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.

     

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.

     

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.

     

On the coefficients of Jacobi sums in prime cyclotomic fields

Let and be prime numbers, and let be a primitive root mod . For , denote by the Jacobi sum . We study the integers such that and . We give a list of properties that characterize these coefficients. Then we show some of their applications to the study of the arithmetic of , in particular to the study of Vandiver's conjecture. For , let be the number of distinct roots of in . We show that . We give closed formulas for the numbers and in terms of quadratic and cubic power residue symbols mod .

     

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 .

     

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.

     

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 the number of terms in the middle of almost split sequences over tame algebras

Let be a finite dimensional tame algebra over an algebraically closed field . It has been conjectured that any almost split sequence with indecomposable modules has and in case , then exactly one of the is a projective-injective module. In this work we show this conjecture in case all the are directing modules, that is, there are no cycles of non-zero, non-iso maps between indecomposable -modules. In case, and are isomorphic, we show that and give precise information on the structure 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 , .

     

Operator valued weights without structure theory

A result of Haagerup, generalizing a theorem of Takesaki, states the following: If are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight if and only if there exist fns weights on and on satisfying . In fact, can be chosen such that ; is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory.

     

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 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 .

     

Causal compactification and Hardy spaces

Let be a irreducible symmetric space of Cayley type. Then is diffeomorphic to an open and dense -orbit in the Shilov boundary of . This compactification of is causal and can be used to give answers to questions in harmonic analysis on . In particular we relate the Hardy space of to the classical Hardy space on the bounded symmetric domain . This gives a new formula for the Cauchy-Szegö kernel for .

     

On Ore's conjecture and its developments

The -component of the index of a number field , , depends only on the completions of at the primes over . More precisely, equals the index of the -algebra . If is normal, then for some normal over and some , and we write for its index. In this paper we describe an effective procedure to compute for all and all normal and tamely ramified extensions of , hence to determine for all Galois number fields that are tamely ramified at . Using our procedure, we are able to exhibit a counterexample to a conjecture of Nart (1985) on the behaviour of .

     

Representing nonnegative homology classes of by minimal genus smooth embeddings

For any nonnegative class in , the minimal genus of smoothly embedded surfaces which represent is given for , and in some cases with , the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus , we prove that it is true for with and for with .

     

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.

     

Homogeneous projective varieties with degenerate secants

The secant variety of a projective variety in , denoted by , is defined to be the closure of the union of lines in passing through at least two points of , and the secant deficiency of is defined by . We list the homogeneous projective varieties with under the assumption that arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety with and , and the -variety is the only homogeneous projective variety with largest secant deficiency . This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.

     

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 .

     

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.

     

A growth dichotomy for o-minimal expansions of ordered groups

Let be an o-minimal expansion of a divisible ordered abelian group with a distinguished positive element . Then the following dichotomy holds: Either there is a -definable binary operation such that is an ordered real closed field; or, for every definable function there exists a -definable with . This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) -definable groups with underlying set .