Zero divisors and

Let be a discrete group, the group ring of over and the Lebesgue space of with respect to Haar measure. It is known that if is torsion free elementary amenable, and , then . We will give a sufficient condition for this to be true when , and in the case we will give sufficient conditions for this to be false when .

     

Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

We analyze the stability of Muckenhoupt's and classes of weights under a nonlinear operation, the -operation. We prove that the dyadic doubling reverse Hölder classes are not preserved under the -operation, but the dyadic doubling classes are preserved for . We give an application to the structure of resolvent sets of dyadic paraproduct operators.

     

Uniqueness for an overdetermined boundary value problem for the p-Laplacian

For set , and let be a measure with compact support. Suppose, for , there are functions and (bounded) domains , both containing the support of with the property that in (weakly) and in the complement of . If in addition is convex, then and .

     

Pseudo-uniform convergence, a nonstandard treatment

We introduce and study the notion of pseudo-uniform convergence which is a weaker variant of quasi-uniform convergence. Applications include the following nonstandard characterization of weak convergence. Let be an infinite set, the Banach space of all bounded real-valued functions on a bounded sequence in and Then the sequence converges weakly to if and only if the convergence is pointwise on and, for each strictly increasing function , each , and each , there is an unlimited such that .

     

Conjugate Hardy's inequalities with decreasing weights

We prove that for a decreasing weight on , the conjugate Hardy transform is bounded on () if and only if it is bounded on the cone of all decreasing functions of . This property does not depend on .

     

Lipschitz images with fractal boundaries and their small surface wrapping

Assume and is a Lipschitz -mapping; and denote the volume and the surface area of . We verify that there exists a figure with , and, of course, , where depends only on the dimension and on . We also give an example when is a square and ; in fact, the boundary of can contain a fractal of Hausdorff dimension exceeding one.

     

On certain character sums over

Let be the finite field with elements and let denote the ring of polynomials in one variable with coefficients in . Let be a monic polynomial irreducible in . We obtain a bound for the least degree of a monic polynomial irreducible in ( odd) which is a quadratic non-residue modulo . We also find a bound for the least degree of a monic polynomial irreducible in which is a primitive root modulo .

     

A classification of all is a Prüfer domain

Let be an integral domain with quotient field . The ring of integer-valued polynomials over is defined by . It is known that if is a Prüfer domain, then is an almost Dedekind domain with all residue fields finite. This condition is necessary and sufficient if is Noetherian, but has been shown to not be sufficient if is not Noetherian. Several authors have come close to a complete characterization by imposing bounds on orders of residue fields of and on normalized values of particular elements of . In this note we give a double-boundedness condition which provides a complete charaterization of all integral domains such that is a Prüfer domain.

     

A generalization of Lyapunov's convexity theorem with applications in optimal stopping

Lyapunov proved that the range of finite measures defined on the same -algebra is compact, and if each measure also is atomless, then the range is convex. Although both conclusions may fail for measures on different -algebras of the same set, they do hold if the -algebras are nested, which is exactly the setting of classical optimal stopping theory.

     

Examples of chain domains

Let be a nonzero ordinal such that for every ordinal . A chain domain (i.e. a domain with linearly ordered lattices of left ideals and right ideals) is constructed such that is isomorphic with all its nonzero factor-rings and is the ordinal type of the set of proper ideals of . The construction provides answers to some open questions.

     

Lifting up an infinite chain of prime ideals to a valuation ring

We prove that for an arbitrary chain of prime ideals in an integral domain, there exists a valuation domain which has a chain of prime ideals lying over .

     

A counterexample to a -analogue of the chromatic splitting conjecture

We prove that, if , the -localization of the -localization map is not a split monomorphism in the stable category by exhibiting spectra for which the map is not injective. If and , we show that may be taken to be a two-cell complex in the sense of -local homotopy theory. The question of whether the map splits was asked by Hovey and is in some sense a -analogue of Hopkins' chromatic splitting conjecture.

     

A uniqueness theorem for harmonic functions

The main result of this note is the following theorem: Theorem 1. Let be a half ball in and . Assume that is in and harmonic in , and that for every positive integer there exists a constant such that

Then .

First we prove it for , and then we show by induction that it holds for all .

     

Stability of the surjectivity of endomorphisms and isometries of

We determine the largest positive number with the property that whenever are endomorphisms, respectively unital isometries of the algebra of all bounded linear operators acting on a separable Hilbert space, holds for every nonzero and is surjective, then so is . It turns out that in the first case we have , while in the second one .

     

Automorphic-differential identities and actions of pointed coalgebras on rings

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let be a ring, its left Martindale quotient ring and a right ideal of having no nonzero left annihilator. (1) Let be a pointed coalgebra which measures such that the group-like elements of act as automorphisms of . If is prime and for , then . Furthermore, if the action of extends to and if such that , then . (2) Let be an endomorphism of given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If is semiprime and , then .

     

A note on Greenberg's conjecture and the abc conjecture

For any totally real number field and any prime number , Greenberg's conjecture for asserts that the Iwasawa invariants and are both zero. For a fixed real abelian field , we prove that the conjecture is ``affirmative' for infinitely many (which split in if we assume the abc conjecture for .

     

A note on Kamenev type theorems for second order matrix differential systems

Some oscillation criteria are given for the second order matrix differential system , where and are real continuous matrix functions with symmetric, . These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation , where is the identity matrix, is a given function on and .

     

Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

On an analogue of Selberg's eigenvalue conjecture for

Let be the homogeneous space associated to the group
. Let where and consider the first nontrivial eigenvalue of the Laplacian on . Using geometric considerations, we prove the inequality . Since the continuous spectrum is represented by the band , our bound on can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

     

Cycle rank of Lyapunov graphs and the genera of manifolds

We show that the cycle-rank of a Lyapunov graph on a manifold satisfies: , where is the genus of . This generalizes a theorem of Franks. We also show that given any integer with , for some Lyapunov graph on .