Root closed function algebras on compacta of large dimension

Let be a Hausdorff compact space and let be the algebra of all continuous complex-valued functions on , endowed with the supremum norm. We say that is (approximately) -th root closed if any function from is (approximately) equal to the -th power of another function. We characterize the approximate -th root closedness of in terms of -divisibility of the first Cech cohomology groups of closed subsets of . Next, for each positive integer we construct an -dimensional metrizable compactum such that is approximately -th root closed for any . Also, for each positive integer we construct an -dimensional compact Hausdorff space such that is -th root closed for any .

     

Equivariant crystalline cohomology and base change

Given a perfect field of characteristic , a smooth proper -scheme , a crystal on relative to and a finite group acting on and , we show that, viewed as a virtual -module, the reduction modulo of the crystalline cohomology of is the de Rham cohomology of modulo . On the way we prove a base change theorem for the virtual -representations associated with -equivariant objects in the derived category of -modules.

     

On frames for countably generated Hilbert -modules

Let be a countably generated Hilbert -module over a -algebra We prove that a sequence is a standard frame for if and only if the sum converges in norm for every and if there are constants such that for every We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert -modules over the -algebra of all compact operators on some Hilbert space is discussed.

     

Restrictions of bounded linear operators: Closed range

Let be a bounded linear operator on a Banach space and let be a subspace of which is a Banach space and invariant. Denote by the restriction of to This paper explores the questions:

If the range of is closed, under what conditions is the range of closed?

If the range of is closed, under what conditions is the range of closed?

     

Whittaker models for , a -adic field

Let be a non-archimedean local field with odd residual characteristic. Let be the group We construct explicit Whittaker models for any supercuspidal representation of with positive level.

     

Isomorphism of Borel full groups

Suppose that and are Polish groups which act in a Borel fashion on Polish spaces and . Let and denote the corresponding orbit equivalence relations, and and the corresponding Borel full groups. Modulo the obvious counterexamples, we show that .

     

On characterization and perturbation of local -semigroups

Let be a -group with generator , and let be a local -semigroup commuting with . Then the operators , , form a local -semigroup. It is proved that if is injective and is the generator of , then is closable and is the generator of . Also proved are a characterization theorem for local -semigroups with not necessarily injective and a theorem about solvability of the abstract inhomogeneous Cauchy problem:

     

Factorization formulae on counting zeros of diagonal equations over finite fields

Let be the number of solutions of the equation over the finite field , and let be the number of solutions of the equation . If , let be the least integer represented by . and play important roles in estimating . Based on a partition of , we obtain the factorizations of and , respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for in some special cases.

     

A condition under which follows from

We will give some sufficient conditions for a -hyponormal operator, , to be normal, and a sufficient condition for a triplet of operators , , with , self-adjoint and unitary such that necessarily satisfies .

     

A prediction problem in

For a nonnegative integrable weight function on the unit circle , we provide an expression for , in terms of the series coefficients of the outer function of , for the weighted distance , where is the normalized Lebesgue measure and ranges over trigonometric polynomials with frequencies in , , . The problem is open for .

     

An obstruction for the mean curvature of a conformal immersion

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature of a conformal immersion satisfies where is a conformal vector field on and where the integration is carried out with respect to the Euclidean volume measure of the image. This identity is analogous to the Kazdan-Warner obstruction that appears in the problem of prescribing the scalar curvature on inside the standard conformal class.

     

Lightface -indescribable cardinals

-absoluteness for forcing means that for any forcing , . `` inaccessible to reals' means that for any real , . To measure the exact consistency strength of `` -absoluteness for forcing and is inaccessible to reals', we introduce a weak version of a weakly compact cardinal, namely, a (lightface) -indescribable cardinal; has this property exactly if it is inaccessible and .

     

The degree of the bicanonical map of a surface with

In this note it is shown that, given a smooth minimal complex surface of general type with , , for which the bicanonical map is a morphism, the degree of is not 3. This completes our earlier results, showing that if is a minimal surface of general type with , such that is free, then the bicanonical map of can have degree 1, 2 or 4.

     

Character degree graphs that are complete graphs

Let be a finite group, and write for the set of degrees of irreducible characters of . We define to be the graph whose vertex set is , and there is an edge between and if . We prove that if is a complete graph, then is a solvable group.

     

Algebras generated by the disc algebra and bounded harmonic functions

Let denote the open unit disc, and let denote the disc algebra. The subsets of such that the inclusion holds for every nonconstant continuous on , or the inclusion holds for every bounded harmonic nonholomorphic function on continuous on , are characterized. In the first case the condition is that has positive measure, and in the second case that has full measure in .

     

Holes in the spectrum of functions generating affine systems

Given a expansive dilation matrix , a measurable set is called a -dilation generator of if is tiled (modulo null sets) by the collection . Our main goal in this paper is to prove certain results relating the support of the Fourier transform of functions generating a wavelet or orthonormal affine system associated with the dilation to an arbitrary set which is a -dilation generator of .

     

On the finiteness properties of extension and torsion functors of local cohomology modules

Let be a commutative Noetherian ring with non-zero identity, and ideals of with , and a finitely generated -module. In this paper, for fixed integers and , we study the finiteness of and in several cases.

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

     

On the Hartshorne-Speiser-Lyubeznik theorem about Artinian modules with a Frobenius action

Let be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik's extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring (associated to and the Frobenius homomorphism , in the indeterminate ) that is both -torsion and Artinian over .

     

A Schottky-type theorem for starlike domains in Banach spaces

We show that if is a starlike domain in a Banach space and is a family of holomorphic functions on that omit two distinct values and is bounded at the origin, then is uniformly bounded on each -bounded set.