Infinitary model theory of abelian groups

The paper is a survey of results in the model theory of abelian groups, dealing with two sorts of problems: finding invariants which classify groups up toL _λκ-equivalence; and determining whether certain classes of groups are definable inL _λκ. Research supported by NSF grant GP 43910     

To solving problems of algebra for two-parameter matrices. II

This paper is one of the series of survey papers dedicated to the development of methods for solving problems of algebra for two-parameter polynomial matrices of general form. The paper considers the AB-algorithm and the ∇V-2 factorization algorithm, together with their applications. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 150–165.     

Extremal K¨ahler Metrics of Toric Manifolds*

This paper is a survey of some recent developments concerning extremal K¨ahler metrics on Toric Manifolds.     

Problems of the theory of bitopological spaces. 2

This paper is the second part of a survey of recent results and new problems in the theory of bitopological spaces. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 5–67. Translated by A. A. Ivanov.     

Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays

The aims of this paper are (i) to present a survey of recent advances in the analysis of superconvergence of collocation solutions for linear Volterra-type functional integral and integro-differential equations with delay functions θ(t) vanishing at the initial point of the interval of integration (with ia(t) = qt (0 < q < 1, t ⩾ 0) being an important special case), and (ii) to point, by means of a list of open problems, to areas in the numerical analysis of such Volterra functional equations where more research needs to be carried out.      

Separating functions and their applications

In the present paper, a survey of applications of separating functions in the theory of representations of quivers and algebras in linear and metric spaces is given. An application of these functions to problems in spectral graph theory is also demonstrated. Bibliography: 27 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 202–212.     

First- and second-order methods for semidefinite programming

 In this paper, we survey the most recent methods that have been developed for the solution of semidefinite programs. We first concentrate on the methods that have been primarily motivated by the interior point (IP) algorithms for linear programming, putting special emphasis in the class of primal-dual path-following algorithms. We also survey methods that have been developed for solving large-scale SDP problems. These include first-order nonlinear programming (NLP) methods and more specialized path-following IP methods which use the (preconditioned) conjugate gradient or residual scheme to compute the Newton direction and the notion of matrix completion to exploit data sparsity. Received: December 16, 2002 / Accepted: May 5, 2003 Published online: May 28, 2003 Key words. semidefinite programming – interior-point methods – polynomial complexity – path-following methods – primal-dual methods – nonlinear programming – Newton method – first-order methods – bundle method – matrix completion The author's research presented in this survey article has been supported in part by NSF through grants INT-9600343, INT-9910084, CCR-9700448, CCR-9902010, CCR-0203113 and ONR through grants N00014-93-1-0234, N00014-94-1-0340 and N00014-03-1-0401. Mathematics Subject Classification (2000): 65K05, 90C06, 90C22, 90C25, 90C30, 90C51     

Optimal Orientations of Graphs and Digraphs: A Survey

 For a graph or digraph G, let be the family of strong orientations of G; and for any , we denote by d(D) the diameter of D. Define . In this paper, we survey the results obtained and state some problems and conjectures for the parameter . Received: October, 2001 Final version received: September 23, 2002     

How to compute the volume in high dimension?

 In some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones. Approximating the volume of a convex n-dimensional body, given by an oracle is one of the areas where this difference can be proved. In general, if we use a deterministic algorithm to approximate the volume, it requires exponentially many oracle questions in terms of n as n→∞. Dyer, Frieze and Kannan gave a randomized polynomial approximation algorithm for the volume of a convex body K⊆ℝ ⁿ , given by a membership oracle. The DKF algorithm was improved in a sequence of papers. The area is full of deep and interesting problems and results. This paper is an introduction to this field and also a survey. Received: January 28, 2003 / Accepted: April 29, 2003 Published online: May 28, 2003     

Global dynamics of delay differential equations

In this survey paper the delay differential equation (t) = −μx(t) + g(x(t − 1)) is considered with μ ≥ 0 and a smooth real function g satisfying g(0) = 0. It is shown that the dynamics generated by this simple-looking equation can be very rich. The dynamics is completely understood only for a small class of nonlinearities. Open problems are formulated. Supported in part by the Hungarian NFSR, Grant No. T049516.     

