Publishing type：Research paper (international conference proceedings)   Publisher：Leibniz International Proceedings in Informatics, LIPIcs  

It is a celebrated fact that a simple random walk on an infinite k-ary tree for k ≥ 2 returns to the initial vertex at most finitely many times during infinitely many transitions; it is called transient. This work points out the fact that a simple random walk on an infinitely growing k-ary tree can return to the initial vertex infinitely many times, it is called recurrent, depending on the growing speed of the tree. Precisely, this paper is concerned with a simple specific model of a random walk on a growing graph (RWoGG), and shows a phase transition between the recurrence and transience of the random walk regarding the growing speed of the graph. To prove the phase transition, we develop a coupling argument, introducing the notion of less homesick as graph growing (LHaGG). We also show some other examples, including a random walk on {0, 1}n with infinitely growing n, of the phase transition between the recurrence and transience. We remark that some graphs concerned in this paper have infinitely growing degrees.

DOI： 10.4230/LIPIcs.SAND.2024.17

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Language：English   Publishing type：Research paper (scientific journal)  

We compute the eigenvalues of up-Laplacians on cubical lattices and derive the torsion-weighted count of spanning acycles in cubical lattices by using the matrix-tree theoremfor simplicial/cell complexes. As a corollary, we show that the torsion-weighted span-ning acycle entropy of a cubical lattice defined on Z^b \times \prod_{j=b+1}^q {0, 1,...,n_j} isexpressed as a linear combination of logarithmic Mahler measures.

DOI： https://doi.org/10.1007/s41468-024-00163-y

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Journal of Applied and Computational Topology  

We compute the eigenvalues of up-Laplacians on cubical lattices and derive the torsion-weighted count of spanning acycles in cubical lattices by using the matrix-tree theorem for simplicial/cell complexes. As a corollary, we show that the torsion-weighted spanning acycle entropy of a cubical lattice defined on Zb×∏j=b+1q{0,1,⋯,nj} is expressed as a linear combination of logarithmic Mahler measures.

DOI： 10.1007/s41468-024-00163-y

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Publishing type：Research paper (scientific journal)   Publisher：Journal of Theoretical Probability  

We are concerned with zeros of random power series with coefficients being a stationary, centered, complex Gaussian process. We show that the expected number of zeros in every smooth domain in the disk of convergence is less than that of the hyperbolic Gaussian analytic function with i.i.d. coefficients. When coefficients are finitely dependent, i.e., the spectral density is a trigonometric polynomial, we derive precise asymptotics of the expected number of zeros inside the disk of radius r centered at the origin as r tends to the radius of convergence, in the proof of which we clarify that the negative contribution to the number of zeros stems from the zeros of the spectral density.

DOI： 10.1007/s10959-022-01203-y

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Language：English   Publishing type：Research paper (scientific journal)  

We give a brief explanation of homology and persistent homology intuitively by using matrix representations of boundary operators and introduce a result of the law of large numbers for persistence diagrams of a stationary ergodic point process. We recall the notion of Bitcoin graphs and chainlets and show an example of how to compute persistence diagrams for chainlet matrices by viewing them as point clouds.

DOI： https://doi.org/10.7566/JPSCP.40.011

Publishing type：Research paper (scientific journal)   Publisher：Modern Stochastics: Theory and Applications  

Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of Čech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.

DOI： 10.15559/22-VMSTA214

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：The Japan Academy  

We consider the Laplace-Beltrami operator $\Delta_g$ on
a smooth, compact Riemannian manifold $(M,g)$
and the determinantal point process $\xila$ on $M$ associated with the spectral
projection of $-\Delta_g$ onto the subspace corresponding to the eigenvalues up to
$\la^2$.
We show that the pull-back of $\xila$ by the exponential map
$\exp_p : T_p^*M \to M$ under a suitable scaling converges
weakly to the universal determinantal point process on
$T_p^* M$ as $\la \to \infty$.

DOI： 10.3792/pjaa.98.018

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： https://doi.org/10.3792/pjaa.98.018

Language：English   Publishing type：Research paper (scientific journal)  

Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of Cˇech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.

DOI： https://doi.org/10.15559/22-VMSTA214

Publisher：Institute of Electrical and Electronics Engineers ({IEEE})  

Disordered complex networks are of fundamental interest as stochastic models
for information transmission over wireless networks. Well-known networks based
on the Poisson point process model have limitations vis-a-vis network
efficiency, whereas strongly correlated alternatives, such as those based on
random matrix spectra (RMT), have tractability and robustness issues. In this
work, we demonstrate that network models based on random perturbations of
Euclidean lattices interpolate between Poisson and rigidly structured networks,
and allow us to achieve the best of both worlds : significantly improve upon
the Poisson model in terms of network efficacy measured by the Signal to
Interference plus Noise Ratio (abbrv. SINR) and the related concept of coverage
probabilities, at the same time retaining a considerable measure of
mathematical and computational simplicity and robustness to erasure and noise.
We investigate the optimal choice of the base lattice in this model,
connecting it to the celebrated problem optimality of Euclidean lattices with
respect to the Epstein Zeta function, which is in turn related to notions of
lattice energy. This leads us to the choice of the triangular lattice in 2D and
face centered cubic lattice in 3D. We demonstrate that the coverage probability
decreases with increasing strength of perturbation, eventually converging to
that of the Poisson network. In the regime of low disorder, we approximately
characterize the statistical law of the coverage function.
In 2D, we determine the disorder strength at which the PTL and the RMT
networks are the closest measured by comparing their network topologies via a
comparison of their Persistence Diagrams . We demonstrate that the PTL network
at this disorder strength can be taken to be an effective substitute for the
RMT network model, while at the same time offering the advantages of greater
tractability.

