Abstract: Summary Integral inequalities have been widely used in stability analysis for systems with time-varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger-based inequality, the Bessel–Legendre inequality, the Wirtinger-based double integral inequality, and the auxiliary function-based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time-varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

Abstract: Polynomial chaos expansion and Gaussian mixture models are combined in a hybrid fashion to propagate state uncertainty for spacecraft with initial Gaussian errors. Polynomial chaos expansion models uncertainty by performing an expansion using orthogonal polynomials. The accuracy of polynomial chaos expansion for a given problem can be improved by increasing the order of the orthogonal polynomial expansion. The number of terms in the orthogonal polynomial expansion increases factorially with dimensionality of the problem, thereby reducing the effectiveness of the polynomial chaos expansion approach for problems of moderately high dimensionality. This paper shows a combination of Gaussian mixture model and polynomial chaos expansion, Gaussian mixture model–polynomial chaos expansion as an alternative form of the multi-element polynomial chaos expansion. Gaussian mixture model–polynomial chaos expansion reduces the overall order required to reach a desired accuracy. The initial distribution is converted to a...

Abstract: Given a matrix polynomial $W(x)$, matrix bi-orthogonal polynomials with respect to the sesquilinear form $\langle P(x),Q(x)\rangle_W=\int P(x) W(x)\operatorname{d}\mu(x)(Q(x))^{\top}$, $P(x),Q(x)\in\mathbb R^{p\times p}[x]$, where $\mu(x)$ is a matrix of Borel measures supported in some infinite subset of the real line, are considered. Connection formulas between the sequences of matrix bi-orthogonal polynomials with respect to $\langle \cdot,\cdot\rangle_W$ and matrix polynomials orthogonal with respect to $\mu(x)$ are presented. In particular, for the case of nonsingular leading coefficients of the perturbation matrix polynomial $W(x)$ we present a generalization of the Christoffel formula constructed in terms of the Jordan chains of $W(x)$. For perturbations with a singular leading coefficient several examples by Duran et al are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy.

Abstract: This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big $q$-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended $\ell^2$ Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrodinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the $q$-Meixner ($q$-Charlier) polynomials do not form a complete basis of the $\ell^2$ Hilbert space, based on the fact that the dual $q$-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the $q$-Meixner polynomials is obtained by constructing the duals of the dual $q$-Meixner polynomials which require the two component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big $q$-Jacobi family and their duals and $q$-Meixner ($q$-Charlier) polynomials.

Abstract: We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz–Segur family of solutions to the Painleve II equation. Our results complement the ones in [33]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painleve II transcendents, and we also prove a new result on the poles of the Ablowitz–Segur solutions to the Painleve II equation. We also highlight applications of our results in random matrix theory.

Abstract: Rankin–Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of $$SL(2,\mathbb {R})$$ . We address a general problem to find explicit formulae for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose, we use a new method (F-method) developed in Kobayashi and Pevzner (Sel. Math. New Ser., (2015). doi: 10.1007/s00029-15-0207-9 ) and based on the algebraic Fourier transform for generalized Verma modules. The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulae of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin–Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.

Abstract: Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials, and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi type identities, and associated Fomin-Greene operators.

Abstract: Hermite polynomial-based functional link artificial neural network FLANN is proposed here to solve the Van der Pol-Duffing oscillator equation. A single-layer hermite neural network HeNN model is used, where a hidden layer is replaced by expansion block of input pattern using Hermite orthogonal polynomials. A feedforward neural network model with the unsupervised error backpropagation principle is used for modifying the network parameters and minimizing the computed error function. The Van der Pol-Duffing and Duffing oscillator equations may not be solved exactly. Here, approximate solutions of these types of equations have been obtained by applying the HeNN model for the first time. Three mathematical example problems and two real-life application problems of Van der Pol-Duffing oscillator equation, extracting the features of early mechanical failure signal and weak signal detection problems, are solved using the proposed HeNN method. HeNN approximate solutions have been compared with results obtained by the well known Runge-Kutta method. Computed results are depicted in term of graphs. After training the HeNN model, we may use it as a black box to get numerical results at any arbitrary point in the domain. Thus, the proposed HeNN method is efficient. The results reveal that this method is reliable and can be applied to other nonlinear problems too.

