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Sample records for legendre polynomials applied

  1. Superiority of legendre polynomials to Chebyshev polynomial in ...

    African Journals Online (AJOL)

    In this paper, we proved the superiority of Legendre polynomial to Chebyshev polynomial in solving first order ordinary differential equation with rational coefficient. We generated shifted polynomial of Chebyshev, Legendre and Canonical polynomials which deal with solving differential equation by first choosing Chebyshev ...

  2. Fast Minimum Variance Beamforming Based on Legendre Polynomials.

    Science.gov (United States)

    Bae, MooHo; Park, Sung Bae; Kwon, Sung Jae

    2016-09-01

    Currently, minimum variance beamforming (MV) is actively investigated as a method that can improve the performance of an ultrasound beamformer, in terms of the lateral and contrast resolution. However, this method has the disadvantage of excessive computational complexity since the inverse spatial covariance matrix must be calculated. Some noteworthy methods among various attempts to solve this problem include beam space adaptive beamforming methods and the fast MV method based on principal component analysis, which are similar in that the original signal in the element space is transformed to another domain using an orthonormal basis matrix and the dimension of the covariance matrix is reduced by approximating the matrix only with important components of the matrix, hence making the inversion of the matrix very simple. Recently, we proposed a new method with further reduced computational demand that uses Legendre polynomials as the basis matrix for such a transformation. In this paper, we verify the efficacy of the proposed method through Field II simulations as well as in vitro and in vivo experiments. The results show that the approximation error of this method is less than or similar to those of the above-mentioned methods and that the lateral response of point targets and the contrast-to-speckle noise in anechoic cysts are also better than or similar to those methods when the dimensionality of the covariance matrices is reduced to the same dimension.

  3. Compression of head-related transfer function using autoregressive-moving-average models and Legendre polynomials

    DEFF Research Database (Denmark)

    Shekarchi, Sayedali; Hallam, John; Christensen-Dalsgaard, Jakob

    2013-01-01

    -moving-average (ARMA) filters whose coefficients are calculated using Prony's method. Such filters are specified by a few coefficients which can generate the full head-related impulse responses (HRIRs). Next, Legendre polynomials (LPs) are used to compress the ARMA filter coefficients. LPs are derived on the sphere...

  4. State-vector formalism and the Legendre polynomial solution for modelling guided waves in anisotropic plates

    Science.gov (United States)

    Zheng, Mingfang; He, Cunfu; Lu, Yan; Wu, Bin

    2018-01-01

    We presented a numerical method to solve phase dispersion curve in general anisotropic plates. This approach involves an exact solution to the problem in the form of the Legendre polynomial of multiple integrals, which we substituted into the state-vector formalism. In order to improve the efficiency of the proposed method, we made a special effort to demonstrate the analytical methodology. Furthermore, we analyzed the algebraic symmetries of the matrices in the state-vector formalism for anisotropic plates. The basic feature of the proposed method was the expansion of field quantities by Legendre polynomials. The Legendre polynomial method avoid to solve the transcendental dispersion equation, which can only be solved numerically. This state-vector formalism combined with Legendre polynomial expansion distinguished the adjacent dispersion mode clearly, even when the modes were very close. We then illustrated the theoretical solutions of the dispersion curves by this method for isotropic and anisotropic plates. Finally, we compared the proposed method with the global matrix method (GMM), which shows excellent agreement.

  5. Random regression models to estimate genetic parameters for milk production of Guzerat cows using orthogonal Legendre polynomials

    Directory of Open Access Journals (Sweden)

    Maria Gabriela Campolina Diniz Peixoto

    2014-05-01

    Full Text Available The objective of this work was to compare random regression models for the estimation of genetic parameters for Guzerat milk production, using orthogonal Legendre polynomials. Records (20,524 of test-day milk yield (TDMY from 2,816 first-lactation Guzerat cows were used. TDMY grouped into 10-monthly classes were analyzed for additive genetic effect and for environmental and residual permanent effects (random effects, whereas the contemporary group, calving age (linear and quadratic effects and mean lactation curve were analized as fixed effects. Trajectories for the additive genetic and permanent environmental effects were modeled by means of a covariance function employing orthogonal Legendre polynomials ranging from the second to the fifth order. Residual variances were considered in one, four, six, or ten variance classes. The best model had six residual variance classes. The heritability estimates for the TDMY records varied from 0.19 to 0.32. The random regression model that used a second-order Legendre polynomial for the additive genetic effect, and a fifth-order polynomial for the permanent environmental effect is adequate for comparison by the main employed criteria. The model with a second-order Legendre polynomial for the additive genetic effect, and that with a fourth-order for the permanent environmental effect could also be employed in these analyses.

  6. Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments

    Directory of Open Access Journals (Sweden)

    Wei Li

    2017-01-01

    Full Text Available We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is, ∫-11Pl′m′xt-1/1+t2-2xtPk′m′(x/(1+t2-2tx(l′+1/2dx, where t∈(0,1. The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument ∫0∞Jn(αx2+z2/(x2+z2nx2m+1dx.

  7. Characterizing the Lyman-alpha forest flux probability distribution function using Legendre polynomials

    Science.gov (United States)

    Cieplak, Agnieszka; Slosar, Anze

    2018-01-01

    The Lyman-alpha forest has become a powerful cosmological probe at intermediate redshift. It is a highly non-linear field with much information present beyond the power spectrum. The flux probability flux distribution (PDF) in particular has been a successful probe of small scale physics. However, it is also sensitive to pixel noise, spectrum resolution, and continuum fitting, all of which lead to possible biased estimators. Here we argue that measuring the coefficients of the Legendre polynomial expansion of the PDF offers several advantages over measuring the binned values as is commonly done. Since the n-th Legendre coefficient can be expressed as a linear combination of the first n moments of the field, this allows for the coefficients to be measured in the presence of noise and allows for a clear route towards marginalization over the mean flux. Additionally, in the presence of noise, a finite number of these coefficients are well measured with a very sharp transition into noise dominance. This compresses the information into a small amount of well-measured quantities. Finally, we find that measuring fewer quasars with high signal-to-noise produces a higher amount of recoverable information.

  8. Characterizing the Lyα forest flux probability distribution function using Legendre polynomials

    Energy Technology Data Exchange (ETDEWEB)

    Cieplak, Agnieszka M.; Slosar, Anže, E-mail: acieplak@bnl.gov, E-mail: anze@bnl.gov [Brookhaven National Laboratory, Bldg 510, Upton, NY, 11973 (United States)

    2017-10-01

    The Lyman-α forest is a highly non-linear field with considerable information available in the data beyond the power spectrum. The flux probability distribution function (PDF) has been used as a successful probe of small-scale physics. In this paper we argue that measuring coefficients of the Legendre polynomial expansion of the PDF offers several advantages over measuring the binned values as is commonly done. In particular, the n -th Legendre coefficient can be expressed as a linear combination of the first n moments, allowing these coefficients to be measured in the presence of noise and allowing a clear route for marginalisation over mean flux. Moreover, in the presence of noise, our numerical work shows that a finite number of coefficients are well measured with a very sharp transition into noise dominance. This compresses the available information into a small number of well-measured quantities. We find that the amount of recoverable information is a very non-linear function of spectral noise that strongly favors fewer quasars measured at better signal to noise.

  9. Random regression models using Legendre polynomials or linear splines for test-day milk yield of dairy Gyr (Bos indicus) cattle.

    Science.gov (United States)

    Pereira, R J; Bignardi, A B; El Faro, L; Verneque, R S; Vercesi Filho, A E; Albuquerque, L G

    2013-01-01

    Studies investigating the use of random regression models for genetic evaluation of milk production in Zebu cattle are scarce. In this study, 59,744 test-day milk yield records from 7,810 first lactations of purebred dairy Gyr (Bos indicus) and crossbred (dairy Gyr × Holstein) cows were used to compare random regression models in which additive genetic and permanent environmental effects were modeled using orthogonal Legendre polynomials or linear spline functions. Residual variances were modeled considering 1, 5, or 10 classes of days in milk. Five classes fitted the changes in residual variances over the lactation adequately and were used for model comparison. The model that fitted linear spline functions with 6 knots provided the lowest sum of residual variances across lactation. On the other hand, according to the deviance information criterion (DIC) and bayesian information criterion (BIC), a model using third-order and fourth-order Legendre polynomials for additive genetic and permanent environmental effects, respectively, provided the best fit. However, the high rank correlation (0.998) between this model and that applying third-order Legendre polynomials for additive genetic and permanent environmental effects, indicates that, in practice, the same bulls would be selected by both models. The last model, which is less parameterized, is a parsimonious option for fitting dairy Gyr breed test-day milk yield records. Copyright © 2013 American Dairy Science Association. Published by Elsevier Inc. All rights reserved.

  10. Solved problems in analysis as applied to gamma, beta, Legendre and Bessel functions

    CERN Document Server

    Farrell, Orin J

    2013-01-01

    Nearly 200 problems, each with a detailed, worked-out solution, deal with the properties and applications of the gamma and beta functions, Legendre polynomials, and Bessel functions. The first two chapters examine gamma and beta functions, including applications to certain geometrical and physical problems such as heat-flow in a straight wire. The following two chapters treat Legendre polynomials, addressing applications to specific series expansions, steady-state heat-flow temperature distribution, gravitational potential of a circular lamina, and application of Gauss's mechanical quadrature

  11. Comment on 'Analytical results for a Bessel function times Legendre polynomials class integrals'

    International Nuclear Information System (INIS)

    Cregg, P J; Svedlindh, P

    2007-01-01

    A result is obtained, stemming from Gegenbauer, where the products of certain Bessel functions and exponentials are expressed in terms of an infinite series of spherical Bessel functions and products of associated Legendre functions. Closed form solutions for integrals involving Bessel functions times associated Legendre functions times exponentials, recently elucidated by Neves et al (J. Phys. A: Math. Gen. 39 L293), are then shown to result directly from the orthogonality properties of the associated Legendre functions. This result offers greater flexibility in the treatment of classical Heisenberg chains and may do so in other problems such as occur in electromagnetic diffraction theory. (comment)

  12. Wavefront reconstruction algorithm based on Legendre polynomials for radial shearing interferometry over a square area and error analysis.

    Science.gov (United States)

    Kewei, E; Zhang, Chen; Li, Mengyang; Xiong, Zhao; Li, Dahai

    2015-08-10

    Based on the Legendre polynomials expressions and its properties, this article proposes a new approach to reconstruct the distorted wavefront under test of a laser beam over square area from the phase difference data obtained by a RSI system. And the result of simulation and experimental results verifies the reliability of the method proposed in this paper. The formula of the error propagation coefficients is deduced when the phase difference data of overlapping area contain noise randomly. The matrix T which can be used to evaluate the impact of high-orders Legendre polynomial terms on the outcomes of the low-order terms due to mode aliasing is proposed, and the magnitude of impact can be estimated by calculating the F norm of the T. In addition, the relationship between ratio shear, sampling points, terms of polynomials and noise propagation coefficients, and the relationship between ratio shear, sampling points and norms of the T matrix are both analyzed, respectively. Those research results can provide an optimization design way for radial shearing interferometry system with the theoretical reference and instruction.

  13. Evaluation of integrals involving powers of (1-x2) and two associated Legendre functions or Gegenbauer polynomials

    International Nuclear Information System (INIS)

    Rashid, M.A.

    1984-08-01

    Integrals involving powers of (1-x 2 ) and two associated Legendre functions or two Gegenbauer polynomials are evaluated as finite sums which can be expressed in terms of terminating hypergeometric function 4 F 3 . The integrals which are evaluated are ∫sub(-1)sup(1)[Psub(l)sup(m)(x)Psub(k)sup(n)(x)]/[(1-x 2 )sup(p+1)]dx and ∫sub(-1)sup(1)Csub(l)sup(α)(x)Csub(k)sup(β)(x)[(1-x 2 )sup[(α+β-3)/2-p

  14. Numerical solutions of integral and integro-differential equations using Legendre polynomials

    Science.gov (United States)

    Khater, A.; Shamardan, A.; Callebaut, D.; Sakran, M.

    2007-11-01

    In this paper, a finite Legendre expansion is developed to solve singularly perturbed integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The error analysis is derived. Numerical results and comparisons with other methods in literature are considered.

  15. Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials.

    Science.gov (United States)

    Sun, Wenqing; Chen, Lei; Tuya, Wulan; He, Yong; Zhu, Rihong

    2013-12-01

    Chebyshev and Legendre polynomials are frequently used in rectangular pupils for wavefront approximation. Ideally, the dataset completely fits with the polynomial basis, which provides the full-pupil approximation coefficients and the corresponding geometric aberrations. However, if there are horizontal translation and scaling, the terms in the original polynomials will become the linear combinations of the coefficients of the other terms. This paper introduces analytical expressions for two typical situations after translation and scaling. With a small translation, first-order Taylor expansion could be used to simplify the computation. Several representative terms could be selected as inputs to compute the coefficient changes before and after translation and scaling. Results show that the outcomes of the analytical solutions and the approximated values under discrete sampling are consistent. With the computation of a group of randomly generated coefficients, we contrasted the changes under different translation and scaling conditions. The larger ratios correlate the larger deviation from the approximated values to the original ones. Finally, we analyzed the peak-to-valley (PV) and root mean square (RMS) deviations from the uses of the first-order approximation and the direct expansion under different translation values. The results show that when the translation is less than 4%, the most deviated 5th term in the first-order 1D-Legendre expansion has a PV deviation less than 7% and an RMS deviation less than 2%. The analytical expressions and the computed results under discrete sampling given in this paper for the multiple typical function basis during translation and scaling in the rectangular areas could be applied in wavefront approximation and analysis.

  16. Legendre and Laguerre polynomial approach for modeling of wave propagation in layered magneto-electro-elastic media.

    Science.gov (United States)

    Bou Matar, Olivier; Gasmi, Noura; Zhou, Huan; Goueygou, Marc; Talbi, Abdelkrim

    2013-03-01

    A numerical method to compute propagation constants and mode shapes of elastic waves in layered piezoelectric-piezomagnetic composites, potentially deposited on a substrate, is described. The basic feature of the method is the expansion of particle displacement, stress fields, electric and magnetic potentials in each layer on different polynomial bases: Legendre for a layer of finite thickness and Laguerre for the semi-infinite substrate. The exponential convergence rate of the method for propagation of Love waves is numerically verified. The main advantage of the method is to directly determine complex wave numbers for a given frequency via a matricial eigenvalue problem, in a way that no transcendental equation has to be solved. Results are presented and the method is discussed.

  17. S4 solution of the transport equation for eigenvalues using Legendre polynomials

    Directory of Open Access Journals (Sweden)

    Öztürk Hakan

    2017-01-01

    Full Text Available Numerical solution of the transport equation for monoenergetic neutrons scattered isotropically through the medium of a finite homogeneous slab is studied for the determination of the eigenvalues. After obtaining the discrete ordinates form of the transport equation, separated homogeneous and particular solutions are formed and then the eigenvalues are calculated using the Gauss-Legendre quadrature set. Then, the calculated eigenvalues for various values of the c0, the mean number of secondary neutrons per collision, are given in the tables.

  18. Theoretical study on the dispersion curves of Lamb waves in piezoelectric-semiconductor sandwich plates GaAs-FGPM-AlAs: Legendre polynomial series expansion

    Science.gov (United States)

    Othmani, Cherif; Takali, Farid; Njeh, Anouar

    2017-06-01

    In this paper, the propagation of the Lamb waves in the GaAs-FGPM-AlAs sandwich plate is studied. Based on the orthogonal function, Legendre polynomial series expansion is applied along the thickness direction to obtain the Lamb dispersion curves. The convergence and accuracy of this polynomial method are discussed. In addition, the influences of the volume fraction p and thickness hFGPM of the FGPM middle layer on the Lamb dispersion curves are developed. The numerical results also show differences between the characteristics of Lamb dispersion curves in the sandwich plate for various gradient coefficients of the FGPM middle layer. In fact, if the volume fraction p increases the phase velocity will increases and the number of modes will decreases at a given frequency range. All the developments performed in this paper were implemented in Matlab software. The corresponding results presented in this work may have important applications in several industry areas and developing novel acoustic devices such as sensors, electromechanical transducers, actuators and filters.

