Universal abelian groups

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:Theorem:For n≧2, there is a purely universal separable p-group in ℵ _n if, and only if, . Partially supported by the United States-Israel Binational Science Foundation. Publication number 455.     

Some variations on the notion of connection

Distributions on manifolds are studied in terms of jets of submanifolds and are interpreted as «pre-connections» or «almost-fibrings»; the associated differential calculus is developed in detail. A comparison with connections on fibred manifolds is analysed. Moreover, «higher order pre-connections», defined as pre-connections dependent on jets of arbitrary order, are introduced and studied. It is shown that infinite jets play an essential role in the associated differential calculus.This work has been performed the visits of Prof. A. M.Vinogradov at the Department of Applied Mathematics, supported by Gruppo Nazionale per la Fisica Matematica of CNR (1989, 1990).This work has been partially supported by funds (40% and 60%) of MURST.     

L-Splines

In this paper, we study the problem of unique interpolation and approximation by a class of spline functions,L-splines, containing as special cases the deficient and generalized spline functions ofAhlberg, Nilson, andWalsh [3, 5, 6], the Chebyshevian spline functions ofKarlin andZiegler [27], and the piecewise Hermite polynomial functions, as considered in [17]. We first give sufficient conditions for unique interpolation byL-spline functions in Section 2. Then, we obtain newL andL ² error estimates for interpolation byL-splines in Section 4, and show that these error estimates are, in a certain sense, sharp. In addition, we make a similar study for theg-splines ofSchoenberg, cf. [44, 3], in Section 5. In Section 6, an application of these new error estimates is made to the analysis of the error made in the use of finite dimensional subspaces ofL-splines andg-splines. in the Rayleigh-Ritz procedure for the class of nonlinear two-point boundary value problems studied in [17].Because of the rapid growth of the number of papers devoted to or connected with the topic of splines, we believe that a compilation of papers on splines for the reader's use is desirable, and such a list is found in the References at the end of this paper.This research was supported in part by NSF Grant GP-5553Papers not specifically concerned with splines are referred to in the text by [1, 2], etc.     

Some Geometric Characterizations of the Hopf Fibrations of the Three-Sphere

The Fréchet manifold of all embeddings (up to orientation preserving reparametrizations) of the circle in S ³ has a canonical weak Riemannian metric. We use the characterization obtained by H. Gluck and F. Warner of the oriented great circle fibrations of S ³ to prove that among all such fibrations π:S ³→B, the manifold B consisting of the oriented fibers is totally geodesic in , or has minimum volume or diameter with the induced metric, exactly when π is a Hopf fibration. Partially supported by foncyt, Antorchas, ciem (conicet) and secyt (unc).     

Selmer groups of abelian varieties in extensions of function fields

Let k be a field of characteristic q, a smooth geometrically connected curve defined over k with function field . Let A/K be a non-constant abelian variety defined over K of dimension d. We assume that q = 0 or >  2d + 1. Let p ≠ q be a prime number and a finite geometrically Galois and étale cover defined over k with function field . Let (τ′, B′) be the K′/k-trace of A/K. We give an upper bound for the -corank of the Selmer group Sel _p (A × _K K′), defined in terms of the p-descent map. As a consequence, we get an upper bound for the -rank of the Lang–Néron group A(K′)/τ′B′(k). In the case of a geometric tower of curves whose Galois group is isomorphic to , we give sufficient conditions for the Lang–Néron group of A to be uniformly bounded along the tower. This work was partially supported by CNPq research grant 305731/2006-8.     

Universal abelian covers of certain surface singularities

Every normal complex surface singularity with -homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations'. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If (X, o) is a rational or minimally elliptic singularity, then its universal abelian cover (Y, o) is an equisingular deformation of an isolated complete intersection singularity (Y₀, o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y₀ and X is an equisingular deformation of the quotient Y₀/G. Dedicated to Professor Jonathan Wahl on his sixtieth birthday. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

A condition for absolute continuity of infinitely divisible distribution functions

Summary Throughout this paper the symbols r.v., d.f., ch.f., and i.d. will stand, respectively, for random variable, distribution function, characteristic function, and infinitely divisible.Let F(x) be an i.d.d.f. Hartman and Wintner [5] and Blum and Rosenblatt [1] have given a condition, necessary and sufficient, for F(x) to be a continuous d.f. In this note a sufficient condition for F(x) to be an absolutely continuous d.f. is given.Research supported by ONR Contract No. NONR-285(46).Research supported in part by a National Science Foundation fellowship.     

