Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a generalized polynomial chaos expansion(GPCE)for statistical moment and reliability analyses associated with the stochastic output and a static reanalysis method to generate the input-output data set.In the reanalysis,we employ substructuring for a structure to isolate its local regions that vary due to random inputs.This allows for avoiding repeated computations of invariant substructures while generating the input-output data set.Combining substructuring with static condensation further improves the computational efficiency of the reanalysis without losing accuracy.Consequently,the GPCE with the static reanalysis method can achieve significant computational saving,thus mitigating the curse of dimensionality to some degree for UQ under high-dimensional inputs.The numerical results obtained from a simple structure indicate that the proposed method for UQ produces accurate solutions more efficiently than the GPCE using full finite element analyses(FEAs).We also demonstrate the efficiency and scalability of the proposed method by executing UQ for a large-scale wing-box structure under ten-dimensional(all-dependent)random inputs.

High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.

Convergence Phenomenon with Fourier Series of tg(x2)and Alike

The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .

RIEMANN-HILBERT CHARACTERIZATION FOR MAIN BESSEL POLYNOMIALS WITH VARYING LARGE NEGATIVE PARAMETERS

In this article, we establish the Bessel polynomials with varying large negative parameters and discuss their orthogonality based on the generalized Bessel polynomials. By using the Riemann-Hilbert boundary value problem on the positive real axis, we get the Riemann-Hilbert characterization of the main Bessel polynomials with varying large negative parameters.

Some Remarks for the Relationships between the Generalized Bernoulli and Euler Polynomials

In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.

SATURATION THEOREMS FOR GENERALIZED BERNSTEIN POLYNOMIALS

In the present paper, we give the explicit formula of the principal part of ∑k=0^n（{k}q-[n]qx）^sxk ∏m=0^n-k-1（1-q^mx） with respect to [n]q for any integer s and q ∈ （0, 1]. And, using the expressions, we obtain saturation theorems for Bn （f , qn,x） approximating to f（x） ∈ C[O, 1], 0 〈 qn ≤ 1, qn → 1.

Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

In this paper some novel integrals associated with the product of classical Hermite＇s polynomials ∫-∞＋∞（x2）mexp（-x2）{Hr（x）}2dx,∫0∞exp（-x2）H2k（x）H2s＋1（x）dx,∫0∞exp（-x2）H2k（x）H2s（x）dx and ∫0∞exp（-x2）H2k＋1（x）H2s＋1（x）dx, are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of spe- cial functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite＇s polynomials by suitable simplifications of arbitrary parameters.

An Investigation on Generalized Eulerian Polynomials and Fractions

This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summarion formula is given

GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO SOBOLEV ORTHOGONAL POLYNOMIALS Ⅱ

We present a definition of general Sobolev spaces with respect to arbitrary measures, Wk,p(Ω,μ) for 1 .≤p≤∞.In [RARP] we proved that these spaces are complete under very light conditions. Now we prove that if we consider certain general types of measures, then Cτ∞(R) is dense in these spaces. As an application to Sobolev orthogonal polynomials, toe study the boundedness of the multiplication operator. This gives an estimation of the zeroes of Sobolev orthogonal polynomials.

Binary Cyclic Codes and Minimal Codewords

Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal codewords in linear codes are widely used in constructing decoding algorithms and studying linear secret sharing scheme. In this paper, we show that in the binary cyclic code all of the codewords are minimal, except 0 and 1. Then, we obtain a result about the number of minimal codewords in the binary cyclic codes.

New method of time-frequency representation for ISAR imaging of ship targets

Inverse synthetic aperture radar （ISAR） imaging of ship targets is very important in the national defense. For the high maneuverability of ship targets, the Doppler frequency shift of the received signal is time-varying, which will degrade the ISAR image quality for the traditional range-Doppler （RD） algorithm. In this paper, the received signal in a range bin is characterized as the multi-component polynomial phase signal （PPS） after the motion compensation, and a new approach of time-frequency represen- tation, generalized polynomial Wigner-Ville distribution （GPWVD）, is proposed for the azimuth focusing. The GPWVD is based on the exponential matched-phase （EMP） principle. Compared with the conventional polynomial Wigner-Ville distribution （PWVD）, the EMP principle transfers the non-integer lag coefficients of the PWVD to the position of the exponential of the signal, and the interpolation can be avoided completely. For the GPWVD, the cross-terms between multi-component signals can be reduced by decomposing the GPWVD into the convolution of Wigner-Ville distribution （WVD） and the spectrum of phase adjust functions. The GPWVD is used in the ISAR imaging of ship targets, and the high quality instantaneous ISAR images can be obtained. Simulation results and measurement data demonstrate the effectiveness of the proposed new method.

