Stability of Order Statistics under Dependence

We precisely evaluate the upper and lower deviations of the expectation of every order statistic from an i.i.d. sample under arbitrary violations of the independence assumption, measured in scale units generated by various central absolute moments of the parent distribution of a single observation. We also determine the distributions for which the bounds are attained. The proof is based on combining the Moriguti monotone approximation of functions with the Hölder inequality applied for proper integral representations of expected order statistics in the independent and dependent cases. The method allows us to derive analogous bounds for arbitrary linear combinations of order statistics.     

On the Norms of Singular Integral Operators on Contours with Intersections

The exact bounds are obtained for the norm of the singular integral operator S on the family of rays originating at the same point. These bounds, with the use of the localization technique, are then extended to the essential norm of S on piecewise smooth curves with finitely many points of self intersection. Submitted: April 13, 2007. Accepted: September 19, 2007.     

Some remarks on the constant in the strengthened CBS inequality: Estimate for hierarchical finite element discretizations of elasticity problems

For a class of two‐dimensional boundary value problems including diffusion and elasticity problems, it is proved that the constants in the corresponding strengthened Cauchy‐Buniakowski‐Schwarz (CBS) inequality in the cases of two‐level hierarchical piecewise‐linear/piecewise‐linear and piecewise‐linear/piecewise‐quadratic finite element discretizations with triangular meshes differ by the factor 0.75. For plane linear elasticity problems and triangulations with right isosceles triangles, formulas are presented that show the dependence of the constant in the CBS inequality on the Poisson's ratio. Furthermore, numerically determined bounds of the constant in the CBS inequality are given for plane linear elasticity problems discretized by means of arbitrary triangles and for three‐dimensional elasticity problems discretized by means of tetrahedral elements. Finally, the robustness of iterative solvers for elasticity problems is discussed briefly. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 469–487, 1999     

A solution for antiplane‐strain local problems using elliptic integrals of Cauchy type

Quasi‐periodic piecewise analytic solutions, without poles, are found for the local antiplane‐strain problems. Such problems arise from applying the asymptotic homogenization method to an elastic problem in a parallel fiber‐reinforced periodic composite that presents an imperfect contact of spring type between the fiber and the matrix. Our methodology consists of rewriting the contact conditions in a complex appropriate form that allow us to use the elliptic integrals of Cauchy type. Several general conditions are assumed including that the fibers are disposed of arbitrary manner in the unit cell, that all fibers present imperfect contact with different constants of imperfection, and that their cross section is smooth closed arbitrary curves. Finally, we obtain a family of piecewise analytic solutions for the local antiplane‐strain problems that depend of a real parameter. When we vary this parameter, it is possible to improve classic bounds for the effective coefficients. Copyright © 2017 John Wiley & Sons, Ltd.     

Height-analysis ofk-dimensional leaf and node height-balanced trees: A new approach

A new general approach is proposed for the height-analysis ofk-dimensional leaf and node height-balanced trees which is expected to be useful for such analysis of other neighbor-supported balanced multidimensional trees as well. The approach leads to upper bounds which are the same as the known upper bounds for the corresponding 1-dimensional trees, to within an additive term.This research is partially supported by a grant from the College of Business Administration, Georgia State University, Atlanta, Georgia.Actually a variant calledalmost 2-dimensional leaf AVL-trees is introduced in order to facilitate the derivation of an upper bound for the height of these trees which is within an additive term to the bound for such 1-dimensional trees.     

Results on the relative perturbation of the singular values of a matrix

A highly accurate computation of the singular values of a matrix is a topic of current interest in the literature. In this paper we develop general bounds on relative perturbation of singular values. These bounds permit slight improvements in a unified derivation of some previous inequalities. The main result is a better criterion to neglect off-diagonal elements in the bidiagonal singular values decomposition.The present paper has been developed under the M. U. R. S. T. 40% National Program and the Interuniversity Center of Numerical Analysis and Computational Mathematics.     

Efficiency Bounds for Product Designs in Linear Models

We provide lower efficiency bounds for the best product design for an additive multifactor linear model. The A-optimality criterion is used to demonstrate that out bounds are better than the conventional bounds. Applications to other criteria, such as IMSE (integrated mean squared error) criterion are also indicated. In all the cases, the best product design appears to perform better when there are more levels in each factor but decreases when more factors are included. Explicit efficiency formulas for non-additive models are also constructed.     

