Approximation of Distributions by Parametric Sets

Let P be a probability distribution on a separable metric space (S,d). We study the following problem of approximation of a distribution P by a set from a given class A2 ^S : W(A,P) _S (d(x,A))P(dx)min _AA , where is a nondecreasing function. A special case where A is a parametric class A={A():T} is considered in detail. Our main interest is to obtain convergence results for sequences {A ^* _n }, where A ^* _n is an optimal set for a measure P _n satisfying P _n P, as n.     

Selection of smoothing parameters in<Emphasis Type="Italic">B</Emphasis>-spline nonparametric regression models using information criteria

We consider the use ofB-spline nonparametric regression models estimated by the maximum penalized likelihood method for extracting information from data with complex nonlinear structure. Crucial points inB-spline smoothing are the choices of a smoothing parameter and the number of basis functions, for which several selectors have been proposed based on cross-validation and Akaike information criterion known as AIC. It might be however noticed that AIC is a criterion for evaluating models estimated by the maximum likelihood method, and it was derived under the assumption that the ture distribution belongs to the specified parametric model. In this paper we derive information criteria for evaluatingB-spline nonparametric regression models estimated by the maximum penalized likelihood method in the context of generalized linear models under model misspecification. We use Monte Carlo experiments and real data examples to examine the properties of our criteria including various selectors proposed previously.     

Characterization of Two-Scale Gradient Young Measures and Application to Homogenization

This work is devoted to the study of two-scale gradient Young measures naturally arising in nonlinear elasticity homogenization problems. Precisely, a characterization of this class of measures is derived and an integral representation formula for homogenized energies, whose integrands satisfy very weak regularity assumptions, is obtained in terms of two-scale gradient Young measures.     

Discrete Approximation of Unbounded Operators and Approximation of their Spectra

Let E be a Banach space over and let the densely defined closed linear operator A: (A)E→E be discretely approximated by the sequence ((A_n, (A_n)))_n of operators A_n where each A_n is densely defined in the Banach space F_n. Let σ_a(A) be the approximate point spectrum of A and let σ(A_n) denote the -pseudospectrum of A_n. Generalizing our own result, we show that σ_a(A)lim inf σ(A_n)=_n ∩_kn σ(A_k) holds for every >0. We deduce that then for every compact set K lim_n dist(σ_a(A)∩K, σ_a(A_n))=0 provided there exists M>0 such that (λ−A_n)⁻¹M dist(λ, σ(A_n))⁻¹ holds for every n and every λ in the resolvent set ρ(A_n) of A_n. We finally treat the problem under which conditions σ_a(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩_>0 ∩_{n kn} σ_, a(A_k)σ_a(A) hold where σ_, a(A) denotes the -approximate pseudospectrum?     

The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)     

Beta Approximation to the Distribution of Kolmogorov-Smirnov Statistic

The distribution of Kolmogorov-Smirnov statistic can be globally approximated by a general beta distribution. The approximation is very simple and accurate. It can be easily implemented in any statistical software. Therefore, we can use a beta distribution to find the practical p-value of a goodness-of-fit test, which is much simpler than existing methods in the literature.     

On two questions of the theory of retracts

We establish that condition (Γ) on brick decomposition is indecomposable. This answers K. Borsuk’s question [1]. We prove that there exist metric spaces X and Y and a point (a, b) ∈ X × Y such that (a, b) is an r-point of the product X × Y; moreover, a is not an r-point of X. This answers A. Kosinski’s question [2].     

Approximation of a function given by its Laurent series

Approximants are defined for a function which is holomorphic in an annulus. They are shown to have good qualitative properties whenf is meromorphic with a fixed number of poles in the annulus. Their denominators are linked to the reverse orthogonal polynomials of dimension 2, or orthogonal polynomials of dimension –2, following the choice of the parameters. Their numerators then follow the same recurrence relation as the denominators.This work was supported by the Human Capital and Mobility Programme of the European Community, Project ROLLS, Contract CHRX-CT93-0416(DG 12 Coma).     

