It is well-known that the renormalized Rauzy map is conservative and ergodic. In this paper, we show that a Darling-Kac set exists for the renormalized Rauzy map. This implies the pointwise dual ergodicity of this map.

Let $\Sigma_{g,b}$ denote a closed oriented surface of genus $g$ with $b$ punctures and let $\mathrm{Mod}(\Sigma_{g,b})$ denote its mapping class group. Kassabov showed that $\mathrm{Mod}(\Sigma_{g,b})$ is generated by 4 involutions if $g>7$ or $g=7$ and $b$ is even, 5 involutions if $g>5$ or $g=5$ and $b$ is even, and 6 involutions if $g>3$ or $g=3$ and $b$ is even. We proved that $\mathrm{Mod}(\Sigma_{g,b})$ is generated by 4 involutions if $g=7$ and $b$ is odd, and 5 involutions if $g=5$ and $b$ is odd.

In this paper, we will give a new proof of the Deuring-Shafarevich formula, which asserts a relation between the $p$-ranks of Jacobi varieties. We analyze the zeta functions of global function fields to prove the formula, without using tools of the algebraic geometry.

In this paper, we give a certain identity between an $\eta$-product of weight 1 and theta series associated with a pair of binary quadratic forms. We also have explicit description of Siegel's theorem by an $\eta$-product. For quadratic forms $Q_1$ and $Q_2$ which are in the same genus, we express the difference $\vartheta_{Q_1}(\tau)-\vartheta_{Q_2}(\tau)$ by an $\eta$-product.

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{@{}ll} u_{tt} - \Delta u + b(t)u_t=f(u)\,, & (t,x) \in {\bf R}^+ \times {\bf R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x)\,, & x \in {\bf R}^N\,. \end{array}\right. {(*)} $$ hen $b(t)=(t+1)^{-\beta}$ with $0\le \beta <1$, the damping is effective and the solution $u$ to ($*$) behaves as that to the corresponding parabolic problem. When $f(u)=O(|u|^{\rho})$ as $u \to 0$ with $1<\rho < \frac{N+2}{[N-2]_+}$(the Sobolev exponent), our main aim is to show the time-global existence of solutions for small data in the supercritical exponent $\rho>\rho_F(N):=1+2/N$. We also obtain some blow-up results on the solution within a finite time, so that the smallness of the data is essential to get global existence in the supercritical exponent case.

We consider real-valued twice differentiable functions defined on an open interval. Our main result states that if a function $f$ satisfies some inequalities then a map $x\mapsto f(x)\exp(-cx)$ is convex, where $c$ is an arbitrary point of $\mathbf{R}$ or of $\mathbf{R}\setminus (c_1,c_2)$ for some real $c_1, c_2$.

We consider Apostol-Vu double zeta values for coordinatewise limits at non-positive integers, and we give some relations among Riemann's zeta values, Euler-Zagier double zeta values and Apostol-Vu double zeta values for all coordinatewise limits at non-positive integers. Using the relations, we also give relations among multiple Bernoulli numbers.

Let $E_k$ be the elliptic curve given by $y^2=x^3-p^k x$, where $p$ is a prime number and $k \in \{1,2,3\}$. In this paper, we first give a necessary and sufficient condition for the rank of $E_k(\mathbf{Q})$ to equal one or two, respectively, and in the rank two case, explicitly describe independent points of free part of the Mordell-Weil group $E_k(\mathbf{Q})$. Secondly, we show several subfamilies of $E_k$ whose integer points and ranks can be completely determined.

A probability distribution $\mu$ on $\mathbf{R}^d$ is selfdecomposable if its characteristic function $\widehat\mu(z), z\in\mathbf{R}^d$, satisfies that for any $b>1$, there exists an infinitely divisible distribution $\rho_b$ satisfying $\widehat\mu(z) = \widehat\mu(b^{-1}z)\widehat\rho_b(z)$. This concept has been generalized to the concept of $\alpha$-selfdecomposability by many authors in the following way. Let $\alpha\in\mathbf{R}$. An infinitely divisible distribution $\mu$ on $\mathbf{R}^d$ is $\alpha$-selfdecomposable, if for any $b>1$, there exists an infinitely divisible distribution $\rho_b$ satisfying $\widehat\mu(z) = \widehat\mu(b^{-1}z)^{b^{\alpha}}\widehat\rho_b(z)$. By denoting the class of all $\alpha$-selfdecomposable distributions on $\mathbf{R}^d$ by $L^{(\alpha)}(\mathbf{R}^d)$, we define in this paper a sequence of nested subclasses of $L^{(\alpha)}(\mathbf{R}^d)$, and investigate several properties of them by two ways. One is by using limit theorems and the other is by using mappings of infinitely divisible distributions.

Let $k$ be a totally real number field and $L$ a CM-field such that $L/k$ is finite and abelian. In this paper, we study a stronger version of Brumer's conjecture that the Stickelberger element times the annihilator of the group of roots of unity in $L$ is in the Fitting ideal of the ideal class group of $L$, and also study the dual version. We mainly study the Teichmüller character component, and determine the Fitting ideal in a certain case. We will see that these stronger versions hold in a certain case. It is known that the stronger version (SB) does not hold in general. We will prove in this paper that the dual version (DSB) does not hold in general, either.

In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.

We update and refine the work of T. Hamada concerning *-Einstein hypersurfaces in $\mathbb{C}\mathbb{P}^n$ and $\mathbb{C}\mathrm{H}^n$. We also address existence questions using the methods of moving frames and exterior differential systems.

In this note, we construct a birational map of a moduli space of stable sheaves on a $K3$ surface induced by a reflection functor.

We shall describe the one-to-one correspondence between the set of pictures and the set of Littlewood-Richardson crystals.

The general hypergeometric functions of confluent type are studied. We establish a link between the general hypergeometric functions defined by 1-dimensional integrals and those defined by multi-dimensional integrals. The key point is to form an intermediate Wronskian determinant for the 1-dimensional ones and to rewrite it into a multi-dimensional integral using the generalized Veronese map.

We extend a well-known result of H. Fujita and H. Morimoto [1] to exterior domains of $\mathbb{R}^m$, with $m\ge 4$.

We introduce new weight classes, and extend the Adams inequality to weighted Morrey spaces. We also investigate the boundedness of the modified fractional integral operator from weighted Morrey spaces to Lipschitz or BMO spaces.

We will derive the asymptotic expansions of the solutions $U(x,t)$ to the heat equation with $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$, $r\geq 1/2$, initial value, where $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$ is the dual space of the Gel'fand-Shilov space $\mathcal{S}^r_r(\mathbf{R}^d$. Moreover, we show that, when $1/2\leq r\leq 1$, these asymptotic expansions satisfy the strong asymptotic condition on some circle $D_R=\{t\in\mathbf{C}\ |\ \mathrm{Re}\ t^{-1}>R^{-1}\}$. Therefore, we find that these asymptotic series for $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$ initial value are Borel summable by means of A. D. Sokal's result on the Borel summability. As an application, we show the asymptotic expansions of the Weyl transform with Planck's constant $\hbar$ in some state, which are refinement of a classical limit of the quantum mechanical expectation values expressed by the Weyl transform.