“Mathematics Days in Sofia”

Section “Numerical Analysis, Operations Research, Probability and Statistics”

Participants

Invited Speakers

  • Geno Nikolov, Sofia University “St. Kliment Ohridski”, Bulgaria

  • George Yanev, University of Texas Rio Grande Valley, USA

Contributors

  • Alessandro Barbiero, Università degli Studi di Milano, Italy

  • Detelina Kamburova, Institute of Mathematics and Informatics, Bulgaria

  • Hristo Sariev, Institute of Mathematics and Informatics, Bulgaria

  • Martin Minchev, Sofia University “St. Kliment Ohridski”, Bulgaria

  • Mohammed Beggas, University of El-Oued, Algeria

  • Pando Georgiev, Institute of Mathematics and Informatics and Sofia University “St. Kliment Ohridski”, Bulgaria

Program and Abstracts

The beautiful refinement of the inequality of the brothers Markov, found by R.J. Duffin and A.C. Schaeffer in 1941, can be viewed as a comparison-type theorem: the assumption that the absolute values of a n-th degree polynomial are majorized by the absolute values of the n-th Chebyshev polynomial at its extreme points in [-1,1] implies inequalities between their uniform norms. I will discuss some comparison theorems of Duffin-Schaeffer type where the role of the extreme polynomial is played by ultraspherical or Jacobi polynomials and the check-points are the zeros of related ultraspherical or Jacobi polynomials. Results of this kind are related to the Markov-type inequalities with a curved majorant, and to estimation of the round-off error in formulae for numerical differentiation.

The problem of characterization of probability distributions can be described as follows. It is known that a family of distributions F possesses certain property P. Is it true, conversely, that a distribution has the property P only if it is a member of F? If so, then P characterizes the family F.

In this talk, we will discuss characterizations of the exponential distribution. More specifically, we will present recent results regarding characterization properties of the exponential distribution involving order statistics, record values, and sums of random variables. Let us point out that the contributions of Bulgarian mathematicians in the area of probability characterizations can be traced back to a 1961 paper by Apostol (a.k.a. Shefa) Obretenov.

A count distribution obtained as a discrete version of the continuous half-logistic distribution is introduced. It is derived by assigning to each non-negative integer value a probability proportional to the corresponding value of the density function of the parent model. Statistical properties of this new distribution, in particular related to its shape, moments, and reliability concepts, are described. Parameter estimation, which can be carried out resorting to different methods including maximum likelihood, is discussed and a numerical comparison between the methods, based on Monte Carlo simulations, is presented. The applicability of the proposed distribution is proved on a real dataset, which has been already fitted by other well-established count distributions. In order to increase the flexibility of this counting model, a generalization is finally suggested, which is obtained by adding a shape parameter to the continuous one-parameter half-logistic and then applying the same discretization technique, based on the mimicking of the density function.

We apply some recent results for supinf problems in completely regular topological spaces and establish a variational principle for saddle points. Well-posedness of saddle point problems is studied as well.

This is a joint work with Rumen Marinov and Nadia Zlateva.

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