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

Drawings as representations of children's conceptions

Drawings are often used to get an idea of children's conceptions. Doing so takes for granted an unambiguous relation between conceptions and their representations in drawings. This study was undertaken to gain knowledge of the relation between children's conceptions and their representation of these conceptions in drawings. A theory of contextualization was the basis for finding out how children related their contextualization of conceptions in conceptual frameworks to their contextualization of drawings in pictorial convention. Eighteen children were interviewed in a semi-structured method while they were drawing the Earth. Audio-recorded interviews, drawings and notes were analysed to find the cognitive and cultural intentions behind the drawings. Also, even children who demonstrated alternative conceptions of the Earth in the interviews still followed cultural conventions in their drawings. Thus, these alternative conceptions could not be deduced from the drawings. The results indicate that children's drawings can be used to grasp children's conceptions only by considering the meaning the children themselves give to their own drawings

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

Chaos and Synchronized Chaos in an Earthquake Model

We show that chaos is present in the symmetric two-block Burridge-Knopoff model for earthquakes. This is in contrast with previous numerical studies, but in agreement with experimental results. In this system, we have found a rich dynamical behavior with an unusual route to chaos. In the three-block system, we see the appearance of synchronized chaos, showing that this concept can have potential applications in the field of seismology.Comment: To appear in Physical Review Letters (13 pages, 6 figures

Quasi-analyticity and determinacy of the full moment problem from finite to infinite dimensions

This paper is aimed to show the essential role played by the theory of quasi-analytic functions in the study of the determinacy of the moment problem on finite and infinite-dimensional spaces. In particular, the quasi-analytic criterion of self-adjointness of operators and their commutativity are crucial to establish whether or not a measure is uniquely determined by its moments. Our main goal is to point out that this is a common feature of the determinacy question in both the finite and the infinite-dimensional moment problem, by reviewing some of the most known determinacy results from this perspective. We also collect some properties of independent interest concerning the characterization of quasi-analytic classes associated to log-convex sequences.Comment: 28 pages, Stochastic and Infinite Dimensional Analysis, Chapter 9, Trends in Mathematics, Birkh\"auser Basel, 201

An argument for the use of Aristotelian method in bioethics

The main claim of this paper is that the method outlined and used in Aristotle's Ethics is an appropriate and credible one to use in bioethics. Here “appropriate” means that the method is capable of establishing claims and developing concepts in bioethics and “credible” that the method has some plausibility, it is not open to obvious and immediate objection. It begins by suggesting why this claim matters and then gives a brief outline of Aristotle's method. The main argument is made in three stages. First, it is argued that Aristotelian method is credible because it compares favourably with alternatives. In this section it is shown that Aristotelian method is not vulnerable to criticisms that are made both of methods that give a primary place to moral theory (such as utilitarianism) and those that eschew moral theory (such as casuistry and social science approaches). As such, it compares favourably with these other approaches that are vulnerable to at least some of these criticisms. Second, the appropriateness of Aristotelian method is indicated through outlining how it would deal with a particular case. Finally, it is argued that the success of Aristotle's philosophy is suggestive of both the credibility and appropriateness of his method.</p

Nowhere Home: The Waiting of Vulnerable Child Refugees

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Lowe syndrome

Lowe syndrome (the oculocerebrorenal syndrome of Lowe, OCRL) is a multisystem disorder characterised by anomalies affecting the eye, the nervous system and the kidney. It is a uncommon, panethnic, X-linked disease, with estimated prevalence in the general population of approximately 1 in 500,000. Bilateral cataract and severe hypotonia are present at birth. In the subsequent weeks or months, a proximal renal tubulopathy (Fanconi-type) becomes evident and the ocular picture may be complicated by glaucoma and cheloids. Psychomotor retardation is evident in childhood, while behavioural problems prevail and renal complications arise in adolescence. The mutation of the gene OCRL1 localized at Xq26.1, coding for the enzyme phosphatidylinositol (4,5) bisphosphate 5 phosphatase, PtdIns (4,5)P2, in the trans-Golgi network is responsible for the disease. Both enzymatic and molecular testing are available for confirmation of the diagnosis and for prenatal detection of the disease. The treatment includes: cataract extraction, glaucoma control, physical and speech therapy, use of drugs to address behavioural problems, and correction of the tubular acidosis and the bone disease with the use of bicarbonate, phosphate, potassium and water. Life span rarely exceeds 40 years

Ruelle-Perron-Frobenius spectrum for Anosov maps

We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem.Comment: 58 pages, LaTe