DOI： 10.1109/TIT.2022.3163604

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arXiv

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Other Link： http://arxiv.org/pdf/2009.08811v1

Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Ltd  

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures [Formula: see text] on a space S with measure [Formula: see text], whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, [Formula: see text], which are assumed to be realized as [Formula: see text]-spaces, [Formula: see text], [Formula: see text], and introduce a bounded linear operator [Formula: see text] and its adjoint [Formula: see text]. We show that if [Formula: see text] is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP [Formula: see text] associated with [Formula: see text]. In addition, if [Formula: see text] is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, [Formula: see text], [Formula: see text]. We also give a practical framework which makes [Formula: see text] and [Formula: see text] satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces S, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ([Formula: see text]) series of infinite DPPs on [Formula: see text] and [Formula: see text] are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

DOI： 10.1142/S2010326322500253

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Language：English   Publishing type：Research paper (scientific journal)  

On an annulus 𝔸𝑞:={𝑧∈ℂ:𝑞<|𝑧|<1} with a fixed 𝑞∈(0,1), we study a Gaussian analytic function (GAF) and its zero set which defines a point process on 𝔸𝑞 called the zero point process of the GAF. The GAF is defined by the i.i.d. Gaussian Laurent series such that the covariance kernel parameterized by 𝑟>0 is identified with the weighted Szegő kernel of 𝔸𝑞 with the weight parameter r studied by McCullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the q-inversion of coordinate 𝑧↔𝑞/𝑧 and the parameter change 𝑟↔𝑞2/𝑟. When 𝑟=𝑞 they are invariant under conformal transformations which preserve 𝔸𝑞. Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of McCullough and Shen but the weight parameter r is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on r of the unfolded 2-correlation function of the PDPP is studied. If we take the limit 𝑞→0, a simpler but still non-trivial PDPP is obtained on the unit disk 𝔻. We observe that the limit PDPP indexed by 𝑟∈(0,∞) can be regarded as an interpolation between the determinantal point process (DPP) on 𝔻 studied by Peres and Virág (𝑟→0) and that DPP of Peres and Virág with a deterministic zero added at the origin (𝑟→∞).

DOI： https://doi.org/10.1007/s00220-022-04365-2

Publishing type：Research paper (scientific journal)   Publisher：Probability Theory and Related Fields  

The article [Absolute continuity and singularity of Palm measures of the Ginibre point process], written by [Hirofumi Osada and Tomoyuki Shirai], was originally published Online First without Open Access. After publication in volume [165], issue [3–4], page [725–770] the author decided to opt for Open Choice and to make the article an Open Access publication. Therefore, the copyright of the article has been changed to © [The Author(s)] [2021] and this article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. Ifmaterial is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The original article has been corrected.

DOI： 10.1007/s00440-021-01094-w

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Publisher：Communications in Mathematical Physics  

On an annulus ${\mathbb{A } }_q :=\{z \in {\mathbb{C } }: q < |z| < 1\}$ with a
fixed $q \in (0, 1)$, we study a Gaussian analytic function (GAF) and its zero
set which defines a point process on ${\mathbb{A } }_q$ called the zero-point
process of the GAF. The GAF is defined by the i.i.d. Gaussian Laurent series
such that the covariance kernel parameterized by $r >0$ is given by the
weighted Szeg\H{o} kernel of ${\mathbb{A } }_q$ with the weight parameter $r$
studied by Mccullough and Shen. The GAF and the zero-point process have
symmetry associated with the $q$-inversion of coordinate $z \leftrightarrow
q/z$ and the parameter change $r \leftrightarrow q^2/r$, and when $r=q$ they
are invariant under conformal transformations which preserve ${\mathbb{A } }_q$.
Conditioning the GAF by giving zeros, new GAFs are induced such that the
covariance kernels are also given by the weighted Szeg\H{o} kernel of
Mccullough and Shen but the weight parameter $r$ is changed depending on the
given zeros. Then we prove that the zero-point process of the GAF provides a
permanental-determinantal point process (PDPP) in which each correlation
function is expressed by a permanent multiplied by a determinant. Dependence on
$r$ of the unfolded 2-correlation function of the PDPP is studied. If we take
the limit $q \to 0$, a simpler but still non-trivial PDPP is obtained on a
punctured unit disk ${\mathbb{D } }^{\times} := {\mathbb{D } } \setminus \{0\}$. In
the further limit $r \to 0$ the present PDPP is reduced to the determinantal
point process on ${\mathbb{D } }$ studied by Peres and Vir\'ag.

DOI： 10.1007/s00220-022-04365-2

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arXiv

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Other Link： http://arxiv.org/pdf/2008.04177v1

Language：English   Publishing type：Research paper (scientific journal)  

