7,978 research outputs found

    Numerical hyperinterpolation over nonstandard planar regions

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    We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes

    Spontaneous magnetisation in the plane

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    The Arak process is a solvable stochastic process which generates coloured patterns in the plane. Patterns are made up of a variable number of random non-intersecting polygons. We show that the distribution of Arak process states is the Gibbs distribution of its states in thermodynamic equilibrium in the grand canonical ensemble. The sequence of Gibbs distributions form a new model parameterised by temperature. We prove that there is a phase transition in this model, for some non-zero temperature. We illustrate this conclusion with simulation results. We measure the critical exponents of this off-lattice model and find they are consistent with those of the Ising model in two dimensions.Comment: 23 pages numbered -1,0...21, 8 figure

    Fragmentation of a Circular Disc by Impact on a Frictionless Plate

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    The break-up of a two-dimensional circular disc by normal and oblique impact on a hard frictionless plate is investigated by molecular dynamics simulations. The disc is composed of numerous unbreakable randomly shaped convex polygons connected together by simple elastic beams that break when bent or stretched beyond a certain limit. It is found that for both normal and oblique impacts the crack patterns are the same and depend solely on the normal component of the impact velocity. Analysing the pattern of breakage, amount of damage, fragment masses and velocities, we show the existence of a critical velocity which separates two regimes of the impact process: below the critical point only a damage cone is formed at the impact site (damage), cleaving of the particle occurs at the critical point, while above the critical velocity the disc breaks into several pieces (fragmentation). In the limit of very high impact velocities the disc suffers complete disintegration (shattering) into many small fragments. In agreement with experimental results, fragment masses are found to follow the Gates-Gaudin-Schuhmann distribution (power law) with an exponent independent of the velocity and angle of impact. The velocity distribution of fragments exhibit an interesting anomalous scaling behavior when changing the impact velocity and the size of the disc.Comment: submitted to J. Phys: Condensed Matter special issue on Granular Medi

    Accelerating Reinforcement Learning by Composing Solutions of Automatically Identified Subtasks

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    This paper discusses a system that accelerates reinforcement learning by using transfer from related tasks. Without such transfer, even if two tasks are very similar at some abstract level, an extensive re-learning effort is required. The system achieves much of its power by transferring parts of previously learned solutions rather than a single complete solution. The system exploits strong features in the multi-dimensional function produced by reinforcement learning in solving a particular task. These features are stable and easy to recognize early in the learning process. They generate a partitioning of the state space and thus the function. The partition is represented as a graph. This is used to index and compose functions stored in a case base to form a close approximation to the solution of the new task. Experiments demonstrate that function composition often produces more than an order of magnitude increase in learning rate compared to a basic reinforcement learning algorithm
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