  19. A Jacobi–Legendre polynomial-based method for the stable solution of a deconvolution problem of the Abel integral equation type

    International Nuclear Information System (INIS)

    Ammari, Amara; Karoui, Abderrazek

    2012-01-01

    In this paper, we build a stable scheme for the solution of a deconvolution problem of the Abel integral equation type. This scheme is obtained by further developing the orthogonal polynomial-based techniques for solving the Abel integral equation of Ammari and Karoui (2010 Inverse Problems 26 105005). More precisely, this method is based on the simultaneous use of the two families of orthogonal polynomials of the Legendre and Jacobi types. In particular, we provide an explicit formula for the computation of the Legendre expansion coefficients of the solution. This explicit formula is based on some known formulae for the exact computation of the integrals of the product of some Jacobi polynomials with the derivatives of the Legendre polynomials. Besides the explicit and the exact computation of the expansion coefficients of the solution, our proposed method has the advantage of ensuring the stability of the solution under a fairly weak condition on the functional space to which the data function belongs. Finally, we provide the reader with some numerical examples that illustrate the results of this work. (paper)

  20. Genetic evaluation and selection response for growth in meat-type quail through random regression models using B-spline functions and Legendre polynomials.

    Science.gov (United States)

    Mota, L F M; Martins, P G M A; Littiere, T O; Abreu, L R A; Silva, M A; Bonafé, C M

    2018-04-01

    The objective was to estimate (co)variance functions using random regression models (RRM) with Legendre polynomials, B-spline function and multi-trait models aimed at evaluating genetic parameters of growth traits in meat-type quail. A database containing the complete pedigree information of 7000 meat-type quail was utilized. The models included the fixed effects of contemporary group and generation. Direct additive genetic and permanent environmental effects, considered as random, were modeled using B-spline functions considering quadratic and cubic polynomials for each individual segment, and Legendre polynomials for age. Residual variances were grouped in four age classes. Direct additive genetic and permanent environmental effects were modeled using 2 to 4 segments and were modeled by Legendre polynomial with orders of fit ranging from 2 to 4. The model with quadratic B-spline adjustment, using four segments for direct additive genetic and permanent environmental effects, was the most appropriate and parsimonious to describe the covariance structure of the data. The RRM using Legendre polynomials presented an underestimation of the residual variance. Lesser heritability estimates were observed for multi-trait models in comparison with RRM for the evaluated ages. In general, the genetic correlations between measures of BW from hatching to 35 days of age decreased as the range between the evaluated ages increased. Genetic trend for BW was positive and significant along the selection generations. The genetic response to selection for BW in the evaluated ages presented greater values for RRM compared with multi-trait models. In summary, RRM using B-spline functions with four residual variance classes and segments were the best fit for genetic evaluation of growth traits in meat-type quail. In conclusion, RRM should be considered in genetic evaluation of breeding programs.

  1. Random Regression Models Using Legendre Polynomials to Estimate Genetic Parameters for Test-day Milk Protein Yields in Iranian Holstein Dairy Cattle.

    Science.gov (United States)

    Naserkheil, Masoumeh; Miraie-Ashtiani, Seyed Reza; Nejati-Javaremi, Ardeshir; Son, Jihyun; Lee, Deukhwan

    2016-12-01

    The objective of this study was to estimate the genetic parameters of milk protein yields in Iranian Holstein dairy cattle. A total of 1,112,082 test-day milk protein yield records of 167,269 first lactation Holstein cows, calved from 1990 to 2010, were analyzed. Estimates of the variance components, heritability, and genetic correlations for milk protein yields were obtained using a random regression test-day model. Milking times, herd, age of recording, year, and month of recording were included as fixed effects in the model. Additive genetic and permanent environmental random effects for the lactation curve were taken into account by applying orthogonal Legendre polynomials of the fourth order in the model. The lowest and highest additive genetic variances were estimated at the beginning and end of lactation, respectively. Permanent environmental variance was higher at both extremes. Residual variance was lowest at the middle of the lactation and contrarily, heritability increased during this period. Maximum heritability was found during the 12th lactation stage (0.213±0.007). Genetic, permanent, and phenotypic correlations among test-days decreased as the interval between consecutive test-days increased. A relatively large data set was used in this study; therefore, the estimated (co)variance components for random regression coefficients could be used for national genetic evaluation of dairy cattle in Iran.

  2. Genetic analysis of milk production traits of Tunisian Holsteins using random regression test-day model with Legendre polynomials.

    Science.gov (United States)

    Ben Zaabza, Hafedh; Ben Gara, Abderrahmen; Rekik, Boulbaba

    2017-08-16

    The objective of this study was to estimate genetic parameters of milk, fat, and protein yields within and across lactations in Tunisian Holsteins using a random regression test-day model. A random regression multiple trait multiple lactation test-day (TD) model was used to estimate genetic parameters in the Tunisian dairy cattle population. Data were TD yields of milk, fat, and protein from the first three lactations. Random regressions were modeled with third-order Legendre polynomials for the additive genetic, and permanent environment effects. Heritabilities, and genetic correlations were estimated by Bayesian techniques using the Gibbs sampler. All variance components tended to be high in the beginning and the end of lactations. Additive genetic variances for milk, fat, and protein yields were the lowest and were the least variable compared to permanent variances. Heritability values tended to increase with parity. Estimates of heritabilities for 305-d yield-traits were low to moderate, 0.14 to 0.2, 0.12 to 0.17, and 0.13 to 0.18 for milk, fat, and protein yields, respectively. Within-parity, genetic correlations among traits were up to 0.74. Genetic correlations among lactations for the yield traits were relatively high and ranged from 0.78 0.01 to 0.82 0.03, between the first and second parities, from 0.73 0.03 to 0.8 0.04 between the first and third parities, and from 0.82 0.02 to 0.84 0.04 between the second and third parities. These results are comparable to previously reported estimates on the same population, indicating that the adoption of a random regression TD model as the official genetic evaluation for production traits in Tunisia, as developed by most Interbull countries, is possible in the Tunisian Holsteins.

  3. Parallel Fast Legendre Transform

    NARCIS (Netherlands)

    Alves de Inda, M.; Bisseling, R.H.; Maslen, D.K.

    1998-01-01

    We discuss a parallel implementation of a fast algorithm for the discrete polynomial Legendre transform We give an introduction to the DriscollHealy algorithm using polynomial arithmetic and present experimental results on the eciency and accuracy of our implementation The algorithms were

  4. Study of the influence of semiconductor material parameters on acoustic wave propagation modes in GaSb/AlSb bi-layered structures by Legendre polynomial method

    Science.gov (United States)

    Othmani, Cherif; Takali, Farid; Njeh, Anouar; Ben Ghozlen, Mohamed Hédi

    2016-09-01

    The propagation of Rayleigh-Lamb waves in bi-layered structures is studied. For this purpose, an extension of the Legendre polynomial (LP) method is proposed to formulate the acoustic wave equation in the bi-layered structures induced by thin film Gallium Antimonide (GaSb) and with Aluminum Antimonide (AlSb) substrate in moderate thickness. Acoustic modes propagating along a bi-layer plate are shown to be quite different than classical Lamb modes, contrary to most of the multilayered structures. The validation of the LP method is illustrated by a comparison between the associated numerical results and those obtained using the ordinary differential equation (ODE) method. The convergency of the LP method is discussed through a numerical example. Moreover, the influences of thin film GaSb parameters on the characteristics Rayleigh-Lamb waves propagation has been studied in detail. Finally, the advantages of the Legendre polynomial (LP) method to analyze the multilayered structures are described. All the developments performed in this work were implemented in Matlab software.

  5. Study of the influence of semiconductor material parameters on acoustic wave propagation modes in GaSb/AlSb bi-layered structures by Legendre polynomial method

    Energy Technology Data Exchange (ETDEWEB)

    Othmani, Cherif, E-mail: othmanicheriffss@gmail.com; Takali, Farid; Njeh, Anouar; Ben Ghozlen, Mohamed Hédi

    2016-09-01

    The propagation of Rayleigh–Lamb waves in bi-layered structures is studied. For this purpose, an extension of the Legendre polynomial (LP) method is proposed to formulate the acoustic wave equation in the bi-layered structures induced by thin film Gallium Antimonide (GaSb) and with Aluminum Antimonide (AlSb) substrate in moderate thickness. Acoustic modes propagating along a bi-layer plate are shown to be quite different than classical Lamb modes, contrary to most of the multilayered structures. The validation of the LP method is illustrated by a comparison between the associated numerical results and those obtained using the ordinary differential equation (ODE) method. The convergency of the LP method is discussed through a numerical example. Moreover, the influences of thin film GaSb parameters on the characteristics Rayleigh–Lamb waves propagation has been studied in detail. Finally, the advantages of the Legendre polynomial (LP) method to analyze the multilayered structures are described. All the developments performed in this work were implemented in Matlab software.

  6. An Algorithm for the Convolution of Legendre Series

    KAUST Repository

    Hale, Nicholas

    2014-01-01

    An O(N2) algorithm for the convolution of compactly supported Legendre series is described. The algorithm is derived from the convolution theorem for Legendre polynomials and the recurrence relation satisfied by spherical Bessel functions. Combining with previous work yields an O(N 2) algorithm for the convolution of Chebyshev series. Numerical results are presented to demonstrate the improved efficiency over the existing algorithm. © 2014 Society for Industrial and Applied Mathematics.

  7. A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula

    KAUST Repository

    Hale, Nicholas

    2014-02-06

    A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(log N)2/ log log N) operations is derived. The fundamental idea of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the discrete cosine transform. Numerical results are provided to demonstrate the efficiency and numerical stability. Since the algorithm evaluates a Legendre expansion at an N +1 Chebyshev grid as an intermediate step, it also provides a fast transform between Legendre coefficients and values on a Chebyshev grid. © 2014 Society for Industrial and Applied Mathematics.

  8. Delannoy numbers and Legendre polytopes

    OpenAIRE

    Hetyei, Gábor

    2008-01-01

    International audience; We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre ...

  9. Generation of leading coefficients of orthogonal polynomials with an application to anisotropic scattering of neutrons

    International Nuclear Information System (INIS)

    Ofek, R.

    1984-01-01

    A simple method to generate leading coefficients for high-order sets of orthogonal polynomials, by derivation of recurrence expression for these coefficients, is developed. The method is applied to Legendre, Hermite, Chebyshev and Laguerre polynomials. The method may be used in calculations of high anisotropic neutron-scattering transfer cross-sections, where the angular distribution of the scattered neutrons is given in the ENDF/B files for most materials as coefficients of an expansion into Legendre polynomials. (author)

  10. Application of a composition of generating functions for obtaining explicit formulas of polynomials

    OpenAIRE

    Kruchinin, Vladimir; Kruchinin, Dmitry

    2012-01-01

    Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials, Bernoulli Polynomials of the Second Kind, Generalized Bernoulli polynomials, Euler Polynomials, Peters polynomials, Narumi polynomials, Humbert polynomials, Lerch polynomials and Mahler polynomials.

  11. Polynomials

    Science.gov (United States)

    Apostol, Tom M. (Editor)

    1991-01-01

    In this 'Project Mathematics! series, sponsored by California Institute for Technology (CalTech), the mathematical concept of polynomials in rectangular coordinate (x, y) systems are explored. sing film footage of real life applications and computer animation sequences, the history of, the application of, and the different linear coordinate systems for quadratic, cubic, intersecting, and higher degree of polynomials are discussed.

  12. Fitting of two and three variant polynomials from experimental data through the least squares method. (Using of the codes AJUS-2D, AJUS-3D and LEGENDRE-2D); Ajuste de polinomios en dos y tres variables independientes por el metodo de minimos cuadrados. (Desarrollo de los codigos AJUS-2D, AJUS-3D y LEGENDRE-2D)

    Energy Technology Data Exchange (ETDEWEB)

    Sanchez Miro, J. J.; Sanz Martin, J. C.

    1994-07-01

    Obtaining polynomial fittings from observational data in two and three dimensions is an interesting and practical task. Such an arduous problem suggests the development of an automatic code. The main novelty we provide lies in the generalization of the classical least squares method in three FORTRAN 77 programs usable in any sampling problem. Furthermore, we introduce the orthogonal 2D-Legendre function in the fitting process. These FORTRAN 77 programs are equipped with the options to calculate the approximation quality standard indicators, obviously generalized to two and three dimensions (correlation nonlinear factor, confidence intervals, cuadratic mean error, and so on). The aim of this paper is to rectify the absence of fitting algorithms for more than one independent variable in mathematical libraries. (Author) 10 refs.

  13. Fitting of two and three variant polynomials from experimental data through the least squares method. (Using of the codes AJUS-2D, AJUS-3D and LEGENDRE-2D)

    International Nuclear Information System (INIS)

    Sanchez Miro, J. J.; Sanz Martin, J. C.

    1994-01-01

    Obtaining polynomial fittings from observational data in two and three dimensions is an interesting and practical task. Such an arduous problem suggests the development of an automatic code. The main novelty we provide lies in the generalization of the classical least squares method in three FORTRAN 77 programs usable in any sampling problem. Furthermore, we introduce the orthogonal 2D-Legendre function in the fitting process. These FORTRAN 77 programs are equipped with the options to calculate the approximation quality standard indicators, obviously generalized to two and three dimensions (correlation nonlinear factor, confidence intervals, cuadratic mean error, and so on). The aim of this paper is to rectify the absence of fitting algorithms for more than one independent variable in mathematical libraries. (Author) 10 refs

  14. Applying polynomial filtering to mass preconditioned Hybrid Monte Carlo

    Science.gov (United States)

    Haar, Taylor; Kamleh, Waseem; Zanotti, James; Nakamura, Yoshifumi

    2017-06-01

    The use of mass preconditioning or Hasenbusch filtering in modern Hybrid Monte Carlo simulations is common. At light quark masses, multiple filters (three or more) are typically used to reduce the cost of generating dynamical gauge fields; however, the task of tuning a large number of Hasenbusch mass terms is non-trivial. The use of short polynomial approximations to the inverse has been shown to provide an effective UV filter for HMC simulations. In this work we investigate the application of polynomial filtering to the mass preconditioned Hybrid Monte Carlo algorithm as a means of introducing many time scales into the molecular dynamics integration with a simplified parameter tuning process. A generalized multi-scale integration scheme that permits arbitrary step-sizes and can be applied to Omelyan-style integrators is also introduced. We find that polynomial-filtered mass-preconditioning (PF-MP) performs as well as or better than standard mass preconditioning, with significantly less fine tuning required.

  15. On the efficient parallel computation of Legendre transforms

    NARCIS (Netherlands)

    Inda, M.A.; Bisseling, R.H.; Maslen, D.K.

    2001-01-01

    In this article, we discuss a parallel implementation of efficient algorithms for computation of Legendre polynomial transforms and other orthogonal polynomial transforms. We develop an approach to the Driscoll-Healy algorithm using polynomial arithmetic and present experimental results on the

  16. On the efficient parallel computation of Legendre transforms

    NARCIS (Netherlands)

    Inda, M.A.; Bisseling, R.H.; Maslen, D.K.

