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.

Semilinear elliptic partial differential equations of the type

1

will be considered throughout real Euclidean N-space, where m ≧ 2 is a positive integer, Δ denotes the N-dimensional Laplacian, and f is a real-valued continuous function in [0, ∞) × (0, ∞). Detailed hypotheses on the structure of f are listed in Section 3.

Our objective is to prove the existence of radially symmetric positive entire solutions u(x) of (1) which are asymptotic to positive constant multiples of |x|2m−2i as |x| → ∞ for every i = 1,…, m, N ≧ 2i + 1.

In this paper we continue our effort [11], [12], [13], [14] to interpret geometrically the harmonic forms on certain locally symmetric spaces constructed by using the theta correspondence. The point of this paper is to prove an integral formula, Theorem 2.1, which will allow us to generalize the results obtained in the above papers to the finite volume case (the previous papers treated only the compact case). We then apply our integral formula to certain finite volume quotients of symmetric spaces of orthogonal groups. The main result obtained is Theorem 4.2 which is described below. We let (,) denote the bilinear form associated to a quadratic form with integer coefficients of signature (p, q). We assume that the fundamental group Γ ⊂ SO(p, q) of our locally symmetric space is the subgroup of the integral isometries of (,) congruent to the identity matrix modulo some integer N. We assume that N is chosen large enough so that Γ is neat (the multiplicative subgroup of C* generated by the eigenvalues of the elements of Γ has no torsion), Borel [2], 17.1 and that every element in Γ has spinor norm 1, Millson-Raghunathan [15], Proposition 4.1. These conditions are needed to ensure that our cycles Cx (see below) are orientable. The methods we will use apply also to unitary and quaternion unitary locally symmetric spaces, see [13].

The q-shifted factorial (a)n or (a; q)n is

and an empty product is interpreted as 1. Recently, Askey and Wilson [6] introduced the polynomials

1.1

where

1.2

and

1.3

We shall refer to these polynomials as the Askey-Wilson polynomials or the orthogonal 4ϕ3 polynomials. They generalize the 6 — j symbols and are the most general classical orthogonal polynomials, [2].

The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus Tn with its algebra, C(Tn), of continuous complex-valued functions under pointwise multiplication. But C(Tn) is the universal C*-algebra generated by n commuting unitary operators. By definition, [15, 16, 50], a non-commutative n-torus is the universal C*-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [8, 10], these algebras have a natural differentiable structure, defined by a natural ergodic action of Tn as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [37] that the K-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK-equivalent to ordinary tori by Corollary 7.5 of [52].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [17].)

Let G be a compact connected Lie group with H a connected subgroup of maximal rank. Suppose there exists a compact connected Lie subgroup K with H ⊂ K ⊂ G. Then there exists a smooth fiber bundle G/H → G/K with K/H as the fiber. (See for example [13].) This can be incorporated into a diagram involving the classifying spaces as follows:

(1)

Here π, ϕ, ϕ1 and ϕ2 denote fibrations. We also know that the homogeneous spaces and the Lie groups, which are homotopy equivalent to the loop spaces of their respective classifying spaces, are homotopy equivalent to connected finite complexes.

Now suppose H is a maximal subgroup. Can there still exist spaces, which we will call BK, K/H, and G/K, and fibrations so that diagram (1) is still valid? This paper will show that in many cases either G/K or K/H will be homo topically trivial.

In this paper we shall compute the cyclic cohomology of a non-commutative torus, i.e., a certain algebra associated with an antisymmetric bicharacter of a finite rank free abelian group G.

The main result is

1.1

where

The method of computation generalises the computation of the cyclic cohomology of the irrational rotation algebras given by Connes in [3]. (Our method works equally well also in the rational case, which was dealt with by a different method by Connes in [3].)

We first describe the Hochschild cohomology of in an explicit way, and then combine this description with the exact sequence of [3]:

1.2

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.

It is a well-known fact (cf., for instance Lemma 7.3.1 of [8], and also [2] and [4] ) that if M and N are closed subspaces of a finite-dimensional Hilbert space, and if M and N are in ‘generic’ position (i.e., any two of the four subspaces M, M⊥, N, N⊥ have trivial intersection), then N is the graph of a linear isomorphism of M onto M⊥ . To be sure, there exist infinite-dimensional versions of this, where one must allow for unbounded operators in case the ‘gap’ between M and N is zero, in the sense of Kato [7]. (There is an extensive literature on pairs of subspaces, [2], [3], [4], [6] and [7], to cite a few; for a fairly extensive bibliography, see [3].)

This paper addresses itself to the case of n (2 ≦ n < ∞) subspaces.

Throughout this article c0, c1, c2, … will denote effectively computable positive absolute constants. Denote the cardinality of a set X by |X|. Let N be a positive integer and let A and B be non-empty subsets of {1, …,N}. Put

In [3], Balog and Sá;rközy proved that if N > c0 and

(1)

then there exist a0 and b0 with a0 ∊ A0 and b0 ∊ B0 and a prime number p such that

and

(2)

If follows from this result that if |A| ≫ N and |B| ≫ N then there exist a in A and b in B and a prime p such that p2|(a + b) with

In [7, b.2, VIII, 128] Pólya and Szegö state the following theorem of A. Cohn:

THEOREM 1. Let dndn−x … d0 be the decimal representation of a prime. Then

is irreducible.

