Let G be a reductive Lie group; Γ a nonuniform lattice in G. Then the theory of Eisenstein series plays a major role in the spectral decomposition of L2(G/Γ) (cf. [5]). One of the most difficult aspects of the subject is the analytic continuation of the Eisenstein series along with its associated c-function. This was originally done by Langlands using some very difficult analysis (cf. [5]). Later Harish-Chandra was able to simplify somewhat the most difficult part of the continuation, the continuation to zero, by the introduction of the Maas-Selberg relation. The purpose of this note is to give a simplified account of this particular part of the theory.

Our chief tool will be the truncation operator of Arthur (cf. [1] and [8]), the systematic utilization of which has the effect of streamlining the earlier accounts, especially in so far as continuation to zero is concerned, which is reduced to an elementary manipulation.

Nous traitons de la régularité d'une fonction holomorphe sur un ouvert de Cn à partir de celle de la fonction ou de la partie réelle connue sur une sous-variété totalement réelle maximale (de dimension n) incluse dans le bord de l'ouvert de définition.

Un théorème classique de Privalov assure qu'une fonction continue sur un disque fermé de C, holomorphe sur le disque ouvert, de partie réelle lipschitzienne d'ordre α sur le bord du disque vérifie une condition de même ordre α sur tout le disque. De façon analogue, le module de continuité d'une fonction holomorphe sur un ouvert borné de Cn, continue sur l'adhérence est contrôlé par celui de la fonction restreinte au bord [2] et [5], Le bord de l'ouvert peut être remplacé par la frontière de Shilov dans le cas d'un polyèdre analytique [4].

E. L. Stout [9] étudie la régularité sur une sous-variété totalement réelle maximale M de classe Cp, d'une fonction holomorphe f définie sur un ouvert Ω de classe Cq dont le bord contient M et telle que M soit l'ensemble sur lequel |f| est maximum; il démontre que f|M est Cp−0 lorsque q > p et Cp−1 lorsque q = p pour p supérieur ou égal à 3.

For an algebraic number field K there is a similarity between the additive characters defined on the Witt ring W(K), [20], [11], [17], [14, p. 131], and the local root numbers associated to a real orthogonal representation of the absolute Galois group of K, [18], [5]. Using results of Deligne and of Serre, [16], we shall derive in (5.3) a formula expressing the value, at a prime in K, of the additive character on a Witt class in terms of the rank modulo 2, the stable Hasse-Witt invariant and the local root number associated to the real quadratic character defined by the square class of the discriminant. Thus we are able to separate out the contributions made to the value of the additive character by each of the standard Witt class invariants.

This paper deals with boundary trajectories of non-smooth control systems and differential inclusions.

Consider a control system

(1.1)

and denote by R(t) its reachable set at time t. Let (z, u*) be a trajectory-control pair. If for every t from the time interval [0, 1], z(t) lies on the boundary of R(t) then z is called a boundary trajectory. It is known that for systems with Lipschitzian in x right-hand side, z is a boundary trajectory if and only if z(1) belongs to the boundary of the set R(1). If z is not a boundary trajectory, that is, z(1) ∊ Int R(1) then the system is said to be locally controllable around z at time 1.

A first-order necessary condition for boundary trajectories of smooth systems comes from the Pontriagin maximum principle, (see e.g. [12]).

Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).

Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.

(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2)

exists a.e. for all.

Let G be a reductive group over a field of characteristic zero. Fix a Borel subgroup B of G which contains a maximal torus T. For each dominant weight X we have an irreducible representation V(X) of G with highest weight X. For two dominant representation X1 and X2 we have a decomposition

This decomposition is determined by the element

of the group ring of the group of characters of T.

The objective of this paper is to compute r(X1, X2) for all pairs X1 and X2 of fundamental weights. This will be used to compute the equations for cones over homogeneous spaces. This problem immediately reduces to the case when G has simple type; An, Bn, Cn, Dn, E6, E7, E8, F4 and G2. We will give complete details for the classical types. For the case An we will work with GLn.

