On the projective-injective modules over cellular algebras

We show that the projective module over a cellular algebra is injective if and only if the socle of coincides with the top of , and this is also equivalent to the condition that the th socle layer of is isomorphic to the th radical layer of for each positive integer . This eases the process of determining the Loewy series of the projective-injective modules over cellular algebras.

     

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 .

     

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 .

     

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).

     

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 linear transformations preserving at least one eigenvalue

Let be an algebraically closed field and be a linear transformation. In this paper we show that if preserves at least one eigenvalue of each matrix, then preserves all eigenvalues of each matrix. Moreover, for any infinite field (not necessarily algebraically closed) we prove that if is a linear transformation and for any with at least an eigenvalue in , and have at least one common eigenvalue in , then preserves the characteristic polynomial.

     

Backward uniqueness for solutions of linear parabolic equations

We address the backward uniqueness property for the equation in . We show that under rather general conditions on and , implies that vanishes to infinite order for all points . It follows that the backward uniqueness holds if and when n/2$">. The borderline case is also covered with an additional continuity and smallness assumption.

     

Cohomology and finite subgroups of profinite groups

We prove two theorems linking the cohomology of a pro- group with the conjugacy classes of its finite subgroups.

The number of conjugacy classes of elementary abelian -subgroups of is finite if and only if the ring is finitely generated modulo nilpotent elements.

If the ring is finitely generated, then the number of conjugacy classes of finite subgroups of is finite.

     

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 .

     

-extensions in characteristic 3

We present an explicit construction of the complete family of Galois extensions of a field of characteristic 3 with Galois group the central product of a double cover of the symmetric group and the quaternion group , containing a given -extension of the field .

     

Symmetric word equations in two positive definite letters

For every symmetric (``palindromic") word in two positive definite letters and for each fixed -by- positive definite and , it is shown that the symmetric word equation has an -by- positive definite solution . Moreover, if and are real, there is a real solution . The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. In some cases, the solution is unique, but, in general, uniqueness is an open question. Applications and methods for finding solutions are also discussed.

     

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.

     

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 .

     

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.

     

Free products in linear groups

Let be a commutative integral domain of characteristic , and let be a finite subgroup of , the projective general linear group of degree over . In this note, we show that if , then also contains the free product , where is the infinite cyclic group generated by the image of a suitable transvection.

     

Arc-analytic roots of analytic functions are Lipschitz

Let be an arc-analytic function (i.e., analytic on every analytic arc) and assume that for some integer the function is real analytic. We prove that is locally Lipschitz; even if is less than the multiplicity of . We show that the result fails if is only a , arc-analytic function (even blow-analytic), . We also give an example of a non-Lipschitz arc-analytic solution of a polynomial equation , where are real analytic functions.

     

Stability of -algebras associated to graphs

We characterize stability of graph -algebras by giving five conditions equivalent to their stability. We also show that if is a graph with no sources, then is stable if and only if each vertex in can be reached by an infinite number of vertices. We use this characterization to realize the stabilization of a graph -algebra. Specifically, if is a graph and is the graph formed by adding a head to each vertex of , then is the stabilization of ; that is, .

     

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.

     

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 .

     

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.