Disordered complex networks are of fundamental interest in statistical physics, and they have attracted recent interest as stochastic models for information transmission over wireless networks. While mathematically tractable, a network based on the regulation Poisson point process model offers challenges vis-a-vis network efficiency. Strongly correlated alternatives, such as networks based on random matrix spectra (the Ginibre network), on the other hand offer formidable challenges in terms of tractability and robustness issues. In this work, we demonstrate that network models based on random perturbations of Euclidean lattices interpolate between Poisson and rigidly structured networks, and allow us to achieve the best of both worlds: significantly improve upon the Poisson model in terms of network efficacy measured by the Signal to Interference plus Noise Ratio (abbrv. SINR) and the related concept of coverage probabilities, at the same time retaining a considerable measure of mathematical and computational simplicity and robustness to erasure and noise. We investigate the optimal choice of the base lattice in this model, connecting it to the celebrated problem optimality of Euclidean lattices with respect to the Epstein Zeta function, which is in turn related to notions of lattice energy. This leads us to the choice of the triangular lattice in 2D and face centered cubic lattice in 3D, whose Gaussian perturbations we consider. We provide theoretical analysis and empirical investigations to demonstrate that the coverage probability decreases with increasing strength of perturbation, eventually converging to that of the Poisson network. In the regime of low disorder, our studies suggest an approximate statistical behaviour of the coverage function near a base station as a log-normal distribution with parameters depending on the Epstein Zeta function of the lattice, and related approximate dependencies for a power-law constant that governs the network coverage probability at large thresholds. In 2D, we determine the disorder strength at which the perturbed triangular lattice (abbrv. PTL) and the Ginibre networks are the closest measured by comparing their network topologies via a comparison of their Persistence Diagrams in the total variation as well as the symmetrized nearest neighbour distances. We demonstrate that, at this very same disorder, the PTL and the Ginibre networks exhibit very similar coverage probability distributions, with the PTL performing at least as well as the Ginibre. Thus, the PTL network at this disorder strength can be taken to be an effective substitute for the Ginibre network model, while at the same time offering the advantages of greater tractability both from theoretical and empirical perspectives.

DOI： doi: 10.1109/TIT.2022.3163604

Language：English   Publishing type：Research paper (scientific journal)  

DOI： https://dx.doi.org/10.1142/S2010326322500253

Language：English   Publishing type：Research paper (scientific journal)  

DOI： https://doi.org/10.1088/1751-8121/abecaa

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

Other Link： http://hdl.handle.net/2433/260647

Language：English   Publishing type：Research paper (scientific journal)  

We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented.

DOI： 10.7494/OpMath.2019.39.3.361

Language：English   Publishing type：Research paper (scientific journal)  

The persistent homology of a stationary point process on R^N is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

DOI： 10.1214/17-AAP1371

Language：English   Publishing type：Research paper (scientific journal)  

This paper studies a higher dimensional generalization of Frieze's ζ(3) -limit theorem on the d-Linial–Meshulam process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in Θ (n^{d-1}).

DOI： 10.1002/rsa.20718

Language：English   Publishing type：Research paper (scientific journal)  

In this note, we show that determinantal point processes on the real line corresponding to de Branges spaces of entire functions are rigid in the sense of Ghosh-Peres and, under certain additional assumptions, quasi-invariant under the group of diffeomorphisms of the line with compact support.

DOI： 10.3792/pjaa.93.1

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1214/ECP.v20-4252

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s00440-015-0644-6

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.2969/jmsj/06720763

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

We introduce certain classes of random point fields, including fermion and boson point processes, which are associated with Fredholm determinants of certain integral operators and study some of their basic properties: limit theorems, correlation functions, Palm measures etc. Also we propose a conjecture on an α-analogue of the determinant and permanent.

DOI： 10.1016/S0022-1236(03)00171-X

Language：English   Publishing type：Research paper (scientific journal)  

We construct and study a family of probability measures on the configuration space over countable discrete space associated with nonnegative definite symmetric operators via determinants. Under a mild condition they turn out unique Gibbs measures. Also some ergodic properties, including the entropy positivity, are discussed in the lattice case.

DOI： 10.1214/aop/1055425789

Language：English   Publishing type：Research paper (scientific journal)  

This paper proposes a simple mathematical model forupstream fish migration along rivers. The model describes the fish migration along a river based on a mixed optimal control approach having swimming velocity, school size, and stopping time of migration as control variables. The optimization problem reduces to a variational inequality. Its explicit “viscosity” solution is presented with the dependence of the fish migration on river environment. To prove uniqueness of the solution to the variational inequality requires a constructive argument not based on the conventional theorems. A novel finite difference scheme for solving the variational inequality is also proposed with its convergence results. An application example of the model discusses the upstream migration of Plecoglossus altivelis (Ayu) in Japan, which evaluates the dependence of the fish migration on the habitat quality and provides recommendations for managing river environment. This is an interdisciplinary research between environmental and mathematical fields.

DOI： 10.13044/j.sdewes.d6.0221

Language：English   Publishing type：Research paper (scientific journal)  

Rock pore geometry has heterogeneous characteristics and is scale dependent. This feature in a geological formation differs significantly from artificial materials and makes it difficult to predict hydrologic and elastic properties. To characterize pore heterogeneity, we propose an evaluation method that exploits the recently developed persistent homology theory. In the proposed method, complex pore geometry is first represented as sphere cloud data using a pore-network extraction method. Then, a persistence diagram (PD) is calculated from the point cloud, which represents the spatial distribution of pore bodies. A new parameter (distance index H) derived from the PD is proposed to characterize the degree of rock heterogeneity. Low H value indicates high heterogeneity. A new empirical equation using this index H is proposed to predict the effective elastic modulus of porous media. The results indicate that the proposed PD analysis is very efficient for extracting topological feature of pore geometry.

DOI： 10.1029/2017WR021864

Language：English   Publishing type：Research paper (international conference proceedings)  

The determinantal point process (DPP) has been receiving increasing attention in machine learning as a generative model of subsets consisting of relevant and diverse items. Recently, there has been a significant progress in developing efficient algorithms for learning the kernel matrix that characterizes a DPP. Here, we propose a dynamic DPP, which is a DPP whose kernel can change over time, and develop efficient learning algorithms for the dynamic DPP. In the dynamic DPP, the kernel depends on the subsets selected in the past, but we assume a particular structure in the dependency to allow efficient learning. We also assume that the kernel has a low rank and exploit a recently proposed learning algorithm for the DPP with low-rank factorization, but also show that its bottleneck computation can be reduced from O(M^2 K) time to O(M K^2) time, where M is the number of items under consideration, and $K$ is the rank of the kernel, which can be set smaller than M by orders of magnitude.

