Existence and extensions of positive linear operators

In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators. In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators. In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied. Dedicated to Prof. Romulus Cristescu on his 80th birthday     

Extension of periodic Tchebycheff Systems

Given a periodic Tchebycheff System {y_i}_{i = 0}²ⁿ it is proved that there exist two functions y_{2n + 1}, y_{2n + 2}, such that also the system {y_i}_{i = 0}^{2n + 2} is a periodic T-System.     

A characterization of Tchebycheff systems

A characterization of Tchebycheff systems is given, in terms of Weak Tchebycheff systems.     

Tchebycheff polynomials on a triangular region

A bivariable polynomial of total degreen that has minimal uniform norm on a triangular region is given explictly.Communicated by Edward B. Saff.     

Existence of parallelisms and projective extensions for strongly n-uniform near affine Hjelmslev planes

Geometriae Dedicata - An affine Hjelmslev plane is a near affine Hjelmslev plane with a parallelism. It is proved that every strongly n-uniform near affine Hjelmslev plane possesses an even...     

On a class of weak Tchebycheff systems

In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formrequiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.     

Multi Agent Collaborative Search based on Tchebycheff decomposition

This paper presents a novel formulation of Multi Agent Collaborative Search, for multi-objective optimization, based on Tchebycheff decomposition. A population of agents combines heuristics that aim at exploring the search space both globally (social moves) and in a neighborhood of each agent (individualistic moves). In this novel formulation the selection process is based on a combination of Tchebycheff scalarization and Pareto dominance. Furthermore, while in the previous implementation, social actions were applied to the whole population of agents and individualistic actions only to an elite subpopulation, in this novel formulation this mechanism is inverted. The novel agent-based algorithm is tested at first on a standard benchmark of difficult problems and then on two specific problems in space trajectory design. Its performance is compared against a number of state-of-the-art multi-objective optimization algorithms. The results demonstrate that this novel agent-based search has better performance with respect to its predecessor in a number of cases and converges better than the other state-of-the-art algorithms with a better spreading of the solutions.     

Primitive complete normal bases: Existence in certain 2-power extensions and lower bounds

The present paper is a continuation of the author’s work (Hachenberger (2001) [3]) on primitivity and complete normality. For certain 2-power extensions E over a Galois field F_q, we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of E/F_q. The main result is as follows: Letq≡3mod4and letm(q)≥3be the largest integer such that2^m(q)dividesq²−1; ifE=F_q^2l, wherel≥m(q)+3, then there exists a primitive element inEthat is completely normal overF_q.Our method not only shows existence but also gives a fairly large lower bound on the number of primitive completely normal elements. In the above case this number is at least 4⋅(q−1)^2l−2. We are further going to discuss lower bounds on the number of such elements in r-power extensions, where r=2 and q≡1mod4, or where r is an odd prime, or where r is equal to the characteristic of the underlying field.     

Ranking efficient DMUs using the Tchebycheff norm

One problem that has been discussed frequently in data envelopment analysis (DEA) literature has been lack of discrimination in DEA applications, in particular when there are insufficient DMUs or the number of inputs and outputs is too high relative to the number of units. This is an additional reason for the growing interest in complete ranking techniques. In this paper a method for ranking extreme efficient decision making units (DMUs) is proposed. The method uses L_∞(or Tchebycheff) Norm, and it seems to have some superiority over other existing methods, because this method is able to remove the existing difficulties in some methods, such as Andersen and Petersen [2] (AP) that it is sometimes infeasible. The suggested model is always feasible.     

Existence of Lyapunov functions of variable sign for linear extensions of dynamical systems on a torus

The problem of the existence of quadratic forms that have a positive definite derivative along the solutions of linear extensions of dynamical systems on a torus is considered. Assuming the existence of quadratic forms whose derivative is positive definite only with respect to part of the variables, conditions ensuring the existence of a quadratic form whose derivative is already positive definite with respect to all variables are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1713–1717, December, 1990.     

Extremal problems for linear functionals on the Tchebycheff spaces

The study of Tchebycheff spaces (generalizing the space of algebraic polynomials) and extremal problems related to them began one and a half centuries ago. Recently, many facts of approximation theory have been understood and reinterpreted from the point of view of general principles of the theory of extremum and convex duality. This approach not only allowed one to prove the previously known results for algebraic polynomials and generalized polynomials in a unified way, but also enabled one to obtain new results. In this paper, we work out this direction with special attention to the optimal recovery problems. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 87–100, 2005.     

Functions of bounded variation with respect to a Tchebycheff system

A characterization of Tchebycheff systems is given, in terms of Weak Tchebycheff systems.     

An interactive weighted Tchebycheff procedure for multiple objective programming

The procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted Tchebycheff metric. Using ‘filtering’ techniques, maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration. The procedure has the advantage that it can converge to non-extreme final solutions. Especially suitable for multiple objective linear programming, the procedure is also applicable to integer and nonlinear multiple objective programs.     

Augmented Lagrangian and Tchebycheff Approaches in Multiple Objective Programming

Relationships between the Tchebycheff scalarization and the augmented Lagrange multiplier technique are examined in the framework of general multiple objective programs (MOPs). It is shown that under certain conditions the Tchebycheff method can be represented as a quadratic weighted-sums scalarization of the MOP, that is, given weight values in the former, the coefficients of the latter can be found so that the same efficient point is selected. Analysis for concave and linear MOPs is included. Resulting applications in multiple criteria decision making are also discussed.     

A Fixed Point Method for Tchebycheff Solution of Inconsistent Linear Equations

The problem of solving an inconsistent set of linear equationsin the Tchebycheff sense is reduced to solving a finite sequenceof inconsistent sets of linear equations in the least-squaressense. The method, which is not a variant of either the simplexmethod or the well known ascent and descent methods, requiresno restrictive assumptions concerning the system of equations.A single solution is obtained, whether or not the problem hasa unique Tchebycheff solution.     

Compromise programming with Tchebycheff norm for discrete stochastic orders

This paper presents a method of decision making with returns in the form of discrete random variables. The proposed method is based on two approaches: stochastic orders and compromise programming used in multi-objective programming. Stochastic orders are represented by stochastic dominance and inverse stochastic dominance. Compromise programming uses the augmented Tchebycheff norm. This norm, in special cases, takes form of the Kantorovich and Kolmogorov probability metrics. Moreover, in the paper we show applications of the presented methodology in the following problems: projects selections, decision tree and choosing a lottery.