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

Virtue and austerity

Virtue ethics is often proposed as a third way in health-care ethics, that while consequentialism and deontology focus on action guidelines, virtue focuses on character; all three aim to help agents discern morally right action although virtue seems to have least to contribute to political issues, such as austerity. I claim: (1) This is a bad way to characterize virtue ethics. The 20th century renaissance of virtue ethics was first proposed as a response to the difficulty of making sense of ‘moral rightness’ outside a religious context. For Aristotle the right action is that which is practically best; that means best for the agent in order to live a flourishing life.There are no moral considerations besides this. (2) Properly characterized, virtue ethics can contribute to discussion of austerity. A criticism of virtue ethics is that fixed characteristics seem a bad idea in ever-changing environments; perhaps we should be generous in prosperity, selfish in austerity. Furthermore, empirical evidence suggests that people indeed do change with their environment. However, I argue that virtues concern fixed values not fixed behaviour; the values underlying virtue allow for different behaviour in different circumstances: in austerity, virtues still give the agent the best chance of flourishing. Two questions arise. (a) In austere environments might not injustice help an individual flourish by, say, obtaining material goods? No, because unjust acts undermine the type of society the agent needs for flourishing. (b) What good is virtue to those lacking the other means to flourish? The notion of degrees of flourishing shows that most people would benefit somewhat from virtue. However, in extreme circumstances virtue might harm rather than benefit the agent: such circumstances are to be avoided; virtue ethics thus has a political agenda to enable flourishing. This requires justice, a fortiori when in austerity

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

Intergenerational justice of what: welfare, resources or capabilities?

An important aspect of intergenerational justice concerns the specification of a 'currency of advantage' that can be used to evaluate distributive outcomes across time. Environmental theorists have introduced several innovative currencies of justice in recent years, such as ecological space and critical natural capital. However they have often downplayed the application of established currencies (such as welfare, resources or capabilities) to issues of futurity. After exploring the merits of a number of rival currencies, it is argued that the currency of 'capabilities to function' provides a promising basis for a theory of justice that takes seriously the rights and duties of intergenerational justice

Educating for Autonomy: Liberalism and Autonomy in the Capabilities Approach

Martha Nussbaum grounds her version of the capabilities approach in political liberalism. In this paper, we argue that the capabilities approach, insofar as it genuinely values the things that persons can actually do and be, must be grounded in a hybrid account of liberalism: in order to show respect for adults, its justification must be political; in order to show respect for children, however, its implementation must include a commitment to comprehensive autonomy, one that ensures that children develop the skills necessary to make meaningful choices about whether or not to exercise their basic capabilities. Importantly, in order to show respect for parents who do not necessarily recognize autonomy as a value, we argue that the liberal state, via its system of public education, should take on the role of ensuring that all children within the state develop a sufficient degree of autonomy

Linear instability criteria for ideal fluid flows subject to two subclasses of perturbations

In this paper we examine the linear stability of equilibrium solutions to incompressible Euler's equation in 2- and 3-dimensions. The space of perturbations is split into two classes - those that preserve the topology of vortex lines and those in the corresponding factor space. This classification of perturbations arises naturally from the geometric structure of hydrodynamics; our first class of perturbations is the tangent space to the co-adjoint orbit. Instability criteria for equilibrium solutions are established in the form of lower bounds for the essential spectral radius of the linear evolution operator restricted to each class of perturbation.Comment: 29 page

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

Quantum correlations and distinguishability of quantum states

A survey of various concepts in quantum information is given, with a main emphasis on the distinguishability of quantum states and quantum correlations. Covered topics include generalized and least square measurements, state discrimination, quantum relative entropies, the Bures distance on the set of quantum states, the quantum Fisher information, the quantum Chernoff bound, bipartite entanglement, the quantum discord, and geometrical measures of quantum correlations. The article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them, and for mathematicians looking for an up-to-date introductory course on these subjects, which are mainly developed in the physics literature.Comment: Review article, 103 pages, to appear in J. Math. Phys. 55 (special issue: non-equilibrium statistical mechanics, 2014