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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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
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