2,135 research outputs found

    Lax pair and super-Yangian symmetry of the non-linear super-Schr\"odinger equation

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    We consider a version of the non-linear Schr\"odinger equation with M bosons and N fermions. We first solve the classical and quantum versions of this equation, using a super-Zamolodchikov-Faddeev (ZF) algebra. Then we prove that the hierarchy associated to this model admits a super-Yangian Y(gl(M|N)) symmetry. We exhibit the corresponding (classical and quantum) Lax pairs. Finally, we construct explicitly the super-Yangian generators, in terms of the canonical fields on the one hand, and in terms of the ZF algebra generators on the other hand. The latter construction uses the well-bred operators introduced recently.Comment: 32 pages, no figur

    Multilinear functional inequalities involving permanents, determinants, and other multilinear functions of nonnegative matrices and M-matrices

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    AbstractMotivated by a proof due to Fiedler of an inequality on the determinants of M-matrices and a recent paper by the authors, we now obtain various inequalities on permanents and determinants of nonsingular M-matrices. This is done by extending the multilinear considerations of Fiedler and, subsequently, of the authors, to fractional multilinear functionals on pairs of nonnegative matrices. Two examples of our results: For an n×n nonsingular M-matrix M (i) we give a sharp upper bound for det(M)+per(M), when M is a nonsingular M-matrix, (ii) we determine an upper bound on the relative error |per(M+E)−per(M)|/|per(M)|, when M+E is a certain componentwise perturbation of M

    Direct and Inverse Computation of Jacobi Matrices of Infinite Homogeneous Affine I.F.S

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    We introduce a new set of algorithms to compute Jacobi matrices associated with measures generated by infinite systems of iterated functions. We demonstrate their relevance in the study of theoretical problems, such as the continuity of these measures and the logarithmic capacity of their support. Since our approach is based on a reversible transformation between pairs of Jacobi matrices, we also discuss its application to an inverse / approximation problem. Numerical experiments show that the proposed algorithms are stable and can reliably compute Jacobi matrices of large order.Comment: 20 pages 6 figure

    Semigroups of distributions with linear Jacobi parameters

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    We show that a convolution semigroup of measures has Jacobi parameters polynomial in the convolution parameter tt if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha-Lukacs-type characterization, and is related to the q=0q=0 case of quadratic harnesses.Comment: v3: the article is merged back together with arXiv:1003.4025. A significant revision following suggestions by the referee. 2 pdf figure

    Laguerre-Hahn orthogonal polynomials on the real line

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    A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coeffcients - the so-called Laguerre-Hahn class. The main goal is to describe analytical aspects, focusing on differential equations for those orthogonal polynomials, difference and differential equations for the recurrence coeffcients, and distributional equations for the corresponding linear functionals.info:eu-repo/semantics/publishedVersio

    Relative trace formulae toward Bessel and Fourier-Jacobi periods of unitary groups

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    We propose a relative trace formula approach and state the corresponding fundamental lemma toward the global restriction problem involving Bessel or Fourier-Jacobi periods of unitary groups Un×Um\mathrm{U}_n\times\mathrm{U}_m, extending the work of Jacquet-Rallis for m=n1m=n-1 (which is a Bessel period). In particular, when m=0m=0, we recover a relative trace formula proposed by Flicker concerning Kloosterman/Fourier integrals on quasi-split unitary groups. As evidence for our approach, we prove the fundamental lemma for Un×Un\mathrm{U}_n\times\mathrm{U}_n in positive characteristics.Comment: 55 page
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