We classify transcendental classical solutions of the third Painlevé equation. This result combined with the list of algebraic solutions in [11] gives a complete table of classical solutions of the third Painlevé equation.

A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

The first part of the paper is concerned, among other things, with a characterization of filter regular sequences in terms of modules of generalized fractions. This characterization leads to a description, in terms of generalized fractions, of the structure of an arbitrary local cohomology module of a finitely generated module over a notherian ring. In the second part of the paper, we establish homomorphisms between the homology modules of a Koszul complex and the homology modules of a certain complex of modules of generalized fractions. Using these homomorphisms, we obtain a characterization of unconditioned strong d-sequences.

On the setting of the half-space of the euclidean n-space, we prove representation theorems and interpolation theorems for harmonic Bergman functions in a constructive way. We also consider the harmonic (little) Bloch spaces as limiting spaces. Our results show that well-known phenomena for holomorphic cases continue to hold. Our proofs of representation theorems also yield a uniqueness theorem for harmonic Bergman functions. As an application of interpolation theorems, we give a distance estimate to the harmonic little Bloch space. In the course of the proofs, pseudohyperbolic balls are used as substitutes for Bergman metric balls in the holomorphic case.

We show that any Killing vector field on the unit tangent sphere bundle with Sasaki metric of a space of constant curvature k ≠ 1 is fiber preserving by studying some property of geodesies on the bundle. As a consequence, any Killing vector field on the unit tangent sphere bundle of a space of constant curvature k ≠ 1 can be extended to a Killing vector field on the tangent bundle.

We show that if E → X is a locally trivial holomorphic fibrations whose fiber is an open Riemann surface and X is a q-complete space, then E is q-complete.

En analyse du bruit blanc (W.N.A) on utilise usuellement le triplet de Gelfand (S(R) ⊂ L2(R, dt) ⊂ S′(R), μ) où S′(R) est l’espace de L. Schwartz des distributions tempérées et μ la mesure Gaussienne donnée par sa fonction caractéristique:

(1)   

Let n > 1, m ≥ 1, g ≥ 3 and γ be given integers. The purpose of this paper is to determine the relations of n, m, g and γ for the existence of the symmetric Riemann surfaces S of type (n, m) with genus g and species γ. If n is an odd prime, the relations are known in [3]. In the case that n is odd, we shall show the analogous result when E(S) is isomorphic to a cyclic group Z2n and when the quotient space S/E(S) is orientable.

We prove a multiplicity result of 2π-periodic solutions for certain weakly coupled system of ordinary differential equations with real parameters. The proofs are based on differential inequalities and coincidence degree.

Dans cet article, on donne la classification des orbites des préhomogènes de type parabolique (g0, g1) correspondants aux paraboliques maximaux, notés (Δ, λ0), suivants: (A2n-1, αn), (Bn, α1) ou (Dn, α1), (Cn, αn), (D2n, α2n) ou (D2n, α2n-1), (E7, α7), (F4, α1), (E6, α2), (E7, α1), (E8, α8) lorsque F est un corps local ou global de caractéristique zéro, à l’aide de formes quadratiques, ceci lorsque le sous-espace radiciel est de dimension un.

In his masterwork Disquisitiones Arithmeticae, Gauss stated an approximate formula for the average of the class number for negative discriminants. In this paper we improve the known estimates for the error term in Gauss approximate formula. Namely, our result can be written as for every ∊ > 0, where H(−n) is, in modern notation, h(−4n). We also consider the average of h(−n) itself obtaining the same type of result.

Proving this formula we transform firstly the problem in a lattice point problem (as probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class number formula to express the error term as a sum of character and exponential sums that can be estimated with techniques introduced in a previous work on the sphere problem.

In this paper we introduce the notion of G-pre-weighted homogeneous map germ, (G is one of Mather’s groups A or K) and show that any G-pre-weight ed homogeneous map germ is G-finitely determined. We also give an explicit “order”, based on the Newton polyhedron of a pre-weighted homogeneous germ of function, such that the topological structure is preserved after perturbations by terms of higher order.

We show that any bounded hyperconvex domain is Bergman complete.