Fusion in the character table

Suppose that is a Sylow -subgroup of a finite -solvable group . If , then the number of -conjugates of in can be read off from the character table of .

     

Optimal estimation of shell thickness in Cutland's construction of Wiener measure

In Cutland's construction of Wiener measure, he used the product of Gaussian measures on , where is an infinite integer. It is mentioned by Cutland and Ng that for the product measure ,

where and with any positive infinite number. We prove here that may be replaced by with any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that . And for the *Lebesgue measure , is finite and not infinitesimal iff with finite, while for the *Lebesgue area of the sphere , should be .

     

Average curvature of convex curves in

A well-known result states that, if a curve in has geodesic curvature less than or equal to one at every point, then is embedded. The converse is obviously not true, but the embeddedness of a curve does give information about the curvature. We prove that, if is a convex embedded curve in , then the average curvature (curvature per unit length) of , denoted , satisfies . This bound on the average curvature is tight as for a horocycle.

     

Neither first countable nor Cech-complete spaces are maximal Tychonoff connected

A connected Tychonoff space is called maximal Tychonoff connected if there is no strictly finer Tychonoff connected topology on . We show that if is a connected Tychonoff space and locally separable spaces, locally \v{C}ech-complete spaces, first countable spaces, then is not maximal Tychonoff connected. This result is new even in the cases where is compact or metrizable.

     

Degrees of high-dimensional subvarieties of determinantal varieties

Let , where is a prime, and . In , let be the variety defined by . We show that any subvariety of of codimension less than must have degree a multiple of . We also show that the bounds on the codimension in our results are strict by exhibiting subvarieties of the appropriate codimension whose degrees are prime to .

     

Microlocal analysis of ultradistributions

The ultradistributional wave front sets of an ultradistribution are characterized by the behaviour of on the boundary of the tube domain , where is the kernel analysed by Hörmander.

     

Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

We analyze the stability of Muckenhoupt's and classes of weights under a nonlinear operation, the -operation. We prove that the dyadic doubling reverse Hölder classes are not preserved under the -operation, but the dyadic doubling classes are preserved for . We give an application to the structure of resolvent sets of dyadic paraproduct operators.

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

A characterization for spaces of sections

The space of smooth sections of a bundle over a compact smooth manifold can be equipped with a manifold structure, called an -manifold, where represents the Fréchet algebra of real valued smooth functions on . We prove that the -manifold structure characterizes the spaces of sections of bundles over and its open subspaces. We also describe the -maps between -manifolds.

     

Quasipositive plumbing (constructions of quasipositive knots and links, V)

A Seifert surface is a fiber surface if a push-off induces a homotopy equivalence; roughly, is quasipositive if pushing into produces a piece of complex plane curve. A Murasugi sum (or plumbing) is a way to fit together two Seifert surfaces to build a new one. Gabai proved that a Murasugi sum is a fiber surface iff both its summands are; we prove the analogue for quasipositive Seifert surfaces. The slice (or Murasugi) genus of a link is the least genus of a smooth surface bounded by . By the local Thom Conjecture, if is quasipositive; we derive a lower bound for for any Seifert surface , in terms of quasipositive subsurfaces of .

     

A counterexample to a question of R. Haydon, E. Odell and H. Rosenthal

We give an example of a compact metric space , an open dense subset of , and a sequence in which is pointwise convergent to a non-continuous function on , such that for every there exists with for all , yet is equivalent to the unit vector basis of the James quasi-reflexive space of order 1. Thus does not embed isomorphically in the closed linear span of . This answers in the negative a question asked by H. Haydon, E. Odell and H. Rosenthal.

     

On an Isomorphism Problem for Endomorphism Near-Rings

Suppose and are endomorphism near-rings generated by
groups of automorphisms containing the inner automorphisms of two respective finite perfect groups and . In this note we show that if and are isomorphic, then and are isomorphic.

     

Continuous multiplicative mappings on

Let and be compact Hausdorff topological spaces, and let and be real Banach algebras of all real-valued continuous functions on and , respectively. The general form of continuous multiplicative mappings is given.

     

Zero divisors and

Let be a discrete group, the group ring of over and the Lebesgue space of with respect to Haar measure. It is known that if is torsion free elementary amenable, and , then . We will give a sufficient condition for this to be true when , and in the case we will give sufficient conditions for this to be false when .

     

Closure ordering and the Kostant-Sekiguchi correspondence

Let be a real semisimple Lie group with Lie algebra . The Kostant-Sekiguchi correspondence is a bijection between nilpotent orbits on and nilpotent orbits on . In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant-Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.

     

Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

Derivations mapping into the socle, II

Let be a unital Banach algebra. Several characterizations of continuous derivations of having only elements with finite spectrum in their ranges are obtained. In particular, if is semisimple, then such derivations map into its socle.

     

On uniqueness of invariant means

The following results on uniqueness of invariant means are shown:

(i) Let be a connected almost simple algebraic group defined over . Assume that , the group of the real points in , is not compact. Let be a prime, and let be the compact -adic Lie group of the -points in . Then the normalized Haar measure on is the unique invariant mean on .

(ii) Let be a semisimple Lie group with finite centre and without compact factors, and let be a lattice in . Then integration against the -invariant probability measure on the homogeneous space is the unique -invariant mean on .

     

Duality and local group cohomology

Let be a group, let be a field, and let be a local system - an upwardly directed collection of subgroups whose union is . In this paper we give a short, elementary proof of the following result: If either is a --bimodule, or else is finite dimensional over its center, then . From this we deduce as easy corollaries some recent results of Meierfrankenfeld and Wehrfritz on the cohomology of a finitary module.

     

Normalizers of nest algebras

For a nest with associated nest algebra , we define , the normalizer of . We develop a characterization of elements of based on certain order homomorphisms of into itself. This characterization enables us to prove several structure theorems.