Quasisimilarity of model contractions with unequal defects

Let T and T be C₁₀ contractions with characteristic functions H (ⁿⁿ⁺¹), H (^mm+1). The fundamental result is: T and T are quasisimilar if and only if The paper contains an analysis of this condition; examples are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 24–37, 1986.     

Gauss sums for symplectic groups over a finite field

For a nontrivial additive character and a multiplicative character of the finite field withq elements, the Gauss sums (trg) overgSp(2n,q) and (detg)(trg) overgGSp(2n, q) are considered. We show that it can be expressed as a polynomial inq with coefficients involving powers of Kloosterman sums for the first one and as that with coefficients involving sums of twisted powers of Kloosterman sums for the second one. As a result, we can determine certain generalized Kloosterman sums over nonsingular matrices and generalized Kloosterman sums over nonsingular alternating matrices, which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.Supported in part by Basic Science Research Institute program, Ministry of Education of Korea, BSRI 95-1414 and KOSEF Research Grant 95-K3-0101 (RCAA)Dedicated to my father, Chang Hong Kim     

Waterman classes and triangular sums of double Fourier series

Let ^a ={nln^a (n+1)}, where a R. The following results are established: For every &fnof ^a BV ((- ]²), the triangular partial sums of its Fourier series are uniformly bounded if a = -1, and converge everywhere if a < -1.For every a>0, there exists &fnof ^a BV ((- ]²) such that the triangular partial sums of its Fourier series are unbounded at the point (0;0).     

Optimization over the efficient set: overview

Over the past several decades, the optimization over the efficient set has seen a substantial development. The aim of this paper is to provide a state-of-the-art survey of the development. Given p linear criteria c ¹x,,c^p x and a feasible region X of R ⁿ, the linear multicriteria problem is to find a point x of X such that no point x' of X satisfies (c¹ x',,c^p x')(c¹ x,,c^p x) and (c¹x',,c^p x')q (c¹ x ,,c^p x). Such a point is called an efficient point. The optimization over the efficient set is the maximization of a given function over the set of efficient points. The difficulty of this problem is mainly due to the nonconvexity of this set. The existing algorithms for solving this problem could be classified into several groups such as adjacent vertex search algorithm, nonadjacent vertex search algorithm, branch-and-bound based algorithm, Lagrangian relaxation based algorithm, dual approach and bisection algorithm. In this paper we review a typical algorithm from each group and compare them from the computational point of view.     

On ℓ p -summability of the characteristic values of intergral operators onL(R N )

LetB _p (L ₂) (0<p<) denote the space of all compact linear operatorsK onL(R ^N ) with the following property: the sum ofp-th powers of all positive eigenvalues (repeated according to their multiplicity) of the nonnegative compact operator (K ^* K)^1/2 converges. We prove aB _p -criterion for an integral operatorK onL(R ^N ), for 0<p2. We require that its kernel have sufficient fractional smoothness and decay at infinity. Our proof is based on a suitable factorization of the operatorK which involves fractional powers (H+I)^– 0, 0<1, of the resolvent of the selfadjoint elliptic operatorH=(–) ^m +M _q ^2m onL(R ^N ).     

Controllability of nonlinear systems to affine manifolds

Controllability to an affine manifold involves controlling a system to a target defined by the generalized boundary condition x=r, where :C ⁿ R ⁿ is a bounded linear operator on the continuous functions, as defined for ordinary differential equations by Kartsatos. In this paper, sufficient conditions are obtained for such controllability for linear systems and for a class of nonlinear perturbations of linear systems.     

Nowhere dense set mappings on the generalized linear continua

A set is free for a set mapping F:XP(X) provided xF(y) for any distinct x, y in A. If F maps the reals R into nowhere dense subsets of R, then Bagemihl proved that there is an everywhere dense free set for F, and assuming CH Hechler showed that such an F does not always admit an uncountable free set. In this paper, we show that Bagemihl's theorem cannot be generalized to the generalized linear continua C for arbitrarily large ordinal , and under GCH we extend Hechler's theorem to C for every .     

