64,079 research outputs found

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

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    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

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    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

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    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

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    We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic nn-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 T1(Hn)T^1(H^n) is mapped into a proper subsphere of the ideal boundary sphere βˆ‚Hn\partial H^n under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in HnH^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 βˆ‚Hn\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
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