We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $\left( X,\,\sum ,\,\mu \right)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group ${{L}^{\infty }}\left( X,G \right)$, which is a Fréchet-Lie group if $G$ is so. We also show that the weak direct product $\prod{_{i\in I}^{*}{{G}_{i}}}$ of an arbitrary family ${{\left( {{G}_{i}} \right)}_{i\in I}}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum ${{\oplus }_{i\in I}}L\left( {{G}_{i}} \right)$ .

Of late there has been considerable interest in the study of analogs of the Frattini subgroup of a finite group and the investigation of their properties, particularly their influence on the structure of the group, see [2-11], [14-16] and [18]. Gaschütz [11] and more recently Bechtell [2] and Rose [18] have considered extensively the intersection of the family of all non-normal, maximal subgroups of a finite group. Deskins [8] has discussed the intersection of the family of all maximal subgroups of a finite group whose indices are not divisible by a given prime. Bhatia [7] considered the intersection of the class of all maximal subgroups of a given group whose indices are composites. In this paper we investigate the intersection of another class of maximal subgroups and its relationship with the structure of the group. The subgroup we consider here contains the Frattini subgroup and also the two subgroups introduced in [8] and [7].

It is known that the following classes of spaces (all spaces in this article are assumed T1) are identical:

1. Images of metric spaces under continuous open maps with compact point inverses.

2. Spaces with uniform bases (in the sense of Alexandrov [1]).

3. Metacompact developable spaces.

4. Spaces with σ-point-finite bases in which closed sets are Gδ's.

We use the eta invariant to show every non-simply connected spherical space form of dimension m ≥ 5 has a countable family of non bordant metrics of positive scalar curvature.

Suppose that two men stand at the same elevation on opposite sides of a mountain range and begin to climb in such a way that their elevations remain equal at all times. Will they ever meet along the way? It is this question, restated in mathematical terms, that we shall consider. We replace the mountain range by the graph of a continuous, real-valued function f(x) defined for x ∈ [0, 1], where f(0) = f(1) = 0, and we ask whether there exist continuous mappings ϕ(t), ψ(t) from [0, 1] into [0, 1] such that

1

2

It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.

Suppose that A and B are C*-algebras, J is a closed two-sided ideal in B, and that η: B →B/J is the quotient map. Given a linear contraction φ : A →B/J, a linear map Ψ: A →B is a lifting of φ if one has a commutative diagram

On dit qu'un nombre entier A est hautement composé si tout nombre M plus petit que A a moins de diviseurs que A. Si l'on définit d(n) = nombre de diviseurs de n, on sait que, si la décomposition en facteurs premiers de n est n = Πi piai, on a:

La définition devient: A est hautement composé si et seulement si:

1

Ramanujan [8] a défini et étudié les nombres hautement composés, démontrant les propriétés suivantes:

Si A = 2a2 3a3 … pkapk est un nombre hautement composé, on a:

et à l'exception de A = 4 et A = 36, on a: apk = 1 [8, §8].

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of ${{L}^{\infty }}\left( \widehat{\mathbb{G}} \right)$ as an operator ${{L}^{1}}\left( \widehat{\mathbb{G}} \right)$-module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $\text{VN}\left( G \right)$ is 1-injective as an operator module over the Fourier algebra $A\left( G \right)$. As an application, we provide a decomposability result for completely bounded ${{L}^{1}}\left( \widehat{\mathbb{G}} \right)$-module maps on ${{L}^{\infty }}\left( \widehat{\mathbb{G}} \right)$, and give a simplified proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

In this paper we study the oscillatory behavior of the even order nonlinear delay differential equation

(1)

where

(i) denotes the order of differentiation with respect to t. The delay terms τi σi are assumed to be real-valued, continuous, non-negative, non-decreasing and bounded by a common constant M on the half line (t0, + ∞ ) for some t0 ≧ 0.

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.

In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.

Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such that

for all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, then

where U is a bounded invertible operator on X and A′ is the adjoint of A.

In this paper we consider the problem of extending the Weierstrass division theorem to quasianalytic local rings of germs of functions of k real variables which properly contain the local ring of germs of analytic functions. After some background material (§ 2) and some technical preliminaries (§ 3), we show by examples that the so-called generic division theorem fails in such rings if k ≧ 1 and that the Weierstrass division theorem fails in such rings if k ≧ 2 (§ 4).

Limit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are q-analogues of the three parameter family of BC type Jacobi polynomials in n variables. The limit transitions will be derived by taking limits of q-difference operators which have these polynomials as eigenfunctions.

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.

In this paper, we develop a new approach to obtain the compactness of the fluctuation processes for Boltzmann dynamics. Our method is applicable to Kac's model, already studied by Uchiyama, but it covers many other cases. A novelty worth mentioning is the use of the weak topology of a Hilbert space.

For a truncated matrix moment problem, we describe in detail the set of positive definite matrices of measures μ in V2n (this is the set of solutions of the problem of degree 2n) for which the polynomials up to degree n are dense in the corresponding space L2(μ). These matrices of measures are exactly the extremal measures of the set Vn.

We investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdös weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), where

α > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < ∞, Δ ∊ ℝ, k > 0. Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then for

to hold for every continuous function ƒ: ℝ —> ℝ satisfying

it is necessary and sufficient that

The problem of the field equations and the equations of motion in general relativity theory is now sufficiently clarified. The equations of motion can be deduced from pure field equations by treating matter as singularities, [2; 3], or from field equations with the energy momentum tensor [4]. Recently two papers appeared in which the problem of the coordinate system was considered [5; 8]. The two papers are in general agreement as far as the role of the coordinate system is concerned. Yet there are some differences which require clarification.

In this paper we study incompressible and injective (see § 2 for definitions) surfaces embedded in M2 × S1, where M2 is a surface and S1 is the 1-sphere. We are able to characterize embeddings which are incompressible in M2 × S1 when M2 is closed and orientable. Namely, a necessary and sufficient condition for the closed surface F to be incompressible in M2 × S1, where M2is closed and orientable, is that there exists an ambient isotopy ht, 0 ≦ t ≦ 1, of M2 × S1onto itself so that either

(i) there is a non-trivial simple closed curve J ⊂ M2 and h1(F) = J × S1, or

(ii) p\h1(F) is a covering projection of h1(F) onto M2, where p is the natural projection of M2 × S1onto M2.

Let G be a subgroup of the general linear group GL(n, Q) over the rational field Q, and consider its action by right multiplication on the vector space Qn of n-tuples over Q. The present paper investigates the question of how we may constructively determine the orbits and stabilizers of this action for suitable classes of groups. We suppose that G is specified by a finite set {x1, …, xr) of generators, and investigate whether there exist algorithms to solve the two problems:

(Orbit Problem) Given u, v ∊ Qn, does there exist x ∊ G such that ux = v; if so, find such an element x as a word in x1, …, xr and their inverses.

(Stabilizer Problem) Given u, v ∊ Qn, describe all words in x1, …, xr and their inverses which lie in the stabilizer