Bulletin of the London Mathematical Society - Advance Access

Spectral characterizations of scalars in a Banach algebra

Braatvedt, G., Brits, R., Raubenheimer, H. — Wed, 18 Nov 2009 05:48:03 PST

For a complex Banach algebra A with unit 1, we give several characterizations of the scalars, that is, multiples of the identity. To a large extent, this work is a continuation and generalization of the work done on characterizations of the radical in Banach algebras. In particular it is shown that if a A has the property that the number of elements in the spectrum of ax is less than or equal to the number of elements in the spectrum of x for all x in an arbitrary neighbourhood of 1, then a is a scalar. Moreover, as a consequence of some of the results, new spectral characterizations of commutative Banach algebras are obtained. In particular, A is commutative if and only if it has the property that the number of elements in the spectrum remains invariant under all permutations of three elements in some neighbourhood of the identity.

Koszul complexes and fully faithful integral functors

Sancho de Salas, F. — Wed, 18 Nov 2009 05:48:03 PST

We characterize those objects in the derived category of a scheme that are a sheaf supported on a closed subscheme in terms of Koszul complexes. This is applied to generalize to arbitrary schemes the fully faithfulness criteria of an integral functor.

Existence and vanishing set of inverse integrating factors for analytic vector fields

Enciso, A., Peralta-Salas, D. — Wed, 18 Nov 2009 05:48:02 PST

In this paper we address the problem of existence of inverse integrating factors for an analytic planar vector field in a neighborhood of its nonwandering sets. It is proved that there always exists a smooth inverse integrating factor in a neighborhood of a limit cycle, obtaining a necessary and sufficient condition for the existence of an analytic one. This condition is expressed in terms of the Ecalle–Voronin modulus of the associated Poincaré map. The existence of inverse integrating factors in a neighborhood of an elementary singularity is also established, and we give the first known examples of analytic vector fields in R² not admitting a C inverse integrating factor in any neighborhood of either a limit cycle or a weak focus. Moreover, it is shown that a C¹ inverse integrating factor of a C¹ planar vector field must vanish identically on the polycycles that are limit sets of its flow, thereby solving a problem posed by García and Shafer (‘Integral invariants and limit sets of planar vector fields’, J. Differential Equations 217 (2005) 363–376).

On integrable solutions of a nonlinear Volterra integral equation under Caratheodory conditions

Banas, J., Chlebowicz, A. — Sun, 01 Nov 2009 22:28:12 PST

In this paper we study the existence of solutions of a nonlinear Volterra integral equation in the space of Lebesgue integrable functions on an unbounded interval. Our existence result is obtained under the assumption that functions involved in the considered integral equation satisfy conditions of Carathéodory type. The main tool used in the investigations of the paper is the combination of the technique of measures of weak noncompactness with the classical Schauder fixed point principle. The obtained result generalizes several ones obtained earlier in many papers and monographs.

Continuous curves from infinite Kempe linkages

Owen, J., Power, S. — Sun, 01 Nov 2009 22:28:12 PST

In 1876 Kempe showed that any algebraic curve in the plane may be realised as the locus of one of the joints of a finite bar-joint linkage. An often cited illustration of this is that there is a linkage that can write a person's signature to any particular accuracy. An infinite analogue is established showing that any continuous curve in the plane is the curve of motion of a joint of an infinite bar-joint linkage. This is curious, as continuous curves can be space filling. Moreover, there is a single infinite linkage that simultaneously traces everybody's signature with no error whatsoever.

Twisted doubles and nonpositive curvature

Aravinda, C. S., Farrell, F. T. — Sun, 01 Nov 2009 22:28:12 PST

We investigate the question of putting nonpositively curved metrics on doubles of finite volume real hyperbolic manifolds M, where we allow the boundary components to be identified via self-diffeomorphisms of M.

Multiplicity of direct sums of operators on Banach spaces

Grivaux, S., Roginskaya, M. — Sun, 01 Nov 2009 22:28:11 PST

Let T be a bounded operator on a complex Banach space X and let T_n be the direct sum T... T of n copies of T acting on X... X. The aim of this paper is to study the sequence (m(T_n))_{n ≥ 1} of the multiplicities of the operators T_n. Answering a question of Atzmon, it is shown that this sequence is either eventually constant or grows to infinity at least as fast as n. Then examples of operators on Hilbert spaces, such that m(T_n) = d for every n ≥ 1, are constructed, where d is an arbitrary positive integer. This answers a question of Herrero and Wogen and characterizes convex sequences that can be realized as a sequence (m(T_n))_{n ≥ 0} for some operator T on a Hilbert space.

Non-commutative Hardy inequalities

Hansen, F. — Wed, 23 Sep 2009 01:34:22 PDT

We extend Hardy's inequality from sequences of non-negative numbers to sequences of positive semi-definite operators if the parameter p satisfies 1 < p ≤ 2, and to operators under a trace for arbitrary p > 1. Applications to trace functions are given. We introduce the tracial geometric mean and generalize Carleman's inequality.

