Probability and Measure

A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

A Theory of Graphs

Ramanujan’s notebooks, by Bruce C. Berndt. Part 1. Pp 357. DM188. 1985. ISBN 3-540-96110-0 (Springer)

Recently the author developed a numerical method for the multidimensional momentconstrained
					maximum entropy problem, which is practically capable of solving maximum entropy
					problems in the two-dimensional domain with moment constraints of order up to 8, in the threedimensional
					domain with moment constraints of order up to 6, and in the four-dimensional domain
					with moment constraints of order up to 4, corresponding to the total number of moment constraints of
					44, 83 and 69, respectively. In this work, the author brings together key algorithms and observations
					from his previous works as well as other literature in an attempt to present a comprehensive exposition
					of the current methods and results for the multidimensional maximum entropy moment problem.

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The multidimensional maximum entropy moment problem: a review of numerical methods

The purpose of this paper is to exploit umbral calculus in order to derive some properties, recurrence relations, and identities related to the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials. Especially, we will express the degenerate r-Bell polynomials as linear combinations of many well-known families of special polynomials.

Degenerate r-Stirling Numbers and r-Bell Polynomials

Bell polynomials and binomial type sequences

In this paper, we show that the r-Stirling numbers of both kinds, the r-Whitney numbers of both kinds, the r-Lah numbers and the r-Whitney-Lah numbers form particular cases of family of polynomials forming a generalization of the partial Bell polynomials. We deduce the generating functions of several restrictions of these numbers. In addition, a new combinatorial interpretations is presented for the r-Whitney numbers and the r-Whitney-Lah numbers.

The partial r-Bell polynomials

Letting B"n(x) the n-th Bell polynomial, it is well known that B"n admit specific integer coordinates in the two following bases {x^i}"i"="0","...","n and {xB"i(x)}"i"="0","...","n"-"1 according, respectively, to Stirling numbers and binomial coefficients. Our aim is to prove that, for r+s=n, the sequence {x^jB"k(x)}"j"="0","...","r"k"="0","...","s is a family of bases of the Q-vectorial space formed by polynomials of Q[X] for which B"n admits a Binomial Recurrence Coefficient.

A generalized recurrence for Bell polynomials: An alternate approach to Spivey and Gould-Quaintance formulas

In this paper, we show that the r-Stirling numbers of both kinds, the r-Whitney numbers of both kinds, the r-Lah numbers and the r-Whitney-Lah numbers form particular cases of a family of polynomials forming a generalization of the partial Bell polynomials. We deduce the generating functions of several restrictions of these numbers. In addition, a new combinatorial interpretations is presented for the r-Whitney numbers and the r-Whitney-Lah numbers.

Miloud Mihoubi

Papers

Bell polynomials and binomial type sequences

The partial r-Bell polynomials

A generalized recurrence for Bell polynomials: An alternate approach to Spivey and Gould-Quaintance formulas

The partial r-Bell polynomials

Some applications of the r-Whitney numbers