E. Mossel and J. Neeman recently provided a heat flow monotonicity proof of Borell’s noise stability theorem. In this note, we develop the argument to include in a common framework noise stability, Brascamp-Lieb inequalities (including hypercontractivity), and even a weak form of Slepian inequalities. The scheme applies furthermore to families of measures with are more log-concave than the Gaus...

For a real-valued nonnegative and log-concave function f defined in R, we introduce a notion of difference function ∆f ; the difference function represents a functional analog on the difference body K + (−K) of a convex body K. We prove a sharp inequality which bounds the integral of ∆f from above, in terms of the integral of f and we characterize equality conditions. The investigation is exten...

We describe competition over space between a competitive shipping industry (truck-barge) and one with market power (the railroad). The latter prices so as to "beat the competition" in equilibrium, or else at the monopoly price, if that is lower. The monopoly price rises more slowly than do the costs of transportation (freight absorption) if the spatial demand at each point is log-concave. With ...

A polynomial is unimodal if its sequence of coefficients are increasing up to an index, and then are decreasing after that index. A polynomial is logconcave if the sequence of the logarithms of the coefficients is concave. We prove that if P (x) is a polynomial with nonnegative non-decreasing coefficients then P (x+z) is unimodal for any natural z. Furthermore, we prove that if P (x) is a log-c...

In this paper, we discuss the properties of the hyperfibonacci numbers F [r] n and hyperlucas numbers L [r] n . We investigate the log-concavity (log-convexity) of hyperfibonacci numbers and hyperlucas numbers. For example, we prove that {F [r] n }n≥1 is log-concave. In addition, we also study the log-concavity (log-convexity) of generalized hyperfibonacci numbers and hyperlucas numbers.

A class of probabilistic constrained programming problems are considered where the probabilistic constraint is of the form P{gi(x, ξ) ≥ 0, i = 1, . . . , r} ≥ p and the functions gi, i = 1, . . . , r are concave. It is shown that the x-function on the left hand side is logarithmic concave provided ξ has a logarithmic concave density. Special cases are mentioned and algorithmic solution of probl...

In this paper the turnpike property is established for convex optimal control problems, involving undiscounted utility function and differential inclusions defined by multi-valued mapping having convex graph. Utility function is concave but not necessarily strictly concave. The turnpike theorem is proved under the main assumption that over any given line segment, either multi-valued mapping is ...

We provide a uniformly efficient and simple random variate generator for the entire parameter range of the generalized inverse gaussian distribution. A general algorithm is provided as well that works for all densities that are proportional to a log-concave function φ, even if the normalization constant is not known. It requires only black box access to φ and its derivative.

Let In,k (respectively, Jn,k) be the number of involutions (respectively, fixed-point free involutions) of {1, . . . , n} with k descents. Motivated by Brenti’s conjecture which states that the sequence In,0, In,1, . . . , In,n−1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such t...

Recently, some combinatorial properties for the the Catalan-Larcombe-French numbers have been proved by Sun and Wu, and Zhao. Recently, Z. W. Sun conjectured that the root of the Catalan-Larcombe-French numbers is log-concave. In this paper, we confirm Sun's conjecture by establishing the lower and upper bound for the ratios of the Catalan-Larcombe-French numbers.