Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures

We consider the set of diffeomorphisms of the 2-torus , provided the conditions that the tangent bundle splits into the directed sum of -invariant subbundles , and there is such that and . Then we prove that the set is the union of Anosov diffeomorphisms and diffeomorphisms approximated by Anosov, and moreover every diffeomorphism approximated by Anosov in the set has no SBR measures. This is related to a result of Hu-Young.

     

On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order

This paper is concerned with the multiplicity of radially symmetric solutions to the Dirichlet problem

on the unit ball with boundary condition on . Here is a positive function and is a function that is superlinear (but of subcritical growth) for large positive , while for large negative we have that , where is the smallest positive eigenvalue for in with on . It is shown that, given any integer , the value may be chosen so large that there are solutions with or less interior nodes. Existence of positive solutions is excluded for large enough values of .

     

Deformations of dihedral -group extensions of fields

Given a -Galois extension of number fields we ask whether it is a specialization of a regular -Galois cover of . This is the ``inverse" of the usual use of the Hilbert Irreducibility Theorem in the Inverse Galois problem. We show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property. We explicitly construct generic extensions for dihedral -groups under certain assumptions on the base field . We also show that dihedral groups of order and have generic extensions over any base field with characteristic different from .

     

Compressions of resolvents and maximal radius of regularity

Suppose that is left invertible in for all , where is an open subset of the complex plane. Then an operator-valued function is a left resolvent of in if and only if has an extension , the resolvent of which is a dilation of of a particular form. Generalized resolvents exist on every open set , with included in the regular domain of . This implies a formula for the maximal radius of regularity of in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by
J. Zemánek is obtained.

     

Galois coverings of selfinjective algebras by repetitive algebras

In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form where is the repetitive algebra of an artin algebra and is an admissible group of automorphisms of . If is of finite global dimension, then the stable module category of finitely generated -modules is equivalent to the derived category of bounded complexes of finitely generated -modules. For a selfinjective artin algebra , an ideal and , we establish a criterion for to admit a Galois covering with an infinite cyclic Galois group . As an application we prove that all selfinjective artin algebras whose Auslander-Reiten quiver has a non-periodic generalized standard translation subquiver closed under successors in are socle equivalent to the algebras , where is a representation-infinite tilted algebra and is an infinite cyclic group of automorphisms of .

     

Extendability of Large-Scale Lipschitz Maps

Let be metric spaces, a subset of , and a large-scale lipschitz map. It is shown that possesses a large-scale lipschitz extension (with possibly larger constants) if is a Gromov hyperbolic geodesic space or the cartesian product of finitely many such spaces. No extension exists, in general, if is an infinite-dimensional Hilbert space. A necessary and sufficient condition for the extendability of a lipschitz map is given in the case when is separable and is a proper, convex geodesic space.

     

On the depth of the tangent cone and the growth of the Hilbert function

For a dimensional Cohen-Macaulay local ring we study the depth of the associated graded ring of with respect to an -primary ideal in terms of the Vallabrega-Valla conditions and the length of , where is a minimal reduction of and . As a corollary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to -primary ideals. We also study the growth of the Hilbert function.

     

On sectional genus of quasi-polarized 3-folds

Let be a smooth projective variety over and a nef-big (resp. ample) divisor on . Then is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that , where is the sectional genus of and is the irregularity of . In general it is unknown whether this conjecture is true or not, even in the case of . For example, this conjecture is true if and . But it is unknown if and . In this paper, we prove if and . Furthermore we classify polarized manifolds with , , and .

     

Forcing minimal extensions of Boolean algebras

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications:

1) We make Fedorchuk's method more flexible, obtaining, for every cardinal of uncountable cofinality, a consistent example of a Boolean algebra whose every infinite homomorphic image is of cardinality and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight ). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra can be strictly less than the minimal size of a homomorphic image of , answering a question of J. D. Monk.

2) We prove that for every cardinal of uncountable cofinality it is consistent that and both and exist.

3) By combining these algebras we obtain many examples that answer questions of J.D. Monk.

4) We prove the consistency of MA + CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangelskii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character.

5) We prove that for any cardinal of uncountable cofinality it is consistent that there is a countably tight Boolean algebra with a distinguished ultrafilter such that for every the algebra is countable and has hereditary character .

     

Classes of singular integrals along curves and surfaces

This paper is concerned with singular convolution operators in , , with convolution kernels supported on radial surfaces . We show that if , then boundedness holds if and only if . This statement can be reduced to a similar statement about the multiplier in . We also construct smooth for which the corresponding operators are bounded for but unbounded for , for given . Finally we discuss some examples of singular integrals along convex curves in the plane, with odd extensions.