Thus, for example, since 1289 is prime, x3 + 2x2 + 8x + 9 is irreducible. Brillhart, Odlyzko, and the author generalized Cohn's Theorem in three different directions. As examples of these types of generalizations, we note the following results, the first two of which are special cases of a result in [1] and the third of a result in [3].

THEOREM 2. Let dndn−x … d0 be the base b representation of a prime where b is an integer ≧2. Then

is irreducible.

THEOREM 3. Let

be such that f(10) is prime and 0 ≦ dj ≦ 167 for j = 0, 1, …, n. Then f(x) is irreducible.

Consider the second order neutral differential equation

1

where the coefficients p and q and the deviating arguments τ and σ are real numbers. The characteristic equation of Eq. (1) is

2

The main result in this paper is the following necessary and sufficient condition for all solutions of Eq. (1) to oscillate.

THEOREM. The following statements are equivalent:

• (a) Every solution of Eq. (1) oscillates.

• (b) Equation (2) has no real roots.

Blow up a sphere in one of the interstices of a lattice until it is held rigidly. There will be no lattice points in the interior and sufficiently many on the boundary so that their convex hull is a solid figure. Such a sphere was called an empty sphere by B. N. Delone in 1924 when he introduced his method for lattice coverings [3, 4]. The circumscribed polytope is called an L-polytope. Our interest in such matters stems from the following result [6, Theorems 2.1 and 2.3]: With a list of the L-polytopes for lattices of dimension ≦n one can give a geometrical description of the possible sets of integer solutions of

where f satisfies the following condition (in which Z denotes the integers):

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.

Given a finite group G, the Frattini subgroup of G, Φ(G) is defined to be the intersection of all the maximal subgroups of G. Of late there have been several attempts to consider generalizations of Φ(G). For example, Gaschutz [7] and Rose [13] have investigated the intersection of all non-normal, maximal subgroups of a finite group. Deskins [6] has discussed the intersection of the family of maximal subgroups of a finite group whose indices are co-prime to a given prime. In [4-5, 12] we have considered the investigation of the family of all maximal subgroups of a finite group whose indices are composite and co-prime to a given prime. We have obtained several results about the family . In this paper which is a sequel to [4] we prove some further results about this family indicating the interesting role it plays especially when G is solvable or p-solvable. First we recall the main definition from [4].

Many of the basic results of HP theory on the disk Δ = {|z| < 1} are proved using the Cauchy-Stieltjes representation

1.1

and the Poisson-Stieltjes representation

1.2

Here, μ:R → C is a complex-valued function of a real variable that is of bounded variation on [0, 2π] such that μ(t + 2π) = μ(t) + μ(2t) — μ(0), t ∊ R,

is the Cauchy kernel, and

is the Poisson kernel. It is therefore natural to generalize these representations in such a way that some of the basic properties and results carry over. Such a generalization occurs when the assumption that μ is of bounded variation on [0, 2μ] is replaced by the requirement that it is measurable and bounded on [0, 2μ] (cf. [9]). The integrals in (1.1) and (1.2) are then defined by a formal integration by parts. After some preliminaries in Section 2, we catalogue a variety of results which remain valid in Section 3.

Le but de ce travail est de définir un certain nombre d'objets de Fq((1/t) et de fonctions

qui sont analogues à des objets classiques de R(ou C) et à des fonctions classiques R → R (ou C → C) : par exemple ζ(l), ζ(2), etc., fractions continues, domaine fondamental du “demi-plan” de Poincaré, séries d'Eisenstein, fonctions multiplement périodiques.

Un tel travail avait été entrepris autrefois par E. Artin [2], mais dans un autre esprit, et la fonction zéta que j'ai définie dans [7] n'est pas celle d'Artin, mais celle de Carlitz [3].

L'intersection de la thèse d'Artin et de ce travail n'est donc pas vide, mais elle n'est pas non plus très étendue bien que je me sois limité aux définitions et aux propriétés les plus élémentaires. En revanche, par son esprit, ce travail se rapproche davantage du point de vue de Carlitz.

Throughout the paper, G will denote an additively written, but not always abelian, group of finite order n; and X = (xij) will denote a square matrix of order n with entries from G and whose rows and columns are numbered 0, 1, …, n − 1. We call X a cartesian array (afforded by G) if

(1.1) The sequence {−xmi + xki, i = 0,…, n – 1} contains all elements of G whenever k ≠ m.

By a theorem of Jungnickel (see Theorem 2.2 in [5]), the transpose of a cartesian array is also a cartesian array. We call G a cartesian group if there is a cartesian array X afforded by G. In this case, we also call (G, X) a cartesian pair.

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]).

Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if “normal” is replaced by “Hermitian”, “nonnegative” or “positive”? Recall that an operator T is nonnegative (resp. positive) if (Tx, x) ≧ 0 (resp. (Tx, x) ≥ 0) for any x ≠ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions.

If the space is finite-dimensional, then necessary and sufficient conditions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest such number (cf. [10]).