The combinatorics of τ-covers

We solve four out of the six open problems concerning critical cardinalities of topological diagonalization properties involving τ-covers, show that the remaining two cardinals are equal, and give a consistency result concerning this remaining cardinal. Consequently, 21 open problems concerning potential implications between these properties are settled. We also give structural results based on the combinatorial techniques.     

Remarks on continuum cardinals on Boolean algebras

We give some results concerning various generalized continuum cardinals. The results answer some natural questions which have arisen in preparing a new edition of 5 . To make the paper self‐contained we define all of the cardinal functions that enter into the theorems here. There are many problems concerning these new functions, and we formulate some of the more important ones.     

On properties of central manifolds of a stationary point

We present results concerning properties of central manifolds of a stationary point. The results are illustrated by examples.     

Partial differential equations of multi-quasi-elliptic type

The theory of multi-quasi-elliptic operators and associated Sobolev spaces is revised in this article and possible developments of the research are indicated concerning operators of principal type and non-linear equations. Moreover some new results concerning operators with anti-Wick symbols are presented. To the memory of Lamberto Cattabriga     

Two commodity network flows and linear programming

This paper shows that the linear programming formulation of the two-commodity network flow problem leads to a direct derivation of the known results concerning this problem. An algorithm for solving the problem is given which essentially consists of two applications of the Ford—Fulkerson max flow computation. Moreover, the algorithm provides constructive proofs for the results. Some new facts concerning feasible integer flows are also given.     

Eine geometrische Kennzeichnung der reellen absoluten Ebenen

A new geometric characterization of the real absolute planes is presented, which is based upon few and simple axioms concerning properties of a congruence relation. The ordering properties are developed from two axioms concerning triangles and circles. We use essential results of reflection geometry in order to prove that the structures under consideration have the well-known representations.     

Jordan isomorphisms of triangular rings

We investigate Jordan isomorphisms of triangular rings and give a sufficient condition under which they are necessarily isomorphisms or anti-isomorphisms. As corollaries we obtain generalizations of two recent results: the one concerning Jordan isomorphisms of triangular matrix algebras by Beidar, Bresar and Chebotar, and the one concerning Jordan isomorphisms of nest algebras by Lu.

     

A projected gradient dynamical system modelling the dynamics of bargaining

We propose a projected gradient dynamical system as a model for a bargaining scheme for an asset for which the two interested agents have personal valuations that do not initially coincide. The personal valuations are formed using subjective beliefs concerning the future states of the world, and the reservation prices are calculated using expected utility theory. The agents are not rigid concerning their subjective probabilities and are willing to update them under the pressure to reach finally an agreement concerning the asset. The proposed projected dynamical system, on the space of probability measures, provides a model for the evolution of the agents, beliefs during the bargaining period and is constructed so that an agreement is reached under the minimum possible deviation of both agents from their initial beliefs. The convergence results are shown using techniques from convex dynamics and Lyapunov function theory.     

On the existence of equilibrium points of vector functions

The equilibrium problem concerning vector functions is formulated and some results on the existence of equilibrium points for this problem are proved and applied to vector optimizations , Nash equilibrium in noncooperative games, variational inequalities etc. concerning vector functions     

On a classification of dynamic systems subject to noise

A classification of alternative mathematical schemes which determine possible impositions of noise on dynamic systems either of a continuous or a discrete formulation in time is presented. We refer to the results obtained concerning the influence of noise on the correlation dimension and on the largest Lyapunov exponent. A method of constructing models for effecting predictions of time series with a finite correlation dimension is presented. Numerical results concerning the influence of various level of additive noise to the Henon attractor are also provided.     

Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: itô's case

In this article we prove new results concerning the long-time behaviour of random fields that are solutions in some generalized sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually converge with probability. one to a global attractor represented by a single random variable whose properties we investigate in detail. We analyze the partial differential equations of this article in light Itô's stochastic calculus and thereby obtain stabilization and stability results which are substantially different from our earlier results concerning their interpretation in the sense of Stratonovitch. In particular, the asymptotic properties of the random fields that we investigate here exhibit no recurrence and oscillatory properties     

On pseudospectra of matrix polynomials and their boundaries

In the first part of this paper (Sections 2-4), the main concern is with the boundary of the pseudospectrum of a matrix polynomial and, particularly, with smoothness properties of the boundary. In the second part (Sections 5-6), results are obtained concerning the number of connected components of pseudospectra, as well as results concerning matrix polynomials with multiple eigenvalues, or the proximity to such polynomials.

     

Interior Point Algorithms in Linear Optimization

This is a survey of the results concerning the development and study of the interior point algorithms. Some families of the direct and dual algorithms are considered. These algorithms entering the domain of feasible solutions take into account the objective function, which makes it possible to obtain the first feasible solution close to the optimal solution. The main results on the theoretical justification of algorithms are given. Recommendations are proposed concerning the advantages of individual variants of algorithms on the basis of the obtained theoretical results, available experimental studies, and experience of using algorithms in the models of energy engineering. Some numerically efficient version of the polynomial optimization algorithm in the cone of the central path is also presented.     

Modular forms

In this survey there are included results of recent years, concerning the theory of modular forms and representations connected with them of adele groups and Galois groups. There is discussed the hypothetical principle of functoriality of automorphic forms and other conjectures of Langlands concerning automorphic forms and the L-functions connected with them.     

The exact condition of the B-spline basis may be hard to determine

The author's conjecture concerning the knot sequence whose associated B-spline sequence has maximum max-norm condition number is disproved. Related condition numbers are explored and the corresponding conjecture concerning the “worst” knot sequence for them is further supported by numerical results.     

Simple groups, permutation groups, and probability

We derive a new bound for the minimal degree of an almost simple primitive permutation group, and settle a conjecture of Cameron and Kantor concerning the base size of such a group. Additional results concern random generation of simple groups, and the so-called genus conjecture of Guralnick and Thompson. Our proofs are based on probabilistic arguments, together with a new result concerning the size of the intersection of a maximal subgroup of a classical group with a conjugacy class of elements.

     

Asymptotic behavior of solutions to nonlinear Volterra integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.     

On a definition of connected sets of a matroid

In this note a definition of connected sets of a matroid is given. Several natural results concerning connected sets are proved with this definition.     

On a generalization of the Richardson extrapolation process

Summary A convergence result for a generalized Richardson extrapolation process is improved upon considerably and additional results of interest are proved. An application of practical importance is also given. Finally, some known results concerning the convergence of Levin's T-transformation are reconsidered in light of the results of the present work.