In this paper, we mainly study operator spaces which have the locally lifting property $\left( \text{LLP} \right)$. The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the $\text{LLP}$. Several homological characterizations of the $\text{LLP}$ and weak expectation property are given. We also prove that for any operator space $V$, ${{V}^{**}}$ has the $\text{LLP}$ if and only if $V$ has the $\text{LLP}$ and ${{V}^{*}}$ is exact.

Motivated by the problem of giving a functorial (or at least uniform) description of the projective representations for wreath products Gʅ Snin terms of those for G, we study a certain binary operation on the class of “cyclic covering groups with parities”. Along with setting up the basic machinery associated to representations graded by (Ζ/2)ℓ , the main result is a description of the irreducibles for in terms of a (tensorlike) product of those for Aand for B.Finally we describe a programme for producing a PSH-algebra theory in this context, analogous to that of Zelevinsky for the case ℓ=0, and that of the author with with Michael Bean (structure) and with John Humphreys (applications) for the case ℓ=1

A space X is said to be Lipschitz 1-connected if every Lipschitz loop 𝛾 : S1 → X bounds a O (Lip(𝛾))-Lipschitz disk f : D2 → X. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.

A semigroup S is a λ- [ρ-, σ-] semigroup if and only if each subsemigroup of S is a left [right, two-sided] ideal of S. Since the concept of ρ-semigroup is the dual of that of λ-semigroup, the results for ρ-semigroups are generally not stated explicitly, σ-semigroups are treated as a special case of λ-semigroups ; in fact, a semigroup S is a σ-semigroup if and only if it is a λ-semigroup and a ρ-semigroup. The purpose of this paper is to determine the structure of λ- [ρ-, σ-] semigroups.

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.

Let T be a (not necessarily bounded) self-adjoint operator on a Hilbert space H with the spectral resolution The set of elements x in H for which ||Etx||2 is absolutely continuous is a subspace, H a, of H which reduces T. (See H almos [1, p. 104]; Kato [2, p. 516].) If H a ≠ 0, the restriction of T to DT ⋂ H a is called the absolutely continuous part of T; in case H = H a, T is said to be absolutely continuous.

If G has nilpotence class c(G) = c, let G = L1(G) > L2 (G)> . . . > Lc+1(G) = 1 and 1 = Z0(G) < Z1(G) < . . . < Zc(G) = G denote the lower central series and upper central series of G respectively. When there is no possibility of confusion we use Li for Li(G) and Zi for Zi(G). Throughout the paper we assume that G is a finite p-group of class greater than two. Let B (c, pr) denote the collection of all G of class c for which Li/Li+i is cyclic of order pr for i = 2, . . . , c and UC(c, pr) the collection of all G of class c for which Zi/Zi-1 is cyclic of order pr for i = 1, . . . , c – 1.

Let PGL(2, ƒ) denote the group of all Moebius transformations

over a field F. The object of this paper is to prove the following theorem.

Theorem 1. G is an S2-group and the centre of G is trivial if and only if G is isomorphic to a group PGL(2, ƒ), char ƒ ≠ 2.

This theorem was proved for finite groups in (1). The present paper extends the result to infinite groups and also improves the method of proof used in that paper. Many of the theorems given there were proved for infinite groups and are used here with appropriate references.

Let F1(x) and F2(x) be two distribution functions, that is, non-decreasing, right-continuous functions such that Fj(— ∞) = 0 and Fj(+ ∞) = 1 (j = 1, 2). We denote their convolution by F(x) so that

the above integrals being defined as the Lebesgue-Stieltjes integrals. Then it is easy to verify (2, p. 189) that F(x) is a distribution function. Let f1(t), f2(t), and f(t) be the corresponding characteristic functions, that is,

Suppose that p(t) > 0, that both p(t) and f(t) are continuous functions on the half-line 0 ≤ t < ∞, and that λ denotes a real parameter. Only real-valued functions will be considered in this paper. Let the differential equation

,

be of the limit-point type (3, p. 238), so that (1) and a linear homogeneous boundary condition

, 0 ≤ α < π,

determine a boundary value problem on 0 ≤ t < ∞ for every fixed α.

Let A = [aij] denote an n-square matrix with entries in the field of complex numbers. Denote by H a subgroup of Sn, the symmetric group on the integers 1, …, n, and by a character of degree 1 on H. Then

is the generalized matrix function of A associated with H and x; e.g., if H = Sn and χ = 1, then the permanent function. If the sequences ω = (ω1, …, ωm) and ϒ = (ϒ1, …, ϒm) are m-selections, m ≦ w, of integers 1, …, n, then A [ω| ϒ] denotes the m-square generalized submatrix [aωiϒj], i, j = 1, …, m, of the n-square matrix A. If ω is an increasing m-combination, then A [ω|ω] is an m-square principal submatrix of A.