Language：English   Publishing type：Research paper (scientific journal)  

We consider a spatial stochastic model of wireless cellular networks, where the base stations (BSs) are deployed according to a simple and stationary point process on R^d, d ≥ 2. In this model, we investigate tail asymptotics of the distribution of signal-to-interference ratio (SIR), which is a key quantity in wireless communications. In the case where the pathloss function representing signal attenuation is unbounded at the origin, we derive the exact tail asymptotics of the SIR distribution under an appropriate sufficient condition. While we show that widely-used models based on a Poisson point process and on a determinantal point process meet the sufficient condition, we also give a counterexample violating it. In the case of bounded path-loss functions, we derive a logarithmically asymptotic upper bound on the SIR tail distribution for the Poisson-based and α-Ginibrebased models. A logarithmically asymptotic lower bound with the same order as the upper bound is also obtained for the Poisson-based model.

DOI： 10.19195/0208-4147.37.2.12

Language：English   Publishing type：Research paper (scientific journal)  

We consider two solvable models with equal entropy on the infinite ladder graph Z×{1,2}: the uniform spanning forest (USF), the abelian sandpile (ASM). We show that the symbolic models (abelian sandpile and spanning forest) are equal entropy symbolic covers of a certain algebraic dynamical system. In the past results of this nature have been established for sandpile models on lattices Z^d. But we present a first example in case of spanning trees.

DOI： 10.1016/j.indag.2016.02.003

Language：English   Publishing type：Research paper (scientific journal)  

Spatial stochastic models have been much used for perfor mance analysis of wireless communication networks. This is due to th fact that the performance of wireless networks depends on the spatial con figuration of wireless nodes and the irregularity of node locations in a rea wireless network can be captured by a spatial point process. Most work on such spatial stochastic models of wireless networks have adopted homo geneous Poisson point processes as the models of wireless node locations While this adoption makes the models analytically tractable, it assume that the wireless nodes are located independently of each other and thei spatial correlation is ignored. Recently, the authors have proposed to adop the Ginibre point process-one of the determinantal point processes-a the deployment models of base stations (BSS) in cellular networks. Th determinantal point processes constitute a class of repulsive point processe and have been attracting attention due to their mathematically interestin properties and efficient simulation methods. In this tutorial, we provide brief guide to the Ginibre point process and its variant, -Ginibre poin process, as the models of BS deployments in cellular networks and sho some existing results on the performance analysis of cellular network mod els with -Ginibre deployed BSS. The authors hope the readers to use suc point processes as a tool for analyzing various problems arising in futur cellular networks.

DOI： 10.1587/transcom.2016NEI0001

Language：English   Publishing type：Research paper (scientific journal)  

Stochastic geometry models for wireless communication networks have recently attracted much attention. This is because the performance of such networks critically depends on the spatial configuration of wireless nodes and the irregularity of the node configuration in a real network can be captured by a spatial point process. However, most analysis of such stochastic geometry models for wireless networks assumes, owing to its tractability, that the wireless nodes are deployed according to homogeneous Poisson point processes. This means that the wireless nodes are located independently of each other and their spatial correlation is ignored. In this work we propose a stochastic geometry model of cellular networks such that the wireless base stations are deployed according to the Ginibre point process. The Ginibre point process is one of the determinantal point processes and accounts for the repulsion between the base stations. For the proposed model, we derive a computable representation for the coverage probability-the probability that the signal-to-interference-plus-noise ratio (SINR) for a mobile user achieves a target threshold. To capture its qualitative property, we further investigate the asymptotics of the coverage probability as the SINR threshold becomes large in a special case. We also present the results of some numerical experiments.

DOI： 10.1239/aap/1409319562

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

We show that the zeros of the random power series with i.i.d. real Gaussian coefficients form a Pfaffian point process. We further show that the product moments for absolute values and signatures of the power series can also be expressed by Pfaffians.

DOI： 10.1214/EJP.v18-2545

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

We show that under the heat-bath Glauber dynamics for the ferromagnetic Ising model on a finite graph, the single spin expectation at a fixed time starting at the all-up configuration is not necessarily an increasing function of coupling constants.

DOI： 10.3792/pjaa.86.33

Language：English   Publishing type：Research paper (scientific journal)  

We consider a Markov chain on a finite state space and obtain an expression of the joint distribution of the cover time and the last point visited by the Markov chain. As a corollary, we obtain the spectral representation of the distribution of the cover time.

DOI： 10.2206/kyushujm.62.281

Language：English   Publishing type：Research paper (other academic)  

Language：English   Publishing type：Research paper (scientific journal)  

We give a probabilistic expression of the α-determinant by using the Wishart distribution on the cone of positive definite matrices. As a corollary, we give a partial answer to a positivity problem for the α-determinant which is equivalent to the existence problem of certain multivariate probability distributions. We also give a concrete example to support our conjecture for the positivity.

DOI： 10.2206/kyushujm.61.169

Language：English   Publishing type：Research paper (scientific journal)  

The sum of the lower bound and the upper one of the spectrum of our discrete Laplacian is less than or equal to 2. The equality holds if a graph is bipartite while the converse does not hold for general infinite graphs. In this paper, we give an estimate of the upper bounds of Dirichlet forms and using this estimate together with an h-transform, we show that the sum is strictly less than 2 for a certain class of infinite graphs.

