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.

     

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 .

     

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 .

     

Turnpike property for extremals of variational problems with vector-valued functions

In this paper we study the structure of extremals of variational problems with large enough , fixed end points and an integrand from a complete metric space of functions. We will establish the turnpike property for a generic integrand . Namely, we will show that for a generic integrand , any small and an extremal of the variational problem with large enough , fixed end points and the integrand , for each the set is equal to a set up to in the Hausdorff metric. Here is a compact set depending only on the integrand and are constants which depend only on and , .

     

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.

     

A theorem on zeta functions associated with polynomials

Let be an -tuple of non-negative integers and be polynomials in such that for all and the series

is absolutely convergent for Re . We consider the zeta functions

All these zeta functions and are analytic functions of when Re is sufficiently large and they have meromorphic analytic continuations in the whole complex plane.

In this paper we shall prove that

As an immediate application, we use it to evaluate the special values of zeta functions associated with products of linear forms as considered by Shintani and the first author.

     

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.

     

Vertex operators for twisted quantum affine algebras

We construct explicitly the -vertex operators (intertwining operators) for the level one modules of the classical quantum affine algebras of twisted types using interacting bosons, where for (), for , for (), and for (). A perfect crystal graph for is constructed as a by-product.

     

On the diophantine equation

In this paper we prove that the equation , , , , , has only the solutions and with is a prime power. The proof depends on some new results concerning the upper bounds for the number of solutions of the generalized Ramanujan-Nagell equations.

     

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 .

     

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 , .

     

Witten-Helffer-Sjöstrand theory for -equivariant cohomology

Given an -invariant Morse function and an -invariant Riemannian metric , a family of finite dimensional subcomplexes , , of the Witten deformation of the -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian as . In fact the spectrum of can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains as the complex of eigenforms corresponding to the small eigenvalues of . This permits us to verify the -equivariant Morse inequalities. Moreover suppose is self-indexing and satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as to a geometric complex which is induced by and calculates the -equivariant cohomology of .

     

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 .

     

Isoperimetric Estimates on Sierpinski Gasket Type Fractals

For a compact Hausdorff space that is pathwise connected, we can define the connectivity dimension to be the infimum of all such that all points in can be connected by a path of Hausdorff dimension at most . We show how to compute the connectivity dimension for a class of self-similar sets in that we call point connected, meaning roughly that is generated by an iterated function system acting on a polytope such that the images of intersect at single vertices. This class includes the polygaskets, which are obtained from a regular -gon in the plane by contracting equally to all vertices, provided is not divisible by 4. (The Sierpinski gasket corresponds to .) We also provide a separate computation for the octogasket (), which is not point connected. We also show, in these examples, that , where the infimum is taken over all paths connecting and , and denotes Hausdorff measure, is equivalent to the original metric on . Given a compact subset of the plane of Hausdorff dimension and connectivity dimension , we can define the isoperimetric profile function to be the supremum of , where is a region in the plane bounded by a Jordan curve (or union of Jordan curves) entirely contained in , with . The analog of the standard isperimetric estimate is . We are particularly interested in finding the best constant and identifying the extremal domains where we have equality. We solve this problem for polygaskets with . In addition, for we find an entirely different estimate for as , since the boundary of has infinite measure. We find that the isoperimetric profile function is discontinuous, and that the extremal domains have relatively simple polygonal boundaries. We discuss briefly the properties of minimal paths for the Sierpinski gasket, and the isodiametric problem in the intrinsic metric.

     

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.

     

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 .

     

On the non-vanishing of cubic twists of automorphic -series

Let be a normalised new form of weight for over and , its base change lift to . A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the -function of . There is an algorithm to check the condition for any given form. The new form of level is used to illustrate our method.

     

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.

     

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.

     

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.