the m branches of the algebraic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let φ(z, w) be a polynomial in z and w defined as φ(z, w) = det L(z, w) and mφ its degree with respect to w, where s0 ≤ mφ ≤ 2s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let α1(z), α2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', αmφ(z) be the mφ branches of the algebraic function w = α(z) defined by the polynomial equation φ(z, w) = 0, counting multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We number the brunches so that they satisfy the following: (1) For every z ∈ ¯∆zmin 1 ,zmax 1 and for every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0}, |αk(z)| ≤ eθ2(log |z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) For every z ∈ ¯∆zmin 1 ,zmax 1 and for every k ∈ {s0 + 1, s0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ}, |αk(z)| ≥ e¯θ2(log |z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3) For every z ∈ [zmin 1 , zmax 1 ], αs0(z) = eθ2(log z) and αs0+1(z) = e¯θ2(log z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This is possible by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 of [11], the G-matrix function of G(z) satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every z ∈ ¯∆zmin 1 ,zmax 1 , the eigenvalues of G(z) are given by α1(z), α2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', αs0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Without loss of generality, we assume that, for some nφ ∈ N and l1, l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lnφ ∈ N, the polynomial φ(z, w) is factorized as φ(z, w) = f1(z, w)l1f2(z, w)l2 · · · fnφ(z, w)lnφ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) where fk(z, w), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', nφ, are irreducible polynomials in z and w and they are relatively prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the field of coefficients of polynomials is C, this factorization is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ}, αk(z) is a branch of the algebraic function w = α(z) defined by the polynomial equation fn(z, w) = 0 for some n ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', nφ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We denote such n by q(k), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', fq(k)(z, αk(z)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since αs0(z) is the Perron-Frobenius eigenvalue of G(z) when z ∈ [zmin 1 , zmax 1 ], the multiplicity of αs0(z) is one and we have lq(s0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set E1 as E1 = nφ � n=1 Ξ(fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 9 Since, for every n, the polynomial fn(z, w) is irreducible and not identically zero, the point set E1 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Every branch αk(z) is analytic in C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set E2 as E2 = {z ∈ C \\ E1 : fn(z, w) = fn′(z, w) = 0 for some n, n′ ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', nφ} such that n ̸= n′ and for some w ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since, for any n, n′ such that n ̸= n′, fn(z, w) and fn′(z, w) are relatively prime, the point set E2 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that every branch αk(z) is analytic in a neighborhood of any z0 ∈ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ} and for every z ∈ C \\ (E1 ∪ E2), the multiplicity of αk(z) as a zero of det L(z, w) is equal to lq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This means that, for every z ∈ ¯∆zmin 1 ,zmax 1 \\ (E1 ∪ E2), the multiplicity of the eigenvalue αk(z) of G(z) is lq(k), which does not depend on z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a positive integer m0 as m0 = s0 � k=1 1 lq(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This m0 is the number of different branches in {αi(z) : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0} when z ∈ C \\ (E1 ∪ E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote the different branches by ˇαk(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, so that ˇαm0(z) = αs0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Instead of using q(k), we define a function ˇq(k) so that lˇq(k) indicates the multiplicity of ˇαk(z) when z ∈ C\\(E1∪E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We always have lˇq(m0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We give the Jordan normal form of G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a domain Ω as Ω = ∆zmin 1 ,zmax 1 \\ (E1 ∪ E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, define a positive integer tk,i as tk,i = min z∈Ω dim Ker (ˇαk(z)I − G(z))i and a point set Gk,i as Gk,i = {z ∈ Ω : dim Ker (ˇαk(z)I − G(z))i > tk,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ˇαk(z) and G(z) are analytic in Ω, we see from the proof of Theorem S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [3] that each Gk,i is an empty set or a set of discrete complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, define a nonnegative integer sk,i as sk,i = 2tk,i − tk,i+1 − tk,i−1, where tk,0 = 0 and tk,lˇq(k)+1 = lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, define a positive integer mk,0 and point set EG k as mk,0 = tk,1, EG k = lˇq(k) � i=1 Gk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' When z ∈ Ω \\ EG k , this mk,0 is the number of Jordan blocks of G(z) with respect to the eigenvalue ˇαk(z) and, for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, sk,i is the number of Jordan blocks whose dimension is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, the Jordan normal form of G(z) takes a common form in z ∈ Ω \\ �m0 k=1 EG k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, denote by mk,i the dimension of the i-th Jordan block of G(z) with respect to the eigenvalue ˇαk(z), where we number the Jordan blocks so that if i ≤ i′, mk,i ≥ mk,i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, they satisfy �mk,0 i=1 mk,i = lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by Jn(λ) the n-dimensional Jordan block of eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For z ∈ Ω \\ �m0 k=1 EG k , the Jordan normal form of G(z), JG(z), is given by JG(z) = diag(Jmk,i(ˇαk(z)), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4) where mm0,0 = 1 and Jmm0,1(ˇαm0(z)) = αs0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that the matrix function JG(z) is defined on C and analytic in C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' An analytic extension of G(z) is given by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 10 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' There exist vector functions: ˇvL k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i, such that they are analytic in C \\ E1 and satisfy for every z ∈ ∆zmin 1 ,zmax 1 \\ (E1 ∪ E0) that G(z) = T L(z)JG(z)(T L(z))−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) where E0 is a set of discrete complex numbers and matrix function T L(z) is defined as T L(z) = �ˇvL k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 is elementary and very lengthy, we give it in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, {ˇvL k,i,j(z)} is the set of the generalized eigenvectors of G(z), but we denote them with superscript L since they are generated from the matrix function L(z, w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set EL T as EL T = {z ∈ C \\ E1 : det T L(z) = 0}, which is an empty set or a set of discrete complex numbers since det T L(z) is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˇG(z) as ˇG(z) = T L(z)JG(z)(T L(z))−1 = T L(z)JG(z) adj(T L(z)) det(T L(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) Then, it is entry-wise analytic in C\\(E1∪EL T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 and the identity theorem for analytic functions, this ˇG(z) is an analytic extension of the matrix function G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, we denote ˇG(z) by G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, G(z) is entry-wise analytic in ∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The following corollary asserts that G(z) is also analytic on the outside boundary of ∆zmin 1 ,zmax 1 except for the point z = zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The extended G-matrix function G(z) is entry-wise analytic on ∂∆zmax 1 \\ {zmax 1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since this corollary can be proved in a manner similar to that used in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7 