Abstract: In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

Abstract: The present paper deals with Bernstein polynomials and Frobenius-Euler numbers and polynomials. We apply the method of generating function and fermionic p-adic integral representation on Zp, which are exploited to derive further classes of Bernstein polynomials and Frobenius-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between Frobenius-Euler numbers and polynomials. Furthermore, we derive an integral representation of Bernstein polynomials of degree n on Zp . Also we deduce a fermionic p-adic integral representation of product Bernstein polynomials of different degrees n1, n2,...on Zp and show that it can be written with Frobenius-Euler numbers which yields a deeper insight into the effectiveness of this type of generalizations. Our applications possess a number of interesting properties which we state in this paper.

Abstract: It is shown how to systematically construct the $XX$ quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST). Sets of orthogonal polynomials (OPs) are in correspondence with such systems. The key observation is that for any admissible one-excitation energy spectrum, the weight function of the associated OPs is uniquely prescribed. This entails the complete characterization of these PST models with the mirror symmetry property arising as a corollary. A simple and efficient algorithm to obtain the corresponding Hamiltonians is presented. A new model connected to a special case of the symmetric $q$-Racah polynomials is offered. It is also explained how additional models with PST can be derived from a parent system by removing energy levels from the one-excitation spectrum of the latter. This is achieved through Christoffel transformations and is also completely constructive in regards to the Hamiltonians.

Abstract: We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.

Abstract: We propose a novel algorithm for the compression of ECG signals, in particular QRS complexes. The algorithm is based on the expansion of signals with compact support into a basis of discrete Hermite functions. These functions can be constructed by sampling continuous Hermite functions at specific sampling points. They form an orthogonal basis in the underlying signal space. The proposed algorithm relies on the theory of signal models based on orthogonal polynomials. We demonstrate that the constructed discrete Hermite functions have important ad- vantages compared to continuous Hermite functions, which have previously been suggested for the compression of QRS complexes. Our algorithm achieves higher compression ratios compared with previously reported algorithms based on continuous Hermite functions, discrete Fourier, cosine, or wavelet transforms.

Abstract: For a class of large linear input-output systems, we present a new model order reduction algorithm based on general orthogonal polynomials in the time domain. The main idea of the algorithm is first to expand the unknown state variables in the space spanned by orthogonal polynomials, then the coefficient terms of polynomial expansion are calculated by a recurrence formula. The basic procedure is to use the coefficient terms to generate a projection matrix. Many classic methods with orthogonal polynomials are special cases of the general approach. The proposed approach has a good computational efficiency and preserves the stability and passivity under certain condition. Numerical experiments are reported to verify the theoretical analysis.

Abstract: We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy2 process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.

Abstract: Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.

Abstract: A methodology is proposed for the development of reduced-order models of finite element approximations of electromagnetic devices exhibiting uncertainty or statistical variability in their input parameters. In this approach, the reduced order system matrices are represented in terms of their orthogonal polynomial chaos expansions on the probability space defined by the input random variables. The coefficients of these polynomials, which are matrices, are obtained through the repeated, deterministic model order reduction of finite element models generated for specific values of the input random parameters. These values are chosen efficiently in a multi-dimensional grid using a Smolyak algorithm. The generated stochastic reduced order model is represented in the form of an augmented system that lends itself to the direct generation of the desired statistics of the device response. The accuracy and efficiency of the proposed method is demonstrated through its application to the reduced-order finite element modeling of a terminated coaxial cable and a circular wire loop antenna.

Abstract: We use the theory of lecture hall partitions to define a generalization of the Eulerian polynomials, for each positive integer $k$. We show that these ${1}/{k}$- Eulerian polynomials have a simple combinatorial interpretation in terms of a single statistic on generalized inversion sequences . The theory provides a geometric realization of the polynomials as the $h^*$-polynomials of $k$- lecture hall polytopes . Many of the defining relations of the Eulerian polynomials have natural ${1}/{k}$-generalizations. In fact, these properties extend to a bivariate generalization obtained by replacing ${1}/{k}$ by a continuous variable. The bivariate polynomials have appeared in the work of Carlitz, Dillon, and Roselle on Eulerian numbers of higher order and, more recently, in the theory of rook polynomials.

Abstract: In the flrst part of this survey paper we present a short account on some impor- tant properties of orthogonal polynomials on the real line, including computational methods for constructing coe-cients in the fundamental three-term recurrence relation for orthogonal polynomials, and mention some basic facts on Gaussian quadrature rules. In the second part we discuss our Mathematica package OrthogonalPolynomials (see (2)) and show some ap- plications to problems with strong nonclassical weights on (0;+1), including a conjecture for an oscillatory weight on (i1;1). Finally, we give some new results on orthogonal polynomials on radial rays in the complex plane.

