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.     

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.     

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.     

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().

---

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().

---

---

Neu-Technikum, Buchs SG     

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.     

Criteria of algebraic independence of complex numbers over a field of finite transcendence type (II)

In the present paper we establish a criterion of algebraic independence of complex numbers ₁, ..., _n over a field of finite transcendence type using a sequence of nonzero polynomials in several variables with integral coefficients, which satisfy simultaneously certain upper and lower estimates in different orders of magnitude at the point ( ₁, ..., _q , ₁, ..., _n ), where { ₁, ..., _q } is a transcendence basis of over .The Project Supported by the National Natural Science Foundation of China     

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     

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.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

---

Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

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.     

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.     

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     

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.     

Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils

Self-adjoint quadratic operator pencilsL()= ² A + B + C with a noninvertible leading operatorA are considered. In particular, a characterization of the spectral points of positive and of negative type ofL is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.     

Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

We consider a selfadjoint and smooth enough operator-valued functionL() on the segment [a, b]. LetL(a)0,L(b)0, and there exist two positive numbers and such that the inequality |(L()f, f)|< ([a, b] f=1) implies the inequality (L'()f, f)>. Then the functionL() admits a factorizationL()=M()(I-Z) whereM() is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL()0 ( [a,b]). For analytic functionL() the result of this paper was obtained in [13].     

On the Gap between the First Eigenvalues of the Laplacian on Functions and p-Forms

We study the first positive eigenvalue ^(p) ₁(g) of the Laplacian on p-forms for a connected oriented closed Riemannianmanifold (M, g) of dimension m. We show that for 2 p m – 2 a connected oriented closed manifold M admits three metrics g _i (i = 1, 2, 3) such that ^(p) ₁(g ₁)> ⁽⁰⁾ ₁(g ₁),^(p) ₁(g ₂) < ⁽⁰⁾ ₁(g ₂) and^(p) ₁(g ₃)= ⁽⁰⁾ ₁(g ₃).Furthermore, if (M, g) admits a nontrivial parallel p-form,then ^(p) ₁ ⁽⁰⁾ ₁ always holds.     

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence = P : of a chain complete posetP represents a canonical procedure to produce a coreP . It has been proved that if the posetP contains no infinite antichain then this coreP is a retract ofP andP has the fixed point property iffP has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.Supported by a grant from The National Natural Science Foundation of China.     

On a theorem of G. Szegö

It is shown that the precise result of G. Szegö on the asymptotic behavior of the Toeplitz determinants D_n(f), generated by the nonnegative summable function off() holds if Inf L₁ (–, ).Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 693–702, June, 1968.     

The Discrete Spectrum Asymptotics with Respect to a Large Coupling Constant under Strong Nonnegative Perturbations

Let A be a self-adjoint operator, let (, ) be an inner gap in the spectrum of the operator A, and let B(t) = A + tW ^* W, where the operator W(A – iI)^-1 is not necessarily bounded. Conditions are obtained under which the spectrum of B(t) in (, ) is discrete. Let N(, A, W, ), (, ), > 0, be the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to . The asymptotics of N(, A, W, ) as + is obtained in terms of the spectral asymptotics of a certain self-adjoint compact operator. Bibliography: 5 titles.     

Left linear theories — A generalization of module theory

In this paper we introduce left linear theories of exponentN (a set) on the setL as mapsL ×L ^N (l, ) l · L such that for alll L and , L ^N the relation (l · ) =l( · ) holds, where · L ^N is given by ( · )(i) = (i),i N. We assume thatL has a unit, that is an element L ^N withl · =l, for alll L, and · = , for all L ^N . Next, left (resp. right)L-modules andL-M-bimodules and their homomorphisms are defined and lead to categoriesL-Mod, Mod-L, andL-M-Mod. These categories are algebraic categories and their free objects are described explicitly. Finally, Hom(X, Y) andX Y are introduced and their properties are investigated.Herrn Professor Dr. D. Pumplün zum 60. Geburtstag gewidmet