Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms

We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled by showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization techniques, and a new observation about intertwined linear dynamics of various SL(m,R)'s contained in SL(k+1,R).Comment: 28 page

Counting integral matrices with a given characteristic polynomial

We give a simpler proof of an earlier result giving an asymptotic estimate for the number of integral matrices, in large balls, with a given monic integral irreducible polynomial as their common characteristic polynomial. The proof uses equidistributions of polynomial trajectories on SL(n,R)/SL(n,Z), which is a generalization of Ratner's theorem on equidistributions of unipotent trajectories. We also compute the exact constants appearing in the above mentioned asymptotic estimate

Islamic law

Abstract Islamic legal system is one of the major legal systems in the world. It is a time-tested system based on over centuries of evolution. But it does not mean that it is a perfect system. Like any other legal system, it has weaknesses, strengths, and contentious or difficult areas with plenty of room for further development

Limiting distributions of curves under geodesic flow on hyperbolic manifolds

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold. Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space $T^1(H^n)$ is mapped into a proper subsphere of the ideal boundary sphere $\partial H^n$ under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in $H^n$ covering the closed immersed submanifold. In particular, if the visual map does not send a lift of the curve into a proper subsphere of $\partial H^n$, then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bundle of the manifold with respect to the normalized natural Riemannian measure. The proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of SO(n,1).Comment: 27 pages, revised version, Proof of Theorem~3.1 simplified, remarks adde

Investigation of practices and procedures in the use of therapeutic diathermy: A study from the physiotherapists' health and safety perspective

Background and Purpose. The safe use of therapeutic diathermy requires practices and procedures that ensure compliance to professional guidelines and clinical evidence. Inappropriate use may expose physiotherapists and other people in the vicinity of operating diathermy devices to stray radiofrequency electromagnetic fields, which can be a source of risk and may lead to adverse health effects. The aim of the present study was to investigate practices and procedures for therapeutic diathermy from a health and safety perspective. Method. A cross-sectional research design was used, this included a postal survey using a self-administered questionnaire and semi-structured observational visits to 46 physiotherapy departments in National Health Service (NHS) hospitals located in the south-east and south-west of England, including Greater London. Results. Microwave diathermy was not available in the departments surveyed. Pulsed shortwave diathermy was available and was used more commonly than continuous shortwave diathermy. There were metallic objects in treatment cubicles used for pulsed shortwave diathermy and continuous shortwave diathermy. Shortwave diathermy devices created electromagnetic interference with a variety of electrical and medical devices. Physiotherapists reported that they did not stay in the treatment cubicle during the entire period of electrotherapy with pulsed shortwave diathermy or continous shortwave diathermy; pregnant physiotherapists reported that they did not use these devices. Electrotherapy with pulsed shortwave diathermy and continuous shortwave diathermy was not always administered on a wooden couch or chair. Electrotherapy was highest in those departments with the fewest physiotherapists. Conclusions. Departments report good practices and procedures regarding the use of therapeutic diathermy devices. However, field observations of practices and procedures, and the working environment, have identified issues with a potential to create health and safety problems, and these should be addressed. Copyright (c) 2007 John Wiley & Sons, Ltd

Systematic literature review of adverse reproductive outcomes associated with physiotherapists' occupational exposures to non-ionising radiation

This article is available through open access and can be accessed at the link below.Objectives: To review empirical research on adverse health and pregnancy outcomes associated with physiotherapists' occupational exposure to radiofrequency electromagnetic fields (RF EMFs) from shortwave (SWD) and microwave (MWD) diathermy devices. Methods: A systematic review of peer reviewed literature published from 1990 to 2010 in the English language searched in eight online bibliographic databases: CINAHL, EBSCOhost, ISI Web of Knowledge, Medline, OSH UPDATE, PubMed Central, ScienceDirect, and Scopus. Results: Findings suggest that physiotherapists' occupational exposure to SWD was statistically significantly associated with delayed time to pregnancy (>6 months), still birth, altered gender ratio (low ratio of boys to girls), congenital malformations and low birth weight (<2,500 g) among physiotherapists' offspring. Physiotherapists' exposure to MWD was also found to be statistically significantly associated with spontaneous abortion. However, causal mechanisms for these statistical associations are unknown. The present systematic review has found these adverse reproductive outcomes and health effects associations with RF EMFs from therapeutic diathermy devices to be inconsistent. Conclusions: A number of studies did not find statistically significant results to replicate associations with such adverse outcomes, and therefore further research, preferably prospective studies of cohorts of physiotherapists, is warranted.This is extended version of a study funded by the Health and Safety Executive (Project No: R47.022