Friedland (1981) showed that for a nonnegative square matrix A, the spectral radius r(eA) is a log-convex functional over the real diagonal matrices D. He showed that for fully indecomposable A, log r(eA) is strictly convex over D1,D2 if and only if D1 −D2 6= c I for any c ∈ R. Here the condition of full indecomposability is shown to be replaceable by the weaker condition that A and A>A be irre...

Abstract We prove monotonicity of a parabolic frequency on static and evolving manifolds without any curvature or other assumptions. These are analogs Almgren’s function. When the manifold is Euclidean space drift operator Ornstein–Uhlenbeck operator, this can been seen to imply Poon’s for ordinary heat equation. self-similarly by Ricci flow, we solutions For Gaussian soliton, gives directly mo...

We introduce a notion of k-convexity and explore polygons in the plane that have this property. Polygons which are k-convex can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of 2-convex polygons, a particularly interesting class, and show how to recognize them in O(n log n) time. A description of their sh...

Motivated by a dynamic location problem for graphs, Chung, Graham and Saks introduced a graph parameter called windex. Graphs of windex 2 turned out to be, in graph-theoretic language, retracts of hypercubes. These graphs are also known as median graphs and can be characterized as partial binary Hamming graphs satisfying a convexity condition. In this paper an O(n 3 2 logn) algorithm is present...

We continue the recent work of [2] and [25] by showing that whenever the Lévy measure of a spectrally negative Lévy process has a density which is log convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Lévy process. Our method app...

Recently, Hermite-Hadamard’s inequality has been the subject of intensive research. In particular, many improvements, generalizations, and applications for the HermiteHadamard’s inequality can be found in the literature 2–20 . Let I ⊆ 0,∞ be an interval; a real-valued function f : I → R is said to be GA-convex concave on I if f xαy1−α ≤ ≥ αf x 1 − α f y for all x, y ∈ I and α ∈ 0, 1 . In 21 , A...

In this paper, we investigate the online non-convex optimization problem which generalizes the classic online convex optimization problem by relaxing the convexity assumption on the cost function. For this type of problem, the classic exponential weighting online algorithm has recently been shown to attain a sub-linear regret of O( √ T log T ). In this paper, we introduce a novel recursive stru...

The Catalan-like numbers cn,0, defined by cn+1,k = rk−1cn,k−1 + skcn,k + tk+1cn,k+1 for n, k ≥ 0, c0,0 = 1, c0,k = 0 for k 6= 0, unify a substantial amount of well-known counting coefficients. Using an algebraic approach, Zhu showed that the sequence (cn,0)n≥0 is log-convex if rktk+1 ≤ sksk+1 for all k ≥ 0. Here we give a combinatorial proof of this result from the point of view of weighted Mot...

This paper proposes a general method to validate the first-order approach for moral hazard problems with hidden saving. I show that strong convexity assumptions both on the agent’s marginal utility of consumption and the distribution function of output arise naturally in this context. The first-order approach is valid given nonincreasing absolute risk aversion (NIARA) utility and log-convex dis...