of [11], we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˇuL m0,1,1(z) the last row of the matrix function (T L(z))−1, and define a diagonal matrix function Js0(z) as Js0(z) = diag � 0 · · 0 αs0(z) � , where αs0(z) = ˇαm0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, since mm0,0 = 1 and mm0,1 = 1, we obtain the following decomposition of G(z) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6): G(z) = G†(z) + αs0(z)ˇvL m0,1,1(z)ˇuL m0,1,1(z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7) where G†(z) = T L(z)(JG(z) − Js0(z))(T L(z))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By the definition, G(z) satisfies, for n ≥ 1, G(z)n = G†(z)n + αs0(z)nˇvL m0,1,1(z)ˇuL m0,1,1(z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8) and G†(z), for z ∈ ¯∆zmin 1 ,zmax 1 , spr(G†(z)) ≤ spr(G†(|z|) < spr(G(|z|)) = αs0(|z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, in a neighborhood of z = zmax 1 , we have spr(G†(z)) < αs0(zmax 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the point z = zmax 1 is a branch point of ˇαm0(z) (= αs0(z)), there exists a function ˜αs0(ζ) being analytic in a neighborhood of ζ = 0 and satisfying ˇαm0(z) = αs0(z) = ˜αs0((zmax 1 − z) 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 11 Let ˜vs0(ζ) be a vector function satisfying L(zmax 1 − ζ2, ˜αs0(ζ))˜vs0(ζ) = 0, where ˜vs0(ζ) is elementwise analytic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˜T(ζ) the matrix function given by replacing the last column of T L(zmax 1 − ζ2) with ˜vs0(ζ) and by ˜us0(ζ) the last row of ˜T(ζ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By the definition, ˜T(ζ) as well as ˜us0(ζ) is entry-wise analytic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a diagonal matrix function ˜Js0(ζ) as ˜Js0(ζ) = diag � 0 · · 0 ˜αs0(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For later use, we give the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' There exists a matrix function ˜G(ζ) being entry-wise analytic in a neighborhood of ζ = 0 and satisfying G(z) = ˜G((zmax 1 −z) 1 2 ) in a neighborhood of z = zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This ˜G(ζ) is represented as ˜G(ζ) = ˜G†(ζ) + ˜αs0(ζ)˜vs0(ζ)˜us0(ζ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) where ˜G†(ζ) is a matrix function being entry-wise analytic in a neighborhood of ζ = 0 and satisfying G†(z) = ˜G†((zmax 1 − z) 1 2 ) in a neighborhood of z = zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In a neighborhood of ζ = 0, spr( ˜G†(ζ)) < ˜αs0(0) = αs0(zmax 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Give ˜G†(ζ) as ˜G†(ζ) = ˜T(ζ)(JG(zmax 1 − ζ2) − Js0(zmax 1 − ζ2)) ˜T(ζ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), we obtain the results of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The following limit with respect to αs0(z) (= ˇαm0(z)) is given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 of [11] (also see Lemma 10 of [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' lim ˜∆zmax 1 ∋z→zmax 1 αs0(zmax 1 ) − αs0(z) (zmax 1 − z) 1 2 = −αs0,1 = √ 2 � −¯ζ1,w2(ζ2(zmax 1 )) > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) where z = ¯ζ1(w) is the larger one of two real solutions to equation χ(z, w) = 1 and ¯ζ1,w2(w) = (d2/dw2) ¯ζ1(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let R(z) be the rate matrix function generated from {Ai,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i, j = −1, 0, 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' for the definition of R(z), see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function N(z) as N(z) = (I − A∗,0(z) − A∗,1(z)G(z))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' N(z) is well defined for every z ∈ ¯∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The extended G(z) satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' lim ˜∆zmax 1 ∋z→zmax 1 G(zmax 1 ) − G(z) (zmax 1 − z) 1 2 = −G1 = −αs0,1N(zmax 1 )vR(zmax 1 )uG s0(zmax 1 ) ≥ O, ̸= O, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='11) where uG s0(zmax 1 ) is the left eigenvector of G(zmax 1 ) with respect to the eigenvalue eθ2(log zmax 1 ) = αs0(zmax 1 ), vR(zmax 1 ) the right eigenvector of R(zmax 1 ) with respect to the eigenvalue e−¯θ2(log zmax 1 ) = e−θ2(log zmax 1 ) and they satisfy uG s0(zmax 1 )N(zmax 1 )vR(zmax 1 ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since this lemma can be proved in a manner similar to that used in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 of [11], we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 G-matrix in the reverse direction and its properties Let A−1, A0 and A1 be square nonnegative matrices with a finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function A∗(z) and matrix Q as A∗(z) = z−1A−1 + A0 + zA1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12) Q = � � � � � A0 A1 A−1 A0 A1 A−1 A0 A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) We assume: (a1) Q is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (a2) The infimum of the maximum eigenvalue of A∗(eθ) in θ ∈ R is less than or equal to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', infθ∈R spr(A∗(eθ)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, there are two real solutions to equation cp(A∗(eθ)) = 1, counting multiplicity, see comments to Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We denote the solutions by θ and ¯θ, where θ ≤ ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The rate matrix and G-matrix generated from the triplet {A−1, A0, A1} also exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' we denote them by R and G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' R and G are the minimal nonnegative solutions to the following matrix quadratic equations: R = R2A−1 + RA0 + A1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='14) G = A−1 + A0G + A1G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15) We have I − A∗(z) = (I − zR)(I − H)(I − z−1G), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) spr(R) = e−¯θ, spr(G) = eθ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='17) where H = A0 + A1G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' see, for example, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We define a rate matrix and G-matrix in the reverse direction generated from the triplet {A−1, A0, A1}, denoted by Rr and Gr, as the minimal nonnegative solutions to the following matrix quadratic equations: Rr = (Rr)2A1 + RrA0 + A−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='18) Gr = A1 + A0Gr + A−1(Gr)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='19) In other words, Rr and Gr are, respectively, the rate matrix and G-matrix generated from the triplet by exchanging A−1 and A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since z−1A1 + A0 + zA−1 = A∗(z−1), we obtain by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='17) that I − A∗(z−1) = (I − zRr)(I − Hr)(I − z−1Gr), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='20) spr(Rr) = eθ, spr(Gr) = e−¯θ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='21) where Hr = A0 + A−1Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We use the following property in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let v be the right eigenvector of G with respect to the eigenvalue eθ and vr that of Gr with respect to the eigenvalue e−¯θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Gv = eθv and Grvr = e−¯θvr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If θ = ¯θ, we have v = vr, up to multiplication by a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='20), we obtain A∗(eθ)v = v, A∗(e ¯θ)vr = A∗(eθ)vr = vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since spr(A∗(eθ)) = 1 and A∗(eθ) is irreducible, the right eigenvector of A∗(eθ) with respect to the eigenvalue of 1 is unique, up to multiplication by a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This implies v = vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 13 4 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 Methodology and outline of the proof Define the vector generating function of the stationary probabilities in direction c ∈ N2, ϕc(z), as ϕc(z) = ∞ � k=0 zkνkc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Also define zmin c and zmax c as zmin c = eθmin c and zmax c = eθmax c , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hereafter, we set c = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In order to obtain the asymptotic function of the stationary tail probability in the