Abstract: This volume contains the proceedings of the 11th International Symposium on Orthogonal Polynomials, Special Functions, and their Applications, held August 29-September 2, 2011, at the Universidad Carlos III de Madrid in Leganes, Spain. The papers cover asymptotic properties of polynomials on curves of the complex plane, universality behavior of sequences of orthogonal polynomials for large classes of measures and its application in random matrix theory, the Riemann-Hilbert approach in the study of Pade approximation and asymptotics of orthogonal polynomials, quantum walks and CMV matrices, spectral modifications of linear functionals and their effect on the associated orthogonal polynomials, bivariate orthogonal polynomials, and optimal Riesz and logarithmic energy distribution of points. The methods used include potential theory, boundary values of analytic functions, Riemann-Hilbert analysis, and the steepest descent method.

Abstract: We generalize Fomin and Zelevinsky's cluster algebras by allowing exchange polynomials to be arbitrary irreducible polynomials, rather than binomials.

Abstract: As the second stage of the project multi-indexed orthogonal polynomials, we present, in the framework of ‘discrete quantum mechanics’ with real shifts in one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from the (q-)Racah polynomials by the multiple application of the discrete analogue of the Darboux transformations or the Crum–Krein–Adler deletion of ‘virtual state’ vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the ‘solutions’ of the matrix Schrodinger equation with negative ‘eigenvalues’, except for one of the two boundary points.

Abstract: In this article we present the classical theory of Chebyshev polynomials starting from the definition of a family of complex polynomials, including both the first and second kind classical Chebyshev ones, which are related to its real and imaginary part. This point of view permits to derive a lot of generating functions and relations between the two kinds Chebyshev families, which are essentially new, as exponential generating functions, bilinear and bilinear exponential generating functions. We also deduce relevant relations of products of Chebyshev polynomials and the related generating functions.

Abstract: We introduce a large class of measures with orthogonal polynomials satisfying higher-order difference equations with coefficients independent of the degree of the polynomials. These measures are constructed by multiplying the discrete classical weights of Charlier, Meixner, Krawtchouk, and Hahn by certain variants of the annihilator polynomial of a finite set of numbers.

Abstract: Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enables the fabrication of freeform optics, or more specifically, optical surfaces for imaging applications that are not rotationally symmetric. Various forms of polynomials for describing freeform optical surfaces exist in optical design and to support fabrication. A popular method is to add orthogonal polynomials onto a conic section. In this paper, recently introduced gradient-orthogonal polynomials are investigated in a comparative manner with the widely known Zernike polynomials. In order to achieve numerical robustness when higher-order polynomials are required to describe freeform surfaces, recurrence relations are a key enabler. Results in this paper establish the equivalence of both polynomial sets in accurately describing freeform surfaces under stringent conditions. Quantifying the accuracy of these two freeform surface descriptions is a critical step in the future application of these tools in both advanced optical system design and optical fabrication.

Abstract: In this paper, we derive some interesting identities involving Gegenbauer polynomials arising from the orthogonality of Gegenbauer polynomials for the inner product space P n Open image in new window with respect to the weighted inner product 〈 p 1 , p 2 〉 = ∫ − 1 1 p 1 ( x ) p 2 ( x ) ( 1 − x 2 ) λ − 1 2 d x Open image in new window.

Abstract: The quantum free particle on the sphere $S_\kappa^2$ ($\kappa>0$) and on the hyperbolic plane $H_\kappa^2$ ($\kappa 0$ then a discrete spectrum is obtained. The wavefunctions, that are related with a $\kappa$-dependent family of orthogonal polynomials, are explicitly obtained.

Abstract: This paper addresses the generation of an enhanced stochastic model of a carbon nanotube interconnect including the effects of process variation. The proposed approach is based on the expansion of the constitutive relations of state-of-the-art deterministic models of nanointerconnects with uncertain parameters in terms of orthogonal polynomials. The method offers comparable accuracy and improved efficiency with respect to conventional methods like Monte Carlo in predicting the statistical behavior of the electrical performance of next-generation data links. An application example involving both the frequency- and time-domain analysis of a realistic nanointerconnect concludes this paper.

Abstract: This study introduces a new set of orthogonal polynomials and moments and the set's application in signal and image processing. This polynomial is derived from two well-known orthogonal polynomials: the Tchebichef and Krawtchouk polynomials. This study attempts to present the following: (i) the mathematical and theoretical frameworks for the definition of this polynomial including the modelling of signals with the various analytical properties it contains, as well as, recurrence relations and transform equations that need to be addressed; and (ii) the results of empirical tests that compare the representational capabilities of this polynomial with those of the more traditional Tchebichef and Krawtchouk polynomials using speech and image signals from different databases. This study attempts to demonstrate that the proposed polynomials can be applied in the field of signal and image processing because of the promising properties of this polynomial especially in its localisation and energy compaction capabilities.

Abstract: We develop the theory of CKP hierarchy introduced in the papers of Kyoto school (Date E., Jimbo M., Kashiwara M., Miwa T., J. Phys. Soc. Japan 50 (1981), 3806{ 3812) (see also (Kac V.G., van de Leur J.W., Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 369{406)). We present appropriate bosonization formulae. We show that in the context of the CKP theory certain orthogonal polynomials appear. These polynomials are polynomial both in even and odd (in Grassmannian sense) variables.