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

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

DOI： 10.1515/ms-2022-0038

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

The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying the Green formula to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, N2aN order, where a is a positive constant less than one and N is the number of collocation points. Furthermore, it is demonstrated that the error tends to 0 in exponential order in the numerical simulations with increasing number of collocation points N.

DOI： 10.1016/j.cam.2021.113795

Web of Science

Scopus

Other Link： https://www.sciencedirect.com/science/article/pii/S0377042721004179?via%3Dihub

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

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n≥r≥s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with n=4,r=2,s=1

DOI： https://doi.org/10.3390/sym13071283

Other Link： https://www.mdpi.com/2073-8994/13/7/1283

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

Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.

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

We explicitly give new group invariant solutions to a class of Riemann-Liouville time fractional evolution systems with variable coefficients. These solutions are derived from every element in an optimal system of Lie algebras generated by infinitesimal symmetries of evolution systems in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions and show that these solutions solve diffusion-wave equations with variable coefficients. These solutions contain previously known solutions as particular cases. Some plots of solutions subject to the order of the fractional derivative are illustrated.

DOI： 10.1063/1.5035392

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

We prove, for graphical models for nearest neighbour interactions, a conjecture stated by Letac and Massam in 2007. Our result is important in the analysis of Wishart distributions on cones related to graphical models and in its statistical applications.

DOI： 10.3792/pjaa.93.16

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

Good parametrizations of affine transformations are essential to interpolation, deformation, and analysis of shape, motion, and animation. It has been one of the central research topics in computer graphics. However, there is no single perfect method and each one has both advantages and disadvantages. In this paper, we propose a novel parametrization of affine transformations, which is a generalization to or an improvement of existing methods. Our method adds yet another choice to the existing toolbox and shows better performance in some applications. A C++ implementation is available to make our framework ready to use in various applications.

DOI： 10.1137/16M1056936

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

This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/effective creation of computer animation. This book, therefore, serves as a good guidepost to mathematics (differential geometry and Lie theory) for students of geometric modeling and animation in computer graphics. Experienced developers and researchers will also benefit from this book, since it gives a comprehensive overview of mathematical approaches that are particularly useful in character modeling, deformation, and animation.

DOI： 10.2200/S00599ED1V01Y201409CGR017

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

DOI： http://dx.doi.org/10.4153/CJM-2013-047-5

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

Language：English   Publishing type：Research paper (bulletin of university, research institution)  

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

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

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

Language：English   Publishing type：Research paper (bulletin of university, research institution)  

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

A special value of the spectral zeta function of the non-commutative harmonic oscillators

Publisher：Optimization Letters  

We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to determine the convergence rate of alternating projections for the positive semidefinite cone and a line. As a consequence, we show that such alternating projections converge at O(k-12), independently of the singularity degree. A sufficient condition for the linear convergence is also obtained. Our method directly analyzes the defining equation for an alternating projection sequence without using error bounds.

DOI： 10.1007/s11590-025-02196-3

Web of Science

Scopus

Web of Science

Publisher：Osaka Journal of Mathematics  

A Fuchsian system of rank 8 in 3 variables with 4 parameters is found. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equations, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions of these ordinary differential equations are found. Their coefficients are sums of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series4 F3 at 1.

Scopus

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

DOI： https://doi.org/10.3390/sym12081244

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

We derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of generalized Wright functions and Fox H-functions. These solutions are invariant solutions of diffusion-wave equations obtained through certain transformations, which are briefly discussed. We show that the solutions given in this work contain previously known results as particular cases.

DOI： 10.1515/fca-2019-0028

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

We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems. We find that the class of systems of differential equations studied is naturally divided into two cases on the basis of the type of a function that they contain. In each case, the dimension of the Lie algebra generated by the infinitesimal symmetries is greater than 2, and for this reason we present the structures and one-dimensional optimal systems of these Lie algebras. The reduced systems corresponding to the optimal systems are also obtained. Explicit group invariant solutions are found for particular cases.

DOI： 10.1016/j.amc.2018.01.056

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

DOI： 10.1145/2897826.2927338

Language：English  

Zeta Functions of Representations

DOI： 10.14992/00010884

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

Let K be a complex quadratic extension of Q and let A(K) denote the adeles of K. We find special values at all of the critical points of twisted tensor L-functions attached to cohomological cuspforms on GL(2)(AK) and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these L-functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these L-functions, such as their functional equations.

