The -exponent of the -local spectrum

Let be a fixed odd prime. In this paper we prove an exponent conjecture of Bousfield, namely that the -exponent of the spectrum is for . It follows from this result that the -exponent of is at least for and , where denotes the -connected cover of .

     

-injective rings and -stable primes

The notion of stability of the highest local cohomology module with respect to the Frobenius functor originates in the work of R. Hartshorne and R. Speiser. R. Fedder and K.-i. Watanabe examined this concept for isolated singularities by relating it to -rationality. The purpose of this note is to study what happens in the case of non-isolated singularities and to show how this stability concept encapsulates a few of the subtleties of tight closure theory. Our study can be seen as a generalization of the work by Fedder and Watanabe. We introduce two new ring invariants, the -stability number and the set of -stable primes. We associate to every ideal generated by a system of parameters and an ideal of multipliers denoted and obtain a family of ideals . The set is independent of and consists of finitely many prime ideals. It also equals prime ideal such that is -stable. The maximal height of such primes defines the -stability number.

     

Modular group algebras of -separable -groups

Under the assumptions of MA and CH, it is proved that if is a field of prime characteristic and is an -separable abelian -group of cardinality , then an isomorphism of the group algebras and implies an isomorphism of and .

     

Discrete spectra of -algebras and complemented submodules in Hilbert -modules

Let be a -algebra and let be a full (right) Hilbert -module. It is shown that if the spectrum of is discrete, then every closed --submodule of is complemented in , and conversely that if is a -space and if every closed --submodule of is complemented in , then is discrete.

     

-hyponormal operators are subscalar

We prove that if are injective, then is subscalar if and only if is subscalar. As corollaries, it is shown that -hyponormal operators and log-hyponormal operators are subscalar; also w-hyponormal operators with Ker Kerand their generalized Aluthge transformations are subscalar.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

-theory of -pseudo-differential algebras

We are concerned with the so-called -pseudo-differential calculus. We describe the spectrum of the unital and commutative -algebra given by the norm closure of the space of -order pseudo-differential operators modulo compact operators; other related algebras are also considered. Finally, their -theory is computed.

     

On the best possible character of the norm in some a priori estimates for non-divergence form equations in Carnot groups

Let be a group of Heisenberg type with homogeneous dimension . For every we construct a non-divergence form operator and a non-trivial solution to the Dirichlet problem: in , on . This non-uniqueness result shows the impossibility of controlling the maximum of with an norm of when . Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as

where is the dimension of the horizontal layer of the Lie algebra and is the symmetrized horizontal Hessian of .

     

On non-measurability of in its second dual

We show that with the weak topology is not an intersection of Borel sets in its Cech-Stone extension (and hence in any compactification). Assuming (CH), this implies that has no continuous injection onto a Borel set in a compact space, or onto a Lindelöf space. Under (CH), this answers a question of Arhangel'ski.

     

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 .

     

The Diophantine equation

Let be a rational prime and a positive rational integer coprime with . Denote by the number of solutions of the equation in rational integers and . In a paper of Le, he claimed that without giving a proof. Furthermore, the statement has been used by Le, Bugeaud and Shorey in their papers to derive results on certain Diophantine equations. In this paper we point out that the statement is incorrect by proving that .

     

Number of Sylow subgroups in -solvable groups

If is a finite group and is a prime number, let be the number of Sylow -subgroups of . If is a subgroup of a -solvable group , we prove that divides .

     

Cancellation of direct sums of countable abelian -groups

Let be abelian groups where is a direct sum of countable -groups. A condition is given on the Ulm-Kaplansky -invariants of and such that .

     

-inner automorphisms of finite groups

We refer to an automorphism of a group as -inner if given any subset of with cardinality less than , there exists an inner automorphism of agreeing with on . Hence is 2-inner if it sends every element of to a conjugate. New examples are given of outer -inner automorphisms of finite groups for all natural numbers .

     

estimate of a parabolic Monge-Ampère equation on

We consider a special type of parabolic Monge-Ampère equation on arising from convex hypersurfaces expansion in Euclidean spaces. We obtained a parabolic estimate of the support functions for the convex hypersurfaces assuming that we have already had a parabolic estimate.

     

Principal eigenvalues for indefinite weight problems in all of

We show the existence of principal eigenvalues of the problem in where is an indefinite weight function. The existence of a continuous family of principal eigenvalues is demonstrated. Also, we prove the existence of a principal eigenvalue for which the principal eigenfunction at .

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

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).

     

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.

     

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.