On the category Q-Mod

In this paper we consider the category Q-Mod of modules over a given quantale Q. The paper is motivated by constructions and results from the category of modules over a ring. We show that the category Q-Mod is monadic, consider its relation to the category Q-Top of Q-topological spaces and generalize a method of completion of partially ordered sets. Received December 20, 2005; accepted in final form December 4, 2006.     

Lower Bounds for the Principal Eigenvalue of the p-Laplacian on the Unit Ball

Abstract

In this paper, by the sub-super solution method we provide explicit lower bounds for the principal eigenvalue of the p-Laplacian on the unit ball. We compare our results with those of earlier studies and also demonstrate that they are in good agreement with numerical results.     

Group inverse and core inverse in Banach and C*-algebras

Abstract

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].     

Rates of approximation in the multidimensional invariance principle for sums of i.i.d. random vectors with finite moments

The aim of this paper is to derive consequences of a result of G?tze and Zaitsev (2008). We show that the i.i.d. case of this result implies a multidimensional version of some results of Sakhanenko (1985). We establish bounds for the rate of strong Gaussian approximation of sums of i.i.d. R ^d -valued random vectors ξ _j having finite moments E IIξ _j II^γ, γ>2. Bibliography: 13 titles.     

On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems

The purpose of this study is to broaden the scope of projective transformation methods in mathematical programming, both in terms of theory and algorithms. We start by generalizing the concept of the analytic center of a polyhedral system of constraints to the w-center of a polyhedral system, which stands for weighted center, where there is a positive weight on the logarithmic barrier term for each inequality constraint defining the polyhedronX. We prove basic results regarding contained and containing ellipsoids centered at the w-center of the systemX. We next shift attention to projective transformations, and we exhibit an elementary projective transformation that transforms the polyhedronX to another polyhedronZ, and that transforms the current interior point to the w-center of the transformed polyhedronZ. We work throughout with a polyhedral system of the most general form, namely both inequality and equality costraints.This theory is then applied to the problem of finding the w-center of a polyhedral systemX. We present a projective transformation algorithm, which is an extension of Karmarkar's algorithm, for finding the w-center of the systemX. At each iteration, the algorithm exhibits either a fixed constant objective function improvement, or converges superlinearly to the optimal solution. The algorithm produces upper bounds on the optimal value at each iteration. The direction chosen at each iteration is shown to be a positively scaled Newton direction. This broadens a result of Bayer and Lagarias regarding the connection between projective transformation methods and Newton's method. Furthermore, the algorithm specializes to Vaidya's algorithm when used with a line-search, and so shows that Vaidya's algorithm is superlinearly convergent as well. Finally, we show how the algorithm can be used to construct well-scaled containing and contained ellipsoids at near-optimal solutions to the w-center problem.This paper is a revision of the two papers Projective transformations for interior point methods, part I: Basic theory and linear programming, O.R. working paper 179-88 and Projective transformations for interior point methods, part II: Analysis of an algorithm for finding the weighted center of a polyhedral system, O.R. working paper 180-88, M.I.T.     

Estimates for the rate of strong approximation in the multidimensional invariance principle

The aim of this paper is to derive simplest consequences of the author's result of the multidimensional invariance principle. We obtain bounds for the rate of strong Gaussian approximation of sums of independent R ^d-valued random variables ξ_j having finite moments of the form EH (‖ξ_j‖), where H (x) is a monotone function growing not slower than x² and not faster than e^cx. A multidimensional version of results of Sakhanenko is obtained. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 37–53.     

KNOPFMACHER EXPANSIONS IN NUMBER THEORY

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

We survey in this paper J. Knopfmacher results on number theoretical expansions as they arised from our common collaboration.     

Inertial methods for fixed point problems and zero point problems of the sum of two monotone mappings

ABSTRACT

The purpose of this paper is to investigate the problem of finding a common element of the set of zero points of the sum of two operators and the fixed point set of a quasi-nonexpansive mapping. We introduce modified forward-backward splitting methods based on the so-called inertial forward-backward splitting algorithm, Mann algorithm and viscosity method. We establish weak and strong convergence theorems for iterative sequences generated by these methods. Our results extend and improve some related results in the literature.     

A characterization of compact convex polyhedra in hyperbolic 3-space

Summary In this paper we study the extrinsic geometry of convex polyhedral surfaces in three-dimensional hyperbolic spaceH ³. We obtain a number of new uniqueness results, and also obtain a characterization of the shapes of convex polyhedra inH ³ in terms of a generalized Gauss map. This characterization greatly generalizes Andre'ev's Theorem.Oblatum 12-XI-1991 & 29-V-1992     

Weaker forms of the Menger property in bitopological spaces

Abstract

In this paper we continue previous investigations on the weaker forms of the Menger property in bitopological spaces. We introduce weakly Menger property and study some topological properties of almost and weakly Menger bitopological spaces. We also consider the almost Hurewicz spaces in a bitopological context.     

