When is a non-self-adjoint Hill operator a spectral operator of scalar type?

We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator $H=?{\mathrm{d}}^2/\mathrm{d}{x}^2+V$ in $L^2(\mathbb{R})$ to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. To cite this article: F. Gesztesy, V. Tkachenko, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

When are Homogeneous Functions Linear?

An R-module V is a ray if for every R-module W, the R-homogeneous functions from V to W are additive. We use properties of the lattice of submodules of $ V, {\cal L}(V) $ , to determine conditions for V to be a ray. We also use the lattice structure of $ {\cal L}(V) $ to further study those rings such that every R-module V is a ray. As a result, we can characterize all semisimple modules which are rays.     

When are stars the largest cross-intersecting families?

We provide a necessary and sufficient condition for stars to be the largest cross-intersecting families.     

When are inner mapping groups generated by conjugation maps?

A subloop of a loop Q is said to be normal if it is stabilized by all maps in the inner mapping group of Q. Here we show that in many cases, the inner mapping group of a Moufang loop is actually generated by conjugation maps. This includes any Moufang loop whose cubes generate either the entire loop or a subloop of index three. Such a result can be an extremely useful tool when proving that certain subloops are indeed normal just by showing that they are stabilized by the conjugation maps.     

When are intermediate processes of the same stochastic order?

Let Z_m⁽ⁿ⁾ represent the mth largest order statistic in a random sample of size n. Here we study the process Z_[mt]⁽ⁿ⁾, t > 0, where m(n) is an intermediate sequence such that m → ∞, m/n → 0 as n → ∞.     

When are solutions to nonlinear elliptic boundary value problems convex?

We extend and apply a concavity maximum principle from [10, 9, 7] to some nonlinear elliptic boundary problems and free boundary problems on convex domains Ω?IRⁿ. In particular we extend "convex dead core' results from n = 2 as in [4 ] to arbitrary n. We also show the convexity of the coincidence set in the obstacle problem under suitable assumptions.     

When are small subgraphs of a random graph normally distributed?

Summary LetG be a graph and letX _n count copies ofG in a random graphK(n,p). The random variable is asymptotically normally distributed if and only ifnp ^m andn ² (1-p), wherem=max {e(H)/|H|:HG}. In addition to, and in connection with this main result we investigate the formula for Var (X _n ) and the Poisson convergence ofX _n .Part of the research was done during the author's stay in Division of Mathematics and Science, St. John's University, Staten Island, New York     

Plates made of piezoelectric materials: When are they really piezoelectric?

This paper aims at presenting in a synthetic way mathematical results that have been rigorously and recently derived by the authors. These results deal with the simplified but accurate modeling of linearly piezoelectric thin plates. It is shown how mathematical tools lead to two different situations linked to either sensors or actuators devices. Moreover we enlighten the fact that for some piezoelectric crystal classes, the coupling between the electrical and the mechanical effects disappear in the plate. Finally, we furnish a detailed example of such models in the case of a plate constituted by a 222 symmetry class material.     

When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?

Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.     

Is the linearized Boltzmann-Enskog operator dissipative?

In this paper, it is shown that the linearized Boltzmann-Enskog collision operator cannot be dissipative in the L₂-space setting contrarily to the linearized Boltzmann operator. Some estimates useful for the spectral theory are given.     

When is tit-for-tat unbeatable?

We characterize the class of symmetric two-player games in which tit-for-tat cannot be beaten even by very sophisticated opponents in a repeated game. It turns out to be the class of exact potential games. More generally, there is a class of simple imitation rules that includes tit-for-tat but also imitate-the-best and imitate-if-better. Every decision rule in this class is essentially unbeatable in exact potential games. Our results apply to many interesting games including all symmetric 2 $\times $ 2 games, and standard examples of Cournot duopoly, price competition, public goods games, common pool resource games, and minimum effort coordination games.     

When is Flatness Coherent?

ABSTRACT

We characterize those small categories with the property that flat (contravariant) functors on them are coherently axiomatized in the language of presheaves on them. They are exactly the categories with the property that every finite diagram into them has a finite set of (weakly) initial cocones.