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Hiroki Yagisita

Publications and source records attributed to Hiroki Yagisita.

11 recordsLinked to original sources
arXiv2025

Heat equation and Schrödinger equation with translation invariance on the infinite-dimensional vector space $\mathbb R^\infty$

The standard Laplacian $-\triangle_{\mathbb R^n}$ in $L^2(\mathbb R^n)$ is self-adjoint and translation invariant on the finite-dimensional linear space $\mathbb R^n$. In this paper, we define a translation invariant operator $-\triangle_{\mathbb R^\infty}$ on $\mathbb R^\infty$ as a non-negative self-adjoint operator in some non-separable Hilbert space $L^2(\mathbb R^\infty)$. The set $L^2(\mathbb R^\infty)$ is a translation invariant subset of the set $CM(\mathbb R^\infty)$ of all complex measures on the product measurable space $\mathbb R^\infty$. Furthermore, we show that for any $f\in L^2(\mathbb R^n)$ and any $u\in L^2(\mathbb R^\infty)$, the separations of variables $e^{\triangle_{\mathbb R^\infty}t}(f\otimes u) =(e^{\triangle_{\mathbb R^n}t}f)\otimes (e^{\triangle_{\mathbb R^\infty}t}u) \ (t\in [0,+\infty))$ and $e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}(f\otimes u) =(e^{\sqrt{-1}\triangle_{\mathbb R^n}t}f)\otimes (e^{\sqrt{-1}\triangle_{\mathbb R^\infty}t}u) \ (t\in (-\infty,+\infty))$ hold. This clearly shows that $-\triangle_{\mathbb R^\infty}$ is an analog of $-\triangle_{\mathbb R^n}$. The starting point for the discussion in this paper is to naturally introduce a translation invariant structure of Hilbert space into $CM(\mathbb R^\infty)$. $L^2(\mathbb R^\infty)$ is a closed linear subspace of $CM(\mathbb R^\infty)$. The inner product of $L^2(\mathbb R^\infty)$ is defined as that of $CM(\mathbb R^\infty)$. For a manifold, Hörmander defined an inner product that does not depend on a particular measure. In fact, the way we introduce the inner product into $CM(\mathbb R^\infty)$ is a generalization of his. Not only is a statistical manifold on $\mathbb R^\infty$ a submanifold of $CM(\mathbb R^\infty)$, but the real inner product $\mathrm{Re}(\langle \cdot, \cdot \rangle_{CM(\mathbb R^\infty)})$ induces Fisher information metric.

math.GM↗
arXiv2023

In Markov process, an extremal reversible measure is an extremal invariant measure

We consider a discrete-time temporally-homogeneous conservative Markov process. We show that extremality of reversible measure implies extremality of invariant measure. Using analogue of Dirichlet form, we modify a proof that in stochastic Ising model (Glauber dynamics), an extreme Gibbs state is an extreme invariant measure.

math.GM↗
arXiv2019

A remark on embedding of a cylinder on a real commutative Banach algebra

Let $A$ be a real commutative Banach algebra with unity. Let $a_0\in A\setminus\{0\}$. Let $\mathbb Z a_0:=\{na_0\}_{n\in \mathbb Z}$. Then, $\mathbb Z a_0$ is a discrete subgroup of $A$. For any $n\in \mathbb Z$, the Frechet derivative of the mapping $$x \, \in \, A \ \ \ \mapsto \ \ \ x+na_0 \, \in \, A$$ is the identity map on $A$ and, especially, an $A$-linear transformation on $A$. So, the quotient group $A/(\mathbb Z a_0)$ is a $1$-dimensional $A$-manifold and the covering projection $$x \, \in \, A \ \ \ \mapsto \ \ \ x+\mathbb Z a_0 \, \in \, A/(\mathbb Z a_0)$$ is an $A$-map. We call $A/(\mathbb Z a_0)$ the $1$-dimensional $A$-cylinder by $a_0$. Let $T$ be a compact Hausdorff space. Suppose that there exist $t_1\in T$ and $t_2\in T$ such that $t_1\not=t_2$ holds. Then, the set $C(T;\mathbb R)$ of all real-valued continuous functions on $T$ is a real commutative Banach algebra with unity and $\mathbb R \, \subsetneq \, C(T;\mathbb R)$ holds. In this paper, we show that there exists $a_0 \, \in \, C(T;\mathbb R)\setminus \mathbb R$ such that for any $k\, \in \, \mathbb N$, the $1$-dimensional $C(T;\mathbb R)$-cylinder $(C(T;\mathbb R))/(\mathbb Z a_0)$ by $a_0$ cannot be embedded in the finite direct product space $(C(T;\mathbb R))^k$ as a $C(T;\mathbb R)$-submanifold.

