Abstract:
A new polynomial approximation operator has been proposed which uses weight-functions of the well-known Bernstein’s Polynomial operator in its probabilistically improved combinatorial structure, achieved through a rather-ingenious ‘Fusion’ of two dual perspectives. These weights are functions of the impugned variable of the unknown function being approximated, and are not mere constants. The new approximation formula has been compared empirically with the simple classical method of polynomial approximation using the well-known “Bernstein Operator”. The percentage absolute relative errors for the proposed approximation formula and that with the “Bernstein Operator” have been computed for certain selected functions and with different number of node points in the interval of approximation. It has been observed that the proposed approximation formula produces exceedingly-significantly better results.

Abstract:
We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen such object has the required properties with positive yet tiny probability. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.

Abstract:
Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Egecioglu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and Shimozono. We discuss how this approach may work in general.

Abstract:
Divided symmetrization of a function $f(x_1,\dots,x_n)$ is symmetrization of the ratio $$DS_G(f)=\frac{f(x_1,\dots,x_n)}{\prod (x_i-x_j)},$$ where the product is taken over the set of edges of some graph $G$. We concentrate on the case when $G$ is a tree and $f$ is a polynomial of degree $n-1$, in this case $DS_G(f)$ is a constant function. We give a combinatorial interpretation of the divided symmetrization of monomials for general trees and probabilistic game interpretation for a tree which is a path. In particular, this implies a result by Postnikov originally proved by computing volumes of special polytopes.

Abstract:
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen structure has the required properties with positive yet tiny probability. Our method allows also to give rather precise estimates on the number of objects of a given size and this is applied to count the number of orthogonal arrays, t-designs and regular hypergraphs. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.

Abstract:
The main object of this paper is to show how we can use classical probabilistic methods such as Maximum Entropy (ME), maximum likelihood (ML) and/or Bayesian (BAYES) approaches to do microscopic and macroscopic data fusion. Actually ME can be used to assign a probability law to an unknown quantity when we have macroscopic data (expectations) on it. ML can be used to estimate the parameters of a probability law when we have microscopic data (direct observation). BAYES can be used to update a prior probability law when we have microscopic data through the likelihood. When we have both microscopic and macroscopic data we can use first ME to assign a prior and then use BAYES to update it to the posterior law thus doing the desired data fusion. However, in practical data fusion applications, we may still need some engineering feeling to propose realistic data fusion solutions. Some simple examples in sensor data fusion and image reconstruction using different kind of data are presented to illustrate these ideas. Keywords: Data fusion, Maximum entropy, Maximum likelihood, Bayesian data fusion, EM algorithm.

Abstract:
Applying techniques similar to Combinatorial Nullstellensatz we prove a lower estimate of $|f(A,B)|$ for finite subsets $A$, $B$ of a field, and polynomial $f(x,y)$ of the form $f(x,y)=g(x)+yh(x)$, where degree of $g$ is greater then degree of $h$.

Abstract:
A new combinatorial-probabilistic diagnostic entropy has been introduced. It describes the pair-wise sum of probabilities of system conditions that have to be distinguished during the diagnosing process. The proposed measure describes the uncertainty of the system conditions, and at the same time complexity of the diagnosis problem. Treating the assumed combinatorial-diagnostic entropy as a primary notion, the information delivered by the symptoms has been defined. The relationships have been derived to facilitate explicit, quantitative assessment of the information of a single symptom as well as that of a symptoms set. It has been proved that the combinatorial-probabilistic information shows the property of additivity. The presented measures are focused on diagnosis problem, but they can be easily applied to other disciplines such as decision theory and classification.

Abstract:
Crucial to an Evolutionary Algorithm's performance is its selection scheme. We mathematically investigate the relation between polynomial rank and probabilistic tournament methods which are (respectively) generalisations of the popular linear ranking and tournament selection schemes. We show that every probabilistic tournament is equivalent to a unique polynomial rank scheme. In fact, we derived explicit operators for translating between these two types of selection. Of particular importance is that most linear and most practical quadratic rank schemes are probabilistic tournaments.