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 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.     

Divisible designs and groups

We study (s, k, ₁, ₂)-translation divisible designs with ₁0 in the singular and semi-regular case. Precisely, we describe singular (s, k, ₁, ₂)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, ₁, ₂)-TDD's (and, more general, for the case ₂>₁) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.     

Lecture Hall Partitions II

For a non-decreasing integer sequence a=(a₁,...,a_n) we define L_a to be the set of n-tuples of integers = (₁,...,_n) satisfying . This generalizes the so-called lecture hall partitions corresponding to a_i=i and previously studied by the authors and by Andrews. We find sequences a such that the weight generating function for these a-lecture hall partitions has the remarkable form In the limit when n tends to infinity, we obtain a family of identities of the kind the number of partitions of an integer m such that the quotient between consecutive parts is greater than is equal to the number of partitions of m into parts belonging to the set P, for certain real numbers and integer sets P. We then underline the connection between lecture hall partitions and Ehrhart theory and discuss some reciprocity results.     

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.     

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.     

On the number of non conjugate subgroups

We shall prove that every group of cardinality ₁ has at least ₁ non conjugate subgroups, and we shall generalize this theorem to many more uncountable cardinalities. For example underGCH for every uncountable cardinal and every groupG of cardinality ,G has at least non conjugate subgroups.Presented by W. Taylor.I would like to thank Rami Grossberg for writing and rewriting this paper, and Wilfrid Hodges for removing many errors and suggesting improvements in presentation; many facts are proved only due to his explicit request.This research was supported by grant (No. 1110) from the United States-Israel Binational Science Foundation.     

Untere und obere Schranken für den tiefsten Eigenwert der elastisch gestützten Membran

Zusammenfassung Es werden untere und obere Schranken für den tiefsten Eigenwert ₁() der elastisch gestützten schwingenden Membran hergeleitet. Die elastische Bindung der Membran am Rande wird durch charakterisiert, und wird als Parameter betrachtet.Die Verwendung des klassischen Rayleigh-Prinzipes liefert obere Schranken, mit Hilfe eines konvexen FunktionalsJ() erhält man obere und untere Schranken. Eine Zerlegungsmethode endlich gibt eine untere Schranke für ₁().

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Summary This article is concerned with the determination of upper and lower bounds for the lowest eigenvalue ₁() of the elastically supported vibrating membrane. The elastic support on the boundary is characterized by which is regarded as a parameter.The classical Rayleigh-Principle gives upper bounds. The use of a convex functionalJ() yields upper and lower bounds for ₁(). A method of decomposition leads to a lower bound for ₁().

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Neu-Technikum, Buchs SG     

Conditions for eigenvectors of a selfadjoint operator-function to form a basis

Consider a functionL() defined on an interval of the real axis whose values are linear bounded selfadjoint operators in a Hilbert spaceH. A point ₀ and a vector ₀ H( ₀ 0) are called eigenvalue and eigenvector ofL() ifL() ifL(₀) ₀ = 0. Supposing that the functionL() can be represented as an absolutely convergent Fourier integral, the interval is sufficiently small and the derivative will be positive at some point, it has been proved that all the eigenvectors of the operator-functionL() corresponding to the eigenvalues from the interval form an unconditional basis in the subspace spanned by them.     

Remarks on the perturbation of analytic matrix functions III

As in [N], [LN] the Newton diagram is used in order to get information about the first terms of the Puiseux expansions of the eigenvalues () of the perturbed matrix pencilT(, )=A()+B(, ) in the neighbourhood of an unperturbed eigenvalue () ofA(). In fact sufficient conditions are given which assure that the orders of these first terms correspond to the partial multiplicities of the eigenvalue ₀ ofA().     

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.     

Difference Sets with n = 2pm

Let D be a (v,k,) difference set over an abelian group G with even n = k - . Assume that t N satisfies the congruences t q _i ^fi (mod exp(G)) for each prime divisor q_i of n/2 and some integer f_i. In [4] it was shown that t is a multiplier of D provided that n > , (n/2, ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > may be removed. As a corollary we obtain that in the case of n= 2p^a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].     

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     

Note on the Rayleigh problem for a visco-elastic fluid

Zusammenfassung Der Mangel an nichtstationären Lösungen für viskoelastische Strömungen hat den Autor veranlasst, das (aufRayleigh zurückgehende) Problem der plötzlich beschleunigten Platte für eine spezielle, vonOldroyd beschriebene viskoelastische Flüssigkeit zu untersuchen.Es wird eine lineare partielle Differentialgleichung erhalten, die, teilweise durch numerische Integretion, für den Bereich 0...1 der Variablen ₂/₁ gelöst wird (₂: Retardationszeit, ₁: Relaxationszeit). Einige Beispiele werden graphisch ausgewertet, und es wird das asymptotische Verhalten angegeben. Zwischen den Fällen ₂/₁=1 (Diffusionsgleichung) und ₂/₁=0 (gedämpfte Wellen) besteht ein kontinuierlicher Übergang. An einer gegebenen Stelle nimmt die Geschwindigkeit anfänglich langsamer und später rascher zu als im viskosen Fall. Dies lässt sich durch Speicherung und Abgabe von elastischer Energie erklären.     

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.     

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.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

Principal part of resolvent and factorization of an increasing nonanalytic operator-function

Let a selfadjoint operator-valued functionL() be given on the interval [a,b] such thatL(a)0,L(b)0,L()0 (ab), andL() has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued functionL() can be reduced to the spectral theory of one operatorZ, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L()=M()(I–Z), where the operator-valued functionM() is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition ofL ^–1() into the sume of its principal and regular parts.