Subordination des résolvantes

Résumé Soit (V ₎₀ une résolvante définie sur un espace mesurable telle que le noyau initial est borné; on trouve une condition nécéssaire et suffisante pour qu'un noyau borné U possède une résolvante (U ₎₀ telle que U V pour tout 0. On donne plusieurs applications de ce résultat.     

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

The Real Period Function of A3 Singularity and Perturbations of the Spherical Pendulum

We prove that the Hessian matrix of the real period function () associated with the real versal deformation f (x)=±x ⁴+₂ x ²+₁ x+₀ of a singularity of type A ₃, is nondegenerate, provided that ³ does not belong to the discriminant set of the singularity. We explain the relation between this result and the perturbations of the spherical pendulum.     

Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim _n ⁿ , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .     

The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves

For a partition ={₁₂₃0} of non-negative integers, we calculate the Euler characteristic of the local system on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some of low degree, we make a guess for the motivic Euler characteristic of using counting curves over finite fields.Mathematics Subject Classification (1991): 14J15, 20B25     

275-01275-01275-01Regularity of the spatial derivatives of the solutions to parabolic systems in divergence form

Sunto Si dimostra che, nelle ipotesi: fⁱ L^2,(Q, ^N), 0<< n+2, i=0,1,..., n, uL^2,(Q,^N)H _–T ^*0,1/2() (Q,^N), D_iuL^2,(Q,^N),i=1,2,...,n, la soluzione v: Q ^N del problema di Cauchy-Dirichlet: ha derivate spaziali D_iv appartenenti a L^2,(Q,^N) e che sussiste la maggiorazione:.

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Lavoro eseguito con contributo finanziario del M.U.R.S.T. e nell'ambito del G.N.A.F.A. del C.N.R.     

Stability of unconditional convergence almost everywhere

We will investigate the properties of series of functions which are unconditionally convergent almost everywhere on [0, 1]. We will establish the following theorem: If the series _k=1 f _k(x) converges unconditionally almost everywhere, then there exists a sequence {_k} ₁ ,_k , such that if _{k k} , k=1, 2,..., the series _{k=1 k}/k(x) converges unconditionally almost every-where.Translated from Mate matte heskie Zametki, Vol. 14, No. 5, pp. 645–654, November, 1973.The author wishes to thank Professor P. L. Ul'yanov for his help.     

Topological influence of the presence of fully-λ-bases in locally convex spaces

This article reveals the topological impact of fully--bases in locally convex spaces where carries either the traditional normal topology or the fairly generalized-topology of Ruckle. It has been established that the generalized nuclearity of plays a significant role in influencing the topology of the space. Further, the equivalence of normal topology and the topology arising out of the fully--base ( being equipped with normal topology or-topology) has been investigated.We acknowledge with thanks the suggestions of the referee.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Approximate computation of the positive eigenvalue of a positive operator with a nonlinear occurrence of a parameter

A method is indicated for the approximate determination of the positive eigenvalue of the problem x–Qx=0, >0, xK, x0, whereK is a cone in Banach space and Q is an operator-valued function positive relative toK.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 37–39, 1976.     

Eigenvalues of convex processes and convergence properties of differential inclusions

In this paper, we study (real) eigenvalues and eigenvectors of convex processes, and provide conditions for the existence of eigenvectors in a given convex coneK ⁿ . It is established that the maximal eigenvalue ofG(·) inK is expressed by (whereK ⁰ is the polar cone ofK) provided that the minimum is attained in intK ⁰. This result is applied to study the asymptotic behaviour of certain differential inclusions{G(x(t)). We extend some known results for the von Neumann-Gale model to our more general framework. We prove that ifx ₀ is the unique eigenvector corresponding to the maximal eigenvalue ₀ ofG(·) inK, then the nonexistence of solutions of a certain special trigonometric form is necessary and sufficient for every viable solutionx(·) to satisfy^- ₀ ^t x(t)cx ₀ ast for somec0. Our method is to study the family of convex conesW =cl{v–x :xK,vG(x) where is any real number. We characterize the maximal eigenvalue ₀ as the minimal for whichW can be separated fromK.The research was supported in part by a grant from the ministry of science and the Maagara special project for the absorption of new immigrants in the Department of Mathematics at Technion.     

