1,287 research outputs found

    A nilpotent IP polynomial multiple recurrence theorem

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    We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.Comment: v4: switch to TeXlive 2016 and biblate

    Hill's Equation with Random Forcing Parameters: Determination of Growth Rates through Random Matrices

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    This paper derives expressions for the growth rates for the random 2 x 2 matrices that result from solutions to the random Hill's equation. The parameters that appear in Hill's equation include the forcing strength and oscillation frequency. The development of the solutions to this periodic differential equation can be described by a discrete map, where the matrix elements are given by the principal solutions for each cycle. Variations in the forcing strength and oscillation frequency lead to matrix elements that vary from cycle to cycle. This paper presents an analysis of the growth rates including cases where all of the cycles are highly unstable, where some cycles are near the stability border, and where the map would be stable in the absence of fluctuations. For all of these regimes, we provide expressions for the growth rates of the matrices that describe the solutions.Comment: 22 pages, 3 figure

    How do random Fibonacci sequences grow?

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    We study two kinds of random Fibonacci sequences defined by F1=F2=1F_1=F_2=1 and for n1n\ge 1, Fn+2=Fn+1±FnF_{n+2} = F_{n+1} \pm F_{n} (linear case) or Fn+2=Fn+1±FnF_{n+2} = |F_{n+1} \pm F_{n}| (non-linear case), where each sign is independent and either + with probability pp or - with probability 1p1-p (0<p10<p\le 1). Our main result is that the exponential growth of FnF_n for 0<p10<p\le 1 (linear case) or for 1/3p11/3\le p\le 1 (non-linear case) is almost surely given by 0logxdνα(x),\int_0^\infty \log x d\nu_\alpha (x), where α\alpha is an explicit function of pp depending on the case we consider, and να\nu_\alpha is an explicit probability distribution on \RR_+ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of pp, since we prove that it is equal to zero for 0<p1/30<p\le1/3. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative

    Do We Invest Less Time in Children? Trends in Parental Time in Selected Industrialized Countries Since the 1960\u27s

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    This paper examines trends in parental time in selected industrialized countries since the 1960s using time-use survey data. Despite the time pressures to which today’s families are confronted, parents appear to be devoting more time to children than they did some 40 years ago. Results also suggest a decrease in the differences between fathers and mothers in time devoted to children. Mothers continue to devote more time to childcare than fathers, but the gender gap has been reduced. These results are observed in several countries and therefore suggest a large global trend towards an increase in parental time investment with their children

    Ramsey numbers and adiabatic quantum computing

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    The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers R(m,n)R(m,n) with m,n3m,n\geq 3, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey numbers R(m,n)R(m,n). We show how the computation of R(m,n)R(m,n) can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evolution. We numerically simulate this adiabatic quantum algorithm and show that it correctly determines the Ramsey numbers R(3,3) and R(2,s) for 5s75\leq s\leq 7. We then discuss the algorithm's experimental implementation, and close by showing that Ramsey number computation belongs to the quantum complexity class QMA.Comment: 4 pages, 1 table, no figures, published versio

    Singularity results for functional equations driven by linear fractional transformations

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    We consider functional equations driven by linear fractional transformations, which are special cases of de Rham's functional equations. We consider Hausdorff dimension of the measure whose distribution function is the solution. We give a necessary and sufficient condition for singularity. We also show that they have a relationship with stationary measures.Comment: 14 pages, Title changed, to appear in Journal of Theoretical Probabilit

    Growth and Structure of Stochastic Sequences

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    We introduce a class of stochastic integer sequences. In these sequences, every element is a sum of two previous elements, at least one of which is chosen randomly. The interplay between randomness and memory underlying these sequences leads to a wide variety of behaviors ranging from stretched exponential to log-normal to algebraic growth. Interestingly, the set of all possible sequence values has an intricate structure.Comment: 4 pages, 4 figure

    Exact Lyapunov Exponent for Infinite Products of Random Matrices

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    In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2×22\times 2 real matrices. All these products are constructed using only two types of matrices, AA and BB, which are chosen according to a stochastic process. The matrix AA is singular, namely its determinant is zero. This formula is derived by using a particular decomposition for the matrix BB, which allows us to write the Lyapunov exponent as a sum of convergent series. Finally, we show with an example that the Lyapunov exponent is a discontinuous function of the given parameter.Comment: 1 pages, CPT-93/P.2974,late

    Covering Partial Cubes with Zones

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    A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite poset. For several such special cases, we give upper and lower bounds on the minimum size of a covering by zones. We also consider the computational complexity of those problems, and establish some hardness results

    Ray splitting in paraxial optical cavities

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    We present a numerical investigation of the ray dynamics in a paraxial optical cavity when a ray splitting mechanism is present. The cavity is a conventional two-mirror stable resonator and the ray splitting is achieved by inserting an optical beam splitter perpendicular to the cavity axis. We show that depending on the position of the beam splitter the optical resonator can become unstable and the ray dynamics displays a positive Lyapunov exponent.Comment: 13 pages, 7 figures, 1 tabl
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