Borel-Cantelli sequences

A sequence $\{x_{n}\}_1^\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if for all non-increasing sequences of positive real numbers $\{a_n\}$ with $\underset{i=1}{\overset{\infty}{\sum}}a_i=\infty$ the set $$\underset{k=1}{\overset{\infty}{\cap}} \underset{n=k}{\overset{\infty}{\cup}} B(x_n, a_n))=\{x\in[0,1)\mid |x_n-x|<a_n \text{for} \infty \text{many}n\geq1\}$$ has full Lebesgue measure. (To put it informally, BC sequences are sequences for which a natural converse to the Borel-Cantelli Theorem holds). The notion of BC sequences is motivated by the Monotone Shrinking Target Property for dynamical systems, but our approach is from a geometric rather than dynamical perspective. A sufficient condition, a necessary condition and a necessary and sufficient condition for a sequence to be BC are established. A number of examples of BC and not BC sequences are presented. The property of a sequence to be BC is a delicate diophantine property. For example, the orbits of a pseudo-Anosoff IET (interval exchange transformation) are BC while the orbits of a "generic" IET are not. The notion of BC sequences is extended to more general spaces.Comment: 20 pages. Some proofs clarifie

Identities for hyperelliptic P-functions of genus one, two and three in covariant form

We give a covariant treatment of the quadratic differential identities satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of genera 1, 2 and 3

Dense Packings of Superdisks and the Role of Symmetry

We construct the densest known two-dimensional packings of superdisks in the plane whose shapes are defined by |x^(2p) + y^(2p)| <= 1, which contains both convex-shaped particles (p > 0.5, with the circular-disk case p = 1) and concave-shaped particles (0 < p < 0.5). The packings of the convex cases with p 1 generated by a recently developed event-driven molecular dynamics (MD) simulation algorithm [Donev, Torquato and Stillinger, J. Comput. Phys. 202 (2005) 737] suggest exact constructions of the densest known packings. We find that the packing density (covering fraction of the particles) increases dramatically as the particle shape moves away from the "circular-disk" point (p = 1). In particular, we find that the maximal packing densities of superdisks for certain p 6 = 1 are achieved by one of the two families of Bravais lattice packings, which provides additional numerical evidence for Minkowski's conjecture concerning the critical determinant of the region occupied by a superdisk. Moreover, our analysis on the generated packings reveals that the broken rotational symmetry of superdisks influences the packing characteristics in a non-trivial way. We also propose an analytical method to construct dense packings of concave superdisks based on our observations of the structural properties of packings of convex superdisks.Comment: 15 pages, 8 figure

Renormalisation scheme for vector fields on T2 with a diophantine frequency

We construct a rigorous renormalisation scheme for analytic vector fields on the 2-torus of Poincare type. We show that iterating this procedure there is convergence to a limit set with a ``Gauss map'' dynamics on it, related to the continued fraction expansion of the slope of the frequencies. This is valid for diophantine frequency vectors.Comment: final versio

mTORC1 activity is essential for erythropoiesis and B cell lineage commitment

Mechanistic target of rapamycin (mTOR) is a serine/threonine protein kinase that mediates phosphoinositide-3-kinase (PI3K)/AKT signalling. This pathway is involved in a plethora of cellular functions including protein and lipid synthesis, cell migration, cell proliferation and apoptosis. In this study, we proposed to delineate the role of mTORC1 in haemopoietic lineage commitment using knock out (KO) mouse and cell line models. Mx1-cre and Vav-cre expression systems were used to specifically target Raptorfl/fl (mTORC1), either in all tissues upon poly(I:C) inoculation, or specifically in haemopoietic stem cells, respectively. Assessment of the role of mTORC1 during the early stages of development in Vav-cre+Raptorfl/fl mice, revealed that these mice do not survive post birth due to aberrations in erythropoiesis resulting from an arrest in development at the megakaryocyte-erythrocyte progenitor stage. Furthermore, Raptor-deficient mice exhibited a block in B cell lineage commitment. The essential role of Raptor (mTORC1) in erythrocyte and B lineage commitment was confirmed in adult Mx1-cre+Raptorfl/fl mice upon cre-recombinase induction. These studies were supported by results showing that the expression of key lineage commitment regulators, GATA1, GATA2 and PAX5 were dysregulated in the absence of mTORC1-mediated signals. The regulatory role of mTOR during erythropoiesis was confirmed in vitro by demonstrating a reduction of K562 cell differentiation towards RBCs in the presence of established mTOR inhibitors. While mTORC1 plays a fundamental role in promoting RBC development, we showed that mTORC2 has an opposing role, as Rictor-deficient progenitor cells exhibited an elevation in RBC colony formation ex vivo. Collectively, our data demonstrate a critical role played by mTORC1 in regulating the haemopoietic cell lineage commitment

Testing Hardy nonlocality proof with genuine energy-time entanglement

We show two experimental realizations of Hardy ladder test of quantum nonlocality using energy-time correlated photons, following the scheme proposed by A. Cabello \emph{et al.} [Phys. Rev. Lett. \textbf{102}, 040401 (2009)]. Unlike, previous energy-time Bell experiments, these tests require precise tailored nonmaximally entangled states. One of them is equivalent to the two-setting two-outcome Bell test requiring a minimum detection efficiency. The reported experiments are still affected by the locality and detection loopholes, but are free of the post-selection loophole of previous energy-time and time-bin Bell tests.Comment: 5 pages, revtex4, 6 figure

Maslov Indices and Monodromy

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the resulting restrictions on the monodromy matrix are derived.Comment: 6 page

What difference does ("good") HRM make?

The importance of human resources management (HRM) to the success or failure of health system performance has, until recently, been generally overlooked. In recent years it has been increasingly recognised that getting HR policy and management "right" has to be at the core of any sustainable solution to health system performance. In comparison to the evidence base on health care reform-related issues of health system finance and appropriate purchaser/provider incentive structures, there is very limited information on the HRM dimension or its impact. Despite the limited, but growing, evidence base on the impact of HRM on organisational performance in other sectors, there have been relatively few attempts to assess the implications of this evidence for the health sector. This paper examines this broader evidence base on HRM in other sectors and examines some of the underlying issues related to "good" HRM in the health sector. The paper considers how human resource management (HRM) has been defined and evaluated in other sectors. Essentially there are two sub-themes: how have HRM interventions been defined? and how have the effects of these interventions been measured in order to identify which interventions are most effective? In other words, what is "good" HRM? The paper argues that it is not only the organisational context that differentiates the health sector from many other sectors, in terms of HRM. Many of the measures of organisational performance are also unique. "Performance" in the health sector can be fully assessed only by means of indicators that are sector-specific. These can focus on measures of clinical activity or workload (e.g. staff per occupied bed, or patient acuity measures), on measures of output (e.g. number of patients treated) or, less frequently, on measures of outcome (e.g. mortality rates or rate of post-surgery complications). The paper also stresses the need for a "fit" between the HRM approach and the organisational characteristics, context and priorities, and for recognition that so-called "bundles" of linked and coordinated HRM interventions will be more likely to achieve sustained improvements in organisational performance than single or uncoordinated interventions

Cyclotomic integers, fusion categories, and subfactors

Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the A_n or D_n Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in \bx, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\"odinger operator with a periodic magnetic potential in all dimensions.Comment: 61 page