Extremal Eigenvalues of the Laplacian in a Conformal Class of Metrics: The `Conformal Spectrum'

Let M be a compact connected manifold of dimension n endowed witha conformal class C ofRiemannian metrics of volume one. For any integer k 0, we consider the conformal invariant _k ^c (C) defined as the supremum of the k-th eigenvalue _k (g) of the Laplace–Beltrami operator _g , where g runs over C.First, we give a sharp universal lower bound for _k ^c (C) extending to all k a result obtained by Friedlander andNadirashvili for k = 1. Then, we show that the sequence \{ _k ^c (C)\}, that we call `conformal spectrum',is strictly increasing and satisfies, k 0, _k+1 ^c (C) ^n/2– _k ^c (C) ^n/2 n ^n/2 _n , where _n is the volume of the n-dimensionalstandard sphere.When M is an orientable surface of genus , we also considerthe supremum _k ^top()of _k (g) over theset of all the area one Riemannian metrics on M, and study thebehavior of _k ^top() in terms of .     

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.     

The application of an interpolational method

We consider an equation of infinite order in generalized derivatives in the sense of A. O. Gel'fond with a characteristic function of finite order L()=L₁()... L_n(). It is explained by a purely analytical method (by the application of an interpolational method) when any solution of the equation is the sum of the solutions of similar equations with the characteristic functions L₁(), ..., L_n(). In the case of an equation in ordinary derivatives, when L() is a function of exponential type, the problem is solved by the application of algebraic and functional methods of V. V. Napalkov [1].Translated from Matematicheskie Zametki, Vol. 18, No. 5, pp. 735–752, November, 1975.     

On the immersion of metrics close to immersible metrics

For a C^r,-immersion z:X E, r 2, 0 < < 1, of an n-dimensional (n 1) simply-connected C^r+2,-manifold X into Euclidean space E, the metric I(z) induced by z has a neighborhood in C^r,-topology in which every metric from a given subbundle of metrics is C^r,-immersible into E. In particular, it is proved that metric ds ₀ ² of the Riemannian product of p spheres of dimensions ₁, , _p 2 has a neighborhood in C^2,-topology from which any conformally equivalent metric to ds ₀ ² , is immersible into E with dimE = ₁ + + _p + p. The proofs are based on the investigation of a varied system of Gauss—Codazzi—Ricci equations for an infinitely small deformation of surface z(X) in E with a prescribed variation of the metric.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 49–67, 1992.     

Reducibility of nonlinear almost periodic systems of difference equations on an infinite-dimensional torus

We establish sufficient conditions for the reducibility of a nonlinear system of difference equationsx(t+1)=x(t)++P(x(t),t+ to a system y(t+1)= y(t)+, wherex, , m and the infinite-dimensional vector function P(x(t),t) is 2p-periodic inx _i i=1,2,...) and almost periodic int with a frequency basis.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1216–1223, September, 1994.     

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.     

Reverse Functional Inequalities and Their Applications to Nonlinear Elliptic Boundary Value Problems

We establish some reverse inequalities. We give applications to nonlinear elliptic boundary value problems containing a parameter which have two branches of solutions u (0) and U (>0) of which the first is continuous at the origin and the second increases indefinitely as 0.     

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.

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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.

     

Transfinite interpolation on the medians of a triangle and best L 1-approximation

Let be a triangle in and let be the set of its three medians. We construct interpolants to smooth functions using transfinite (or blending) interpolation on The interpolants are of type f(₁)+g(₂)+h(₃), where (₁,₂,₃) are the barycentric coordinates with respect to the vertices of . Based on an error representation formula, we prove that the interpolant is the unique best L¹-approximant by functions of this type subject the function to be approximated is from a certain convexity cone in C³().Received: 17 December 2003     

The dynamics of perturbations of the contracting Lorenz attractor

We show here that by modifying the eigenvalues ₂ < ₃ < 0 < ₁ of the geometric Lorenz attractor, replacing the usualexpanding condition ₃+₁ > 0 by acontracting condition ₃+₁ < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space. Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them. On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects.     