DOI： 10.4153/CJM-2013-047-5

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

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

Let G be a connected reductive algebraic group over C. We denote by K = (G^θ)₀the identity component of the fixed points of an involutive automorphism θ of G. The pair (G, K) is called a symmetric pair. Let Q be a parabolic subgroup of K. We want to find a pair of parabolic subgroups P ₁, P ₂of G such that (i) P ₁∩ P ₂= Q and (ii) P ₁ P ₂is dense in G. The main result of this article states that, for a simple group G, we can find such a pair if and only if (G, K) is a Hermitian symmetric pair. The conditions (i) and (ii) imply that the K -orbit through the origin (eP ₁, eP ₂) of G/P ₁× G/P ₂is closed and it generates an open dense G -orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed K -orbits on G/P ₁× G/P ₂.

DOI： 10.2206/kyushujm.68.113

Language：English  

This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/effective creation of computer animation. This book, therefore, serves as a good guidepost to mathematics (differential geometry and Lie theory) for students of geometric modeling and animation in computer graphics. Experienced developers and researchers will also benefit from this book, since it gives a comprehensive overview of mathematical approaches that are particularly useful in character modeling, deformation, and animation.

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

DOI： 10.1145/2614028.2615386

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

DOI： 10.1145/2614217.2630575

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

While many technical terms, such as Euler angle, quaternion, and affine transformation, now become quite popular in computer graphics, their graphical meanings are sometimes a bit far from the original mathematical entities, which might cause misunderstanding or misuse of the mathematical techniques. This course presents an intuitive introduction to several mathematical concepts that are quite useful for describing motion and deformation of geometric objects. The concepts are inherited mostly from differential geometry and Lie theory, and now commonly used in various aspects of computer graphics, including curve/surface editing, deformation and animation of geometric objects. The objective of this course is to fill the gap between the original mathematical concepts and the practical meanings in computer graphics without assuming any prior knowledge of pure mathematics. We then illustrate practical usefulness of deep understanding of the mathematical concepts. Moreover this course demonstrates our current/ongoing research work, which is benefited from our mathematical formulation.

DOI： 10.1145/2542266.2542268

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

Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ⊂ G or Q ⊂ K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.

DOI： 10.1007/s00031-013-9243-8

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

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

We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.

DOI： 10.3792/pjaa.89.75

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

We give a combinatorial formula of the dimension of global solutions to a generalization of Gauss-Aomoto-Gelfand hypergeometric system, where the quadratic differential operators are replaced by higher order operators. We also derive a polynomial estimate of the dimension of global solutions for the case in 3 × 3 variables.

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

DOI： 10.2996/kmj/1383660691

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

This is a survey on the non-commutative harmonic oscillator, which is a generalization of usual (scalar) harmonic oscillators to the system introduced by Parmeggiani and Wakayama.With the definitions and the basic properties, we summarize the positivity of several related operators with s^l
₂ interpretations. We also mention some unsolved questions, in order to clarify the current status of the problems and expected further development.

DOI： 10.1007/978-1-4471-4863-0_19

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

The visual simulation of fluids has become an important element in many applications, such as movies and computer games. In these applications, large-scale fluid scenes, such as fire in a village, are often simulated by repeatedly rendering multiple small-scale fluid flows. In these cases, animators are requested to generate many variations of a small-scale fluid flow. This paper presents a method to help animators meet such requirements. Our method enables the user to generate flow field variations from a single simulated dataset obtained by fluid simulation. The variations are generated in both the frequency and spatial domains. Fluid velocity fields are represented using Laplacian eigenfunctions which ensure that the flow field is always incompressible. In generating the variations in the frequency domain, we modulate the coefficients (amplitudes) of the basis functions. To generate variations in the spatial domain, our system expands or contracts the simulation space, then the flow is calculated by solving a minimization problem subject to the resized velocity field. Using our method, the user can easily create various animations from a single dataset calculated by fluid simulation. 2013 Copyright held by the Owner/Author.

