The behavior of solutions of singular initial value problems is studied for a second order ordinary differential equation. The main purpose of this paper is to obtain sharp sufficient conditions so that any solution has a finite number of zeros or infinitely many zeros. We treat them systematically and generalize previous results by using the Pohozaev identity. As an application, we investigate the number of zeros of radially symmetric solutions to generalized Laplace equations.

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A homogeneous space of a reductive group is called weakly spherical if the action of some proper parabolic subgroup is prehomogeneous. We associate Dirichlet series with weakly spherical homogeneous spaces defined over the rational number field and prove their functional equations in the case where the space under consideration is a homogeneous space of the general linear group.

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In this paper, we consider an n-species almost periodic Lotka-Volterra competition system with dominated infinite delays. By constructing suitable Lyapunov functionals, we are able to show that, under a set of algebraic conditions, the system has a unique positive almost periodic solution which is globally attractive.

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A fundamental solution for the free Weyl equation is easily constructed using the Clifford relation of the Pauli matrices. But, we insist on Feynman's idea of representing a fundamental solution using classical objects. To do this, we first reformulate the usual matrix-valued Weyl equation on the ordinary Euclidian space to the "non-commutative scalar"-valued equation on the superspace, called the super Weyl equation. Then, we may find the classical mechanics corresponding to that super Weyl equation. Using analysis on the superspace, we may associate the classical Hamiltonian with that super Weyl equation. From this mechanics, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi and continuity equations, respectively. Moreover, they are exactly solvable. Then, we define a Fourier integral operator with phase and amplitude given by those functions, which gives a solution to the initial value problem of that super Weyl equation. The method and idea developped here, may be applied not only to the Pauli, Weyl or Dirac equations but also to any system of P.D.E's.

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We give a criterion, in terms of pluri-genera, for a normal surface singularity over the complex number field to be a simple elliptic or cusp singularity (resp. quotient singularity, log-canonical singularity).

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Bieberbach, Fatou et Picard avaient étudié une application biholomorphe de l'espace entier de n variables complexes sur son domaine qui a un point extérieur. En appliquant ces résultats au problème de Cauchy pour l'opérateur différentiel à coefficients des fonctions entières, nous donnons une remarque sur le domaine d'holomorphie de la solution.
Bieberbach, Fatou and Picard studied a biholomorphic map from the entire space of n complex variables onto its domain that has an exterior point. By applying these results to the Cauchy problem for the differential operator with coefficients of entire functions, we give a remark on the domain of holomorphy of the solution.

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The homogeneous coordinate ring of a toric variety was first introduced by Cox. In this paper, we study that of a toric variety with enough invariant effective Cartier divisors in detail. Here a toric variety is said to have enough invariant effective Cartier divisors if, for each nonempty affine open subset stable under the action of the torus, there exists an effective Cartier divisor whose support equals its complement. Both quasi-projective toric varieties and simplicial toric varieties have enough invariant effective Cartier divisors. In terms of the homogeneous coordinate ring, we describe the data needed to specify a morphism from a scheme to such a toric variety. As a consequence, we generalize a result of Cox, one of Oda and Sankaran, and one of Guest concerning data on morphisms.

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