We consider a class of graphs G such that the height of the edge ideal I(G) is half of the number #V(G) of the vertices. We give Cohen-Macaulay criteria for such graphs.

As Fuks [3] stated, every domain of holomorphy or meromorphy over Cn is analytically convex in the sense of Hartogs. Oka [6] proved that every domain over Cn analytically convex in the sense of Hartogs is a domain of holomorphy. Therefore a domain of meromorphy over Cn coincides with a domain of holomorphy over Cn.

Let H1, H2,…,Hq be hyperplanes in PN (ℂ) in general position. Previously, the author proved that, in the case where q ≥ 2N + 3, the condition ν(f,Hj) = ν(g, Hj) imply f = g for algebraically nondegenerate meromorphic maps f, g: ℂn → PN(ℂ), where ν(f, Hj) denote the pull-backs of Hj through f considered as divisors. In this connection, it is shown that, for q ≥ 2N + 2, there is some integer ℓ0 such that, for any two nondegenerate meromorphic maps f, g: ℂn → PN(ℂ) with min(ν(f, Hj),ℓ0) = min(ν(g, Hj), ℓ0) the map f × g into PN(ℂ) × PN(ℂ) is algebraically degenerate. He also shows that, for N = 2 and q = 7, there is some ℓ0 such that the conditions min(ν(f, Hj), ℓ0) = min(ν(g, Hj), ℓ0) imply f = g for any two nondegenerate meromorphic maps f, g into P2(ℂ) and seven generic hyperplanes Hj’s.

A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T. T is an abelian surface if and only if its associated Kummer surface is algebraic. Kummer surfaces are among classical examples of K3-surfaces (which are simply-connected smooth surfaces with a nowhere-vanishing holomorphic 2-form), and play a crucial role in the theory of K3-surfaces. In a sense, all Kummer surfaces (resp. algebraic Kummer surfaces) form a 4 (resp. 3)-dimensional subset in the 20 (resp. 19)-dimensional family of K3-surfaces (resp. algebraic K3 surfaces).

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of . Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

Soit un domaine d’intégrité intégralement clos dans son corps des quotients, et soit m un idéal premier de . Soit un domaine d’intégrité intégralement clos dans son corps des quotients et dont est sous-anneau; soit un idéal premier de tel que

In der Differentialgleichung w″+a1(z)w‣w‣+a0(z)w=0 seien die Koeffizienten aj(z) ganze Funktionen. Dann ist jede Lösung w(z) eine ganze Funktion und genau dann von endlicher Ordnung, wenn die Koeffizienten Polynome sind. In diesem Fall gilt für die Ordnung

die Beziehung λ(w)= m/2, m ≧ 0 ganzzahlig.

This paper is a continuation of “Stable vector bundles on algebraic surfaces” [10]. For simplicity we deal with non-singular projective varieties over the field of complex numbers. Let W be a variety whose fundamental group is solvable, let H be an ample line bundle on W, and let f: V → W be an unramified covering. Then we show in section 1 that if E is an f*H-stable vector bundle on V then f*E is a direct sum of H-stable vector bundles. In particular f*L is a direct sum of simple vector bundles if L is a line bundle on V.

Let M be a connected metric space and H an isometry group of M which leaves fixed a point p in M. M is said H isotropic at p when, for any two points q and r of M at the same distance from p, there exists an isometry in H which carries q to r. When H coincides with the maximum isometry group leaving p fixed, M is said merely isotropic at p.

A d-dimensional normal projective variety X is called a Q-Fano d-fold if it has only terminal singularities and if the anti-canonical Weil divisor – Kx is ample. The singularity index I = I(X) of X is defined to be the smallest positive integer such that – IKX is Cartier. Then there is a positive integer r and a Cartier divisor H such that – IKX ~ rH. Taking the largest number of such r, we call r/I the Fano index of X.

In discussing convergence of Feynman path integrals [2], we need a stationary phase method of oscillatory integrals over a space of large dimension. More precisely, we have to know how the remainder term behaves when the dimension of the space goes to ∞ (cf. [2], [3] and [5]). The aim of the present note is to give answer to this question under rather mild assumptions. Application to the Feynman path integrals is discussed in [3] and [5].

In this paper we will discuss p-adic Artin L-functions. The existence of these functions is a simple consequence of a theorem of Deligne and Ribet [4]. One can formulate a “p-adic Artin conjecture” for these functions. Our primary purpose here is to relate this conjecture to the “main conjecture” discussed by Coates in [3]. We will describe the precise formulations of these conjectures that we will use later. Our main result will be that in fact the main conjecture implies the p-adic Artin conjecture.

In 1958 D. Puppe developed the theory of Abbildungsfolge and applied it to the study of Hilton formula and his sphärenähnliche Mannigfaltigkeiten. We shall be concerned here with a dual situation.

In this paper we define some lifts of tensor fields of types (1, k) and (0, k) as well as connections to a product preserving functor . We study algebraic properties of introduced lifts and we apply these lifts to prolongation of geometric structures from a manifold M to (M). In particular cases of the tangent bundle of pr -velocities and the tangent bundle of infinitesimal near points our constructions contain all constructions due to Morimoto (see [20]-[23]). In the cases of the tangent bundle our definitions coincide with the definitions of Yano and Kobayashi (see [31]). To construct our lifts and to study its properties we use only general properties of product preserving functors. All lifts verify so-called the naturality condition. It means that for a smooth mapping φ:M → N and for two φ-related geometric objects defined on M and N its lifts to (M) and (N) respectively are (φ)-related. We explain later the term φ-related for considered geometric objects.

The purpose of the present paper is to give the details of the results announced in the C. R. notes [7], [8] and [9].

In I which corresponds to [7], we shall develop a method for the study of affine transformations of a Riemannian manifold and prove Lemma 5 which is fundamental for the results in I and II. This part of our work has been motivated by a result of K. Yano [12] stating that the largest connected group of affine transformations of a compact orientable Riemannian manifold consists of isometries. Our method will have further applications to some other problems concerning or involving the connected group of isometries of a Riemannian manifold (for example, [10]).

Modular forms for the Weil representation of SL2 (ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems and to the Leech lattice.

Let (=,F) be a filtered, noetherian ring. A sequence x = x1 …, xn in B is called super regular if the sequence of initial forms

ξ = L(x1, …, ξn = L(xn)

is a regular sequence in grF(B).

Let q (x) be real and continuous in the infinite open interval (- ∞, ∞) and let y1(x, λ), y2(x, λ) be the solutions of

(1.1)

with the initial conditions

(1.2) .

The stability of the systems given by ordinary differential equations or functional-differential equations has been studied by many mathematicians. The most powerful tool in this field seems to be the Liapunov’s second method (see, for example [6]).

Let k be an algebraically closed field of arbitrary characteristic. Let T be an n-dimensional algebraic torus, i.e. T = Gm × · · · × Gm n-times), where Gm = Spec (k[t, t-1]) is the multiplicative group.