Solitary wave and other solutions for nonlinear heat equations

A number of explicit solutions for the heat equation with a polynomial non-linearity and for the Fisher equation is presented. An extended class of non-linear heat equations admitting solitary wave solutions is described. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity.     

Variational Inequalities for Arbitrary Multivariate Distributions

Upper bounds for the total variation distance between two arbitrary multivariate distributions are obtained in terms of the correspondingw-functions. The results extend some previous inequalities satisfied by the normal distribution. Some examples are also given.     

On Martin Bordemann's proof of the existence of projectively equivariant quantizations

The paper explains the notion of projectively equivariant quantization. It gives a sketch of Martin Bordemann's proof of the existence of projectively equivariant quantization on arbitrary manifolds.     

The rate of convergence for spectra of GUE and LUE matrix ensembles

We obtain optimal bounds of order O(n ⁻¹) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively. Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, by RFBR grant N 02-01-00233, and by RFBR-DFG grant N 04-01-04000.     

A Comparison of Finite Element Error Bounds for Poisson's Equation

Computable finite element error bounds are developed for approximationby: (1) piecewise bilinear elements over rectangles and (2) piecewise linear elements over triangles. These theoretical bounds are then compared with numerical valuesobtained by solving a variety of test problems. For the energysemi-norm, the bounds are comparable to the actual errors, whichdemonstrates the usefulness of these results.     

Estimation of systems with statistically-constrained inputs

This paper discusses the estimation of a class of discrete-time linear stochastic systems with statistically-constrained unknown inputs (UI), which can represent an arbitrary combination of a class of un-modeled dynamics, random UI with unknown covariance matrix and deterministic UI. In filter design, an upper bound filter is explored to compute, recursively and adaptively, the upper bounds of covariance matrices of the state prediction error, innovation and state estimate error. Furthermore, the minimum upper bound filter (MUBF) is obtained via online scalar parameter convex optimization in pursuit of the minimum upper bounds. Two examples, a system with multiple piecewise UIs and a continuous stirred tank reactor (CSTR), are used to illustrate the proposed MUBF scheme and verify its performance.     

Finiteness of the strong global dimension of radical square zero algebras

The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.     

Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian

We prove that the eigenvalues of the Laplacian acting on functions converge to those of the limit manifold for a special collapsing family of closed Riemannian manifolds without curvature bounds. The proof uses L ²-analysis.Dedicated to Professor Hajime Urakawa on his sixtieth birthday.The author is partially supported by the Grant-in-Aid for Scientific Research No. 16740026 of the Japan Society for the Promotion of Science.     

Linear best approximation using a class ofk-majorl p norms

Consideration is given to problems of linear best approximation using a variant of the usuall _p norms referred to ask-majorl _p norms, for the cases when 1<p<. The underlying problem is the minimization of a piecewise smooth function. Best approximations are characterized, and a descent algorithm is developed.     

Tight convex underestimators for {{\mathcal C}^2}-continuous problems: I. univariate functions

A novel method for the convex underestimation of univariate functions is presented in this paper. The method is based on a piecewise application of the well-known αBB underestimator, which produces an overall underestimator that is piecewise convex. Subsequently, two algorithms are used to identify the linear segments needed for the construction of its -continuous convex envelope, which is itself a valid convex underestimator of the original function. The resulting convex underestimators are very tight, and their tightness benefits from finer partitioning of the initial domain. It is theoretically proven that there is always some finite level of partitioning for which the method yields the convex envelope of the function of interest. The method was applied on a set of univariate test functions previously presented in the literature, and the results indicate that the method produces convex underestimators of high quality in terms of both lower bound and tightness over the whole domain under consideration.     

Meromorphic singular foliations on complex projective surfaces

Here we show that many numerical computations and bounds on the degrees of the algebraic leaves for singular meromorphic foliations on ² may be extended to large classes of foliations and complex projective surfaces.The author was partially supported by MURST and GNSAGA of CNR (Italy)     

On a family of vector space categories

In continuation of our earlier work [2] we describe the indecomposable representations and the Auslander-Reiten quivers of a family of vector space categories playing an important role in the study of domestic finite dimensional algebras over an algebraically closed field. The main results of the paper are applied in our paper [3] where we exhibit a wide class of almost sincere domestic simply connected algebras of arbitrary large finite global dimensions and describe their Auslander-Reiten quivers.     

Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics

In this paper, we establish lower bounds for n-term approximations in the metric of L ²⁽I ² ) of characteristic functions of plane convex subsets of the square I ² with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than .