     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

On minimal parabolic functions and time-homogeneous parabolic -transforms

Does a minimal harmonic function remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes of variable width and minimal harmonic functions corresponding to the boundary point of ``at infinity.' Suppose is the width of the tube units away from its endpoint and is a Lipschitz function. The answer to the question is affirmative if and only if . If the test fails, there exist parabolic -transforms of space-time Brownian motion in with infinite lifetime which are not time-homogenous.

     

A classification of Baire-1 functions

In this paper we give some topological characterizations of
bounded Baire-1 functions using some ranks. Kechris and Louveau classified the Baire-1 functions to the subclasses for every (where is a compact metric space). The first basic result of this paper is that for , iff there exists a sequence of differences of bounded semicontinuous functions on with pointwise and (where ``' denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for . We also show that the result fails for . The second basic result of the paper involves the introduction of a new ordinal-rank on sequences , called the -rank, which is smaller than the convergence rank . This result yields the following characterization of iff there exists a sequence of continuous functions with pointwise and if , resp. if .

     

Exact Hausdorff measure and intervals of maximum density for Cantor sets

Consider a linear Cantor set , which is the attractor of a linear iterated function system (i.f.s.) , , on the line satisfying the open set condition (where the open set is an interval). It is known that has Hausdorff dimension given by the equation , and that is finite and positive, where denotes Hausdorff measure of dimension . We give an algorithm for computing exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When (or more generally, if and are commensurable), the algorithm also gives an interval that maximizes the density . The Hausdorff measure is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters , it is possible to choose the translation parameters in such a way that , so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.

     

Combinatorial families that are exponentially far from being listable in Gray code sequence

Let be a collection of subsets of . In this paper we study numerical obstructions to the existence of orderings of for which the cardinalities of successive subsets satisfy congruence conditions. Gray code orders provide an example of such orderings. We say that an ordering of is a Gray code order if successive subsets differ by the adjunction or deletion of a single element of . The cardinalities of successive subsets in a Gray code order must alternate in parity. It follows that if is the difference between the number of elements of having even (resp. odd) cardinality, then is a lower bound for the cardinality of the complement of any subset of which can be listed in Gray code order. For , the collection of -blockfree subsets of is defined to be the set of all subsets of such that if and . We will construct a Gray code order for . In contrast, for we find the precise (positive) exponential growth rate of with as . This implies is far from being listable in Gray code order if is large. Analogous results for other kinds of orderings of subsets of are proved using generalizations of . However, we will show that for all , one can order so that successive elements differ by the adjunction and/or deletion of an integer from . We show that, over an -letter alphabet, the words of length which contain no block of consecutive letters cannot, in general, be listed so that successive words differ by a single letter. However, if and or if and , such a listing is always possible.

     

Knot invariants from symbolic dynamical systems

If is the group of an oriented knot , then the set of representations of the commutator subgroup into any finite group has the structure of a shift of finite type , a special type of dynamical system completely described by a finite directed graph. Invariants of , such as its topological entropy or the number of its periodic points of a given period, determine invariants of the knot. When is abelian, gives information about the infinite cyclic cover and the various branched cyclic covers of . Similar techniques are applied to oriented links.

     

The diagonal subring and the Cohen-Macaulay property of a multigraded ring

Let be a multigraded ring defined over a local ring . This paper deals with the question how the Cohen-Macaulay property of is related to that of its diagonal subring . In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of . If are ideals of positive height, we can then compare the Cohen-Macaulay property of the multi-Rees algebra with the Cohen-Macaulay property of the usual Rees algebra . We also obtain a bound for the joint reduction numbers of two -primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.

     

Second-order subgradients of convex integral functionals

The purpose of this work is twofold: on the one hand, we study the second-order behaviour of a nonsmooth convex function defined over a reflexive Banach space . We establish several equivalent characterizations of the set , known as the second-order subdifferential of at relative to . On the other hand, we examine the case in which is the functional integral associated to a normal convex integrand . We extend a result of Chi Ngoc Do from the space to a possible nonreflexive Banach space . We also establish a formula for computing the second-order subdifferential .

     

On Vassiliev knot invariants induced from finite type 3-manifold invariants

We prove that the knot invariant induced by a -homology 3-sphere invariant of order in Ohtsuki's sense, where , is of order . The method developed in our computation shows that there is no -homology 3-sphere invariant of order 5.

     

Multi-separation, centrifugality and centripetality imply chaos

Let be an interval. need not be compact or bounded. Let be a continuous map, and be a trajectory of with or . Then there is a point such that . A point is called a centripetal point of relative to if or , and is centrifugal if or . In this paper we prove that if there exist centripetal points of in , then has periodic points of some odd () period . In addition, we also prove that if ) is multi-separated by Fix(), or there exists a centrifugal point of in , then is turbulent and hence has periodic points of all periods.