The Levy-Steinitz rearrangement theorem for duals of metrizable spaces

Extending the Levy-Steinitz rearrangement theorem in ℝ ⁿ , which in turn extended Riemann’s theorem, Banaszczyk proved in 1990/93 that a metrizable, locally convex space is nuclear if and only if the domain of sums of every convergent series (i.e. the set of all elements in the space which are sums of a convergent rearrangement of the series) is a translate of a closed subspace of a special form. In this paper we present an apparently complete analysis of the domains of sums of convergent series in duals of metrizable spaces or, more generally, in (DF)-spaces in the sense of Grothendieck. The research of the first author was partially supported by DGES Project PB97-0333, and the second one by a grant of the Ministerio de Educatión y Cultura of Spain (Ref: Sab 1995-0736).     

Mechanical theorem proving in differential geometry

An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used to simplify one of Chem’s theorems: “The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces.” Some other theorems are also tested by this method. The proofs are generally simpler than those in differential geometry textbooks. Project supported partially by the National Natural Science Foundation of China.     

Central Value Of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindel?f hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting. The first author was partially supported by NSF grant DMS-0070762. The second author was partially supported by NSF grant DMS-0355285. Received: July 2005 Accepted: August 2005     

Extensions of covariantly finite subcategories

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated categories; we apply Gentle–Todorov’s theorem to obtain short proofs of a classical result by Ringel and a recent result by Krause and Solberg. This project is partially supported by China Postdoctoral Science Foundation (No.s 20070420125 and 200801230). The author also gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong.     

Frames and sampling theorem

The sampling problem on a closed subspace V_o of L²(ℝ) is studied. For the regular sampling, a necessary and sufficient condition on the sampling theorem is presented and the sampling problem proposed by Walter is completely solved. Moreover, a representation formula for the derived function is given. The irregular sampling is studied. The results improve the ones of Walter and Liu’s. Project supported by the National Natural Science Foundation of China (Grant No. 16971047).     

Equidistribution of expanding translates of curves and Dirichlet’s theorem on diophantine approximation

We show that for almost all points on any analytic curve on ℝ ^k which is not contained in a proper affine subspace, the Dirichlet’s theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in ℝ ^k+1. The proof involves Ratner’s theorem on ergodic properties of unipotent flows on homogeneous spaces. Dedicated to my inspiring teacher Professor A.R. Rao (VASCSC, Ahmedabad) on his 100th birthday. Research supported in part by Swarnajayanti Fellowship.     

A Half-Space Approach to Order Dimension

The aim of the present paper is to investigate the half-spaces in the convexity structure of all quasiorders on a given set and to use them in an alternative approach to classical order dimension. The main result states that linear orders can almost always be replaced by half-space quasiorders in the definition of the dimension of a partially ordered set. The work of the first named author was partially supported by the European Community’s Marie Curie Program (contract MTKD-CT-2004-003006). The second named author’s work was supported by OTKA of Hungary No. T043034.     

Minkowski’s Second Theorem over a Simple Algebra

We generalize the notion of successive minima, Minkowski’s second theorem and Siegel’s lemma to a free module over a simple algebra whose center is a global field. The author was partly supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science.     

A combination theorem for Veech subgroups of the mapping class group

In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein–Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups are amalgamated along a parabolic subgroup. As a corollary, we construct subgroups of the mapping class group (for all genera at least 2), which are isomorphic to non-abelian closed surface groups in which all but one conjugacy class (up to powers) is pseudo-Anosov. Received: October 2004 Revision: April 2005 Accepted: April 2005 C.J.L.’s work was partially supported by an NSF postdoctoral fellowship. A.W.R’s work was partially supported by an NSF grant.     

Families of vector measures of uniformly bounded variation

We characterize families of vector measures of uniformly bounded variation and semivariation in terms of additivity properties. A classical theorem of Rickart and a simplified proof of Nikodym’s boundedness theorem follow. Received: 3 January 2006; Revised: 31 March 2006 The second named author was supported by Estonian Science Foundation Grant 5704.     

Ordering in mechanical geometry theorem proving

Ordering in mechanical geometry theorem proving is studied from geometric viewpoint and some new ideas are proposed. For Thebault’s theorem which is the most difficult theorem that has ever been proved by Wu’s method, a very simple proof using Wu’s method under a linear order is discovered. Project supported by the National Natural Science Foundation of China.     

A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

Cartan decomposition of the moment map

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.First author partially supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft and the DFG Schwerpunk program Globale Methoden in der komplexen Geometrie.Second author partially supported by NSA grant H98230–04–01–0070.     

Remarks on the Regularity to 3-D Ideal Magnetohydrodynamic Equations

In this paper we are interested in the sufficient conditions which guarantee the regularity of solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval [0,T]. Five sufficient conditions are given. Our results are motivated by two main ideas: one is to control the accumulation of vorticity alone; the other is to generalize the corresponding geometric conditions of 3-D Euler equations to 3-D ideal magnetohydrodynamic equations.     

The polarized Ramsey’s theorem

We study the effective and proof-theoretic content of the polarized Ramsey’s theorem, a variant of Ramsey’s theorem obtained by relaxing the definition of homogeneous set. Our investigation yields a new characterization of Ramsey’s theorem in all exponents, and produces several combinatorial principles which, modulo bounding for formulas, lie (possibly not strictly) between Ramsey’s theorem for pairs and the stable Ramsey’s theorem for pairs. We are grateful to D. Hirschfeldt, A. Montalbán, and R. Soare for making our collaboration possible and for helpful comments and suggestions. We thank J. Schmerl for first bringing the subject of polarized partitions to our attention and J. Mileti for his generous insights. We also thank one anonymous referee for valuable observations and corrections. The first author was partially supported by an NSF Graduate Research Fellowship.     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.     

On some developments in the representation theory of cylindric-like algebras

In this survey paper we recall the representation problem of certain classes of algebras of relations. Two specific recent research directions in algebraic logic are surveyed: the one is connected with the representation theory of cylindric relativized set algebras including Resek’s representation theorem and merry-go-round axioms, the other discusses the representation problem in some non-well-founded set theories where the axiom of foundation is replaced by Boffa’s weak anti-foundation axiom. This paper is dedicated to Walter Taylor. Received September 8, 2005; accepted in final form January 12, 2006. The first author was supported by Hungarian National Foundation for Scientific Research grants T43242 and T035192. The second author was supported by Hungarian National Foundation for Scientific Research grants D042177, T43242 and T035192.     

Nilpotent subgroups of the group of self-homotopy equivalences

In this paper, we study subgroups of self-homotopy equivalences associated to generalized homology theories. We generalize Dror-Zabrodsky’s nilpotency theorem on the group of self-homotopy equivalences. Dedicated to Professor Akio Hattori on his sixtieth birthday The first-named author was partially supported by Grant-in-Aid for the Science Research of the Ministry of Education, 02740012.     

Periodic Orbits for Some Systems of Delay Differential Equations

We provide sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay. We extend Kaplan-Yorke＇s method for finding periodic orbits from a delay differential equation with several delays to a system of delay differential equations with a unique delay.     

Minimizing weak solutions for calabi’s extremal metrics on toric manifolds

In this paper, we discuss Donaldson’s version of the modified K-energy associated to the Calabi’s extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified K-energy. The second author was partially supported by NSF10425102 in China and the Huo Y-T Fund.