We investigate the location and nature of the spectrum of the fourth-order self-adjoint equation

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subject to certain asymptotic assumptions on the coefficients. The main tools are the theory of asymptotic integration and the Titchmarsh–Weyl M-matrix. Asymptotic integration yields asymptotic formulae for the solutions of the differential equation which are then used to derive properties of the M-matrix. The characterisation of spectral properties in terms of the boundary behaviour of M leads to the desired results.

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

Let f and g be two analytic function germs without common branches. We define the Jacobian quotients of (g, f), which are ‘first order invariants’ of the discriminant curve of (g, f), and we prove that they only depend on the topological type of (g, f). We compute them with the help of the topology of (g, f). If g is a linear form transverse to f, the Jacobian quotients are exactly the polar quotients of f and we affirm the results of D. T. Lê, F. Michel and C. Weber.

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.

This paper concerns a wide class of singular perturbation problems arising from such diverse fields as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. The corresponding elliptic equations in a bounded domain without any symmetry assumptions are studied. It is assumed that the mean curvature of the boundary has M isolated, non-degenerate critical points. Then it is shown that for any positive integer M[les ]M there exists a stationary solution with M local peaks which are attained on the boundary and which lie close to these critical points. The method is based on Lyapunov–Schmidt reduction.

We consider the appearance of discrete spectrum in spectral gaps of magnetic Schrödinger operators with electric background field under strong, localised perturbations. We show that for compactly supported perturbations the asymptotics of the counting function of the occurring eigenvalues in the limit of a strong perturbation does not depend on the magnetic field nor on the background field.

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.

Estimates for the Dirichlet eigenfunctions near the boundary of an open, bounded set in euclidean space are obtained. It is assumed that the boundary satisfies a uniform capacitary density condition.

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.

We prove Tauberian theorems from Jp-summability methods of power series type to ordinary convergence, respectively Mp-summability methods of weighted means. Particular cases are the Abel and Cesàro, as well as logarithmic and harmonic summability. Besides numerical series, we also consider orthogonal series with coefficients from [lscr ]2. In the latter case, it turns out that one of our Tauberian conditions is satisfied almost everywhere on the underlying measure space, thereby proving the claim stated in the title.

An asymptotically perturbed cylinder is a manifold which, to the exterior of a compact set, is of the form R×M, where M is a compact manifold (with or without boundary), and for which the metric approaches the product metric as the axial variable tends to infinity. The properties of eigenfunctions of the Laplacian on asymptotically perturbed cylinders, with either Dirichlet or Neumann boundary conditions if the boundary is non-empty, are studied. For a large class of asymptotic perturbations, the eigenfunctions are proved to decay faster than the reciprocal of any polynomial as the axial variable tends to infinity. The decay estimates are then used to prove that the eigenvalues are of finite multiplicity, and can accumulate at the thresholds only from below. The results are shown to apply to a large class of acoustic and quantum waveguides.

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.

In this paper we shall be dealing with best constants characterising the embedding of a Sobolev space of L2-type Hl=Wl2 into the space of bounded continuous functions when l>n/2. More specifically, we are interested in the value of the best constant cM(p, l) in the inequality

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where M stands for Euclidean space Rn or the n-sphere Sn (in the latter case f is assumed to have zero average (f, 1)=0). Accordingly, Δ is either the classical Laplace operator or the Laplace–Beltrami operator acting on the surface of Sn[ratio ]Δf(s)= Δf(x/[mid ]x[mid ])[mid ]x=s, s∈Sn, n[ges ]2 (on S1, of course, Δf=f″).

Throughout ∥·∥ is the L2-norm, p and l are real numbers satisfying

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and θ=(2l−n)/(2(l−p)), 1−θ=(n−2p)/(2(l−p)).

Before describing the contents of the paper we recall the well-known references [3, 10, 11, 16] and the survey [18] where best constants and corresponding extremal functions of the Sobolev embeddings in Rn were dealt with.

For each permutation f of the set of positive integers, all triples s, t, u are determined such that t and u are the lower and upper limits of the sequence of partial sums of the ‘f-rearrangement’ [sum ]af(n) of some real series [sum ]an with sum s.

Given a smooth, compactly supported hypersurface S in ℝn that does not pass through the origin, and denoting by tS the surface dilated by a factor t>0, we can consider the averaging operator defined for functions f∈[Sscr ], the Schwartz class of functions, by

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where dσ is Lebesgue measure on S. We can now define a maximal average operator,

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In the case where S is Sn−1, the sphere of unit radius in ℝn, we are looking at Stein's spherical maximal function, as treated in his paper [10]. Stein proved that M is a bounded operator on the Lp spaces if and only if p>n/(n−1) when n>3. Subsequently, Bourgain [2] showed that if S is any compactly supported smooth curve with non-vanishing Gaussian curvature, then M will be bounded on Lp, if and only if p>2, thus dealing with the case of the circular maximal function in the plane. (For related results, see also [6] and [7]). In the case where S is a curve whose curvature vanishes to order at most m−2 at a single point, Iosevich [4] showed that M is bounded on Lp for p>m, and unbounded if p = m. If we study curves given by Γ(s) = (s, γ(s)+1), s∈[0, 1], for some suitably smooth γ, where γ(0) = γ′(0) = … = γ(m−1)(0) ≠ γ(m)(0)>0, then we can reinterpret his results as follows. Define

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for Schwartz functions f. Iosevich proved that

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for p>2. If we note that κ(s), the curvature of the curve Γ(s) is approximately 2−k(m−2) whenever s∈[2−k, 21−k], then we have that the operator

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is bounded on Lp for some p>2, if σ is sufficiently large, since

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which is finite so long as σ>(m/p−1) (m−2)−1. If we want to choose σ independent of m>2, the type of the curve, such that Mσ is bounded on Lp for some fixed p>2, then clearly we can take σ = 1/p. In this paper we show that Mσ will be bounded on Lp for p>max{σ−1, s} for a class of infinitely flat, convex curves in the plane. Counterexamples will show that this is the best possible result, in that there exist flat curves for which Mσ is unbounded for 2<p[les ]σ−1.

We give necessary and sufficient conditions on the parameters s1, s2, …, sm, p1, p2, …, pm such that the Jacobian determinant extends to a bounded operator from [Hscr ]s1p1× [Hscr ]s2p2×…× [Hscr ]smpm into [Zscr ]′. Here all spaces are defined on ℝn and 2[les ]m[les ]n. In almost all cases the regularity of the Jacobian determinant is calculated exactly.

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.

Part of the Choquet theory of integral representations on a universal cap is presented here in a slightly improved form. This makes possible a rapid proof of a generalised Bochner theorem for non-separable locally compact groups and of two extensions to such groups of the classical Lévy–Cramér continuity theorem concerning the convergence of sequences of probabilities.

Several differential geometric aspects of the space of representations of amenable Banach algebras are studied: differential structure, natural connection, geodesics, reductive structure, and so on. As an application we get a characterisation of amenable groups in terms of the existence of a reductive structure in that space.

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.