This paper completes a study, begun in [7], of conditions under which a generalized Hankel conjugate transformation H ƛ is bounded between a pair of μα -rearrangement invariant function spaces, the measure μα being defined by dμα(t) = t α∼1dt. Examples of such spaces are the Lp(μα) of Lebesgue and generalizations of them due respectively to Orlicz and Lorentz.

We give upper bounds on the modulus of the values at $s\,=\,1$ of Artin $L$ -functions of abelian extensions unramified at all the infinite places. We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for $\text{CM}$ -fields. For example, we will reduce the determination of all the non-abelian normal $\text{CM}$ -fields of degree 24 with Galois group $\text{S}{{\text{L}}_{\text{2}}}\left( {{F}_{3}} \right)$ (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of 23 such $\text{CM}$ -fields.

Explicit solutions to the recurrence relation for associated continuous Hahn polynomials are derived using 3 F 2 contiguous relations. These solutions are used to obtain a new continued fraction and the associated absolutely continuous measure. An exceptional case is shown to yield entry 33 in Chapter 12 of Ramanujan's second notebook.

We study the Bishop-Phelps-Bollobàs property $\left( \text{BPBp} \right)$ for compact operators. We present some abstract techniques that allow us to carry the $\text{BPBp}$ for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$ and $Y$ be Banach spaces. If $\left( {{c}_{0}},Y \right)$ has the $\text{BPBp}$ for compact operators, then so do $\left( {{C}_{0}}\left( L \right),Y \right)$ for every locally compact Hausdorff topological space $L$ and $\left( X,\,Y \right)$ whenever ${{X}^{*}}$ is isometrically isomorphic to ${{\ell }_{1}}$ . If ${{X}^{*}}$ has the Radon-Nikodým property and $\left( {{\ell }_{1}}\left( X \right),\,Y \right)$ has the $\text{BPBp}$ for compact operators, then so does $\left( {{L}_{1}}\left( \mu ,X \right),\,\,Y \right)$ for every positive measure $\mu $ ; as a consequence, $\left( {{L}_{1}}\left( \mu ,X \right),\,\,Y \right)$ has the $\text{BPBp}$ for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X={{c}_{0}}$ or $X={{L}_{p}}\left( v \right)$ for any positive measure $v$ and $1\,<\,p\,<\,\infty $ . For $1\,\le p\,<\,\infty$ , if $\left( X,{{l}_{p}}(Y) \right)$ has the $\text{BPBp}$ for compact operators, then so does $\left( X,{{L}_{p}}\left( \mu ,\,Y \right) \right)$ for every positive measure $\mu $ such that ${{L}_{1}}\left( \mu\right)$ is infinite-dimensional. If $\left( X,\,Y \right)$ has the $\text{BPBp}$ for compact operators, then so do $\left( X,\,{{L}_{\infty }}\left( \mu ,\,\,Y \right) \right)$ for every $\sigma $ -finite positive measure $\mu $ and $\left( X,\,C\left( K,\,Y \right) \right)$ for every compact Hausdorff topological space $K$ .

In the theory of hyperbolic differential equations a mixed boundary value problem involves two types of auxiliary conditions which may be described as initial and boundary conditions respectively. The problem of Cauchy, in which only initial conditions are present, has been studied in great detail, starting with the early work of Riemann and Volterra, and the well-known monograph of Hadamard (4). A modern treatment of great generality has been given by Leray (7).

C* -convex sets in matrix algebras are convex sets of matrices in which matrix-valued convex coefficients are admitted along with the usual scalar-valued convex coefficients. A Carathéodory-type theorem is developed for C *-convex hulls of compact sets of matrices, and applications of this theorem are given to the theory of matricial ranges. If T is an element in a unital C*-algebra , then for every n ∈ N, the n x n matricial range Wn (T) of T is a compact C * -convex set of n x n matrices. The basic relation W 1(T) = conv σ-(T) is well known to hold if T exhibits the normal-like quality of having the spectral radius of β T + μ 1 coincide with the norm ||β T + μ 1|| for every pair of complex numbers β and μ. An extension of this relation to the matrix spaces is given by Theorem 2.6: Wn (T) is the C *-convex hull of the n x n matricial spectrum σn(T) of T if, for every B,M ∈ ℳn, the norm of T ⊗ B + 1 ⊗ M in ⊗ ℳn is the maximum value in {||∧⊗B + 1 ⊗M|| : Λ ∈ σn (T)}. The spatial matricial range of a Hilbert space operator is the analogue of the classical numerical range, although it can fail to be convex if n > 1. It is shown in § 3 that if T has a normal dilation N with σ (N) ⊂ σ (T), then the closure of the spatial matricial range of T is convex if and only if it is C *-convex.

The fact that the perimeter S(a, b) of an ellipse is not an elementary function of its semiaxes a, b has led to many suggested approximations of S in finite form.

