923 research outputs found

    Connected Lie groups and property RD

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    For a locally compact group, property RD gives a control on the convolution norm of any compactly supported measure in terms of the L2L^2-norm of its density and the diameter of its support. We give a complete classification of those Lie groups with property RD.Comment: 29 page

    Scattering of dislocated wavefronts by vertical vorticity and the Aharonov-Bohm effect II: Dispersive waves

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    Previous results on the scattering of surface waves by vertical vorticity on shallow water are generalized to the case of dispersive water waves. Dispersion effects are treated perturbatively around the shallow water limit, to first order in the ratio of depth to wavelength. The dislocation of the incident wavefront, analogous to the Aharonov-Bohm effect, is still observed. At short wavelengths the scattering is qualitatively similar to the nondispersive case. At moderate wavelengths, however, there are two markedly different scattering regimes according to wether the capillary length is smaller or larger than 3\sqrt{3} times depth. The dislocation is characterized by a parameter that depends both on phase and group velocity. The validity range of the calculation is the same as in the shallow water case: wavelengths small compared to vortex radius, and low Mach number. The implications of these limitations are carefully considered.Comment: 30 pages, 11 figure

    On the divine clockwork: the spectral gap for the correspondence limit of the Nelson diffusion generator for the atomic elliptic state

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    The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson's stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane z=0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that in the infinite time limit the density for the limiting Nelson diffusion will converge to its invariant measure. We also include a summary of the Cheeger and Poincare inequalities both of which are used in our proof of the existence of the spectral gap.Comment: 30 pages, 5 figures, submitted to J. Math. Phy

    Asymptotic singularities of planar parallel 3-RPR manipulators

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    We study the limits of singularities of planar parallel 3-RPR manipulators as the lengths of their legs tend to infinity, paying special attention to the presence of cusps. These asymptotic singularities govern the kinematic behavior of the manipulator in a rather large portion of its workspace

    Properties making a chaotic system a good Pseudo Random Number Generator

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    We discuss two properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers: high value of Kolmogorov-Sinai entropy and high-dimensionality. We propose the multi dimensional Anosov symplectic (cat) map as a Pseudo Random Number Generator. We show what chaotic features of this map are useful for generating Pseudo Random Numbers and investigate numerically which of them survive in the discrete version of the map. Testing and comparisons with other generators are performed.Comment: 10 pages, 3 figures, new version, title changed and minor correction

    Approximation of conformal mappings by circle patterns

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    A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in (0,π)(0,\pi). Two sequences of circle patterns are employed to approximate a given conformal map gg and its first derivative. For the domain of gg we use embedded circle patterns where all circles have the same radius decreasing to 0 and which have uniformly bounded intersection angles. The image circle patterns have the same combinatorics and intersection angles and are determined from boundary conditions (radii or angles) according to the values of gâ€Čg' (∣gâ€Č∣|g'| or arg⁥gâ€Č\arg g'). For quasicrystallic circle patterns the convergence result is strengthened to C∞C^\infty-convergence on compact subsets.Comment: 36 pages, 7 figure

    Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights

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    This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.Comment: 12 pages. Fourth of 4 papers. Important revision: improvement of main result by eliminating use of Poincar\'e inequalities replaced by the weaker Gaussian keat kernel bound
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