An active index algorithm for the nearest point problem in a polyhedral cone

We consider the problem of finding the nearest point in a polyhedral cone C={x∈R ⁿ :D x≤0} to a given point b∈R ⁿ , where D∈R ^m×n . This problem can be formulated as a convex quadratic programming problem with special structure. We study the structure of this problem and its relationship with the nearest point problem in a pos cone through the concept of polar cones. We then use this relationship to design an efficient algorithm for solving the problem, and carry out computational experiments to evaluate its effectiveness. Our computational results show that our proposed algorithm is more efficient than other existing algorithms for solving this problem.     

Weak convergence theorem for Lipschizian pseudocontraction semigroups in Banach spaces

The purpose of this paper is to study the weak convergence problems of the irnplicity iteration process for Lipschitzian pseudocontraction semigroups in general Banach spaces. The results presented in this paper extend and improve the corresponding results of Zhou [Nonlinear Anal., 68, 2977-2983 （2008）], Chen, et ah [J. Math. Anal. Appl., 314, 701 709 （2006）], Xu and Ori [Numer. Funct. Anal. Optim, 22, 767-773 （2001）] and Osilike [J. Math. Anal. Appl., 294, 73-81 （2004）]. Keywords     

On the estimates of the order of finitelya k -determined map-germs

In this paper we deal with the estimates of the order of determinacy of finitely determined map-germs. This problem has been noted by mathematicians for more than ten years. But the results given by some authors for the estimates of the order of determinacy are stated with respect toa, while those given in this paper are stated with respect toa _k , wherek may be any non-negative integer, and we havea ₀=a in particular. This paper was completed during the author's visit to the Institute of Mathematics, Now I'd like to express my heart-felt thanks to the Institute for its financial support and Prof. Li Peixin for his instruction.     

Property A in an Infinite-Dimensional Space. I

   Abstract. We prove that an infinite-dimensional space of piecewise polynomial functions of degree at most n-1 with infinitely many simple knots, n ≥ 2 , satisfies Property A. Apart from its independent interest, this result allows us to solve an open classical problem (n ≥ 3 ) in theory of best approximation: the uniqueness of best L ₁ -approximation by n -convex functions to an integrable, continuous function defined on a bounded interval. In this first part of the paper we prove the case n=2 and give key results in order to complete the general proof in the second part.     

A graphical tool for selecting the number of slices and the dimension of the model in SIR and SAVE approaches

Sliced inverse regression (SIR) and related methods were introduced in order to reduce the dimensionality of regression problems. In general semiparametric regression framework, these methods determine linear combinations of a set of explanatory variables X related to the response variable Y, without losing information on the conditional distribution of Y given X. They are based on a “slicing step” in the population and sample versions. They are sensitive to the choice of the number H of slices, and this is particularly true for SIR-II and SAVE methods. At the moment there are no theoretical results nor practical techniques which allows the user to choose an appropriate number of slices. In this paper, we propose an approach based on the quality of the estimation of the effective dimension reduction (EDR) space: the square trace correlation between the true EDR space and its estimate can be used as goodness of estimation. We introduce a na?ve bootstrap estimation of the square trace correlation criterion to allow selection of an “optimal” number of slices. Moreover, this criterion can also simultaneously select the corresponding suitable dimension K (number of the linear combination of X). From a practical point of view, the choice of these two parameters H and K is essential. We propose a 3D-graphical tool, implemented in R, which can be useful to select the suitable couple (H, K). An R package named “edrGraphicalTools” has been developed. In this article, we focus on the SIR-I, SIR-II and SAVE methods. Moreover the proposed criterion can be use to determine which method seems to be efficient to recover the EDR space, that is the structure between Y and X. We indicate how the proposed criterion can be used in practice. A simulation study is performed to illustrate the behavior of this approach and the need for selecting properly the number H of slices and the dimension K. A short real-data example is also provided.     

Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.     

HIGHER DERIVATIONS OF SEMIPRIME RINGS

ABSTRACT

In this paper we study higher derivations of prime and semiprime rings satisfying linear relations. We extend several results known for algebraic derivations, and we prove some other results.     

On the justification of the Reynolds equation,describing isentropic compressible flows through a tiny pore

Abstract We consider the isentropic compressible flow through a tiny pore. Our approach is to adapt the recent results by N. Masmoudi on the homogenization of compressible flows through porous media to our situation. The major difference is in the a priori estimates for the pressure field. We derive the appropriate ones and then Masmoudi’s results allow to conclude the convergence. In this way the compressible Reynolds equation in the lubrication theory is rigorously justified. Keywords: Compressible Navier-Stokes equations, Lubrication, Pressure estimates Mathematics Subject Classification (2000): 35B27, 76M50, 35D05     

Hidden Markov Filter Estimation of the Occurrence Time of an Event in a Financial Market

Abstract

In this paper, we use filtering techniques to estimate the occurrence time of an event in a financial market. The occurrence time is being viewed as a Markov stopping time with respect to the σ-field generated by a hidden Markov process. We also generalize our result to the Nth occurrence time of that event.