    1999-01-01

    In this article we discuss a parallel implementation of efficient algorithms for computation of Legendre polynomial transforms and other orthogonal polynomial transforms. We develop an approach to the Driscoll-Healy algorithm using polynomial arithmetic and present experimental results on the

  17. Orthogonal polynomials

    CERN Document Server

    Szegő, G

    1939-01-01

    The general theory of orthogonal polynomials was developed in the late 19th century from a study of continued fractions by P. L. Chebyshev, even though special cases were introduced earlier by Legendre, Hermite, Jacobi, Laguerre, and Chebyshev himself. It was further developed by A. A. Markov, T. J. Stieltjes, and many other mathematicians. The book by Szegő, originally published in 1939, is the first monograph devoted to the theory of orthogonal polynomials and its applications in many areas, including analysis, differential equations, probability and mathematical physics. Even after all the

  18. Further Insight and Additional Inference Methods for Polynomial Regression Applied to the Analysis of Congruence

    Science.gov (United States)

    Cohen, Ayala; Nahum-Shani, Inbal; Doveh, Etti

    2010-01-01

    In their seminal paper, Edwards and Parry (1993) presented the polynomial regression as a better alternative to applying difference score in the study of congruence. Although this method is increasingly applied in congruence research, its complexity relative to other methods for assessing congruence (e.g., difference score methods) was one of the…

  19. Method of applying single higher order polynomial basis function over multiple domains

    CSIR Research Space (South Africa)

    Lysko, AA

    2010-03-01

    Full Text Available M) with higher-order polynomial basis functions, and applied to a surface form of the electrical field integral equation, under thin wire approximation. The main advantage of the proposed method is in permitting to reduce the required number of unknowns when...

  20. Method of applying single higher order polynomial basis function over multiple domains

    CSIR Research Space (South Africa)

    Lysko, AA

    2010-03-01

    Full Text Available A novel method has been devised where one set of higher order polynomial-based basis functions can be applied over several wire segments, thus permitting to decouple the number of unknowns from the number of segments, and so from the geometrical...

  1. Coherent orthogonal polynomials

    International Nuclear Information System (INIS)

    Celeghini, E.; Olmo, M.A. del

    2013-01-01

    We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis (|x〉), for an alternative countable basis (|n〉). The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L 2 functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L 2 and, in particular, generalized coherent polynomials are thus obtained. -- Highlights: •Fundamental characteristic of orthogonal polynomials (OP): existence of a Lie algebra. •Differential recurrence relations of OP determine a unitary representation of a non-compact Lie group. •2nd order Casimir originates a 2nd order differential equation that defines the

  2. Genetic parameters for test day milk yield of first lactation Holstein cows estimated by random regression using Legendre polynomials Parâmetros genéticos para a produção de leite do dia do controle da primeira lactação de vacas da raça Holandesa estimadas por regressão aleatória com polinômios de Legendre

    Directory of Open Access Journals (Sweden)

    Claudio Napolis Costa

    2008-04-01

    Full Text Available Data comprising 263,390 test-day (TD records of 32,448 first parity cows calving in 467 herds between 1991 and 2001 from the Brazilian Holstein Association were used to estimate genetic and permanent environmental variance components in a random regression animal model using Legendre polynomials (LP of order three to five by REML. Residual variance was assumed to be constant in all or in some classes of lactation periods for each LP. Estimates of genetic and permanent environmental variances did not show any trend due to the increase in the LP order. Residual variance decreased as the order of LP increased when it was assumed constant, and it was highest at the beginning of lactation and relatively constant in mid lactation when assumed to vary between classes. The range for the estimates of heritability (0.27 - 0.42 was similar for all models and was higher in mid lactation. There were only slight differences between the models in both genetic and permanent environmental correlations. Genetic correlations decreased for near unity between adjacent days to values as low as 0.24 between early and late lactation. A five parameter LP to model both genetic and permanent environmental effects and assuming a homogeneous residual variance would be a parsimonious option to fit TD yields of Holstein cows in Brazil.Um total de 263.390 registros de produção de leite do dia do controle (PC de 32.448 primeiras lactações de vacas da raça Holandesa com partos entre 1991 e 2001, disponibilizados pela Associação Brasileira de Criadores de Bovinos da Raça Holandesa, foi usado para estimar componentes de variância para os efeitos genético e de ambiente permanente com modelos de regressão aleatória usando polinômios de Legendre (PL de ordens 3 a 5 por REML. A variância residual foi assumida como constante em todo ou em algumas classes do período de lactação para cada PL. As estimativas dos efeitos genético e permanente de ambiente não apresentaram

  3. Quadratic Lagrangians and Legendre transformation

    International Nuclear Information System (INIS)

    Magnano, G.

    1988-01-01

    In recent years interest is grown about the so-called non-linear Lagrangians for gravitation. In particular, the quadratic lagrangians are currently believed to play a fundamental role both for quantum gravity and for the super-gravity approach. The higher order and high degree of non-linearity of these theories make very difficult to extract physical information out of them. The author discusses how the Legendre transformation can be applied to a wide class of non-linear theories: it corresponds to a conformal transformation whenever the Lagrangian depends only on the scalar curvature, while it has a more general form if the Lagrangian depends on the full Ricci tensor

  4. Solution of the neutron point kinetics equations with temperature feedback effects applying the polynomial approach method

    Energy Technology Data Exchange (ETDEWEB)

    Tumelero, Fernanda, E-mail: fernanda.tumelero@yahoo.com.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Petersen, Claudio Z.; Goncalves, Glenio A.; Lazzari, Luana, E-mail: claudiopeteren@yahoo.com.br, E-mail: gleniogoncalves@yahoo.com.br, E-mail: luana-lazzari@hotmail.com [Universidade Federal de Pelotas (DME/UFPEL), Capao do Leao, RS (Brazil). Instituto de Fisica e Matematica

    2015-07-01

    In this work, we present a solution of the Neutron Point Kinetics Equations with temperature feedback effects applying the Polynomial Approach Method. For the solution, we consider one and six groups of delayed neutrons precursors with temperature feedback effects and constant reactivity. The main idea is to expand the neutron density, delayed neutron precursors and temperature as a power series considering the reactivity as an arbitrary function of the time in a relatively short time interval around an ordinary point. In the first interval one applies the initial conditions of the problem and the analytical continuation is used to determine the solutions of the next intervals. With the application of the Polynomial Approximation Method it is possible to overcome the stiffness problem of the equations. In such a way, one varies the time step size of the Polynomial Approach Method and performs an analysis about the precision and computational time. Moreover, we compare the method with different types of approaches (linear, quadratic and cubic) of the power series. The answer of neutron density and temperature obtained by numerical simulations with linear approximation are compared with results in the literature. (author)

  5. Chudnovsky-Ramanujan Type Formulae for the Legendre Family

    OpenAIRE

    Chen, Imin; Glebov, Gleb

    2017-01-01

    We apply the method established in our previous work to derive a Chudnovsky-Ramanujan type formula for the Legendre family of elliptic curves. As a result, we prove two identities for $1/\\pi$ in terms of hypergeometric functions.

  6. Applying Semigroup Property of Enhanced Chebyshev Polynomials to Anonymous Authentication Protocol

    Directory of Open Access Journals (Sweden)

    Hong Lai

    2012-01-01

    Full Text Available We apply semigroup property of enhanced Chebyshev polynomials to present an anonymous authentication protocol. This paper aims at improving security and reducing computational and storage overhead. The proposed scheme not only has much lower computational complexity and cost in the initialization phase but also allows the users to choose their passwords freely. Moreover, it can provide revocation of lost or stolen smart card, which can resist man-in-the-middle attack and off-line dictionary attack together with various known attacks.

  7. Bell Polynomials Approach Applied to (2 + 1-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

    Directory of Open Access Journals (Sweden)

    Wen-guang Cheng

    2014-01-01

    Full Text Available The bilinear form, bilinear Bäcklund transformation, and Lax pair of a (2 + 1-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived through Bell polynomials. The integrable constraint conditions on variable coefficients can be naturally obtained in the procedure of applying the Bell polynomials approach. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota bilinear method. Finally, the infinite conservation laws of this equation are obtained by decoupling binary Bell polynomials. All conserved densities and fluxes are illustrated with explicit recursion formulae.

  8. Optimal polynomial theory applied to 0--350 MeV pp scattering

    International Nuclear Information System (INIS)

    Bergervoet, J.R.; van Campen, P.C.; Rijken, T.A.; de Swart, J.J.

    1988-01-01

    The optimal polynomial theory of Cutkosky, Deo, and Ciulli has been tested for its use in a multienergy phase-shift analysis of pp scattering data below T/sub lab/ = 350 MeV. The power of the optimal polynomial theory to predict higher partial wave phase parameters is investigated also for a realistic potential model; the Nijmegen potential. It is seen that the optimal polynomial theory has indeed predictive power whenever the phase parameters do not decrease too rapidly as a function of the orbital angular momentum. For a high-quality phase-shift analysis, the optimal polynomial theory does not predict F and G waves well enough. Therefore, these have to be parametrized. The predictive power of the optimal polynomial theory is then only used for higher partial waves. It appears that the Nijmegen potential tail contains valuable physical information beyond the optimal polynomial theory

  9. The Gibbs Phenomenon for Series of Orthogonal Polynomials

    Science.gov (United States)

    Fay, T. H.; Kloppers, P. Hendrik

    2006-01-01

    This note considers the four classes of orthogonal polynomials--Chebyshev, Hermite, Laguerre, Legendre--and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same…

  10. Comparative assessment of orthogonal polynomials for wavefront reconstruction over the square aperture.

    Science.gov (United States)

    Ye, Jingfei; Gao, Zhishan; Wang, Shuai; Cheng, Jinlong; Wang, Wei; Sun, Wenqing

    2014-10-01

    Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials, 2D Legendre polynomials, Zernike square polynomials and Numerical polynomials. They are all orthogonal over the full unit square domain. 2D Chebyshev polynomials are defined by the product of Chebyshev polynomials in x and y variables, as are 2D Legendre polynomials. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness. Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.

  11. Legendre's and Kummer's Theorems Again

    Indian Academy of Sciences (India)

    mathematical education and mathematical contests. Dorel Mihet». Some results related to Legendre's Theorem and ... mentioned theorems in problem solving. We will see that many olympiad-type problems as: `If f(m) denotes the greatest k such that 2k divides m, prove that there are infinite many numbers m such that ...

  12. Legendre's and Kummer's Theorems Again

    Indian Academy of Sciences (India)

    RESONANCE December 2010. GENERAL ARTICLE. 1 The formula commonly called Legendre's formula, appears in the second edition of 'Essai sur la la théorie des nombres' [1]. However, it may have been discovered in- dependently by various persons. For example, in [3] this formula is named De Polignac's ...

  13. Orthogonal polynomials in transport theories

    International Nuclear Information System (INIS)

    Dehesa, J.S.

    1981-01-01

    The asymptotical (k→infinity) behaviour of zeros of the polynomials gsub(k)sup((m)(ν)) encountered in the treatment of direct and inverse problems of scattering in neutron transport as well as radiative transfer theories is investigated in terms of the amplitude antiwsub(k) of the kth Legendre polynomial needed in the expansion of the scattering function. The parameters antiwsub(k) describe the anisotropy of scattering of the medium considered. In particular, it is shown that the asymptotical density of zeros of the polynomials gsub(k)sup(m)(ν) is an inverted semicircle for the anisotropic non-multiplying scattering medium

  14. Polynomial Similarity Transformation Theory: A smooth interpolation between coupled cluster doubles and projected BCS applied to the reduced BCS Hamiltonian

    Energy Technology Data Exchange (ETDEWEB)

    Degroote, M. [Rice Univ., Houston, TX (United States); Henderson, T. M. [Rice Univ., Houston, TX (United States); Zhao, J. [Rice Univ., Houston, TX (United States); Dukelsky, J. [Consejo Superior de Investigaciones Cientificas (CSIC), Madrid (Spain). Inst. de Estructura de la Materia; Scuseria, G. E. [Rice Univ., Houston, TX (United States)

    2018-01-03

    We present a similarity transformation theory based on a polynomial form of a particle-hole pair excitation operator. In the weakly correlated limit, this polynomial becomes an exponential, leading to coupled cluster doubles. In the opposite strongly correlated limit, the polynomial becomes an extended Bessel expansion and yields the projected BCS wavefunction. In between, we interpolate using a single parameter. The e ective Hamiltonian is non-hermitian and this Polynomial Similarity Transformation Theory follows the philosophy of traditional coupled cluster, left projecting the transformed Hamiltonian onto subspaces of the Hilbert space in which the wave function variance is forced to be zero. Similarly, the interpolation parameter is obtained through minimizing the next residual in the projective hierarchy. We rationalize and demonstrate how and why coupled cluster doubles is ill suited to the strongly correlated limit whereas the Bessel expansion remains well behaved. The model provides accurate wave functions with energy errors that in its best variant are smaller than 1% across all interaction stengths. The numerical cost is polynomial in system size and the theory can be straightforwardly applied to any realistic Hamiltonian.

  15. Legendre polynomial modeling for vibrations of guided Lamb waves modes in [001]c, [011]c and [111]c polarized (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (x = 0.29 and 0.33) piezoelectric plates: Physical phenomenon of multiple intertwining of An and Sn modes

    Science.gov (United States)

    Othmani, Cherif; Takali, Farid; Njeh, Anouar

    2017-12-01

    Guided wave devices have recently become one of the most important applications in the industry because such waves are directly related to applications in sensor technology, chemical sensing, agricultural science, fields of bio-sensing and surface acoustic wave (SAW) devices that are used in electronic filters and signal processing. On that account, this numerical investigation aims to study the propagation behavior of guided Lamb waves in a (1-x)Pb(Mg1/3Nb2/3)O3- x PbTiO3 [PMN- x PT] ( x=0.29 or 0.33) piezoelectric single crystal plate. In fact, the PMN- xPT ( x=0.29 or 0.33) piezoelectric crystals are being polarized along [001]c, [011]c and [111]c of the cubic reference directions so that the macroscopic symmetries are tetragonal 4 mm, orthogonal mm2 and rhombohedral 3 m, respectively. Both open- and short-circuit conditions are considered. Here, the Legendre polynomial method is proposed to solve the guided Lamb waves equations. The validity of the proposed method is illustrated by comparison with the ordinary differential equation (ODE). The convergence of this method is discussed. Consequently, the converged results are obtained with very low truncation order M . This constitutes a major advantage of the present method when compared with the other matrix methods. There is cross-crossings among multiple modes for both symmetric ( Sn) and the anti-symmetric ( An) guided Lamb waves propagation. A displacement field has been illustrated to judge whether Sn and An modes cross with each other. Moreover, electric displacement, stress field and electric potential for the open-circuit case were presented for both S0 and A0 Lamb modes.

  16. On the derivative of the Legendre function of the first kind with respect to its degree

    International Nuclear Information System (INIS)

    Szmytkowski, Radoslaw

    2006-01-01

    We study the derivative of the Legendre function of the first kind, P ν (z), with respect to its degree ν. At first, we provide two contour integral representations for ∂P ν (z)/∂ν. Then, we proceed to investigate the case of [∂P ν (z)/∂ν] ν=n , with n being an integer; this case is met in some physical and engineering problems. Since it holds that [∂P ν' (z)/∂ν'] ν'==ν-1 -[∂P ν' (z0/∂ν'] ν'=ν , we focus on the sub-case of n being a non-negative integer. We show that ∂P ν (z)/∂ν vertical bar ν=n = P n (z) ln((z+1)/2) + R n (z) (n element of N) where R n (z) is a polynomial in z of degree n. We present alternative derivations of several known explicit expressions for R n (z) and also add some new. A generating function for R n (z) is also constructed. Properties of the polynomials V n (z) = [R n (z) + (-1) n R n (-z)]/2 and W n-1 (z) = -[R n (z) - (-1) n R n (-z)]/2 are also investigated. It is found that W n-1 (z) is the Christoffel polynomial, well known from the theory of the Legendre function of the second kind, Q n (z). As examples of applications of the results obtained, we present non-standard derivations of some representations of Q n (z), sum to closed forms some Legendre series, evaluate some definite integrals involving Legendre polynomials and also derive an explicit representation of the indefinite integral of the Legendre polynomial squared

  17. Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method

    Science.gov (United States)

    Sohaib, Muhammad; Haq, Sirajul; Mukhtar, Safyan; Khan, Imad

    2018-03-01

    An efficient method is proposed to approximate sixth order boundary value problems. The proposed method is based on Legendre wavelet in which Legendre polynomial is used. The mechanism of the method is to use collocation points that converts the differential equation into a system of algebraic equations. For validation two test problems are discussed. The results obtained from proposed method are quite accurate, also close to exact solution, and other different methods. The proposed method is computationally more effective and leads to more accurate results as compared to other methods from literature.

