11,696 research outputs found

    Around supersymmetry for semiclassical second order differential operators

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    Let P(h),h∈]0,1]P(h),h\in]0,1] be a semiclassical scalar differential operator of order 22. The existence of a supersymmetric structure given by a matrix G(x;h)G(x;h) was exhibited in \cite{HeHiSj13} under rather general assumptions. In this note we give a sufficient condition on its coefficient so that the matrix G(x;h)G(x;h) enjoys some nice estimates with respect to the semiclassical parameter

    On Kronecker's density theorem, primitive points and orbits of matrices

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    We discuss recent quantitative results in connexion with Kronecker's theorem on the density of subgroups in R^n and with Dani and Raghavan's theorem on the density of orbits in the spaces of frames. We also propose several related problems. The case of the natural linear action of the unimodular group SL_2(Z) on the real plane is investigated more closely. We then establish an intriguing link between the configuration of (discrete) orbits of primitive points and the rate of density of dense orbits

    On inhomogeneous Diophantine approximation and Hausdorff dimension

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    Let Γ=ZA+Zn\Gamma = Z A +Z^n be a dense subgroup with rank n+1n+1 in RnR^n and let ω(A)\omega(A) denote the exponent of uniform simultaneous rational approximation to the point AA. We show that for any real number v≥ω(A)v\ge \omega(A), the Hausdorff dimension of the set BvB_v of points in RnR^n which are vv-approximable with respect to Γ\Gamma, is equal to 1/v1/v

    Spectral analysis of random walk operators on euclidian space

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    We study the operator associated to a random walk on Rd\R^d endowed with a probability measure. We give a precise description of the spectrum of the operator near 11 and use it to estimate the total variation distance between the iterated kernel and its stationary measure. Our study contains the case of Gaussian densities on Rd\R^d.Comment: 19 page

    Approximation to points in the plane by SL(2,Z)-orbits

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    Let x be a point in R^2 with irrational slope and let \Gamma denote the lattice SL(2,Z) acting linearly on R^2. Then, the orbit \Gamma x is dense in R^2. We give efective results on the approximation of a point y in R^2 by points of the form \gamma x, where \gamma belongs to \Gamma, in terms of the size of \gamma

    Exponents of Diophantine Approximation and Sturmian Continued Fractions

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    Let x be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w_n(x) and w_n^*(x) defined by Mahler and Koksma. We calculate their six values when n=2 and x is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction x by quadratic surds.Comment: 25 page

    Graph classes and forbidden patterns on three vertices

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    This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page
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