The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

In the Kourovka Notebook [7] Khukhro posed a conjecture on the structure of finite p-groups admitting an automorphism of order p.

Let g be a primitive root modulo a prime p. It is proved that the triples (gx, gy, gxy), x, y = 1, …, p−1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε>0 be fixed. Then

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uniformly for any integers a, b, c with gcd(a, b, c, p) = 1. Incomplete sums are estimated as well.

The question is motivated by the assumption, often made in cryptography, that the triples (gx, gy, gxy) cannot be distinguished from totally random triples in feasible computation time. The results imply that this is in any case true for a constant fraction of the most significant bits, and for a constant fraction of the least significant bits.

The non-existence set forth in the title is proved. It is known that for numerical Campedelli surfaces the algebraic fundamental group is of order [les ]9 and that the dihedral group of order 8 cannot occur. Therefore the quaternionic group is the only non-abelian algebraic fundamental group in this range.

One of the most useful results in modular representation theory of finite groups is Green's indecomposability theorem [4]. In order to state it in a simple form, let us fix a complete discrete valuation ring [Oscr ] of characteristic 0 with algebraically closed residue field [ ] of characteristic p≠0. Unless stated otherwise, all our modules will be free of finite rank over [Oscr ] or [ ], respectively. In its most popular form, Green's theorem says the following.

The aim of this paper is to determine the minimal polynomials of p-elements in irreducible representations of quasi-simple groups with a cyclic Sylow p-subgroup. We consider both the characteristic 0 and characteristic p cases; we have no results for representations over fields of other characteristics. It suffices to describe the cases where the degree of the polynomial is less than the order of the element.

An interpretation is given of the Shintani descent from the points of a connected algebraic group over a finite field to its group of points over a subfield in terms of a Shintani descent between two different rational structures over the same finite field, using restriction of scalars. From this a generalization is deduced of a result of Gyoja's on the Shintani descent of Deligne–Lusztig characters of a finite group of Lie type.

The fine topology on ℝn (n[ges ]2) is the coarsest topology for which all superharmonic functions on ℝn are continuous. We refer to Doob [11, 1.XI] for its basic properties and its relationship to the notion of thinness. This paper presents several theorems relating the fine topology to limits of functions along parallel lines. (Results of this nature for the minimal fine topology have been given by Doob – see [10, Theorem 3.1] or [11, 1.XII.23] – and the second author [15].) In particular, we will establish improvements and generalizations of results of Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihl and Seidel [6], Schneider [21], Berman [7], and Armitage and Nelson [4], and will also solve a problem posed by the latter authors.

An early version of our first result is due to Evans [12, p. 234], who proved that, if u is a superharmonic function on ℝ3, then there is a set E⊆ℝ2×{0}, of two-dimensional measure 0, such that u(x, y,·) is continuous on ℝ whenever (x, y, 0)∉E. We denote a typical point of ℝn by X=(X′ x), where X′∈ℝn−1 and x∈ℝ. Let π:ℝn→ℝn−1×{0} denote the projection map given by π(X′, x) = (X′, 0). For any function f:ℝn→[−∞, +∞] and point X we define the vertical and fine cluster sets of f at X respectively by

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and

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Sets which are open in the fine topology will be called finely open, and functions which are continuous with respect to the fine topology will be called finely continuous. Corollary 1(ii) below is an improvement of Evans' result.

Working over an algebraically closed, complete, non-Archimedean, nontrivially valued field of characteristic zero, it is shown that (i) any subanalytic subset of a one-dimensional semialgebraic set is semialgebraic, (ii) any one-dimensional subanalytic set is semianalytic, and (iii) any subanalytic subset of a two-dimensional semianalytic set is semianalytic.

Consider an analytic germ f:(Cm, 0)→ (C, 0) (m[ges ]3) whose critical locus is a 2-dimensional complete intersection with an isolated singularity (icis). We prove that the homotopy type of the Milnor fiber of f is a bouquet of spheres, provided that the extended codimension of the germ f is finite. This result generalizes the cases when the dimension of the critical locus is zero [8], respectively one [12]. Notice that if the critical locus is not an icis, then the Milnor fiber, in general, is not homotopically equivalent to a wedge of spheres. For example, the Milnor fiber of the germ f:(C4, 0)→(C, 0), defined by f(x1, x2, x3, x4) = x1x2x3x4 has the homotopy type of S1×S1×S1. On the other hand, the finiteness of the extended codimension seems to be the right generalization of the isolated singularity condition; see for example [9–12, 17, 18].