  18. Some further results on Legendre numbers

    Directory of Open Access Journals (Sweden)

    Paul W. Haggard

    1988-01-01

    Full Text Available The Legendre numbers, Pnm, are expressed in terms of those numbers, Pkm−1, in the previous column down to Pnm and in terms of those, Pkm, above but in the same column. Other results are given for numbers close to a given number. The limit of the quotient of two consecutive non-zero numbers in any one column is shown to be −1. Bounds for the Legendre numbers are described by circles centered at the origin. A connection between Legendre numbers and Pascal numbers is exhibited by expressing the Legendre numbers in terms of combinations.

  19. Numerical Performances of Two Orthogonal Polynomials in the Tau ...

    African Journals Online (AJOL)

    In this work, efforts are made at comparing the numerical effectiveness of the two most accurate orthogonal polynomials; Chebyshev and Legendre polynomials. Although the two have different weight functions, but they most time give close results especially when they are considered in the same interval. This work has ...

  20. Lagrange polynomial interpolation method applied in the calculation of the J({xi},{beta}) function

    Energy Technology Data Exchange (ETDEWEB)

    Fraga, Vinicius Munhoz; Palma, Daniel Artur Pinheiro [Centro Federal de Educacao Tecnologica de Quimica de Nilopolis, RJ (Brazil)]. E-mails: munhoz.vf@gmail.com; dpalma@cefeteq.br; Martinez, Aquilino Senra [Universidade Federal do Rio de Janeiro (UFRJ), RJ (Brazil). Coordenacao dos Programas de Pos-graduacao de Engenharia (COPPE) (COPPE). Programa de Engenharia Nuclear]. E-mail: aquilino@lmp.ufrj.br

    2008-07-01

    The explicit dependence of the Doppler broadening function creates difficulties in the obtaining an analytical expression for J function . The objective of this paper is to present a method for the quick and accurate calculation of J function based on the recent advances in the calculation of the Doppler broadening function and on a systematic analysis of its integrand. The methodology proposed, of a semi-analytical nature, uses the Lagrange polynomial interpolation method and the Frobenius formulation in the calculation of Doppler broadening function . The results have proven satisfactory from the standpoint of accuracy and processing time. (author)

  1. Lagrange polynomial interpolation method applied in the calculation of the J(ξ,β) function

    International Nuclear Information System (INIS)

    Fraga, Vinicius Munhoz; Palma, Daniel Artur Pinheiro; Martinez, Aquilino Senra

    2008-01-01

    The explicit dependence of the Doppler broadening function creates difficulties in the obtaining an analytical expression for J function . The objective of this paper is to present a method for the quick and accurate calculation of J function based on the recent advances in the calculation of the Doppler broadening function and on a systematic analysis of its integrand. The methodology proposed, of a semi-analytical nature, uses the Lagrange polynomial interpolation method and the Frobenius formulation in the calculation of Doppler broadening function . The results have proven satisfactory from the standpoint of accuracy and processing time. (author)

  2. ROBUST REGULATION FOR SYSTEMS WITH POLYNOMIAL NONLINEARITY APPLIED TO RAPID THERMAL PROCESSES

    Directory of Open Access Journals (Sweden)

    S. V. Aranovskiy

    2014-07-01

    Full Text Available Abstract. A problem of output robust control for a system with power nonlinearity is considered. The considered problem can be rewritten as a stabilization problem for a system with polynomial nonlinearity by introducing the error term. The problem of temperature regulation is considered as application; the rapid thermal processes in vapor deposition processing are studied. Modern industrial equipment uses complex sensors and control systems; these devices are not available for laboratory setups. The limited amount of available sensors and other technical restrictions for laboratory setups make it an actual problem to design simple low-order output control laws. The problem is solved by the consecutive compensator approach. The paper deals with a new type of restriction which is a combination of linear and power restrictions. It is shown that the polynomial nonlinearity satisfies this restriction. Asymptotical stability of the closed-loop system is proved by the Lyapunov functions approach for the considered nonlinear function; this contribution extends previously known results. Numerical simulation of the vapor deposition processing illustrates that the proposed approach results in zero-mean tracking error with standard deviation less than 1K.

  3. Nonclassicality Generated by Applying Hermite-Polynomials Photon-Added Operator on the Even/Odd Coherent States

    Science.gov (United States)

    Ren, Gang; Du, Jian-ming; Zhang, Wen-hai; Yu, Hai-jun

    2017-05-01

    We examine nonclassical properties of the quantum state generated by applying Hermite polynomials photon-added operator on the even/odd coherent state (HPECS/HPOCS). Explicit expressions for its nonclassical properties, such as quantum statistical properties and squeezing phenomenon, are obtained. It is interesting to find that the HPECS/HPOCS exhibits sub-Poissonian distribution, anti-bunching effects and negative values of the Wigner function. Thus, we confirm the HPPECS/HPPOCS is a new nonclassical state. Finally, we reveal that the HPPECS/HPPOCS is a novel intelligent state by its squeezing effects in position distribution and quadrature squeezing.

  4. Infinitesimal Legendre symmetry in the Geometrothermodynamics programme

    Energy Technology Data Exchange (ETDEWEB)

    García-Peláez, D., E-mail: dgarciap@up.edu.mx [Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.P. 70-543, 04510 México D.F., México (Mexico); Universidad Panamericana, Tecoyotitla 366. Col. Ex Hacienda Guadalupe Chimalistac, 01050 México D.F., México (Mexico); López-Monsalvo, C. S., E-mail: cesar.slm@correo.nucleares.unam.mx [Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.P. 70-543, 04510 México D.F., México (Mexico)

    2014-08-15

    The work within the Geometrothermodynamics programme rests upon the metric structure for the thermodynamic phase-space. Such structure exhibits discrete Legendre symmetry. In this work, we study the class of metrics which are invariant along the infinitesimal generators of Legendre transformations. We solve the Legendre-Killing equation for a K-contact general metric. We consider the case with two thermodynamic degrees of freedom, i.e., when the dimension of the thermodynamic phase-space is five. For the generic form of contact metrics, the solution of the Legendre-Killing system is unique, with the sole restriction that the only independent metric function – Ω – should be dragged along the orbits of the Legendre generator. We revisit the ideal gas in the light of this class of metrics. Imposing the vanishing of the scalar curvature for this system results in a further differential equation for the metric function Ω which is not compatible with the Legendre invariance constraint. This result does not allow us to use Quevedo's interpretation of the curvature scalar as a measure of thermodynamic interaction for this particular class.

  5. Improved polynomial remainder sequences for Ore polynomials.

    Science.gov (United States)

    Jaroschek, Maximilian

    2013-11-01

    Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.

  6. Polynomials to model the growth of young bulls in performance tests.

    Science.gov (United States)

    Scalez, D C B; Fragomeni, B O; Passafaro, T L; Pereira, I G; Toral, F L B

    2014-03-01

    The use of polynomial functions to describe the average growth trajectory and covariance functions of Nellore and MA (21/32 Charolais+11/32 Nellore) young bulls in performance tests was studied. The average growth trajectories and additive genetic and permanent environmental covariance functions were fit with Legendre (linear through quintic) and quadratic B-spline (with two to four intervals) polynomials. In general, the Legendre and quadratic B-spline models that included more covariance parameters provided a better fit with the data. When comparing models with the same number of parameters, the quadratic B-spline provided a better fit than the Legendre polynomials. The quadratic B-spline with four intervals provided the best fit for the Nellore and MA groups. The fitting of random regression models with different types of polynomials (Legendre polynomials or B-spline) affected neither the genetic parameters estimates nor the ranking of the Nellore young bulls. However, fitting different type of polynomials affected the genetic parameters estimates and the ranking of the MA young bulls. Parsimonious Legendre or quadratic B-spline models could be used for genetic evaluation of body weight of Nellore young bulls in performance tests, whereas these parsimonious models were less efficient for animals of the MA genetic group owing to limited data at the extreme ages.

  7. Reduction of the number of parameters needed for a polynomial random regression test-day model

    NARCIS (Netherlands)

    Pool, M.H.; Meuwissen, T.H.E.

    2000-01-01

    Legendre polynomials were used to describe the (co)variance matrix within a random regression test day model. The goodness of fit depended on the polynomial order of fit, i.e., number of parameters to be estimated per animal but is limited by computing capacity. Two aspects: incomplete lactation

  8. Legendre Elliptic Curves over Finite Fields

    NARCIS (Netherlands)

    Auer, Roland; Top, Jakob

    2002-01-01

    We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the

  9. Discrete fractional solutions of a Legendre equation

    Science.gov (United States)

    Yılmazer, Resat

    2018-01-01

    One of the most popular research interests of science and engineering is the fractional calculus theory in recent times. Discrete fractional calculus has also an important position in fractional calculus. In this work, we acquire new discrete fractional solutions of the homogeneous and non homogeneous Legendre differential equation by using discrete fractional nabla operator.

  10. Nonnegativity of uncertain polynomials

    Directory of Open Access Journals (Sweden)

    Šiljak Dragoslav D.

    1998-01-01

    Full Text Available The purpose of this paper is to derive tests for robust nonnegativity of scalar and matrix polynomials, which are algebraic, recursive, and can be completed in finite number of steps. Polytopic families of polynomials are considered with various characterizations of parameter uncertainty including affine, multilinear, and polynomic structures. The zero exclusion condition for polynomial positivity is also proposed for general parameter dependencies. By reformulating the robust stability problem of complex polynomials as positivity of real polynomials, we obtain new sufficient conditions for robust stability involving multilinear structures, which can be tested using only real arithmetic. The obtained results are applied to robust matrix factorization, strict positive realness, and absolute stability of multivariable systems involving parameter dependent transfer function matrices.

  11. Chebyshev polynomials

    CERN Document Server

    Mason, JC

    2002-01-01

    Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods. Yet no book dedicated to Chebyshev polynomials has been published since 1990, and even that work focused primarily on the theoretical aspects. A broad, up-to-date treatment is long overdue.Providing highly readable exposition on the subject''s state of the art, Chebyshev Polynomials is just such a treatment. It includes rigorous yet down-to-earth coverage of the theory along with an in-depth look at the properties of all four kinds of Chebyshev polynomials-properties that lead to a range of results in areas such as approximation, series expansions, interpolation, quadrature, and integral equations. Problems in each chapter, ranging in difficulty from elementary to quite advanced, reinforce the concepts and methods presented.Far from being an esoteric subject, Chebysh...

  12. On equiconvergence of ultraspherical polynomials solution of one-speed neutron transport equation

    International Nuclear Information System (INIS)

    Yilmazer, Ayhan; Tombakoglu, Mehmet

    2006-01-01

    Ultraspherical polynomial approximation is used in slab criticality calculations for strongly anisotropic scattering. A unique and general formulation is developed for slab criticality condition whose sub-cases are spherical harmonics approximation, Chebyshev polynomial approximation of first and second kind. Since Legendre polynomials, Chebyshev Polynomials of first and second kinds are special cases of ultraspherical polynomials; our formulation inherently covers all these approximations and lets one to employ any other ultraspherical polynomial approximation in the solution of one-speed neutron transport equation. Our calculations showed that solution of one-speed neutron transport equation for various degrees of anisotropy and cross-section parameters is almost insensitive to the choice of ultraspherical polynomial with the present days' computing capabilities. In other words, as much as high order ultraspherical polynomial approximation is used the solution converges to the same value for a specified problem regardless the type of the ultraspherical polynomial assigned in the solution as equiconvergence theorem of Jacobi polynomials states

  13. Shifted Legendre method with residual error estimation for delay linear Fredholm integro-differential equations

    Directory of Open Access Journals (Sweden)

    Şuayip Yüzbaşı

    2017-03-01

    Full Text Available In this paper, we suggest a matrix method for obtaining the approximate solutions of the delay linear Fredholm integro-differential equations with constant coefficients using the shifted Legendre polynomials. The problem is considered with mixed conditions. Using the required matrix operations, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. Additionally, error analysis for the method is presented using the residual function. Illustrative examples are given to demonstrate the efficiency of the method. The results obtained in this study are compared with the known results.

  14. Polynomial Asymptotes

    Science.gov (United States)

    Dobbs, David E.

    2010-01-01

    This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic asymptotes. Prerequisites include the division algorithm for polynomials with coefficients in the field of…

  15. Blurred image recognition by legendre moment invariants

    Science.gov (United States)

    Zhang, Hui; Shu, Huazhong; Han, Guo-Niu; Coatrieux, Gouenou; Luo, Limin; Coatrieux, Jean-Louis

    2010-01-01

    Processing blurred images is a key problem in many image applications. Existing methods to obtain blur invariants which are invariant with respect to centrally symmetric blur are based on geometric moments or complex moments. In this paper, we propose a new method to construct a set of blur invariants using the orthogonal Legendre moments. Some important properties of Legendre moments for the blurred image are presented and proved. The performance of the proposed descriptors is evaluated with various point-spread functions and different image noises. The comparison of the present approach with previous methods in terms of pattern recognition accuracy is also provided. The experimental results show that the proposed descriptors are more robust to noise and have better discriminative power than the methods based on geometric or complex moments. PMID:19933003

  16. Generalizations of orthogonal polynomials

    Science.gov (United States)

    Bultheel, A.; Cuyt, A.; van Assche, W.; van Barel, M.; Verdonk, B.

    2005-07-01

    We give a survey of recent generalizations of orthogonal polynomials. That includes multidimensional (matrix and vector orthogonal polynomials) and multivariate versions, multipole (orthogonal rational functions) variants, and extensions of the orthogonality conditions (multiple orthogonality). Most of these generalizations are inspired by the applications in which they are applied. We also give a glimpse of these applications, which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role: moment problems, numerical quadrature, rational approximation, linear algebra, recurrence relations, and random matrices.

  17. Legendre transformations and Clairaut-type equations

    Energy Technology Data Exchange (ETDEWEB)

    Lavrov, Peter M., E-mail: lavrov@tspu.edu.ru [Tomsk State Pedagogical University, Kievskaya St. 60, 634061 Tomsk (Russian Federation); National Research Tomsk State University, Lenin Av. 36, 634050 Tomsk (Russian Federation); Merzlikin, Boris S., E-mail: merzlikin@tspu.edu.ru [National Research Tomsk Polytechnic University, Lenin Av. 30, 634050 Tomsk (Russian Federation)

    2016-05-10

    It is noted that the Legendre transformations in the standard formulation of quantum field theory have the form of functional Clairaut-type equations. It is shown that in presence of composite fields the Clairaut-type form holds after loop corrections are taken into account. A new solution to the functional Clairaut-type equation appearing in field theories with composite fields is found.

  18. The Trigonometric Polynomial Like Bernstein Polynomial

    Science.gov (United States)

    2014-01-01

    A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given. PMID:25276845

  19. The Trigonometric Polynomial Like Bernstein Polynomial

    Directory of Open Access Journals (Sweden)

    Xuli Han

    2014-01-01

    Full Text Available A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given.

