arXiv subjects

Takayuki Okuda

Publications and source records attributed to Takayuki Okuda.

At least 19 recordsLinked to original sources

arXiv2026

Some categorical remarks on coarse subspaces of coarse spaces

In this paper, we provide a categorical framework for understanding coarse subspaces of coarse spaces. First, we introduce the notion of a controlled total relation between coarse spaces and show that the category whose morphisms are closeness classes of controlled total relations is isomorphic to the conventional category of coarse spaces defined using closeness classes of controlled maps. Next, we show that the assignment associating to each coarse space the finite-join partially ordered set of its coarse subspaces is functorial, and prove that this partially ordered set is naturally isomorphic to the poset of subobjects in the category of coarse spaces. Furthermore, we formulate asymptotic disjointness between coarse subspaces and show that mono-morphisms preserve this relation. These results provide a categorical interpretation of the framework of coarse subspaces introduced by Leitner--Vigolo [Lecture Notes in Math.~(2023)] and characterize coarse subspaces as objects intrinsic to the category of coarse spaces. They also provide a foundation for a coarse-geometric interpretation of the properness criterion established by Kobayashi [Math.~Ann.~(1989); J.~Lie Theory (1996)] and Benoist [Ann.~of Math.~(1996)] (cf.~Nagaya--Ogawa--Okuda [Proc.~Japan Acad.~Ser.~A (2025)]).

math.CT↗

arXiv2026

Invariant statistical connections on the multivariate centered Gaussian model and their moduli spaces

We study invariant statistical connections on the space $\mathcal{N}_0^n$ of zero-mean multivariate normal distributions (the multivariate centered Gaussian model) equipped with the Fisher metric $g^F$. We introduce moduli spaces of invariant statistical connections on homogeneous Riemannian manifolds via two natural equivalence relations arising from a categorical viewpoint, and apply this framework to $(\mathcal{N}_0^n, g^F)$. We explicitly determine the $GL(n,\mathbb{R})$-invariant and $\mathrm{Isom}(\mathcal{N}_0^n, g^F)$-invariant statistical connections, with particular emphasis on the dually flat case, and describe the corresponding moduli spaces.

math.DG↗

arXiv2026

The moduli spaces of left-invariant statistical structures on Lie groups

In the context of information geometry, the concept known as left-invariant statistical structure on Lie groups is defined by Furuhata--Inoguchi--Kobayashi (Inf Geom 4(1):177--188, 2021). In this paper, we introduce the notion of the moduli space of left-invariant statistical structures on a Lie group. We study the moduli spaces for three particular Lie groups, each of which has a moduli space of left-invariant Riemannian metrics that is a singleton. As applications, we classify left-invariant conjugate symmetric statistical structures and left-invariant dually flat structures (which are equivalent to left-invariant Hessian structures) on these three Lie groups. A characterization of the Amari--Chentsov $α$-connections on the Takano Gaussian space is also given.

math.DG↗

arXiv2025

The intersection of two real flag manifolds in a complex flag manifold

We give a necessary and sufficient condition for two real flag manifolds, which are not necessarily congruent, in a complex flag manifold to intersect transversally in terms of the symmetric triad. Then we show that the intersection of two real flag manifolds is antipodal. As an application, we prove that any real flag manifold in a complex flag manifold is a globally tight Lagrangian submanifold.

math.DG↗

arXiv2025

On Invariant Conjugate Symmetric Statistical Structures on the Space of Zero-Mean Multivariate Normal Distributions

By the results of Furuhata--Inoguchi--Kobayashi [Inf. Geom. (2021)] and Kobayashi--Ohno [Osaka Math. J. (2025)], the Amari--Chentsov $α$-connections on the space $\mathcal{N}$ of all $n$-variate normal distributions are uniquely characterized by the invariance under the transitive action of the affine transformation group among all conjugate symmetric statistical connections with respect to the Fisher metric. In this paper, we investigate the Amari--Chentsov $α$-connections on the submanifold $\mathcal{N}_0$ consisting of zero-mean $n$-variate normal distributions. It is known that $\mathcal{N}_0$ admits a natural transitive action of the general linear group $GL(n,\mathbb{R})$. We establish a one-to-one correspondence between the set of $GL(n,\mathbb{R})$-invariant conjugate symmetric statistical connections on $\mathcal{N}_0$ with respect to the Fisher metric and the space of homogeneous cubic real symmetric polynomials in $n$ variables. As a consequence, if $n \geq 2$, we show that the Amari--Chentsov $α$-connections on $\mathcal{N}_0$ are not uniquely characterized by the invariance under the $GL(n,\mathbb{R})$-action among all conjugate symmetric statistical connections with respect to the Fisher metric. Furthermore, we show that any invariant statistical structure on a Riemannian symmetric space is necessarily conjugate symmetric.

