Simple real rank zero algebras with locally Hausdorff spectrum

Let be a unital, simple, separable -algebra with real rank zero, stable rank one, and weakly unperforated ordered group. Suppose, also, that can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then is tracially approximately finite dimensional (i.e., has tracial rank zero).

Hence, is an -algebra with bounded dimension growth and is determined by -theoretic invariants.

The above result also gives the first proof for the locally case.

     

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).

     

Nikolskii-type inequalities for shift invariant function spaces

Let be a vectorspace of complex-valued functions defined on of dimension over . We say that is shift invariant (on ) if implies that for every , where on . In this note we prove the following.

Theorem. Let be a shift invariant vectorspace of complex-valued functions defined on of dimension over . Let . Then

for every and

     

Universality of uniform Eberlein compacta

We prove that if for some regular , then there is no family of less than c-algebras of size which are jointly universal for c-algebras of size . On the other hand, it is consistent to have a cardinal as large as desired and satisfying and , while there are c-algebras of size that are jointly universal for c-algebras of size . Consequently, by the known results of M. Bell, it is consistent that there is as in the last statement and uniform Eberlein compacta of weight such that at least one among them maps onto any Eberlein compact of weight (we call such a family universal). The only positive universality results for Eberlein compacta known previously required the relevant instance of to hold. These results complete the answer to a question of Y. Benyamini, M. E. Rudin and M. Wage from 1977 who asked if there always was a universal uniform Eberlein compact of a given weight.

     

Traces and Sobolev extension domains

Assume that is a bounded domain and its boundary is -regular, . We show that if there exists a bounded trace operator , and , and -Hölder continuous functions are dense in , , then the domain is a -extension domain.

     

Free modules of relative invariants and some rings of invariants that are Cohen-Macaulay

Let be a faithful representation of a finite group and a linear character. We study the module of -relative invariants. We prove a modular analogue of result of R. P. Stanley and V. Reiner in the case of nonmodular reflection groups to the effect that these modules are free on a single generator over the ring of invariants . This result is then applied to show that the ring of invariants for is Cohen-Macaulay. Since the Cohen-Macaulay property is not an issue in the nonmodular case (it is a consequence of a theorem of Eagon and Hochster), this would seem to be a new way to verify the Cohen-Macaulay property for modular rings of invariants. It is known that the Cohen-Macaulay property is inherited when passing from the ring of invariants of to that of a pointwise stabilizer of a subspace . In a similar vein, we introduce for a subspace the subgroup of elements of having as an eigenspace, and prove that Cohen-Macaulay implies is also.

     

Isomorphic -subspaces in Orlicz-Lorentz sequence spaces

Given a decreasing weight and an Orlicz function satisfying the -condition at zero, we show that the Orlicz-Lorentz sequence space contains an -isomorphic copy of , if and only if the Orlicz sequence space does, that is, if , where and are the Matuszewska-Orlicz lower and upper indices of , respectively. If does not satisfy the -condition, then a similar result holds true for order continuous subspaces and of and , respectively.

     

Binomial coefficients and quadratic fields

Let be a real quadratic field with discriminant where is an odd prime. For we determine modulo in terms of a Lucas sequence, the fundamental unit and the class number of .

     

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 Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions

Let be a sequence of positive real numbers. We define as the space of functions which are analytic in the unit disc , continuous on and such that

where is the Fourier coefficient of the restriction of to the unit circle . Let be a closed subset of . We say that is a Beurling-Carleson set if

where denotes the distance between and . In 1980, A. Atzmon asked whether there exists a sequence of positive real numbers such that for all and that has the following property: for every Beurling-Carleson set , there exists a non-zero function in that vanishes on . In this note, we give a negative answer to this question.

     

Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

Poly-log diameter bounds for some families of finite groups

Fix a prime and an integer with . Define the family of finite groups

for . We will prove that there exist two positive constants and such that for any and any generating set ,

when is the diameter of the finite group with respect to the set of generators . It is defined as the maximum over of the length of the shortest word in representing .

This result shows that these families of finite groups have a poly-logarithmic bound on the diameter with respect to any set of generators. The proof of this result also provides an efficient algorithm for finding such a poly-logarithmic representation of any element.

     

Super solutions of the dynamical Yang-Baxter equation

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical matrices. A super dynamical matrix satisfies the zero weight condition if

    for all 

In this paper we classify super dynamical matrices with zero weight.

     

Asymptotic cones and Assouad-Nagata dimension

We prove that the dimension of any asymptotic cone over a metric space does not exceed the asymptotic Assouad-Nagata dimension of . This improves a result of Dranishnikov and Smith (2007), who showed for all separable subsets of special asymptotic cones , where is an exponential ultrafilter on natural numbers.

We also show that the Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.

     

On exceptional eigenvalues of the Laplacian for

An explicit Dirichlet series is obtained, which represents an analytic function of in the half-plane except for having simple poles at points that correspond to exceptional eigenvalues of the non-Euclidean Laplacian for Hecke congruence subgroups by the relation for . Coefficients of the Dirichlet series involve all class numbers of real quadratic number fields. But, only the terms with for sufficiently large discriminants contribute to the residues of the Dirichlet series at the poles , where is the multiplicity of the eigenvalue for . This may indicate (I'm not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on .

     

Universal absolute extensors in extension theory

Let be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension contains a universal element which is an absolute extensor in dimension . Our main result shows that is quasi-finite.

     

Resolutions of ideals of fat points with support in a hyperplane

Let be a fat point subscheme of , and let be a linear form such that some power of vanishes on (i.e., the support of lies in the hyperplane defined by , regarded as ). Let , where is the subscheme of residual to ; note that is a fat points subscheme of . In this paper we give a graded free resolution of the ideal over , in terms of the graded minimal free resolutions of the ideals . We also give a criterion for when the resolution is minimal, and we show that this criterion always holds if .

     

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.

     

Hodge structures on posets

Let be a poset with unique minimal and maximal elements and . For each , let be the vector space spanned by -chains from to in . We define the notion of a Hodge structure on which consists of a local action of on , for each , such that the boundary map intertwines the actions of and according to a certain condition.

We show that if has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of into Hodge pieces.

We consider the case where is , the poset of subsets of with cardinality divisible by is fixed, and is a multiple of . We prove a remarkable formula which relates the characters of acting on the Hodge pieces of the homologies of the to the characters of acting on the homologies of the posets of partitions with every block size divisible by .

     

The rank of elliptic curves with rational 2-torsion points over large fields

Let be a number field, an algebraic closure of , the absolute Galois group , the maximal abelian extension of and an elliptic curve defined over . In this paper, we prove that if all 2-torsion points of are -rational, then for each , has infinite rank, and hence has infinite rank.