Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

Nonparametric Inference for a Class of Stochastic Partial Differential Equations II

Consider the stochastic partial differential equationdu (t,x) = (t)u (t, x)dt + dW _Q(t,x), 0 t T where = ₂/x ₂, and is a class of positive valued functions. We obtain an estimator for the linear multiplier (t) and establish the consistency, rate of convergence and asymptotic normality of this estimator as 0.     

Joint convergence of conditional expectations

Let P_n, nIN{0}, be probability measures on a-fieldA; f_n, nIN{0}, be a family of uniformly boundedA-measurable functions andA _n, nIN, be a sequence of sub--fields ofA, increasing or decreasing to the-fieldA _o. It is shown in this paper that the conditional expectations converge in P_o-measure to with k, n, m , if P_n|A, nIN, converges uniformly to P_n|A and f_n, nIN, converges in P_o-measure to f_o.     

Aufeinanderfolgende Elemente in multiplikativen Zahlenmengen

LetM be a multiplicative set with 1M andmnM if and only ifmM,nM for (m,n)=1. It is shown by elementary means that there exists the asymptotic density of the setM(M–1) for every multiplicative setM. The density is positive if and only ifM possesses a positive density and 2M for some . This result is slightly generalized to sums over multiplicative functionsf with |f|1.     

The Logarithmic Frequency of Values of Additive Functions

We consider the weak convergence of distribution functions (_mx 1/ m)^-1 _{m x},f_x(m)x is a set (x 2) of strongly additive functions such that f_x(p){0,1} for each prime number p.     

On the complete solution of ∈y″=y 3

In Ref. 1, the author claimed that the problem y=y ³ is soluble only for a certain range of the parameter . An analytic approach, as adopted in the following contribution, reveals that a unique solution exists for any positive value of . The solution is given in closed form by means of Jacobian elliptic functions, which can be numerically computed very efficiently. In the limit 0+, the solutions exhibit boundary-layer behavior at both endpoints. An easily interpretable approximate solution for small is obtained using a three-variable approach.     

Une approche transformationnelle a l'algebre universelle

In this paper we propose an axiomatization of the notion of system of terms of a theory by means of which we obtain a representation of equational classes (or varieties) of algebras. We define analgebraic transformational system (S.T.A.) as a quadruple (T,v,S,+) satisfying the axioms, where T is a set containing the variables v(n), n, and having operators S(): TT, . In addition there are operations Q⁺ on T commuting with the operators. A notion of morphism between S.T.A. 's is defined to obtain the category which is shown to be equivalent to the dual of the category of equational classes. In the last section we establish the equivalence between and Lawvere's category of algebraic theories in which every definable constant is present.

---

---

Extrait de la Thèse de doctorat de l'auteur, Université de Montréal, 1971.     

OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC ²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III₁.     

Almost Stationary ∈-Equilibria in Zero-Sum Stochastic Games

We show the existence of almost stationary -equilibria, for all > 0, in zero-sum stochastic games with finite state and action spaces. These are -equilibria with the property that, if neither player deviates, then stationary strategies are played forever with probability almost 1. The proof is based on the construction of specific stationary strategy pairs, with corresponding rewards equal to the value, which can be supplemented with history-dependent -optimal strategies, with small > 0, in order to obtain almost stationary -equilibria.     

Connected and alternating vectors: Polyhedra and algorithms

Given a graphG = (V, E), leta ^S, S L, be the edge set incidence vectors of its nontrivial connected subgraphs.The extreme points of = {x R ^E: a^sx |V(S)| - |S|, S L} are shown to be integer 0/± 1 and characterized. They are the alternating vectorsb ^k, k K, ofG. WhenG is a tree, the extreme points ofB 0,b ^kx 1,k K} are shown to be the connected vectors ofG together with the origin. For the four LP's associated with andA, good algorithms are given and total dual integrality of andA proven.On leave from Swiss Federal Institute of Technology, Zurich.     

On the Representation of Linear Continuous Functionals in Spaces of Analytic Functions of the Hardy–Sobolev Type in a Polydisk

We describe dual spaces of classes of the Hardy–Sobolev type of functions holomorphic in a polydisk for 0 < p 1 and q (0, ) and for p (1, ) and q = 1.     

Some minimax problems of vector-valued functions

The concepts of cone extreme points, cone saddle points, and cone saddle values are introduced. The relation of inclusion among the sets mini _xX max _yY f(x, y), maxi _yY min _xX f(x, y), and the set of all weak cone saddle values is investigated in the case where the image space ⁿ off is ordered by an acute convex cone.The author is grateful for the useful suggestions and comments given by Prof. K. Tanaka, Niigata University, Niigata, Japan.The author would like to thank the referees for their valuable suggestions on the original draft.     

Efficiency of Proximal Bundle Methods

We give efficiency estimates for proximal bundle methods for finding f_*min_Xf, where f and X are convex. We show that, for any accuracy <0, these methods find a point x^kX such that f(x^k)–f^* after at most k=O(1/³) objective and subgradient evaluations.     

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.     

An Averaging Method for Singularly Perturbed Systems of Semilinear Differential Inclusions with C0-Semigroups

We consider a system of two semilinear differential inclusions with infinitesimal generators of C ₀-semigroups. The nonlinear terms are of high frequency with respect to time and periodic with a specified period. Moreover, they are condensing in the state variables (x,y) with respect to a suitable measure of noncompactness. The goal of the paper is to give sufficient conditions to guarantee, for >0 sufficiently small, the existence of periodic solutions and to study their behaviour as 0. The main tool to achieve this is the topological degree theory for uppersemicontinuous, condensing vector fields.     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

The Solution Set to BVP for Some Functional Differential Inclusions

We consider a multivalued BVP x'(t) A(t)x((t))+ F(t,x((t))), Lx= . Under appropriate assumptions on A, L and F, we prove that for sufficiently small the set of solutions to this problem is a nonempty infinite dimensional AR-space (Theorem 4).     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

On Domination Number of 4-Regular Graphs

Let G be a simple graph. A subset S V is a dominating set of G, if for any vertex v V – S there exists a vertex u S such that uv E(G). The domination number, denoted by (G), is the minimum cardinality of a dominating set. In this paper we prove that if G is a 4-regular graph with order n, then (G) ⁴/₁₁ n