Minimization Problems with Singular Data

This work contains a survey of some results on minimization problems with singular data and some new contributions (not previously published) presented in my lecture at the “Seminario Matematico e Fisico di Milano”. Lecture held in the Seminario Matematico e Fisico on October 10, 2005 Received: June 2006     

Positive definite probability densities and probability distributions

In the first short survey article [H.-J. Rossberg,Wiss. Z. Karl Marx Univ.,37, 366–374 (1988)] the author indicated that positive definite probability densities deserve attention and that the investigation of this class leads to serious new mathematical problems. Later [R. Riedel and H.-J. Rossberg,Metrika, (submitted for publication)] it turned out that it is worthwhile to consider an even larger class of distribution functions, namely those having characteristic function f≥0; they were first briefly considered in [P. Lévy,C. R. Acad. Sci.,265, 249–252 (1967)]. We survey the basic ideas and the present state of the developing new theory; moreover, we formulate open problems. Some new results are proved. For the sake of brevity we do not enter into a discussion of continuation problems and limit theorems; for these subjects see [H.-J. Rossberg,Appendix to Gnedenko B. W., Einführung in die Wahrscheinlichkeitstheorie, 9th ed., Akademie-Verlag, Berlin (1991)], [H.-J. Rossberg,Math. Nachr.,141, 227–232 (1989)], and [H.-J. Rossberg,Lect. Notes Math.,1412, 296–308 (1989)]. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Online algorithms: a survey

 During the last 15 years online algorithms have received considerable research interest. In this survey we give an introduction to the competitive analysis of online algorithms and present important results. We study interesting application areas and identify open problems. Received: December 1, 2002 / Accepted: April 28, 2003 Published online: May 28, 2003 Mathematics Subject Classification (1991): 68W25, 68W40     

λ-Topologies on Function Spaces

This paper is devoted to the spaces C _λ(X) of all continuous real-valued functions on X endowed with arbitrary λ-topologies. This is a fairly complete survey of the results obtained by the author in the following domains of the theory of λ-topologies: cardinal functions; locally convex properties; weak and strong topologies; dual spaces; lattices of λ-topologies; completeness. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 3–56, 2003.     

Uniqueness of Positive Solutions of Semilinear Elliptic Equations and Related Eigenvalue Problems

In this paper we survey some recent results about the uniqueness of the solution of some semilinear elliptic Dirichlet problems in bounded domains. The presentation aims to emphasize the role of the geometrical properties of the second eigenfunction of the linearized problem in the study of the above question. This motivates the analysis of the asymptotic behaviour of these eigenfunctions and of the relative eigenvalues when the nonlinear term is a power with exponent close to the critical Sobolev exponent. Research supported by MIUR, project “Variational Methods and Nonlinear Differential Equations”. Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005     

Solutions to open problems in Neeb’s recent survey on infinite-dimensional Lie groups

We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb (Towards a Lie theory of locally convex groups, Jpn. J. Math. 1 (2006), 291–468). Moreover, we prove a result by the author announced in the survey, which answers a question posed in an earlier, unpublished version thereof.      

The initial boundary-value problem with a free surface condition for the penalized equations of aqueous solutions of polymers

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (ℝ⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω⊂ℝ³. Bibliography: 25 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 185–207. Translated by N. A. Karazeeva.     

Semicontinuity of Solution Mappings of Parametric Generalized Vector Equilibrium Problems

In this paper, we study the ε-generalized vector equilibrium problem (ε-GVEP) and the ε-extended vector equilibrium problem (ε-EVEP), which can be regarded as approximate problems to the generalized vector equilibrium problems (GVEP). Existence results for ε-GVEP and ε-EVEP are established. We investigate also the continuity of the solution mappings of ε-GVEP and ε-EVEP. In particular, two results concerning the lower semicontinuity of the solution mappings of ε-GVEP and ε-EVEP are presented. This research was partially supported by Grant NSC 95-2811-M-110-010.     

Progress on perfect graphs

 A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first part of the paper we survey the main aspects of perfect graphs and their relevance. In the second part we outline our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such an odd cycle. Received: December 19, 2002 / Accepted: April 29, 2003 Published online: May 28, 2003 Key words. Berge graph – perfect graph – skew partition Mathematics Subject Classification (1991): 05C17     

Robust solutions to uncertain linear complementarity problems

In this paper,we adopt the robust optimization method to consider linear complementarity problems in which the data is not specified exactly or is uncertain,and it is only known to belong to a prescribed uncertainty set.We propose the notion of the p- robust counterpart and the p-robust solution of uncertain linear complementarity problems.We discuss uncertain linear complementarity problems with three different uncertainty sets,respectively,including an unknown-but-bounded uncertainty set,an ellipsoidal uncertainty set and an intersection-of-ellipsoids uncertainty set,and present some sufficient and necessary(or sufficient) conditions which p- robust solutions satisfy.Some special cases are investigated in this paper.