  20. Modified Legendre Wavelets Technique for Fractional Oscillation Equations

    Directory of Open Access Journals (Sweden)

    Syed Tauseef Mohyud-Din

    2015-10-01

    Full Text Available Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques.

  1. On computation and use of Fourier coefficients for associated Legendre functions

    Science.gov (United States)

    Gruber, Christian; Abrykosov, Oleh

    2016-06-01

    The computation of spherical harmonic series in very high resolution is known to be delicate in terms of performance and numerical stability. A major problem is to keep results inside a numerical range of the used data type during calculations as under-/overflow arises. Extended data types are currently not desirable since the arithmetic complexity will grow exponentially with higher resolution levels. If the associated Legendre functions are computed in the spectral domain, then regular grid transformations can be applied to be highly efficient and convenient for derived quantities as well. In this article, we compare three recursive computations of the associated Legendre functions as trigonometric series, thereby ensuring a defined numerical range for each constituent wave number, separately. The results to a high degree and order show the numerical strength of the proposed method. First, the evaluation of Fourier coefficients of the associated Legendre functions has been done with respect to the floating-point precision requirements. Secondly, the numerical accuracy in the cases of standard double and long double precision arithmetic is demonstrated. Following Bessel's inequality the obtained accuracy estimates of the Fourier coefficients are directly transferable to the associated Legendre functions themselves and to derived functionals as well. Therefore, they can provide an essential insight to modern geodetic applications that depend on efficient spherical harmonic analysis and synthesis beyond [5~× ~5] arcmin resolution.

  2. Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils.

    Science.gov (United States)

    Mahajan, Virendra N

    2012-06-20

    In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, L(l)(x)L(m)(y), where l and m are positive integers (including zero) and L(l)(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial L(l)(x)L(m)(y), there is a corresponding orthonormal polynomial L(l)(y)L(m)(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as

  3. On the Atkin Polynomials

    OpenAIRE

    El-Guindy, Ahmad; Ismail, Mourad E. H.

    2013-01-01

    We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials. This shows that co-recursive polynomials may lead to interesting sets of orthogonal polynomials.

  4. Discrete-Time Filter Synthesis using Product of Gegenbauer Polynomials

    Directory of Open Access Journals (Sweden)

    N. Stojanovic

    2016-09-01

    Full Text Available A new approximation to design continuoustime and discrete-time low-pass filters, presented in this paper, based on the product of Gegenbauer polynomials, provides the ability of more flexible adjustment of passband and stopband responses. The design is achieved taking into account a prescribed specification, leading to a better trade-off among the magnitude and group delay responses. Many well-known continuous-time and discrete-time transitional filter based on the classical polynomial approximations(Chebyshev, Legendre, Butterworth are shown to be a special cases of proposed approximation method.

  5. Irreducible multivariate polynomials obtained from polynomials in ...

    Indian Academy of Sciences (India)

    over K(X)? We provided some methods to construct irreducible multivariate polynomials over an arbi- trary field, starting from arbitrary irreducible polynomials in fewer variables, of which we mention the following two results: Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ∈ K[X] ...

  6. Irreducible multivariate polynomials obtained from polynomials in ...

    Indian Academy of Sciences (India)

    We provided some methods to construct irreducible multivariate polynomials over an arbi- trary field, starting from arbitrary irreducible polynomials in fewer variables, of which we mention the following two results: Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ∈ K[X] with deg a0 ...

  7. Fast and Accurate Computation of Gauss--Legendre and Gauss--Jacobi Quadrature Nodes and Weights

    KAUST Repository

    Hale, Nicholas

    2013-03-06

    An efficient algorithm for the accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton\\'s root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The n-point quadrature rule is computed in O(n) operations to an accuracy of essentially double precision for any n ≥ 100. © 2013 Society for Industrial and Applied Mathematics.

  8. Composite Gauss-Legendre Quadrature with Error Control

    Science.gov (United States)

    Prentice, J. S. C.

    2011-01-01

    We describe composite Gauss-Legendre quadrature for determining definite integrals, including a means of controlling the approximation error. We compare the form and performance of the algorithm with standard Newton-Cotes quadrature. (Contains 1 table.)

  9. Physically elastic analysis of a cylindrical ring as a unit cell of a complete composite under applied stress in the complex plane using cubic polynomials

    Science.gov (United States)

    Monfared, Vahid

    2018-03-01

    Elastic analysis is analytically presented to predict the behaviors of the stress and displacement components in the cylindrical ring as a unit cell of a complete composite under applied stress in the complex plane using cubic polynomials. This analysis is based on the complex computation of the stress functions in the complex plane and polar coordinates. Also, suitable boundary conditions are considered and assumed to analyze along with the equilibrium equations and bi-harmonic equation. This method has some important applications in many fields of engineering such as mechanical, civil and material engineering generally. One of the applications of this research work is in composite design and designing the cylindrical devices under various loadings. Finally, it is founded that the convergence and accuracy of the results are suitable and acceptable through comparing the results.

  10. Bound-preserving Legendre-WENO finite volume schemes using nonlinear mapping

    Science.gov (United States)

    Smith, Timothy; Pantano, Carlos

    2017-11-01

    We present a new method to enforce field bounds in high-order Legendre-WENO finite volume schemes. The strategy consists of reconstructing each field through an intermediate mapping, which by design satisfies realizability constraints. Determination of the coefficients of the polynomial reconstruction involves nonlinear equations that are solved using Newton's method. The selection between the original or mapped reconstruction is implemented dynamically to minimize computational cost. The method has also been generalized to fields that exhibit interdependencies, requiring multi-dimensional mappings. Further, the method does not depend on the existence of a numerical flux function. We will discuss details of the proposed scheme and show results for systems in conservation and non-conservation form. This work was funded by the NSF under Grant DMS 1318161.

  11. Investigation of snow single scattering properties based on first order Legendre phase function

    Science.gov (United States)

    Eppanapelli, Lavan Kumar; Casselgren, Johan; Wåhlin, Johan; Sjödahl, Mikael

    2017-04-01

    Angularly resolved bidirectional reflectance measurements were modelled by approximating a first order Legendre expanded phase function to retrieve single scattering properties of snow. The measurements from 10 different snow types with known density and specific surface area (SSA) were investigated. A near infrared (NIR) spectrometer was used to measure reflected light above the snow surface over the hemisphere in the wavelength region of 900-1650 nm. A solver based on discrete ordinate radiative transfer (DISORT) model was used to retrieve the estimated Legendre coefficients of the phase function and a correlation between the coefficients and physical properties of different snow types is investigated. Results of this study suggest that the first two coefficients of the first order Legendre phase function provide sufficient information about the physical properties of snow where the latter captures the anisotropic behaviour of snow and the former provides a relative estimate of the single scattering albedo of snow. The coefficients of the first order phase function were compared with the experimental data and observed that both the coefficients are in good agreement with the experimental data. These findings suggest that our approach can be applied as a qualitative tool to investigate physical properties of snow and also to classify different snow types.

  12. Criterion for polynomial solutions to a class of linear differential equations of second order

    International Nuclear Information System (INIS)

    Saad, Nasser; Hall, Richard L; Ciftci, Hakan

    2006-01-01

    We consider the differential equations y-prime = λ 0 (x)y' + s 0 (x)y, where λ 0 (x), s 0 (x) are C ∞ -functions. We prove (i) if the differential equation has a polynomial solution of degree n > 0, then δ n = λ n s n-1 - λ n-1 s n = 0, where λ n λ' n-1 + s n-1 + λ 0 λ n-1 ands n = s' n-1 + s 0 λ k-1 , n = 1, 2, .... Conversely (ii) if λ n λ n-1 ≠ 0 and δ n = 0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first and second kinds), Gegenbauer and the Hypergeometric type, etc obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations

  13. A new Arnoldi approach for polynomial eigenproblems

    Energy Technology Data Exchange (ETDEWEB)

    Raeven, F.A.

    1996-12-31

    In this paper we introduce a new generalization of the method of Arnoldi for matrix polynomials. The new approach is compared with the approach of rewriting the polynomial problem into a linear eigenproblem and applying the standard method of Arnoldi to the linearised problem. The algorithm that can be applied directly to the polynomial eigenproblem turns out to be more efficient, both in storage and in computation.

  14. Polynomial Regressions and Nonsense Inference

    Directory of Open Access Journals (Sweden)

    Daniel Ventosa-Santaulària

    2013-11-01

    Full Text Available Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis and psychology, just to mention a few examples. In many cases, the data employed to estimate such specifications are time series that may exhibit stochastic nonstationary behavior. We extend Phillips’ results (Phillips, P. Understanding spurious regressions in econometrics. J. Econom. 1986, 33, 311–340. by proving that an inference drawn from polynomial specifications, under stochastic nonstationarity, is misleading unless the variables cointegrate. We use a generalized polynomial specification as a vehicle to study its asymptotic and finite-sample properties. Our results, therefore, lead to a call to be cautious whenever practitioners estimate polynomial regressions.

  15. Recurrence Relations for Orthogonal Polynomials on Triangular Domains

    Directory of Open Access Journals (Sweden)

    Abedallah Rababah

    2016-04-01

    Full Text Available In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w : u , v , w ≥ 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation.

  16. Higher-Order Hierarchical Legendre Basis Functions in Applications

    DEFF Research Database (Denmark)

    Kim, Oleksiy S.; Jørgensen, Erik; Meincke, Peter

    2007-01-01

    degree of orthogonality. The basis functions are well-suited for solution of complex electromagnetic problems involving multiple homogeneous or inhomogeneous dielectric regions, metallic surfaces, layered media, etc. This paper presents real-life complex antenna radiation problems modeled...... with electromagnetic simulation tools based on the higher-order hierarchical Legendre basis functions....

  17. Composite Gauss-Legendre Formulas for Solving Fuzzy Integration

    Directory of Open Access Journals (Sweden)

    Xiaobin Guo

    2014-01-01

    Full Text Available Two numerical integration rules based on composition of Gauss-Legendre formulas for solving integration of fuzzy numbers-valued functions are investigated in this paper. The methods' constructions are presented and the corresponding convergence theorems are shown in detail. Two numerical examples are given to illustrate the proposed algorithms finally.

  18. On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations

    Science.gov (United States)

    Cima, A.; Gasull, A.; Mañosas, F.

    2017-12-01

    In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.

  19. Method of moments solution of volume integral equations using higher-order hierarchical Legendre basis functions

    DEFF Research Database (Denmark)

    Kim, Oleksiy S.; Jørgensen, Erik; Meincke, Peter

    2004-01-01

    An efficient higher-order method of moments (MoM) solution of volume integral equations is presented. The higher-order MoM solution is based on higher-order hierarchical Legendre basis functions and higher-order geometry modeling. An unstructured mesh composed of 8-node trilinear and/or curved 27...... of magnitude in comparison to existing higher-order hierarchical basis functions. Consequently, an iterative solver can be applied even for high expansion orders. Numerical results demonstrate excellent agreement with the analytical Mie series solution for a dielectric sphere as well as with results obtained...

  20. Computing derivative-based global sensitivity measures using polynomial chaos expansions

    International Nuclear Information System (INIS)

    Sudret, B.; Mai, C.V.

    2015-01-01

    In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol' indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The derivative-based global sensitivity measures (DGSMs) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. Closed-form expressions for Hermite, Legendre and Laguerre polynomial expansions are given. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis. - Highlights: • Derivative-based global sensitivity measures (DGSM) have been developed for screening purpose. • Polynomial chaos expansions (PC) are used as a surrogate model of the original computational model. • From a PC expansion the DGSM can be computed analytically. • The paper provides the derivatives of Hermite, Legendre and Laguerre polynomials for this purpose

  1. A Polynomial Term Structure Model with Macroeconomic Variables

    Directory of Open Access Journals (Sweden)

    José Valentim Vicente

    2007-06-01

    Full Text Available Recently, a myriad of factor models including macroeconomic variables have been proposed to analyze the yield curve. We present an alternative factor model where term structure movements are captured by Legendre polynomials mimicking the statistical factor movements identified by Litterman e Scheinkmam (1991. We estimate the model with Brazilian Foreign Exchange Coupon data, adopting a Kalman filter, under two versions: the first uses only latent factors and the second includes macroeconomic variables. We study its ability to predict out-of-sample term structure movements, when compared to a random walk. We also discuss results on the impulse response function of macroeconomic variables.

  2. Block orthogonal polynomials: II. Hermite and Laguerre standard block orthogonal polynomials

    International Nuclear Information System (INIS)

    Normand, Jean-Marie

    2007-01-01

    The standard block orthogonal (SBO) polynomials P i;n (x), 0 ≤ i ≤ n are real polynomials of degree n which are orthogonal with respect to a first Euclidean scalar product to polynomials of degree less than i. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. Applying the general results obtained in a previous paper, we determine and investigate these polynomials when the first scalar product corresponds to Hermite (resp. Laguerre) polynomials. These new sets of polynomials, which we call Hermite (resp. Laguerre) SBO polynomials, provide a basis of functional spaces well suited for some applications requiring to take into account special linear constraints which can be recast into an Euclidean orthogonality relation

  3. Chaos, Fractals, and Polynomials.

    Science.gov (United States)

    Tylee, J. Louis; Tylee, Thomas B.

    1996-01-01

    Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)

  4. Branched polynomial covering maps

    DEFF Research Database (Denmark)

    Hansen, Vagn Lundsgaard

    2002-01-01

    A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...... set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere. (C) 2001 Elsevier Science B.V. All rights reserved....

  5. Branched polynomial covering maps

    DEFF Research Database (Denmark)

    Hansen, Vagn Lundsgaard

    1999-01-01

    A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...... set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere....

  6. Branched polynomial covering maps

    DEFF Research Database (Denmark)

    Hansen, Vagn Lundsgaard

    2002-01-01

    A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch ...

  7. Relations between zeros of special polynomials associated with the Painleve equations

    International Nuclear Information System (INIS)

    Kudryashov, Nikolai A.; Demina, Maria V.

    2007-01-01

    A method for finding relations of roots of polynomials is presented. Our approach allows us to get a number of relations between the zeros of the classical polynomials as well as the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. These are: the Hermite polynomials, the Laguerre polynomials, the Yablonskii-Vorob'ev polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as analogues of generalized Stieltjes relations

  8. Polynomial Regressions and Nonsense Inference

    DEFF Research Database (Denmark)

    Ventosa-Santaulària, Daniel; Rodríguez-Caballero, Carlos Vladimir

    Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis, and psychology, just to mention a few examples. In many cases, the data employed to estimate such estimations are time series that may exhibit stochastic nonstationary ...

  9. On Parameter Differentiation for Integral Representations of Associated Legendre Functions

    Directory of Open Access Journals (Sweden)

    Howard S. Cohl

    2011-05-01

    Full Text Available For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for associated Legendre functions of the first and second kind with respect to the degree are evaluated at odd-half-integer degrees, for general complex-orders, and derivatives with respect to the order are evaluated at integer-orders, for general complex-degrees. We also discuss the properties of the complex function f: C{−1,1}→C given by f(z=z/((z+1^{1/2}(z−1^{1/2}.

  10. Lapped Block Image Analysis via the Method of Legendre Moments

    Directory of Open Access Journals (Sweden)

    El Fadili Hakim

    2003-01-01

    Full Text Available Research investigating the use of Legendre moments for pattern recognition has been performed in recent years. This field of research remains quite open. This paper proposes a new technique based on block-based reconstruction method (BBRM using Legendre moments compared with the global reconstruction method (GRM. For alleviating the blocking artifact involved in the processing, we propose a new approach using lapped block-based reconstruction method (LBBRM. For the problem of selecting the optimal number of moment used to represent a given image, we propose the maximum entropy principle (MEP method. The main motivation of the proposed approaches is to allow fast and efficient reconstruction algorithm, with improvement of the reconstructed images quality. A binary handwritten musical character and multi-gray-level Lena image are used to demonstrate the performance of our algorithm.

