Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

A characterization of semimartingales on nuclear spaces

Summary This work is devoted to prove the following fact: Suppose that is a nuclear space whose dual is nuclear under the strong topology. IfX is a weakly adapted mapping with values in such that for any,X'() has a modification which is a semimartingale then there exists a unique projective system of Hubert space-valued semimartingales indexed by the Hilbert-Schmidt neighbourhood base of the dual space whose projective limit isX.In the last part we study in detail a semimartingale defined as the convolution of a distribution by a random Dirac measure whose support is determined by the trajectories of a real-valued semimartingale.     

Generalized convex functions and vector variational inequalities

In this paper, (, ,Q)-invexity is introduced, where :X ×X intR _m ⁺ , :X ×X X,X is a Banach space,Q is a convex cone ofR ^m . This unifies the properties of many classes of functions, such asQ-convexity, pseudo-linearity, representation condition, null space condition, andV-invexity. A generalized vector variational inequality is considered, and its equivalence with a multi-objective programming problem is discussed using (, ,Q)-invexity. An existence theorem for the solution of a generalized vector variational inequality is proved. Some applications of (, ,Q)-invexity to multi-objective programming problems and to a special kind of generalized vector variational inequality are given.The author is indebted to Dr. V. Jeyakumar for his constant encouragement and useful discussion and to Professor P. L. Yu for encouragement and valuable comments about this paper.     

Fourth Order Linear Recurrences Satisfied by Wythoff Pairs

A result of Stolarsky is extended to show that there are infinitely many irreducible fourth order linear recurrences satisfied by sequences of pairs , where i is a natural number and is the golden ratio . It is also proved that the characteristic polynomial of every such recurrence factorizes non-trivially if is adjoined to the rationals.     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD     

Linear cyclic codes of wordlength v over GF(q s ) which are also linear cyclic codes of wordlength sv over GF(q)

Any nonsingular linear transformation : GF(q^s) GF(q^s) can be used to treat a linear cyclic code of wordlength v over GF(q^s) as a linear code () of Wordlength sv over GF(q). This paper determines those linear cyclic codes and transformations for which the resulting linear code () is also cyclic.     

Two limit theorems on ARIMA models

Let the time series {X(t), t=1, 2, ...} satisfy (B)(1–B) ^d X(t)=(B)e(t), whereB is a backward shift operator, defined byBX(t)=X(t–1), and (z)=1+₁ z+...+ _p z ^p , (z)=1+₁ z+...+ _q z ^q , and all the roots of (z) lie outside the unit circle; {e(t)} is a sequence of iid random variables with mean zero andE|e(t)|^4+r < (r>0). In this paper, the limit properties of , where the integerd1, have been considered.     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.     

Complete hyperbolicity of Hartogs domain

Let X be a complex space and an upper semicontinuous function on X. Consider the Hartogs domain (X) given by (X)={(z, w)X×C: |w| < e ^–(z) }. In this article, some necessary and sufficient conditions on the complete hyperbolicity of (X) are established. Mathematics Subject Classification (2000):32A10, 32C10, 32H20, 32A17     

Asymptotic formulas for Toeplitz and Wiener-Hopf operators

For certain analytic functions f, the expression trace {f(T_n[]) – f(T_n[]T_n[])} is computed asymptotically. Here T_n[] is the finite Toeplitz matrix generated by the function . The analogous expression for Wiener-Hopf operators is also computed asymptotically. These results in turn yield information concerning the asymptotic behavior of determinants of finite Toeplitz and Wiener-Hopf operators with discontinuous generating function.     

The Groups H3(F(X)/FM) and CH2(X) for Generic Splitting Varieties of Quadratic Forms

Let F be a field of characteristic different from 2 and be a quadratic form over F. Let X be an arbitrary projective homogeneous generic splitting variety of . For example, we can take X to be equal to the variety X_,m of totally isotropic m-dimensional subspaces of V, where V is the quadratic space corresponding to and < dim V. In this paper, we study the groups CH²(X) and H³(F(X)/F) = ker(H ³(F) H ³(F(X))). One of the main results of this paper claims that the group Tors CH²(X) is always zero or isomorphic to . In many cases we prove that Tors CH²(X) = 0 and compute the group H ³(F(X)/F) completely. As an application of the main results, we give a criterion of motivic equivalence of eight-dimensional forms except for the case where the Schur indices of their Clifford algebras equal 4.     

Nonlinear programming,approximation, and optimization on infinitely differentiable functions

A nonnegative, infinitely differentiable function defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ₀ ¹ (t)dt=1. In this article, the following problem is considered. Determine _k =inf ₀ ¹ |^(k)(t)|dt,k=1, 2, ..., where ^(k) denotes thekth derivative of and the infimum is taken over the set of all mollifier functions , which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. The problem is reducible to three equivalent problems, a nonlinear programming problem, a problem on the functions of bounded variation, and an approximation problem involving Tchebycheff polynomials. One of the results of this article shows that _k =k!2^2k–1,k=1, 2, .... The numerical values of the optimal solutions of the three problems are obtained as a function ofk. Some inequalities of independent interest are also derived.This research was supported in part by the National Science Foundation, Grant No. GK-32712.     

A Foliation of the Space of Conjugacy Classes of Representations of a Surface Group

Let be the fundamental group of a closed orientable surface of genus g 1, and let R(, G)/G be the space of conjugacy classes of representations of into a connected real reductive Lie group G. Motivated by the theory of geometric quantization, we define a map ¯ on R(, G)/G and investigate whether the fibres of ¯ are isotropic with respect to the natural symplectic structure on R(, G)/G. If g = 2 and G = SU(2), then the foliation given by the fibres of ¯ is equivalent to a real polarization defined by Weitsman, and we reprove his result that the fibres are isotropic in this case. If g = 1 then the fibres of ¯ are also isotropic, but we give an example to show that in general they are not.     

On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

On a bounded C ²-domain we consider the singular boundary-value problem 1/2u=f(u) in D, u _D =, where d3, f:(0,)(0,) is a locally Hölder continuous function such that f(u) as u0 at the rate u ^–, for some (0,1), and is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in . Such solutions are shown to satisfy a boundary Harnack principle.     

On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

Weak Variation of Gaussian Processes

Let X(t) (tR) be a real-valued centered Gaussian process with stationary increments. We assume that there exist positive constants ₀, C ₁, and c ₂ such that for any tR and hR with |h|₀ and for any 0r<min{|t|, ₀} where is regularly varying at zero of order (0 < < 1). Let be an inverse function of near zero such that (s)=(s) log log(1/s) is increasing near zero. We obtain exact estimates for the weak -variation of X(t) on [0,a].     

Grid structure of a ballistic function and a computing instrument with linear potentiometers for its generation

Zusammenfassung Die Arbeit befasst sich mit der ballistischen Beziehung zwischen Schrägdistanzs, Lagewinkel und Geschossflugzeitt zum Ziel im Vakuum.Der 1. Teil behandelt die Gewebestruktur dieser Beziehung. Es zeigt sich, dass sie annähernd ein Sechseckgewebe ist. Eine topologische TransformationS(s), (),T(t) wird so gefunden, dass eine Zerlegung in Funktionen einer Variablen in der FormT(t)=S(s)+() die Beziehungt(s,) mit befriedigender Genauigkeit darstellt (der maximale relative Fehler ist weniger als 1%).Im 2. Teil wird auf dieser Grundlage ein Rechengerät zur Darstellung der Funktiont(s, ) konstruiert, das lineare Potentiometer als Schaltelemente benutzt.