DOI： 10.1007/s11118-006-9027-z

Language：English   Publishing type：Research paper (scientific journal)  

For the fermion point process on the whole complex plane associated with the exponential kernel e^zw̄, we show the central limit theorem for the random variable ξ(D _r , the number of points inside the ball D _r of radius r, as r → ∞ and we establish the large deviation principle for the random variables {r ^-2ξ (D _r ), r > 0}.

DOI： 10.1007/s10955-006-9026-x

Language：English   Publishing type：Research paper (other academic)  

Language：English   Publishing type：Research paper (international conference proceedings)  

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

For a certain class of reversible random walks possibly with drift on an abelian covering graph of a finite graph, using the technique of twisted transition operator, we obtain the asymptotic behavior of the n-step transition probability p_n(x, y) as n → − give an expression of the constant which appears in the asymptotics.

DOI： 10.2748/tmj/1113246940

Language：English   Publishing type：Research paper (scientific journal)  

We construct a Glauber dynamics on {0, 1}^R, R a discrete space, with infinite range flip rates, for which a fermion point process is reversible. We also discuss the ergodicity of the corresponding Markov process and the log-Sobolev inequality.

DOI： 10.1017/S0027763000008412

Language：English   Publishing type：Research paper (scientific journal)  

For a magnetic Schrödinger operator on a graph, which is a generalization of classical Harper operator, we study some spectral properties: the Bloch property and the behaviour of the bottom of the spectrum with respect to magnetic fields. We also show some examples which have interesting properties.

DOI： 10.1017/S0027763000022157

Language：English   Publishing type：Research paper (scientific journal)  

For simple random walks (Pⁿ _G) on a homogeneous graph G and (Pⁿ _L(G)) on its line graph L(G), we obtain the relationship between the asymptotic behavior of the n-step transition probability Pⁿ _G(x, x) and that of Pⁿ _L(G)(x, x) as n→∞.

DOI： 10.2969/jmsj/05210099

Language：English   Publishing type：Research paper (scientific journal)  

Let G be an infinite d-regular graph and L(G) its line graph. We consider discrete Laplacians on G and L(G), and show the exact relation between the spectrum of -Δc and that of -Δ_L(G), Our method is also applicable to (d₁,d₂)-seiniregular graphs, subdivision graphs and para-line graphs.

Language：English   Publishing type：Research paper (scientific journal)  

For discrete magnetic Schrödinger operators on covering graphs of a finite graph, we investigate two spectral properties: (1) the relationship between the spectrum of the operator on the covering graph and that on a finite graph, (2) the analyticity of the bottom of the spectrum with respect to magnetic flow. Also we compute the second derivative of the bottom of the spectrum and represent it in terms of geometry of a graph.

DOI： 10.1006/jfan.1999.3478

Language：English   Publishing type：Research paper (other academic)  

Language：English   Publishing type：Research paper (scientific journal)  

Ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition. In the course of our argument, we employ perturbation theory for the maximal eigenvalues of twisted transition operator. As a result, an asymptotic of the probabilityp(n,x,y) that a particle starting atxreachesyat timenasngoes to infinity is established.

DOI： 10.1006/jfan.1998.3322

Language：English   Publishing type：Research paper (scientific journal)  

We consider the expectation of the determinant del(λ - X)^-1 for Im λ > 0 associated with some random NxN matrices and factorize it into N Stieltjes transforms of probability measures. Moreover, using this factorization, we investigate the limiting behavior of the logarithm of the quantity as N → ∞.

Language：English   Publishing type：Research paper (scientific journal)  

We discuss two types of trace formula which arise from the inverse spectral problem for discrete Schrödinger operators as L = -Δ + V (x) where V is a bounded potential. One is the relationship between a potential and spectral data, and another is the one between the green function of L and periodic orbits of a state space.

DOI： 10.2977/prims/1195144826

Total pages：xi, 258 p.   Language：English  

CiNii Books

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Event date： 2015.2

Language：English   Presentation type：Oral presentation (general)  

Venue：Tohoku University   Country：Japan  

Event date： 2014.11

Language：English   Presentation type：Oral presentation (general)  

Venue：University of Tokyo   Country：Japan  

Event date： 2014.9

Language：English   Presentation type：Oral presentation (general)  

Venue：University of Warwick   Country：Japan  

Event date： 2012.6

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：理化学研究所   Country：Japan  

Event date： 2011.12

Presentation type：Oral presentation (invited, special)  

Venue：関西大学   Country：Japan  

Event date： 2011.12

Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2010.12

Presentation type：Oral presentation (general)  

Country：Japan  

Random analytic functions and their zeros

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：九州大学伊都キャンパス   Country：Japan  

Event date： 2025.3

Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Event date： 2024.12

Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Event date： 2024.11

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 2024.10

Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Event date： 2024.9

Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Event date： 2024.9

Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Event date： 2024.8

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 2024.7

Language：English  

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Event date： 2024.7

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Chuo University   Country：Japan  

The zeros of random power series with i.i.d. complex Gaussian
coefficients form the determinantal point process associated with the Bergman
kernel. As a natural generalization of this model, we are concerned with Gaussian
power series with coefficients being stationary, centered, complex Gaussian
process. We focus on the zeros of such analytic Gaussian power series and discuss
the asymptotic behavior of their density as the window approaching to the
convergence circle.

