This paper continues the study of stochastic integrals in abstract Wiener space previously given in [14]. We will present, among other things, the detailed discussion and proofs of the results announced in [16]. Let H ⊂ B be an abstract Wiener space.

In a recent note bearing the same title (SIAM Journ. of Numerical Analysis, Vol. 3, 1966), I showed that there existed Weierstrass zeta functions having a vanishing pseudo-period. Cf. line 8, p. 376 of Saks-Zygmund, Analytic Functions. Professor Peter Henrici kindly pointed out to me that the mere existence of a Weierstrass zeta function with a vanishing pseudo-period was readily concluded from a classical formula of the theory of elliptic functions (Hurwitz-Courant, Funktionentheorie, 5th ed., p. 210) which reduces the question to the study of the zeros of

Soient k un corps de nombres, p un nombre premier, et L la p-tour de Hilbert de k; G = Gal(L/k). Dans ce papier, on se propose de donner un raffinement du critère de Golod-Safarevic appliqué à G, construit en filtrant le groupe de nombres Λ = {x ∈ k×/(x) = }. Ce raffinement permettra alors de montrer qu’un corps quadratique réel dont le groupe des classes contient un sous-groupe de la forme (4, 4, 4, 4), possède une 2-tour de Hilbert infinie.

We study the Saito-Ikeda infinitesimal invariant of the cycle defined by curves in their Jacobians using rank k + 1 vector bundles. We give a criterion for which the higher cycle class map is not trivial. When k = 2, this turns out to be strictly linked to the Petri map for vector bundles. In this case we can improve a result of Ikeda: an explicit construction on a curve of genus g ≥ 10 shows the existence of a non trivial element in the higher Griffiths group.

In this article we continue the investigations on invariant metrics on a certain class of weakly pseudoconvex domains which we began in [H 1]. While in that paper the differential metrics of Caratheodory and Kobayashi were estimated precisely, the present paper contains a sharp estimate of the singularity of the Bergman kernel and metric on domains belonging to that class.

Cohomology theory for (associative) algebras was first established in general higher dimensionalities by G. Hochschild [3], [4], [5]. Algebras with vanishing 1-cohomology groups are separable semisimple algebras ([3], Theorem 4.1). On extending and refining our recent results [6], [8], [12], we establish in the present paper the following:

Let n ≧ 2. Let A be an (associative) algebra (of finite rank) possessing a unit element 1 over a field Ω, and N be its radical.

The Lévy Brownian motion with multidimensional parameter was introduced and discussed in his book [1] and it is known as the most important random field. Many approaches have been made to the investigation of the Lévy Brownian motion by H.P. McKean [7], Yu. A. Rozanov and others, by using various techniques.

The flow of Bingham type through a domain Ω in the d-th dimensional space Rd (d ≥ 2) during the time (0, T) is a flow of an incompressible visco-plastic fluid governed by the equations for a velocity vector u = (u1,…,ud) and a stress tensor

The low density limit in the Boson Fock case has been investigated in [1] where also the physical meaning and their motivations have been explained (cf. also [0]). From these papers one knows how the number processes can be obtained from a quantum Hamiltonian model via a certain limit procedure.

For a finite set S = {a1,…, aq}, consider the polynomial PS(w) = (w – a1)(w – a2) … (w – aq) and assume that has distinct k zeros. Suppose that PS(w) is a uniqueness polynomial for entire functions, namely that, for any nonconstant entire functions ɸ and ψ, the equality PS(ɸ) = cPS(ψ) implies ɸ = ψ, where c is a nonzero constant which possibly depends on ɸ and ψ. Then, under the condition q > k + 2, we prove that, for any given nonconstant entire function g, there exist at most (2q-2)/(q – k – 2) nonconstant entire functions f with f*(S) = g*(S), where f*(S) denotes the pull-back of S considered as a divisor. Moreover, we give some sufficient conditions of uniqueness polynomials for entire functions.

Denote by H3 the 3-dimensional hyperbolic space with sectional curvatures equal to – 1, and let g be a geodesic in H3 Let {ψt} be the translation along g (see § 2) and let {φt} be the one-parameter subgroup of isometries of H3 whose orbits are circles centered on g. Given any α ∊ R, one can show that λ = {λt} = ψt ∘ φαt} is a one-parameter subgroup of isometries of H3 (see § 2) which is called a helicoidal group of isometries with angular pitch α. Any surface in H3 which is λ-invariant is called a helicoidal surface.

In this work we prove some results concerning minimal helicoidal surfaces in H3. The first one reads:

An equivalence relation is defined and studied on the set of B-stable ideals in the nilradical of the Lie algebra of a Borel subgroup B. Techniques are developed to compute the equivalence relation and these are carried out in the exceptional groups. There is a natural partial order on equivalence classes coming from inclusion of one ideal in another. A main theorem is that this partial order is a refinement of the closure ordering on nilpotent orbits.

We prove the invariance of plurigenera under smooth projective deformations in full generality.

Assuming the Riemann hypothesis, we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Ω-term, we show that our result is essentially the best possible.

In [5], H. Maass showed that the dimension of a space of generalized Whittaker functions satisfying certain system of differential equations on Siegel’s upper half space H2 of degree 2 is three. First of all, we shall investigate the structure of a space of generalized Whittaker functions which are eigen functions for the algebra of invariant differential operators on H2. The theory of generalized Whittaker functions is discussed in Yamashita [12], [13], [14], [15] with full generality. But, we will get an outlook of the space of generalized Whittaker functions by using elementary calculus instead of representation theory of Lie groups.

A line bundle, L, on a smooth, connected projective surface, S, is defined [7] to be k-very ample for a non-negative integer, k, if given any 0-dimensional sub-scheme with length , it follows that the restriction map is onto. L is 1-very ample (respectively 0-very ample) if and only if L is very ample (respectively spanned at all points by global sections). For a smooth surface, S, embedded in projective space by | L | where L is very ample, L being k-very ample is equivalent to there being no k-secant Pk−1 to S containing ≥ k + 1 points of S.

We prove new cases of the direct summand conjecture using fundamental theorems in p-adic Hodge theory due to Faltings. The cases tackled include the ones when the ramification locus lies entirely in characteristic p.

We use a method of Berndtsson to obtain a simplification of Ohsawa’s result concerning extension of L2-holomorphic functions. We also study versions of the Ohsawa-Takegoshi theorem for some unbounded pseudoconvex domains, with an application to the theory of Bergman spaces. Using these methods we improve some constants, that arise in related inequalities.

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

Let S be a minimal algebraic surface of general type defined over the complex number field C, and let K denote the canonical bundle. According to [10], we call S a canonical surface if the rational map ФK associated with | K | induces a birational map of S onto the image X. We denote by Q (X) the intersection of all hyperquadrics through X.