We consider an oscillatory integral of the form

Here each xj, j = 0, 1,…, L, runs in Rd, ν > 1 is a constant and tj, j = 1,…,L, are positive constants. Fujiwara [5] discussed this integral for L large and developed the stationary phase method with an estimate of the remainder term for the phase function S(xL,…, x0) coming from the action integral for a particle in an electric field. But his results cannot be applied to the integral which naturally arises in the discussion of quantum mechanics of a charged particle moving in a magnetic field. In this paper we extend his results to the case for the phase function involving both electric and magnetic fields.

One of the most powerful tools in homotopy theory is the homotopy groups of a triad introduced by Blakers and Massey in [1]. Our aim here is to develop systematically the formal, elementary aspects of the theory of a generalized triad and the mapping track associated with it. This will be used in §5 to deduce a result (Theorem 5.5) which seems to be closely related to an exact sequence established by Brown [2].

As an extension of a result due to W. D. Barcus and J. P. Meyer [4], T. Ganea [8] has recently proved a theorem concerning the fibre of the extension E∪CF→B of a fibre map p: E→B to the cone CF erected over the fibre F. In this paper we shall establish a generalized Ganea theorem which asserts that the homotopy type of the fibre of a canonical extension ξ′ of a triad (cf. [13]) is determined by those of f and g (see Theorem 3.4).

Our objective is to prove that certain Dirichlet series (in our variable q−s), which are defined by infinite sums, can be expressed as a product of an explicit rational function in q−s times an unknown polynomial M in q−s Moreover we show that M(q−s) is 1 if a simple condition is met. The Dirichlet series appear in the Euler products of Fourier coefficients for Eisenstein series. The series discussed below generalize the functions α0(N, q−s) used by Shimura in [12], and the theorem is an extension of Kitaoka’s result [5].

R. Bowen and P. Walters [2] have defined expansive flows on metric spaces which generalized the similar notion by D. Anosov [1]. On the other hand, P. Walters [4] investigated continuous transformations of metric spaces with discrete centralizers and unstable centralizers and proved that expansive homeomorphisms have unstable centralizers.

In this paper, we study the composition series of certain principal series representations of the three-fold metaplectic covering group of SL(2, K), where K is a non-archimedean local field. These representations are parametrized by unramified characters μ(x) = |x|s of K× and characters ω of the group of third roots of unity.

In a recent paper [2], an operator valued function space integral was defined by the authors as follows.

Dans toute la suite X désignera un groupe abélien localement compact et dénombrable à l’infini; ξ sera sa mesure de Haar. Un noyau de convolution N sur X signifie une mesure de Radon positive dans X, et, pour une mesure de Radon réelle μ dans X, N * μ s’appelle le N-potentiel de μ dès que cette convolution a un sens.

The following two related problems in Commutative Algebra are certainly very popular:

1. 1. Study the permanence of properties from a ring A to an idealadic completion of A; and

2. 2. Study the lifting of properties from A/I to A where A is a ring which is complete and separated in the I-adic topology.

Let p be a prime number and k an algebraic number field of finite degree d. Manin [14] showed that there exists an integer n = n(k,p) (≧0) which satisfies the condition

Let Ω be a bounded, decoupled pseudo-convex domain of finite type in ℂn with smooth boundary. In this paper, we generalize results of Bonami-Grellier [BG] and Bonami-Chang-Grellier [BCG] to study weighted Bergman projections for weights which are a power of the distance to the boundary. We define a class of operators of Bergman type for which we develop a functional calculus. Then we may obtain Sobolev and Lipschitz estimates, both of isotropic and anisotropic type, for these projections.

It is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable into ℙn by a CR mapping with an arbitrarily high, though finite, order of regularity.

A convergent power series f(z) in the open unit disk D is called Hadamard lacunary if it is expressed as follows:

In [5], the author proved that the Gauss map of a nonflat complete minimal surface immersed in R3 can omit at most four points of the sphere, and in [7] he revealed some relations between this result and the defect relation in Nevanlinna theory on value distribution of meromorphic functions. Afterwards, Mo and Osserman obtained an improvement of these results in their paper [11], which asserts that if the Gauss map of a nonflat complete minimal surface M immersed in R3 takes on five distinct values only a finite number of times, then M has finite total curvature. The author also gave modified defect relations for holomorphic maps of a Riemann surface with a complete conformai metric into the n-dimensional complex projective space Pn(C) and, as its application, he showed that, if the (generalized) Gauss map G of a complete minimal surface M immersed in Rm is nondegenerate, namely, the image G(M) is not contained in any hyperplane in Pm − 1(C), then it can omit at most m(m + 1)/2 hyperplanes in general position ([8]). Here, the number m(m + 1)/2 is best-possible for arbitrary odd numbers and some small even numbers m (see [6]). Recently, Ru showed that the “nondegenerate” assumption of the above result can be dropped ([13]). In this paper, we shall introduce a new definition of modified defect and prove a refined Modified defect relation. As its application, we shall give some improvements of the above-mentioned results in [5], [7], [8], [11] and [13].

(Anti-)self-dual metrics are 4-dimensional Riemannian metrics whose Weyl conformai tensor W half vanishes. The Weyl conformai tensor W of an arbitrary metric on an oriented 4-manifold has in general the self-dual part W+ and the anti-self-dual part W− with respect to the Hodge star operator * and one says that a metric is self-dual or anti-self-dual if W− = 0 or W+ = 0, respectively.

Our dream is to revive the ideal theory in partially ordered sets from a viewpoint of commutative algebra.

Historically, the concept of ideals in commutative algebra was first studied by Dedekind, who considered the ring of algebraic integers in an algebraic number field.

In this paper, we deal with the first part of an application of our Riemann-Roch type inequalities (cf. [13], [23]) toward deformations of polarized normal varieties. In Chapter I, we discuss the problem of eliminating fixed components from complete linear systems defined by multiples of a given divisor. Let Un be a complete normal variety and Y0 an ample Cartier divisor on U. The main result of the chapter is that there is a positive integer c0, predicted by the first two leading coefficients of the polynomial X(U, O(m Y0)), such that the complete linear system Λ(rc0Y0) has no fixed component whenever r is a positive integer (which is essentially contained in Theorem 1.1 (cf. [13], Lemma 5.2)). An easy consequence of this result is that when n = 2, we can find another positive integer c1 predicted by the same coefficients as above, such that rc1c0Y0 is very ample on U whenever r is a positive integer. Even though this has been generalized to n = 3 by J. Kollár (cf. [12]), we have included this in Section 3 since it is very simple.

The present notes provide a proof of i* (ℂP + ℂ(I - P); ℂQ + ℂ(I - Q)) = – χorb(P,Q) for any pair of projections P,Q with τ(P) = τ(Q) = 1/2. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

In this paper, we prove that every proper holomorphic self-map of a smoothly bounded pseudoconvex circular or Hartogs domain of finite type in C2 is biholomorphic.

New simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the contribution of “central” unipotent elements in the dimension formula of Siegel cusp forms of any degree. These results lead us to a conjecture on explicit values of dimensions of Siegel cusp forms of any torsion-free principal congruence subgroups of the symplectic groups of general degree.