direction c = (1, 1), we apply the following lemma to the vector generating function ϕc(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 (Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 of Flajolet and Sedgewick [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let f be a generating function of a sequence of real numbers {an, n ∈ Z+}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', f(z) = �∞ n=0 anzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If f(z) is singular at z = z0 > 0 and analytic in ˜∆z0(ε, θ) for some ε > 0 and some θ ∈ [0, π/2) and if it satisfies lim ˜∆z0∋z→z0 (z0 − z)αf(z) = c0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) for α ∈ R \\ {0, −1, −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='} and some nonzero constant c0 ∈ R, then lim n→∞ �nα−1 Γ(α) z−n 0 �−1 an = c (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) for some real number c, where Γ(z) is the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This means that the asymptotic function of the sequence {an} is given by nα−1z−n 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For the purpose, we prove the following propositions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1 ≤ 1/¯η′ 2(θ∗ 1), the vector function ϕc(z) is elementwise analytic in ˜∆zmax c (ε, θ) for some ε > 0 and some θ ∈ [0, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1 ≤ 1/¯η′ 2(θ∗ 1), there exist a vector function ˜ϕc(ζ) being meromorphic in a neighborhood of ζ = 0 and satisfying ϕc(z) = ˜ϕc((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) < −1 < 1/¯η′ 2(θ∗ 1), the point ζ = 0 is a pole of ˜ϕc(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1 or ¯η′ 2(θ∗ 1) = −1, it is a pole of ˜ϕc(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2, if ¯η′ 1(θ∗ 2) < −1 < 1/¯η′ 2(θ∗ 1), the Puiseux series of ϕc(z) is represented as ϕc(z) = ∞ � k=−1 ϕc 1,k(zmax c − z) k 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) where {ϕc 1,k} is a series of coefficient vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1 or ¯η′ 2(θ∗ 1) = −1, it is represented as ϕc(z) = ∞ � k=−2 ϕc 2,k(zmax c − z) n 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4) where {ϕc 2,k} is a series of coefficient vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let l be a positive integer such that ϕc 1,l−2 ̸= 0 and ϕc 1,k−2 = 0 for all positive integer k less than l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3), we obtain hc(k) = k− 1 2 (2l−1)(zmax c )−k = k− 1 2 (2l−1)e−θmax c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the former half of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) = −1 or ¯η′ 2(θ∗ 1) = −1, ϕc(z) satisfies the following property, which will be proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 14 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, we have, for some positive vectors uc 1 and uc 2, lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc(z) = � � � uc 1 if ¯η′ 1(θ∗ 2) = −1 and ¯η′ 2(θ∗ 1) < −1, uc 2 if ¯η′ 1(θ∗ 2) < −1 and ¯η′ 2(θ∗ 1) = −1, uc 1 + uc 2 if ¯η′ 1(θ∗ 2) = ¯η′ 2(θ∗ 1) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) Hence, ϕc 2,−2 is positive, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, we obtain hc(k) = (zmax c )−k = e−θmax c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the latter half of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1 and ¯η′ 1(θ∗ 2) < −c1/c2 = −1 < 1/¯η′ 2(θ∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If the vector function ϕc(z) diverges at z = zmax c , the coefficient vector ϕc 1,−1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) must be nonzero and , by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, we have hc(k) = k− 1 2 (zmax c )−k = k− 1 2 e−θmax c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 Proof of Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 Recall that the direction vector c is set as c = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Notation of this subsection follows [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by Φ{1,2} = (Φ{1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) the fundamental matrix (potential matrix) of P {1,2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Φ{1,2} = �∞ n=0(P {1,2})n, where P {1,2} = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) is the transition probability matrix of the induced MA-process {Y {1,2} n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For x ∈ Z2, define the matrix generating function of the blocks of Φ{1,2} in direction c, Φc x,∗(z), as Φc x,∗(z) = ∞ � k=−∞ zkΦ{1,2} x,kc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' According to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) of [14], we divide ϕc(z) into three parts as follows: ϕc(z) = ϕc 0(z) + ϕc 1(z) + ϕc 2(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) where ϕc 0(z) = � i1,i2∈{−1,0,1} ν(0,0)(A∅ i1,i2 − A{1,2} i1,i2 )Φc (i1,i2),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7) ϕc 1(z) = ∞ � k=1 � i1,i2∈{−1,0,1} ν(k,0)(A{1} i1,i2 − A{1,2} i1,i2 )Φc (k+i1,i2),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8) ϕc 2(z) = ∞ � k=1 � i1,i2∈{−1,0,1} ν(0,k)(A{2} i1,i2 − A{1,2} i1,i2 )Φc (i1,k+i2),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) According to [14], we focus on ϕc 2(z) and consider another skip-free MA-process generated from {Y {1,2} n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The MA-process is { ˆY n} = {( ˆXn, ˆJn)} = {( ˆX1,n, ˆX2,n), ( ˆRn, ˆJn)}, where ˆX1,n = X{1,2} 1,n , ˆX2,n and ˆRn are the quotient and remainder of X{1,2} 2,n − X{1,2} 1,n divided by 2, respectively, and ˆJn = J{1,2} n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The state space of { ˆY n} is Z2 × {0, 1} × S0 and the additive part { ˆXn} is skip free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From the definition, if ˆXn = (x1, x2) and ˆRn = r in the new MA-process, it follows that X{1,2} 1,n = x1, X{1,2} 2,n = x1 + 2x2 + r in the original MA-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, ˆY n = (k, 0, 0, j) means 15 Y {1,2} n = (k, k, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˆP = ( ˆPx,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x′ ∈ Z2) the transition probability matrix of { ˆY n},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' which is given as ˆPx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='x′ = � ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} x′−x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if x′ − x ∈ {−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 1}2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' where ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 = � O O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 O � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 = �O O O O � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 = �O O O O � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 = � O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 O O � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆA{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 = � A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='0 O A{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2} 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˆΦ = (ˆΦx,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) the fundamental matrix of ˆP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', ˆΦ = �∞ n=0( ˆP)n, and for x = (x1, x2) ∈ Z2, define a matrix generating function ˆΦx,∗(z) as ˆΦx,∗(z) = ∞ � k=−∞ zk ˆΦx,(k,0) = � Φc (x1,x1+2x2),∗(z) Φc (x1,x1+2x2−1),∗(z) Φc (x1,x1+2x2+1),∗(z) Φc (x1,x1+2x2),∗(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) We consider analytic properties of the matrix function Φc (x1,x1+2x2),∗(z) through ˆΦ(x1,x2),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define blocks ˆA{2} i1,i2, i1, i2 ∈ {−1, 0, 1}, as ˆA{2} −1,1 = ˆA{2} −1,0 = ˆA{2} −1,−1 = O and ˆA{2} 0,1 = � O O A{2} 0,1 O � , ˆA{2} 0,0 = � A{2} 0,0 A{2} 0,1 A{2} 0,−1 A{2} 0,0 � , ˆA{2} 0,−1 = � O A{2} 0,−1 O O � , ˆA{2} 1,1 = �O O O O � , ˆA{2} 1,0 = � A{2} 1,1 O A{2} 1,0 A{2} 1,1 � , ˆA{2} 1,−1 = � A{2} 1,−1 A{2} 1,0 O A{2} 1,−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i1, i2 ∈ {−1, 0, 1}, define the following matrix generating functions: ˆA{1,2} ∗,i2 (z) = � i∈{−1,0,1} zi ˆA{1,2} i,i2 , ˆA{1,2} i1,∗ (z) = � i∈{−1,0,1} zi ˆA{1,2} i1,i , ˆA{2} ∗,i2(z) = � i∈{0,1} zi ˆA{2} i,i2, ˆA{2} i1,∗(z) = � i∈{−1,0,1} zi ˆA{2} i1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a vector generating function