(1) The symbols p and q stand for prime numbers and throughout the paper we assume that p is fixed and contained in {3, 5, 7}. Let L be an algebraic number field (i.e., L is a finite extension of Q). Then prime divisors of L dividing p (resp. q) are denoted by (resp. ). The completion of L with respect to is denoted by . Let S be a finite set of prime numbers, and let M/L be a Galois extension with abelian Galois group of exponent p.

Definition. M/L is said to be little ramified outside S if for primes q ∉ S and all one has

with k ∊ N and . Here ζp is a pth root of unity, u1, …, uk are elements in and is the normed valuation belonging to . In particular M/L is unramified at all divisors of primes q ∉ S ∪ {p}.

Nonsmooth analysis has provided important new mathematical tools for the study of problems in optimization and other areas of analysis [1, 2, 6-12, 28]. The basic building blocks of this subject are local approximations to sets called tangent cones.

Definition 1.1. Let E be a real, locally convex, Hausdorff topological vector space (abbreviated l.c.s.). A tangent cone (on E) is a mapping A:2E × E → 2E such that A(C, x) is a (possibly empty) cone for all nonempty C in 2E and x in E.

In the sequel, we will say that a tangent cone has a certain property (e.g. “A is closed” or “A is convex“) if A(C, x) has that property for all non-empty sets C and all x in C. (If A(C, x) is empty, it will be counted as having the property trivially.)

In connection with the translation equation

(T)

three differential equations arise together with a differential initial condition. They are satisfied by the differentiable solutions of (T) and of the initial condition

(I)

These equations are attributed in [9] to E. Jabotinsky who seems to have been the first who treated these equations in connection with the theory of analytic iteration (see [6], cf. [7, 8], but see also [1, 2, 3]).

Gronau (see [9]) asked whether, conversely, it is true that all solutions of each of these “Jabotinsky differential equations”, possibly with some further initial conditions added, are also solutions of the translation equation. In this paper we give counter examples but also partial positive answers to these questions. First we show how one obtains the Jabotinsky equations from (T) and (I).

The complex maximal theorem of Hardy and Little-wood states:

(Mp). For 0 < p < ∞, there exists a positive constant Cp such that if f is holomorphic in the unit disc U of the complex plane then

where

The corresponding statement to the limiting case p = 0 can be stated as follows:

(M0) There exists a positive constant C0 such that if f is holomorphic in U

where log+t = max(log t, 0).

The statement (M0) is false as the following example shows.

In this paper we consider a question concerning the conjugacy class of an arbitrary ergodic automorphism σ of a sigma finite Lebesgue space (X, , μ) (i.e., a is a ju-preserving bimeasurable bijection of (X, , μ). Specifically we prove

THEOREM 1. Let τ, σ be any pair of ergodic automorphisms of an infinite sigma finite Lebesgue space (X, , μ). Let F be any measurable set such that

Then there is some conjugate σ' of σ such that σ'(x) = τ(x) for μ-almost every x in F.

The requirement that F ∪ τF has a complement of infinite measure is, for example, satisfied when F has finite measure, and in that case, the theorem was proved by Choksi and Kakutani ([7], Theorem 6).

Conjugacy theorems of this nature have proved to be very useful in proving approximation results in ergodic theory. These conjugacy results all assert the denseness of the conjugacy class of an ergodic (or antiperiodic) automorphism in various topologies and subspaces.

Convergence of exponential sums

that interpolate to Laplace transforms

(1.1)

have been studied by several authors [3, 6, 8, 15]. For rational functions that interpolate to Markov functions (also called Hamburger or Stieltjes Series or Hilbert Transforms)

(1.2)

far more detailed convergence results are available (see [10, 11, 16] and references therein). Both (1.1) and (1.2) are special cases of the transform

(1.3)

where K(x, t) is a strictly totally positive kernel.