A rigged Hilbert space formalism is introduced to study Fock space operators. The symbols of continuous operators on a rigged Fock space are characterized in terms of Bargmann-Segal spaces and complex Gaussian integrals. In particular, characterizations of bounded operators and of operators of Hilbert-Schmidt class on the middle Fock space are obtained. As an application we establish an operator version of chaotic expansion (Wiener-Itô expansion) and describe a relation to the Fock expansion in terms of the Wick exponential of the number operator. As another application we discuss regularity property of a solution to a normal-ordered white noise differential equation generalizing a quantum stochastic differential equation.

In this paper we prove several necessary and sufficient conditions for a harmonic function in the unit ball belong to $H^p\left(B\right)$-Hardy harmonic space.

For a topological dynamical system $\Sigma$$=(X,\sigma )$ where $\sigma$ is a homeomorphism in an arbitrary compact Hausdorff space $X$, we consider the noncommutative hulls and kernels with respect to the action $\sigma$ in the associated $C\text{*}-$ algebra $A\left(\Sigma \right)$. We show that several ideals important for the structure of $A\left(\Sigma \right)$ have the form of such kernels and give topological characterizations of their hulls from the behavior of orbits in the dynamical system.

Optimal energy decay estimates will be derived for weak solutions to the Cauchy problem in $R^N(N=1,2,3)$ of dissipative wave equations, which have lower power nonlinearities $|u{|}^{p-1}u$ satisfying .

In general, it is difficult to determine the dimension of the space of Siegel modular forms with low weight. In this paper, we consider the spaces of modular forms belonging to the principal congruence subgroup of level 3 as the representation spaces of the finite symplectic group to calculate their dimensions.

This paper is devoted to the study of the extention of the existence time of solutions in Sobolev spaces to the Cauchy problem for Kirchhoff equations.

The first purpose is to characterize the existence of well-behaved $*$-representations of locally convex $*$-algebras by unbounded $C^{*}$-seminorms. The second is to define the notion of spectral $*$-representations and to characterize the existence of spectral well-behaved $*$-representations by unbounded $C^{*}$-seminorms.

In the preceding paper we gave an analogue of the Capelli identity for relative invariants in Hermitian symmetric settings, and the analogue was constructed on scalar generalized Verma modules. In this paper we give an analogue of the Capelli identity of lower degrees. This analogue contains non-principal minors in contrast with the original Capelli identity.

In this paper, we study $n$-dimensional complete immersed submanifolds in a Euclidean space $E^{n+p}$. We prove that if $M^n$ is an $n$-dimensional compact connected immersed submanifold with nonzero mean curvature $H$ in $E^{n+p}$ and satisfies either:

(1)$S\le \frac{n^2{H}^2}{n-1},$ or

(2)$n^2{H}^2\le \frac{(n-1)R}{n-2}$,

then $M^n$ is diffeomorphic to a standard $n$-sphere, where $S$ and $R$ denote the squared norm of the second fundamental form of $M^n$ and the scalar curvature of $M^n$, respectively.

On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere $S^n\left(c\right)$, the totally geodesic Euclidean space $E^n$, and the generalized cylinder $S^{n-1}\left(c\right)\times E^1$ are only $n$-dimensional $(n>2)$ complete connected submanifolds $M^n$ with constant mean curvature $H$ in $E^{n+p}$ if $S\le n^2{H}^2/(n-1)$ holds.

We discuss the behaviour of (transfinite) dimension functions, in particular, Cmp and trInd, on finite and countable unions of closed subsets in separable metrizable spaces.

In the present paper, we introduce $L^2$-torsion invariants ${\tau }_k(k\ge 1)$ for surface bundles over the circle and investigate them from the view point of the mapping class group of a surface. It is conjectured that they converge to the $L^2$-torsion for the regular representation of the fundamental group. Further we give an explicit and computable formula of the first two invariants by using the Mahler measure.

For a Brownian motion with a constant drift $X$ and its maximum process $M,$$M-X$ and $2M-X$ are diffusion processes by the extensions of Lévy's and Pitman's theorems. We show that $cM-X$ is not a Markov process if $c$∊$R\setminus \{0,1,2\}$. We also give other elementary proofs of Lévy's and Pitman's theorems.

Cone-parameter Lévy processes and convolution semigroups on $R^d$ are defined. Here, cone-parameter Lévy processes have stationary independent increments along increasing sequences on the cone. This property ensures that subordination of a cone-parameter Lévy process by an independent cone-valued cone-parameter Lévy process yields a new cone-parameter Lévy process. It is shown that a cone-parameter Lévy process induces a cone-parameter convolution semigroup. The converse statement, that any convolution semigroup appears in this way, is however not true. In particular we show that there is no Brownian motion with parameter in the set of nonnegative-definite symmetric $d\times d$ matrices. The question when a given cone-parameter convolution semigroup is generated by a Lévy process is studied. It is shown that this is the case if one of the following three conditions is satisfied: $d=1$; the convolution semigroup is purely non-Gaussian; or $K$ is isomorphic to $R_{+}^N$.

In this paper we investigate the structure of the subsemigroup generated by the inner automorphisms in $Ext(\mathrm{L},K)$. As an application, we give a new point of view to the example of J. Plastiras, which are two $C^{*}$-algebras $\mathfrak{U}$ and a satisfying $\mathfrak{U}\ncong \mathfrak{B}$ and $M_2\otimes \mathfrak{U}\cong M_2\otimes \mathfrak{B}$.

We study the positive blowing-up solutions of the semilinear parabolic system: $u_t-\Delta u=v^p+u^r,$$v_t-\Delta v=u^q+v^s$, where $t∊(0,T),$$x∊R^N$ and $p,$$q,$$r,$$s>1$. We prove that if $r>q+1$ or $s>p+1$ then one component of a blowing-up solution may stay bounded until the blow-up time, while if and this cannot happen. We also investigate the blow up rates of a class of positive radial solutions. We prove that in some range of the parameters $p,$$q,$$r$ and $s$, solutions of the system have an uncoupled blow-up asymptotic behavior, while in another range they have a coupled blow-up behavior.

In this paper we study the Cauchy problem to the linear damped wave equation $u_{tt}-\Delta u+2a{u}_t=0$ in $(0,\infty )$$\times R^n(n\ge 2)$. It has been asserted that the above equation has the diffusive structure as $t\to \infty$. We give the precise interpolation of the diffusive structure, which is shown by $L^p\text{-}{L}^q$ estimates. We apply the above $L^p\text{-}{L}^q$ estimates to the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u+$$2a{u}_t=|u{|}^{\sigma }u$ in $(0,\infty )$$\times R^n(2\le n\le 5)$. If the power $\sigma$ is larger than the critical exponent $2/n$(Fujita critical exponent) and it satisfies $\sigma \le 2/(n-2)$ when $n\ge 3$, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.

In this paper, we will construct a family of full compact oriented minimal surfaces of genus 3 with degenerate Gauss map in 4-dimensional flat tori, which are not holomorphic with respect to any complex structure of the tori.

Let $H$ be the Hilbert class field of an imaginary quadratic field $K$. An elliptic curve $E$ over $H$ with complex multiplication by $K$ is called a $Q$-curve if $E$ is isogenous over $H$ to all its Galois conjugates. We classify $Q$-curves over $H$, relating them with the cohomology group $H^2(H/Q,\pm 1)$. The structures of the abelian varieties over $Q$ obtained from $Q$-curves by restriction of scalars are investigated.