In this note, the real analyticity of the solutions for non-linear elliptic equations will be proved by the method of Friedman [5] and Kato [7] using Faá di Bruno's formula.

Using the Mehler kernel $E(x,\xi,t)$, we show that the solution of the Hermite heat equation $({\partial}_{t} - \triangle + |x|^{2})U(x,t) = 0$ in $\mathbf{R}^{n}\times (0,T)$ satisfying $\sup_{x\in{\mathbf{R}}^{n}}|U(x,t)|\leq C(1+ t^{-N})$ for some constants $C$ and $N$ can be expressed as $U(x,t) = \langle u(\xi), E(x,\xi,t)\rangle$ for unique $u$ in ${\mathcal S}^{'}(\mathbf{R}^{n})$. This is a parallel result with the one in (Theorem 1.2, T. Matsuzawa, {\it A calculus approach to hyperfunctions} III, Nagoya Math. J. {\bf 118} (1990), 133--153). Moreover we represent the tempered distributions as initial values of solution of the Hermite heat equation and apply it to generalize a theorem by Strichartz [Theorem 3.2, Trans. Amer. Math. Soc. {\bf 338} (1993), 971--979] in the space of tempered distributions.

In this paper, we study the Mordell-Weil ranks of elliptic curves defined over the maximal abelian extension of the rational number field, assuming several conjectures on the Hasse-Weil $L$-functions. We prove that an elliptic curve defined over an abelian field with odd degree has infinite rank over the maximal abelian extension of the rational number field. This result gives affirmative evidence for 'the largeness' (in the sense of Pop) of the maximal abelian extension of the rational number field.

We prove that the monomial curve $(t^{17}, t^{19}, t^{25}, t^{27})$ is set-theoretic complete intersection.

The relationship between mixed multiplicities of arbitrary ideals in local rings and Hilbert-Samuel multiplicities was solved by Viêt in [8]. In this paper, we extend some important results of Viêt in [8, 9, 10, 11] to modules. We build the concept of (FC)-sequences of modules and use this notion to study reductions of ideals with respect to modules, mixed multiplicities of modules and multiplicities of Rees modules.

After certain completions, we define adjoint groups of extended affine Lie algebras with nullity $2$. Then we show that such groups have Tits systems with affine Weyl groups (Part I). This idea allows us to consider linear groups over some completed quantum tori. By the same argument, we can prove that these linear groups also have Tits systems with affine Weyl groups. Using this fact we will study their universal central extensions as well as associated $K_1$-groups and $K_2$-groups (Part II). We will discuss some relationship among our groups constructed here (Part III).

In this paper we study linear elasticity equations in an infinite elastic strip with a semi-infinite crack. We find the derivative of the energy functional as the crack shifts with an angle. Then we obtain the formula given by surface force and the angle.

We prove a relationship on the rank of the Floer homology groups of integral homology handles. Moreover we make a conjecture on a certain difference between these ranks.

Let $Q$ be a parameter ideal in a Noetherian local ring $A$ with the maximal ideal $\frak{m}$. Then $A$ is a regular local ring and $\frak{m}/Q$ is cyclic, if $\rm{depth}\ A > 0$ and $Q^n$ is $\frak{m}$-full for some integer $n \geq 1$. Consequently, $A$ is a regular local ring and all the powers of $Q$ are integrally closed in $A$ once $Q^n$ is integrally closed for some $n \geq 1$.

In this paper, we introduce $q$-analogues of the Barnes multiple zeta functions. We show that these functions can be extended meromorphically to the whole plane, and moreover, tend to the Barnes multiple zeta functions when $q \uparrow 1$ for all complex numbers.

Let $\mathcal{A}$ be a central arrangement of hyperplanes in $\mathbb{C}^n$ defined by the homogeneous polynomial $d_{\mathcal{A}}$. Let $D_n$ be the Weyl algebra of rank $n$ over $\mathbb{C}$ and let $P=\mathbb{C}[x_1,\ldots ,x_n,d_{\mathcal{A}}^{-1}]$ be the algebra of rational functions on the variety $Y_{\mathcal{A}}=\mathbb{C}^n\setminus \bigcup_{H\in \mathcal{A}}H$. Studying the structure of $P$ as a $D_n$-module we obtain a sequence of new $D_n$-modules. These modules allow us to define useful complexes that determine the De Rham cohomology of $Y_{\mathcal{A}}=\mathbb{C}^n\setminus \bigcup_{H\in \mathcal{A}}H$. Finally we compute the Poincaré series of $P$.

Let $f:({\bf R}^n, 0)\to ({\bf R}^p, 0)$ be a $C^{\infty}$ map-germ. We are interested in whether the number modulo 2 of stable singular points of codimension $n$ that appear near the origin in a generic perturbation of $f$ is a topological invariant. In this paper we concentrate on investigating the problem when $p$ is $2n- 1$, where stable singular points of codimension $n$ are only Whitney's umbrellas, and give a positive answer to the problem.

The aim of the paper is to construct the fundamental solution of an arbitrary complex power of the Dirac operator, these powers being defined as convolution operators with a kernel expressed in terms of specific distributions in Euclidean space. The desired fundamental solution is found, at least formally, in terms of the same families of distributions as those arising in the kernel of the corresponding operator. Clearly, in order to prove these results in a rigorous way, we first have to investigate the definition and properties of both the convolution and the product of arbitrary elements of the families of distributions under consideration, leading to a very attractive pattern of mutual relations between them.

Graph divide links were introduced as an extension of the class of A'Campo's divide links. We show that some of such links cannot be represented as the links of any graph divides including less circles or less cycles.