DISCRETE GALERKIN METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

In this article, we develop a fully Discrete Galerkin（DG） method for solving ini- tial value fractional integro-differential equations（FIDEs）. We consider Generalized Jacobi polynomials（CJPs） with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.

An Identity with Skew Derivations on Lie Ideals

Let R be a 2-torsion free prime ring and L a noncommutative Lie ideal of R. Suppose that （d,σ） is a skew derivation of R such that xsd（x）xt = 0 for all x ∈ L, where s, t are fixed non-negative integers. Then d = 0.

Quadrature Formula of Singular Integral Based on Rational Interpolation

We construct a quadrature formula of the singular integral with the Chebyshev weight of the second kind by using Lagrange interpolation based on the rational system {1/(x?a 1), 1/(x?a 2), …}, and both the remainder and convergence of the quadrature formula established here are discussed. Our results extend some classical ones.

Annihilator on Co-commutators with Derivations on Lie Ideals in Prime Rings

Let R be a prime ring of characteristic different from 2, d and 9 two derivations of R at least one of which is nonzero, L a non-central Lie ideal of R, and a ∈ R. We prove that if a（d（u）u - ug（u）） = 0 for any u ∈ L, then either a = O, or R is an sa-ring, d（x） = [p, x], and g（x） = -d（x） for some p in the Martindale quotient ring of R.

New useful special function in quantum optics theory

By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e.,min（m,n）∑l=0^min（m,n）n!m!（-1）^l/l!（n-1）!（m-l）!/Hn-l（x）y^m-l≡ n,m（x,y）.By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators（IWOP） we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.

On a generalized homogeneous Hahn polynomial

We investigate a generalized homogeneous Hahn polynomial in some detail. This polynomial includes as special cases the homogeneous Hahn polynomial and the homogeneous Rogers-Szeg o polynomial.A generating function, which contains a known generating function as a special case, is given. We also give a finite series generating function. Some results on the asymptotic expansion for this polynomial are derived.Certain results on zeros are also obtained. We deduce several results on zeros of certain entire functions involving this generalized Hahn polynomial. As results, one of Zhang(2017)’s results as well as others is obtained. Finally,we derive several general results on q-congruences of the generalized q-Apéry polynomials, from which two qcongruences involving the generalized homogeneous Hahn polynomial are deduced.

Knot Insertion Algorithms for ECT B-spline Curves

Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local prop- erties of B-spline curve and the control flexibility of its shape can be fiu＇ther improved, also the segmentation of the curve can be rea- lized. ECT spline curve is drew by the multi-knots spline curve with associated matrix in ECT spline space; Muehlbach G and Tang Y and many others have deduced the existence and uniqueness of the ECT spline function and developed many of its important properties .This paper mainly focuses on the knot insertion algorithm of ECT B-spline curve.It is the widest popularization of B-spline Behm algorithm and theory. Inspired by the Behm algorithm, in the ECT spline space, structure of generalized P61ya poly- nomials and generalized de Boor Fix dual functional, expressing new control points which are inserted after the knot by linear com- bination of original control vertex the single knot, and there are two cases, one is the single knot, the other is the double knot. Then finally comes the insertion algorithm of ECT spline curve knot. By application of the knot insertion algorithm, this paper also gives out the knot insertion algorithm of four order geometric continuous piecewise polynomial B-spline and algebraic trigonometric spline B-spline, which is consistent with previous results.

Existence of Multiple Positive Periodic Solutions for Second Order Differential Equations

Let α be a nonzero endomorphism of a ring R, n be a positive integer and T_n（R, α） be the skew triangular matrix ring. We show that some properties related to nilpotent elements of R are inherited by T_n（R, α）. Meanwhile, we determine the strongly prime radical, generalized prime radical and Behrens radical of the ring R[x; α]/（x^n）, where R[x; α] is the skew polynomial ring.

Convolution of Absolute Value Sum of Coefficients on Chebyshev Polynomials of the Second Kind

The main purpose of this paper is in using the generating function of generalized Fibonacci polynomials and its partial derivative to study the convolution evaluation of the second-kind Chebyshev polynomials, and give an interesting formula.