Other Link： https://sites.google.com/g.chuo-u.ac.jp/event-website/home

Event date： 2024.5

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 2024.3

Language：English   Presentation type：Oral presentation (general)  

Venue：Leiden University, Netherlands   Country：Netherlands  

Event date： 2024.3

Language：English   Presentation type：Oral presentation (general)  

Venue：IHES, France   Country：France  

Event date： 2023.12

Language：English   Presentation type：Oral presentation (general)  

Venue：National University of Singapore   Country：Singapore  

Event date： 2023.10

Language：English   Presentation type：Oral presentation (general)  

Venue：RIMS, Kyoto University   Country：Japan  

Event date： 2023.8

Language：English   Presentation type：Oral presentation (general)  

Venue：Chuo University   Country：Japan  

Event date： 2023.3

Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

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Event date： 2022.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：Nishijin Plaza, Kyushu University, Fukuoka   Country：Japan  

Event date： 2022.12

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 2022.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：Tohoku University, Sendai.   Country：Japan  

Event date： 2022.11

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：Hokkaido University, Sapporo.   Country：Japan  

Event date： 2022.11

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 2022.8 - 2022.9

Language：English   Presentation type：Oral presentation (invited, special)  

Country：Singapore  

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Event date： 2022.8

Language：English   Presentation type：Oral presentation (general)  

Venue：TOKYO ELECTRON House of Creativity, Tohoku University   Country：Japan  

Event date： 2022.8

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 2022.6

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：中央大学   Country：Japan  

Event date： 2022.6

Language：English   Presentation type：Oral presentation (general)  

Venue：金沢大学   Country：Japan  

Event date： 2022.6

Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Event date： 2022.1

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：統計数理研究所.   Country：Japan  

Event date： 2021.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2021.11

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：滋賀大学彦根キャンパス講堂多目的室I   Country：Japan  

一般的なガウス型解析関数の零点の性質，および 独立同分布なガウス確率変数列を係数とするべき級数として定義される 双曲型のガウス型解析関数の零点の著しい性質について簡単なサーベイをした後， 係数を定常ガウス過程に置き換えた場合に， 零点の個数の期待値がその定常ガウス過程にどのように影響されるかについて論じる． (野田航平氏との共同研究)

Event date： 2021.3

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：Zoom online   Country：Japan  

Event date： 2020.6

Language：English   Presentation type：Oral presentation (general)  

Venue：I2CNER, Kyushu University   Country：Japan  

Event date： 2020.6

Language：English   Presentation type：Oral presentation (general)  

Venue：KTH Royal Institute of Technology   Country：Sweden  

Event date： 2019.5

Language：English   Presentation type：Oral presentation (general)  

Country：Korea, Republic of  

Event date： 2019.4

Language：Japanese  

Country：Japan  

Event date： 2019.4

Language：English   Presentation type：Oral presentation (general)  

Venue：Marrakesh, Morocco   Country：Morocco  

Event date： 2018.6 - 2020.6

Language：English   Presentation type：Oral presentation (general)  

Venue：National University of Singapore   Country：Singapore  

Event date： 2018.2

Language：English  

Venue：New Orleans   Country：United States  

The determinantal point process (DPP) has been receiving increasing attention in machine learning as a generative model of subsets consisting of relevant and diverse items. Recently, there has been a significant progress in developing efficient algorithms for learning the kernel matrix that characterizes a DPP. Here, we propose a dynamic DPP, which is a DPP whose kernel can change over time, and develop efficient learning algorithms for the dynamic DPP. In the dynamic DPP, the kernel depends on the subsets selected in the past, but we assume a particular structure in the dependency to allow efficient learning. We also assume that the kernel has a low rank and exploit a recently proposed learning algorithm for the DPP with low-rank factorization, but also show that its bottleneck computation can be reduced from O(M² K) time to O(M K²) time, where M is the number of items under consideration, and K is the rank of the kernel, which can be set smaller than M by orders of magnitude.

Event date： 2017.9

Language：English   Presentation type：Oral presentation (general)  

Venue：TU Kaiserslautern, Germany.   Country：Germany  

The persistent homology of a stationary point process on R^N is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence dia- gram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to in- finity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

Event date： 2017.8

Language：English   Presentation type：Oral presentation (general)  

Venue：Nagoya University   Country：Japan  

Event date： 2016.5

Language：English  

Venue：Tempe   Country：United States  

We consider the spatial stochastic model of single-tier downlink cellular networks, where the wireless base stations are deployed according to a general stationary point process on the Euclidean plane with general i.i.d. propagation effects. Recently, Ganti & Haenggi (2016) consider the same general cellular network model and, as one of many significant results, derive the tail asymptotics of the signal-to-interference ratio (SIR) distribution. However, they do not mention any conditions under which the result holds. In this paper, we compensate their result for the lack of the condition and expose a sufficient condition for the asymptotic result to be valid. We further illustrate some examples satisfying such a sufficient condition and indicate the corresponding asymptotic results for the example models. We give also a simple counterexample violating the sufficient condition.

Event date： 2015.5

Language：English  

Venue：Mumbai   Country：India  

Recently, spatial stochastic models based on deter-minantal point processes (DPP) are studied as promising models for analysis of cellular wireless networks. Indeed, the DPPs can express the repulsive nature of the macro base station (BS) configuration observed in a real cellular network and have many desirable mathematical properties to analyze the network performance. However, almost all the prior works on the DPP based models assume the Rayleigh fading while the spatial models based on Poisson point processes have been developed to allow arbitrary distributions of fading/shadowing propagation effects. In order for the DPP based model to be more promising, it is essential to extend it to allow non-Rayleigh propagation effects. In the present paper, we propose the downlink cellular network model where the BSs are deployed according to the Ginibre point process, which is one of the main examples of the DPPs, over Nakagami-m fading. For the proposed model, we derive a numerically computable form of the coverage probability and reveal some properties of it numerically and theoretically.