math.DG↗

arXiv2025

Zariski dense discontinuous surface groups for reductive symmetric spaces

Let $G/H$ be a homogeneous space of reductive type with non-compact $H$. The study of deformations of discontinuous groups for $G/H$ was initiated by T.~Kobayashi. In this paper, we show that a standard discontinuous group $Γ$ admits a non-standard small deformation as a discontinuous group for $G/H$ if $Γ$ is isomorphic to a surface group of high genus and its Zariski closure is locally isomorphic to $SL(2,\mathbb{R})$. Furthermore, we also prove that if $G/H$ is a symmetric space and admits some non virtually abelian discontinuous groups, then $G$ contains a Zariski-dense discrete surface subgroup of high genus acting properly discontinuously on $G/H$. As a key part of our proofs, we show that for a discrete surface subgroup $Γ$ of high genus contained in a reductive group $G$, if the Zariski closure of $Γ$ is locally isomorphic to $SL(2,\mathbb{R})$, then $Γ$ admits a small deformation in $G$ whose Zariski closure is a reductive subgroup of the same real rank as $G$.

math.DG↗

arXiv2024

Educational Effects in Mathematics: Conditional Average Treatment Effect depending on the Number of Treatments

This study examines the educational effect of the Academic Support Center at Kogakuin University. Following the initial assessment, it was suggested that group bias had led to an underestimation of the Center's true impact. To address this issue, the authors applied the theory of causal inference. By using T-learner, the conditional average treatment effect (CATE) of the Center's face-to-face (F2F) personal assistance program was evaluated. Extending T-learner, the authors produced a new CATE function that depends on the number of treatments (F2F sessions) and used the estimated function to predict the CATE performance of F2F assistance.

stat.ME↗

arXiv2024

Some remarks on real minimal nilpotent orbits and symmetric pairs

For a non-compact simple Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, we denote by $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}}$ the unique complex nilpotent orbit in $\mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$ containing all minimal real nilpotent orbits in $\mathfrak{g}$. In this paper, we give a complete classification of symmetric pairs $(\mathfrak{g},\mathfrak{h})$ such that $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}} \cap \mathfrak{g}^d = \emptyset$, where $\mathfrak{g}^d$ denotes the dual Lie algebra of $(\mathfrak{g},\mathfrak{h})$. Furthermore, for symmetric pairs $(G,H)$ with real simple Lie group $G$, we apply our classification to theorems given by T. Kobayashi [J. Lie Theory (2023)], and study bounded multiplicity properties of restrictions on $H$ of infinite-dimensional irreducible $G$-representations with minimum Gelfand--Kirillov dimension.

math.RT↗

arXiv2023

A proof of Kobayashi's properness criterion from a viewpoint of metric geometry

Let $G$ be a locally-compact group and $(H,L)$ a pair of closed subgroups of $G$. For the cases where $G$ is a real linear reductive Lie group, T. Kobayashi [Math. Ann. '89, J. Lie Theory '96] established a criterion for properness of the $L$-action on the homogeneous space $G/H$ in terms of Cartan's KAK-decomposition of $G$. In this paper, we show that a similar theorem also holds if $G$ is a locally-compact group admitting a suitable isometric action on a metric space, and give a proof of Kobayashi's criterion in terms of CAT(0) metric geometry on non-compact Riemannian symmetric spaces.

math.DG↗

arXiv2022

Functoriality of Bose-Mesner algebras and profinite association schemes

We show that taking the set of primitive idempotents of commutative association schemes is a functor from the category of commutative association schemes with surjective morphisms to the category of finite sets with surjective partial functions. We then consider projective systems of commutative association schemes consisting of surjections (which we call profinite association schemes), for which Bose-Mesner algebra is defined, and describe a Delsarte theory on such schemes. This is another method for generalizing association schemes to those on infinite sets, related with the approach by Barg and Skriganov. Relation with $(t,m,s)$-nets and $(t,s)$-sequences is studied. We reprove some of the results of Martin-Stinson from this viewpoint.

math.CO↗

arXiv2022

On the spectrum and linear programming bound for hypergraphs

The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander graphs. In this paper, we extend previous work done for graphs and bipartite graphs and present a linear programming method for obtaining an upper bound on the order of a regular uniform hypergraph with prescribed distinct eigenvalues. Furthermore, we obtain a general upper bound on the order of a regular uniform hypergraph whose second eigenvalue is bounded by a given value. Our results improve and extend previous work done by Feng-Li (1996) on Alon-Boppana theorems for regular hypergraphs and by Dinitz-Schapira-Shahaf (2020) on the Moore or degree-diameter problem. We also determine the largest order of an $r$-regular $u$-uniform hypergraph with second eigenvalue at most $θ$ for several parameters $(r,u,θ)$. In particular, orthogonal arrays give the structure of the largest hypergraphs with second eigenvalue at most $1$ for every sufficiently large $r$. Moreover, we show that a generalized Moore geometry has the largest spectral gap among all hypergraphs of that order and degree.

math.CO↗

arXiv2021

Quantum circuits for exact unitary $t$-designs and applications to higher-order randomized benchmarking

A unitary $t$-design is a powerful tool in quantum information science and fundamental physics. Despite its usefulness, only approximate implementations were known for general $t$. In this paper, we provide for the first time quantum circuits that generate exact unitary $t$-designs for any $t$ on an arbitrary number of qubits. Our construction is inductive and is of practical use in small systems. We then introduce a $t$-th order generalization of randomized benchmarking ($t$-RB) as an application of exact $2t$-designs. We particularly study the $2$-RB in detail and show that it reveals self-adjointness of quantum noise, a new metric related to the feasibility of quantum error correction (QEC). We numerically demonstrate that the $2$-RB in one- and two-qubit systems is feasible, and experimentally characterize background noise of a superconducting qubit by the $2$-RB. It is shown from the experiment that interactions with adjacent qubits induce the noise that may result in an obstacle toward the realization of QEC.

quant-ph↗

arXiv2020

Explicit construction of exact unitary designs

The purpose of this paper is to give explicit constructions of unitary $t$-designs in the unitary group $U(d)$ for all $t$ and $d$. It seems that the explicit constructions were so far known only for very special cases. Here explicit construction means that the entries of the unitary matrices are given by the values of elementary functions at the root of some given polynomials. We will discuss what are the best such unitary $4$-designs in $U(4)$ obtained by these methods. Indeed we give an inductive construction of designs on compact groups by using Gelfand pairs $(G,K)$. Note that $(U(n),U(m) \times U(n-m))$ is a Gelfand pair. By using the zonal spherical functions for $(G,K)$, we can construct designs on $G$ from designs on $K$. We remark that our proofs use the representation theory of compact groups crucially. We also remark that this method can be applied to the orthogonal groups $O(d)$, and thus provides another explicit construction of spherical $t$-designs on the $d$ dimensional sphere $S^{d-1}$ by the induction on $d$.

math.CO↗

arXiv2020

Approximation of integration over finite groups, difference sets and association schemes

Let $G$ be a finite group and $f:G \to {\mathbb C}$ be a function. For a non-empty finite subset $Y\subset G$, let $I_Y(f)$ denote the average of $f$ over $Y$. Then, $I_G(f)$ is the average of $f$ over $G$. Using the decomposition of $f$ into irreducible components of ${\mathbb C}^G$ as a representation of $G\times G$, we define non-negative real numbers $V(f)$ and $D(Y)$, each depending only on $f$, $Y$, respectively, such that an inequality of the form $|I_G(f)-I_Y(f)|\leq V(f)\cdot D(Y)$ holds. We give a lower bound of $D(Y)$ depending only on $\#Y$ and $\#G$. We show that the lower bound is achieved if and only if $\#\{(x,y)\in Y^2 \mid x^{-1}y \in [a]\}/\#[a]$ is independent of the choice of the conjugacy class $[a]\subset G$ for $a \neq 1$. We call such a $Y\subset G$ as a pre-difference set in $G$, since the condition is satisfied if $Y$ is a difference set. If $G$ is abelian, the condition is equivalent to that $Y$ is a difference set. We found a non-trivial pre-difference set in the dihedral group of order 16, where no non-trivial difference set exists. The pre-difference sets in non-abelian groups of order 16 are classified. A generalization to commutative association schemes is also given.

math.CO↗

arXiv2016

Abundance of nilpotent orbits in real semisimple Lie algebras

We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace of the longest Weyl group element. The result is used to the study of fundamental groups of non-Riemannian locally symmetric spaces.

math.RT↗

arXiv2016

Proper SL(2,R)-actions on homogeneous spaces

We study the existence problem of proper actions of SL(2,R) on homogeneous spaces G/H of reductive type. Based on Kobayashi's properness criterion [Math. Ann. (1989)], we show that G/H admits a proper SL(2,R)-action via G if a maximally split abelian subspace of Lie H is included in the wall defined by a restricted root of Lie G. We also give a number of examples of such G/H.

math.GR↗

arXiv2015

Singular fibers in barking families of degenerations of elliptic curves

Takamura established a theory on splitting families of degenerations of complex curves. He introduced a powerful method for constructing a splitting family, called a barking family, in which there appear not only a singular fiber over the origin but also singular fibers over other points, called subordinate fibers. In this paper, for the case of degenerations of elliptic curves, we determine the types of these subordinate fibers.

math.GT↗

arXiv2015

Relation between spherical designs through a Hopf map

Cohn--Conway--Elkies--Kumar [Experiment. Math. (2007)] described that one can construct a family of designs on $S^{2n-1}$ from a design on $\mathbb{CP}^{n-1}$. In this paper, we prove their claim for the case where $n=2$. That is, we give an algorithm to construct $2t$-designs on $S^{3}$ as products through a Hopf map $S^3 \rightarrow S^2$ of a $t$-design on $S^2$ and a $2t$-design on $S^1$.

math.MG↗