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

Gabor Frames on Local Fields of Positive Characteristic

Gabor frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. Finding general and verifiable conditions which imply that the Gabor systems are Gabor frames is among the core problems in time-frequency analysis. In this paper, we give some simple and sufficient conditions that ensure a Gabor system ${M_{u(m)b}T_{u(n)a}g:m,n\in \mathbb N_{0}}$ to be a frame for L^2(K). The conditions proposed are stated in terms of the Fourier transforms of the Gabor system's generating functions.Comment: 11. arXiv admin note: text overlap with arXiv:1312.0443, arXiv:1103.0090 by other author

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

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

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

Availability and use of electrotherapy devices: a survey

This item is published and the copyright holder of this article is the International Journal of Therapy and Rehabilitation © 2010 MA Healthcare Limited, http://www.ijtr.co.uk/. The article is available here with the permission of the copyright holder. Any use of the article from this site for personal use is permitted; however, if it is to be used for any other purpose, or reproduced in part or in full, the copyright holder must be contacted.Electrophysical agents such as radiofrequency electromagnetic fields (shortwave and microwave), ultrasound, laser and electrical stimulation are used for therapeutic purpose in physiotherapy departments. They are primarily used for treating a wide range of musculoskeletal injuries. This study investigated the availability and use of therapeutic diathermy, ultrasound, laser, transcutaneous electrical nerve stimulation (TENS) and interferential equipment in 46 physiotherapy departments in NHS hospitals in the south of England, using a self-administered questionnaire. Results indicated that therapeutic ultrasound was the most commonly available and most often used modality by surveyed departments. Pulsed shortwave diathermy, interferential, and laser were available to a lesser degree and also used less often. Continuous shortwave diathermy was used rarely and only in larger departments. Microwave diathermy was not available in any of the surveyed departments. The level of non-use of equipment despite availability was highest for continuous shortwave diathermy followed by pulsed shortwave diathermy and then laser. The rare use and total non-use of some of the modalities, despite availability of equipment, may have implications for purchasers of this expensive equipment. While the findings of this study show a regional trend in NHS physiotherapy departments, this may not be generalizable to a national level.This study was funded by the Health and Safety Executive, UK, (Grant No. 4371/R47.022