Let E be a quadratic extension of a number field F with Galois conjugation σ, G’ the quasi-split unitary group in three variables, G the group GL(3, E). We let S be the space of the matrices s in G such that σ(s)s = e. One conjectures a comparison identity between the relative Kuznietsov trace formula for the symmetric space S and the ordinary Kuznietsov trace formula for the group G’ (See [10]). We prove the corresponding “fundamental lemmas”.

If a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or non-compact simple. The latter case is described

We compute the homology of limn->∞(Gn ≀ X), where (Gn) is a system of subgroups of Σpn containing a p-Sylow subgroup (Σpn p) and satisfying certain properties. We show that H*(limn->∞(Gn, ≀ X);Z/pZ) is built naturally over homology operations related to (Gn). We describe this family of operations using modular coinvariants.

Birelative K2-groups are computed for the fiber squares needed to study K2 and K3 of ℤG when G is a group of square-free order.

The good weights for the one-sided Hardy-Littlewood operators have been characterized by conditions . In this paper we introduce a new condition which is analogous to A∞. We show several characterizations of . For example, we prove that the class of weights is the union of classes. We also give a new characterization of weights. Finally, as an application of condition, we characterize the weights for one-sided fractional integrals and one-sided fractional maximal operators.

We display a number with a surprising continued fraction expansion and show that we may explain that expansion as a specialisation of the continued fraction expansion of a formal series: A series ΣchX-h has a continued fraction expansion with partial quotients polynomials in X of positive degree (other, perhaps than the 0-th partial quotient). Simple arguments, let alone examples, demonstrate that it is noteworthy if those partial quotients happen to have rational integer coefficients only. In that special case one may replace the variable X by an integer ≥ 2; that is: one may 'specialise' and thereby proceed to obtain the regular continued fraction expansion of values of the series. And that is significant because, generally, it is difficult to obtain the explicit continued fraction expansion of a number presented in different shape. Our example leads to a series with a specialisable continued fraction expansion and, a little surprisingly, our arguments suggest that the phenomenon of specialisability for series of the kind appearing here may be reserved to just the special subclass of series we happen to have stumbled upon.

The notion of expansive homeomorphism is important in topological dynamics and continuum theory. In this paper, a new kind of homeomorphism will be introduced and studied, namely the continuum-wise expansive homeomorphism. The class of continuum-wise expansive homeomorphisms is much larger than the one of expansive homeomorphisms. In fact, the class of continuum-wise expansive homeomorphisms contains many important homeomorphisms which often appear in "chaotic" topological dynamics and continuum theory, but which are not expansive homeomorphisms. For example, the shift maps of Knaster's indecomposable chainable continua are continuum-wise expansive homeomorphisms, but they are not expansive homeomorphisms. Also, there is a continuum-wise expansive homeomorphism on the pseudoarc. We study several properties of continuum-wise expansive homeomorphisms. Many theorems concerning expansive homeomorphisms will be generalized to the case of continuum-wise expansive homeomorphisms.

Consider the n-th order (n ≥ 1 ) neutral differential equation where σ1 < σ 2 < ∞ and μ and η are increasing real-valued functions on [Ƭ1, Ƭ2] and [σ1, σ2] respectively. The function μ is assumed to be not constant on [Ƭ1, Ƭ2] and [Ƭ1, Ƭ2] for every Ƭ ∈ (Ƭ1, Ƭ2) similarly, for each σ ∈ (σ1, σ2), it is supposed that r\ is not constant on [σ1 , σ] and [σ, σ2]. Under some mild restrictions on Ƭ1,- and σ1, (ι = 1,2), it is proved that all solutions of (E) are oscillatory if and only if the characteristic equation of (E) has no real roots.

Let be independent Ornstein-Uhlenbeck processes and In this paper the law of iterated logarithm for X(t, n)is considered. The results obtained improve those of Csorgő and Lin(1988) and Schmuland(1987).

In this paper we characterize the bounded harmonic functions ƒ and g on the unit disk for which the Toeplitz operators Tƒ and Tg defined on the Bergman space of the unit disk are essentially commuting.