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 .     

Properties of eigenvalues of positive operators depending nonlinearly on a parameter

It is proved that in a Banach space with a cone, the nonlinear (with respect to the spectral parameter ) eigenvalue problem (here is a scalar function and M is an operator-function, positive with respect to the cone for 0) admits a simple positive eigenvalue ₁. The number ₁ is characterized by variational principles. The obtained results are illustrated on the example of the quadratic pencil y+y+²B²x²y=0, which arises in the theory of structural analysis (see Ref. Zh. 1974, 9B1134).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 177–181, 1976.The author expresses his deep gratitude to D. F. Kharazov and Yu. Sh. Abramov for the interest shown and the advice given during the preparation of this paper.     

Asymptotic expansion of the spectral function of elliptic operators inR n

Let {E} be the spectral family of the operator corresponding to a self-adjoint second-adjoint second-order elliptic differential equation or to a first-order elliptic system inR ⁿ . The coefficients of the equations are assumed to be constant in a neighborhood of infinity and trapping rays are absent. We obtain the full asymptotic expansion as of the kernel e ¹ (x, y) of the operator dE/d in terms of rays going from the point y into the point x.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 75–87, 1987.     

Excesses of systems of exponential functions

Conditions on the closeness of real sequences {_n} and {_n} are studied which imply the equality of the excesses of the systems {exp(i_nx)} and {exp(i_nx)} in the space L²(–a, a). A theorem is formulated in terms of the difference of the sequences {_n} and {_n} enumerating the functions. In the corollaries of the theorem, conditions are given in terms of the behavior of the difference _n–_n0. An example is constructed showing that the condition _n–_n0 alone is not sufficient for equality of the excesses.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 803–814, December, 1977.     

Asymptotic behavior of the spectrum of the Maxwell operator

For the distribution functions of the positive and the negative eigenvalues of the operator in a domain with a smooth boundary, one obtains the asymptotic formula N^±()=(3²)^–1 mes ·³+0(²). Under additional assumptions on the properties of the geodesic billiard in , one shows that N±()= (3²)^–1 mes ·³+0(²).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 169–180, 1983.     

The existence of some resolvable block designs with divisibility into groups

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; ₁, ₂) without repeated blocks and with arbitrary parameters such that ₁ = k, (v–1)/(k–1) ₂ v^k–2 (and also ₁ k/2, (v–1)/(2(k–1)) ₂ v^k–2 in case k is even) k 4 andp=1 (mod k–1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, ) without repeated blocks is deduced with X = k (and also with = k/2 in case of even k) k , where a is a natural number if k is a prime power and=1 if k is a composite number.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 623–634, April, 1976.     

Isolated point of spectrum ofP-hyponormal, log-hyponormal operators

LetT B(H) be a bounded linear operator on a complex Hilbert spaceH. Let ₀ (T) be an isolated point of (T) and let be the Riesz idempotent for ₀. In this paper, we prove that ifT isp-hyponormal or log-hyponormal, thenE is self-adjoint andE H=ker(H–₀)=ker(H–₀ ^*.This research was supported by Grant-in-Aid Research 1 No. 12640187.     

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.     

The domain of partial attraction of a Poisson law

Groshev gave a characterization of the union of domains of partial attraction of all Poisson laws in 1941. His classical condition is expressed by the underlying distribution function and disguises the role of the mean of the attracting distribution. In the present paper we start out from results of the recent probabilistic approach and derive characterizations for any fixed >0 in terms of the underlying quantile function. The approach identifies the portion of the sample that contributes the limiting Poisson behavior of the sum, delineates the effect of extreme values, and leads to necessary and sufficient conditions all involving . It turns out that the limiting Poisson distributions arise in two qualitatively different ways depending upon whether >1 or <1. A concrete construction, illustrating all the results, also shows that in the boundary case when =1 both possibilities may occur.