Reflexive Cones

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \ell_{1}. Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in a Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the Schur spaces by the properties of reflexive cones is given.Comment: 23 page

Spectrahedral cones generated by rank 1 matrices

Let ${\cal S}_+^n \subset {\cal S}^n$ be the cone of positive semi-definite matrices as a subset of the vector space of real symmetric $n \times n$ matrices. The intersection of ${\cal S}_+^n$ with a linear subspace of ${\cal S}^n$ is called a spectrahedral cone. We consider spectrahedral cones $K$ such that every element of $K$ can be represented as a sum of rank 1 matrices in $K$. We shall call such spectrahedral cones rank one generated (ROG). We show that ROG cones which are linearly isomorphic as convex cones are also isomorphic as linear sections of the positive semi-definite matrix cone, which is not the case for general spectrahedral cones. We give many examples of ROG cones and show how to construct new ROG cones from given ones by different procedures. We provide classifications of some subclasses of ROG cones, in particular, we classify all ROG cones for matrix sizes not exceeding 4. Further we prove some results on the structure of ROG cones. We also briefly consider the case of complex or quaternionic matrices. ROG cones are in close relation with the exactness of semi-definite relaxations of quadratically constrained quadratic optimization problems or of relaxations approximating the cone of nonnegative functions in squared functional systems.Comment: Version 2: section on complex and quaternionic case added, many sections completely rewritte

PARENTAL EFFECTS ON OFFSPRING REACTION NORMS: CONSEQUENCES FOR COMPLEX PHENOTYPES IN VARIABLE ENVIRONMENTS

Organismal traits all exhibit some degree of environmental sensitivity, and both the strength and direction of these phenotypically plastic responses to the environment can evolve in adaptive ways. For example, parents can use information about their own environment to precondition the traits of their offspring so that they thrive in their future environment. This transgenerational plasticity can also alter the plasticity of offspring, but explicit investigations of this specific phenomenon are rare. I begin with a review of the literature and provide a quantitative genetic framework to investigate this phenomenon, which I then explore empirically using avian embryos. The metabolic rate of avian embryos is highly plastic to temperature, and thus, cold conditions cause costly developmental delays. However, evidence from other taxa suggests that ectotherms in cold conditions can modify their metabolic plasticity to temperature. Consequently, I predict that (1) parental effects, such as lay date and incubation behavior, modify avian embryonic metabolic reaction norms through their effects on early thermal conditions, and (2) that these modifications of embryonic metabolism affect growth. First, I analyzed whether parental effects alter embryonic metabolic reaction norms in free-living house sparrows (Passer domesticus). Indeed, both the incubation environment and maternal age influenced embryonic reaction norms and growth. Next, I experimentally tested the effect of incubation temperature on embryonic reaction norms and growth in pekin ducks (Anas platyrhynchos domesticus), which readily develop in laboratory conditions. Contrary to expectations, the incubation environment did not alter embryonic reaction norms. Instead, genetic and/or pre-laying parental effects produced individual differences in metabolic plasticity, which related to post-hatching growth. Finally, I investigated the factors affecting optimal parental incubation by developing a simulation model. I found that optimal incubation was plastic to many environmental factors but was strongly modified by species ecophysiology. Together, these results demonstrate that plastic parental effects do alter offspring reaction norms and fitness. These investigations provide a conceptual and analytical framework for future investigations of this phenomenon. More work is needed to understand how these interactions between parental and offspring environments affect the evolution of plastic traits in an environmentally variable world