Abstract: We derive a new matrix representation for higher-order Daehee numbers and polynomials, higher-order λ-Daehee numbers and polynomials, and twisted λ-Daehee numbers and polynomials of order k This helps us to obtain simple and short proofs of many previous results on higher-order Daehee numbers and polynomials Moreover, we obtain recurrence relations, explicit formulas, and some new results for these numbers and polynomials Furthermore, we investigate the relation between these numbers and polynomials and Stirling, Norlund, and Bernoulli numbers of higher-order Some numerical results and program are introduced using Mathcad for generating higher-order Daehee numbers and polynomials The results of this article generalize the results derived very recently by El-Desouky and Mustafa (Appl Math Sci 9(73):3593-3610, 2015)

Abstract: In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials $${\Phi_{n}^{(\alpha)}(x, u)}$$ of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers E n , Euler polynomials E n (x), generalized Euler numbers E n (a, b), generalized Euler polynomials E n (x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials $${{_HE}_n(x,y)}$$ of Dattoli et al. and $${{_HE}_n^{(\alpha)} (x,y)}$$ of Pathan are generalized to the one $${ {_HE}_n^{(\alpha)}(x,y,a,b,c)}$$ which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between E n , E n (x), E n (a, b), E n (x; a, b, c) and $${{}_HE_n^{(\alpha)}(x,y;a,b,c)}$$ are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions.

Abstract: We propose a novel algorithm to solve a general class of linear ill-posed inverse problems. For our numerical tests, we consider ill-posed problems on the sphere as they appear in the geosciences. Based on an iterative greedy algorithm, called the orthogonal matching pursuit, the signal is expanded in terms of trial functions which are picked from a large redundant set of functions, the so-called dictionary. The method is able to combine arbitrary trial functions which is a great advantage to former approximation algorithms. In particular, we combine orthogonal polynomials (such as spherical harmonics in the case of the sphere) of low degrees with localized trial functions such as wavelets and/or scaling functions for the reconstruction of global trends and regional details of the signal, respectively. Since we deal with ill-posed problems, we use a Tikhonov-type regularization with a penalty term based on a (spherical) Sobolev norm. There is no need to solve any system of equations or any integration pro...

Abstract: We study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simple manner from the inverse of the empirical moment matrix. In fact, this SOS polynomial is directly related to orthogonal polynomials and the Christoffel function. This allows to generalize and interpret extremality properties of orthogonal polynomials and to provide a mathematical rationale for the observed phenomenon. Among diverse potential applications, we illustrate the relevance of our results on a network intrusion detection task for which we obtain performances similar to existing dedicated methods reported in the literature.

Abstract: In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions (PCE) to quantify uncertainty in models subject to uncertain input parameters. The standard sampling approach for recovering sparse polynomials is to use Monte Carlo (MC) sampling of the density of orthogonality. However MC methods result in poor function recovery when the polynomial degree is high. Here we propose a general algorithm that can be applied to any admissible weight function on a bounded domain and a wide class of exponential weight functions defined on unbounded domains. Our proposed algorithm samples with respect to the weighted equilibrium measure of the parametric domain, and subsequently solves a preconditioned $\ell^1$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided that demonstrate that our proposed Christoffel Sparse Approximation algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermite specific algorithms.

Abstract: The R\'enyi entropies $R_{p}[\rho]$, $p>0, eq 1$ of the highly-excited quantum states of the $D$-dimensional isotropic harmonic oscillator are analytically determined by use of the strong asymptotics of the orthogonal polynomials which control the wavefunctions of these states, the Laguerre polynomials. This Rydberg energetic region is where the transition from classical to quantum correspondence takes place. We first realize that these entropies are closely connected to the entropic moments of the quantum-mechanical probability $\rho_n(\vec{r})$ density of the Rydberg wavefunctions $\Psi_{n,l,\{\mu\}}(\vec{r})$; so, to the $\mathcal{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the asymptotics $n\to\infty$ of these norms by use of modern techniques of approximation theory based on the strong Laguerre asymptotics. Finally, we determine the dominant term of the R\'enyi entropies of the Rydberg states explicitly in terms of the hyperquantum numbers ($n,l$), the parameter order $p$ and the universe dimensionality $D$ for all possible cases $D\ge 1$. We find that (a) the R\'enyi entropy power decreases monotonically as the order $p$ is increasing and (b) the disequilibrium (closely related to the second order R\'enyi entropy), which quantifies the separation of the electron distribution from equiprobability, has a quasi-Gaussian behavior in terms of $D$.