ˆϕ2(z) as ˆϕ2(z) = �ˆϕ2,1(z) ˆϕ2,2(z) � = ∞ � k=1 � i1,i2∈{−1,0,1} ˆν(0,k)( ˆA{2} i1,i2 − ˆA{1,2} i1,i2 )ˆΦ(i1,k+i2),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='11) where, for x = (x1, x2) ∈ Z2 +, ˆνx = � ν(x1,x1+2x2) ν(x1,x1+2x2+1) � and hence, for k ≥ 0, ˆν(0,k) = � ν(0,2k) ν(0,2k+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) of [14], ϕc 2(z) is represented as ϕc 2(z) = ˆϕ2,1(z) + � i1,i2∈{−1,0,1} ν(0,1)(A{2} i1,i2 − A{1,2} i1,i2 )Φc (i1,i2+1),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12) 16 Hence, we consider analytic properties of the vector function ϕc 2(z) through ˆϕc 2(z) and ˆΦx,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆG0,∗(z) be the G-matrix function generated from the triplet { ˆA{1,2} ∗,−1 (z), ˆA{1,2} ∗,0 (z), ˆA{1,2} ∗,1 (z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) of [14], we have, for x2 ≥ 0, ˆΦ(x1,x2),∗(z) = zx1 ˆG0,∗(z)x2 ˆΦ(0,0),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) and this leads us to ˆϕ2(z) = ∞ � k=1 � i2∈{−1,0,1} ˆν(0,k)( ˆA{2} ∗,i2(z) − ˆA{1,2} ∗,i2 (z)) ˆG0,∗(z)k+i2 ˆΦ(0,0),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='14) Hence, analytic properties of the vector function ˆϕ2(z) as well as the matrix function ˆΦx,∗(z) can be clarified through ˆG0,∗(z) and ˆΦ(0,0),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='14), ˆϕ2(z) is represented as ˆϕ2(z) = ˆa(z, ˆG0,∗(z))ˆΦ(0,0),∗(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15) where ˆa(z, w) = ∞ � k=1 ˆν(0,k) ˆD(z, ˆG0,∗(z))wk−1, ˆD(z, w) = ˆA{2} ∗,−1(z) + ˆA{2} ∗,0 (z)w + ˆA{2} ∗,1 (z)w2 − Iw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' First, we consider ˆΦ(0,0),∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆGr 0,∗(z) be the G-matrix function in the reverse direction generated from the triplet { ˆA{1,2} ∗,−1 (z), ˆA{1,2} ∗,0 (z), ˆA{1,2} ∗,1 (z)}, which means that ˆGr 0,∗(z) is the G-matrix function generated from the triplet by exchanging ˆA{1,2} ∗,−1 (z) and ˆA{1,2} ∗,1 (z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˆU(z) as ˆU(z) = ˆA{1,2} ∗,−1 (z) ˆGr 0,∗(z) + ˆA{1,2} ∗,0 (z) + ˆA{1,2} ∗,1 (z) ˆG0,∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) Then, ˆΦ(0,0),∗(z) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15) is given as ˆΦ(0,0),∗(z) = ∞ � n=0 ˆU(z)n = (I − ˆU(z))−1 = adj(I − ˆU(z)) det(I − ˆU(z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='17) Recall that zmin c = eθmin c and zmax c = eθmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For θ ∈ [θmin c , θmax c ], let (ηR c,1(θ), ηR c,2(θ)) and (ηL c,1(θ), ηL c,2(θ)) be the two real roots of the simultaneous equations: spr(A{1,2} ∗,∗ (eθ1, eθ2)) = 1, θ1 + θ2 = θ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='18) counting multiplicity, where ηL c,1(θ) ≤ ηR c,1(θ) and ηL c,2(θ)) ≥ ηR c,2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that ηL c,1(θmax c ) = ηR c,1(θmax c ) and ηL c,2(θmax c ) = ηR c,2(θmax c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='32) of [14], we have spr( ˆG0,∗(eθ)) = e2ηR c,2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='19) Since the eigenvalues of ˆGr 0,∗(z) are coincide with those of the rate matrix function generated from the same triplet { ˆA{1,2} ∗,−1 (z), ˆA{1,2} ∗,0 (z), ˆA{1,2} ∗,1 (z)}, we have spr( ˆGr 0,∗(eθ)) = e−2ηL c,2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='20) By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, ˆG0,∗(z) and ˆGr 0,∗(z) satisfy the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 17 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (1) The extended G-matrix functions ˆG0,∗(z) and ˆGr 0,∗(z) are entry-wise an- alytic in ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The point z = zmax c is a common branch point of ˆG0,∗(z) and ˆGr 0,∗(z) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) There exist matrix functions ˜G0,∗(ζ) and ˜Gr 0,∗(ζ) being analytic in a neighborhood of ζ = 0 and satisfying ˆG0,∗(z) = ˜G0,��((zmax c − z) 1 2 ) and ˆGr 0,∗(z) = ˜Gr 0,∗((zmax c − z) 1 2 ), respectively, in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In order to investigate singularity of ˆΦ(0,0),∗(z) at z = zmax c , we give the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The maximum eigenvalue of ˆU(zmax c ) is 1, and it is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='30) of [14], we have spr( ˆA{1,2} ∗,∗ (zmax c , e2ηR c,2(θmax c ))) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let v be the right eigenvector of ˆA{1,2} ∗,∗ (zmax c , e2ηR c,2(θmax c )) with respect to eigenvalue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since spr( ˆG0,∗(zmax c )) = e2ηR c,2(θmax c ) and spr( ˆGr 0,∗(zmax c )) = e−2ηL c,2(θmax c ) = e−2ηR c,2(θmax c ), we have, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6, ˆG0,∗(zmax c )v = e2ηR c,2(θmax c )v, ˆGr 0,∗(zmax c )v = e−2ηR c,2(θmax c )v, Hence, ˆU(zmax c )v = ˆA{1,2} ∗,∗ (zmax c , e2ηR c,2(θmax c ))v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This means that the value of 1 is an eigenvalue of ˆU(zmax c ), and we obtain spr( ˆU(zmax c )) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Suppose spr( ˆU(zmax c )) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, since spr( ˆU(eθ)) is convex in θ ∈ R, there exist a positive θ0 < θmax c such that spr( ˆU(eθ0)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For this θ0, ˆΦ(0,0),∗(z) diverges at z = eθ0 < zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [14], which asserts that ˆΦ(0,0),∗(z) absolutely convergent in z ∈ ∆zmin c ,zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, spr( ˆU(zmax c )) ≤ 1, and this implies the maximum eigenvalue of ˆU(zmax c ) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ˆU(zmax c ) is irreducible, it is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆλU(z) be the eigenvalue of ˆU(z) satisfying ˆλU(z) = spr( ˆU(z)) for z ∈ [zmin c , zmax c ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆuU(z) and ˆvU(z) be the left and right eigenvectors of ˆU(z) with respect to the eigenvalue ˆλU(z), respectively, satisfying ˆuU(z)ˆvU(z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˜U(ζ) as ˜U(ζ) = ˆA{1,2} ∗,−1 (zmax c − ζ2) ˜Gr 0,∗(ζ) + ˆA{1,2} ∗,0 (zmax c − ζ2) + ˆA{1,2} ∗,1 (zmax c − ζ2) ˜G0,∗(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, ˜U(ζ) is entry-wise analytic in a neighborhood of ζ = 0 and satisfies ˆU(z) = ˜U((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˜Φ(0,0),∗(ζ) as ˜Φ(0,0),∗(ζ) = (I − ˜U(ζ))−1 = adj(I − ˜U(ζ)) det(I − ˜U(ζ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='21) ˆΦ(0,0),∗(z) and ˜Φ(0,0),∗(ζ) satisfy the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (1) The matrix function ˆΦ(0,0),∗(z) is entry-wise analytic in ∆zmin c ,zmax c ∪∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) ˜Φ(0,0),∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0, and the point ζ = 0 is a pole of ˜Φ(0,0),∗(ζ) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˆΦ(0,0),∗(z) is represented as ˆΦ(0,0),∗(z) = ˜Φ(0,0),∗((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 18 (3) ˆΦ(0,0),∗(z) satisfies lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 ˆΦ(0,0),∗(z) = ˆgΦˆvU(zmax c )ˆuU(zmax c ) > O, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='22) where both ˆvU(zmax c ) and ˆuU(zmax c ) are positive, ˆgΦ = − � ˆuU(zmax c )( ˆA{1,2} ∗,−1 (zmax c ) ˆGr 0,∗,1 + ˆA{1,2} ∗,1 (zmax c ) ˆG0,∗,1)ˆvU(zmax c ) �−1 > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='23) and ˆGr 0,∗,1 and ˆG0,∗,1 are the limits of ˆGr 0,∗(z) and ˆG0,∗(z), respectively, given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='16) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, ˆU(z) is entry-wise analytic in ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='17), ˆΦ(0,0),∗(z) is entry-wise meromorphic in the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Recall that, under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2, the induced MA-process {Y {1,2} n } is irreducible and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, in a manner similar to that used in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11], we obtain by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 that, for every z ∈ ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }, spr( ˆU(z)) < spr( ˆU(|z|)) < spr( ˆU(zmax c )) = 1, and this leads us to det(I − ˆU(z)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='21), ˜Φ(0,0),∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ˜U(0) = ˆU(zmax c ), we see by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 