Approximation of smooth functions on compact two-point homogeneous spaces

Estimates of Kolmogorov n-widths and linear n-widths , (1q∞) of Sobolev's classes , (r>0, 1p∞) on compact two-point homogeneous spaces (CTPHS) are established. For part of (p,q)[1,∞]×[1,∞], sharp orders of or were obtained by Bordin et al. (J. Funct. Anal. 202(2) (2003) 307). In this paper, we obtain the sharp orders of and for all the remaining (p,q). Our proof is based on positive cubature formulas and Marcinkiewicz–Zygmund-type inequalities on CTPHS.     

On Convex Limit Sets and Brownian Motion

We prove limsup results for nonnegative functionals of convex sets determined by normalized Brownian paths in Banach spaces. This continues the interesting investigation of D. Khoshnevisan into this area, and relates to some classical unsolved isoperimetric problems for the convex hull of curves in ^d. Section 4 contains the solution of a problem similar to these classical problems.     

A Weighted Goodness-of-Fit Test for GARCH(1,1) Specification

Suppose that (j) is the lag-j autocorrelation of the squared residuals computed from a realization of length n under the assumption that the observations follow a GARCH(1,1) model. We study the asymptotic distribution of the statistics of the form , where the _j are nonnegative summable weights and the matrix , can be estimated from the data. We show that, under weak assumptions on model errors, the statistic Q _n converges in distribution to , where the N _i are iid standard normal. We discuss choices of the weights _j for which the distribution of Q is tabulated. Our results lead to and provide a rigorous justification for Portmanteau goodness-of-fit tests for GARCH(1,1) specification.     

A large deviation principle for (r,p)-capacities on the Wiener space

Summary We formulate and prove a large deviation principle for the (r, p)-capacity on an abstract Wiener space. As an application, we obtain a sharpening of Strassen's law of the iterated logarithm in terms of the capacity.     

Testing Appearance of Polynomial Trend

For deciding whether a gradual polynomial trend appears in mean of a sequence of independent random variables a trimmed maximum-type statistic may be used. As the number of observations tends to infinity the suggested statistic converges in distribution to a maximum of a differentiable ²-process. The method for asymptotic critical values approximation based on high level exceedence probability was developed.     

A Stationary Rho-Mixing Markov Chain Which Is Not “Interlaced” Rho-Mixing

A strictly stationary, countable-state Markov chain is constructed which is -mixing (with arbitrarily fast mixing rate) but fails to be *-mixing (interlaced-mixing).     

Iterative Schemes for Solving Mixed Variational-Like Inequalities

In the present paper, we introduce the concept of -cocoercivity of a map and develop some iterative schemes for finding the approximate solutions of mixed variational-like inequalities. We use the concept of -cocoercivity to prove the convergence of the approximate solutions to the exact solution of mixed variational-like inequalities.     

The ε c -Product of a Schwartz b-Space by a Quotient Banach Space and Applications

We define the -product of a -space by a quotient Banach space. We give conditions under which this -product will be monic. Finally, we define the _c -product of a Schwartz b-space by a quotient Banach space and we give some examples of applications.     

Limit Theorems for Logarithmic Averages of Fractional Brownian Motions

We prove almost sure invariance principles for logarithmic averages of fractional Brownian motions.Research supported byResearch supported by     

On integration in complete bornological locally convex spaces

A generalization of I. Dobrakov's integral to complete bornological locally convex spaces is given.     

A Numerical Comparison of Some Modified Controlled Random Search Algorithms

In this paper we propose a new version of the Controlled Random Search(CRS) algorithm of Price. The new algorithmhas been tested on thirteen global optimization test problems. Numericalexperiments indicate that the resulting algorithm performs considerablybetter than the earlier versions of the CRS algorithms. The algorithm,therefore, could offer a reasonable alternative to many currently availablestochastic algorithms, especially for problems requiring direct searchtype methods. Also a classification of the CRS algorithms is made based onglobal technique – local technique and the relative performance ofclasses is numerically explored.     

Existence theorems for games of survival

A two-person, zero-sum differential game of survival with general type phase constraints is investigated. The dynamics of both players is governed by a system of differential inclusions. Player II can choose any strategy in the Varaiya-Lin sense, while player I can select any lower -strategy (Ref. 1, p. 400). The existence of a value and an optimal strategy for player II is proved under the assumptions that the set of all player II's trajectories is compact in the Banach space of all continuous mappings and that some capturability condition is fulfilled.     

Controllability and extremality in nonconvex differential inclusions

LetF:[0, T]×R ⁿ 2 ^{R n} be a set-valued map with compact values; let :R ⁿ R ^m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion . We prove sufficient conditions for (z(T))intR and establish necessary conditions in maximum principle form for (z(T))(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.The author is grateful to an anonymous referee for his valuable remarks and comments which have helped to improve the paper.The paper was written while the author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.     

L ∞ Algebra Representations

It is shown that the direct sum of an L algebra and a module over it has a natural L algebra structure.     

Spatial shape‐preserving interpolation using ν‐splines

We present a global iterative algorithm for constructing spatial G ²continuous interpolating splines, which preserve the shape of the polygonal line that interpolates the given points. Furthermore, the algorithm can handle data exhibiting two kinds of degeneracy, namely, coplanar quadruples and collinear triplets of points. The convergence of the algorithm stems from the asymptotic properties of the curvature, torsion and Frénet frame of splines for large values of the tension parameters, which are thoroughly investigated and presented. The performance of our approach is tested on two data sets, one of synthetic nature and the other of industrial interest.     

Study of Noetherian Γ-Semirings via Operator Semirings

In this paper we study Noetherian -semirings and obtain Cohens theorem for a special class of -semirings. Weak primary decomposition theorem for a particular type of -semirings is also obtained.Presently Lecturer in Mathematics, University of Burdwan, GOLAPBAG, W.B. INDIA.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Robustness of minimax controllers to nonlinear perturbations

We study the robustness of minimax controllers, originally designed for nominal linear or nonlinear systems, to unknown static nonlinear perturbations in the state dynamics, measurement equation, and performance index. When the nominal system is linear, we consider both perfect state measurements and general imperfect state measurements; in the case of nominally nonlinear systems, we consider perfect state measurements only. Using a differential game theoretic approach, we show for the former class that, as the perturbation parameter (say, >0) approaches zero, the optimal disturbance attenuation level for the overall system converges to the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy certain prescribed growth conditions. We also show that anH -controller, designed based on a chosen performance level for the nominal linear system, achieves the same performance level when the parameter is smaller than a computable threshold, except for the finite-horizon imperfect state measurements case. For that case, we show that the design of the nominal controller must be based on a decreased confidence level of the initial data, and a controller thus designed again achieves a desired performance level in the face of nonlinear perturbations satisfying a computable norm bound. In the case of nominally nonlinear systems, and assuming that the nominal system is solvable, we obtain sufficient conditions such that the nominal controller achieves a desired performance in the face of perturbations satisfying computable norm bounds. In this way, we provide a characterization of the class of uncertainties that are tolerable for a controller designed based on the nominal system. The paper also presents two numerical examples; in one of these, the nominal system is linear; in the other one, it is nonlinear.This research was supported in part by the US Department of Energy, Grant DE-FG-02-88-ER-13939 and in part by the National Science Foundation, Grant ECS-91-13153.An abridged version was presented at the 32nd IEEE Conference on Decision and Control, San Antonio, Texas, December 15–17, 1993, and it appeared in the Conference Proceedings.