In two previous papers [1, 2] we investigated the zeros of certain arithmetic functions. Using units of cubic fields, we also succeeded to construct, almost by accident, and as a by-product so to speak, entirely new and comparatively complicated combinatorial identities. In an interesting paper combinatorialist L. Carlitz [10] proved those identities in an elementary way. In a/m, we had to prove that the units used were fundamental ones.

Encouraged by these results, we took a closer look at this method that had led to the construction of combinatorial identities. Since the latter are such an important tool in mathematics, we thought it would be “einer Messe wert“ to generalize these results and lay the theoretical foundations of a new method for the construction of highly sophisticated combinatorial identities.

We study families of metrics on automorphic vector bundles associated with representations of the modular group. These metrics are defined using an Eisenstein series construction. We show that in certain cases, the residue of these Eisenstein metrics at their rightmost pole is a harmonic metric for the underlying representation of the modular group. The last section of the paper considers the case of a family of representations that are indecomposable but not irreducible. The analysis of the corresponding Eisenstein metrics, and the location of their rightmost pole, is an open question whose resolution depends on the asymptotics of matrix-valued Kloosterman sums.

If A = (aij) is a matrix whose elements are 0's and l's, more briefly a (0, l)-matrix, the trace of A, defined as the sum of the diagonal elements aii and denoted by Tr A, is the number of l's on the main diagonal. Matrices which can be obtained from A by permutation of its rows or of its columns, or both, may be expressed as PA, AQ, or PAQ, respectively, where P and Q are permutation matrices.

Let V denote a ring of subsets of an abstract space X, let R denote the real numbers, and let N denote the positive integers. Denote by a(V, R) (respectively ca(V, R)) the space of real valued, finitely additive (respectively countably additive) functions on the ring V and denote by ab(V, R) the subspace consisting of those members of the space a(V, R) with finite variation on each set in the ring V. Members of the space a(V, R) are referred to as charges and members of the space ab(V, R) are referred to as locally bounded charges. We denote by cab(V, R) the intersection of the spaces ab(V, R) and ca(V, R).

Let D be the unit disk, |z| < 1, and H(D) the Fréchet space of holomorphic functions on D, provided with the topology of uniform convergence on compact subsets of D. If f is meromorphic in D, we denote by

We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field $K$ of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine algebraic group $G$ over $K$, which is discrete in the finite-dimensional case and linearly compact in general. As an example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show that the closure of $\left[ L,\,L \right]$ in $\mathcal{L}(G)$ is algebraic in $\mathcal{L}(G)$.

We also discuss the Hopf algebra of representative functions $H(L)$ of a residually finite dimensional Lie algebra $L$. As an example, we show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$ is connected, then the canonical Hopf algebra morphism from $K\left[ G \right]$ into $H(L)$ is injective if and only if $L$ is algebraically dense in $\mathcal{L}(G)$.

Let us consider the following problem. Let there be v elements x1 , . . . , xv and v sets S1, . . . , Sv such that every set contains exactly k distinct elements and every pair of sets has exactly λ distinct elements in common. To avoid trivial situations we shall in general assume that 0 < λ < k < v — 1. This is known as a v, k, λconfiguration or design. We can give an equivalent characterization of a configuration in terms of a matrix A = [aij], called its incidence matrix, by writing the elements x1 , . . . , xv row and the sets S1, . . . , Sv in a column and setting aij = 1 if xj is in Si and aij = 0 if xj is not.

Upper and lower bounds for the capacity of planar Cantor-like sets are presented. Chebichev polynomials are the principal tool employed in the derivation of these estimates. A necessary and sufficient condition for certain planar Cantor-like sets to have positive capacity is obtained. Related one-sided capacitary estimates for more general Cantor-like sets can be found in [3, pp. 106-109]. Techniques analogous to those used in this paper yield similar results for linear Cantor-like sets which are well-known [2, pp. 150-161]. The use of Chebichev polynomials to obtain these results provides an alternate, possibly more elementary, approach to these linear problems.

In a mixed problem one is required to find a solution of a system of partial differential equations when the values of certain combinations of the derivatives are given on two or more distinct intersecting surfaces. If the differential equations arise from some physical process, the correct boundary conditions are usually apparent. Many particular problems of this type have been solved by special methods such as separation of variables and the method of images. However, no general criterion has been given for what constitutes a correctly set mixed problem. In fact such problems have usually been formulated in connection with hyperbolic differential equations with data prescribed on two surfaces (called the initial and boundary surfaces).