  11. Location of zeros of Wiener and distance polynomials.

    Science.gov (United States)

    Dehmer, Matthias; Ilić, Aleksandar

    2012-01-01

    The geometry of polynomials explores geometrical relationships between the zeros and the coefficients of a polynomial. A classical problem in this theory is to locate the zeros of a given polynomial by determining disks in the complex plane in which all its zeros are situated. In this paper, we infer bounds for general polynomials and apply classical and new results to graph polynomials namely Wiener and distance polynomials whose zeros have not been yet investigated. Also, we examine the quality of such bounds by considering four graph classes and interpret the results.

  12. N-Level Quantum Systems and Legendre Functions

    OpenAIRE

    Mazurenko, A. S.; Savva, V. A.

    2001-01-01

    An excitation dynamics of new quantum systems of N equidistant energy levels in a monochromatic field has been investigated. To obtain exact analytical solutions of dynamic equations an analytical method based on orthogonal functions of a real argument has been proposed. Using the orthogonal Legendre functions we have found an exact analytical expression for a population probability amplitude of the level n. Various initial conditions for the excitation of N-level quantum systems have been co...

  13. Study of a method to solve the one speed, three dimensional transport equation using the finite element method and the associated Legendre function

    International Nuclear Information System (INIS)

    Fernandes, A.

    1991-01-01

    A method to solve three dimensional neutron transport equation and it is based on the original work suggested by J.K. Fletcher (42, 43). The angular dependence of the flux is approximated by associated Legendre functions and the finite element method is applied to the space components is presented. When the angular flux, the scattering cross section and the neutrons source are expanded in associated Legendre functions, the first order neutron transport equation is reduced to a coupled set of second order diffusion like equations. These equations are solved in an iterative way by the finite element method to the moments. (author)

  14. Poly-Frobenius-Euler polynomials

    Science.gov (United States)

    Kurt, Burak

    2017-07-01

    Hamahata [3] defined poly-Euler polynomials and the generalized poly-Euler polynomials. He proved some relations and closed formulas for the poly-Euler polynomials. By this motivation, we define poly-Frobenius-Euler polynomials. We give some relations for this polynomials. Also, we prove the relationships between poly-Frobenius-Euler polynomials and Stirling numbers of the second kind.

  15. Tutte polynomial in functional magnetic resonance imaging

    Science.gov (United States)

    García-Castillón, Marlly V.

    2015-09-01

    Methods of graph theory are applied to the processing of functional magnetic resonance images. Specifically the Tutte polynomial is used to analyze such kind of images. Functional Magnetic Resonance Imaging provide us connectivity networks in the brain which are represented by graphs and the Tutte polynomial will be applied. The problem of computing the Tutte polynomial for a given graph is #P-hard even for planar graphs. For a practical application the maple packages "GraphTheory" and "SpecialGraphs" will be used. We will consider certain diagram which is depicting functional connectivity, specifically between frontal and posterior areas, in autism during an inferential text comprehension task. The Tutte polynomial for the resulting neural networks will be computed and some numerical invariants for such network will be obtained. Our results show that the Tutte polynomial is a powerful tool to analyze and characterize the networks obtained from functional magnetic resonance imaging.

  16. A model-based 3D phase unwrapping algorithm using Gegenbauer polynomials.

    Science.gov (United States)

    Langley, Jason; Zhao, Qun

    2009-09-07

    The application of a two-dimensional (2D) phase unwrapping algorithm to a three-dimensional (3D) phase map may result in an unwrapped phase map that is discontinuous in the direction normal to the unwrapped plane. This work investigates the problem of phase unwrapping for 3D phase maps. The phase map is modeled as a product of three one-dimensional Gegenbauer polynomials. The orthogonality of Gegenbauer polynomials and their derivatives on the interval [-1, 1] are exploited to calculate the expansion coefficients. The algorithm was implemented using two well-known Gegenbauer polynomials: Chebyshev polynomials of the first kind and Legendre polynomials. Both implementations of the phase unwrapping algorithm were tested on 3D datasets acquired from a magnetic resonance imaging (MRI) scanner. The first dataset was acquired from a homogeneous spherical phantom. The second dataset was acquired using the same spherical phantom but magnetic field inhomogeneities were introduced by an external coil placed adjacent to the phantom, which provided an additional burden to the phase unwrapping algorithm. Then Gaussian noise was added to generate a low signal-to-noise ratio dataset. The third dataset was acquired from the brain of a human volunteer. The results showed that Chebyshev implementation and the Legendre implementation of the phase unwrapping algorithm give similar results on the 3D datasets. Both implementations of the phase unwrapping algorithm compare well to PRELUDE 3D, 3D phase unwrapping software well recognized for functional MRI.

  17. Recurrence relations for orthogonal polynomials for PDEs in polar and cylindrical geometries.

    Science.gov (United States)

    Richardson, Megan; Lambers, James V

    2016-01-01

    This paper introduces two families of orthogonal polynomials on the interval (-1,1), with weight function [Formula: see text]. The first family satisfies the boundary condition [Formula: see text], and the second one satisfies the boundary conditions [Formula: see text]. These boundary conditions arise naturally from PDEs defined on a disk with Dirichlet boundary conditions and the requirement of regularity in Cartesian coordinates. The families of orthogonal polynomials are obtained by orthogonalizing short linear combinations of Legendre polynomials that satisfy the same boundary conditions. Then, the three-term recurrence relations are derived. Finally, it is shown that from these recurrence relations, one can efficiently compute the corresponding recurrences for generalized Jacobi polynomials that satisfy the same boundary conditions.

  18. Orthogonal polynomials from Hermitian matrices. II

    Science.gov (United States)

    Odake, Satoru; Sasaki, Ryu

    2018-01-01

    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 ℓ2 Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrödinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of the 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 ℓ2 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.

  19. Arabic text classification using Polynomial Networks

    Directory of Open Access Journals (Sweden)

    Mayy M. Al-Tahrawi

    2015-10-01

    Full Text Available In this paper, an Arabic statistical learning-based text classification system has been developed using Polynomial Neural Networks. Polynomial Networks have been recently applied to English text classification, but they were never used for Arabic text classification. In this research, we investigate the performance of Polynomial Networks in classifying Arabic texts. Experiments are conducted on a widely used Arabic dataset in text classification: Al-Jazeera News dataset. We chose this dataset to enable direct comparisons of the performance of Polynomial Networks classifier versus other well-known classifiers on this dataset in the literature of Arabic text classification. Results of experiments show that Polynomial Networks classifier is a competitive algorithm to the state-of-the-art ones in the field of Arabic text classification.

  20. Improved polynomial remainder sequences for Ore polynomials☆

    Science.gov (United States)

    Jaroschek, Maximilian

    2013-01-01

    Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients. PMID:26523087

  1. Incomplete q-Chebyshev Polynomials

    OpenAIRE

    Ercan, Elif; Firengiz, Mirac Cetin; Tuglu, Naim

    2016-01-01

    In this paper, we get the generating functions of q-Chebyshev polynomials using operator. Also considering explicit formulas of q-Chebyshev polynomials, we give new generalizations of q-Chebyshev polynomials called incomplete q-Chebyshev polynomials of the first and second kind. We obtain recurrence relations and several properties of these polynomials. We show that there are connections between incomplete q-Chebyshev polynomials and the some well-known polynomials. Keywords: q-Chebyshev poly...

  2. AMDLIBAE, IBM 360 Subroutine Library, Special Function, Polynomials, Differential Equation

    International Nuclear Information System (INIS)

    Wang, Jesse Y.

    1980-01-01

    Description of problem or function: AMDLIBAE is a subset of the IBM 360 Subroutine Library at the Applied Mathematics Division at Argonne National Laboratory. This subset includes library categories A-E: Identification/Description: A152S A MPA: Mult. prec. floating point arith. package; B156S A ARSIN: Arcsine, arccosine; B158S A DSIN/DCOS: DP sine, cosine; B159S A DTAN/DCOT: DP tangent, cotangent; B252S A SINH/COSH: Hyperbolic sine, cosine; B353S A ALOG: SP logarithm; B354S A DEXP: DP exponential; B355S A DLOG: DP logarithm; B456S A DCUBRT: DP cube root; B457S A ARGPOWER: X Y ; B458S A ARGFDXPD: DP X Y ; C150S F DQD: Q. D. algorithm applied to a power series; C151S F DCONF1: Eval. cont. fract. Q. D. of power series; C250S F CUBIC: Roots of cubic polynomial equation; C251S F QUARTIC: Roots of quartic polynomial equation; C252S F RSSR: All roots of poly eqs. with real coef.; C253S F POLDRV: Driver for C254S; C254S F CPOLY: Roots arb. poly. Jenkins-Traub algorithm; C353S F1 CLEBSH: Ang. mom. coef. - Clebsch, Racah, Wigner; C365S A ALGAMA: Logarithm of the gamma function; C366S A DGAMMA/DLGAMA: DP gamma and log(gamma) functions; C368S F EONE: Exponential integral E1; C370S F BESJY: Bessel functions J and Y; C371S F BESIK: Bessel functions I and K; C372S F CHIPRB: Chi-square integral; C380S F DRZETA: Long precision zeta, zeta-1 functions; C382S F DCGAM: Long precision complex gamma; C383S A DERF/DERFC: DP error function; C384S F BFJ1: Bessel function J1; C385S F COULMB: Regular Coulomb wave functions; C386S F1 DSGMAL: Coulomb phase shift; C387S F BFJYR: Bessel functions J0,J1,Y0,Y1; C388S F IRCOUL: LP irregular Coulomb wave functions; C389S F GAMIN: Incomplete gamma function; C390S F LQ: Assoc. Legendre functions of 2. kind; C392S A DAERF: Inverse error function; C393S F CDEONE: Modified complex exponential integral; D153S F DROMB: Two-dimensional Romberg quadrature; D153S P DROMBP: Two-dimensional Romberg quadrature; D158S F ANC4: Adap. quad. using 4. order Newton

  3. Certain non-linear differential polynomials sharing a non zero polynomial

    Directory of Open Access Journals (Sweden)

    Majumder Sujoy

    2015-10-01

    functions sharing a nonzero polynomial and obtain two results which improves and generalizes the results due to L. Liu [Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl., 56 (2008, 3236-3245.] and P. Sahoo [Uniqueness and weighted value sharing of meromorphic functions, Applied. Math. E-Notes., 11 (2011, 23-32.].

  4. Polynomial Graphs and Symmetry

    Science.gov (United States)

    Goehle, Geoff; Kobayashi, Mitsuo

    2013-01-01

    Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or…

  5. Weierstrass polynomials for links

    DEFF Research Database (Denmark)

    Hansen, Vagn Lundsgaard

    1997-01-01

    There is a natural way of identifying links in3-space with polynomial covering spaces over thecircle. Thereby any link in 3-space can be definedby a Weierstrass polynomial over the circle. Theequivalence relation for covering spaces over thecircle is, however, completely different from...

  6. Spherical space Bessel-Legendre-Fourier mode solver for Maxwell's wave equations

    Science.gov (United States)

    Alzahrani, Mohammed A.; Gauthier, Robert C.

    2015-02-01

    For spherically symmetric dielectric structures, a basis set composed of Bessel, Legendre and Fourier functions, BLF, are used to cast Maxwell's wave equations into an eigenvalue problem from which the localized modes can be determined. The steps leading to the eigenmatrix are reviewed and techniques used to reduce the order of matrix and tune the computations for particular mode types are detailed. The BLF basis functions are used to expand the electric and magnetic fields as well as the inverse relative dielectric profile. Similar to the common plane wave expansion technique, the BLF matrix returns the eigen-frequencies and eigenvectors, but in BLF only steady states, non-propagated, are obtained. The technique is first applied to a air filled spherical structure with perfectly conducting outer surface and then to a spherical microsphere located in air. Results are compared published values were possible.

  7. Extended biorthogonal matrix polynomials

    Directory of Open Access Journals (Sweden)

    Ayman Shehata

    2017-01-01

    Full Text Available The pair of biorthogonal matrix polynomials for commutative matrices were first introduced by Varma and Tasdelen in [22]. The main aim of this paper is to extend the properties of the pair of biorthogonal matrix polynomials of Varma and Tasdelen and certain generating matrix functions, finite series, some matrix recurrence relations, several important properties of matrix differential recurrence relations, biorthogonality relations and matrix differential equation for the pair of biorthogonal matrix polynomials J(A,B n (x, k and K(A,B n (x, k are discussed. For the matrix polynomials J(A,B n (x, k, various families of bilinear and bilateral generating matrix functions are constructed in the sequel.

  8. On Symmetric Polynomials

    OpenAIRE

    Golden, Ryan; Cho, Ilwoo

    2015-01-01

    In this paper, we study structure theorems of algebras of symmetric functions. Based on a certain relation on elementary symmetric polynomials generating such algebras, we consider perturbation in the algebras. In particular, we understand generators of the algebras as perturbations. From such perturbations, define injective maps on generators, which induce algebra-monomorphisms (or embeddings) on the algebras. They provide inductive structure theorems on algebras of symmetric polynomials. As...

  9. A new class of generalized polynomials associated with Hermite and Bernoulli polynomials

    Directory of Open Access Journals (Sweden)

    M. A. Pathan

    2015-05-01

    Full Text Available In this paper, we introduce a new class of generalized  polynomials associated with  the modified Milne-Thomson's polynomials Φ_{n}^{(α}(x,ν of degree n and order α introduced by  Derre and Simsek.The concepts of Bernoulli numbers B_n, Bernoulli polynomials  B_n(x, generalized Bernoulli numbers B_n(a,b, generalized Bernoulli polynomials  B_n(x;a,b,c of Luo et al, Hermite-Bernoulli polynomials  {_HB}_n(x,y of Dattoli et al and {_HB}_n^{(α} (x,y of Pathan  are generalized to the one   {_HB}_n^{(α}(x,y,a,b,c which is called  the generalized  polynomial depending on three positive real parameters. Numerous properties of these polynomials and some relationships between B_n, B_n(x, B_n(a,b, B_n(x;a,b,c and {}_HB_n^{(α}(x,y;a,b,c  are established. 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 generalized Bernoulli numbers and polynomials

  10. Darboux polynomials and first integrals of natural polynomial Hamiltonian systems

    International Nuclear Information System (INIS)

    Maciejewski, Andrzej J.; Przybylska, Maria

    2004-01-01

    We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals

  11. Many-body orthogonal polynomial systems

    International Nuclear Information System (INIS)

    Witte, N.S.

    1997-03-01

    The fundamental methods employed in the moment problem, involving orthogonal polynomial systems, the Lanczos algorithm, continued fraction analysis and Pade approximants has been combined with a cumulant approach and applied to the extensive many-body problem in physics. This has yielded many new exact results for many-body systems in the thermodynamic limit - for the ground state energy, for excited state gaps, for arbitrary ground state avenges - and are of a nonperturbative nature. These results flow from a confluence property of the three-term recurrence coefficients arising and define a general class of many-body orthogonal polynomials. These theorems constitute an analytical solution to the Lanczos algorithm in that they are expressed in terms of the three-term recurrence coefficients α and β. These results can also be applied approximately for non-solvable models in the form of an expansion, in a descending series of the system size. The zeroth order order this expansion is just the manifestation of the central limit theorem in which a Gaussian measure and hermite polynomials arise. The first order represents the first non-trivial order, in which classical distribution functions like the binomial distributions arise and the associated class of orthogonal polynomials are Meixner polynomials. Amongst examples of systems which have infinite order in the expansion are q-orthogonal polynomials where q depends on the system size in a particular way. (author)

  12. Context-free grammars for several polynomials associated with Eulerian polynomials

    OpenAIRE

    Ma, Shi-Mei; Ma, Jun; Yeh, Yeong-Nan; Zhu, Bao-Xuan

    2016-01-01

    In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.

