4,342 research outputs found

    Rational Approximations to Certain Algebraic Numbers

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    W.M.Schmit[11] conjectured that for any  θ\;\theta with deg  θ3,\;\theta\geq 3, there is no constant  C=C(θ)\;C=C(\theta) so that  pqθ>Cq1\;|p-q\theta|>Cq^{-1} for every rationa  p/q.\;p/q. [12,p26] states that the computations of the first several thousand partial quotients for such numbers as  23\;\sqrt[3]{2} and  33\;\sqrt[3]{3} support the conjecture that the sequence of partial quotients is unbounded. In this paper, applying Dirichlet's approximation theorem to certain algebraic numbers  θ,\;\theta, e.g.  θ=dn,dN,n3,d>0;\;\theta=\sqrt[n]{d},d\in N,n\geq 3,d>0;   θ3+b1θb0=0,b0>0;\;\theta^{3}+b_{1}\theta-b_{0}=0,b_{0}>0;   θ4+b2θ2b0=0,  b0>0.\;\theta^{4}+b_{2}\theta^{2}-b_{0}=0,\;b_{0}>0. We proved that there exists a effective constant  C=C(θ)\;C=C(\theta) such that  pqθ>Cq1\;|p-q\theta|>Cq^{-1} for all  p/q.\;p/q. Our theorem shows their sequence of partial quotients can not be unbounded

    The Bane of Low-Dimensionality Clustering

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    In this paper, we give a conditional lower bound of nΩ(k)n^{\Omega(k)} on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in low-dimensional Euclidean space of dimension four, assuming the Exponential Time Hypothesis (ETH). We also consider k-median (and k-means) with penalties where each point need not be assigned to a center, in which case it must pay a penalty, and extend our lower bound to at least three-dimensional Euclidean space. This stands in stark contrast to many other geometric problems such as the traveling salesman problem, or computing an independent set of unit spheres. While these problems benefit from the so-called (limited) blessing of dimensionality, as they can be solved in time nO(k11/d)n^{O(k^{1-1/d})} or 2n11/d2^{n^{1-1/d}} in d dimensions, our work shows that widely-used clustering objectives have a lower bound of nΩ(k)n^{\Omega(k)}, even in dimension four. We complete the picture by considering the two-dimensional case: we show that there is no algorithm that solves the penalized version in time less than no(k)n^{o(\sqrt{k})}, and provide a matching upper bound of nO(k)n^{O(\sqrt{k})}. The main tool we use to establish these lower bounds is the placement of points on the moment curve, which takes its inspiration from constructions of point sets yielding Delaunay complexes of high complexity

    Recurrence and algorithmic information

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    In this paper we initiate a somewhat detailed investigation of the relationships between quantitative recurrence indicators and algorithmic complexity of orbits in weakly chaotic dynamical systems. We mainly focus on examples.Comment: 26 pages, no figure
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