that det(I − ˜U(0)) = 0 and the multiplicity of zero of det(I − ˜U(ζ)) at ζ = 0 is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by the identity theorem for analytic functions, det(I − ˜U(ζ)) is nonzero in a neighborhood of ζ = 0 except for the point ζ = 0 and the point ζ = 0 is a pole of ˜Φ(0,0),∗(ζ) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (2) since the representation of ˆΦ(0,0),∗(z) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a function f(λ, z) as f(λ, z) = det(λI − ˆU(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Corollary 2 of Seneta [15] and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 (also see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='11 of [11]), adj(I − ˆU(zmax c )) = fλ(1, zmax c )ˆvU(zmax c )ˆuU(zmax c ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='24) where fλ(λ, z) = ∂ ∂λf(λ, z) and both ˆvU(zmax c ) and ˆuU(zmax c ) are positive since ˆU(zmax c ) is irre- ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, in a manner similar to that used in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9 of [11], we obtain lim ˜∆zmax c ∋z→zmax c (zmax c − z)− 1 2 f(1, z) = −c0fλ(1, zmax c ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='25) where c0 = ˆuU(zmax c )( ˆA{1,2} ∗,−1 (zmax c ) ˆGr 0,∗,1 + ˆA{1,2} ∗,1 (zmax c ) ˆG0,∗,1)ˆvU(zmax c ) < 0 since, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5, both ˆG0,∗,1 and ˆGr 0,∗,1 are nonzero and nonpositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='21), this completes the proof of statement (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let αs0(z) be the eigenvalue of ˆG0,∗(z) that satisfies, for z ∈ [zmin c , zmax c ], αs0(z) = spr( ˆG0,∗(z)) = e2ηR c,2(log z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˆuG(z) and ˆvG(z) be the left and right eigenvectors of ˆG0,∗(z) with respect to the eigenvalue αs0(z), satisfying ˆuG(z)ˆvG(z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3, ˜G0,∗(ζ) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 19 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' There exists a matrix function ˜G† 0,∗(ζ) entry-wise analytic in a neighborhood of ζ = 0 such that ˜G0,∗(ζ) is represented as ˜G0,∗(ζ) = ˜G† 0,∗(ζ) + ˜αs0(ζ)˜vG(ζ)˜uG(ζ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='26) where function ˜αs0(ζ), row vector function ˜uG(ζ) and column vector ˜vG(ζ) are elementwise analytic in a neighborhood of ζ = 0 and satisfying αs0(z) = ˜αs0((zmax c − z) 1 2 ), ˆuG(z) = ˜uG((zmax c − z) 1 2 ) and ˆvG(z) = ˜vG((zmax c − z) 1 2 ), respectively, in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In a neighborhood of ζ = 0, ˜G† 0,∗(ζ) satisfies spr( ˜G† 0,∗(ζ)) < αs0(zmax c ) = e2ηR c,2(θmax c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Furthermore, ˜G0,∗(ζ) satisfies, for n ≥ 1, ˜G0,∗(ζ)n = ˜G† 0,∗(ζ)n + ˜αs0(ζ)n˜vG(ζ)˜uG(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='27) Let ˆν(0,∗)(z) be the generating function of {ˆν(0,k)} defined as ˆν(0,∗)(z) = �∞ k=1 zkˆν(0,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function ˆU2(z) as ˆU2(z) = ˆA{2} 0,∗ (z) + ˆA{2} 1,∗ (z) ˆG∗,0(z), and let ˆuU 2 (z) and ˆvU 2 (z) be the left and right eigenvectors of ˆU2(z) with respect to the maximum eigenvalue of ˆU2(z), satisfying ˆuU 2 (z)ˆvU 2 (z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [11] (also see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5 of [14]), ˆν(0,∗)(z) satisfies the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (1) The vector function ˆν(0,∗)(z) is elementwise analytic in ¯∆e2θ∗ 2 \\ {e2θ∗ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) If θ∗ 2 < θmax 2 , ˆν(0,∗)(z) is elementwise meromorphic in a neighborhood of z = e2θ∗ 2 and the point z = e2θ∗ 2 is a pole of ˆν(0,∗)(z) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' It satisfies, for some positive constant ˆg2, lim ˜∆ e2θ∗ 2 ∋z→e2θ∗ 2 (e2θ∗ 2 − z)ˆϕ2(z) = ˆg2ˆuU 2 (e2θ∗ 2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='28) where ˆuU 2 (e2θ∗ 2) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a vector function ˜a(ζ, w) as ˜a(ζ, w) = ∞ � k=1 ˆν(0,k) ˆD(zmax c − ζ2, ˜G0,∗(ζ))wk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='29) Then, the vector functions ˆa(z, ˆG0,∗(z)) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15) and ˜a(ζ, ˜G0,∗(ζ)) satisfy the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (1) If ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1, the vector function ˆa(z, ˆG0,∗(z)) is elementwise analytic in ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (2) If ¯η′ 1(θ∗ 2) < −1, ˜a(ζ, ˜G0,∗(ζ)) is elementwise analytic in a neighborhood of ζ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1, it is elementwise meromorphic in a neighborhood of ζ = 0 and the point ζ = 0 is a pole of it with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The vector function ˆa(z, ˆG0,∗(z)) is represented as ˆa(z, ˜G0,∗(z)) = ˜a((zmax c − z) 1 2 , ˜G0,∗((zmax c − z) 1 2 )) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 20 (3) If ¯η′ 1(θ∗ 2) = −1, ˆa(z, ˆG0,∗(z)) satisfies, for a positive constant ˆga 2, lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 ˆa(z, ˆG0,∗(z)) = ˆga 2 ˆuG(zmax c ) ≥ 0⊤, ̸= 0⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='30) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11], if ¯η′ 1(θ∗ 2) ≤ −1, we have for z ∈ ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c } that |αs0(z)| < αs0(zmax c ) = e2ηR c,2(θmax c ) ≤ e2θ∗ 2, and this implies spr( ˆG0,∗(z)) < e2θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 of [11] and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8, vector series ˆa(z, ˆG0,∗(z)) elementwise converges absolutely in ∆zmin c ,zmax c ∪ ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7, we have ˜a(ζ, ˜G0,∗(ζ)) = ˜a(ζ, ˜G† 0,∗(ζ)) + (˜αs0(ζ)−1ˆν(0,∗)(˜αs0(ζ)) − ˆν(0,1)) ˆD(zmax c − ζ2, ˜αs0(ζ))˜vG(ζ)˜uG(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='31) If ¯η′ 1(θ∗ 2) ≤ −1, spr( ˜G† 0,∗(ζ)) < e2ηR c,2(θmax c ) ≤ e2θ∗ 2 in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, vec- tor series ˜a(ζ, ˜G† 0,∗(ζ)) is elementwise convergent absolutely and analytic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) < −1, ˜αs0(0) = αs0(zmax c ) = e2ηR c,2(θmax c ) < e2θ∗ 2, and this implies |˜αs0(ζ)| < e2θ∗ 2 in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8, the vector function ˜a(ζ, ˜G0,∗(ζ)) as well as ˆν(0,∗)(˜αs0(ζ)) is elementwise analytic in a neighborhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) = −1, ˜αs0(0) = e2ηR c,2(θmax c ) = e2θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8, the vector function ˜a(ζ, ˜G0,∗(ζ)) as well as ˆν(0,∗)(˜αs0(ζ)) is meromorphic in a neighborhood of ζ = 0 and the point ζ = 0 is a pole of it with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) = −1, αs0(zmax c ) = e2ηR c,2(θmax c ) = e2θ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8, we have lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 ˆν(0,∗)(αs0(z)) = lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 αs0(zmax c ) − αs0(z)(αs0(zmax c ) − αs0(z))ˆν(0,∗)(αs0(z)) = (−αs0,1)−1ˆg2ˆuU 2 (e2θ∗ 2), where αs0,1 is the limit of αs0(z) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) and it is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This leads us to lim ˜∆zmax c ∋z→zmax c (zmax c − z) 1 2 ˆa(z, ˆG0,∗(z)) = (−αs0,1)−1ˆg2e−2θ∗ 2 ˆuU 2 (e2θ∗ 2)D(zmax c , e2θ∗ 2)ˆvG(zmax c )ˆuG(zmax c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='32) From this, we see that ˆga 2 ˆuG(zmax c ) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='30) is given by the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ˆuU 2 (e2θ∗ 2) is positive, ˆga 2 is also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of statement (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Finally, we give the proof of Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1 and ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1 ≤ 1/¯η′ 2(θ∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since ϕc(z) is a probability vector generating function, it is automatically analytic elementwise in ∆zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, we prove it is elementwise analytic on ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For the purpose, we use equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9, ˆG0,∗(z), ˆa(z, ˆG0,∗(z)) and ˆΦ(0,0),∗(z) are elementwise analytic on ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15), ˆΦx,∗(z) and ˆϕ2(z) are also analytic elementwise on ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10), the analytic property of ˆΦx,∗(z) implies that Φc x,∗(z) is entry-wise 21 analytic on ∂∆zmax c \\{zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12), ϕc 2(z) is elementwise analytic on ∂∆zmax c \\{zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the same way, we can see that if ¯η′ 2(θ∗ 1) ≤ −1, ϕc 1(z) is elementwise analytic on ∂∆zmax c \\{zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), the analytic property of Φc x,∗(z) implies that ϕc 0(z) is elementwise analytic on ∂∆zmax c \\{zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' As a result, we see by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) that ϕc(z) is elementwise analytic on ∂∆zmax c \\ {zmax c }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assuming Type 1 and ¯η′ 1(θ∗ 2) ≤ −c1/c2 = −1 ≤ 1/¯η′ 2(θ∗ 1), we also use equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10),(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' First, we consider about Φc x,∗(z) and ϕc 0(z), where x = (x1, x2) ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define ˜Φ(x1,x2),∗(ζ) as ˜Φ(x1,x2),∗(ζ) = (zmax c − ζ2)x1 ˜G0,∗(ζ)x2 ˜Φ(0,0),∗(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, by Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6, the matrix function ˜Φx,∗(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0 and satisfies ˆΦx,∗(z) = ˜Φ(x1,x2),∗((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The point ζ = 0 is a pole of ˜Φx,∗(ζ) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10), there exists a matrix function ˜Φc x,∗(ζ) being entry-wise meromorphic in a neighborhood of ζ = 0 and satisfying Φc x,∗(z) = ˜Φc x,∗((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The point ζ = 0 is a pole of ˜Φc x,∗(ζ) with order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define ˜ϕc 0(z) as ˜ϕc 0(ζ) = � i1,i2∈{−1,0,1} ν(0,0)(A∅ i1,i2 − A{1,2} i1,i2 )˜Φc (i1,i2),∗(ζ), which satisfies the same analytic property as ˜Φc x,∗(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' It also satisfies ϕc 0(z) = ˜ϕc 0((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Next, we consider about ϕc 2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define ˜ϕ2(ζ) as ˜ϕ2(ζ) = ˜a(ζ, ˜G0,∗(ζ))˜Φ(0,0),∗(ζ) By Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='15), ˜ϕ2(ζ) is entry-wise meromorphic in a neighborhood of ζ = 0 and satisfying ˆϕ2(z) = ˜ϕ2((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) < −1, the point ζ = 0 is a pole of ˜ϕ2(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1, it is a pole of ˜ϕ2(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Represent ˜ϕ2(ζ) in block form as ˜ϕ2(ζ) = �˜ϕ2,1(ζ) ˜ϕ2,2(ζ) � and define ˜ϕc 2(ζ) as ˜ϕc 2(ζ) = ˜ϕ2,1(ζ) + � i1,i2∈{−1,0,1} ν(0,1)(A{2} i1,i2 − A{1,2} i1,i2 )˜Φc (i1,i2+1),∗(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, the vector function ˜ϕc 2(ζ) is elementwise meromorphic in a neighborhood of ζ = 0, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12), it satisfies ϕc 2(z) = ˜ϕc 2((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 1(θ∗ 2) < −1, the point ζ = 0 is a pole of ˜ϕc 2(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) = −1, it is a pole of ˜ϕc 2(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Finally, we consider about ϕc(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' In the same way as that used for ϕc 2(z), we can see that there exists a vector function ˜ϕc 1(ζ) being elementwise meromorphic in a neighborhood of ζ = 0 and satisfying ϕc 1(z) = ˜ϕc 1((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ¯η′ 2(θ∗ 1) < −1, the point ζ = 0 is a pole of ˜ϕc 1(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 2(θ∗ 1) = −1, it is a pole of ˜ϕc 1(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define ˜ϕc(ζ) as ˜ϕc(ζ) = ˜ϕc 0(ζ) + ˜ϕc 1(ζ) + ˜ϕc 2(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, the vector function ˜ϕc(ζ) is elementwise meromorphic in a neighborhood of ζ = 0, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6), it satisfies ϕc(z) = ˜ϕc((zmax c − z) 1 2 ) in a neighborhood of z = zmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If ˜η′ 1(θ∗ 2) < −c1/c2 = −1 < 1/˜η′ 2(θ∗ 1), the point ζ = 0 is a pole of ˜ϕc(ζ) with at most order one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ˜η′ 1(θ∗ 2) = −1 or ˜η′ 2(θ∗ 1) = −1, it is a pole of ˜ϕc(ζ) with at most order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 22 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume Type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 and equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='13), lim ˜∆zmax c ∋z→zmax c (zmax c − z)Φc x,∗(z) = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='33) Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7), lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 0(z) = 0⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='34) If ¯η′ 1(θ∗ 2) = −1, by Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9 and equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='34), representing ˆuU(zmax c ) in block form as ˆuU(zmax c ) = � ˆuU 1 (zmax c ) ˆuU 2 (zmax c ) � , we obtain lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 2(z) = uc 2 = ˆga 2ˆgΦˆuG(zmax c )ˆvU(zmax c )ˆuU 1 (zmax c ) > 0⊤, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='35) where ˆuG(zmax c ) is nonzero and nonnegative and other terms on the right-hand side of the equation are positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' if ¯η′ 1(θ∗ 2) < −1, we have lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 2(z) = 0⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='36) In a manner similar to that used for ϕc 2(z), we can see that if ¯η′ 2(θ∗ 1) = −1, then for some positive vector uc 1, lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 1(z) = uc 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='37) and if ¯η′ 1(θ∗ 2) < −1, lim ˜∆zmax c ∋z→zmax c (zmax c − z)ϕc 1(z) = 0⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='38) As a result, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='35), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='36), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='37) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='38), we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 5 Concluding remarks We consider another topic, which relates to the singularity of the vector generating function ϕc(z) at z = zmax c = eθmax c , where c ∈ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Recall that P {1,2} = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) is the transition probability matrix of the induced MA-process {Y {1,2} n } and Φ{1,2} = (Φ{1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2) the fundamental matrix (potential matrix) of P {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let hΦ c (k) be the asymptotic decay function of the matrix sequence {Φ{1,2} x,kc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', for some positive matrix C, lim k→∞ Φ{1,2} x,kc /hΦ c (k) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6, we obtain hΦ c (k) = k− 1 2 e−θmax c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) Furthermore, recall that P + is a partial matrix of P {1,2} given by restricting the state space of the level to the positive quadrant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', P + = (P {1,2} x,x′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' P + is also a partial matrix of the transition probability matrix of the original 2d-QBD process, P = (Px,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ Z2 +), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', P + = (Px,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let ˜Q = ( ˜Qx,x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' x, x′ ∈ N2) be the fundamental matrix of P +, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', 23 ˜Q = �∞ n=0(P +)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For j, j′ ∈ S0, denote by ˜q(x,j),(x′,j′) the (j, j′)-entry of ˜Qx,x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' The entries of ˜Q are called an occupation measure in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [13], the asymptotic decay rate of the matrix sequence { ˜Qx,kc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N} is given by eθmax c , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', − lim k→∞ 1 k log ˜q(x,j),(kc,j′) = θmax c , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) which coincides with that of the matrix sequence {Φ{1,2} x,kc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' One question, therefore, arises: Does the asymptotic decay function of the matrix sequence { ˜Qx,kc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N} coincide with that of the matrix sequence {Φ{1,2} x,kc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' k ∈ N}?