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

Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents

We are concerned with the semilinear polyharmonic model problem (–)K v = v +v|v| ^s–1 inB,D v|B = 0 for ¦|<-K – 1. HereK ,B is the unit ball in ⁿ,n >2K, is the critical Sobolev exponent. Let ₁ denote the first Dirichlet eigenvalue of (-)^K inB. The existence of a positive radial solutionv is shown for

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.     

Bifurcation of periodic solutions to variational inequalities in R κ based on Alexander-Yorke theorem

Variational inequalities are studied, where K is a closed convex cone in , 3, B is a × matrix, G is a small perturbation, a real parameter. The assumptions guaranteeing a Hopf bifurcation at some ₀ for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some _{I 0}. Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at ₀ constructed on the basis of the Alexander-Yorke theorem as global bifurcation branches of a certain enlarged system.     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

---

Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

A new class of symmetric (v,k, λ)-designs

We construct new families of symmetric (v, k, )-designs with parameters

Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes

A compound Poisson process is of the form where Z, Z ₁, Z ₂, are arbitrary i.i.d. random variables and N is an independent Poisson random variable with parameter . This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate . The truncation level introduced depends only on and Z and not on the overall exceedance level a.     

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 the existence of strongly normal ideals overP κ λ

For every uncountable regular cardinal and any cardinal,P denotes the set . Furthermore, < denotes=" the=" binary=" operation=" defined=">P byx<> iffxy¦x<>.By anideal over P we mean a proper, non-principal,-complete ideal overP extending the ideal dual to the filter generated by . For any idealI overP ,I ⁺ denotes the setP –I, andI ^* the filter dual toI.     

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

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 2-extension of the field ℚ of rational numbersof rational numbers

It is proved that the rational number field has one, and only one, normal 2-extension (_{2, t8)}/with group isomorphic to .If is the maximal subfield of a real-closed field, which does not contain ,then the algebraic closure of is isomorphic to the field .Bibliography: 7titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 192–196.     

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 < .     

On the Existence of Solutions for Some Nondegenerate Nonlinear Wave Equations of Kirchhoff Type

Let be a bounded domain in #x211D;ⁿ with a smooth boundary . In this work we study the existence of solutions for the following boundary value problem:

On positive solutions of quasilinear elliptic systems

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems where –_p is the p-Laplace operator, p > 1 and is a C ^1,-domain in . We prove an analogue of [7, 16] for the eigenvalue problem with and obtain a non-existence result of positive solutions for the general systems.     

On the Existence of Positive Eigenvalues for Semilinear Elliptic Equation on all of R d

In this paper, we consider a problem of the form

Some stability properties for analytic operator functions

Let be a connected, finite-dimensional, complex analytic manifold; let T() be an analytic function defined on , whose values are J-biexpanding operators on a J-space H. Let (A) denote the range of A. The following assertions are proved: 1. The lineals and do not depend on . 2. For arbitrary we have Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 511–520, October, 1976.     

Identities of an algebra of triangular matrices

This paper deals with the ideals of identities of certain associative algebras over a field F of characteristic zero. An algebra W of matrices of the form ,,,M, where and , are F-algebras with unity and M is a (,)-bimodule, is considered. Under certain natural restrictions on M one obtains the equality of ideals of identities T(W)=T()T(), if [[x₁,x₂], x₃[x₄,x₅]]T().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 7–27, 1982.     

A Multiplicity Result for the p-Laplacian Involving a Parameter

We study existence and multiplicity of positive solutions for the following problem