Convergence of theQR algorithm

TheQR algorithm ofJ. G. F. Francis is used in computing matrix eigenvalues. The convergence proof given here is an analogue ofRutishauser's proof of the convergence of theLR algorithm, but our proof covers the case of disorder of the eigenvalues.The work presented in this paper was supported by the AEC Computing and Applied Mathematics Center, Courant Institute of Mathematical Sciences, New York University, under contract AT(30-1)-1480 with the U.S.Atomic Energy Commission.     

Cylindrically orthotropic circular plates with large deflections

Zusammenfassung Es werden die Verschiebungs- und Spannungszustände in einer Kreisplatte mit grossen Durchbiegungen und zylindrischer Orthotropie mit Hilfe zweier Methoden untersucht, nämlich der Methode des kleinen Parameters vonPoincaré und einer modifizierten Methode vonGalerkin.Ein numerisches Beispiel für gleichmässige Belastung der Platte, die Poissonsche Zahlv =1/4 und die Beziehungv :v _r =4 wird gegeben.

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This work has been sponsored by the United States Army under contract No. DA-11-002-ORD-2059     

More consistency results in partition calculus

This paper has two principal aims. The first is to supply a proof of Theorem 6 of [ShSt1]:Theorem:If ZFC+ “there are c ⁺ measurable cardinals” is consistent, then so is ZFC+ “ ℵ _c +is not a strong limit cardinal and ”. This is done in sections 1 and 2. See the introduction for a discussion of the evolution of the proof and of some interesting questions which remain open, related to obstacles encountered in obtaining maximum freedom in arranging for any desired cardinal exponentiation in Theorems 4 and 6 of [ShSt1]. The method is quite generally applicable in partition calculus and variants of it have in fact been applied in recent work of the authors, see [ShSt2]. first, a preservation result is proved for the game-theoretic properties of the filters considered in [ShSt1]. Then, it is shown that the existence of a system of such filters yields a canonization-style result. Finally, it is shown that the canonization property gives the positive partition relation. The second aim makes the title of this paper slightly inaccurate (but we suspect this will be pardoned): we supply a (straightforward) proof of a result which shows that Theorem 2 of [ShSt1] in some sense is best possible. This is done in section 3. Research of both authors partially supported by NSF grants. Partially supported by the BSF. Paper number 293.     

Numerical study of the representation of a totally positive quadratic integer as the sum of quadratic integral squares

Resume Altogether a total of almost 200,000 totally positive numbers from different fields were decomposed into squares; in no case were more than five squares required, although in many cases no number of squares sufficed. For some quadratic fields, from our evidence it would seem safe to completely characterize the couples for whichQ=5 or 0, particularly whenm13; and furthermore, form=17 or 33 it seems possible to characterize all cases whereQ=4, 5, or 0.The whole calculation seems to be pointed toward the result that three squares are sufficient except for special cases. Incidentally, the analytic methods ofSiegel andMaass run parallel to the calculation in that these methods involve the third, fourth, and fifth power of a theta-function. The numberical evidence would therefore suggest that their methods point to an analytic (or even a purely algebraic) proof of the futility of using more than five squares in any case.The work was supported in part by the U. S. National Science Foundation Grant G-4222 and the computer services were contributed by the Argonne National Laboratory of the U. S. Atomic Energy Commission during the summer of 1958. The coding was performed by Mr.Alan V. Lemmon with remarkable economy of length of program and running time.The deepest debt of gratitude is owed to the lateDonald A. (Moll) Flanders whose contributions to the logical design ofGeorge had made the rapid execution of the program possible and whose personal interest made possible the availability of the computer for this work. Donald A. Flanders in Memoriam (1900–1958).     

Locally compact groups acting on trees and propertyT

FollowingKazhdan, a separable locally compact groupG is said to have propertyT if the trivial representation is isolated in the dual space,, of equivalence classes of continuous irreducible unitary representations ofG. We generalize results ofMargulis—Tits by showing that groups which have propertyT can not be amalgams.Research supported by NSF.     

