Here we give an estimation of the modular convergence of translated equidis- tant Riemann sums to the integral of a function belonging to a modular space. Thus we extend some previous results by Fominykh and Kaminska-Musielak.

Vallee-Poussin sums are introduced and their approximation properties for classes of continuous 2$\pi$-periodic functions are studied.

This paper is concerned with the investigation of two closely related questions. The first question is: What is the $\lambda$-basic subgroup of the group of normed units $V(RG)$ in an abelian group ring $RG$ with identity of prime characteristic $p$ if $B$ is a $\lambda$-basic subgroup of the $p$-primary abelian group $G$ ? In this way it is shown that if $B$ is a $\lambda$-basic subgroup of the $p$–torsion $G$ and $R$ is perfect, then $1+I(RG;B)$ is a $\lambda$-basic subgroup of $V(RG)$, where $\lambda$ is a countable limit ordinal. Moreover, B is a direct factor of $1+I(RG;B)$ provided that it is $\lambda$-basic. This generalizes results due to Nachev (1996) and to the author (1995). The second question is the following: What is the criterion illustrated $V(RG)$ to be a $C_{\lambda}$-group when $G$ is an abelian p-group and $R$ is an unitary commutative ring with prime characteristic p ? In this direction it is proved that $V(RG)$ is a p-primary $C_{\lambda}$-group if and only if $G$ is a p-primary $C_{\lambda} group, provided R is perfect and $\lambda\leq\Omega$. Besides, if $R$ is perfect and $G$ is a p-group which is a $C_{\lambda}$-group, then the same holds for $V(RG)/G$, provided $\lambda\leq\Omega$. Moreover, if $G$ is a p-torsion $C_{\lambda}$-group of countable length $\lambda$ and $R$ is perfect without nilpotents, then $V(RG)/G$ is totally projective and so $G$ is a direct factor of $V(RG)$. The last extends in some aspect a result of May (1979, 1988).

We construct examples where graphs evolved as curves by a generalized surface diffusion law lose the graph property in finite time, although they remain to exist as smooth curves.

The purpose of this paper is an extension of the results of Simon [13] on the asymptotic behavior of the trace of the heat kernel for the Schrödinger operator. We discuss the case where the operator has compact resolvents in spite of the fact that the electric potential is degenerate on some submanifold. According to the degree of the degeneracy, we obtain non-classical asy mptotics.

We consider the asymptotic behavior of solutions to a generalized crystalline motion which describes evolution of plane curves driven by nonsmooth interfacial energy. Our main results say that solution polygonal curves expand to infinity or shrink to a single point depending on the size of initial data and the sign of the driving force term. In the expanding case, we show that any rescaled solution polygon converges to the boundary of the Wulff shape for the driving force term and hence if the driving force term is a constant, then any solution polygon approaches to an expanding regular polygon even if the motion is anisotropic. We also give lower and upper bounds of the extinction time for the shrinking case. In the appendix, we shall explain the notion of a discrete curvature and crystalline curvature from a numerical point of view.

Let $M$ be a complete, simply connected Riemannian manifold with positive constant scalar curvature, and $TM$ its tangent bundle with the complete lift metric. Assume that $TM$ admits an essential infinitesimal conformal transformation, then $M$ is isometric to the standard sphere.

First, we give a necessary and sufficient condition for a torsion-free rank-one subgroup of an arbitrary abelian group to be purifiable in a given group and show that all pure hulls of a purifiable torsion-free rank-one subgroup are isomorphic. Next, we show that if a $T(G)$-high subgroup A of an abelian group $G$ is purifiable in $G$, then there exists a subgroup $T'$ of $T(G)$ such that $G=H\oplus T'$ for every pure hull $H$ of $A$ in $G$. An abelian group $G$ is said to be a strongly ADE decomposable group if there exists a purifiable $T(G)$-high subgroup of $G$. We present an example $G$ such that not all $T(G)$-high subgroups of a strongly ADE decomposable group G are purifiable in $G$. Moreover, we characterize the strongly ADE decomposable groups of torsion-free rank 1. Finally, we use previous results to give a necessary and sufficient condition for an abelian group of torsion-free rank 1 to be splitting.

We compute the cohomology of currents invariant by some elementary Kleinian groups. The result is used to compute the cohomology of harmonic coclosed automorphic forms on the hyperbolic space of even dimension for such a group. This proves that for these groups the cohomology of these forms is isomorphic to the cohomol- ogy of the quotient space. This question is related to the Borel conjecture ( cf. Corollaries 2.14 and 3.9).

Let $\Gamma$ be a strongly regular graph with parameters $(k, \lambda, \mu)=(q^{2}+1,0,2)$ admitting $G(\cong PGL(2, q)2)$ as one point stabilizer for odd prime power $q$. We show that if $G$ stabilizes a vertex $x$ of $\Gamma$ and acts on $\Gamma_{2}(x)$ transitively, then $q=3$ holds and $\Gamma$ is the Gewirtz graph. Moreover it is shown that an antipodal double cover whose diameter 4 of a strongly regular graph with parameters $(k, 0, 2)$ is reconstructed from a symmetric association scheme of class 6 with parameters $p_{j.k}^{i}(0\leq i, j, k\leq 6)$ in the Section 3.

In this paper, we prove the global existence of a small solution to the Cauchy problem for the nonlinear Schrödinger equation with a class of cubic nonlinearities in one space dimension. Moreover, we also consider the asymptotic behavior in large time of the solution. Our results says that two cubic nonlinearities given in this paper can be considered as nonlinearities of higher degree (more precisely, of degree 5).