  13. Classifying polynomials and identity testing

    Indian Academy of Sciences (India)

    m = nO(log n). [2]. This provides a excellent classification of easy polynomials. Hence, any polynomial that cannot be written as a determinant of a small sized matrix is not easy. A simple counting argument shows that there exist many such polynomials. However, prov- ing an explicitly given polynomial to be hard has.

  14. Chromatic polynomials for simplicial complexes

    DEFF Research Database (Denmark)

    Møller, Jesper Michael; Nord, Gesche

    2016-01-01

    In this note we consider s s -chromatic polynomials for finite simplicial complexes. When s=1 s=1 , the 1 1 -chromatic polynomial is just the usual graph chromatic polynomial of the 1 1 -skeleton. In general, the s s -chromatic polynomial depends on the s s -skeleton and its value at r...

  15. Additive and polynomial representations

    CERN Document Server

    Krantz, David H; Suppes, Patrick

    1971-01-01

    Additive and Polynomial Representations deals with major representation theorems in which the qualitative structure is reflected as some polynomial function of one or more numerical functions defined on the basic entities. Examples are additive expressions of a single measure (such as the probability of disjoint events being the sum of their probabilities), and additive expressions of two measures (such as the logarithm of momentum being the sum of log mass and log velocity terms). The book describes the three basic procedures of fundamental measurement as the mathematical pivot, as the utiliz

  16. Fast methods for resumming matrix polynomials and Chebyshev matrix polynomials

    International Nuclear Information System (INIS)

    Liang Wanzen; Baer, Roi; Saravanan, Chandra; Shao Yihan; Bell, Alexis T.; Head-Gordon, Martin

    2004-01-01

    Fast and effective algorithms are discussed for resumming matrix polynomials and Chebyshev matrix polynomials. These algorithms lead to a significant speed-up in computer time by reducing the number of matrix multiplications required to roughly twice the square root of the degree of the polynomial. A few numerical tests are presented, showing that evaluation of matrix functions via polynomial expansions can be preferable when the matrix is sparse and these fast resummation algorithms are employed

  17. Approximations of orthogonal polynomials in terms of Hermite polynomials

    NARCIS (Netherlands)

    J.L. López; N.M. Temme (Nico)

    1999-01-01

    textabstractSeveral orthogonal polynomials have limit forms in which Hermite polynomials show up. Examples are limits with respect to certain parameters of the Jacobi and Laguerre polynomials. In this paper we are interested in more details of these limits and we give asymptotic representations of

  18. Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

    NARCIS (Netherlands)

    Koornwinder, T.H.; Bouzeffour, F.

    2011-01-01

    Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2 × 2 matrix-valued operator. As a new result made

  19. Cyclotomy and Cyclotomic Polynomials

    Indian Academy of Sciences (India)

    Home; Journals; Resonance – Journal of Science Education; Volume 4; Issue 12. Cyclotomy and Cyclotomic Polynomials - The Story of how Gauss Narrowly Missed Becoming a Philologist. B Sury. General Article Volume 4 Issue 12 December 1999 pp 41-53 ...

  20. Computing the Alexander Polynomial Numerically

    DEFF Research Database (Denmark)

    Hansen, Mikael Sonne

    2006-01-01

    Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically.......Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically....

  1. Interpolation and Polynomial Curve Fitting

    Science.gov (United States)

    Yang, Yajun; Gordon, Sheldon P.

    2014-01-01

    Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…

  2. transformation of independent variables in polynomial regression ...

    African Journals Online (AJOL)

    Ada

    preferable when possible to work with a simple functional form in transformed variables rather than with a more complicated form in the original variables. In this paper, it is shown that linear transformations applied to independent variables in polynomial regression models affect the t ratio and hence the statistical ...

  3. Time-dependent generalized polynomial chaos

    International Nuclear Information System (INIS)

    Gerritsma, Marc; Steen, Jan-Bart van der; Vos, Peter; Karniadakis, George

    2010-01-01

    Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan-Orszag problem with favorable results.

  4. Fourier series in orthogonal polynomials

    CERN Document Server

    Osilenker, Boris

    1999-01-01

    This book presents a systematic course on general orthogonal polynomials and Fourier series in orthogonal polynomials. It consists of six chapters. Chapter 1 deals in essence with standard results from the university course on the function theory of a real variable and on functional analysis. Chapter 2 contains the classical results about the orthogonal polynomials (some properties, classical Jacobi polynomials and the criteria of boundedness).The main subject of the book is Fourier series in general orthogonal polynomials. Chapters 3 and 4 are devoted to some results in this topic (classical

  5. Generalized q-Hermite polynomials

    International Nuclear Information System (INIS)

    Berg, C.; Ruffing, A.

    2001-01-01

    We consider two operators A and A + in a Hilbert space of functions on the exponential lattice {q n ,-q n vertical stroke n element of Z}, where 0 0 and that A and A + are ladder operators for the sequence ψ n =(A + ) n ψ 0 . The sequence (ψ n /ψ 0 ) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. (Orig.) (orig.)

  6. Complex Polynomial Vector Fields

    DEFF Research Database (Denmark)

    vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...... of parameter spaces into structurally stable domains, and a description of the bifurcations. For this reason, the talk will focus on these questions for complex polynomial vector fields.......The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...

  7. General Reducibility and Solvability of Polynomial Equations ...

    African Journals Online (AJOL)

    General Reducibility and Solvability of Polynomial Equations. ... Unlike quadratic, cubic, and quartic polynomials, the general quintic and higher degree polynomials cannot be solved algebraically in terms of finite number of additions, ... Galois Theory, Solving Polynomial Systems, Polynomial factorization, Polynomial Ring ...

  8. A Characterization of Polynomials

    DEFF Research Database (Denmark)

    Andersen, Kurt Munk

    1996-01-01

    Given the problem:which functions f(x) are characterized by a relation of the form:f[x1,x2,...,xn]=h(x1+x2+...+xn), where n>1 and h(x) is a given function? Here f[x1,x2,...,xn] denotes the divided difference on n points x1,x2,...,xn of the function f(x).The answer is: f(x) is a polynomial of degr...

  9. Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

    International Nuclear Information System (INIS)

    Hampton, Jerrad; Doostan, Alireza

    2015-01-01

    Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ 1 -minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy

  10. Comprehensive gravitational modeling of the vertical cylindrical prism by Gauss-Legendre quadrature integration

    Science.gov (United States)

    Asgharzadeh, M. F.; Hashemi, H.; von Frese, R. RB

    2018-01-01

    Forward modeling is the basis of gravitational anomaly inversion that is widely applied to map subsurface mass variations. This study uses numerical least-squares Gauss-Legendre quadrature (GLQ) integration to evaluate the gravitational potential, anomaly and gradient components of the vertical cylindrical prism element. These results, in turn, may be integrated to accurately model the complete gravitational effects of fluid bearing rock formations and other vertical cylinder-like geological bodies with arbitrary variations in shape and density. Comparing the GLQ gravitational effects of uniform density, vertical circular cylinders against the effects calculated by a number of other methods illustrates the veracity of the GLQ modeling method and the accuracy limitations of the other methods. Geological examples include modeling the gravitational effects of a formation washout to help map azimuthal variations of the formation's bulk densities around the borehole wall. As another application, the gravitational effects of a seismically and gravimetrically imaged salt dome within the Laurentian Basin are evaluated for the velocity, density and geometric properties of the Basin's sedimentary formations.

  11. Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

    Science.gov (United States)

    Hoque, Md. Fazlul; Marquette, Ian; Post, Sarah; Zhang, Yao-Zhong

    2018-04-01

    We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schrödinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.

  12. Bernoulli numbers and polynomials from a more general point of view

    International Nuclear Information System (INIS)

    Dattoli, G.; Cesarano, C.; Lorenzutta, S.

    2000-01-01

    In this work it is applied the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials [it

  13. Study of Zernike polynomials of an elliptical aperture obscured with an elliptical obscuration: comment.

    Science.gov (United States)

    Díaz, José Antonio; Mahajan, Virendra N

    2013-08-20

    Recently, Hasan and Shaker published a set of orthonormal polynomials for an annular elliptical pupil obtained by the Gram-Schmidt orthogonalization of the Zernike circle polynomials [Appl. Opt.51, 8490 (2012)]. However, the expressions for many of the polynomials are incorrect, apparently due to wrong usage of the Gram-Schmidt orthogonalization process. We provide the correct equations for the orthogonalization process and the expressions for the orthonormal polynomials obtained by applying them.

  14. Moments, positive polynomials and their applications

    CERN Document Server

    Lasserre, Jean Bernard

    2009-01-01

    Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP) . This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones,

  15. Fuzzy Morphological Polynomial Image Representation

    Directory of Open Access Journals (Sweden)

    Chin-Pan Huang

    2010-01-01

    Full Text Available A novel signal representation using fuzzy mathematical morphology is developed. We take advantage of the optimum fuzzy fitting and the efficient implementation of morphological operators to extract geometric information from signals. The new representation provides results analogous to those given by the polynomial transform. Geometrical decomposition of a signal is achieved by windowing and applying sequentially fuzzy morphological opening with structuring functions. The resulting representation is made to resemble an orthogonal expansion by constraining the results of opening to equate adapted structuring functions. Properties of the geometric decomposition are considered and used to calculate the adaptation parameters. Our procedure provides an efficient and flexible representation which can be efficiently implemented in parallel. The application of the representation is illustrated in data compression and fractal dimension estimation temporal signals and images.

  16. Application of Legendre spectral-collocation method to delay differential and stochastic delay differential equation

    Science.gov (United States)

    Khan, Sami Ullah; Ali, Ishtiaq

    2018-03-01

    Explicit solutions to delay differential equation (DDE) and stochastic delay differential equation (SDDE) can rarely be obtained, therefore numerical methods are adopted to solve these DDE and SDDE. While on the other hand due to unstable nature of both DDE and SDDE numerical solutions are also not straight forward and required more attention. In this study, we derive an efficient numerical scheme for DDE and SDDE based on Legendre spectral-collocation method, which proved to be numerical methods that can significantly speed up the computation. The method transforms the given differential equation into a matrix equation by means of Legendre collocation points which correspond to a system of algebraic equations with unknown Legendre coefficients. The efficiency of the proposed method is confirmed by some numerical examples. We found that our numerical technique has a very good agreement with other methods with less computational effort.

  17. Identification of chaotic memristor systems based on piecewise adaptive Legendre filters

    International Nuclear Information System (INIS)

    Zhao, Yibo; Zhang, Xiuzai; Xu, Jin; Guo, Yecai

    2015-01-01

    Memristor is a nonlinear device, which plays an important role in the design and implementation of chaotic systems. In order to be able to understand in-depth the complex nonlinear dynamic behaviors in chaotic memristor systems, modeling or identification of its nonlinear model is very important premise. This paper presents a chaotic memristor system identification method based on piecewise adaptive Legendre filters. The threshold decomposition is carried out for the input vector, and also the input signal subintervals via decomposition satisfy the convergence condition of the adaptive Legendre filters. Then the adaptive Legendre filter structure and adaptive weight update algorithm are derived. Final computer simulation results show the effectiveness as well as fast convergence characteristics.

  18. Polynomial intelligent states

    Energy Technology Data Exchange (ETDEWEB)

    Milks, Matthew M; Guise, Hubert de [Department of Physics, Lakehead University, Thunder Bay, ON, P7B 5E1 (Canada)

    2005-12-01

    The construction of su(2) intelligent states is simplified using a polynomial representation of su(2). The cornerstone of the new construction is the diagonalization of a 2 x 2 matrix. The method is sufficiently simple to be easily extended to su(3), where one is required to diagonalize a single 3 x 3 matrix. For two perfectly general su(3) operators, this diagonalization is technically possible but the procedure loses much of its simplicity owing to the algebraic form of the roots of a cubic equation. Simplified expressions can be obtained by specializing the choice of su(3) operators. This simpler construction will be discussed in detail.

  19. Primitive polynomials selection method for pseudo-random number generator

    Science.gov (United States)

    Anikin, I. V.; Alnajjar, Kh

    2018-01-01

    In this paper we suggested the method for primitive polynomials selection of special type. This kind of polynomials can be efficiently used as a characteristic polynomials for linear feedback shift registers in pseudo-random number generators. The proposed method consists of two basic steps: finding minimum-cost irreducible polynomials of the desired degree and applying primitivity tests to get the primitive ones. Finally two primitive polynomials, which was found by the proposed method, used in pseudorandom number generator based on fuzzy logic (FRNG) which had been suggested before by the authors. The sequences generated by new version of FRNG have low correlation magnitude, high linear complexity, less power consumption, is more balanced and have better statistical properties.

  20. Roots of the Chromatic Polynomial

    DEFF Research Database (Denmark)

    Perrett, Thomas

    The chromatic polynomial of a graph G is a univariate polynomial whose evaluation at any positive integer q enumerates the proper q-colourings of G. It was introduced in connection with the famous four colour theorem but has recently found other applications in the field of statistical physics....... In this thesis we study the real roots of the chromatic polynomial, termed chromatic roots, and focus on how certain properties of a graph affect the location of its chromatic roots. Firstly, we investigate how the presence of a certain spanning tree in a graph affects its chromatic roots. In particular we prove...... of certain minor-closed families of graphs. Later, we study the Tutte polynomial of a graph, which contains the chromatic polynomial as a specialisation. We discuss a technique of Thomassen using which it is possible to deduce that the roots of the chromatic polynomial are dense in certain intervals. We...

  1. Non polynomial B-splines

    Science.gov (United States)

    Laksâ, Arne

    2015-11-01

    B-splines are the de facto industrial standard for surface modelling in Computer Aided design. It is comparable to bend flexible rods of wood or metal. A flexible rod minimize the energy when bending, a third degree polynomial spline curve minimize the second derivatives. B-spline is a nice way of representing polynomial splines, it connect polynomial splines to corner cutting techniques, which induces many nice and useful properties. However, the B-spline representation can be expanded to something we can call general B-splines, i.e. both polynomial and non-polynomial splines. We will show how this expansion can be done, and the properties it induces, and examples of non-polynomial B-spline.

  2. Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation

    Directory of Open Access Journals (Sweden)

    S. Balaji

    2014-01-01

    Full Text Available A Legendre wavelet operational matrix method (LWM is presented for the solution of nonlinear fractional-order Riccati differential equations, having variety of applications in quantum chemistry and quantum mechanics. The fractional-order Riccati differential equations converted into a system of algebraic equations using Legendre wavelet operational matrix. Solutions given by the proposed scheme are more accurate and reliable and they are compared with recently developed numerical, analytical, and stochastic approaches. Comparison shows that the proposed LWM approach has a greater performance and less computational effort for getting accurate solutions. Further existence and uniqueness of the proposed problem are given and moreover the condition of convergence is verified.

  3. The application of Legendre-tau approximation to parameter identification for delay and partial differential equations

    Science.gov (United States)

    Ito, K.

    1983-01-01

    Approximation schemes based on Legendre-tau approximation are developed for application to parameter identification problem for delay and partial differential equations. The tau method is based on representing the approximate solution as a truncated series of orthonormal functions. The characteristic feature of the Legendre-tau approach is that when the solution to a problem is infinitely differentiable, the rate of convergence is faster than any finite power of 1/N; higher accuracy is thus achieved, making the approach suitable for small N.

  4. A new Fortran 90 program to compute regular and irregular associated Legendre functions (new version announcement)

    Science.gov (United States)

    Schneider, Barry I.; Segura, Javier; Gil, Amparo; Guan, Xiaoxu; Bartschat, Klaus

    2018-04-01

    This is a revised and updated version of a modern Fortran 90 code to compute the regular Plm (x) and irregular Qlm (x) associated Legendre functions for all x ∈(- 1 , + 1) (on the cut) and | x | > 1 and integer degree (l) and order (m). The necessity to revise the code comes as a consequence of some comments of Prof. James Bremer of the UC//Davis Mathematics Department, who discovered that there were errors in the code for large integer degree and order for the normalized regular Legendre functions on the cut.