DOI： 10.1145/2542355.2542371

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

Differential operators on Siegel modular forms which behave well under the restriction of the domain are essentially intertwining operators of the tensor product of holomorphic discrete series to its irreducible components. These are characterized by polynomials in the tensor of pluriharmonic polynomials with some invariance properties. We give a concrete study of such polynomials in the case of the restriction from Siegel upper half space of degree 2n to the product of degree n. These generalize the Gegenbauer polynomials which appear for n = 1. We also describe their radial parts parametrization and differential equations which they satisfy, and show that these differential equations give holonomic systems of rank 2 ⁿ.

DOI： 10.2969/jmsj/06410273

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

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

Differential operators on Siegel modular forms which behave well under the restriction of the domain are essentially intertwining operators of the tensor product of holomorphic discrete series to its irreducible components. These are characterized by polynomials in the tensor of pluriharmonic polynomials with some invariance properties. We give a concrete study of such polynomials in the case of the restriction from Siegel upper half space of degree 2n to the product of degree n. These generalize the Gegenbauer polynomials which appear for n = 1. We also describe their radial parts parametrization and differential equations which they satisfy, and show that these differential equations give holonomic systems of rank 2(n).

DOI： 10.2969/jmsj/06410273

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

Knowing the number of solutions for a Diophantine equation is an important step to study various arithmetic problems. Igusa originated the study of Igusa zeta functions associated to local Diophantine problems. Multiplying all these local Igusa zeta functions we obtain the global version in the natural way. Unfortunately, investigations on global Igusa zeta functions are rare up to now. Reformulating the global Igusa zeta function via the number of morphisms between algebraic systems we discover a new aspect: the multivariable Euler product of Igusa type and its applications. A purpose of this paper is to encourage experts for further studies on global Igusa zeta functions by treating a simple interesting example.

DOI： 10.1016/j.jnt.2008.10.008

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

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

DOI： 10.4064/aa132-2-1

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

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

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function.

DOI： 10.1007/s11139-007-9065-1

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

DOI： 10.4064/aa135-3-5

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

We find a symmetry for the reflection groups in the double shuffle space of depth three. The space was introduced by Ihara, Kaneko and Zagier and consists of polynomials in three variables satisfying certain identities which are connected with the double shuffle relations for multiple zeta values. Goncharov has defined a space essentially equivalent to the double shuffle space and has calculated the dimension. In this paper we relate the structure among multiple zeta values of depth three with the invariant theory for the reflection groups and discuss the dimension of the double shuffle space in this view point.

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

Let V be a neighborhood of a singular locus of a hyperbolic 3-cone-manifold, which is a quotient space of the 3-dimensional hyperbolic space. In this paper we give an explicit expression of a harmonic vector field v on the hyperbolic manifold V in terms of hypergeometric functions. The expression is obtained by solving a system of ordinary differential equations which is induced by separation of the variables in the vector-valued partial differential equation (Δ + 4)τ = 0, where Δ is the Laplacian of V and τ is the dual 1-form of v. We transform this system of ordinary differential equations to single-component differential equations by elimination of unknown functions and solve these equations. The most important step in solving them consists of two parts, decomposing their differential operators into differential operators of the type appearing in Riemann's P-equation in the ring of differential operators and then describing the projections to the components of this decomposition in terms of differential operators that are also of the type appearing in Riemann's P-equation.

DOI： 10.2977/prims/1201012040

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

Lot V be a neighborhood of a singular locus of a hyperbolic 3-cone-manifold, which is a quotient space of the 3-dimensional hyperbolic space. In this paper we give an explicit expression of a harmonic vector field v on the hyperbolic manifold V in terms of hypergeometric functions. The expression is obtained by solving a system of ordinary differential equations which is induced by separation of the variables in the vector-valued partial differential equation (A + 4)tau = 0, where Delta is the Laplacian of V and tau is the dual 1-form of v. We transform this system of ordinary differential equations to single-component differential equations by elimination of unknown functions and solve these equations. The most important step in solving them consists of two parts, decomposing their differential operators into differential operators of the type appearing in Riemann's P-equation in the ring of differential operators and then describing the projections to the components of this decomposition in terms of differential operators that are also of the type appearing in Riemann's P-equation.