On graphical partitions

An integer partition {₁,₂,..., _v } is said to be graphical if there exists a graph with degree sequence _i . We give some results corcerning the problem of deciding whether or not almost all partitions of even integer are non-graphical. We also give asymptotic estimates for the number of partitions with given rank.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

Conjugation for polynomial mappings

We consider Keller's functions, namely polynomial functionsf:C ⁿ C ⁿ with detf(x)=1 at allx C ⁿ. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh , > 1, defined by power series in a ball with center at the origin, such thath(0)=I andh (f(x))=h (x). So eachh conjugates f to its linear part I in a ball where it is injective.We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s ^dg(x),g(x)ⁿ=0,xC ⁿ,sC the conjugationh for f is anentire function.     

On the lowest eigenvalue of the Laplacian for the intersection of two domains

IfA andB are two bounded domains in _n and (A), (B) are the lowest eigenvalues of – with Dirichlet boundary conditions then there is some translate,B _x, ofB such that (AB _x)<(A)+(B). A similar inequality holds for .There are two corollaries of this theorem: (i) A lower bound for sup _x {volume (AB _x)} in terms of (A), whenB is a ball; (ii) A compactness lemma for certain sequences inW ^1,p ( _n ).Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15     

J-pseudo-spectral andJ-inner-pseudo-outer factorizations for matrix polynomials

For a comonic polynomialL() and a selfadjoint invertible matrixJ the following two factorization problems are considered: firstly, we parametrize all comonic polynomialsR() such that . Secondly, if it exists, we give theJ-innerpseudo-outer factorizationL()=()R(), where () isJ-inner andR() is a comonic pseudo-outer polynomial. We shall also consider these problems with additional restrictions on the pole structure and/or zero structure ofR(). The analysis of these problems is based on the solution of a general inverse spectral problem for rational matrix functions, which consists of finding the set of rational matrix functions for which two given pairs are extensions of their pole and zero pair, respectively.The work of this author was supported by the USA-Israel Binational Science Foundation (BSF) Grant no. 9400271.     

Resolution of the Symmetric Nonnegative Inverse Eigenvalue Problem for Matrices Subordinate to a Bipartite Graph

There is a symmetric nonnegative matrix A, subordinate to a given bipartite graph G on n vertices, with eigenvalues _{12 n} if and only if, ₁ + _n 0, ₂ + _n-10,..., _m + _{n - m + 1}0, _{m + 1}0,..., _{n - m} 0, in which m is the matching numberof G. Other observations are also made about the symmetric nonnegative inverse eigenvalue problem with respect to a graph     

Symmetric designs and geometroids

A-setS in a symmetric 2-(v, k, ) design is a subset which every block meets in 0, 1 or points such that for any point ofS there is a unique block meetingS at that point only. Ovoids in three-dimensional projective spaces are examples of-secs. It is shown that if has a-set then is a geometroid withv=u ²+u+1 andk=u+1, whereu–1. The cases whenu is–1, and+1 are investigated and some open problems discussed.     

Characteristic function and a model of a linear operator pencil

Let w() be an operator function, holomorphic in the domain , with a reproducing kernel that is Hermitian positive with respect to the semiplane. Then w() coincides with the characteristic function (c.f.) of a pencil A+B of bounded operators, where the pencil is determined up to unitary equivalence. A connection between the invariants of the pencil and the divisors of its characteristic function is established. A universal model of an arbitrary pencil and a triangular model under the condition AB^*+BA^* _w are constructed.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 98–111, 1986.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .