Set Of Orthogonal Polynomials Research Articles

On the dynamic response of non-linear systems with parameter uncertainties

This paper presents a procedure for obtaining the dynamic response of non-linear systems with parameter uncertainties. Consideration is given to systems with polynomial non-linearity subjected to deterministic excitation. The uncertain parameters are modeled as time-independent random variables. The set of orthogonal polynomials associated with the probability density function is used as the solution basis, and the response variables are expanded in terms of a finite sum of these polynomials. A set of deterministic non-linear differential equations is derived using the weighted residual method. The discrete-time solutions to the equation set are evaluated numerically using a step-by-step time-integration scheme and the response statistics are determined. Application of the proposed method is illustrated through the analysis of non-linear single-degree-of-freedom structural systems exhibiting uncertain stiffnesses. Both hardening and softening stiffness characteristics are examined. The accuracy of the results is validated by direct numerical integration.

Neural Network for Invariant Image Classification

In this paper, a Neural Network (NN) based approach for classification of images represented by translation, scale and rotation-invariant features is presented. The utilized network is a multi-layer perception (MLP) classifier with one hidden layer. The back-propagation learning is used for its training. The translation and scale invariant feautres are obtained by Fourier-Modified Direct Mellin Transform (F-MDMT). The rotation invariances are obtained by Zernike moments. Zernike moments are the mapping of the image onto a set of complex orthogonal polynomials. The performance of the MLP classifier is compared to three other traditional statistical classifiers, namely: Bayes, nearest-neighbour and minimum-mean-distance. Through extensive experiments with noiseless as well as noisy binary images of all Oriya characters (49 classes) and comparing these with usual geometrical invariants, it is reported that the neural network (MLP) has got a high degree of performance as a classifier.

Quantitative measurement of roughness of fractured rubber surfaces by an image processing technique

In this paper, an image processing technique has been employed to quantitatively analyse various fractured surfaces of rubber and measure roughness. The image surfaces are represented in terms of a closed set of orthogonal polynomials. The significant orthogonal effects are measured and combined to represent the local texture, called pronum. The frequency of occurrence of the pronums is the prospectrum, a global descriptor. A few statistical parameters have been calculated from the prospectrum and correlated to the roughness of the fractured surfaces. Using the image processing technique, the laborious procedure involved in quantification, especially of irregular microfeatures, has been shown to be overcome.

Basis set dependence of the locality of the kinetic energy operator

We have analyzed the locality of matrices of the kinetic energy operator in even-tempered Gaussian and Slater-type (STO) bases, as well as in several three-dimensional, one-dimensional, and pseudo-one-dimensional basis sets of orthogonal polynomials and trigonometric functions. We find that the locality of the kinetic energy matrix in Gaussian bases is dependent upon the basis function parameters, while STO bases are found to always produce kinetic energy matrices which are very nearly equal to the matrices of local operators. In addition, it is observed that in the limit of completeness, the locality measure for all of the one-dimensional basis sets appears to converge to a common value. In the case of a basis set of particle on a ring eigenfunctions, an exact value for this limit has been determined analytically. Three-dimensional and l=0 pseudo-one-dimensional basis sets of orthogonal polynomials give kinetic energy matrices that do not behave similarly in the limit of completeness. It is found that all of these basis sets result in kinetic energy matrices which clearly exhibit nonlocal behavior, except for those involving exponentials. Such bases, like the STO bases, appear always to yield kinetic energy matrices that remain nearly local regardless of the basis function parameters and the number of basis functions used.

Octabasic Laguerre polynomials and permutation statistics

A set of orthogonal polynomials with eight independent “ q's” is defined which generalizes the Laguerre polynomials. The moments of the measure for these polynomials are the generating functions for permutations according to eight different statistics. Specializing these statistics gives many other well-known sets of combinatorial objects and relevant statistics. The specializations are studied, with applications to classical orthogonal polynomials and equidistribution theorems for statistics.

Vibration studies on moderately thick doubly-curved elliptic shallow shells

Based on a refined first-order shear deformation theory, the vibratory characteristics of doubly-curved shallow shells of elliptical planform are investigated. Integral expressions incorporating the effects of shear deformation and rotary inertia for the strain and kinetic energies are derived. The transverse shear strain components are obtained as linear functions of thickness in contrast to constant strains available in the current literature. With the use of the extremum energy principle, a governing eigen-matrix equation is formulated which is subsequently solved to extract the frequencies and mode shapes. The shallow shells considered here are the spherical, cylindrical and hyperbolic paraboloidal shells. The effects of various geometric parameters and boundary constraints on the resonant frequencies and mode shapes are examined. The solution method employs displacement functions comprising of a set of two-dimensional orthogonal polynomials and a basic function for each degree of freedom: three orthogonal displacement components and two rotations. The convergence of eigenvalues is examined. The validity of the present results is verified by comparing, if possible, with the values from the literature.

Quasi-exactly solvable systems and orthogonal polynomials

This paper shows that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials {Pn}. The quantum-mechanical wave function is the generating function for the Pn(E), which are polynomials in the energy E. The condition of quasi-exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value J. The zeros of the critical polynomial PJ(E) are the quasi-exact energy eigenvalues of the system.

Combinatorial orthogonal expansions

The linearization coefficients for a set of orthogonal polynomials are given explicitly as a weighted sum of combinatorial objects. Positivity theorems of Askey and Szwarc are corollaries of these expansions.

Orthogonal polynomials and laurent polynomials related to the Hahn-Extonq-Bessel function

Laurent polynomials related to the Hahn-Extonq-Bessel function, which areq-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurentq-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials, the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurentq-Lommel polynomials and related functions is used to obtain estimates for the latter.

Computing the moments of the density of zeros for orthogonal polynomials

A representation formula (by means of the generalized Lucas Polynomials of first kind) for the moments of the density of zeros of Orthogonal Polynomial Sets defined by a three-term recurrence relation is used to derive information directly from the entries of the Jacobi matrix. Numerical computations are explicitly developed in some particular case.

Quantitative measurement of dispersion of carbon black in rubber by an image processing technique

AbstractIn this article, the quantification of dispersion of carbon black in rubber is done by an image‐processing technique. Surfaces, having different dispersion ratings, are mostly textured. These textured surfaces are digitized and the images are analyzed for quantification. Textured images are suitably represented by a closed set of orthogonal polynomials. The presence of texture has been identified by suitably measuring the orthogonal effects. The textured surfaces are described locally by pronum and globally by prospectrum. Mean, variance, and Fisher distance were computed from the prospectrums and are correlated to various dispersion ratings. There is a linear relationship between the calculated mean and the dispersion rating. © 1995 John Wiley & Sons, Inc.

Eigenvalue integrodifferential equations for orthogonal polynomials on the real line

The one-dimensional harmonic oscillator wave functions are solutions to a Sturm–Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm–Liouville problem on the whole real line. In this paper it is shown how to characterize an arbitrary set of polynomials orthogonal on (−∞,∞) in terms of a system of integrodifferential equations of the Hartree–Fock type. This system replaces and generalizes the linear differential equation associated with a Sturm–Liouville problem. Our results are demonstrated for the special case of continuous Hahn polynomials.

Synthesis of extremal wavelet-generating filters using Gaussian quadrature

Orthogonal wavelets can be generated from finite impulse response quadrature mirror filters; these filters are also used in perfect reconstruction filter banks. This paper addresses the problem of efficiently synthesizing such filters. A class of extremal filters is defined by the property that their magnitude spectrum maximizes an integral criterion. It is found that these filters are characterized by their zeros on the unit circle, which frequently can be obtained from a set of orthogonal polynomials. A family of filters is constructed that minimize the subband aliasing energy and can generate wavelets with an arbitrary number of vanishing moments. The algorithm for generating these filters makes use of the Levinson recursions, Gaussian quadrature and a fast version of Euclid's algorithm. Similar to other methods for constructing quadrature mirror filters, the spectral factorization of a polynomial is the computationally expensive part of this algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new statistical approach for micro texture description

A new statistical approach using a complete set of orthogonal polynomials is proposed here for texture description. The method tests the presence of micro texture in an image in terms of the significance of orthogonal effects and then represents a local descriptor called pronum for the texture. The frequency of occurrence of pronums which is called the prospectrum is used to represent the global descriptor of texture. The proposed method has been used for describing textured regions. The primary advantage of this scheme is that the spatial variation resulting from textural characteristics of the image has been effectively separated out for identification of textured regions.

Vibration of mindlin plates using boundary characteristic orthogonal polynomials

Abstract The vibration analysis of shear deformable plates formulated on the basis of first order Mindlin theory is presented. The displacement and rotational functions of the plates are approximated by sets of boundary characteristic orthogonal polynomials. The ease of generation and manipulation of these polynomial functions greatly enhances the computational efficiency of the numerical method. The energy functional of the shear deformable plates derived from the Mindlin plate theory is minimized in the Ritz procedure to arrive at the governing eigenvalue equation. Corresponding natural frequencies and mode shapes can be obtained by solving the eigenvalue equation. Computed frequency results for various boundary conditions, aspect ratios a/b and thickness ratios t/b are presented to demonstrate the effects of each factor on the vibration frequencies of moderately thick plates. Vibration mode shapes in the form of contour plots are presented for a thickness ratio of t/b = 0.1. These plots are believed to be the first known in the open literature.

Quantitative Fractography of Rubber by Image Processing Technique

Abstract In this paper, an image processing technique to quantitatively analyze abraded, torn and fatigue failed surfaces is described. The image surfaces were represented in terms of a closed set of orthogonal polynomials. The significant orthogonal effects were measured and combined to represent the local texture, called pronum. The frequency of occurrence of the pronums is the prospectrum, a global descriptor. Various statistical parameters were calculated from the prospectrum and correlated to the ridge spacings on abraded surfaces. The statistical closeness between various textured surfaces was also quantified. Using the image processing technique, the laborious procedure involved for quantification especially of irregular microfeatures has been shown to be overcome. Author to whom correspondence should be addressed.

Tracking the amplitude versus offset (AVO) by using orthogonal polynomials1

AbstractInversion for S‐wave velocities from the amplitude variation with offset of P‐wave data is far from being a standard routine in the seismic processing sequence. However, the need for tracking the amplitude versus offset (AVO) occurs in several situations, for example in order to estimate the zero‐offset amplitude, to reveal areas with particular AVO characteristics, or to compress the AVO so that it is more easily obtainable at a later stage of the seismic processing. Furthermore, weak reflections can occasionally, due to the effect of the angle‐dependent reflectivity, have a polarity‐shift with offset, resulting in a very poor, or even vanishing, stack response. In such cases, the reflection event has to be represented by some other property than its mean amplitude or stack value.We outline how the AVO of seismic data may be extracted and classified by the use of orthogonal polynomials. The main advantage of this method compared to a general polynomial fit is that the AVO may be classified by a unique Spectrum of polynomial coefficients. This is in analogy to Fourier coefficients where the orthogonal basis is harmonic functions. The set of orthogonal polynomials is constructed entirely from the set of offset coordinates, and these polyno‐mials are defined only on the offset window considered. Compared to a Fourier transform, this is a major advantage since there is no effect of a limited spatial bandwidth.The AVO of normal‐moveout corrected data may be represented by a data gather where the orthogonal polynomial coefficients are given as time traces with each trace revealing a certain AVO characteristic. For instance, the stack is proportional to the zeroth‐order coefficient, the mean gradient is given by the firstorder coefficient, while the second‐order coefficient indicates whether the AVO increases and then decreases, or vice versa.

A discrete element model for vibration analysis of mixed edge plates

In this paper, the free vibration analysis of plates with mixed edge boundaries is presented. A discrete element model is developed for the present study. In the solution process, the original domain is discretised into small subdomains with the admissible functions of each subdomain represented by sets of orthogonal polynomials. Higher-order polynomials in the set are generated using the Gram-Schmidt recurrence process. Continuity matrices derived from the geometric compatibilities between subdomains are used to couple the eigenvectors of adjacent subdomains. The global stiffness and mass matrices are considered to be the sum of the stiffness and mass matrices of the constitutive subdomains after pre- and post-multiplied by the respective continuity matrices. Convergence and comparison studies were carried out to demonstrate the accuracy and efficiency of the proposed method in solving this class of problems.

On The Use Of The Domain Decomposition Method For Vibration Of Symmetric Laminates Having Discontinuities At The Same Edge

A free vibration analysis of symmetrically laminated plates with mixed edge boundary conditions is presented. A highly efficient and accurate domain decomposition method is employed. To establish the model, the original domain is first assumed to be an assemblage of small subdomains, with the admissible functions of each subdomain represented by sets of orthogonal polynomials. These polynomials are generated by using the Gram-Schmidt procedure. The continuity matrices resulting from the geometric compatibilities between the subdomains are used to couple the eigenvectors of adjacent subdomains. The stiffness and mass matrices of each subdomain, after pre- and post-multiplying by the respective continuity matrices, are assembled to form the global stiffness and mass matrices. To demonstrate this method, the free vibrations of symmetrically laminated plates of various types of mixed edge boundaries are calculated.

Orthogonal polynomials in phase space and classical analogs to quantum eigenstates with an application to the transmission flux for the potential step

We introduce a set of orthogonal polynomials in phase space with similar properties, and which can be reduced to the usual Hermite polynomials. Based on these polynomials, we develop a method that allows the finding of classical analogs to quantum eigenstates. This method is applied to the free particle scattered by a potential step and the transmission flux is calculated and compared with the quantum one.