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.

     

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.

     

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 .

     

On the invariance of classes under composition

The necessary and sufficient condition for to be in the class for every of that class whose range is in the domain of is that be in .

     

A concrete description of -spaces as -spaces and its applications

We prove that for a compact Hausdorff space without isolated points, and are isometrically Riesz isomorphic spaces under a certain topology on . Moreover, is a closed subspace of . This provides concrete examples of compact Hausdorff spaces such that the Dedekind completion of is (= the set of all bounded real-valued functions on ) since the Dedekind completion of is ( and spaces as Banach lattices).

     

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 .

     

Spectrally bounded operators on simple -algebras

A linear mapping from a subspace of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant such that for all , where denotes the spectral radius. We prove that every spectrally bounded unital operator from a unital purely infinite simple -algebra onto a unital semisimple Banach algebra is a Jordan epimorphism.

     

-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.

     

Polynomial approximation on real-analytic varieties in

We prove: Let be a compact real-analytic variety in . Assume (i) is polynomially convex and (ii) every point of is a peak point for . Then . This generalizes a previous result of the authors on polynomial approximation on three-dimensional real-analytic submanifolds of .

     

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 .

     

Banach spaces embedding isometrically into when

For we give examples of Banach spaces isometrically embedding into but not into any with

     

Bundles with periodic maps and mod Chern polynomial

Suppose that is a vector bundle with a linear periodic map of period ; the map is assumed free on the outside of the -section. A polynomial , called a mod Chern polynomial of , is defined. It is analogous to the Stiefel-Whitney polynomial defined by Dold for real vector bundles with the antipodal involution. The mod Chern polynomial can be used to measure the size of the periodic coincidence set for fibre preserving maps of the unit sphere bundle of into another vector bundle.

     

Numerical radius distance-preserving maps on

Let be a complex Hilbert space, be the algebra of all bounded linear operators on , be the subset of all selfadjoint operators in and or . Denote by the numerical radius of . We characterize surjective maps that satisfy for all without the linearity assumption.

     

Weakly sequential completeness of the projective tensor product

D. R. Lewis (1977) proved that for a Banach space and , , the projective tensor product of and , is weakly sequentially complete whenever is weakly sequentially complete. In this note, we give a short proof of Lewis's result, based on our sequential representation (2001) of .

     

Transferred Chern classes in Morava -theory

Let be a complex -plane bundle over the total space of a cyclic covering of prime index . We show that for the -th Chern class of the transferred bundle differs from a certain transferred class of by a polynomial in the Chern classes of the transferred bundle. The polynomials are defined by the formal group law and certain equalities in .

     

On -analogues of the Euler constant and Lerch's limit formula

We introduce and study a -analogue of the Euler constant via a suitably defined -analogue of the Riemann zeta function. We show, in particular, that the value is irrational. We also present a -analogue of the Hurwitz zeta function and establish an analogue of the limit formula of Lerch in 1894 for the gamma function. This limit formula can be regarded as a natural generalization of the formula of .

     

A note on the support of a Sobolev function on a -cell

It is shown that a -cell (the homeomorphic image of a closed ball in ) in , , cannot support a function in if [\frac{k+1}{2}]$">, the greatest integer in .

     

The number of Hall -subgroups of a -separable group

We observe a simple formula to compute the number of Hall -subgroups of a -separable finite group in terms of only the action of a fixed Hall -subgroup of on a set of normal -sections of . As a consequence, we obtain that divides whenever is a subgroup of a finite -separable group . This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro's result.

     

On an approximate automorphism on a -algebra

It is shown that for an approximate algebra homomorphism on a Banach -algebra , there exists a unique algebra -homomorphism near the approximate algebra homomorphism. This is applied to show that for an approximate automorphism on a unital -algebra , there exists a unique automorphism near the approximate automorphism. In fact, we show that the approximate automorphism is an automorphism.

     

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 .