Tensor Products, Positive Linear Operators, and Delay-Differential Equations

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\dot x(t)=-\alpha(t)x(t)-\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta\beta(t)\ge 0$ with $\delta\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha(t)$ and $\beta(t)$ are periodic of the same period, and $\beta(t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$-positivity of the exterior product is investigated when $\beta(t)$ satisfies a uniform sign condition.Comment: 84 page

Taricha torosa

Number of Pages: 4Integrative BiologyGeological Science

The horofunction boundary of the Hilbert geometry

We investigate the horofunction boundary of the Hilbert geometry defined on an arbitrary finite-dimensional bounded convex domain D. We determine its set of Busemann points, which are those points that are the limits of `almost-geodesics'. In addition, we show that any sequence of points converging to a point in the horofunction boundary also converges in the usual sense to a point in the Euclidean boundary of D. We prove that all horofunctions are Busemann points if and only if the set of extreme sets of the polar of D is closed in the Painleve-Kuratowski topology.Comment: 24 pages, 2 figures; minor changes, examples adde

Existence of positive solutions of a superlinear boundary value problem with indefinite weight

We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change its sign. We assume that the function $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, so as the case $g(s)=s^{p}$, with $p>1$, is covered. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.Comment: 12 pages, 4 PNG figure

A Metric Inequality for the Thompson and Hilbert Geometries

There are two natural metrics defined on an arbitrary convex cone: Thompson's part metric and Hilbert's projective metric. For both, we establish an inequality giving information about how far the metric is from being non-positively curved.Comment: 15 pages, 0 figures. To appear in J. Inequalities Pure Appl. Mat

In defence of global egalitarianism

This essay argues that David Miller's criticisms of global egalitarianism do not undermine the view where it is stated in one of its stronger, luck egalitarian forms. The claim that global egalitarianism cannot specify a metric of justice which is broad enough to exclude spurious claims for redistribution, but precise enough to appropriately value different kinds of advantage, implicitly assumes that cultural understandings are the only legitimate way of identifying what counts as advantage. But that is an assumption always or almost always rejected by global egalitarianism. The claim that global egalitarianism demands either too little redistribution, leaving the unborn and dissenters burdened with their societies' imprudent choices, or too much redistribution, creating perverse incentives by punishing prudent decisions, only presents a problem for global luck egalitarianism on the assumption that nations can legitimately inherit assets from earlier generations – again, an assumption very much at odds with global egalitarian assumptions

Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

Ten myths about character, virtue and virtue education – plus three well-founded misgivings

Initiatives to cultivate character and virtue in moral education at school continue to provoke sceptical responses. Most of those echo familiar misgivings about the notions of character, virtue and education in virtue – as unclear, redundant, old-fashioned, religious, paternalistic, anti-democratic, conservative, individualistic, relative and situation dependent. I expose those misgivings as ‘myths’, while at the same time acknowledging three better-founded historical, methodological and practical concerns about the notions in question