Some remarks on automorphisms of bounded displacement and bounded cocycles

Some time agoTits has considered the question of characterizingB(G), and more generally, of finding the automorphisms of bounded displacement ofG, particularly whenG is a connected real Lie group. Our purpose here is to extend these results to various other cases as well as to deal with the analogous questions for 1-cocycles. We concern ourselves, among other things, with the question of sufficient conditions forB(G)=Z(G), the center, or more generally, forG to have no non-trivial automorphisms of bounded displacement. The significance of such conditions can be seen in work of the author together withF. Greenleaf andL. Rothschild where, for example, the Selberg form of the Borel density theorem is considerably generalized. These conditions are therefore closely related to, but not identical with, sufficient conditions for the Zariski density of a closed subgroupH ofG withG/H having finite volume, see [11]. For this reason it is enlightening to compare these results with those of [11]. On the cocycle level, we give sufficient conditions for the points with bounded orbit under a linear representation to be fixed and more generally, for a bounded 1-cocycle to be identically zero. These conditions actually play a role in [11]; they are among the sufficient conditions necessary to establish Zariski density ofH inG. We also deal with certain converse questions and applications to homogeneous spaces of finite volume. For example, ifG/H has finite volume and is an automorphism ofG leavingH pointwise fixed, then has bounded displacement. If is a 1-cocycle and /H is trivial, then is itself bounded.Research partially supported by NSF Grant=MPS 75-08268.     

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

Summary A system ofN particles inR ^d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by extending the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL ₂(R ^d ,dr), wheredr is the Lebesgue measure, the solution will have densities inL ₂(R ^d ,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.This research was supported by NSF grant DMS92-11438 and ONR grant N00014-91J-1386     

Additive Schwarz Methods for Elliptic Mortar Finite Element Problems

Summary. Two variants of the additive Schwarz method for solving linear systems arising from the mortar finite element discretization on nonmatching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed. The methods are defined on subdomains without overlap, and they use special coarse spaces, resulting in algorithms that are well suited for parallel computation. The condition number estimate for the preconditioned system in each method is proportional to the ratio H/h, where H and h are the mesh sizes, and it is independent of discontinuous jumps of the coefficients. For one of the methods presented the choice of the mortar (nonmortar) side is independent of the coefficients.This work has been supported in part by the Norwegian Research Council, grant 113492/420This work has been supported in part by the National Science Foundation, grant NSF-CCR-9732208 and in part by the Polish Science Foundation, grant 2P03A02116 Mathematics Subject Classification (2000):65N55     

On dual-projectively flat affine connections

Given a pseudo Riemannian metrich and a torsion-free affine connection on a smoothn-manifold M,a dual geodesic curve of is defined as a curve whose tangent 1-form is parallel along the curve. The corresponding dual-projective group is defined as a group of transformations of connections preserving dual-geodesic curves. The class of connections semi-compatible with the metrich and pairs of semi-conjugate connections are defined using the relations between their geodesics and dual-geodesics. The dual-projective curvature tensor for a connection semi-compatible withh is determined as an invariant of the dual projective group. Dual-projectively flat connections semi-compatible withh are characterized as connections with vanishing dual-projective curvature tensor. As an application we recover the fundamental theorem for non-degenerate hypersurface immersions.Research partialy supported by Contract MM 18/1991 with the Ministry of Science and Education of Bulgaria and by Contract with the University of Sofia.     

On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions

Recently, Mandelbrot has encountered and numerically investigated a probability densityp _d (t) on the nonnegative reals, where, 0D<1. this=" density=" has=" fourier=" transform=">f _d (-is), wheref _d (z)=–Dz ^d (–D, z) and (·.·) is an incomplete gamma function. Previously, Darling had met this density, but had not studied its form. We expressf _d (z) as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholz. Via properties of Laplace transforms, we approximatep _d (t) asymptotically ast0+ and +, then note some implications asD0+ and 1–.Communicated by Mourad Ismail.     

Some results on surface measures in calculus of variations

We study some relations between the concepts of perimeter, Hausdorff measure, and Minkowsky content, when R ^N is endowed with a convex Finsler metric depending in a continuous way on the position. We show some connections with the theory of -convergence and with the anisotropic motion of a smooth hypersurface by mean curvature.This work was partially supported by NSF Grant DMS-9008999, and by MURST (Progetto Nazionale «Equazioni di Evoluzione e Applicazioni Fisico-Matematiche» and «Analisi Numerica e Matematica Computazionale») and CNR (IAN and Contracts 92.00833.01, 93.00564.01) of Italy.     

Intersection queries in sets of disks

In this paper we develop some new data structures for storing a set of disks that can answer different types of intersection queries efficiency. If the disks are non-intersecting we obtain a linear size data structure that can report allk disks intersecting a query line segment in timeO(n ⁺ +k), wheren is the number of disks,=log₂(1+5)–1 0.695, and is an arbitrarily small positive constant. If the segment is a full line, the query time becomesO(n +k). For intersecting disks we obtain anO(n logn) size data structure that can answer an intersection query in timeO(n ^2/3 log² n+k). We also present a linear size data structure for ray shooting queries, whose query time isO(n ).The research of the first two authors was supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). The work of the third author was supported byDimacs (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center — NSF-STC88-09648.