We construct a one dimensional stochastic process with reflecting barrier of jump type and investigate a limit distribution and stationary distribution of the process.

We estimate stable rank for $C^*$-tensor products with the Jiang-Su algebra. We also estimate real rank for them as well.

Let $T = U|T|$ be the polar decomposition of $T$. For a semi-hyponormal operator $T=U|T|$ with partial isometry $U$, if $|T| - |T^*| \in \mathcal{C}$, then we give the trace formula for the polar decomposition of $T$.

We study a way of coding of irrational rotations, by which Denjoy systems are represented as subshifts. First, we state the subshift generated by a coding sequence is conjugate to a Denjoy system. Next, by using an adic model of a Denjoy system we give a sequence of substitutions to generate the coding sequence.

S. Ferenczi and L. F. C. da Rocha introduced an algorithm which is slightly different form of the negative slope algorithm as the normalized multiplicative algorithm deduced from three interval exchange transformations. It has the form that the ceiling value is taken in the case of \( x+y > 1 \). We call this algorithm as the ``modified negative slope algorithm''. In this paper, the author shows that the modified negative slope algorithm is weak Bernoulli with respect to the absolutely continuous invariant measure and gives an algebraic characterization of periodic orbits of this algorithm using the natural extension method.

Extension of two known facts concerning subordination is made. The first fact is that, in subordination of $1$-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on $\mathbf{R}^d$, selfdecomposability is inherited from subordinator to subordinated. This is extended to semi-selfdecomposability.

We study maps between uniformly closed algebras of complex-valued continuous functions which vanish at infinity on locally compact Hausdorff spaces. Without assuming linearity nor multiplicativity on the maps we show that they are isometrical isomorphisms as Banach space operators if they satisfy that the peripheral range of the product of the images of any two elements coincides with the peripheral range of the product of those elements. Furthermore, if the underlying algebras contain approximate identities, then they are isometrically isomorphic as Banach algebras, which is a generalization of a recent result of Luttman and Tonev for the case of uniform algebras. On the other hand it is not the case without assuming the existence of approximate identities; An example is given.

We review a Laplace operator on the cochain complex of the Lie algebra of the formal Poisson vector fields to obtain an analogy of the Hodge theorem for its cohomology. We present intermediate results of the computation on the plane.

In this paper we examine the question: given $n>1$, find a function $f:\mathbf{N}\rightarrow \mathbf{N}$ where $m=f(n)$ is the smallest integer such that $K_m$ is intrinsically $n$-linked. We prove that for $n>1$, every embedding of $K_{\lfloor \frac{7}{2}n\rfloor}$ in $\mathbf{R}^3$ contains a non-splittable link of $n$ components. We also prove an asymptotic result, that there exists a function $f(n)$ such that $ \lim_{n\to \infty}\frac{f(n)}{n}=3$ and, for every $n,$ $K_{f(n)}$ is intrinsically $n$-linked.

In this paper, we complete the lists of Satake diagrams and restricted root systems (including signatures of roots) for all classical semisimple pseudo-Riemannian symmetric spaces, which were classified by M. Berger. We also complete the list of the cohomogeneities of the linear isotropy representations of the spaces.

Let $p_n/q_n$ be the $n$-th convergent of the continued fraction expansion of a real number $\alpha$. It is known that $|p_n-q_n\alpha|$ is very small tending to $0$ as $n$ tends to infinity. In this paper we establish a method how to express $p_n-q_n\alpha$ in terms of integrals when $\alpha$ is an $e$-type real number and its continued fraction expansion is quasi-periodic.

We show a global representation formula for a certain kind of Weingarten surface in hyperbolic three-space, which is based on the formula due to Gálvez, Martínez and Milán. As an application of the representation formula, we also investigate surfaces with harmonic-mean curvature one (HMC-1 surfaces). We allow them to have certain kinds of singularities, and discuss some global properties.

We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular $K3$ surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over $\mathbf{Q}(\sqrt{5})$ and are conjugate to each other by the action of $\text{Gal}(\mathbf{Q}(\sqrt{5})/\mathbf{Q})$.

We consider the conditions for the multiplication by a weight $w(t)$ that make the system $\{w(t)e^{i\lambda_{n}t}\}$ a conditional basis for $L^2[-\pi,\pi]$. Furthermore, some stability result is investigated on such bases.

Khoury solved Nowicki's conjecture on the kernel of an elementary derivation of a polynomial ring using Gröbner basis theory. In this paper, we give a simple new proof of the conjecture.

This paper discusses the situation that the large deviation rate functional has two distinct minimizers, for a model described by Wiener measures with certain densities involving a scaling. The motivation comes from the study of the so-called $\nabla\varphi$ interface model with weak self potentials. The pinned Wiener measures case was discussed by [3].