Log-concavity and discrete degrees of freedom Jakimiuk, Jacek; Murawski, Daniel; Nayar, Piotr ...
Discrete mathematics,
June 2024, 2024-06-00, Volume:
347, Issue:
6
Journal Article
Peer reviewed
We develop the notion of discrete degrees of freedom of a log-concave sequence and use it to prove that geometric distribution minimises Rényi entropy of order infinity under fixed variance, among ...all discrete log-concave random variables in Z. We also show that the quantity P(X=EX) is maximised, among all ultra-log-concave random variables with fixed integral mean, for a Poisson distribution.
We show that the exterior algebra ΛRα1,⋯,αn, which is the cohomology of the torus T=(S1)n, and the polynomial ring Rt1,…,tn, which is the cohomology of the classifying space B(S1)n=(CP∞)n, are ...Sn-equivariantly log-concave. We do so by explicitly giving the Sn-representation maps on the appropriate sequences of tensor products of polynomials or exterior powers and proving that these maps satisfy the hard Lefschetz theorem. Furthermore, we prove that the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann bilinear relations, holds on the corresponding sequences in an equivariant setting.
Polynomial sequence {Pm}m≥0 is q-logarithmically concave if Pm2−Pm+1Pm−1 is a polynomial with nonnegative coefficients for any m≥1. We introduce an analogue of this notion for formal power series ...whose coefficients are nonnegative continuous functions of a parameter. Four types of such power series are considered where the parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of q-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions. In the last section of the paper we present new inequalities for the Kummer function, the ratio of the Gauss functions and the generalized hypergeometric function obtained as direct applications of the general theorems.
Ehrhart polynomials of rank two matroids Ferroni, Luis; Jochemko, Katharina; Schröter, Benjamin
Advances in applied mathematics,
10/2022, Volume:
141
Journal Article
Peer reviewed
Open access
Over a decade ago De Loera, Haws and Köppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding h⁎-polynomials form ...a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater than or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhart polynomials of minimal and uniform matroids. We furthermore address the second conjecture by proving that h⁎-polynomials of matroid polytopes of sparse paving matroids of rank two are real-rooted and therefore have log-concave and unimodal coefficients. In particular, this shows that the h⁎-polynomial of the second hypersimplex is real-rooted, thereby strengthening a result of De Loera, Haws and Köppe.
Log-concavity of B-splines Floater, Michael S.
Journal of approximation theory,
June 2024, 2024-06-00, Volume:
300
Journal Article
Peer reviewed
Open access
Curry and Schoenberg showed that a B-spline is log-concave in its support by applying Brunn’s theorem to a simplex. In this note we provide an alternative, ‘analytic’ proof of the log-concave ...property using only recursion formulas for B-splines and their first and second derivatives.
Log‐concavity of multivariate distributions is an important concept in general and has a very special place in the field of Reliability Theory. An attempt has been made in this paper to study ...preservation results for (i) the discrete version of multivariate log‐concavity for multistate series and multistate parallel systems consisting of n$$ n $$ independent components, states of both components and systems being represented by elements in a subset S2={0,1,2}$$ {S}_2=\left\{0,1,2\right\} $$ of SM={0,1,2,…,M},$$ {S}_M=\left\{0,1,2,\dots, M\right\}, $$ and (ii) the continuous version of multivariate log‐concavity under multistate series and multistate parallel systems made up of n$$ n $$ independent components and states of both, systems and components, taking values in the set SM$$ {S}_M $$. These results for discrete and continuous versions of log‐concavity have also been extended to systems that are formed using both multistate series and multistate‐parallel systems. Further, the results in (ii) have been used to obtain important and useful bounds on joint probabilities related to times spent by multistate components, multistate series, multistate parallel systems, and the combinations thereof.
In this paper, we provide a new proof of the preservation of the log concavity by Bernstein operator. It is based on the bivariate characterization of the likelihood ratio order. In addition, we give ...new conditions under which the previous property leads to preservation of ageing properties in coherent systems with independent and identically distributed components.
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