A strong hot spot theorem

A real number is said to be -normal if every -long string of digits appears in the base- expansion of with limiting frequency . We prove that is -normal if and only if it possesses no base- ``hot spot'. In other words, is -normal if and only if there is no real number such that smaller and smaller neighborhoods of are visited by the successive shifts of the base- expansion of with larger and larger frequencies, relative to the lengths of these neighborhoods.

     

Global dominated splittings and the Newhouse phenomenon

We prove that given a compact -dimensional boundaryless manifold , , there exists a residual subset of the space of diffeomorphisms such that given any chain-transitive set of , then either admits a dominated splitting or else is contained in the closure of an infinite number of periodic sinks/sources. This result generalizes the generic dichotomy for homoclinic classes given by Bonatti, Diaz, and Pujals (2003).

It follows from the above result that given a -generic diffeomorphism , then either the nonwandering set may be decomposed into a finite number of pairwise disjoint compact sets each of which admits a dominated splitting, or else exhibits infinitely many periodic sinks/sources (the `` Newhouse phenomenon"). This result answers a question of Bonatti, Diaz, and Pujals and generalizes the generic dichotomy for surface diffeomorphisms given by Mañé (1982).

     

Boundary structure of hyperbolic -manifolds admitting annular fillings at large distance

We show that if a hyperbolic -manifold with a union of tori admits two annular Dehn fillings at distance , then is bounded by at most three tori.

     

On maximal operators on -spheres in

A. Magyar's result on -bounds for a family of operators on -spheres () in is improved to match the corresponding theorem for -spheres.

     

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.

     

The sufficiency of arithmetic progressions for the Conjecture

Define by if is odd and if is even. The Conjecture states that the -orbit of every positive integer contains . A set of positive integers is said to be sufficient if the -orbit of every positive integer intersects the -orbit of an element of that set. Thus to prove the Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets are sufficient for and asked if is also sufficient for larger values of . We answer this question in the affirmative by proving the stronger result that is sufficient for any nonnegative integers and with i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.

     

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 .

     

The length of a shortest closed geodesic and the area of a -dimensional sphere

Let be a Riemannian manifold homeomorphic to . The purpose of this paper is to establish the new inequality for the length of a shortest closed geodesic, , in terms of the area of . This result improves previously known inequalities by C.B. Croke (1988), by A. Nabutovsky and the author (2002) and by S. Sabourau (2004).

     

Extending into isometries of

In this paper we generalize a result of Hopenwasser and Plastiras (1997) that gives a geometric condition under which into isometries from to have a unique extension to an isometry in . We show that when and are separable reflexive Banach spaces having the metric approximation property with strictly convex and smooth and such that is a Hahn-Banach smooth subspace of , any nice into isometry has a unique extension to an isometry in .

     

The number of Hall -subgroups of a -separable group

We observe a simple formula to compute the number of Hall -subgroups of a -separable finite group in terms of only the action of a fixed Hall -subgroup of on a set of normal -sections of . As a consequence, we obtain that divides whenever is a subgroup of a finite -separable group . This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro's result.

     

On stable equivalences induced by exact functors

Let and be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence between and such that is induced by exact functors. We present a nice correspondence between indecomposable modules over and . As a consequence, we have the following: (1) If is a self-injective algebra, then so is ; (2) If and are finite dimensional algebras over an algebraically closed field , and if is of finite representation type such that the Auslander-Reiten quiver of has no oriented cycles, then and are Morita equivalent.

     

Borel classes of sets of extreme and exposed points in

It is known that the sets of extreme and exposed points of a convex Borel subset of are Borel. We show that for there exist convex subsets of such that the sets of their extreme and exposed points coincide and are of arbitrarily high Borel class. On the other hand, we show that the sets of extreme and of exposed points of a convex set of additive Borel class are of ambiguous Borel class . For proving the latter-mentioned results we show that the union of the open and the union of the closed segments of are of the additive Borel class if is a convex set of additive Borel class .

     

A characterisation of in terms of forcing

We show that ``saturation' of the universe with respect to forcing over with partial orders on is equivalent to the existence of .

     

Deformations and derived equivalences

Suppose and are block algebras of finite groups over a complete local commutative Noetherian ring whose residue field is a field of positive characteristic. We prove that a split-endomorphism two-sided tilting complex (as introduced by Rickard) for the derived categories of bounded complexes of finitely generated modules over , resp. , preserves the versal deformation rings of bounded complexes of finitely generated modules over , resp. .

     

Fiber products, Poincaré duality and -ring spectra

For a Poincaré duality space and a map , consider the homotopy fiber product . If is orientable with respect to a multiplicative cohomology theory , then, after suitably regrading, it is shown that the -homology of has the structure of a graded associative algebra. When is the diagonal map of a manifold , one recovers a result of Chas and Sullivan about the homology of the unbased loop space .

     

All -cotilting modules are pure-injective

We prove that all -cotilting -modules are pure-injective for any ring and any . To achieve this, we prove that is a covering class whenever is an -module such that is closed under products and pure submodules.

     

is a free semigroup algebra

We provide a simplified version of a construction of Charles Read. For any , there are isometries with orthogonal ranges with the property that the nonselfadjoint weak--closed algebra that they generate is all of .

     

On -adic Hermitian Eisenstein series

In this paper we generalize the notion of -adic modular form to the Hermitian modular case and prove a formula that shows a coincidence between certain -adic Hermitian Eisenstein series and the genus theta series associated with Hermitian matrix with determinant .

     

Topology of three-manifolds with positive -scalar curvature

Consider an -dimensional smooth Riemannian manifold with a given smooth measure on it. We call such a triple a Riemannian measure space. Perelman introduced a variant of scalar curvature in his recent work on solving Poincaré's conjecture , where and is the scalar curvature of . In this note, we study the topological obstruction for the -stable minimal submanifold with positive -scalar curvature in dimension three under the setting of manifolds with density.

     

Semiprime smash products and -stable prime radicals for PI-algebras

Assume that is a finite-dimensional Hopf algebra over a field and that is an -module algebra satisfying a polynomial identity (PI). We prove that if is semisimple and is -semiprime, then is semiprime. If is cosemisimple, we show that the prime radical of is -stable.