Algebras of differentiable functions on Riemannian manifolds

Garrido, I., Jaramillo, J. A., Rangel, Y. C. — Wed, 23 Sep 2009 01:34:22 PDT

For an infinite-dimensional Riemannian manifold M we denote by C¹_b(M) the space of all real bounded functions of class C¹ on M with bounded derivative. In this paper we shall see how the natural structure of normed algebra on C¹_b(M) characterizes the Riemannian structure of M, for the special case of the so-called uniformly bumpable manifolds. For that we need, among other things, to extend the classical Myers–Steenrod theorem on the equivalence between metric and Riemannian isometries, to the setting of infinite-dimensional Riemannian manifolds.

Koecher-Maass series for positive definite Fourier coefficients of real analytic Siegel-Eisenstein series of degree 2

Mizuno, Y. — Tue, 22 Sep 2009 06:02:36 PDT

It is shown that the Koecher–Maass series for positive definite Fourier coefficients of a real analytic Siegel–Eisenstein series of degree 2 has a meromorphic continuation and a simple functional equation.

Smooth partitions and Chebyshev polynomials

Mansour, T. — Tue, 22 Sep 2009 06:02:36 PDT

A partition of the set [n] = {1, 2, ..., n} is a collection {B₁, ..., B_k} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. A partition of [n] is said to be smooth if i B_s implies that i + 1 B_s–1 B_s B_{s + 1} for all i [n–1] (B₀ = B_{k + 1} = ). This paper presents the generating function for the number of k-block, smooth partitions of [n], written in terms of Chebyshev polynomials of the second kind. There follows a formula for the number of k-block, smooth partitions of [n] written in terms of trigonometric sums. Also, by first establishing a bijection between the set of smooth partitions of [n] and a class of symmetric Dyck paths of semilength 2n–1, we prove that the counting sequence for smooth partitions of [n] is Sloane's A005773.

Maximal totally complex submanifolds of HPn: homogeneity and normal holonomy

Bedulli, L., Gori, A., Podesta, F. — Mon, 21 Sep 2009 02:54:37 PDT

We prove that a maximal totally complex submanifold N²ⁿ of the quaternionic projective space HPⁿ (n ≥ 2) is a parallel submanifold, provided that one of the following conditions is satisfied: (1) N is the orbit of a compact Lie group of isometries; (2) the restricted normal holonomy is a proper subgroup of U(n).

A note on Larsen's conjecture and ranks of elliptic curves

Dokchitser, T., Dokchitser, V. — Sun, 13 Sep 2009 22:02:23 PDT

Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that, for any finitely generated subgroup G of Gal(K/K), the Mordell–Weil rank of E is unbounded in number fields fixed by G. We prove that the conjecture holds over K = Q for both the analytic rank and the p-Selmer rank of E for every odd prime p. For arbitrary E/K, we show that Larsen's conjecture follows from the standard conjectures for ranks of elliptic curves, provided that K has a real place or E has non-integral j-invariant.

On the structure of steps of three-term arithmetic progressions in a dense set of integers

Candela, P. — Sun, 13 Sep 2009 22:02:22 PDT

We use recent results in quadratic Fourier analysis to examine the additive structure of the set of steps (or common differences) of three-term arithmetic progressions in a general subset of [N]={1, 2, ..., N} of fixed positive density. In particular, combining the decomposition results of Gowers and Wolf with the recurrence results of Green and Tao, we show that if A [N] has density > 0, then, for some positive constant c = c(), the set of steps of three-term arithmetic progressions in A contains an arithmetic progression of length at least c(log log N)^c. This improves on the estimate of shape (log log log log log N) that one can obtain by a straightforward application of Gowers’ bounds for Szemerédi's theorem.

Universal convex coverings

Bacher, R. — Thu, 03 Sep 2009 03:49:09 PDT

In every dimension d ≥ 1, we establish the existence of a constant v_d > 0 and of a discrete subset U_d of R^d such that the following holds: C + U_d = R^d for every convex set C R^d of volume at least v_d and such that U_d contains at most log (r)^d–1r^d points at distance at most r from the origin, for every large r.

Hardy inequality and Lp estimates for the torsion function

van den Berg, M., Carroll, T. — Thu, 03 Sep 2009 03:49:09 PDT

It is shown that the torsion function for an open set D in Euclidean space R^m is in L(D) if and only if the spectrum of the Dirichlet Laplacian in D is bounded away from 0. For 1 ≤ p ≤ , it is shown that the torsion function for D is in L^p(D) precisely when the distance to the boundary function is in L^2p(D), if it is assumed that the Dirichlet Laplacian acting in L²(D) satisfies a strong Hardy inequality.

Heat-flow monotonicity related to the Hausdorff-Young inequality

Bennett, J., Bez, N., Carbery, A. — Thu, 03 Sep 2009 03:49:08 PDT

It is known that if q is an even integer, then the L^q(R^d) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres ‘simultaneously slide’ to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if q > 2 is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff–Young inequality.