We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes.

As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

1. In this note a definite integral involving the product of Meijer's G-function and a generalized hypergeometric function is evaluated with the help of finite difference operators E. As the generalized hypergeometric function and Meijer's G-function may be converted into a number of polynomials and higher transcendental functions, the integral leads to a generalization of many results. The results given by Agrawal ((l), integral 4), Sharma ((7), integral 1) in these proceedings, Bhonsle ((2), p. 188) and Saxena ((6), p. 198) follow as particular cases of our formula. In what follows v, μ, m and h are positive integers and either of v and μ may be zero. The symbol Δ(m, a) denotes a set of m parameters a/m, (a + 1 )/m, …, (a + m − 1 )/m.

For C*-algebras A1, A2 the map (I1, I2) → ker(qI1 ⊗ qI2) from Id′(A1) × Id′(A2) into Id′(A1 ⊗minA2) is a homeomorphism onto its image which is dense in the range. Here, for a C*-algebra A, the space of all proper closed two sided ideals endowed with an adequate topology is denoted Id′(A) and qI is the quotient map of A onto A/I. This result is used to show that any continuous function on Prim(A1) × Prim(A2) with values into a T1 topological space can be extended to Prim(A1 ⊗minA2). This enlarges the scope of [7, corollary 3·5] that dealt only with scalar valued functions. A new proof for a result of Archbold [3] about the space of minimal primal ideals of A1 ⊗minA2 is obtained also by using the homeomorphism mentioned above. New proofs of the equivalence of the property (F) of Tomiyama for A1 ⊗minA2 with certain other properties are presented.

Necessary and sufficient conditions for sequence-to-sequence or sequence-to-function summability method to include (R, λ, α), when 1 < α ≤ 2, are given. Also, for suitably restricted sequences λ, necessary and sufficient conditions for a series-to-sequence or series-to-function summability method to include (R, λ, α) for 1 < α ≤ 2 are given. These results are obtained by showing that a certain sequence {δj} (j ≥ 0) is a Schauder-basis in Rλα(N) for each α, 1 < α ≤ 2.

This paper shows that the localization theory for algebras of smooth functions and Fréchet modules provides a basis for a theory of differential spaces of finite type, as well as the usual localization theory for commutative rings and modules grounds the theory of schemes. Moreover, most of the elementary results and proofs in the theory of schemes remain valid for differential spaces.

Suppose that λ > − 1 and that

A point in Baire space is found for which the first derived $\omega$-limit set is not Borel, whilst the second is empty. A second point is found for which the sequence of derived $\omega$-limit sets does not stabilise until the first uncountable ordinal. The two points are recursive.

Summary. The main result of the present paper is contained in the following: Theorem ℜ α ≠ −1, −2, −3,…

where2F1[.,.;.;.] and 1f1[.;.;.] (l).

In this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$. We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if $K_1,K_2, \phi$ are as above then there exists an open set S so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.

The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial.

In this paper we directly relate the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. We do this by moreover relating the location of zeros to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios.

We study intermediate-scale statistics for the fractional parts of the sequence $\left(\alpha a_{n}\right)_{n=1}^{\infty}$, where $\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}\!\left(L,\alpha\right)$ in a random interval of length $L/N$, where $L=O\!\left(N^{1-\epsilon}\right)$, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $\alpha$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in $\alpha\in\mathbb{R}$ when $L=O\!\left(N^{1/2-\epsilon}\right)$. For slowly growing L, we further prove a central limit theorem for $S_{N}\!\left(L,\alpha\right)$ which holds for almost all $\alpha\in\mathbb{R}$.

We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type $E_8$ , $F_4$ , and $G_2$ . In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly $\textrm{SL}_{2n+1}(\mathbb{R})$ , $\textrm{SL}_{2n+1}(\mathbb{C})$ , $\textrm{SL}_n(\mathbb{H})$ , or groups of type $E_6$ .

Let C be a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation defining C up to isomorphism. In this paper we consider genus one equations of degree n defining C, namely a (generalised) binary quartic when n = 2, a ternary cubic when n = 3 and a pair of quaternary quadrics when n = 4. In general, minimal genus one equations of degree n are not unique up to isomorphism. We explain how the number of these equations varies according to the Kodaira symbol of the Jacobian of C. Then we count these equations up to isomorphism over a number field of class number 1.

The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, ‖ ‖E) is 2-convex, then there is another Banach sequence space (F, ‖ ‖F) such that ‖x;‖ = ‖x2‖F for all x ε E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.

The Abel sum of the series can be written in the form

where

This suggests that we should define the “Abel limit”, as u → ∞, of any function A(u) as being given by the expression (1) whenever this exists. We shall, however, in this paper, restrict ourselves to functions A(u) which are bounded in any finite interval. Since we are concerned only with the behaviour of A(u) as u → ∞, this does not involve any serious loss of generality, while we avoid difficulties arising from the divergence of integrals at finite points. We note that the expression (1) can be written in the form

where

A1(u) may conveniently be described as the “Abel transform” of A(u).

The current controversy over the interpretation of radio-stellar counts shows, in my view, that in large-scale physics as well as on the very small scale the idea of size, as applied to distributions of matter, has broken down. I shall expand this view.

1. Introduction: In what follows, X will be a real inner-product linear space (inner product (x, y), norm |x| = (x, x)½); En will be n-dimensional real Euclidean space. We shall be dealing with sets {as: 1 ≤ s ≤ rk; r ≥ 1, k ≥ 2} of elements of X; we write a for the vector (a1; …, ark), and put . We suppose that |as| ≤ ∈ (< 0) for every s.

In Part I of this series it was shown that in a metal with overlapping energy bands, one of which is nearly full and the other nearly empty, there exists a critical temperature below which spontaneous magnetization will be present, Tc being given approximately by the equations

where kθ is the exchange energy term postulated by Stoner, b is the ratio of excess vacancies in the d-band to the maximum possible number of parallel spins, and ζd0 is the equivalent energy depth of the vacant levels in the d-band.