math.FA↗
arXiv2019

A remark on locally direct product subsets in a topological Cartesian space

Let $X$ and $Y$ be topological spaces. Let $C$ be a path-connected closed set of $X\times Y$. Suppose that $C$ is locally direct product, that is, for any $(a,b)\in X\times Y$, there exist an open set $U$ of $X$, an open set $V$ of $Y$, a subset $I$ of $U$ and a subset $J$ of $V$ such that $(a,b) \in U\times V$ and $$C\cap (U\times V)=I\times J$$ hold. Then, in this memo, we show that $C$ is globally so, that is, there exist a subset $A$ of $X$ and a subset $B$ of $Y$ such that $$C=A\times B$$ holds. The proof is elementary. Here, we note that one might be able to think of a (perhaps, open) similar problem for a fiber product of locally trivial fiber spaces, not just for a direct product of topological spaces. In Appendix, we mentioned a simple example of a $C([0,1];\mathbb R)$-manifold that cannot be embedded in the direct product $(C([0,1];\mathbb R))^n$ as a $C([0,1];\mathbb R)$-submanifold. In addition, we introduce the concept of topological 2-space, which is locally the direct product of topological spaces and an analog of homotopy category for topological 2-space. Finally, we raise a question on the existence of an $\mathbb R^n$-Morse function and the existence of an $\mathbb R^n$-immersion in a finite-dimensional $\mathbb R^n$-Euclidean space. Here, we note that the problem of defining the concept of an $\mathbb R^n$-handle body may also be considered.

math.GN↗
arXiv2019

Holomorphic differential forms of complex manifolds on commutative Banach algebras and a few related problems

Let $A$ be a commutative Banach algebra. Let $M$ be a complex manifold on $A$ (an $A$-manifold). Then, we define an $A$-holomorphic vector bundle $(\wedge^kT^*)(M)$ on $M$. For an open set $U$ of $M$, $ω$ is said to be an $A$-holomorphic differential $k$-form on $U$, if $ω$ is an $A$-holomorphic section of $(\wedge^kT^*)(M)$ on $U$. So, if the set of all $A$-holomorphic differential $k$-forms on $U$ is denoted by $Ω_{M}^k(U)$, then $\{Ω_{M}^k(U)\}_{U}$ is a sheaf of modules on the structure sheaf $O_M$ of the $A$-manifold $M$ and the cohomology group $H^l(M,Ω_{M}^k)$ with the coefficient sheaf $\{Ω_{M}^k(U)\}_{U}$ is an $O_M(M)$-module and therefore, in particular, an $A$-module. There is no new thing in our definition of a holomorphic differential form. However, this is necessary to get the cohomology group $H^l(M,Ω_{M}^k)$ as an $A$-module. Furthermore, we try to define the structure sheaf of a manifold that is locally a continuous family of $\mathbb C$-manifolds (and also the one of an analytic family). Directing attention to a finite family of $\mathbb C$-manifolds, we mentioned the possibility that Dolbeault theorem holds for a continuous sum of $\mathbb C$-manifolds. Also, we state a few related problems. One of them is the following. Let $n\in \mathbb N$. Then, does there exist a $\mathbb C^n$-manifold $N$ such that for any $\mathbb C$-manifolds $M_1, M_2, \cdots, M_{n-1}$ and $M_n$, $N$ can not be embedded in the direct product $M_1\times M_2 \times \cdots \times M_{n-1} \times M_n$ as a $\mathbb C^n$-manifold ? So, we propose something that is likely to be a candidate for such a $\mathbb C^2$-manifold $N$.

