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.

     

Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .

     

Spherical functions and conformal densities on spherically symmetric -spaces

Let be a -space which is spherically symmetric around some point and whose boundary has finite positive dimensional Hausdorff measure. Let be a conformal density of dimension on . We prove that is a weak limit of measures supported on spheres centered at . These measures are expressed in terms of the total mass function of and of the dimensional spherical function on . In particular, this result proves that is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.

     

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 .

     

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 , .

     

Conjugacy Classes of

Let be a quadratic extension of a global field , of characteristic not two, and the integral closure in of a Dedekind ring of -integers in . Then is isomorphic to the spinorial kernel for an indefinite quadratic -lattice of rank 4. The isomorphism is used to study the conjugacy classes of unitary groups of primitive odd binary hermitian matrices under the action of .

     

The problem on domains with piecewise smooth boundaries with applications

Let be a bounded domain in such that has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation

where is a smooth -closed form with coefficients up to the bundary of , and . In particular, Equation (0.1) is solvable with smooth up to the boundary (for appropriate degree if satisfies one of the following conditions:

i)

is the transversal intersection of bounded smooth pseudoconvex domains.

ii)

where is the union of bounded smooth pseudoconvex domains and is a pseudoconvex convex domain with a piecewise smooth boundary.

iii)

where is the intersection of bounded smooth pseudoconvex domains and is a pseudoconvex domain with a piecewise smooth boundary.

The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for on domains with piecewise smooth boundaries in a pseudoconvex manifold.

     

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 .

     

Derivatives of Wronskians with applications to families of special Weierstrass points

Let be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over . On every such a family, suitable derivatives along the fibers" (in the sense of Lax) of the relative wronskian, as defined by Laksov and Thorup, are constructed. They are sections of suitable jets extensions of the -th tensor power of the relative canonical bundle of the family itself.

The geometrical meaning of such sections is discussed: the zero schemes of the -th derivative () of a relative wronskian correspond to families of Weierstrass Points (WP's) having weight at least .

The locus in , the coarse moduli space of smooth projective curves of genus , of curves possessing a WP of weight at least , is denoted by . The fact that has the expected dimension for all was implicitly known in the literature. The main result of this paper hence consists in showing that has the expected dimension for all . As an application we compute the codimension Chow (-)class of for all , the main ingredient being the definition of the -th derivative of a relative wronskian, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension Chow (-)classes in (), corresponding to varieties of curves having a point with a suitable prescribed Weierstrass Gap Sequence, relating to previous work of Lax.

     

Dehn surgery on arborescent links

This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link is sufficiently complicated, in the sense that it is composed of at least rational tangles with all , and none of its length 2 tangles are of the form , then all complete surgeries on produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let be a tangle with a closed circle, and let . We will show that if and mod , then remains incompressible after all nontrivial surgeries on . Two bridge links are a subclass of arborescent links. For such a link , most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless has a partial fraction decomposition of the form , in which case it does admit non-laminar surgeries.

     

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 .

     

The ideal structure of some analytic crossed products

We study the ideal structure of a class of some analytic crossed products. For an -discrete, principal, minimal groupoid , we consider the analytic crossed product , where is given by a cocycle . We show that the maximal ideal space of depends on the asymptotic range of , ; that is, is homeomorphic to for finite, and consists of the unique maximal ideal for . We also prove that is semisimple in both cases, and that is invariant under isometric isomorphism.

     

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.

     

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.

     

Quasitriangular + small compact = strongly irreducible

Let be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let be a positive number. In this article, we prove that the perturbation of by a compact operator with can be strongly irreducible if is a quasitriangular operator with the spectrum connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero.

Suppose that is a bounded linear operator acting on a separable infinite dimensional Hilbert space with connected. Let be given. Is there a compact operator with such that is strongly irreducible?

     

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.

     

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 .

     

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 .

     

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.