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.     

Equivalent conditions for representing analytic functions by exponential series

Let L() be an entire function of exponential type with simple zeros ₁, ₂, ...; let ¯D be the smallest closed convex set which contains all of the singularities of the function which is associated with L() in the sense of Borel. In [1] there are necessary and sufficient conditions on L() under which a function f(z) which is analytic in ¯D can be represented in D by a Dirichlet series with exponents ₁, ₂, ... We obtain new equivalent conditions on L().Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 91–104, July, 1976.     

Asymptotic Behaviour of λ-Convex Sets in the Hyperbolic Plane

It is known that the limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal 1, and exactly 1 when the sets considered are convex with respect to horocycles. We consider geodesics and horocycles as particular cases of curves of constant geodesic curvature with 0 1 and we study the above limit Area/Length as a function of the parameter .     

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.

     

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.     

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.     

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.     

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.     

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.     

Existence of Incomplete Transversal Designs with Block Size Five and Any Index λ

The basic necessary condition for the existence of a TD(5, ; v)-TD(5, ; u), namely v 4u, is shown to be sufficient for any 1, except when (v, u) = (6, 1) and = 1, and possibly when (v, u) = (10, 1) or (52, 6) and = 1. For the case = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

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.     

On polynomial EkP matrices

In this paper the relation betweenEP--matrices andE ^k P--matrices over an arbitrary filedF is studied. Further, conditions for the product ofE ^k P--matrices to be anE ^k P--matrix and for the reverse order law to hold for the polynomial Moore-Penrose inverse of the product ofE ^k P--matrices are determined     

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.     

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 .     

A Remark on Real Parameter Global Bifurcation

We study the nonlinear eigenvalue problem F(x,) = L()x +R(x,) = 0 where F : X × R X with X a Hilbert space. IfL() is a polynomial in , then it is shown that 0> 0 is a global bifurcation point of the eigenvalue problem provided astandard transversality condition is satisfied, the dimension of the nullspace of L(0) is an odd number and L() is composed of asequence of positive operators on the finite dimensional null space ofL(0).     

Nonlinear eigenvalue approximation

Summary For each in some domainD in the complex plane, letF() be a linear, compact operator on a Banach spaceX and letF be holomorphic in . Assuming that there is a so thatI–F() is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue . In the Newton method the smallest eigenvalue of the operator pencil [I–F(),F()] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I–F(),I] with algebraic multiplicitym. For fixed leth() denote the arithmetic mean of them eigenvalues of the pencil [I–F(),I] which are closest to 0. Thenh is holomorphic in a neighborhood of andh()=0. Under suitable hypotheses the classical Muller's method applied toh converges locally with order approximately 1.84.     

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.     

A note on the problem −Δu=λu+u|u|2*−2

The dual approach to the problem –u=u+u|u|^2*–2, uþe ₀ ¹ (gW), permits a simple proof of a recent existence result [5] and allows extensions of this result to similar problems also with asymmetric mnonlinearities.Supported by min P.I., Gruppo Naz. (40%) Calcolo delle Variazioni...