65,797 research outputs found

    Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

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    We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra U_{q,p}(\hat{sl}_2). The algebra U_{q,p}(\hat{sl}_2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra U_q(\hat{sl}_2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical representations and their parallel structures to U_q(\hat{sl}_2). In particular we give a classification theorem of the finite-dimensional irreducible pseudo-highest weight representations stated in terms of an elliptic analogue of the Drinfeld polynomials. We also investigate a structure of the tensor product of two evaluation representations and derive an elliptic analogue of the Clebsch-Gordan coefficients. We show that it is expressed by using the very-well-poised balanced elliptic hypergeometric series 12V11.Comment: 42 page

    A Lanczos Method for Approximating Composite Functions

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    We seek to approximate a composite function h(x) = g(f(x)) with a global polynomial. The standard approach chooses points x in the domain of f and computes h(x) at each point, which requires an evaluation of f and an evaluation of g. We present a Lanczos-based procedure that implicitly approximates g with a polynomial of f. By constructing a quadrature rule for the density function of f, we can approximate h(x) using many fewer evaluations of g. The savings is particularly dramatic when g is much more expensive than f or the dimension of x is large. We demonstrate this procedure with two numerical examples: (i) an exponential function composed with a rational function and (ii) a Navier-Stokes model of fluid flow with a scalar input parameter that depends on multiple physical quantities

    Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy

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    A D\mathbb{D}-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator D\mathbb{D}. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the D\mathbb{D}-semi-classical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the D\mathbb{D}-semi-classical class it is entirely natural to define a generalisation of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first non-trivial deformation of the Askey-Wilson orthogonal polynomial system defined by the qq-quadratic divided-difference operator, the Askey-Wilson operator, and derive the coupled first order divided-difference equations characterising its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the E7(1) E^{(1)}_7 qq-Painlev\'e system.Comment: Submitted to Duke Mathematical Journal on 5th April 201

    Square-rich fixed point polynomial evaluation on FPGAs

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    Polynomial evaluation is important across a wide range of application domains, so significant work has been done on accelerating its computation. The conventional algorithm, referred to as Horner's rule, involves the least number of steps but can lead to increased latency due to serial computation. Parallel evaluation algorithms such as Estrin's method have shorter latency than Horner's rule, but achieve this at the expense of large hardware overhead. This paper presents an efficient polynomial evaluation algorithm, which reforms the evaluation process to include an increased number of squaring steps. By using a squarer design that is more efficient than general multiplication, this can result in polynomial evaluation with a 57.9% latency reduction over Horner's rule and 14.6% over Estrin's method, while consuming less area than Horner's rule, when implemented on a Xilinx Virtex 6 FPGA. When applied in fixed point function evaluation, where precision requirements limit the rounding of operands, it still achieves a 52.4% performance gain compared to Horner's rule with only a 4% area overhead in evaluating 5th degree polynomials
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