6,969 research outputs found

    Curious Continued Fractions, Nonlinear Recurrences and Transcendental Numbers

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    We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also integers), appear interlaced in the continued fraction expansion of the sum of the reciprocals of the terms. Using the rapid (double exponential) growth of the terms, for each sequence it is shown that the sum of the reciprocals is a transcendental number

    Diophantine non-integrability of a third order recurrence with the Laurent property

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    We consider a one-parameter family of third order nonlinear recurrence relations. Each member of this family satisfies the singularity confinement test, has a conserved quantity, and moreover has the Laurent property: all of the iterates are Laurent polynomials in the initial data. However, we show that these recurrences are not Diophantine integrable according to the definition proposed by Halburd. Explicit bounds on the asymptotic growth of the heights of iterates are obtained for a special choice of initial data. As a by-product of our analysis, infinitely many solutions are found for a certain family of Diophantine equations, studied by Mordell, that includes Markoff's equation

    Laurent Polynomials and Superintegrable Maps

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    This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations

    Non-standard discretization of biological models

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    We consider certain types of discretization schemes for differential equations with quadratic nonlinearities, which were introduced by Kahan, and considered in a broader setting by Mickens. These methods have the property that they preserve important structural features of the original systems, such as the behaviour of solutions near to fixed points, and also, where appropriate (e.g. for certain mechanical systems), the property of being volume-preserving, or preserving a symplectic/Poisson structure. Here we focus on the application of Kahan's method to models of biological systems, in particular to reaction kinetics governed by the Law of Mass Action, and present a general approach to birational discretization, which is applied to population dynamics of Lotka-Volterra type

    Corporate Political Spending: Why Shareholders Must Weigh In

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    This article focuses upon the growing problem confronting companies and their shareholders: the use of general treasury (i.e., shareholder money) to propagate political agendas which are not only contrary to companies’ policies of employment, but are committed without the input or knowledge of the shareholder, leading to an aura of distrust, alienation, and diminution of both shareholder value and principled leadership

    Algebraic curves, integer sequences and a discrete Painleve transcendent

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    We consider some bilinear recurrences that have applications in number theory. The explicit solution of a general three-term bilinear recurrence relation of fourth order is given in terms of the Weierstrass sigma function for an associated elliptic curve. The recurrences can generate integer sequences, including the Somos 4 sequence and elliptic divisibility sequences. An interpretation via the theory of integrable systems suggests the relation between certain higher order recurrences and hyperelliptic curves of higher genus. Analogous sequences associated with a q-discrete Painlev\'e I equation are briefly considered

    Symplectic Maps from Cluster Algebras

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    We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map

    Car Travel Time Variability on Links of a Radial Route in London: Results

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    This working paper describes the results of a study of the variability of travel times and its causes on links of a secti.on of the A41 radial route in north London in the spring and summer of 1987. The objectives were to estimate the extent of variability of travel times of private car users and to explain the observed variability by means of models incorporating a range of traffic factors, including traffic flow, and incorporating seasonal differences. In general the spring was slower and showed more travel time variation between time periods than the summer. slower and more variable links in the spring tended to behave similarly in the summer. The models produced explained around two thirds of the travel time variation between periods, but the explanatory power and explanatory variables differed between links. Blocking of the downstream exit from links was the single variable which was significant in affecting traffic times on most links

    Defining relations for Lie algebras of vector fields

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    AbstractWe calculated defining relations of the graded nilpotent positive part of Lie algebras of vector fields. These calculations suggest that for Wn,ln and Kn (n sufficiently large) there are only tri relations, i.e. relations of degree 2. ForHn however, we prove that there are non-trivial relations of degree 3, which form a standard module

    The Capture and Escape of Stars

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    The shape of galaxies depends on their orbital populations. These populations change through capture into and escape from resonance. Capture problems fall into distinct cases depending upon the shape of the potential well. To visualise the effective potential well for orbital capture, a diagrammatic approach to the resonant perturbation theory of Born is presented. These diagrams we call equiaction sections. To illustrate their use, we present examples drawn from both galactic and Solar System dynamics. The probability of capture for generic shapes of the potential well is calculated. A number of predictions are made. First, there are barred galaxies that possess two outer rings of gas and stars (type R ′ 1R ′ 2). We show how to relate changes in the pattern speed and amplitude of the bar to the strength of the two rings. Secondly, under certain conditions, small disturbances can lead to dramatic changes in orbital shape. This can be exploited as a mechanism to pump counter-rotating stars and gas into the nuclei of disk galaxies. Tidal resonant forcing of highly inclined orbits around a central mass causes a substantial increase in the likelihood of collision. Thirdly, the angular momentum of a potential well is changed by the passage of stars across or capture into the well. This can lead to the creation of holes, notches and high velocity tails in the stellar distribution function, whose form we explicitly calculate.Peer reviewe
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