A componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

In this paper we study the paracomplex analogues of almost contact structures, and we introduce and study the notion of parahodge structures on manifolds. In particular, we construct new examples of paracomplex manifolds and we find all simply connected parahermitian symmetric coset spaces, which are the adjoint orbits of noncompact simple Lie groups, with parahodge structures induced by the Killing forms. This is done by (i) observing that a version of the results of A. Morimoto [4] on almost contact structures can be formulated and proved for almost paracontact structures, and by (ii) the methods of geometric quantization [3] applied to parahermitian symmetric triples [1] in conjunction with results of [7]. Two of the main results are Theorem 2.5 (which ties together the above structures) and Corollary 3.9.

Recently G. Shimura [1] constructed modular forms of integral weight from the forms of half integral weight. His construction is rather indirect. Indeed, he proved that the Dirichlet series, obtained from a form of half integral weight, multiplied by a certain L-function, corresponds to a modular form of an integral weight by means of the characterization of modular forms due to Weil.

Last year when I was preparing for course lectures the work of Ahlfors [1] which establishes that the Bloch constant is at least as large as it appeared to me that the resources of the theory of metrics of negative curvature offered rich possibilities from a function-theoretic point of view. The parallelism between certain properties of subharmonic functions and those of the metrics introduced by Ahlfors [1] is so striking that we are led to ask whether one can introduce a class of metrics including the metrics of Ahlfors for which not only does a Schwarz-Pick-Ahlfors lemma hold, but also requirements of differentiability disappear, as in the modern theory of subharmonic functions.

A. Soit G un groupe de Lie complexe et connexe et soit K un sous-groupe compact maximal de G. Soient g et les algèbres de Lie de G et K respectivement. est une sous-algèbre réelle de l’algèbre complexe g. Soit (i2 = –1). Alors est une sous-algèbre complexe de g.

Throughout this paper, A denotes a noetherian local ring with maximal ideal m and M a finitely generated A-module with d: = dim M≥1.

It has recently been found that my previous paper “On generalized semi-primary rings and Krull-Remak-Schmidt’s theorem” Jap. Journ. Math. 19 (1949) — referred to as S. K. — contained in its Theorems 19 and 20 some errors. Nevertheless the writer has been able to correct them in suitable forms so that most parts of both theorems hold, even under a weaker assumption, and also subsequent theorems remain valid. These will be, together with some supplementary remarks, shown in the present note.

We study the function and show that the poly-Bernoulli numbers introduced in our previous paper are expressed as special values at negative arguments of certain combinations of these functions. As a consequence of our study, we obtain a series of relations among multiple zeta values.

Let R be a ring with unit, and e be an idem potent in R such that (1 - e)Re = 0. In this note we shall explore the relationships between homological properties of R and those of its subring eRe.

Examples of such rings are abundant, the most common being perhaps the ring R of all two-by-two upper triangular matrices over a field, where—

The invariant affine connection over a coset space G/J of a Lie group G have been discussed by various authors. Recently, Nomizu [8] gave a systematic study of this problem when J is reductible in G. Among other results, he established a 1-1 correspondence between the invariant affine connections and certain multilinear mappings, and calculated the torsion and curvature. For canonical affine connection of the second kind, the holonomy group was also given.

0. Introduction. The purpose of this paper is to consider the following question: if R is a regular Noetherian ring and S ⊃ R is a module-finite R-algebra, is R a direct summand of S as R-modules? An affirmative answer is given if R contains a field, and it is shown that if the completions of the local rings of S possess maximal Cohen-Macaulay modules in the sense of § 1 of [6] then the conclusion is valid in this case too. Hence, if Conjecture E of [6] is true then the question raised here has an affirmative answer without further restriction on the regular Noetherian ring R, and it will be shown here that only a greatly weakened version of Conjecture E is needed.

Let k be a quadratic field and E an elliptic curve defined over k. The authors [8, 12, 13] [23] discussed the k-rational points on E of prime power order. For a prime number p, let n = n(k, p) be the least non negative integer such that

for all elliptic curves E defined over a quadratic field k ([15]).

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

In our earlier paper [4] we developed the basic sheaftheoretic and cohomological properties of ample vector bundles. These generalize the corresponding well-known results for ample line bundles. The numerical properties of ample vector bundles are still poorly understood. For line bundles, Nakai’s criterion characterizes ampleness by the positivity of certain intersection numbers of the associated divisor with subvarieties of the ambient variety. For vector bundles, one would like to characterize ampleness by the numerical positivity of the Chern classes of the bundle (and perhaps of its restrictions to subvarieties and their quotients). Such a result, like the Riemann-Roch theorem, giving an equivalence between cohomological and numerical properties of a vector bundle, may be quite subtle. Some progress has been made by Gieseker [2], by Kleiman [8], and in the analytic case, by Griffiths [3].

The object of the present paper is to prove some theorems concerning the existence and the uniqueness of the solution of the initial value problem for the evolution equation

(E).

In this paper we study Frobenius algebras and quasi-Frobenius rings with particular emphasis on their cohomological dimensions. For definitions of these cohomological dimensions we refer the reader to Cartan-Eilenberg [3] or Eilenberg [4].

The fourth Painlevé equation PIV is known to have symmetry of the affine Weyl group of type with respect to the Bäcklund transformations. We introduce a new representation of PIV, called the symmetric form, by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of PIV is given in terms of this representation. Through the symmetric form, it turns out that PIV is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions PIV, called Okamoto polynomials, are expressible in terms of the 3-reduced Schur functions.

Consider a diffusion process in Rd satisfying the stochastic differential equation

.

In this paper we consider a class of nonanalytic automorphic functions which were first mentioned to A. Selberg by C. L. Siegel. These functions have Fourier coefficients which are closely connected with the Fourier coefficients of analytic automorphic forms, and they are also eigenfunctions of the Laplace operator derived from the hyperbolic metric. We shall show how this latter property gives new results in the classical theory of automorphic forms.

In this paper we prove the local existence and uniqueness of C1+γ solutions of the Boussinesq equations with initial data υ0, θ0 ∈ C1+γ, ω0, ∇θ0 ∈ Lq for 0 < γ < 1 and 1 < q < 2. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar θ controls the breakdown of C1+γ solutions of the Boussinesq equations.