A handlebody-link is a disjoint union of handlebodies embedded in the 3-sphere $S^3$. A multiple conjugation quandle is an algebraic system whose axioms are derived from the Reidemeister moves for handlebody-link diagrams. In this paper, we introduce the notion of a presentation of a multiple conjugation quandle and define the fundamental multiple conjugation quandle of a handlebody-link. We also see that the fundamental multiple conjugation quandle is an invariant of handlebody-links.

In this paper, we introduce and study the notion of linkage of modules by reflexive homomorphisms. This notion unifies and generalizes several known concepts of linkage of modules and enables us to study the theory of linkage of modules over Cohen–Macaulay rings rather than the more restrictive Gorenstein rings. It is shown that several known results for Gorenstein linkage are still true in the more general case of module linkage over Cohen–Macaulay rings. We also introduce the notion of colinkage of modules and establish an adjoint equivalence between the linked and colinked modules.

In a previous report, the second and third authors gave general theorems for unique strong solutions of infinite-dimensional stochastic differential equations (ISDEs) describing the dynamics of infinitely many interacting Brownian particles. One of the critical assumptions is the “IFC” condition. The IFC condition requires that, for a given weak solution, the scheme consisting of the finite-dimensional stochastic differential equations (SDEs) related to the ISDEs exists. Furthermore, the IFC condition implies that each finite-dimensional SDE has unique strong solutions. Unlike other assumptions, the IFC condition is challenging to verify, and so the previous report only verified it for solutions given by quasi-regular Dirichlet forms. In the present paper, we provide a sufficient condition for the IFC requirement in more general situations. In particular, we prove the IFC condition without assuming the quasi-regularity or symmetry of the associated Dirichlet forms. As an application of the theoretical formulation, the results derived in this paper are used to prove the uniqueness of Dirichlet forms and the dynamical universality of random matrices.

We study a topologically exact, negative Schwarzian unimodal map without neutral periodic points whose critical point is non-recurrent and flat. Assuming that the critical order is polynomial or logarithmic, we establish the large deviation principle and provide a partial description of the minimizers of the rate function. We apply our main results to a certain parametrized family of unimodal maps in the same topological conjugacy class, and determine the sets of minimizers.

This paper presents our study of the conditions under which the monodromy group for Lauricella's hypergeometric function $F_C (a,b,c;x)$ is finite irreducible. We provide these conditions in terms of the parameters $a, b, c$. In addition, we discuss the structure of the finite irreducible monodromy group.

We determine the long time behavior and the exact order of the tail probability for the maximal displacement of a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of the associated Schrödinger type operator. To establish our results, we show a sharp and locally uniform growth order of the Feynman–Kac semigroup.

We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$. The dispersive estimate $\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_{c}$ is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.

We consider a homogenization problem for symmetric jump-diffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. In this paper, we study sharp endpoint $L^{p}$-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation $i \partial_{t} u + L u = 0$ and show that for all $f \in L^{p}(X)$, $1 < p < \infty$, $\| e^{itL} (I+L)^{-{\sigma n}} f\|_{p} \leq C(1 + |t|)^{\sigma n} \| f \|_{p}$, $t \in \mathbb{R}$, $\sigma \geq |1/2-1/p|$, where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound.