Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rate-balanced. Sharp bounds on in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallest in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,-balanced for0.06.     

Asymptotic behavior of the Jost function of a two-dimensional Schrödinger operator

We find the asymptotic behavior of the Jost function(Z,) of a two-dimensional Schrödinger operator for arbitrary and Z/|Z|S¹ as |Z| We discuss consequences of the asymptotic formulas for the inverse scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 96—103, 1988.     

On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile

A model spectral problem of the form -i)y+xy= y on the finite interval [-1,1] with the Dirichlet boundary conditions is considered. Here is the spectral parameter and is positive. The behavior of the spectrum of this problem as 0 is completely investigated. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.     

Minimization of the Conformal Radius under Circular Cutting of a Domain

Let D be a simply connected domain on the complex plane such that 0 D. For r > 0 , let D _r be the connected component of D {z : |z| < r} containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(D _r, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local -extension E (a) of E among all continua E with given logarithmic capacity. Here, E (a) = E {z : |z – a| }, a E, > 0. Bibliography: 12 titles.     

Construction of ɛ-nets

LetS be a set ofn points in the plane and let be a real number, 0<<1. We give a deterministic algorithm, which in timeO(n ^–2 log(1/)+ ^–8) (resp.O(n ^–2 log(1/)+ ^–10) constructs an-netNS of sizeO((1/) (log(1/))²) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).We also prove that given a setL ofn lines in the plane, we can cut the plane intoO( ^–2) triangles in such a way that no triangle is intersected by more thann lines ofL. We give a deterministic algorithm for this with running timeO(n ^–2 log(1/)). This has numerous applications in various computational geometry problems.     

Über Extremalpunkte des numerischen Wertebereichs eines linearen Operators

For a bounded linear operator T in a Hilbert space denote by (T) the set of the extreme points of its numerical range W(T), and by K(T) the subset of the points of (T) which lie on the boundary of a disc being a spectral set for T in the sense of J. von Neumann. Each corner of W(T) belongs to K(T). This implies: K(T) is dense in (T) if T is compact and the convex hull of its spectrum contains W(T). If Re T0 and o (T) then a necessary and sufficient condition is given that O is a eigenvalue of T. This is applied to show that W(T) (T) is a subset of the point spectrum of T for a class of operators containing the hyponormal operators.     

ɛ-duality theorem of nondifferentiable nonconvex multiobjective programming

Necessary Kuhn-Tucker conditions up to precision without constraint qualification for -Pareto optimality of multiobjective programming are derived. This article suggests the establishment of a Wolfe-type -duality theorem for nondifferentiable, nonconvex, multiobjective minimization problems. The -vector Lagrangian and the generalized -saddle point for Pareto optimality are studied.     

Hermitian K-theory on a Theorem of Giffen

Let be an exact category with duality. In [1] a category () was introduced and the authors asserted that the loop space of the topological realization of () is homotopy equivalent to Karoubis U-theory space of when = (R), the category of finitely generated projective modules over a ring R with an involution if 2 is invertible in R. Unfortunately, their proof contains a mistake. We present a different proof which avoids their argument.Mathematics Subject Classifications (1991): 19DO6, 19G38, 11E70.     

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

Bin packing games

We consider bin packing games introduced by Faigle and Kern (1993) and we restrict ourselves to the subclass of games for which all bins have unit capacity and all items are larger than 1/3. We adopt the taxation model of Faigle and Kern and we prove that for a tax-rate of = sk_7/1 the -core is always non empty. The bound is sharp, since for every < sk_7/1 there exist instances of the bin packing game within our sublass with an empty -core.     

Controllability in Dimensions One and Two of Sobolev-Type Equations in Banach Spaces

We establish conditions necessary for -controllability in dimension one of first-order singular linear differential equation in Banach spaces. This result generalizes similar results for regular equations. For this class of equations, we show that the notion of -controllability in dimension two is more natural, and moreover, analogous necessary conditions are sufficient in the case of dimension two. Using an abstract approach, we derive sufficient conditions for the -controllability in dimension two of the Cauchy--Dirichlet problem for the Barenblatt--Zheltov--Kochina equation.     

Rational Chebyshev approximation on the unit disk

Summary In a recent paper we showed that error curves in polynomial Chebyshev approximation of analytic functions on the unit disk tend to approximate perfect circles about the origin [23]. Making use of a theorem of Carathéodory and Fejér, we derived in the process a method for calculating near-best approximations rapidly by finding the principal singular value and corresponding singular vector of a complex Hankel matrix. This paper extends these developments to the problem of Chebyshev approximation by rational functions, where non-principal singular values and vectors of the same matrix turn out to be required. The theory is based on certain extensions of the Carathéodory-Fejér result which are also currently finding application in the fields of digital signal processing and linear systems theory.It is shown among other things that iff(z) is approximated by a rational function of type (m, n) for >0, then under weak assumptions the corresponding error curves deviate from perfect circles of winding numberm+n+1 by a relative magnitudeO( ^{m + n + 2} as 0. The CF approximation that our method computes approximates the true best approximation to the same high relative order. A numerical procedure for computing such approximations is described and shown to give results that confirm the asymptotic theory. Approximation ofe ^z on the unit disk is taken as a central computational example.     

Probabilities of moderate deviations

Let X₁, X₂, ... be a sequence of independent, identically distributed random variables, and let. The rate of convergence of probabilities of the form andis studied for any > ₀ and some r and ₀. Moreover, necessary and sufficient conditions are given that the relations be satisfied uniformly with respect to x in the region 0 x clog n, where and c are some positive constants, and. Local limit theorems are also presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 85, pp. 6–16, 1979.     

Speed of convergence as a function of given accuracy for random search methods

The essence of this article lies in a demonstration of the fact that for some random search methods (r.s.m.) of global optimization, the number of the objective function evaluations required to reach a given accuracy may have very slow (logarithmic) growth to infinity as the accuracy tends to zero. Several inequalities of this kind are derived for some typical Markovian monotone r.s.m. in metric spaces including thed-dimensional Euclidean space ^d and its compact subsets. In the compact case, one of the main results may be briefly outlined as a constructive theorem of existence: if is a first moment of approaching a good subset of-neighbourhood ofx ₀=arg maxf by some random search sequence (r.s.s.), then we may choose parameters of this r.s.s. in such a way that E c(f) In² . Certainly, some restrictions on metric space and functionf are required.     

On the central limit theorem for geometrically ergodic Markov chains

Let X₀,X₁,... be a geometrically ergodic Markov chain with state space and stationary distribution . It is known that if h: R satisfies (|h|²⁺)< for some >0, then the normalized sums of the X_is obey a central limit theorem. Here we show, by means of a counterexample, that the condition (|h|²⁺)< cannot be weakened to only assuming a finite second moment, i.e., (h²)<.Reasearch supported by the Swedish Research Council.     

Validity of the multiple scale method for very long intervals

For a class of nonlinear oscillation problems containing a small parameter, it is known that a two-scale method using timest and t gives results valid to any desired order for time (1/). We ask when results can be obtained which are valid for (1/²) or for allt > 0. We show that there is an obstruction to introducing a third time scale ² t, and give an example in which this obstruction does not vanish, so that a third scale cannot be introduced, even though the solution exists for all time. The obstruction does vanish if the first order averaged equation vanishes, in which case the three-scale solution actually involves onlyt and ² t and is valid for time (1/²). The obstruction also vanishes if a certain contracting or dissipative condition is met, but in this case the two-scale solution is already valid for all time and the third scale is not needed. These results correspond to known results for the method of averaging, but are here proved for the multiple scale method without use of averaging.     

Simple modules over the multiparameter quantum function algebra at roots of 1

We construct essentially all the irreducible modules for the multiparameter quantum function algebraF [G], whereG is a simple simply connected complex algebraic group, and is a root of unity.     

Asymptotic bounds for the eigenvalues and eigenvectors of a perturbed linear non-self-conjugate operator

We obtain asymptotic bounds for the perturbed eigenvalues and eigenvectors of a perturbed linear bounded operator A(), in a Hilbert space under the assumption that A() is holomorphic at the point =₀ and the eigenvalue ₀= gl(₀) of the operator A(₀) is isolated and of finite multiplicity. We study certain cases of high degeneracy in the limiting problem, i.e., the case when there are generalized associated vectors.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 403–412, October, 1972.The author wishes to express his sincere gratitude to his scientific director T. Sabirov for valuable observations and advice.     

Representation of distributions with compact support

Distributions having compact support are represented as the boundary value of Cauchy and Poisson integrals corresponding to tubular radial domains T^C in ⁿ where C is an open convex cone. The Cauchy integral of U is shown to be an analytic function in T^C which satifies a certain boundedness condition. All analytic functions in T^C having this boundedness condition have a distributional boundary value which can be used to determine an distribution. The results are extended to vector valued distributions.