DOI： 10.2977/prims/1201012040

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

DOI： 10.2996/kmj/1183475511

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

We show that the fake projective planes that are constructed from dyadic discrete subgroups discovered by Cartwright, Mantero, Steger, and Zappa are realized as connected components of certain unitary Shimura surfaces. As a corollary we show that these fake projective planes have models defined over the number field Q (sqrt(-3), sqrt(5)).

DOI： 10.1016/j.jalgebra.2006.07.019

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

We consider a reductive dual pair (G, G′) in the stable range with G′ the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent K′_ℂ-orbits, where K′ is a maximal compact subgroup of G′ and we describe the precise K _ℂ-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair (G, K). As an application, we prove sphericality and normality of the closure of certain nilpotent K _ℂ-orbits obtained in this way. We also give integral formulas for their degrees.

DOI： 10.1090/S0002-9947-05-03826-2

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

We prove existence of scaling limits of sequences of functions defined by the recursion relation w′_n+1(ϰ)= -w_n(ϰ)². which is a successive approximation to w′(ϰ)= -w_n(ϰ)² a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n→ √ in an asymptotically conformal way, [formula omitted], for a sequence of numbers {qn} and a function [formula omitted]. We also discuss implication of the results in terms of random sequential bisections of a rod.

DOI： 10.1619/fesi.49.291

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

We investigate the structure of invariant distributions on a non-isotropic non-Riemannian symmetric space of rank one. Especially, the J-criterion related to the generalized Gelfand pair is shown for this space without imposing the condition on the eigenfuction of the Laplace-Bertrami operator.

DOI： 10.1016/S0019-3577(05)80043-6

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

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.

DOI： 10.1007/s11005-004-4292-5

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

The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.

DOI： 10.1142/S0129167X0400248X

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

DOI： 10.1619/fesi.46.297

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

We give a formula for the coefficients of the Yablonskii-Vorob'ev polynomial. Also the reduction modulo a prime number of the polynomial is studied.

DOI： 10.2969/jmsj/1191418760

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

We study the structure of the twisted homology groups attached to weighted lines in the plane with resonant singular points, and find possible intersection pairings. As a typical example, we treat homology groups attached to 2-dimensional Selberg-type integrals.

DOI： 10.1002/mana.200310105

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

The spectral problem for non-commutative harmonic oscillators is shown to be equivalent to solve Fuchsian ordinary differential equations with four regular singular points in a complex domain.

DOI： 10.1007/s002200100362

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

Let G̃ be the metaplectic double cover of Sp(2n,ℝ), U(p,q) or O*(2p). we study the Bernstein degrees and the associated cycles of the irreducible unitary highest weight representations of G̃, by using the theta correspondence of dual pairs. The first part of this article is a summary of fundamental properties and known results of the Bernstein degrees and the associated cycles. Our first result is a comparison theorem between the K-module structures of the following two spaces; one is the theta lift of the trivial representation and the other is the ring of regular functions on its associated variety. Secondarily, we obtain the explicit values of the degrees of some small nilpotent K_C-orbits by means of representation theory. The main result of this article is the determination of the associated cycles of singular unitary highest weight representations, which are the theta lifts of irreducible representations of certain compact groups. In the proofs of these results, the multiplicity free property of spherical subgroups and the stability of the branching coefficients play important roles.

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

We give a formula for the degrees of orbits of the irreducible representations with multiplicity-free action. In particular, we obtain the Bernstein degree and the associated cycle of the irreducible unitary highest weight modules of the scalar type for arbitrary hermitian Lie algebras.

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

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

The p-adic norms of the Taylor coefficients of the function exp(x + x^p/p) are expressed in terms of a p-adic analytic function for p ≤ 23.

DOI： 10.14492/hokmj/1351001078

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

DOI： 10.3792/pjaa.75.9

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

We give quotientsV//G=Spec(C[V]^G) for some prehomogeneous vector spaces (G×H,V).

