Asymptotically harmonic spaces in dimension 3

Let be a Hadamard manifold of dimension whose sectional curvature satisfies and whose curvature tensor satisfies for suitable constants and . We show that is of constant sectional curvature provided is asymptotically harmonic. This was previously only known if admits a compact quotient.

     

Intersections of conjugacy classes and subgroups of algebraic groups

We show that if is a reductive group, then th roots of conjugacy classes are a finite union of conjugacy classes, and that if is an algebraic overgroup of , then the intersection of with a conjugacy class of is a finite union of -conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.

     

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 .

     

(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).

     

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.

     

Wreath products and Kaluzhnin-Krasner embedding for Lie algebras

The wreath product of groups is one of basic constructions in group theory. We construct its analogue, a wreath product of Lie algebras.

Consider Lie algebras and over a field . Let be the universal enveloping algebra. Then has the natural structure of a Lie algebra, where the multiplication is defined via the comultiplication in . Also, acts by derivations on via the (left) coregular action. The semidirect sum we call the wreath product and denote by . As a main result, we prove that an arbitrary extension of Lie algebras can be embedded into the wreath product .

     

Hilbert-Samuel functions of modules over Cohen-Macaulay rings

For a finitely generated, non-free module over a CM local ring , it is proved that for the length of is given by a polynomial of degree . The vanishing of is studied, with a view towards answering the question: If there exists a finitely generated -module with such that the projective dimension or the injective dimension of is finite, then is regular? Upper bounds are provided for beyond which the question has an affirmative answer.

     

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 .

     

Non-cupping and randomness

Let be Martin-Löf-random. Then there is a promptly simple set such that for each Martin-Löf-random set , . When , one obtains a c.e. non-computable set which is not weakly Martin-Löf cuppable. That is, for any Martin-Löf-random set , if , then .

     

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?

     

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 .

     

Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions of the Bunimovich stadium , satisfying . Write where is the central rectangle and denotes the ``wings,' i.e., the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in as . We obtain a lower bound on the mass of in , assuming that itself is -normalized; in other words, the norm of is controlled by times the norm in . Moreover, if is an quasimode, the same result holds, while for an quasimode we prove that the norm of is controlled by times the norm in . We also show that the norm of may be controlled by the integral of along , where is a smooth factor on vanishing at . These results complement recent work of Burq-Zworski which shows that the norm of is controlled by the norm in any pair of strips contained in , but adjacent to .

     

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 .

     

Realizability of the Adams-Novikov spectral sequence for formal -modules

We show that the formal -module Adams-Novikov spectral sequence of Ravenel does not naturally arise from a filtration on a map of spectra by examining the case . We also prove that when is the ring of integers in a nontrivial extension of , the map of Hopf algebroids, classifying formal groups and formal -modules respectively, does not arise from compatible maps of -ring spectra .

     

Minimal rank and reflexivity of operator spaces

Let be an -dimensional space of linear operators between the linear spaces and over an algebraically closed field . Improving results of Larson, Ding, and Li and Pan we show the following.

Theorem. Let be a basis of . Assume that every nonzero operator in has rank larger than . Then a linear operator belongs to if and only if for every , is a linear combination of .

     

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.

     

Algebraic reflexivity of linear transformations

Let be the set of all linear transformations from to , where and are vector spaces over a field . We show that every -dimensional subspace of is algebraically -reflexive, where denotes the largest integer not exceeding , provided is less than the cardinality of .

     

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 .

     

Henselian valuations and orderings of a commutative ring

The purpose of this paper is to investigate the interplay between henselian valuations and orderings (or semiorderings) of a ring. As a main result, it is proved that for a henselian valuation on a ring , the following statements are equivalent: (1) is compatible with every semiordering of ; (2) is compatible with every ordering of ; (3) Every real prime ideal of is contained in the core of .