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Complexity Analysis of an Interior Point Algorithm for the Semidefinite Optimization Based on a Kernel Function with a Double Barrier Term

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Acta mathematica Sinica,English Series 2015年第3期31卷 543-556页

作者： Mohamed ACHACHE laboratoire de mathématiques fondamentales et Numériques Faculté des Sciences Université Ferhat Abbas Sétif 1Algérie

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization（SDO） problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 〉 q2 〉 1, the algorithm has O（（q1 ＋ 1） nq1＋1/2（q1-q2）logn/ε）and O（（q1 ＋ 1）2（q1-q2）^3q1-2q2＋1√n logn/c） complexity results for large- and small-update methods, respectively.

关键词： Semidefinite optimization kernel functions primal-dual interior point methods large andsmall-update algorithms complexity of algorithms

ESSENTIAL APPROXIMATE POINT SPECTRA FOR UPPER TRIANGULAR MATRIX OF LINEAR RELATIONS

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Acta mathematica Scientia 2013年第4期33卷 1187-1201页

作者： Souhir ELLEUCH Maher MNIF Département de mathématiques Université de Sfax Faculté des Sciences de SfaxLaboratoire de Physiques Mathématiques

When A E ∈LR（H） and B E ∈LR（K） are given, for C E∈LR（K, H） we denoteby Mc the linear relation acting on the infinite dimensional separable Hilbert space H Kof the formIn this paper, we give the necessary and sufficient conditionson A and B for wh{ch Mc is upper semi-Fredholm with negative index or Weyl for some C C ∈LR（K, H）.

关键词： linear relation Fredholm and semi-Fredholm relation essential approximatepoint spectra for linear relation

Optimal Control of a Frictional Contact Problem

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Acta mathematicae Applicatae Sinica 2015年第4期31卷 991-1000页

作者： Arezki TOUZALINE laboratoire de Systèmes Dynamiques Faculté de MathématiquesUSTHB

We consider a mathematical model which describes a contact between a deformable body and a foundation. The contact is bilateral and modelled with Tresca＇s friction law. The goal of this paper is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. We state an optimal control problem which admits at least one solution. We also introduce the regularized control problem for which we study the convergence when the regularization parameter tends to zero. Finally, an optimally condition is established for this problem.

关键词： optimal control of variational inequalities elastic materials Tresca's friction, weak solution

Sanya Lectures: Geometry of Spherical Varieties

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Acta mathematica Sinica,English Series 2018年第3期34卷 371-416页

作者： Nicolas PERRIN laboratoire de mathématiques de Versailles UVSQ CNRS Université Paris-Saclay

These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety.

关键词： Spherical varieties orbits local structure divisors curves

Some Uniqueness Results for Functional Damped Semilinear Differential Equations in Fréchet Spaces

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Acta mathematica Sinica,English Series 2008年第1期24卷 95-106页

作者： Abdelghani OUAHAB laboratoire de mathématiques Universitéde Sidi Bel Abbès BP 89

A recent nonlinear alternative for contraction maps in Frechet spaces due to Frigon and Granas （Resultats de type Leray-Schauder pour des contractions sur des espaces de Frechet, Ann. Sci. math. Quebec 22, （2）, 161-168 （1998））, combined with semigroup theory, is used to investigate the existence and uniqueness of mild solutions for first- and second-order functional semi linear and neutral damped differential equations in Frechet space.

关键词： differential equations damped semigroup family cosinus mild solution fixed point uniqueness Frechet spaces contraction

Weak Solutions for the Vlasov-Poisson Initial-Boundary Value Problem with Bounded Electric Field

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Chinese Annals of mathematics,Series B 2007年第4期28卷 389-420页

作者： Mihai BOSTAN laboratoire de mathématiques Université de Franche-Comté 16 route de Gray F-25030 BesanconFance.

The aim of this work is to construct weak solutions for the three dimensional Vlasov-Poisson initial-boundary value problem with bounded electric field. The main in-gredient consists of estimating the change in momentum along characteristics of regular electric fields inside bounded spatial domains. As direct consequences, the propagation of the momentum moments and the existence of weak solution satisfying the balance of total energy are obtained.

关键词： Vlasov-Poisson equations Vlasov-Maxwell equations Weak solutions

Some Notes on Large deviations of Markov Processes

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Acta mathematica Sinica,English Series 2000年第3期16卷 369-394页

作者： Liming Wu laboratoire de mathématiques Appliquées,CNRS-UMR 6620 UniversitéBlaise Pascal,63177,A UBIERE,France and department of mathematics,Wuhan University Wuhan,430072 P.R China laboratoire de mathématiques Appliquées CNRS-UMR 6620 Université Blaise Pascal AUBIERRE France

In this paper we shall characterize the large deviation principles(abbreviated to LDP) of Donsker-Varadhan of a Markov process both for the weak convergence topology and for theτ- topology,by means of a hyper-exponential recurrence property.A Lyapunov criterion for this type of recurrence property is *** results are applied to countable Markov chains,unidimensional diffusions,elliptic or hypoelliptic diffusions on Riemannian *** counter-examples are equally presented.

关键词： Large deviations hyper-exponential recurrence T-topology uniformly integrable operators

Analysis of a quasistatic contact problem with adhesion and nonlocal friction for viscoelastic materials

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Applied mathematics and Mechanics(English Edition) 2010年第5期31卷 623-634页

作者： Arezki TOUZALINE laboratoire de Systèmes Dynamiques Facult de Mathmatiques Universit des Sciences et de la Technologie Houari Boumediene

A mathematical model is established to describe a contact problem between a deformable body and a foundation. The contact is bilateral and modelled with a nonlocal friction law, in which adhesion is taken into account. Evolution of the bonding field is described by a first-order differential equation. The materials behavior is modelled with a nonlinear viscoelastic constitutive law. A variational formulation of the mechanical problem is derived, and the existence and uniqueness of the weak solution can be proven if the coefficient of friction is sufficiently small. The proof is based on arguments of time-dependent variational inequalities, differential equations, and the Banach fixed-point theorem.

关键词： viscoelastic materials adhesion nonlocal friction fixed point weak solution

A New Elliptic Measure on Lower Dimensional Sets

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Acta mathematica Sinica,English Series 2019年第6期35卷 876-902页

作者： Guy DAVID Joseph FENEUIL Svitlana MAYBORODA Université Paris-Sud Laboratoire de Mathématiques department of mathematics Temple University School of mathematics University of Minnesota

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n-1 dimensional sets across analysis, geometric measure theory, and Pdes. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new Pde approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.

关键词： Elliptic measure in higher codimension degenerate elliptic operators absolute continuity Dahlberg’s theorem Dirichlet solvability