DOI： 10.1006/jabr.1996.6979

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

Commuting differential operators with symbols ∂²₁ + ∂²₂, ∂²₁∂²₂ are related to the functional equation V₁(x₁)U₊(x₁ + x₂) + V₁(x₁)U_-(x₁ - x₂) + V₂(x₂)U₊(x₁ + x₂) - V₂(x₂)U_-(x₁ - x₂) = F_-(x₁ + x₂) + F_-(x₁ - x₂) + G₁(x₁) + G₂(x₂). We discuss several properties, such as a meromorphic extension of solutions and periodicity, of this functional equation and commuting differential operators.

DOI： 10.1016/0019-3577(96)85093-2

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

DOI： 10.2969/jmsj/04540583

Language：English   Book type：Scholarly book

Selected papers from the proceeding of MEIS2013

Event date： 2024.7

Venue：Banff International Research Station for Mathematical Innovation and Discovery, Keowna, Canada  

Event date： 2024.7

Language：English   Presentation type：Oral presentation (general)  

Venue：National University of Mongolia   Country：Mongolia  

Event date： 2024.6

Venue：東京工業大学  

Event date： 2024.3

Venue：静岡大学  

Event date： 2024.3

Venue：北海道大学  

Event date： 2023.9

Venue：宮崎大学  

Event date： 2023.9

Venue：九州大学  

Event date： 2023.2

Language：English   Presentation type：Oral presentation (general)  

Venue：Kyushu University   Country：Japan  

Event date： 2023.2

Language：Japanese  

Venue：オンライン   Country：Japan  

Event date： 2023.2

Language：English   Presentation type：Oral presentation (general)  

Venue：東京大学国際高等研究所カブリ数物連携宇宙研究機構   Country：Japan  

Event date： 2022.8

Language：Japanese  

Venue：パシフィコ横浜ノース   Country：Japan  

リアルやアニメの3Dのコンピュータゲームの制作を行っているプログラマやエンジニアに向けた数学の専門家からのわかりやすい解説と応用を与えた。IMIとして、鍛冶教授、溝口教授とも共同で、何年にもわたって続けている。

Event date： 2022.3

Language：Japanese  

Venue：九州大学   Country：Japan  

Event date： 2021.12

Language：English   Presentation type：Oral presentation (general)  

Venue：東京国際フォーラム   Country：Japan  

Event date： 2021.8

Language：Japanese  

Venue：パシフィコ横浜   Country：Japan  

Event date： 2019.10 - 2020.10

Language：English   Presentation type：Oral presentation (general)  

Venue：Johns Hopkins University   Country：United States  

Event date： 2018.12 - 2019.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：城西大学   Country：Japan  

Event date： 2017.9

Language：English   Presentation type：Oral presentation (general)  

Venue：ホテルアベスト八方アルディラ   Country：Japan  

Event date： 2017.8

Language：English   Presentation type：Oral presentation (general)  

Venue：名古屋大学   Country：Japan  

Event date： 2016.10

Language：English   Presentation type：Oral presentation (general)  

Venue：Seoul National University   Country：Korea, Republic of  

保形形式に作用する共変な微分作用素を多変数の場合に超幾何関数を用いて記述した。

Event date： 2016.7

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

Venue：Anaheim Convention Center   Country：United States  

Event date： 2016.2

Language：English   Presentation type：Oral presentation (general)  

Venue：Weta Digital, NZ   Country：New Zealand  

Event date： 2015.11

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

Venue：静岡県伊豆の国市   Country：Japan  

Event date： 2015.9

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

Venue：京都産業大学   Country：Japan  

Event date： 2015.9

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：鹿児島大学理学部   Country：Japan  

Event date： 2015.8

Language：English   Presentation type：Oral presentation (general)  

Venue：早稲田大学   Country：Japan  

Event date： 2015.7

Language：English   Presentation type：Oral presentation (general)  

Venue：東京大学Kavli IPMU   Country：Japan  

Event date： 2014.8

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

Venue：Vancouver   Country：Canada  

This is a course lecture on mathematical basics to graphics community

Event date： 2013.8

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：パシフィコ横浜,   Country：Japan  

ゲーム開発者を中心とした実業界で活躍するエキスパートに向けて、数学的手法や思想、特に行列やリー群を活用した、運動の記述や補間について、分かりやすく解説した。

Event date： 2013.3

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

Venue：京都大学   Country：Japan  

Event date： 2012.3

Language：English   Presentation type：Oral presentation (general)  

