54,620 research outputs found
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
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 -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 is mapped into a proper
subsphere of the ideal boundary sphere under the visual map.
This proper subsphere can be realized as the ideal boundary of an isometrically
embedded hyperbolic subspace in covering the closed immersed submanifold.
In particular, if the visual map does not send a lift of the curve into a
proper subsphere of , 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
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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 Β© 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
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