Abstract: We propose a simple and robust phase demodulation algorithm for two-shot fringe patterns with random phase shifts. Based on a smoothness assumption, the phase to be recovered is decomposed into a linear combination of finite terms of orthogonal polynomials, and the expansion coefficients and the phase shift are exhaustively searched through global optimization. The technique is insensitive to noise or defects, and is capable of retrieving phase from low fringe-number (less than one) or low-frequency interferograms. It can also cope with interferograms with very small phase shifts. The retrieved phase is continuous and no further phase unwrapping process is required. The method is expected to be promising to process interferograms with regular fringes, which are common in optical shop testing. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

Abstract: We extend the family non-symmetric Macdonald polynomials and define general-basement Macdonald polynomials. We show that these also satisfy a triangularity property with respect to the monomials bases and behave well under the Demazure-Lusztig operators. The symmetric Macdonald polynomials $J_\lambda$ are expressed as a sum of general-basement Macdonald polynomials via an explicit formula. By letting $q=0$, we obtain $t$-deformations of key polynomials and Demazure atoms and we show that the Hall--Littlewood polynomials expand positively into these. This generalizes a result by Haglund, Luoto, Mason and van Willigenburg. As a corollary, we prove that Schur polynomials decompose with non-negative coefficients into $t$-deformations of general Demazure atoms and thus generalizing the $t=0$ case which was previously known. This gives a unified formula for the classical expansion of Schur polynomials in Hall-Littlewood polynomials and the expansion of Schur polynomials into Demazure atoms.

Abstract: In this paper we introduce a new concept of key polynomials for a given valuation $ u$ on $K[x]$. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaquie, for instance, that they are irreducible and that the truncation of $ u$ associated to each key polynomial is a valuation. Moreover, we prove that every valuation $ u$ on $K[x]$ admits a sequence of key polynomials that completely determines $ u$ (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent sequences defined by Kaplansky.

Abstract: Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive Fiedler-comrade pencils for two bases that are extremely important in practical applications: the Chebyshev polynomials of the first and second kind. The new approach allows one to construct linearizations having limited bandwidth: a Chebyshev analogue of the pentadiagonal Fiedler pencils in the monomial basis. Moreover, our theory allows for linearizations of square matrix polynomials expressed in the Chebyshev basis (and in other bases), regardless of whether the matrix polynomial is regular or singular, and for recovery formulas for eigenvectors, and minimal indices and bases.

Abstract: We study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simple manner from the inverse of the empirical moment matrix. In fact, this SOS polynomial is directly related to orthogonal polynomials and the Christoffel function. This allows to generalize and interpret extremality properties of orthogonal polynomials and to provide a mathematical rationale for the observed phenomenon. Among diverse potential applications, we illustrate the relevance of our results on a network intrusion detection task for which we obtain performances similar to existing dedicated methods reported in the literature.

Abstract: We compute the leading asymptotics as $N\to\infty$ of the maximum of the field $Q_N(q)= \log\det|q- A_N|$, $q\in \mathbb{C}$, for any unitarily invariant Hermitian random matrix $A_N$ associated to a non-critical real-analytic potential. Hence, we verify the leading order in a conjecture of Fyodorov and Simm formulated for the GUE. The method relies on a classical upper-bound and a more sophisticated lower-bound based on a variant of the second-moment method which exploits the hyperbolic branching structure of the field $Q_N(q)$, $q$ in the upper half plane. Specifically, we compare $Q_N$ to an idealized Gaussian field by means of exponential moments. In principle, this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments. For unitarily invariant ensembles, we show that these assumptions follow from the Fyodorov-Strahov formula and asymptotics of orthogonal polynomials derived by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou.

Abstract: In this paper, we introduce a new class of degenerate Hermite poly-Bernoulli polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of degenerate Hermite poly-Bernoulli numbers and polynomials. Mathematics subject classification (2010): 11B68, 11B73, 11B83, 33C45.

Abstract: The purpose of this paper is to construct generating functions for the family of the Fibonacci and Jacobsthal polynomials. Using these generating functions and their functional equations, we investigate some properties of these polynomials. We also give relationships between the Fibonacci, Jacobsthal, Chebyshev polynomials and the other well known polynomials. Finally, we give some infinite series applications related to these polynomials and their generating functions.

Abstract: In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

Abstract: We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of GL$(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the set of points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.

Abstract: It is shown that the tridiagonalization of the hypergeometric operator $L$ yields the generic Heun operator $M$. The algebra generated by the operators $L,M$ and $Z=[L,M]$ is quadratic and a one-parameter generalization of the Racah algebra. The new Racah-Heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators $L$ and $M$. An interpretation in terms of the Racah problem for $su(1,1)$ algebras and separation of variables in a superintegrable system are discussed.