In the last few years different types of ‘bouquet theorems’ have appeared. Some of them deal with germs f:(X, x)→(C, 0) where f defines an isolated singularity. In some cases, similarly to the Milnor case [8], F has the homotopy type of a bouquet of (dim X−1)-spheres, for example when X is an icis [2], or X is a complete intersection [5]. Moreover, in [13] Siersma proved that F has a bouquet decomposition F∼F0∨Sn∨…∨Sn (where F0 is the complex link of (X, x)), provided that both (X, x) and f have an isolated singularity. Actually, Siersma conjectured and Tibăr proved [16] a more general bouquet theorem for the case when (X, x) is a stratified space and f defines an isolated singularity (in the sense of the stratified spaces). In this case F∼∨iFi, where the Fi are repeated suspensions of complex links of strata of X. (If (X, x) has the ‘Milnor property’, then the result has been proved by Lê; for details see [6].)

In our situation, the space-germ (X, x) is smooth, but f has big singular locus. Surprisingly, for dim Sing f−1(0)[les ]2, the Milnor fiber is again a bouquet (actually, a bouquet of spheres, maybe of different dimensions). This result is in the spirit of Siersma's paper [12], where dim Sing f−1(0) = 1. In that case, there is only a rather small topological obstruction for the Milnor fiber to be homotopically equivalent to a bouquet of spheres (as explained in Corollary 2.4). In the present paper, we attack the dim Sing f−1(0) = 2 case. In our investigation some results of Zaharia are crucial [17, 18].

The positivity of the partial sums of the series of Legendre polynomials is a classical result of Fejér. The paper presents corresponding results for both even and odd Legendre polynomials with best possible leading constants. These results have applications to demonstrating positivity of Cotes numbers at Jacobi abscissas.

Spatial concavity properties of non-negative weak solutions of the filtration equations with absorption ut = (φ(u))xx−ψ(u) in Q = R×(0, ∞), φ′[ges ]0, ψ[ges ]0 are studied. Under certain assumptions on the coefficients φ, ψ it is proved that concavity of the pressure function is a consequence of a ‘weak’ convexity of travelling-wave solutions of the form V(x, t) = θ(x−λt+a). It is established that the global structure of a so-called proper set [Bscr ] = {V} of such particular solutions determines a property of B-concavity for more general solutions which is preserved in time. For the filtration equation ut = (φ(u))xx a semiconcavity estimate for the pressure, vxx[les ](t+τ)−1θ″(ξ), due to the B-concavity of the solution to the subset [Bscr ] of the explicit self-similar solutions θ(x/√(t+τ)) is proved.

The analysis is based on the intersection comparison based on the Sturmian argument of the general solution u(x, t) with subsets [Bscr ] of particular solutions. Also studied are other aspects of the B-concavity/convexity with respect to different subsets of explicit solutions.

Let us suppose that

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with

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converges for s>0. If the series converges to sum A for s = 0, then f(s)→A as s→0+, but the converse will not hold without an additional, Tauberian, hypothesis. The usual form of such a hypothesis is a condition on {an}, but in the ‘high-indices’ theorem of Hardy and Littlewood [4], the hypothesis is a condition on {λn}. They showed that if

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then the existence and finiteness of lims→0+f(s) imply the convergence of [sum ]an.

The condition (1.3) is usually referred to as the ‘Hadamard gap condition’ and a series (1.1) such that {λn} satisfies (1.3) is called a lacunary series, although this terminology is occasionally used for series satisfying other gap conditions.

Zygmund has shown [15] that if the function f(s), defined as in (1.1) by a lacunary series, is of bounded variation in (0, ∞), then [sum ]an converges absolutely. Clearly, the converse implication holds.

The result of Zygmund has the form

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an uppercase letter with subscripts here denoting, as it will throughout this paper, a constant determined by the indicated subscripts.