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If the answer to the question is yes, we can indicate that the vector generating function ϕc(z) diverges at z = eθmax c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' References [1] Bini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Latouche, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' and Meini, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Oxford University Press, Oxford (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' [2] Flajolet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' and Sedgewick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Analytic Combinatorics, Cambridge University Press, Cam- bridge (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' [3] Gohberg, I.' metadata={'source': 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coordinate and diagonal directions, Matrix-Analytic Methods in Stochastic Models (2013), 145-185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' [5] Latouche, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' and Ramaswami, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' [6] Malyshev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1007/s11134-020-09673-9) 24 [14] Ozawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', Tail Asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process, Queueing Systems 102 (2022), 227–267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1007/s11134-022-09860-w) [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Seneta: Non-negative Matrices and Markov Chains, revised printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Springer-Verlag, New York (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' A Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 First, we give the generalized eigenvectors of G(z) for z ∈ ∆zmain 1 ,zmax 1 \\E1, then analytically extend them to z ∈ C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for each z ∈ Ω \\ �m0 k=1 EG k , since the Jordan normal form of G(z) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4), there exist linearly independent vectors called the generalized eigenvectors of G(z) with respect to the eigenvalue ˇαk(z), ˇvk,i,j(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i, satisfying (ˇαk(z)I − G(z))ˇvk,i,j(z) = ˇvk,i,j+1(z), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1) where ˇvk,i,mk,i+1(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each i, ˇvk,i,j(z), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i, are called a Jordan sequence of the generalized eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Using the Jordan sequences, we define lˇq(k) × 1 block vectors, vk,i,j(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i, as vk,i,j(z) = vec �ˇvk,i,j(z) ˇvk,i,j+1(z) · · ˇvk,i,mk,i(z) 0 · · 0� , where, for a matrix A = � a1 a2 · · an � , vec(A) is the column vector given by vec(A) = � � � � � a1 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' an � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We also define a vector space VG k (z) as VG k (z) = span {vk,i,j(z) : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Note that the generalized eigenvectors ˇvk,i,j(z) are not unique but VG k (z) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since the generalized eigenvectors are linearly independent, vk,i,j(z) are also linearly independent and we have dim VG k (z) = mk,0 � i=1 mk,i = lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, define an lˇq(k) × lˇq(k) block matrix function ΛG k (z) as ΛG k (z) = � � � � � � � ˇαk(z)I − G(z) −I ˇαk(z)I − G(z) −I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' ˇαk(z)I − G(z) −I ˇαk(z)I − G(z) � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We give the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for each z ∈ Ω \\ �m0 k=1 EG k , Ker ΛG k (z) = VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume v ∈ VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, by the definition of VG k (z), we have ΛG k (z)v = 0 and v ∈ Ker ΛG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For v = vec �v1 v2 · · vlˇq(k) � , assume ΛG k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If there exists an index i such that vi = 0, then by the assumption, for every j such that i ≤ j ≤ lˇq(k), we have vj = 0, and this implies v ∈ VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Theorem S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [3], since the matrix function ΛG k (z) is entry-wise analytic in ∆zmin 1 ,zmax 1 \\E1, there exist lˇq(k) vector functions vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k), that are elementwise analytic and linearly independent in ∆zmin 1 ,zmax 1 \\ E1 and satisfy ΛG k (z)vG k,i(z) = 0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, for each z ∈ Ω \\ �m0 k=1 EG k , vG k,i(z) ∈ VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We select the vectors composed of the Jordan sequences from {vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Represent each vG k,i(z) in block form as vG k,i(z) = vec � vG k,i,1(z) vG k,i,2(z) · · vG k,i,lˇq(k)(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' From the proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1, we see that, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, there exists a positive integer µk,i such that vG k,i,j(z) ̸= 0 for every j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', µk,i} and vG k,i,j(z) = 0 for every j ∈ {µk,i + 1, µk,i + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Renumber the elements of {vG k,i(z)} so that if i ≤ i′, then µk,i ≥ µk,i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a set of vector functions, ˇVk, according to the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (S1) Set ˇVk = ∅ and i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (S2) If vG k,i,µk,i(z) is linearly independent of {vG k,i′,µk,i′(z) : vG k,i′(z) ∈ ˇVk}, append vG k,i(z) to ˇVk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (S3) If i = lˇq(k), stop the procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' otherwise add 1 to i and go to (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, the number of elements of ˇVk is mk,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, (ˇαk(z)I−G(z))vG k,i,µk,i = 0 and dim Ker (ˇαk(z)I−G(z)) = mk,0, the number of elements of ˇVk is less than or equal to mk,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' If it is strictly less than mk,0, we have dim Ker ΛG k (z) = dim span {vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)} < dim VG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This contradicts (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2), and we see that the number of elements of ˇVk is just mk,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Denote by ˇvG k,1(z), ˇvG k,2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', ˇvG k,mk,0(z) the elements of ˇVk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='mk,0}, define ˇµk,i in a manner similar to that used for defining µk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We assume ˇvG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, are numbered so that if i ≤ i′, then ˇµk,i ≥ ˇµk,i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, ˇµk,i = mk,i Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, {ˇvG k,i,1(z), ˇvG k,i,2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', ˇvG k,i,µk,i(z)} is a Jordan sequence of the generalized eigenvectors of G(z) with respect to the eigenvalue ˇαk(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, considering the procedure defining ˇvG k,i(z), we see that, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, ˇµk,i ≤ mk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Suppose there exists some i0 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0} such that ˇµk,i = mk,i for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', i0 −1} and ˇµk,i0 < mk,i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, there exists a vector v = vec �v1 v2 · · vmk,i0 0 · · 0� in VG k (z) such that vi ̸= 0 for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i0} and