An Improvement of an Inequality of Fiedler Leading to a New Conjecture on Nonnegative Matrices

Suppose that A is an n × n nonnegative matrix whose eigenvalues are = (A), ₂, ..., _n. Fiedler and others have shown that \det( I -A) ⁿ - ⁿ, for all > with equality for any such if and only if A is the simple cycle matrix. Let a _i be the signed sum of the determinants of the principal submatrices of A of order i × i, i=1, ..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to: for all . We use this inequality to derive the inequality that: . In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of A: If ₁ = (A), ₂,...,_k are (all) the nonzero eigenvalues of A, then . We prove this conjecture for the case when the spectrum of A is real.     

On sequences of Dirichlet polynomials uniformly bounded on half planes

Given and a sequence of Dirichlet polynomials estimates for the coefficientsa _n are proved if {_n} is uniformly bounded on a region containing a half plane. Thereby a result is obtained which is an analogue of a known result for polynomials, that is for theA-transforms of the geometric sequence; moreover a Jentzsch type theorem for {_n(z)} is derived.     

Reconstruction of a rational nonsquare matrix function from local data

In many problems the local zero-pole structure (i.e. locations of zeros and poles together with their orders) of a scalar rational functionw is a key piece of structure. Knowledge of the order of the pole or zero of the rational functionw at the point is equivalent to knowledge of the -module (where is the space of rational functions analytic at ). For the more intricate case of a rationalp×m matrix functionW, we consider the structure of the module as the appropriate analogue of zero-pole structure (location of zeros and poles together with directional information), where is the set of column vectors of heightm with entries equal to rational functions which are analytic at . Modules of the form in turn can be explicitly parametrized in terms of a collection of matrices (C ,A ,B ,B , ) together with a certain row-reduced(p–m)×m matrix polynomialP(z) (which is independent of ) which satisfy certain normalization and consistency conditions. We therefore define the collection (C ,A ,Z ,B , ,P(z)) to be the local spectral data set of the rational matrix functionW at . We discuss the direct problem of how to compute the local spectral data explicitly from a realizationW(z)=D+C(z–A) ^–1 B forW and solve the inverse problem of classifying which collections (C ,A ,Z ,B , ,P(z)) satisfying the local consistency and normalization conditions arise as the local spectral data sets of some rational matrix functionW. Earlier work in the literature handles the case whereW is square with nonzero determinant.     

On T 3-Topological Space Omitting Many Cardinals

We prove that for every (infinite cardinal) there is a T ₃-space X with clopen basis, points such that every closed subspace of cardinality < has cardinality < .     

Asymptotic behavior of the spectrum of a pseudodifferential operator with periodic bicharacteristics

Let _j be the eigenvalues of a positive elliptic pseudodifferential operator of order m > 0 on a closed compact d-dimensional C-manifold and let N()=#{j:_j^m}. It is shown that for each > 0 we have

Décroissance rapide de la distribution fλ

Let X be an open subset of ⁿ and (f₁, ...,f_p): X ^p be a holomorphic mapping. We prove that if (x⁰,0, ⁰) T^* × ^p does not belong to the characteristic variety of the _X []-module _X[]f, then there exists a conic neighborhood V × of (x⁰, ⁰) such the function is rapidely decreasing in | Im | for with Re bounded, for any (n,n)-form of class C with compact support in V. The following partial converse of this result is also established: if for all (n,n)-forms of class C with compact support in X, then .     

Sets with a Prescribed Number of Power Invariants

Let A₁,...,A_n be points in , let be a fixed point, let p be a positive integer, and let ₁,...,_n be positive real numbers. If the does not depend on the position of M on a sphere with center O, then one says that the point system {A₁,...,A_n} has an invariant of degree p with weight system {,...,_n}. It is proved that for arbitrary positive integers d and N there exists a finite point system having invariants of degrees p=1,...,N with common positive weight system {₁,...,_n}. Bibliography: 2 titles.     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

О росте вдоль прямой ц елых функций экспоне нциального типа с заданными нулями

The following result is proved. Let=_n} be a sequence of complex numbers with ¦Re _n¦¦ _n ¦, >0, and letg be an entire function of exponential type with a sequence of zeros which satisfies the same condition. There exists an entire function of exponential typef0 such thatf()=0 and ¦f(iy)¦¦g(iy)¦,yR, if and only if there exists a constantM such that for all numbersr andR, 0rR<>, we have

A Limit Result for U-Statistics of Binary Variables

Define , where is a symmetric U-type statistic, H _k() is the Hermite polynomial of degree k, and {X, X _n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that according as EX=0 or EX0, respectively.     

