Automorphic-differential identities and actions of pointed coalgebras on rings

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let be a ring, its left Martindale quotient ring and a right ideal of having no nonzero left annihilator. (1) Let be a pointed coalgebra which measures such that the group-like elements of act as automorphisms of . If is prime and for , then . Furthermore, if the action of extends to and if such that , then . (2) Let be an endomorphism of given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If is semiprime and , then .

     

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 .

     

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 .

     

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 .

     

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.

     

A weak-type inequality of subharmonic functions

We prove the weak-type inequality , , between a non-negative subharmonic function and an -valued smooth function , defined on an open set containing the closure of a bounded domain in a Euclidean space , satisfying , and , where is a constant. Here is the harmonic measure on with respect to 0. This inequality extends Burkholder's inequality in which and , a Euclidean space.

     

On the Deleted Product Criterion For Embeddability in

For a space let . Let act on and on by exchanging factors and antipodes respectively. We present a new short proof of the following theorem by Weber: For an -polyhedron and , if there exists an equivariant map , then is embeddable in . We also prove this theorem for a peanian continuum and . We prove that the theorem is not true for the 3-adic solenoid and .

     

On the suspension order of

It is shown that the suspension order of the -fold cartesian product of real projective -space is less than or equal to the suspension order of the -fold symmetric product of and greater than or equal to , where and satisfy and . In particular has suspension order , and for fixed the suspension orders of the spaces are unbounded while their stable suspension orders are constant and equal to .

     

groups and quasi-equivalence

A torsion-free abelian group is if every map from a pure subgroup of into lifts to an endomorphism of The class of groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous groups, those homogeneous groups such that, for pure in every has a lifting to a quasi-endomorphism of An irreducible group is if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A group is if and only if every endomorphism of is an integral multiple of an automorphism. A group has minimal test for quasi-equivalence ( if whenever and are quasi-isomorphic pure subgroups of then and are equivalent via a quasi-automorphism of For homogeneous groups, we show that in almost all cases the and properties coincide.

     

-integral bases of a cubic field

A -integral basis of a cubic field is determined for each rational prime , and then an integral basis of and its discriminant are obtained from its -integral bases.

     

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.

     

-ideals of compact operators are separably determined

We prove that the space of compact operators on a Banach space is an -ideal in the space of bounded operators if and only if has the metric compact approximation property (MCAP), and is an -ideal in for all separable subspaces of having the MCAP. It follows that the Kalton-Werner theorem characterizing -ideals of compact operators on separable Banach spaces is also valid for non-separable spaces: for a Banach space is an -ideal in if and only if has the MCAP, contains no subspace isomorphic to and has property It also follows that is an -ideal in for all Banach spaces if and only if has the MCAP, and is an -ideal in .

     

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.

     

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.

     

Remarks on the results by Koskela concerning the radial uniqueness for Sobolev functions

In this note we aim to complete the results by Koskela concerning the radial uniqueness for Sobolev functions.

Let be a positive nonincreasing function on the interval , and let denote the unit ball of . Consider a -precise function on such that

where . We give conditions on which assure that whenever has vanishing fine boundary limits on a set of positive -capacity.

We are also concerned with the sharpness.

     

Pseudo-uniform convergence, a nonstandard treatment

We introduce and study the notion of pseudo-uniform convergence which is a weaker variant of quasi-uniform convergence. Applications include the following nonstandard characterization of weak convergence. Let be an infinite set, the Banach space of all bounded real-valued functions on a bounded sequence in and Then the sequence converges weakly to if and only if the convergence is pointwise on and, for each strictly increasing function , each , and each , there is an unlimited such that .

     

Character degrees and local subgroups of -separable groups

Let be a finite -solvable group for different primes and . Let and be such that . We prove that every of -degree has -degree if and only if and .

     

Improving the metric in an open manifold with nonnegative curvature

The soul theorem states that any open Riemannian manifold with nonnegative sectional curvature contains a totally geodesic compact submanifold such that is diffeomorphic to the normal bundle of . In this paper we show how to modify into a new metric so that:

1. has nonnegative sectional curvature and soul .
2. The normal exponential map of is a diffeomorphism.
3. splits as a product outside of a compact set.

As a corollary we obtain that any such is diffeomorphic to the interior of a convex set in a compact manifold with nonnegative sectional curvature.

     

Remarks on the non-Cohen-Macaulay locus of Noetherian schemes

In this paper we give a notion of polynomial type of a Noetherian scheme and define the function by for all Then we show that if admits a dualizing complex and is equidimensional, is (lower) semicontinuous; moreover, in that case, the non-Cohen-Macaulay locus nCM is not Cohen-Macaulay} is biequidimensional iff is constant on nCM

     

The Schatten space -algebra

For any , let denote the classical -Schatten space of operators on the Hilbert space . It was shown by Varopoulos (for ) and by Blecher and the author (full result) that for any equipped with the Schur product is an operator algebra. Here we prove that (and thus for any ) is actually a -algebra, which means that it is isomorphic to some quotient of a uniform algebra in the Banach algebra sense.