Event date： 2014.8

Language：English   Presentation type：Oral presentation (general)  

Venue：Kansai Unversity   Country：Japan  

Event date： 2014.7

Language：English  

Venue：Quebec City   Country：Canada  

Policy gradient reinforcement learning (PGRL) has been receiving substantial attention as a mean for seeking stochastic policies that maximize cumulative reward. However, the learning speed of PGRL is known to decrease substantially when PGRL explores the policies that give the Markov chains having long mixing time. We study a new approach of regularizing how the PGRL explores the policies by the use of the hitting time of the Markov chains. The hitting time gives an upper bound on the mixing time, and the proposed approach improves the learning efficiency by keeping the mixing time of the Markov chains short. In particular, we propose a method of temporal-difference learning for estimating the gradient of the hitting time. Numerical experiments show that the proposed method outperforms conventional methods of PGRL.

Event date： 2014.5

Language：English  

Venue：Hammamet   Country：Tunisia  

Coverage probability is one of the most important metrics for evaluating the performance of wireless networks. However, the spatial stochastic models for which a computable expression of the coverage probability is available are restricted (such as the Poisson based or α-Ginibre based models). Furthermore, even if it is available, the practical numerical computation may be time-consuming (in the case of α-Ginibre based model). In this paper, we propose the application of Padé approximation to the coverage probability in the wireless network models based on general spatial stationary point processes. The required Maclaurin coefficients are expressed in terms of the moment measures of the point process, so that the approximants are expected to be available for a broader class of point processes. Through some numerical experiments for the cellular network model, we demonstrate that the Padé approximation is effectively applicable for evaluating the coverage probability.

Event date： 2013.3

Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：Technische Universität München   Country：Japan  

Event date： 2013.1

Language：English  

Venue：京都大理学部   Country：Japan  

Event date： 2012.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：京都大数理解析研究所   Country：Japan  

Event date： 2012.8

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：小山高専   Country：Japan  

Event date： 2012.3

Presentation type：Oral presentation (general)  

Venue：湘南国際村センター   Country：Japan  

Event date： 2011.12

Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2011.11

Presentation type：Oral presentation (general)  

Venue：KKR 鹿児島 敬天閣   Country：Japan  

Event date： 2011.10

Presentation type：Oral presentation (general)  

Venue：奈良女子大学   Country：Japan  

Event date： 2011.9

Presentation type：Oral presentation (general)  

Venue：IBM Tokyo   Country：Japan  

Event date： 2011.9

Presentation type：Oral presentation (general)  

Venue：IBM Tokyo   Country：Japan  

Event date： 2011.6

Presentation type：Oral presentation (general)  

Venue：九州大学伊都キャンパス   Country：Japan  

Event date： 2011.3

Presentation type：Oral presentation (general)  

Venue：Ito Campus, Kyushu University   Country：Japan  

Event date： 2011.2

Presentation type：Oral presentation (general)  

Venue：Tohoku University   Country：Japan  

Event date： 2011.1

Presentation type：Oral presentation (general)  

Venue：Komaba Campus, The University of Tokyo   Country：Japan  

Event date： 2011.1

Presentation type：Oral presentation (general)  

Venue：Ito Campus, Kyushu University   Country：Japan  

Event date： 2009.1

Presentation type：Oral presentation (general)  

Country：Korea, Republic of  

Event date： 2009.1

Presentation type：Oral presentation (general)  

Venue：Chungbuk National University   Country：Korea, Republic of  

Event date： 2008.10

Presentation type：Oral presentation (general)  

Venue：Nishijin Plaza, Kyushu University   Country：Japan  

Presentation type：Oral presentation (general)  

Venue：Levico   Country：Italy  

Presentation type：Oral presentation (general)  

Venue：Bedlevo   Country：Poland  

Presentation type：Oral presentation (general)  

Venue：University of Washington   Country：United States  

Presentation type：Oral presentation (general)  

Venue：University of Wahington   Country：United States  

Presentation type：Oral presentation (general)  

Venue：Nishijin Plaza, Fukuoka   Country：Japan  

Other Link： http://stokhos.shinshu-u.ac.jp/salsis2007/index.html

Presentation type：Oral presentation (general)  

Country：Japan  

Other Link： http://www2.math.kyushu-u.ac.jp/~shirai/numprob2007/index.html

Presentation type：Oral presentation (general)  

Venue：Taikanso, Matsushima   Country：Japan  

Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

Language：Japanese   Publishing type：Book review, literature introduction, etc.  

Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

Random point configurations are said to be in hyperuniform states, if density
fluctuations are anomalously suppressed in large-scale. Typical examples are
found in Coulomb gas systems in two dimensions especially called log-gases in
random matrix theory, in which points are repulsively correlated by long-range
potentials. In infertile lands like deserts continuous survival competitions
for water and nutrition will cause long-ranged repulsive interactions among
plants. We have prepared digital data of spatial configurations of
center-of-mass for bushes weighted by bush sizes which we call masses. Data
analysis shows that such ecological point configurations do not show
hyperuniformity as unmarked point processes, but are in hyperuniform states as
marked point processes in which mass distributions are taken into account. We
propose the non-equilibrium statistical-mechanics models to generate marked
point processes having hyperuniformity, in which iterations of random thinning
of points and coalescing of masses transform initial uncorrelated point
processes into non-trivial point processes with hyperuniformity. Combination of
real data-analysis and computer simulations shows the importance of strong
correlations in probability law between spatial point configurations and mass
distributions of individual points to realize hyperuniform marked point
processes.

arXiv

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Other Link： http://arxiv.org/pdf/2501.12807v1

We analyze Gaussian analytic functions (GAFs) defined as power series with
coefficients modeled by discrete stationary Gaussian processes, utilizing their
spectral measures. We revisit some limit theorems for random analytic functions
and examine some examples of GAFs through numerical computations. Furthermore,
we provide an integral representation of the n-point correlation functions of
the zero sets of GAFs in terms of the spectral measures of the underlying
coefficient Gaussian processes.