Asymptotic behavior of general M-estimates for regression and scale with random carriers

Summary Let (xini, y _i be a sequence of independent identically distributed random variables, where x _i R ^p and y _i R, and let R ^p be an unknown vector such that y _i =x _i +u _i (^*), where u _i is independent of x _i and has distribution function F(u/), where >0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, ( ^*,^*), defined as solutions of the system: , where r= (y _i –x _i ^1*/)^*, with R ^p ×RR and RR. This class contains estimators of (, ) proposed by Huber, Mallows and Krasker and Welsch. The consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on and and assuming the joint distribution of (x _i , y _i ) to fulfill the model (^*) only approximately.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Stable solvability of Padé difference schemes for parabolic equations in Hölder spaces

We study Padé difference scheme for the approximate solution of the Cauchy problem for parabolic equations generated by the Padé fractions R_j,, of exponential approximation. We establish an estimate of the coerciveness of the difference schemes for j=–2, –1, or even j= in a smaller space than C₀ (E).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1466–1476, November, 1992.     

Oscillation and Nonoscillation of Higher Order Self-Adjoint Differential Equations

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation (-1)ⁿ(t^alphay⁽ⁿ⁾)⁽ⁿ⁾+q(t)y = 0 (*) are established. In these criteria, equation (*) is viewed as a perturbation of the conditionally oscillatory equation (-1)ⁿ(t^alphay⁽ⁿ⁾)⁽ⁿ⁾ - µ,t^2n-y = 0, where _n, is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.     

The self-similar solutions to a fast diffusion equation

A quasilinear equation u -x·u/2+f(u)=0 is studied, wheref(u)=–u+u , > 0, 0<. <1, >1 andx R ⁿ. The equation arises from the study of blow-up self-similar solutions of the heat equation _t =+. We prove the existence and non-existence of ground state for various combination of , and . In particular, we prove that when / < forn=1,2 or / < (n + 2) /(n – 2) forn 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where / > (n + 2)/(n – 2) there exists an infinite number of non-constant positive radial self-similar solutions.     

Boundary control of parabolic systems: Regularity of optimal solutions

Boundary control problems for the linear, parabolic equations and a quadratic performance index are considered. The controls are allowed to be in the spaceL ²[OT;L²()], where is a boundary. Exploiting the semigroup approach, it is shown that optimal control belongs toL ²[OT;H^1/2()] and, as a consequence, optimal trajectory belongs toL ¹[OT;H¹()]. This result is obtained for two kinds of domains. The first are the domains withC -boundary and the second are the domains being the parallelepipeds.     

More on P-Stable Convex Sets in Banach Spaces

We study the asymptotic behavior and limit distributions for sums S _n =b_n ^-1 _i=1 ⁿ _i,where _i, i 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes S _n(t) =b_n ^-1 _i=1 ^[nt] _i, t[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where _i are segments, the limit of S _n is proved to be countable zonotope. Furthermore, if B = R ^d , the singularity of distributions of two countable zonotopes Y_{p 1}, ₁,Y_{p 2}, ₂, corresponding to values of exponents p ₁, p ₂ and spectral measures ₁, ₂, is proved if either p ₁ p ₂ or _{1 2}; (iv) Some new simple estimates of parameters of stable laws in R ^d , based on these results are suggested.     

Acceleration of convergence for finite element solutions of the Poisson equation

Summary Letu _h be the finite element solution to–u=f with zero boundary conditions in a convex polyhedral domain . Fromu _h we calculate for eachz and ||1 an approximationu _h ^– (z) toD u(z) with |D u(z)–u _h ^– (z)|=O(h ^2k–2) wherek is the order of the finite elements. The same superconvergence order estimates are obtained also for the boundary flux. We need not work on a regular mesh but we have to compute averages ofu _h where the diameter of the domain of integration must not depend onh.     

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.     

Ein beweis des Grauertschen bildgarbensatzes nach ideen von B. Malgrange

In 1960, H. Grauert proved the following coherence theorem [2]: Let X, Y be complex spaces and : X Y a proper holomorphic map. Then, for every coherent analytic sheaf J on X, all direct image sheaves Rⁿ*J are coherent. We give a new proof of this theorem, based on ideas of B. Malgrange. This proof does not use induction on the dimension of the base space Y and can be generalized to relative-analytic spaces X Y where Y belongs to a bigger category of ringed spaces, which contains in particular all complex spaces and differentiable manifolds.     

Noncommutative graded domains with quadratic growth

Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z^–1; ], for some automorphism of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when ||= and thatR is a finite module over its centre when ||<. IfR is not generated in degree one, thenR will still be Noetherian and primitive if has infinite order, butR need not be Noetherian when has finite order.Dedicated to the memory of Shimshon AmitsurOblatum 5-XI-1994 & 28-III-1995This research was supported in part by NSF grants