Inequality for second characteristic values of positive operators of certain classes

A homogeneous additive operator A, positive on a cone K of a Banach space E partially ordered by K, is investigated. It is assumed that K is a reproducing cone in E and that A has a characteristic vector u₀: Au₀ = ₀u₀ in K. It is proved that if for some 1, then any other characteristic value of A satisfies the inequality ||<(-1)/(+1) ₀. This is the best possible upper bound in the class of operators considered.Translated from Matematicheskie Zametki, Vol. 19,No. 1, pp. 27–33, January, 1971.The author takes this opportunity to thank M. A. Krasnosel'skii for his interest in this work.     

Barrelledness of Generalized Sums of Normed Spaces

Let (E _i ) _iI be a family of normed spaces and a space of scalar generalized sequences. The -sum of the family (E _i ) _iI of spaces is

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this work, we study algebraic and analytic properties for the polynomials { Q _n } _{n 0}, which are orthogonal with respect to the inner product where , R such that – ² > 0.     

Relations of Borel Type for Generalizations of Exponential Series

We prove that the condition is necessary and sufficient for the validity of the relation ln F() ln (, F), +, outside a certain set for every function from the class . Here, H(, f) is the class of series that converge for all 0 and have a form

Trace formulae for the eigenvalues of the Laplacian

Summary The trace function where { _m} _m=1 are the eigenvalues of the Laplacian is studied for a variety of domains. The dependence of(t) on the connectivity of a domain and the boundary conditions is analysed. Particular attention is given to annular domains.

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Résumé Pour divers domaines, on étudie la fonction trace , où ₁, ₂, ₃, sont les valeurs propres du laplacien. On analyse comment(t) dépend du domaine et des conditions aux limites. On considère notamment le cas de couronnes circulaires.

     

A sharper stability bound of Fourier frames

Given a real sequence {_n}_n. Suppose that is a frame for L²[–, ] with bounds A, B. The problem is to find a positive constant L such that for any real sequence {_n}_n with ¦_n –_n¦ <L, is also a frame for L²[–, ]. Balan [1] obtained arcsin . This value is a good stability bound of Fourier frames because it covers Kadec's 1/4-theorem and is better than (see Duffin and Schaefer [3]). In this paper, a sharper estimate is given.     

Transformation de Fourier et temps d'occupation browniens

Summary In this paper, we study oscillatory stochastic integrals of the form where is a non zero parameter andg a square integrable function. We study integrability properties of () and its behavior as a function of , using stochastic calculus techniques: martingale theory, representation of Itô for a random variable of the Wiener space, lemma of Garsia-Rodemich-Rumsey .... We also obtain limit theorems in law related to the variables () based upon an asymptotic version of a theorem of Knight on orthogonal continuous martingales.We consider the random measure, image by the Brownian motion of the unbounded measure 1_[0,] (s)g(s) ds; we prove the existence and the continuity of an occupation time density.Finally, under a stronger integrability condition ong, we show the existence of a density for the law of (), using Malliavin's calculus.     

Numerical solution of eigentuple-eigenvector problems in Hilbert spaces by a gradient method

Summary A gradient technique previously developed for computing the eigenvalues and eigenvectors of the general eigenproblemAx=Bx is generalized to the eigentuple-eigenvector problem . Among the applications of the latter are (1) the determination of complex (,x) forAx=Bx using only real arithmetic, (2) a 2-parameter Sturm-Liouville equation and (3) -matrices. The use of complex arithmetic in the gradient method is also discussed. Computational results are presented.This research was partially supported by NSF Grants MPS74-13332 and MCS76-09172     

Isometric dilations in nest algebras

LetT be a contraction acting in a separable Hilbert space and leaving invariant a nest of subspaces of . We answer the question: when doesT have an isometric extension to which leaves invariant the nest = {N N :N ;}.     

Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.