math.DG↗
arXiv2019

Cartan-Thullen theorem for a $\mathbb C^n$-holomorphic function and a related problem

Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${\mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f \, (\, =(f_1,f_2,\cdots,f_n)\, )$ is a $\mathbb C^n$-valued function on an open set $U$ of $\mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$, $f$ is said to be $\mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i\not=j$ implies $\frac{\partial f_i}{\partial z_j}=0$. Here, $(z_1,z_2,\cdots,z_n) \in \mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$ holds. We note that a $\mathbb C^n$-analytic mapping and a $\mathbb C^n$-analytic manifold can be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for a $\mathbb C^n$-analytic function. For $n=1$, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open $\mathbb C^n$-holomorphically convex set $U$ exist such that $U$ is not the direct product of any holomorphically convex sets $U_1, U_2, \cdots, U_{n-1}$ and $U_n$ ? As a corollary of our generalization, we give the following partial result. If $U$ is convex, then $U$ is the direct product of some holomorphically convex sets. Also, $f$ is said to be $\mathbb C^n$-triangular, if $f$ is complex analytic and for any $i$ and $j$, $i<j$ implies $\frac{\partial f_i}{\partial z_j}=0$. Kasuya suggested that a $\mathbb C^n$-analytic manifold and a $\mathbb C^n$-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation.

math.CV↗
arXiv2018

Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds

An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the finite direct product of the complex number field. On the other hand, when A is a commutative Banach algebra, Lorch gave a definition that an A-valued function on an open set of A is holomorphic. The definition of a holomorphic function by Lorch can be straightforwardly generalized to an A-valued function on an open set of the finite direct product of A. Therefore, a manifold modeled on the finite direct product of A (an n-dimensional A-manifold) is easily defined. However, in my opinion, it seems that so many nontrivial examples were not known (including the case of n=1, that is, Riemann surfaces). By the way, if X is a compact Hausdorff space, then the algebra C(X) of all complex valued continuous functions on X is the most basic example of a commutative Banach algebra (furthermore, a commutative C*-algebra). In this note, we see that if the set of all continuous cross sections of a continuous family M of compact complex manifolds (a topological deformation M of compact complex analytic structures) on X is denoted by G(M), then the structure of a C(X)-manifold modeled on the C(X)-modules of all continuous cross sections of complex vector bundles on X is introduced into G(M). Therefore, especially, if X is contractible, then G(M) is a finite-dimensional C(X)-manifold.

math.DG↗
arXiv2016

Existence of traveling waves for a nonlocal monostable equation: an abstract approach

We consider a nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure for the convolution is absolutely continuous. In order to show the main result, we modify a recursive method for abstract monotone discrete dynamical systems by Weinberger. We note that the monotone semiflow generated by the equation does not have compactness with respect to the compact-open topology. At the end, we propose a discrete model that describes the measurement process.

math.AP↗
arXiv2012

Remarks on space-time behavior in the Cauchy problems of the heat equation and the curvature flow equation with mildly oscillating initial values

We study two initial value problems of the linear diffusion equation and a nonlinear diffusion equation, when Cauchy data are bounded and oscillate mildly. The latter nonlinear heat equation is the equation of the curvature flow, when the moving curves are represented by graphs. By using an elementary scaling technique, we show some formulas for space-time behavior of the solution. Keywords: scaling argument, self-similar solution, nonstabilizing solution, nontrivial dynamics, nontrivial large-time behavior, irregular behavior.

math.AP↗
arXiv2008

Existence of traveling wave solutions for a nonlocal bistable equation: an abstract approach

We consider traveling fronts to the nonlocal bistable equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We show that there is a traveling wave solution with monotone profile. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.

math.AP↗