Sturm Sequences and Random Eigenvalue Distributions

This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation of eigenvalue distributions of random matrix ensembles. We first explore the use of Sturm sequences to efficiently compute histograms of eigenvalues for symmetric tridiagonal matrices and apply these ideas to random matrix ensembles such as the β-Hermite ensemble. Using our techniques, we reduce the time to compute a histogram of the eigenvalues of such a matrix from O(n ²+m) to O(mn) time where n is the dimension of the matrix and m is the number of bins (with arbitrary bin centers and widths) desired in the histogram (m is usually much smaller than n). Second, we derive analytic formulas in terms of iterated multivariate integrals for the eigenvalue distribution and the largest eigenvalue distribution for arbitrary symmetric tridiagonal random matrix models. As an example of the utility of this approach, we give a derivation of both distributions for the β-Hermite random matrix ensemble (for general β). Third, we explore the relationship between the Sturm sequence of a random matrix and its shooting eigenvectors. We show using Sturm sequences that assuming the eigenvector contains no zeros, the number of sign changes in a shooting eigenvector of parameter λ is equal to the number of eigenvalues greater than λ. Finally, we use the techniques presented in the first section to experimentally demonstrate a O(log n) growth relationship between the variance of histogram bin values and the order of the β-Hermite matrix ensemble. This research was supported by NSF Grant DMS–0411962.     

Pattern Search Method for Discrete <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>–Approximation

We propose a pattern search method to solve a classical nonsmooth optimization problem. In a deep analogy with pattern search methods for linear constrained optimization, the set of search directions at each iteration is defined in such a way that it conforms to the local geometry of the set of points of nondifferentiability near the current iterate. This is crucial to ensure convergence. The approach presented here can be extended to wider classes of nonsmooth optimization problems. Numerical experiments seem to be encouraging. This work was supported by M.U.R.S.T., Rome, Italy.     

The Weighted Transience and Recurrence of Markov Processes

Transience and recurrence are among the most important concepts in Markov processes. In this paper, we study the transience and recurrence for right processes with a given weight function, and characterize them by potentials, excessive functions, first hitting times and last exit times of the process. We also study the properties of recurrent states.     

ε-Value Function and Dynamic Programming

In this note, we develop a dynamic programming approach for an ε-optimal control problem of Bolza. We prove that each Lipschitz continuous function satisfying the Hamilton-Jacobi inequality (less than zero and greater than −ε) is an ε-value function.     

Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables

In this paper, we establish a Rosenthal-type inequality of the maximum of partial sums for ρ^- -mixing random fields. As its applications we get the Hájeck -Rènyi inequality and weak convergence of sums of ρ^- -mixing sequence. These results extend related results for NA sequence and p^＊ -mixing random fields,     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

Higher order Lipschitz classes of functions and absolutely convergent Fourier series

We introduce the higher order Lipschitz classes Λ _r (α) and λ _r (α) of periodic functions by means of the rth order difference operator, where r = 1, 2, ..., and 0 < α ≦ r. We study the smoothness property of a function f with absolutely convergent Fourier series and give best possible sufficient conditions in terms of its Fourier coefficients in order that f belongs to one of the above classes. This research was supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

Plateau of α-function and c-minimal homoclinic orbits

We consider the construction of the plateau of the α-function in a hyperbolic and positive definite Lagrangian system, and link the boundries of the α-function’s plateau with the distribution of c-minimal homoclinic orbits to Aubry sets.     

Some Limit Theorems on the Increments of l^p-valued Multi-Parameter Gaussian Processes

In this paper, we establish some limit theorems on the increments of an l^p-valued multiparameter Gaussian process under weaker conditions than those of Csoergoe-Shao theorems published in Ann. Probab. (1993).     

Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Let E be a real Banach space with property (α) and let W _Γ be an E-valued Brownian motion with distribution Γ. We show that a function is stochastically integrable with respect to W _Γ if and only if Γ-almost all orbits Ψx are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of Γ-measurable linear extensions of γ-radonifying operators with values in spaces of γ-radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution equations driven by an E-valued Brownian motion. The first named author gratefully acknowledges the support by a ‘VIDI subsidie’ in the ‘Vernieuwingsimpuls’ programme of The Netherlands Organization for Scientific Research (NWO) and the Research Training Network HPRN-CT-2002–00281. The second named author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1–1).     

Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Output Dynamic Observer-Based Control Design of Uncertain Neutral Systems

In this paper, the robust H _∞ control problem of output dynamic observer-based control for a class of uncertain neutral systems is considered. The linear matrix inequality optimization approach is used to design the new H _∞ output dynamic controls. Three classes of H _∞ observer-based controls are proposed. The minimal H _∞-norm bound and the maximal perturbed bound are given. Based on the result of this paper, the constraint of matrix equality is not necessary for designing the H _∞ observer-based controls. A numerical example is given to stress the usefulness of the proposed results. Communicated by C. T. Leondes The research reported here was supported by the National Science Council of Taiwan, ROC under Grant NSC 94-2213-E-507-002.