In the study of ordinary differential equations, Malgrange ([Ma]) and Ramis ([R1], [R2]) established index theorem in (formal) Gevrey spaces, and the notion of irregularity was nicely defined for the study of irregular points. In their studies, a Newton polygon has a great advantage to describe and understand the results in visual form. From this point of view, Miyake ([M2], [M3], [MH]) studied linear partial differential operators on (formal) Gevrey spaces and proved analogous results, and showed the validity of Newton polygon in the study of partial differential equations (see also [Yn]).

Let M be a d-dimensional Riemannian manifold of class C∞ with Riemannian metric and let D be a connected domain in M having a non-empty smooth boundary and a compact closure . Suppose that are given and that converges uniformly to on D′ as for some neighborhood D′ of D. Consider the diffusion process on D′ with a small parameter generated by

(1.1)

Using vector fields on logarithmic jet spaces we obtain some new positive results for the logarithmic Kobayashi conjecture about the hyperbolicity of complements of curves in the complex projective plane. We are interested here in the cases where logarithmic irregularity is strictly smaller than the dimension. In this setting, we study the case of a very generic curve with two components of degrees d1 ≤ d2 and prove the hyperbolicity of the complement if the degrees satisfy either d1 ≥ 4, or d1 = 3 and d2 ≥ 5, or d1 = 2 and d2 ≥ 8, or d1 = 1 and d2 ≥ 11. We also prove that the complement of a very generic curve of degree d at least equal to 14 in the complex projective plane is hyperbolic, improving slightly, with a new proof, the former bound obtained by El Goul.

The goal of this paper is to characterize the operating functions on modulation spaces $M^{p,1}(\mathbb{R})$ and Wiener amalgam spaces $W^{p,1}(\mathbb{R})$. This characterization gives an affirmative answer to the open problem proposed by Bhimani (Composition Operators on Wiener amalgam Spaces, arXiv: 1503.01606) and Bhimani and Ratnakumar (J. Funct. Anal. 270 (2016), pp. 621–648).

About the behavior of brownian motion at time point oo, there are many results by P. Lévy, A. Khintchine etc. P. Lévy cited a theorem by A. Kolmogoroff as the most precise result in his famous book “Processus stochastiques et mouvement brownian” without proof. In this paper we shall prove this theorem, using the similar result about the random sequence by W. Feller, and then, applying the theorem of projective invariance by P. Lévy, we shall find also the behavior of brownian motion at time point 0 from the above theorem.

A field k is called quasi-finite if it is perfect and if Gk≈Ż where Gk is the Galois group of the algebraic closure kc over k and Ż is the completion of the additive group of the rational integers. The classical reciprocity law on the local field with finite residue field is well-known to hold on local fields with quasi-finite residue field ([4] [5]). Thus it is natural to ask if the global reciprocity law should hold in the ordinary sense (see § 1 below) on the function-fields of one variable over quasi-finite field. We consider here two basic prototypes of non-finite quasi-finite fields:

The purpose of this paper is to get a certain explicit expression of automorphic factors, formulated rather differently than usual, of the classically well known theta function

(1)

We show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m,d, n) from the corresponding partition for G(m,1,n). This confirms, in the case W = G(m,d,n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic–geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes.

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

Let A be a Noetherian local ring and p a prime ideal in A. Let

and call them, respectively, the Rees ring and the form ring of p. The purpose of this paper is to clarify, provided p is an almost complete intersection in A (cf. (2.1) for definition), when the rings R(p) and G(p) are Gorenstein.

When a Lie group G operates on a differentiable manifold M as a Lie transformation group, the orbit of a point p in M under G, or the G-orbit of p, is by definition the submanifold G(p) = {G(p); g∈G}. The purpose of this paper is to characterize the structure of a non-compact manifold M such that there exists a compact orbit of dimension (n — 1), n — dim M, under a connected Lie transformation group G, which is assumed to be compact or an isometry group of a Riemannian metric on M.

The aim of this paper is to point out a number-theoretical property of a certain product of values taken by an abelian function at points of finite order of the abelian variety on which the abelian function is considered We use the theory of power residues and the theory of complex multiplication, and the result is somewhat similar to the so-called Stickelberger’s relation in the theory of Gauss sums. A related investigation was previously made by the author in a special case of elliptic curves [2].

Let $G$ be an orthogonal, symplectic or unitary group over a non-archimedean local field of odd residual characteristic. This paper concerns the study of the “wild part” of an irreducible smooth representation of $G$, encoded in its “semisimple character”. We prove two fundamental results concerning them, which are crucial steps toward a complete classification of the cuspidal representations of $G$. First we introduce a geometric combinatorial condition under which we prove an “intertwining implies conjugacy” theorem for semisimple characters, both in $G$ and in the ambient general linear group. Second, we prove a Skolem–Noether theorem for the action of $G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of $G$ which have the same characteristic polynomial must be conjugate under an element of $G$ if there are corresponding semisimple strata which are intertwined by an element of $G$.

The concept of semi-local rings was introduced by C. Chevalley [1], which the writer has generalized in a recent paper [7] by removing the chain condition. The present paper aims mainly at the study of completions of semi-local rings. First in § 1 we investigate semi-local rings which are subdirect sums of semi-local rings, and we see in § 2 that a Noetherian semi-local ring R is complete if (and only if) R/p is complete for every minimal prime divisor p of zero ideal, together with some other properties. Further we consider in § 3 subrings of the completion of a semi-local ring. § 4 gives some supplementary remarks to [7], Chapter II, Proposition 8.

For moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface, we pursue analogy to some results known for moduli spaces of sheaves on a projective plane. We define an invariant height, introduced by Drezet in the projective plane case, for moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface and describe the structure of moduli spaces of height zero. Then we study rational maps of moduli spaces of positive height to moduli spaces of representation of quivers, effective cones of moduli spaces, and strange duality for height-zero moduli spaces.

Let k be an algebraic number field, let K be a Galois extension of k of finite degree, and let OK, Ok be the maximal orders of K, k, respectively. We consider the conjugate operation: for a given quadratic lattice M over OK equipped with a bilinear form B and for an automorphism σ ∈ G(K/k), we define a new quadratic lattice Mσ over OK. Here Mσ has the same underlying abelian group as M, but a new OK-action ; the new bilinear form Bσ on Mσ is defined by

Il y a quelques années, CHOQUET a démontré [5], en théorie classique du potentiel (ou pour certains noyaux) que l’ensemble E des points où un ensemble e est effilé est contenu dans un ouvert ω, tel que ω∩e soit de capacité arbitrairement petite. Cela contient les résultats-elefs, connus depuis longtemps que E∩e est polaire et que si e est ouvert, p. ex. borné, E ∩ ∂e est de mesure harmonique nulle en tout point de e. La partie nouvelle du résultat général est que les points de Ce où e est effilé peuvent être enfermés dans un ouvert ω tel que ω∩e soit de capacité arbitrairement petite et l’effilement n’intervient plus pour un ensemble qu’aux points du complémentaire, d’une façon équivalente à la notion de topologie fine.

Let V be an affine of dimension l over some field K. An arrangement A is a finite collection of affine hyperplanes in V. We call A an l-arrangement when we want to emphasize the dimension of V. We use [6] as a general reference. Choose an arbitrary point of V and fix it throughout this paper. We will use it as the origin.

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.