A compact Riemannian manifold (M, g) of dimension 3 or higher admits a metric of constant (positive or negative) sectional curvature if the following conditions hold: the diameter is bounded from above, the part of the Ricci curvature which lies below some fixed negative number is bounded in LP norm for p > n/2, and the metric is almost spherical or almost hyperbolic in the LP sense. The idea of the proof is to obtain stronger (i.e. L∞) pinching by deforming the initial metric using the Ricci flow, thus reducing the problem to the theorems of Gromov in the case rg < 0 and of Grove, Karcher and Ruh in the case rg > 0. The reduced curvature tensor changes along the flow according to the heat equation, which implies a weak nonlinear parabolic inequality for its norm. The iteration method of De Giorgi, Nash and Moser is applied to obtain the estimate for the maximum norm of the reduced curvature tensor. The crucial step in the iteration consists of controlling the Sobolev constant of the appropriate imbedding (which also changes along the flow, but behaves well) by the isoperimetric constant, which, in turn, can be bounded in terms independent of the particular manifold.

After introducing the q-analogue of the enveloping algebra of the Witt algebra, we construct q-analogues of the module of tensor fields over the Witt algebra and prove a partial q-analogue of Kaplansky's Theorem concerning this module of tensor fields.

We answer a 1975 question of G R Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds

(I) X is metrizable, (II) dim X = 0 or (III) X is a union of finitely many pairwise disjoint generalized simple closed curves.

We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta.

It is well known that the length Zn of the longest head run observed in n tosses with a fair coin is approximately equal to log2n with a stochastically bounded remainder term. Though — log2n does not converge in law, in the present paper it is shown to have almost sure limit distribution in the sense of the a. s. central limit theorem having been studied recently. The results are formulated and proved in a general setup covering other interesting problems connected with patterns and runs such as the longest monotone block or the longest tube of a random walk.

An identity of L. Carlitz for a bibasic hypergeometric series is used to find some summation formulas for series in which the bases are either q and q2 or q and q3,0 < q < 1 In general, these series are neither balanced nor very-well-poised in the usual sense

On sait que la méthode classique de Schneider (en une variable) permet de minorer des combinaisons linéaires de deux logarithmes de nombres algébriques avec des coefficients algébriques. Nous généralisons cette méthode en plusieurs variables pour minorer des combinaisons linéaires de plusieurs logarithmes.

Consider the second order nonlinear differential equation

y" + a(t)f(y) = 0

where a(t) ∈ C[0,∞),f(y) GC1 (-∞, ∞),ƒ'(y) ≥ 0 and yf(y) > 0 for y ≠ 0. Furthermore, f(y) also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function f(y) = |γ|γ sgny with γ > 1 and 0 < γ < 1 known as the Emden-Fowler case. The coefficient a(t) is allowed to be negative for arbitrarily large values of t. Oscillation criteria involving integral averages of a(t) due to Wintner, Hartman, and recently Butler, Erbe and Mingarelli for the linear equation are shown to remain valid for the general equation, subject to certain nonlinear conditions on f(y). In particular, these results are therefore valid for the Emden-Fowler equation.

In this paper we study approximations of convex functions by twice Gâteaux differentiate convex functions. We prove that convex functions (respectively norms) can be approximated by twice Gâteaux differentiate convex functions (respectively norms) in separable Banach spaces which have the Radon-Nikody m property and admit twice Gâteaux differentiable bump functions. New characterizations of spaces isomorphic to Hilbert spaces are shown. Locally uniformly rotund norms that are limits of Ck-smooth norms are constructed in separable spaces which admit Ck-smooth norms.

We give a modulus of continuity for solutions of the wave equation with a noise term:

utt = Δu + a(u) + b(u)G, x ∈ ℝ3

where G is a Gaussian noise. This case is more difficult than in lower dimensions because the fundamental solution of the wave equation is singular.

In this paper we use a particular kind of weighted Sobolev space and pseudo-differential operators to study H3s propagation of singularities for the solution u ∊ Hs of the equations with second order.