A review of the basic discrete q-Distributions

Consider a sequence of independent Bernoulli trials and assume that the probability (or odds) of success (or the probability (or odds) of failure) at a trial varies (increases or decreases) geometrically, with rate q, either with the number of trials or with the number of successes. Let X n be the number of successes up the nth trial and W n (or T k ) be the number of failures (or trials) until the occurrence of the nth (or kth) success. The distributions of these random variables turned out to be q-analogues of the binomial and negative binomial (or Pascal) distributions. The Heine and Euler distributions, which are q-analogues of the Poisson distribution, are obtained as limiting distributions of q-binomial distributions (or negative q-binomial distributions), as the number of trials (or the number of successes) tends to infinity. Also, introducing the notion of a q-drawing of a ball from an urn containing balls of various kinds, a q-analogue of the Pólya urn model is constructed and q-Pólya and inverse q-Pólya distributions are examined. Finally, considering a stochastic model that is developing in time or space, in which events (successes) may occur at continuous points, a Heine and an Euler stochastic processes are presented. © Springer Nature Switzerland AG 2019

New Upper Bounds for the Density of Translative Packings of Three-Dimensional Convex Bodies with Tetrahedral Symmetry

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.Comment: 30 pages, 6 tables, 3 figures, (v3) comments of referees incorporate