The result of Zygmund and that of Hardy and Littlewood are the extreme cases, corresponding to m = 1 and m = ∞ of the inequality

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which holds for m[ges ]1 and which, with more general results, is to be found in the first part of this paper [8].

In all of these results, the interval of integration can be reduced to a finite interval (0, Bq,λ1).

In this paper we study sequence spaces that arise from the concept of strong weighted mean summability. Let q = (qn) be a sequence of positive terms and set Qn = [sum ]nk=1qk. Then the weighted mean matrix Mq = (ank) is defined by

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It is well known that Mq defines a regular summability method if and only if Qn→∞. Passing to strong summability, we let 0<p<∞. Then

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are the spaces of all sequences that are strongly Mq-summable with index p to 0, strongly Mq-summable with index p and strongly Mq-bounded with index p, respectively. The most important special case is obtained by taking Mq = C1, the Cesàro matrix, which leads to the familiar sequence spaces

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respectively, see [4, 21]. We remark that strong summability was first studied by Hardy and Littlewood [8] in 1913 when they applied strong Cesàro summability of index 1 and 2 to Fourier series; orthogonal series have remained the main area of application for strong summability. See [32, §6] for further references.

When we abstract from the needs of summability theory certain features of the above sequence spaces become irrelevant; for instance, the qk simply constitute a diagonal transform. Hence, from a sequence space theoretic point of view we are led to study the spaces

In a previous paper, the authors laid the foundations of a theory of Schatten–von Neumann classes [Gfr ]p (0<p[les ]∞) of trilinear forms in Hilbert space. This paper continues that research. In the n-dimensional case, it is shown that the best constant dˆ that relates the Hilbert–Schmidt norm of a form with its bounded norm behaves like n. Some results are also obtained in the quasi-Banach case (0<p<1), and for two-bounded forms. Finally, the domination problem is investigated in the trilinear set-up.

It is proved that a cosine operator function C(·), with generator A, is locally of bounded semivariation if and only if u″(t) = Au(t)+f(t), t>0, u(0), u′(0)∈D(A), has a strong solution for every continuous function f, if and only if the function ∫t0∫t−s0C(τ) f(s)dτds, t>0, is twice continuously differentiable for every continuous function f, that is, C(·) has the maximal regularity property if and only if A is a bounded operator. Some other characterisations of bounded generators of cosine operator functions are also established in terms of their local semivariations.

Let [Lscr ]1 and [Lscr ]2 be finite distributive subspace lattices on real or complex Banach spaces. It is shown that every rank-preserving algebraic isomorphism of Alg[Lscr ]1 onto Alg[Lscr ]2 is quasi-spatially induced. If the algebraic isomorphism in question is known only to preserve the rank of rank one operators, then it induces a lattice isomorphism between [Lscr ]1 and [Lscr ]2.

Two main results are shown: (1) a characterisation of the symmetric sequence spaces which embed into C(ωω), and (2) a characterisation, in terms of the Orlicz function M, of the Orlicz sequence spaces hM which embed into C(K) for some countable compact Hausdorff space K. The proofs make use of the Schreier families of sets, which have found wide applications in Banach space theory.

It is shown that for certain discrete p-perfect groups G, in particular for all p-perfect finite groups and certain groups of finite virtual cohomological dimension, the loop space on the p-completed classifying space ΩBG∧p is a retract of the loop space on the p-completion of a certain finite complex. For those finite virtual cohomological dimension groups where the author's methods apply, a bound on the dimension of such a complex is provided.

In 1903 Minkowski showed that, given pairwise different unit vectors μ1, …, μm in Euclidean n-space ℝn which span ℝn, and positive reals μ1, …, μm such that [sum ]mi=1μiμi = 0, there exists a polytope P in ℝn, unique up to translation, with outer unit facet normals μ1, …, μm and corresponding facet volumes μ1, …, μm. This paper deals with the computational complexity of the underlying reconstruction problem, to determine a presentation of P as the intersection of its facet halfspaces. After a natural reformulation that reflects the fact that the binary Turing-machine model of computation is employed, it is shown that this reconstruction problem can be solved in polynomial time when the dimension is fixed but is #ℙ-hard when the dimension is part of the input.

The problem of ‘Minkowski reconstruction’ has various applications in image processing, and the underlying data structure is relevant for other algorithmic questions in computational convexity.