Let G be a locally compact group and M(G) the space of finite regular Borel measures on G. If μ and v are in M(G), their convolution is defined by

Thus, if f is a continuous bounded function on G,

μ is central if μ(Ex) = μ(xE) for all x ∈ G and all measurable sets E. μ is idempotent if μ * μ = μ.

The idempotent measures for abelian groups have been classified by Cohen [1]. In this paper we will show that for a certain class of compact groups, containing the unitary groups, the central idempotents can be characterized. The method consists of showing that, in these cases, the central idempotents arise from idempotents on abelian groups and applying Cohen's result.

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].

We construct an automorphic realization of the global minimal representation of quaternionic exceptional groups, using the theory of Eisenstein series, and use this for the study of theta correspondences.

In a recent paper [5], general methods were described for the dissection of a square into a finite number n of unequal non-overlapping squares. In this note, examples of such perfect squares are given in which the sides and elements are relatively small integers; in particular, a dissection of a square into 24 different elements, which is believed to be the squaring of least order known at the present time.

In this paper we will be concerned with the determinants of matrices whose elements are 0, 1 or —1, 1. Accordingly, let Sn,k be the set of n X n 0, 1 matrices with exactly k ones in each row and column; and let Hn be the set of n X n — 1, 1 matrices. Let J = Jn denote (as usual) the n X n matrix all of whose elements are one.

Let G be a finite p-group, having a faithful character χ of degree f. The object of this paper is to bound the number, d(G), of generators in a minimal generating set for G in terms of χ and in particular in terms of f. This problem was raised by D. M. Goldschmidt, and solved by him in the case that G has nilpotence class 2.

We solve two problems in representation theory for the periplectic Lie superalgebra $\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$ into indecomposable blocks.

To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$ for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.

Let G be a finite group and A a Z[G] module.

Definition (1.1). A simply connected CW complex X is of type (A, n) if G operates on X cellularly, and modules.

1. Introduction. It is the object of this paper to investigate the function γ(m), the number of representations of m in the form

(1)

where . It is shown that γ(m) is always equal to the number of odd divisors of m, so that for example γ(2k ) = 1, this representation being the number 2k itself. From this relationship the average order of γ(m) is deduced ; this result is given in Theorem 2. By a method due to Kac [2], it is shown in §3 that the number of positive integers for which γ(m) does not exceed a rather complicated function of n and ω, a real parameter, is asymptotically nD(ω), where D(ω) is the probability integral

Our aim is to study the mixing sequences of a weak mixing transformation. An ergodic measure preserving transformation is weak mixing if and only if for each pair of sets there exists a sequence of density one on which the transformation mixes the sets [9]. An unpublished result of S. Kakutani implies there actually exists a single sequence of density one on which the transformation is mixing for all sets (see Section 3). This result motivated the general définition of a transformation being mixing on a sequence, as well as mixing of higher order on a sequence. Given a weak mixing transformation, there exist sequences along which it is mixing of all degrees. In particular, this is the case for an eventually independent sequence [7].

In Section 3 it will be shown that if T is weak mixing but not mixing, then a sequence on which T is two-mixing must have upper density zero.

A function f, analytic in the unit disk, is said to have finite Dirichlet integral if

1

Geometrically, this is equivalent to f mapping the disk onto a Riemann surface of finite area. The class of Dirichlet integrable functions will be denoted by . The condition above can be restated in terms of Taylor coefficients; if f(z) = Σanzn , then if and only if Σn|an| 2 < ∞. Thus, is contained in the Hardy class H 2.

In particular, every such function has boundary values

almost everywhere and log |f(eiθ)| ∊ L 1(dθ).

The zeros zn of a function must satisfy the Blaschke condition

and f(s) = B(z)F(z), where F(z) has no zeros and

is the Blaschke product with zeros zn ; see (5).

One aspect of the study of 3-manifolds is to determine what finite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action? In the case of periodic transformations of homology spheres, Smith [18] classified the types of fixed point sets which could occur. For homology 3-spheres the fixed point set will be ∅, S 0, S 1, or S 2. Fox [4] looked at periodic transformations of the three sphere which leave a knot invariant and, using Smith's classification of fixed point sets, determined that there were eight types of transformations according to how the fixed point set met the knot. For convenience we shall say a knot is (a, b)-periodic if there is a periodic transformation of S 3 leaving the knot invariant with fixed point set homeomorphic to a and with the fixed point set meeting the knot in a set homeomorphic to b.

Cet article contient une classification détaillée des symétries des C*-algèbres uniformément hyperfinies. Soient une symétrie de ; on associe à θ un entier généralisé N(θ), un réel Tr (θ) ∈ [0, 1], et on a les résultats suivants:

Il existe une symétrie unique à conjugaison près qui possède une suite centrale (U n) d'unitaires vérifiant ‖θ(U n) + U n ‖ → 0(n → ∞). Cette symétrie est la seule dont l'algèbre des points fixes est uniformément hyperfinie.