Goursat distributions are subbundles (of codimension at least 2) in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing -- very slowly -- always by 1. This defining condition is rather strong, implying local polynomial pseudo-normal forms for them (proposed in 1981 by Kumpera and Ruiz) featuring only real parameters of ${\it à}$ ${\it priori}$ unknown status, many of them reducible by further diffeomorphisms of the base manifold. We show that in the local ${\rm C}^\infty$ and ${\rm C}^{\omega}$ classifications of Goursat distributions genuine continuous moduli appear already in codimension 2. First examples of such moduli were given in codimension 3; in codimensions 0 and 1 the local classification is known and discrete.

A resource conscious functional language is introduced based a hypergraph rewriting scheme described by higher dimensional hypergraphs. Its operational and denotational semantics are given.

Let $g$ be an analytic function on the open unit disk $D$ in the complex plane $C$. We will study the following operator \[\ I_{g}(h) (z) := \int_0^z h'( \zeta )g( \zeta) d \zeta \ , \ J_g(h) (z) := \int_0^z h( \zeta )g'( \zeta) d \zeta\] on the Bloch space. In this paper, we will study the boundedness and compactness of $I_g$ on the $\alpha$-Bloch space , and the boundedness and compactness of products of $I_g$ and $J_g$ defined on the $\alpha$-Bloch space. And we will get the relationship of multiplication operator $M_g$ and the operators $I_g$, $J_g$ defined on the $\alpha$-Bloch space.

We study the singular semilinear elliptic equation $\Delta u + f(.,u)$ $= 0$ in ${\cal D}'({\mathbb R}^N)$, $N \geq 3$. $f: {\mathbb R}^N \times (0,\infty) \to [0,\infty)$ is such that $f(.,u) \in L^1({\mathbb R}^N)$ for $u > 0$ and $u \to f(x,u)$ is continuous and nonincreasing for a.e. $x$ in ${\mathbb R}^N$. We assume that there exists a subset $\Omega \subset {\mathbb R}^N$ with positive measure such that $f(x,u) > 0$ for $x \in \Omega$ and $u > 0$ and that $\int_{\rl^N}f(x,c|x|^{2-N})dx < \infty$ for some $c > 0$. Then we show that there exists a unique solution $u$ in the Marcinkiewicz space $M^{N/(N-2)}({\mathbb R}^N)$ such that $\Delta u \in L^1({\mathbb R}^N)$,$u > 0$ a.e. in ${\mathbb R}^N$.

Considered is a quantum system of $N(\ge2)$ charged particles moving in the plane ${\mathbb R}^2$ under the influence of a perpendicular magnetic field concenrated on the positions where the particle exsists. The gauge potential which gives this magnetic field is called a winding gauge potential. Properties of the momentum operators with a winding gauge potential are investigated. The momentum operators with a winding gauge potential are represented by the fibre direct integral of Arai's momentum operators [1]. Using this fibre direct integral decomposition, commutation properties of the momentum operators are investigated. A notion of local quantization of the magnetic flux is introduced to characterize the strong commutativity of the momentum operators. Aspects of the representation of the canonical commutation relations (CCR) are discussed. There is an interesting relation between the representation of the CCR with respect to this system and Arai's representation. Some applications of those results are also discussed.

Considered is a quantum system of $N(\ge2)$ charged particles moving in the plane ${\mathbb R}^2$ under the influence of a perpendicular magnetic field. Each particle feels the magnetic field concenrated on the positions of the other particles. The gauge potential which gives this magnetic field is called a winding gauge potential. Properties of the Dirac-Weyl operators with a winding gauge potential are investigated. Notions of local quantization and partial quantization are introduced to determine them. Especially, it is proven that existence of the zero-energy states of the Dirac-Weyl operators with a winding gauge potential is well determined by the local quantization and the partial quantization.

This paper gives examples of globally hypoelliptic operators on $S^3$, $S^7$, and $S^{15}$ which are sums of squares of real vector fields. These operators fail to satisfy the infinitesimal transitivity condition (the bracket condition) at any point and therefore they are not hypoelliptic in any subdomain.

Let $G$ be an arbitrary group. A subgroup $A$ of $G$ is purifiable in $G$ if, among the pure subgroups of $G$ containing $A$, there exists a minimal one. We studied purifiable subgroups of abelian groups in [4]. In this note, we give simple proofs of [4,Theorem 4.6], [4,Theorem 4.7], and [4,Theorem 4.8].

In this papar we consider an important class of first order partial differential equations (or, holonomic systems). The notion of general Clairaut type equations is one of the generalized notions of classical Clairaut equations. We give a generic classification of holonomic systems of general Clairaut type as an application of the theory of complete Legendrian unfoldings.

In this paper, we study the existence and the regularity of local strong solutions for the Cauchy problem of nonlinear Schröodinger equations with time-dependent potentials and magnetic fields. We consider these equations when the nonlinear term is the critical and/or power type which is, for example, equal to $\lambda |u|^{p-1} u$ with some $1 \le p < \infty$, $\lambda \in {\bf C}$. We prove local well-posedness of strong solutions under the additional assumption $1 \le p < \infty$ for space dimension $n = 4$, $1 \le p \le 1+4/(n-4)$ for $n \ge 5$, and local smoothing effects of it under the additional assumption $1 \le p \le 1+2/(n-4)$ when $n \ge 5$ without any restrictions on $n$.

Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinear terms is studied. By making use of Besov spaces we can improve the regularity assumption on the initial data. When the nonlinear term is $c_1u^2+c_2\bar{u}^2$, our results are as follows: When $d=1$ or $2$, for any initial data $u_0\in H^{-3/4}({\mathbb R}^d)$ there exists a unique local-in-time solution $u\in B_{2,(2,1),-|\xi|^2}^{(-3/4,1/2)}({\mathbb R}^d\times I_T)$. When $d\ge 3$, for any small data $u_0\in H^{\,\rho}({\mathbb R}^d)$, where $\rho(z)=z^{d/2-2}\log (2+z)$, there exists a unique local-in-time solution $u\in B_{2,(2,1),-|\xi|^2}^{(\,\rho,1/2)}({\mathbb R}^d\times I_T)$, and for any $u_0\in H^{s}({\mathbb R}^d)$, $s>d/2-2$, there exists a unique local-in-time solution $u\in B_{2,(2,1),-|\xi|^2}^{(s,1/2)}({\mathbb R}^d\times I_T)$. Here $I_T=(-T,T)$. We also have results for the equation with the nonlinear term $c_3u\bar{u}$.

Let $D$ be a Jordan domain in $\overline R^2$ and $\Gamma=\partial D$ be the boundary of $D$. Then $D$ is a quasidisk if and only if $\Gamma$ is a bounded circular distortion curve.