Weighted pseudo-almost periodic solutions to some differential equations

The paper considers some new classes of functions called weighted pseudo-almost periodic functions, which implement in a natural fashion the classical pseudo-almost periodic functions due to Zhang. Properties of these weighted pseudo-almost periodic functions are discussed, including a composition result for weighted pseudo-almost periodic functions. The results obtained are subsequently utilized to study the existence and uniqueness of a weighted pseudo-almost periodic solution to the heat equation with Dirichlet conditions.     

A local Paley-Wiener theorem for compact symmetric spaces

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M=U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.     

The obstacle problem for nonlinear elliptic equations with variable growth and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-data

The aim of this paper is twofold: to prove, for L ¹-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy-Stampacchia inequalities to the general framework of L ¹. Current address: Manel Sanchón, Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, 08007 Barcelona, Spain; e-mail: msanchon@maia.ub.es Authors’ addresses: J. F. Rodrigues, CMUC, Department of Mathematics, University of Coimbra, and FCUL/Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal; M. Sanchón and J. M. Urbano, CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal     

Parametrization of Pythagorean triples by a single triple of polynomials

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials.     

Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities

Multiple critical points theorems for non-differentiable functionals are established. Applications both to elliptic variational-hemivariational inequalities and eigenvalue problems with discontinuous nonlinearities are then presented.     

Gaps in Convex Disc Packings with an Application to 1-Steiner Minimum Trees

 We show that if six translates of a convex disc C all touch C, and no two of the translates have interior points in common, then there are never more than two gaps, i.e., consecutive non-touching pairs of translates. We also characterize the configurations where there are two, one or no gaps. This result is then applied to show that the Steiner point in a 1-Steiner Minimum Tree in a normed plane has degree at most five if the unit ball is not an affine regular hexagon (where Steiner points of degree six exist). (Received 23 July 1999)     

原子阱,氪-81,撒哈拉的地下水

自放射性碳年龄测定法在1949年首次在实验中得以证实以来,长寿命的由宇宙射线产生的放射性核素痕量分析这一领域,无论是在分析技术还是在可利用的核素种类上,都取得了长足的进步.它所产生的影响已经渗透到广泛的科学和技术领域.文章介绍了一种新的痕量分析方法——原子阱痕量分析(atomtraptraceanalysis,ATTA).这种方法已经用于分析环境样品中的氪-81(半衰期为23万年;同位素含量约为1×10-12)和氪-85(半衰期为10.8年;同位素含量约为2×10-11)核素.氪-81主要由宇宙射线产生于大气层上部.它是理想的测定年代范围从5万年到100万年的水和冰样品的放射性核素.另一方面,氪-85是铀-235和钋-239的核裂变产物.对氪-85的同位素含量测定可以用于监督国际核公约的执行情况.在ATTA中,一个由激光光束和磁场共同产生的原子阱可以有选择地捕陷特定核素的原子.被囚禁的原子可以通过观察其荧光来进行探测.作为ATTA技术的第一次实际应用,一个国际合作小组确定了撒哈拉沙漠地区Nubian含水土层中地下水的寿命.利用ATTA对地下水寿命的成功测定,标志着放射性氪年龄测定法在地球科学中的广泛应用已经具备了可行性.     

Differential eigenforms

The aim of this paper is to show how differential characters of Abelian varieties (in the sense of [A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995) 309-340]) can be used to construct differential modular forms of weight 0 and order 2 (in the sense of [A. Buium, Differential modular forms, Crelle J. 520 (2000) 95-167]) which are eigenvectors of Hecke operators. These differential modular forms will have “essentially the same” eigenvalues as certain classical complex eigenforms of weight 2 (and order 0).     

On Connected [g,f +1]-Factors in Graphs

Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G), and g and f two positive integral functions from V (G) to Z⁺-{1} such that g(v) ≤ f(v) ≤ d_G(v) for all v ∈V (G), where d_G(v) is the degree of the vertex v. It is shown that every graph G, including both a [g,f]-factor and a hamiltonian path, contains a connected [g,f +1]-factor. This result also extends Kano’s conjecture concerning the existence of connected [k,k+1]-factors in graphs. * The work of this author was supported by NSFC of China under Grant No. 10271065, No. 60373025. † The work of these authors was also supported in part by the US Department of Energy’s Genomes to Life program (http://doegenomestolife.org/) under project, “Carbon Sequestration in Synechococcus sp.: From Molecular Machines to Hierarchical Modeling” (www.genomes2life.org) and by National Science Foundation (NSF/DBI-0354771,NSF/ITR-IIS-0407204).     

Some function spaces on spaces of homogeneous type

We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations, the validity of the John–Nirenberg theorem, Poincarè and Sobolev inequalities, Sobolev's embedding theorem and estimates on the pointwise behavior of Sobolev-type mappings. Received: 4 December 2000 / Revised version: 5 July 2001