Venue：名古屋大学大学院多元数理科学研究科   Country：Japan  

Event date： 2012.1

Language：English  

Country：Japan  

Event date： 2011.9

Presentation type：Oral presentation (general)  

Venue：東京大学大学院数理科学研究科   Country：Japan  

Kostant-Sekiguchi correspondence, so called, is the bijective correspondence between the set of nilpotent (co)adjoint orbits on a semisimple Lie algebra and the set of nilpotent KC-orbits on the complexified tangent space of the corresponding Riemannian symmetric space.
From the beginning of the theory, Sekiguchi has the correspondence on the context of non-Riemannian semisimple symmetric spaces. I will review this Sekiguchi's work, as well as an application/ interpretation to the geometric invariants of representation theory.

Other Link： http://www.gem.aoyama.ac.jp/~rtweb/workshop/sekiguchi60/program_sekiguchi60.html

Event date： 2011.7

Presentation type：Oral presentation (general)  

Venue：東京大学大学院数理科学研究科   Country：Japan  

Other Link： http://sites.google.com/site/infiniteanalysis2011/program

Event date： 2010.11

Language：English   Presentation type：Oral presentation (general)  

Venue：Lorentz Center, Leiden University   Country：Netherlands  

Event date： 2010.6

Language：English   Presentation type：Oral presentation (general)  

Venue：Chern Institute of Mathematics in Nankai, 天津   Country：China  

Event date： 2009.12

Presentation type：Oral presentation (general)  

Venue：National University of Singapore,   Country：Singapore  

12/14 と 12/16 の２コマの連続講演

Event date： 2009.11

Presentation type：Oral presentation (general)  

Venue：九州大学西新プラザ   Country：Japan  

Event date： 2009.1

Presentation type：Oral presentation (general)  

Venue：駒場キャンパス数理棟会議場   Country：Japan  

An algebraic transformation of Gauss hypergeometric function

Event date： 2024.12

Venue：JST東京別館会議室  

Event date： 2024.10

Venue：九重研修所  

Event date： 2024.10

Venue：四谷  

Event date： 2024.9

Venue：Redrock resort Gorkhi-Terelj National Park  

Event date： 2023.8

Venue：早稲田大学  

Event date： 2023.8

Venue：早稲田大学  

Event date： 2023.6

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

Venue：モンゴル国立大学   Country：Mongolia  

Event date： 2023.3

Venue：統計数理研究所  

Event date： 2023.3

Venue：中央大学  

Event date： 2023.3

Venue：岡山理科大学  

Event date： 2023.3

Presentation type：Oral presentation (general)  

Venue：金沢大学  

Event date： 2019.3

Language：English   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2018.3

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：神戸大学   Country：Japan  

Event date： 2016.2

Language：English   Presentation type：Oral presentation (general)  

Venue：Victoria University of Wellington, NZ   Country：Japan  

Event date： 2013.2

Language：Japanese   Presentation type：Oral presentation (general)  

Country：Japan  

Event date： 2012.10

Language：English  

Venue：Inha University, Korea   Country：Korea, Republic of  

Event date： 2012.6

Language：English   Presentation type：Oral presentation (general)  

Venue：Chern Institute of Mathematics, Nankai University, China   Country：China  

Event date： 2012.3

Presentation type：Oral presentation (general)  

Venue：九州大学、稲盛ホール   Country：Japan  

In this talk, I will talk about the relation between the spectral
problem of the non-commutative harmonic oscillator of size two in the
positive elliptic case and the second order ordinary differential
equation on the complex projective line with an accessory parameter, so
called Heun's equation. I also mention the multiplicity of the discrete
spectrum of the non-commutative harmonic oscillator in the `odd sector'.

Event date： 2011.11

Presentation type：Oral presentation (invited, special)  

Venue：国民宿舎　紀州路みなべ   Country：Japan  

簡約代数群 G の部分旗多様体と, その対称部分群 K の部分旗多様体の直積を
対称対の２重旗多様体と呼ぶことにする.
２重旗多様体への K の対角的な作用が有限個の軌道を持つのはどんな部分旗多様体の組のときか,
また, その場合に軌道分解はどうなるか, という問題を考える.
この講演は, 西山享, 近藤健介の講演へと引き続く, 概説的な部分を担当する.
この講演の内容は, 上記２名ならびに, 谷口健二, Xuhua He との共同研究の内容に基づく.

