In [1] S. Kobayashi showed that the connected components of the set of zeros of a Killing vector field on a Riemannian manifold (Mn,g) are totally geodesic submanifolds of (Mn,g) of even codimension including the case of isolated singular points. The purpose of this short note is to give a simple proof of the corresponding result for conformal vector fields on compact Riemannian manifolds. In particular we prove the following

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

It is known (Pursell and Shanks [9]) that an isomorphism between Lie algebras of infinitesimal automorphisms of C∞ structures with compact support on manifolds M and M′ yields an isomorphism between C∞ structures of M and M’.

Omori [5] proved that this is still true for some other structures on manifolds. More precisely, let M and M′ be Hausdorff and finite dimensional manifolds without boundary. Let α be one of the following structures:

We prove the deformation invariance of Kodaira dimension and of certain plurigenera and the existence of canonical models for log surfaces which are smooth over an integral Noetherian scheme $S$.

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

Consider the function with variables z1…,zm

where υ and λj (j = 1,…,m) are complex numbers and.Г is a suitably chosen integral domain. In case m = 2, if we set [z1 z2]n as Г, it is the Selberg integral [22]. Our function can be regarded as an extention of it; so we may call (0.1) a Selberg type integral It is known that (0.1) satisfies a Gauss-Manin system, i.e. a system of rationally holonomic differential equations [3], [21].

Dans la première partie de cet article, on généralise une construction donnée dans [2], conduisant à l’homologie des espaces classifiants de groupe Γ. Cette construction utilise au départ un espace fibre principal Y de groupe Γ, le cas traité dans [2] étant celui où Y = Γ. Elle permet de définir dans le module de cohomologie de la base X de Y une filtration par une suite de sous-modules dont l’intersection est le module des classes caractéristiques. Dans la seconde partie, la filtration précédente est étendue à Hl(X, G) où G est le faisceau des germes d’applications continues de X dans un groupe topologique G.

Soit X un groupe abélien localement compact séparé et dénombrable á l’infini. On désignera par ξ la mesure de Haar sur X. Un noyau de convolution réel N (c’est-à-dire, une mesure de Radon réelle) sur X sera dit de type logarithmique s’il existe un semi-groupe (αt)t≥0 vaguement continu, markovien et récurrent des noyaux de convolution (non-négatifs) tel que limt → ∞ (vaguement) et, pour toute (pour la notation, voir § 2),

Let (R,m) be a local ring, and let M and N be finite R-modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules ExtiR(M,N/mnN). A number of corollaries are given, and more general filtrations are also considered.

In [1] L. Fuchs and I. Halperin have proved that a regular ring R is isomorphic to a two-sided ideal of a regular ring with identity. ([1] Theorem 1). Their methed is to imbed the regular ring R in the ring of all pairs (a, p) with a ∊ R and p from a suitable commutative regular ring S with identity such that R is an algebra over S. Thus S may be seen as the ring of R — R endomorphisms of the additive group of R. The following question is naturally raised: Is it true that the ring of all R-R endomorphisms of a rugular ring is a commutative regular ring? The main purpose of this paper is to answer this question affirmatively. (Theorem 1). After established this theorem we can follow the method in [1] to solve the problem in the title.

Let Ω be a locally compact separable metric space and let Ф be a positive symmetric kernel. Then the inner and outer capacities of subsets of Ω are defined by means of Ф-potentials of positive measures in the following manner. We define the capacity c(K) of a compact set K in a certain manner by means of Ф-potentials.

In this paper we deal with the problem (C) in § 4 of [4]. Let Ik be the Shimura mapping in [4] of Sk(4N, χ) into k-1(N′ χ2) (see p. 458). The problem (C) can be stated as follows: I3(f) is a cusp form if and only if ‹f, h› = 0 for all h ∈ U, where U is the vector space spanned by every theta series of S3(4N, χ) associated with some Dirichlet character.

We consider the decay of the local energy for the following equation in three dimension:

(0.1) utt + but − Δu = 0

u(0, x) = f(x) and ut(0, x) = g(x).

Divide the ideal boundary of a noncompact Riemannian manifold M into two parts δ0 and δ1 Viewing that M is surrounded by two conducting electrodes δ0 and δ1 we ask whether (M; δ0, δ1) functions as a condenser in the sense that the unit electrostatic potential difference between two electrodes is produced by putting a charge of finite energy on one electrode when the other is grounded. The generalized condenser problem asks whether there exists a subdivision δ0 ∪ δ1 of the ideal boundary of M such that (M; δ0, δ1 functions as a condenser.

In the papers [11] and [18] Rohlin and Thom have introduced an equivalence relation into the set of compact orientable (not necessarily connected) differentiable manifolds, which, roughly speaking, is described in the following manner: two differentiable manifolds are equivalent (cobordantes), when they together form the boundary of a bounded differentiable manifold. The equivalence classes can be added and multiplied in a natural way and form a graded algebra Ω relative to the addition, the multiplication and the dimension of manifolds. The precise structures of the groups of cobordism Ωk of dimension k are not known thoroughly. Thom [18] has determined the free part of Ω and also calculated explicitly Ωk for 0 ≦ k ≦ 7.

Theorem 5 of Azumaya’s recent article can be formulated in the following generalized form:

Let R be a ring. Let m be a finitely generated right-module of R such that mR = m. Assume that m = u1r + ubr + … + umr for every generating system of u1, u2, …, um of m and for every maximal right-ideal r of R. Then m = 0.

Hartman proved that a diffeomorphism is topologically conjugate to a linear map on a neighbourhood of a hyperbolic fixed point ([3]). In this paper we study the topological conjugacy problem of a diffeomorphism on a neighbourhood of a hyperbolic set, and prove that for any hyperbolic set there is an arbitrarily slight extension to which a sub-shift of finite type is semi-conjugate.

One of the most important problems in the potential theory is the one of capacitability, that is, whether the inner capacity of an arbitrary borelian subset B is equal to the outer capacity of B. As for the capacities induced by the Newtonian potentials and other classical potentials, Choquet [5] has shown that every borelian and, more generally, every analytic set are capacitable. He goes on as follows : first he shows that, for the Newtonian capacity f, the inequality of strong subadditivity holds, that is,

and then, using this inequality, he shows that the outer capacity f* has the analogous property to one of the outer measure, more precisely, if an increasing sequence {An} of arbitrary subsets converges to A, then f*(A) = limf*(An)- This property plays an important role in his proof.

Representation of a Gaussian process in terms of a Brownian motion is a powerful tool in the investigation of its structure. Among various representations is the canonical representation which is viewed as the best one from the viewpoint of the prediction theory. We have discovered some significance of non-canonical representations and discuss their optimality in an information theoretical approach.

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $(R,\mathfrak{m},k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\mathfrak{m}$-adic topology.