Being motivated by the work of Cochran et al. on the Gel'fand triple E] (L 2) E] , we are led to nd elementary functions to replace the exponential generating functions G and G 1== for the characterization of generalized and test functions. We deene the Legendre transformù for u in C +;log and the L-function Lu when u 2 C +;log is (log, exp)-convex. We show that u is equivalent to Lu. The dual ...

A simple proof is given for the convexity of log det(I +KX) in the positive definite matrix variable X ≻ 0 with a given positive semidefinite K 0. Convexity of functions of covariance matrices often plays an important role in the analysis of Gaussian channels. For example, suppose Y and Z are independent complex Gaussian nvectors with Y ∼ N(0,K) and Z ∼ N(0,X). Then, I(Y;Y + Z) = log det(I +KX)...

The Catalan-Larcombe-French sequence {Pn}n≥0 arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence {P2 n – Pn–1Pn+1}n≥1 are strictly log-con...

Let Pn,k be the number of permutations π on [n] = {1, 2, . . . , n} such that the length of the longest increasing subsequences of π equals k, and let M2n,k be the number of matchings on [2n] with crossing number k. Define Pn(x) = ∑ k Pn,kx k and M2n(x) = ∑ k M2n,kx . We propose some conjectures on the log-concavity and q-log-convexity of the polynomials Pn(x) and M2n(x).

In this paper we will develop a systematic method to answer the questions (Q1)(Q2)(Q3)(Q4) (stated in Section 1) with complete generality. As a result, we can solve the difficulties (D1)(D2) (discussed in Section 1) without uncertainty. For these purposes we will introduce certain classes of growth functions u and apply the Legendre transform to obtain a sequence which leads to the weight seque...

We consider a system of two reaction-diffusion equations coming out reversible chemistry. When the reaction happens on totality domain, it is known that exponential convergence to equilibrium holds (with explicit rate). show in this paper also when only given open set ball, thanks an observation estimate deduced by logarithmic convexity.

Despite recent interest in hyperbolic discounting, there has been little discussion of exactly what property of time preferences is instantiated by hyperbolic or quasi-hyperbolic functional forms. The paper revives an earlier proposal in Prelec (1989) that the key property is Pratt– Arrow convexity of the log of the discount function, which corresponds to decreasing impatience (DI) at the level...

Filling operations are procedures which are used in Discrete Tomography for the reconstruction of lattice sets having some convexity constraints. Many algorithms have been published giving fast implementations of these operations, and the best running time ([7]) is O(N2 log N) time, where N is the size of projections. In this paper we improve this result by providing an implementation of the fi...