‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...

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...

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.