arXiv

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Other Link： http://arxiv.org/pdf/2501.03704v1

We study the zeros of random power series with stationary complex Gaussian
coefficients, whose spectral measure is absolutely continuous. We analyze the
precise asymptotic behavior of the radial density of zeros near the boundary of
the circle of convergence. The dependence of the coefficients generally reduces
the density of zeros compared with that of the hyperbolic Gaussian analytic
function (the i.i.d. coefficients case), where the spectral density and its
zeros plays a crucial role in this reduction. We also show the relationship
between the support of the spectral measure and the analytic continuation at
the boundary of the circle of convergence.

arXiv

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Other Link： http://arxiv.org/pdf/2501.02586v2

We consider nonnormal matrix-valued dynamical systems with discrete time. For
an eigenvalue of matrix, the number of times it appears as a root of the
characteristic polynomial is called the algebraic multiplicity. On the other
hand, the geometric multiplicity is the dimension of the linear space of
eigenvectors associated with that eigenvalue. If the former exceeds the latter,
then the eigenvalue is said to be defective and the matrix becomes
nondiagonalizable by any similarity transformation. The discrete-time of our
dynamics is identified with the geometric multiplicity of the zero eigenvalue
$\lambda_0=0$. Its algebraic multiplicity takes about half of the matrix size
at $t=1$ and increases stepwise in time, which keeps excess to the geometric
multiplicity until their coincidence at the final time. Our model exhibits
relaxation processes from far-from-normal to near-normal matrices, in which the
defectivity of $\lambda_0$ is recovering in time. We show that such processes
are realized as size reductions of pseudospectrum including $\lambda_0$. Here
the pseudospectra are the domains on the complex plane which are not
necessarily exact spectra but in which the resolvent of matrix takes extremely
large values. The defective eigenvalue $\lambda_0$ is sensitive to perturbation
and the eigenvalues of the perturbed systems are distributed densely in the
pseudospectrum including $\lambda_0$. By constructing generalized eigenspace
for $\lambda_0$, we give the Jordan block decomposition for the resolvent of
matrix and characterize the pseudospectrum dynamics. Numerical study of the
systems perturbed by Gaussian random matrices supports the validity of the
present analysis.

arXiv

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Other Link： http://arxiv.org/pdf/2411.06472v2

The non-commutative harmonic oscillators (NcHO) and 2p-quantum Rabi models
(2pQRM) are extensions of harmonic oscillators. The purpose of this paper is to
give a relationship between NcHO and 2pQRM, and the fiber decomposition of NcHO
by 2pQRM is shown. We also construct Feynman-Kac formulas of NcHO and 2pQRM.
Then asymptotic behaviors of the spectral zeta function of 2pQRM is considered.

DOI： 10.1063/5.0233587

arXiv

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Other Link： http://arxiv.org/pdf/2408.04239v2

The renormalized spectral zeta function of the quantum Rabi Hamiltonian is
considered. It is shown that the renormalized spectral zeta function converges
to the Riemann zeta function as the coupling constant goes to infinity.
Moreover the path measure associated with the ground state of the quantum Rabi
Hamiltonian is constructed on a discontinuous path space, and several
applications are shown.

arXiv

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Other Link： http://arxiv.org/pdf/2405.09158v2

We introduce two kinds of matrix-valued dynamical processes generated by
nonnormal Toeplitz matrices with the additive rank 1 perturbations $\delta J$,
where $\delta \in {\mathbb{C } }$ and $J$ is the all-ones matrix. For each
process, first we report the complicated motion of the numerically obtained
eigenvalues. Then we derive the specific equation which determines the motion
of non-zero simple eigenvalues and clarifies the time-dependence of degeneracy
of the zero-eigenvalue $\lambda_0=0$. Comparison with the solutions of this
equation, it is concluded that the numerically observed non-zero eigenvalues
distributing around $\lambda_0$ are the exact eigenvalues not of the original
system, but of the system perturbed by uncontrolled rounding errors of
computer. The complex domain in which the eigenvalues of randomly perturbed
system are distributed is identified with the pseudospectrum including
$\lambda_0$ of the original system with $\delta J$. We characterize the
pseudospectrum processes using the symbol curves of the corresponding nonnormal
Toeplitz operators without $\delta J$. We report new phenomena in our second
model such that at each time the outermost closed simple curve cut out from the
symbol curve is realized as the exact eigenvalues, but the inner part of symbol
curve is reduced in size and embedded in the pseudospectrum including
$\lambda_0$. Such separation of exact simple eigenvalues and a degenerated
eigenvalue associated with pseudospectrum will be meaningful for numerical
analysis, since the former is stable and robust, but the latter is highly
sensitive and unstable with respective to perturbations. The present study will
be related to the pseudospectra approaches to non-Hermitian systems developed
in quantum physics

arXiv

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Other Link： http://arxiv.org/pdf/2401.08129v4

We obtain first order linear partial differential equations which are
satisfied by exponential generating functions of two variables for the number
of connected bipartite graphs with given Betti number. By solving these
equations inductively, we obtain the explicit form of generating functions and
derive the asymptotic behavior of their coefficients. We also introduce a
family of basic graphs to classify connected bipartite graphs and give another
expression of the generating functions as the sum over basic graphs of rational
functions of those for the number of labeled bipartite rooted spanning trees.

arXiv

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Other Link： http://arxiv.org/pdf/2208.03996v2

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：2

Type：Scientific advice/Review 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：2

Type：Academic society, research group, etc. 

Type：Scientific advice/Review 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：9

Type：Academic society, research group, etc. 

Type：Competition, symposium, etc. 

Number of participants：150