Event date： 2011.6

Presentation type：Oral presentation (general)  

Venue：伊藤極限プラズマ研究連携センター   Country：Japan  

Event date： 2011.3

Presentation type：Oral presentation (general)  

Venue：Johns Hopkins University, 日米数学研究所   Country：United States  

Event date： 2010.9

Presentation type：Oral presentation (general)  

Venue：京都大学数理解析研究所   Country：Japan  

Event date： 2010.6

Language：English   Presentation type：Oral presentation (general)  

Venue：京都大学数理解析研究所   Country：Japan  

Event date： 2010.2

Presentation type：Oral presentation (general)  

Venue：Inha University, Korea   Country：Korea, Republic of  

2/17と2/19 の２コマの連続講演

Event date： 2009.12

Presentation type：Oral presentation (general)  

Venue：鳥取市、とりぎん文化会館,   Country：Japan  

Event date： 2009.11

Presentation type：Oral presentation (general)  

Venue：フェストーネ、宜野湾市,   Country：Japan  

Event date： 2009.9

Presentation type：Oral presentation (general)  

Venue：名古屋大学   Country：Japan  

Higher Mahler measures

Event date： 2009.7

Presentation type：Oral presentation (general)  

Venue：名古屋大学   Country：Japan  

Separation of variables

Language：English  

We show that the fake projective planes that are constructed from dyadic discrete subgroups discovered by Cartwright, Mantero, Steger, and Zappa are realized as connected components of certain unitary Shimura surfaces. As a corollary we show that these fake projective planes have models defined over the number field Q(root-3,root 5). (c) 2006 Published by Elsevier Inc.

DOI： 10.1016/j.jalgebra.2006.07.019

Language：English  

We prove existence of scaling limits of sequences of functions defined by the recursion relation w(n+1)(1) (x) = -w(n)(x)(2). which is a successive approximation to w(1) (x) = -w(x)(2), a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n -> infinity in an asymptotically conformal way, w(n)(x) asymptotic to q(n)w(-)(q(n)-x), for a sequence of numbers {q(n)} and a function w(-). We also discuss implication of the results in terms of random sequential bisections of a rod.

DOI： 10.1619/fesi.49.291

Language：English  

We investigate the structure of invariant distributions on a non-isotropic non-Riemannian symmetric space of rank one. Especially, the J-criterion related to the generalized Gelfand pair is shown for this space without imposing the condition on the eigenfuction of the Laplace-Bertrami operator. © 2005 Royal Netherlands Academy of Arts and Sciences.

DOI： 10.1016/S0019-3577(05)80043-6

Language：English  

We investigate the structure of invariant distributions on a non-isotropic non-Riemannian symmetric space of rank one. Especially, the J-criterion related to the generalized Gelfand pair is shown for this space without imposing the condition on the eigenfuction of the Laplace-Bertrami operator.

DOI： 10.1016/S0019-3577(05)80043-6

Language：English  

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.

DOI： 10.1007/s11005-004-4292-5

Language：English  

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than I and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.

DOI： 10.1007/s11005-004-4292-5

Language：English  

The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.

DOI： 10.1142/S0129167X0400248X

Language：English  

The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.

DOI： 10.1142/S0129167X0400248X

Type：Competition, symposium, etc. 

Type：Competition, symposium, etc. 

Type：Scientific advice/Review 

Audience：Infants,　Schoolchildren,　Junior students,　High school students

Type：Seminar, workshop

Audience：General,　Scientific,　Company,　Civic organization,　Governmental agency

Type：Lecture

Audience：Infants,　Schoolchildren,　Junior students,　High school students

Type：Seminar, workshop

名著・大著を読む「判別式、終結式、行列式」Gelfand, Kapranov, Zelevinsky

４月号「てらかんと歩む」

「現代数学の巨大パズルを解く／デービット・ヴォーガン(David Vogan) 氏インタビュー」

Aperitif「講演を聴く」

Aperitif「ただいま建設中」

「表現論と特殊関数」