v is linearly independent of {vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By the same reason as that used in the proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2, this contradicts (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2) and, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}, ˇµk,i must be mk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 26 From this proposition, we see that, for z ∈ Ω \\ �m0 k=1 EG k , {ˇvG k,i,j(z) : k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i} is the set of generalized eigenvectors corresponding to the Jordan normal form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function T G(z) as T G(z) = �ˇvG k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i � , which is entry-wise analytic in ∆zmin 1 ,zmax 2 \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a point set EG T as EG T = {z ∈ ∆zmin 1 ,zmax 2 \\ E1 : det T G(z) = 0}, which is an empty set or a set of discrete complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, for z ∈ Ω \\ (�m0 k=1 EG k ∪ EG T ), we obtain the Jordan decomposition of G(z) as G(z) = T G(z)JG(z)(T G(z))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) Since G(z) is entry-wise analytic in ∆zmin 1 ,zmax 1 , we see by the identity theorem for analytic functions that the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3) is also entry-wise analytic in the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Next, we analytically extend ˇvG k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define matrix functions F1(z, w) and F2(z) as F1(z, w) = z(I − A∗,0(z) − 2wA∗,1(z)), F2(z) = zA∗,1(z), where F1(z, w) is entry-wise analytic on C2 and F2(z) on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='2), we have L(z, w) = F1(z, w)(wI − G(z)) + F2(z)(wI − G(z))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4) For k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}, define a lˇq(k) × lˇq(k) block matrix function ΛL k,n(z) as ΛL k (z) = � � � � � � � � � L(z, ˇαk(z)) −F1(z, ˇαk(z)) −F2(z) L(z, ˇαk(z)) −F1(z, ˇαk(z)) −F2(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' L(z, ˇαk(z)) −F1(z, ˇαk(z)) −F2(z) L(z, ˇαk(z)) −F1(z, ˇαk(z)) L(z, ˇαk(z)) � � � � � � � � � , which is entry-wise analytic in C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0} and for every z ∈ ¯∆zmin 1 ,zmax 1 , Ker ΛL k (z) = Ker ΛG k (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5) Before proving this proposition, we give another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0} and z ∈ ∆zmin 1 ,zmax 1 , F1(z, αk(z)) + F2(z)(αk(z)I − G(z)) = z (I − A∗,0(z) − αk(z)A∗,1(z) + A∗,1(z)G(z)) is regular (invertible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let R(z) be the rate matrix function generated from {Ai,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' i, j = −1, 0, 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' for the definition of R(z), see Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='3 of [11], nonzero eigenvalues of R(z) are given by αk(z)−1, k = s0 + 1, s0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since, for every k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', s0}, k′ ∈ {s0 + 1, s0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mφ} and z ∈ ∆zmin 1 ,zmax 1 , |αk(z)| ≤ αs0(|z|) < |αk′(z)|, I − αk(z)R(z) is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function H(z) as H(z) = A∗,0(z) + A∗,1(z)G(z), then by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11], I − H(z) is regular in ∆zmin 1 ,zmax 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [11], we have I − A∗,0(z) − αk(z)A∗,1(z) − A∗,1(z)G(z) = (I − αk(z)R(z))(I − H(z)), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='6) and this implies the assertion of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 27 Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume a vector v = vec �v1 v2 · · vlˇq(k) � satisfies ΛL k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, we have for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)} that L(z, ˇαk(z))vi = F1(z, ˇαk(z))vi+1 + F2(z)vi+2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='7) where vlˇq(k)+1 = vlˇq(k)+2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' We prove by induction that this v satisfies, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, (ˇαk(z)I − G(z))vi = vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let i0 be the maximum integer less than or equal to lˇq(k) that satisfies, for every i ∈ {i0 + 1, i0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, vi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, we have L(z, ˇαk(z))vi0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4), we have L(z, ˇαk(z)) = (F1(z, ˇαk(z)) + F2(z)(ˇαk(z)I − G(z)))(ˇαk(z)I − G(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8) Hence, by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5, we obtain (ˇαk(z)I − G(z))vi0 = 0 = vi0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Assume the assumption of induction holds for a positive integer i less than or equal to i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, L(z, ˇαk(z))vi−1 = F1(z, ˇαk(z))vi + F2(z)vi+1 = (F1(z, ˇαk(z)) + F2(z)(ˇαk(z)I − G(z)))vi, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) and by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='9) and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5, we obtain (ˇαk(z)I − G(z))vi−1 = vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, v satisfies, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, (ˇαk(z)I − G(z))vi = vi+1, and this leads us to ΛG k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Next, assume a vector v = vec �v1 v2 · · vlˇq(k) � satisfies ΛG k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Then, we have for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)} that (ˇαk(z)I − G(z))vi = vi+1, where vlˇq(k)+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='8), this v satisfies, for every i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, L(z, ˇαk(z))vi = F1(z, ˇαk(z))vi+1 + F2(z)(ˇαk(z)I − G(z))vi+1 = F1(z, ˇαk(z))vi+1 + F2(z)vi+2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='10) and this implies ΛL k (z)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Let k be an arbitrary integer in {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Propositions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, we have dim Ker ΛL k (z) = lˇq(k), except for some discrete points in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by Theorem S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1 of [3], since the matrix function ΛL k (z) is entry-wise analytic in C\\E1, there exist lˇq(k) vector functions vL k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k), that are elementwise analytic and linearly independent in C \\ E1 and satisfy ΛL k (z)vL k,i(z) = 0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='4, for each i, vL k,i(z) also satisfies ΛG k (z)vL k,i(z) = 0 for every z ∈ ∆zmin 1 ,zmax 1 \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Hence, by the identity theorem, we see that vL k,i(z) is an analytic extension of vG k,i(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' By the same procedure as that used for selecting {ˇvG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0} from {vG k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)}, we select mk,0 vectors from {vL k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', lˇq(k)} and denote them by {ˇvL k,i(z), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' For each i, ˇvL k,i(z) is represented in block form as ˇvL k,i(z) = vec � ˇvL k,i,1(z) ˇvL k,i,2(z) · · ˇvL k,i,mk,i(z) 0 · · 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Define a matrix function T L(z) as T L(z) = �ˇvL k,i,j(z), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', m0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=', mk,i � , which is entry-wise analytic in C \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Since each ˇvL k,i,j(z) is an analytic extension of ˇvG k,i,j(z), we have for z ∈ Ω \\ (�m0 k=1 EG k ∪ EG T ) that G(z) = T L(z)JG(z)(T L(z))−1, which is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' Set E0 as E0 = E2 ∪ (�m0 k=1 EG k ) ∪ EG T , then E0 is a set of discrete complex numbers and we have Ω\\(�m0 k=1 EG k ∪EG T ) = ∆zmin 1 ,zmax 1 \\(E1 ∪E0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' This completes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE0T4oBgHgl3EQfhgHx/content/2301.02434v1.pdf'}