-bounding and -induction

Working in the base theory of , we show that for all , the bounding principle for -formulas ( ) is equivalent to the induction principle for -formulas ( ). This partially answers a question of J. Paris.

     

Computational estimation of the constant

The paper describes a computational estimation of the constant characterizing the bounds of . It is known that as

with , while the truth of the Riemann hypothesis would also imply that . In the range , two sets of estimates of are computed, one for increasingly small minima and another for increasingly large maxima of . As increases, the estimates in the first set rapidly fall below and gradually reach values slightly below , while the estimates in the second set rapidly exceed and gradually reach values slightly above . The obtained numerical results are discussed and compared to the implications of recent theoretical work of Granville and Soundararajan.

     

The nonarchimedean theta correspondence for

In this paper we consider the theta correspondence between the sets and when is a nonarchimedean local field and . Our main theorem determines all the elements of that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of that occur in the theta correspondence between and . We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.

     

A generalization of Furstenberg's Diophantine Theorem

We obtain a generalization of Furstenberg's Diophantine Theorem on non-lacunary multiplicative semigroups. For example we show that the sets of sums and are dense in the circle for all irrational , where are distinct pairs of multiplicatively independent integers for .

     

On the complexity of the description of -algebra representations by unbounded operators

We study the complexity of the problem to describe, up to unitary equivalence, representations of -algebras by unbounded operators on a Hilbert space. A number of examples are developed in detail.

     

On Λ<Subscript>m</Subscript>-sets and related topological spaces

Summary We introduce and investigate three topological spaces <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(X,\Lambda_m)$, $(X,\Lambda_{mc}^*)$ and $(X,\Lambda_{g\Lambda_m})$ by using $\Lambda_m$-sets, $(\Lambda, m)$-closed sets and generalized $\Lambda_m$-sets, respectively. Especially, we study properties of weak separation axioms on these topological spaces. The investigation enables us to obtain a unified theory of notions related to $\Lambda$-sets [21], semi-$\Lambda$-sets [5] and pre-$\Lambda$-sets [15] in topological spaces.     

A separable non-remainder of

We prove that there is a compact separable continuum that (consistently) is not a remainder of the real line.

     

Maximum excursion and stopping time record-holders for the problem: Computational results

This paper presents some results concerning the search for initial values to the so-called problem which give rise either to function iterates that attain a maximum value higher than all function iterates for all smaller initial values, or which have a stopping time higher than those of all smaller initial values. Our computational results suggest that for an initial value of , the maximum value of the function iterates is bounded from above by , with either a constant or a very slowly increasing function of . As a by-product of this (exhaustive) search, which was performed up to , the conjecture was verified up to that same number.

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

     

Specialized Runge-Kutta methods for index differential-algebraic equations

We consider the numerical solution of systems of semi-explicit index differential-algebraic equations (DAEs) by methods based on Runge-Kutta (RK) coefficients. For nonstiffly accurate RK coefficients, such as Gauss and Radau IA coefficients, the standard application of implicit RK methods is generally not superconvergent. To reestablish superconvergence projected RK methods and partitioned RK methods have been proposed. In this paper we propose a simple alternative which does not require any extra projection step and does not use any additional internal stage. Moreover, symmetry of Gauss methods is preserved. The main idea is to replace the satisfaction of the constraints at the internal stages in the standard definition by enforcing specific linear combinations of the constraints at the numerical solution and at the internal stages to vanish. We call these methods specialized Runge-Kutta methods for index DAEs (SRK-DAE).

     

A criterion for satellite 1-genus 1-bridge knots

Let be a knot in a closed orientable irreducible 3-manifold . Suppose admits a genus 1 Heegaard splitting and we denote by the splitting torus. We say is a -genus -bridge splitting of if intersects transversely in two points, and divides into two pairs of a solid torus and a boundary parallel arc in it. It is known that a -genus -bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If and the slope of the loop is longitudinal in one of the solid tori, then is obtained by twisting a component of a -bridge link along the other component. We give a criterion for determining whether a given -genus -bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.

     

Local automorphisms and derivations on certain -algebras

It is shown that continuous -local derivations on -algebras are derivations and surjective -local *-automorphisms on prime -algebras or on -algebras such that the identity element is properly infinite are *-automorphisms.

     

Products of images

Let be the \u{C}ech-Stone remainder . We show that there exists a large class of images of such that whenever is a subset of of cardinality at most the continuum, then is again an image of . The class contains all separable compact spaces, all compact spaces of weight at most and all perfectly normal compact spaces.

     

A Hahn-Banach theorem for integral polynomials

We study the problem of extendibility of polynomials over Banach spaces: when can a polynomial defined over a Banach space be extended to a polynomial over any larger Banach space? To this end, we identify all spaces of polynomials as the topological duals of a space spanned by evaluations, with Hausdorff locally convex topologies. We prove that all integral polynomials over a Banach space are extendible. Finally, we study the Aron-Berner extension of integral polynomials, and give an equivalence for non-containment of .

     

Global boundary regularity for the $ \bar \partial $‐equation on q‐pseudoconvex domains

We introduce a notion of q ‐pseudoconvex domain of new type for a bounded domain of ?ⁿ and prove that for given a ‐closed (p, r)‐form, r ≥ q, that is smooth up to the boundary, there exists a (p, r – 1)‐form smooth up to the boundary which is a solution of ‐equation on a bounded q ‐pseudoconvex domain. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Dolbeault complex in infinite dimensions II

We study the equation on a ball , and prove that it is solvable if is a Lipschitz continuous, closed -form.

     

Ore‐type condition for the existence of connected factors

In this article, we obtain some Ore‐type sufficient conditions for a graph to have a connected factor with degree restrictions. Let α and k be positive integers with if and if . Let G be a connected graph with a spanning subgraph F, each component of which has order at least α. We show that if the degree sum of two nonadjacent vertices is greater than then G has a connected subgraph in which F is contained and every vertex has degree at most . From the result, we derive that a graph G has a connected ‐factor if the degree sum of two nonadjacent vertices is at least . © Wiley Periodicals, Inc. J. Graph Theory 56: 241–248, 2007     

The optimality of James's distortion theorems

A renorming of , explored here in detail, shows that the copies of produced in the proof of the Kadec-Pelczynski theorem inside nonreflexive subspaces of cannot be produced inside general nonreflexive spaces that contain copies of . Put differently, James's distortion theorem producing one-plus-epsilon-isomorphic copies of inside any isomorphic copy of is, in a certain sense, optimal. A similar renorming of shows that James's distortion theorem for is likewise optimal.

     

Self delta-equivalence of cobordant links

Self -equivalence is an equivalence relation for links, which is stronger than the link-homotopy defined by J. Milnor. It is known that cobordant links are link-homotopic and that they are not necessarily self -equivalent. In this paper, we will give a sufficient condition for cobordant links to be self -equivalent.

     

A global approach to fully nonlinear parabolic problems

We consider the general initial-boundary value problem

(1)        
(2)        
(3)        
where is a bounded open set in with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and

The resulting problem is then reduced to the problem where the operator satisfies Condition  This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces.  The local and global solvability of the operator equation are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.