Monotone matrix functions of successive orders

This paper extends a result obtained by Wigner and von Neumann. We prove that a non-constant real-valued function, , in where is an interval of the real line, is a monotone matrix function of order on if and only if a related, modified function is a monotone matrix function of order for every value of in , assuming that is strictly positive on .

     

Bounding edge degrees in triangulated -manifolds

In this note we prove that every closed orientable -manifold has a triangulation in which each edge has degree , or .

     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

Eigenvalue estimates for operators on symmetric Banach sequence spaces

Using abstract interpolation theory, we study eigenvalue distribution problems for operators on complex symmetric Banach sequence spaces. More precisely, extending two well-known results due to König on the asymptotic eigenvalue distribution of operators on -spaces, we prove an eigenvalue estimate for Riesz operators on -spaces with , which take values in a -concave symmetric Banach sequence space , as well as a dual version, and show that each operator on a -convex symmetric Banach sequence space , which takes values in a -concave symmetric Banach sequence space , is a Riesz operator with a sequence of eigenvalues that forms a multiplier from into . Examples are presented which among others show that the concavity and convexity assumptions are essential.

     

Robust transitivity and topological mixing for -flows

We prove that nontrivial homoclinic classes of -generic flows are topologically mixing. This implies that given , a nontrivial -robustly transitive set of a vector field , there is a -perturbation of such that the continuation of is a topologically mixing set for . In particular, robustly transitive flows become topologically mixing after -perturbations. These results generalize a theorem by Bowen on the basic sets of generic Axiom A flows. We also show that the set of flows whose nontrivial homoclinic classes are topologically mixing is not open and dense, in general.

     

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

Let be a nonempty closed convex subset of a real Banach space and be a Lipschitz pseudocontractive self-map of with . An iterative sequence is constructed for which as . If, in addition, is assumed to be bounded, this conclusion still holds without the requirement that Moreover, if, in addition, has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then the sequence converges strongly to a fixed point of . Our iteration method is of independent interest.

     

-adic formal series and primitive polynomials over finite fields

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the Hansen-Mullen conjecture that there exists a primitive polynomial over of degree with the -th ( coefficient fixed in advance except when if is odd and when if is even.

     

Mansfield's imprimitivity theorem for arbitrary closed subgroups

Let be a nondegenerate coaction of on a -algebra , and let be a closed subgroup of . The dual action is proper and saturated in the sense of Rieffel, and the generalised fixed-point algebra is the crossed product of by the homogeneous space . The resulting Morita equivalence is a version of Mansfield's imprimitivity theorem which requires neither amenability nor normality of .

     

Zero product preserving maps of operator-valued functions

Let be locally compact Hausdorff spaces and , be Banach algebras. Let be a zero product preserving bounded linear map with dense range. We show that is given by a continuous field of algebra homomorphisms from into if is irreducible. As corollaries, such a surjective arises from an algebra homomorphism, provided that is a -algebra and is a semi-simple Banach algebra, or both and are -algebras.

     

On Berry-Esseen bounds of summability transforms

Let , , , be a collection of random variables, where for each , , , are independent. Let be a regular summability method. We provide some rates of convergence (Berry-Esseen type bounds) for the weak convergence of summability transform . We show that when is the classical Cesáro summability method, the rate of convergence of the resulting central limit theorem is best possible among all regular triangular summability methods with rows adding up to one. We further provide some summability results concerning -negligibility. An application of these results characterizes the rate of convergence of Schnabl operators while approximating Lipschitz continuous functions.

     

Continuous-trace groupoid crossed products

Let be a second countable, locally compact groupoid with Haar system, and let be a bundle of -algebras defined over the unit space of on which acts continuously. We determine conditions under which the associated crossed product is a continuous trace -algebra.

     

Critical exponents of discrete groups and -spectrum

Let be a noncompact semisimple Lie group and an arbitrary discrete, torsion-free subgroup of . Let be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space , and let be the exponent of growth of . If has rank , then these quantities are related by a well-known formula due to Elstrodt, Patterson, Sullivan and Corlette. In this note we generalize that relation to the higher rank case by estimating from above and below by quadratic polynomials in . As an application we prove a rigiditiy property of lattices.

     

Monoid of self-equivalences and free loop spaces

Let be a simply-connected closed oriented -dimensional manifold. We prove that for any field of coefficients there exists a natural homomorphism of commutative graded algebras where is the loop algebra defined by Chas and Sullivan. As usual denotes the monoid of self-equivalences homotopic to the identity, and the space of based loops. When is of characteristic zero, yields isomorphisms where denotes the Hodge decomposition on .

     

Control of radii of convergence and extension of subanalytic functions

Let : denote a real analytic function on an open subset of , and let denote the points where does not admit a local analytic extension. We show that if is semialgebraic (respectively, globally subanalytic), then is semialgebraic (respectively, subanalytic) and extends to a semialgebraic (respectively, subanalytic) neighbourhood of . (In the general subanalytic case, is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series centred at points in the image of an analytic mapping , in terms of the radii of convergence of at points , where denotes the Taylor expansion of at .

     

Minimal polynomials of elements of order in -modular projective representations of alternating groups

Let be an algebraically closed field of characteristic 0$"> and let be a quasi-simple group with . We describe the minimal polynomials of elements of order in irreducible representations of over . If , we determine the minimal polynomials of elements of order in -modular irreducible representations of , , , , , and .

     

Real rank and squaring mappings for unital -algebras

It is proved that if is a compact Hausdorff space of Lebesgue dimension , then the squaring mapping , defined by , is open if and only if . Hence the Lebesgue dimension of can be detected from openness of the squaring maps . In the case it is proved that the map , from the selfadjoint elements of a unital -algebra into its positive elements, is open if and only if is isomorphic to for some compact Hausdorff space with .

     

Exponential nonnegativity

Let be a Banach algebra, , the spectrum of and the spectral abscissa of . If , then we show that there exists an algebra cone such that is exponentially nonnegative with respect to and the spectral radius is increasing on .

     

Tight wavelet frames generated by three symmetric -spline functions with high vanishing moments

In this note, we show that one can derive from any -spline function of order ( ) an MRA tight wavelet frame in that is generated by the dyadic dilates and integer shifts of three compactly supported real-valued symmetric wavelet functions with vanishing moments of the highest possible order .

     

Self-normalizing Sylow subgroups

Using the classification of finite simple groups we prove the following statement: Let 3$"> be a prime, a group of automorphisms of -power order of a finite group , and a -invariant Sylow -subgroup of . If is trivial, then is solvable. An equivalent formulation is that if has a self-normalizing Sylow -subgroup with 3$"> a prime, then is solvable. We also investigate the possibilities when .

     

-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.