  5. Global Monte Carlo Simulation with High Order Polynomial Expansions

    International Nuclear Information System (INIS)

    William R. Martin; James Paul Holloway; Kaushik Banerjee; Jesse Cheatham; Jeremy Conlin

    2007-01-01

    The functional expansion technique (FET) was recently developed for Monte Carlo simulation. The basic idea of the FET is to expand a Monte Carlo tally in terms of a high order expansion, the coefficients of which can be estimated via the usual random walk process in a conventional Monte Carlo code. If the expansion basis is chosen carefully, the lowest order coefficient is simply the conventional histogram tally, corresponding to a flat mode. This research project studied the applicability of using the FET to estimate the fission source, from which fission sites can be sampled for the next generation. The idea is that individual fission sites contribute to expansion modes that may span the geometry being considered, possibly increasing the communication across a loosely coupled system and thereby improving convergence over the conventional fission bank approach used in most production Monte Carlo codes. The project examined a number of basis functions, including global Legendre polynomials as well as 'local' piecewise polynomials such as finite element hat functions and higher order versions. The global FET showed an improvement in convergence over the conventional fission bank approach. The local FET methods showed some advantages versus global polynomials in handling geometries with discontinuous material properties. The conventional finite element hat functions had the disadvantage that the expansion coefficients could not be estimated directly but had to be obtained by solving a linear system whose matrix elements were estimated. An alternative fission matrix-based response matrix algorithm was formulated. Studies were made of two alternative applications of the FET, one based on the kernel density estimator and one based on Arnoldi's method of minimized iterations. Preliminary results for both methods indicate improvements in fission source convergence. These developments indicate that the FET has promise for speeding up Monte Carlo fission source convergence

  6. Classifying polynomials and identity testing

    Indian Academy of Sciences (India)

    hard to compute [3,4]! Therefore, the solution to. PIT problem has a key role in our attempt to com- putationally classify polynomials. In this article, we will focus on this connection between PIT and polynomial classification. We now formally define arithmetic circuits and the identity testing problem. 1.1 Problem definition.

  7. Polynomial methods in combinatorics

    CERN Document Server

    Guth, Larry

    2016-01-01

    This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book. Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accompl...

  8. Complex Polynomial Vector Fields

    DEFF Research Database (Denmark)

    Dias, Kealey

    The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... or meromorphic (allowing poles as singularities) functions. There already exists a well-developed theory for iterative holomorphic dynamical systems, and successful relations found between iteration theory and flows of vector fields have been one of the main motivations for the recent interest in holomorphic...... vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...

  9. On the limit from q-Racah polynomials to big q-Jacobi polynomials

    NARCIS (Netherlands)

    Koornwinder, T.H.

    2011-01-01

    A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials.

  10. A stabilized Runge–Kutta–Legendre method for explicit super-time-stepping of parabolic and mixed equations

    International Nuclear Information System (INIS)

    Meyer, Chad D.; Balsara, Dinshaw S.; Aslam, Tariq D.

    2014-01-01

    Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge–Kutta-like time-steps to advance the parabolic terms by a time-step that is s 2 times larger than a single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge–Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems – a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very useful

  11. Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System

    Directory of Open Access Journals (Sweden)

    Zhenyi Ji

    2013-01-01

    Full Text Available We propose an approach for constructing Lyapunov function in quadratic form of a differential system. First, positive polynomial system is obtained via the local property of the Lyapunov function as well as its derivative. Then, the positive polynomial system is converted into an equation system by adding some variables. Finally, numerical technique is applied to solve the equation system. Some experiments show the efficiency of our new algorithm.

  12. Bounds and asymptotics for orthogonal polynomials for varying weights

    CERN Document Server

    Levin, Eli

    2018-01-01

    This book establishes bounds and asymptotics under almost minimal conditions on the varying weights, and applies them to universality limits and entropy integrals.  Orthogonal polynomials associated with varying weights play a key role in analyzing random matrices and other topics.  This book will be of use to a wide community of mathematicians, physicists, and statisticians dealing with techniques of potential theory, orthogonal polynomials, approximation theory, as well as random matrices. .

  13. P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation

    International Nuclear Information System (INIS)

    Duran, Antonio J; Gruenbaum, F Alberto

    2006-01-01

    The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein

  14. Parallel Construction of Irreducible Polynomials

    DEFF Research Database (Denmark)

    Frandsen, Gudmund Skovbjerg

    Let arithmetic pseudo-NC^k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial in (n + p). We show that the problem of constructing an irreducible polynomial of specified degree over GF(p) ...... of polynomials is in arithmetic NC^3. Our algorithm works over any field and compared to other known algorithms it does not assume the ability to take p'th roots when the field has characteristic p....

  15. Sum rules for zeros of polynomials and generalized Lucas polynomials

    Science.gov (United States)

    Ricci, Paolo Emilio

    1993-10-01

    A representation formula in terms of generalized Lucas polynomials of the second kind [see formula (4.3)], for the sum rules Js(i) introduced by Case [J. Math. Phys. 21, 702 (1980)] and studied by Dehesa et al. [J. Math. Phys. 26, 1547 (1985); 26, 2729 (1985)] in order to obtain information about the zeros' distribution of eigenfunctions of a class of ordinary polynomial differential operators, is derived.

  16. An introduction to orthogonal polynomials

    CERN Document Server

    Chihara, Theodore S

    1978-01-01

    Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study. Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some

  17. Julia Sets of Orthogonal Polynomials

    DEFF Research Database (Denmark)

    Christiansen, Jacob Stordal; Henriksen, Christian; Petersen, Henrik Laurberg

    2018-01-01

    For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials fPng to properties of the support. More precisely we relate the Julia set of Pn to the outer boundary of the support, the lled Julia...... set to the polynomial convex hull K of the support, and the Green's function associated with Pn to the Green's function for the complement of K....

  18. Stochastic Estimation via Polynomial Chaos

    Science.gov (United States)

    2015-10-01

    processes. As originally developed by Norbert Weiner, a polynomial chaos represents key properties of a stochastic process through the application of...polynomial chaos method for representing the properties of second order stochastic processes. As originally developed by Norbert Weiner, a...any real or complex functional ][xF in 2L converges to ][xF in the 2L sense with Wiener measure.[3] In the same reference, the orthogonality of

  19. On Modular Counting with Polynomials

    DEFF Research Database (Denmark)

    Hansen, Kristoffer Arnsfelt

    2006-01-01

    For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Z_m of degree O(\\sqrt n) that is a generalized representation (commonly also called weak representation) of the MODl...... function. We give a detailed study of the case when m has exactly two distinct prime factors, and classify the minimum possible degree for a symmetric representing polynomial....

  20. Numerical Simulation of Polynomial-Speed Convergence Phenomenon

    Science.gov (United States)

    Li, Yao; Xu, Hui

    2017-11-01

    We provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for Markov processes. The hybrid method that we introduce is based on the coupling technique and renewal theory. We propose to replace some estimates in classical results about the ergodicity of Markov processes by numerical simulations when the corresponding analytical proof is difficult. After that, all remaining conclusions can be derived from rigorous analysis. Then we apply our results to seek numerical justification for the ergodicity of two 1D microscopic heat conduction models. The mixing rate of these two models are expected to be polynomial but very difficult to prove. In both examples, our numerical results match the expected polynomial mixing rate well.

  1. Log-concavity of the genus polynomials of Ringel Ladders

    Directory of Open Access Journals (Sweden)

    Jonathan L Gross

    2015-10-01

    Full Text Available A Ringel ladder can be formed by a self-bar-amalgamation operation on a symmetric ladder, that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which linear chains of copies of one or more graphs have log-concave genus polyno- mials. Herein we establish Ringel ladders as the first significant non-linear infinite family of graphs known to have log-concave genus polynomials. We construct an algebraic representation of self-bar-amalgamation as a matrix operation, to be applied to a vector representation of the partitioned genus distribution of a symmetric ladder. Analysis of the resulting genus polynomial involves the use of Chebyshev polynomials. This paper continues our quest to affirm the quarter-century-old conjecture that all graphs have log-concave genus polynomials.

  2. Scattering theory and orthogonal polynomials

    International Nuclear Information System (INIS)

    Geronimo, J.S.

    1977-01-01

    The application of the techniques of scattering theory to the study of polynomials orthogonal on the unit circle and a finite segment of the real line is considered. The starting point is the recurrence relations satisfied by the polynomials instead of the orthogonality condition. A set of two two terms recurrence relations for polynomials orthogonal on the real line is presented and used. These recurrence relations play roles analogous to those satisfied by polynomials orthogonal on unit circle. With these recurrence formulas a Wronskian theorem is proved and the Christoffel-Darboux formula is derived. In scattering theory a fundamental role is played by the Jost function. An analogy is deferred of this function and its analytic properties and the locations of its zeros investigated. The role of the analog Jost function in various properties of these orthogonal polynomials is investigated. The techniques of inverse scattering theory are also used. The discrete analogues of the Gelfand-Levitan and Marchenko equations are derived and solved. These techniques are used to calculate asymptotic formulas for the orthogonal polynomials. Finally Szego's theorem on toeplitz and Hankel determinants is proved using the recurrence formulas and some properties of the Jost function. The techniques of inverse scattering theory are used to calculate the correction terms

  3. Bannai-Ito polynomials and dressing chains

    OpenAIRE

    Derevyagin, Maxim; Tsujimoto, Satoshi; Vinet, Luc; Zhedanov, Alexei

    2012-01-01

    Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito polynomials -- are shown to arise in the framework of the SDG maps.

  4. A 'missing' family of classical orthogonal polynomials

    International Nuclear Information System (INIS)

    Vinet, Luc; Zhedanov, Alexei

    2011-01-01

    We study a family of 'classical' orthogonal polynomials which satisfy (apart from a three-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl type. These polynomials can be obtained from the little q-Jacobi polynomials in the limit q = -1. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q = -1.

  5. Birth-death processes and associated polynomials

    NARCIS (Netherlands)

    van Doorn, Erik A.

    2003-01-01

    We consider birth-death processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birth-death polynomials. The sequence of associated polynomials linked with a sequence of birth-death polynomials and its orthogonalizing measure can be used in the analysis

  6. On Multiple Polynomials of Capelli Type

    Directory of Open Access Journals (Sweden)

    S.Y. Antonov

    2016-03-01

    Full Text Available This paper deals with the class of Capelli polynomials in free associative algebra F{Z} (where F is an arbitrary field, Z is a countable set generalizing the construction of multiple Capelli polynomials. The fundamental properties of the introduced Capelli polynomials are provided. In particular, decomposition of the Capelli polynomials by means of the same type of polynomials is shown. Furthermore, some relations between their T -ideals are revealed. A connection between double Capelli polynomials and Capelli quasi-polynomials is established.

  7. A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

    Directory of Open Access Journals (Sweden)

    E. H. Doha

    2014-01-01

    Full Text Available A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

  8. Doubling (Dual) Hahn Polynomials: Classification and Applications

    Science.gov (United States)

    Oste, Roy; Van der Jeugt, Joris

    2016-01-01

    We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester-Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models.

  9. Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate

    International Nuclear Information System (INIS)

    Tajvidi, T.; Razzaghi, M.; Dehghan, M.

    2008-01-01

    A numerical method for solving the classical Blasius' equation is proposed. The Blasius' equation is a third order nonlinear ordinary differential equation , which arises in the problem of the two-dimensional laminar viscous flow over a semi-infinite flat plane. The approach is based on a modified rational Legendre tau method. The operational matrices for the derivative and product of the modified rational Legendre functions are presented. These matrices together with the tau method are utilized to reduce the solution of Blasius' equation to the solution of a system of algebraic equations. A numerical evaluation is included to demonstrate the validity and applicability of the method and a comparison is made with existing results

  10. Consistency of the structure of Legendre transform in thermodynamics with the Kolmogorov–Nagumo average

    Energy Technology Data Exchange (ETDEWEB)

    Scarfone, A.M., E-mail: antoniomaria.scarfone@cnr.it [Istituto dei Sistemi Complessi (ISC-CNR) c/o Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino (Italy); Matsuzoe, H. [Department of Computer Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555 (Japan); Wada, T. [Department of Electrical and Electronic Engineering, Ibaraki University, Nakanarusawacho, Hitachi 316-8511 (Japan)

    2016-09-07

    We show the robustness of the structure of Legendre transform in thermodynamics against the replacement of the standard linear average with the Kolmogorov–Nagumo nonlinear average to evaluate the expectation values of the macroscopic physical observables. The consequence of this statement is twofold: 1) the relationships between the expectation values and the corresponding Lagrange multipliers still hold in the present formalism; 2) the universality of the Gibbs equation as well as other thermodynamic relations are unaffected by the structure of the average used in the theory. - Highlights: • The robustness of the Legendre structure has been shown within the KN average. • The relationships between the expectation values and the Lagrange multipliers still hold in the present formalism. • The universality of the Gibbs equation and other thermodynamic relations are unaffected by the structure of the average used.

  11. Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

    Science.gov (United States)

    Ito, K.; Teglas, R.

    1984-01-01

    The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

  12. Legendre-tau approximation for functional differential equations. II - The linear quadratic optimal control problem

    Science.gov (United States)

    Ito, Kazufumi; Teglas, Russell

    1987-01-01

    The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

  13. Solution of volume-surface integral equations using higher-order hierarchical Legendre basis functions

    DEFF Research Database (Denmark)

    Kim, Oleksiy S.; Meincke, Peter; Breinbjerg, Olav

    2007-01-01

    The problem of electromagnetic scattering by composite metallic and dielectric objects is solved using the coupled volume-surface integral equation (VSIE). The method of moments (MoM) based on higher-order hierarchical Legendre basis functions and higher-order curvilinear geometrical elements...... with the analytical Mie series solution. Scattering by more complex metal-dielectric objects are also considered to compare the presented technique with other numerical methods....

  14. Chromatic polynomials of random graphs

    International Nuclear Information System (INIS)

    Van Bussel, Frank; Fliegner, Denny; Timme, Marc; Ehrlich, Christoph; Stolzenberg, Sebastian

    2010-01-01

    Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.

  15. Fitting of two and three variate polynomials from experimental data through the least squares method

    International Nuclear Information System (INIS)

    Sanchez-Miro, J.J.; Sanz-Martin, J.C.

    1994-01-01

    Obtaining polynomial fittings from observational data in two and three dimensions is an interesting and practical task. Such an arduous problem suggests the development of an automatic code. The main novelty we provide lies in the generalization of the classical least squares method in three FORTRAN 77 programs usable in any sampling problem. Furthermore, we introduce the orthogonal 2D-Legendre function in the fitting process. These FORTRAN 77 programs are equipped with the options to calculate the approximation quality standard indicators, obviously generalized to two and three dimensions (correlation nonlinear factor, confidence intervals, cuadratic mean error, and so on). The aim of this paper is to rectify the absence of fitting algorithms for more than one independent variable in mathematical libraries

  16. All-Pole Recursive Digital Filters Design Based on Ultraspherical Polynomials

    Directory of Open Access Journals (Sweden)

    N. Stojanovic

    2014-09-01

    Full Text Available A simple method for approximation of all-pole recursive digital filters, directly in digital domain, is described. Transfer function of these filters, referred to as Ultraspherical filters, is controlled by order of the Ultraspherical polynomial, nu. Parameter nu, restricted to be a nonnegative real number (nu ≥ 0, controls ripple peaks in the passband of the magnitude response and enables a trade-off between the passband loss and the group delay response of the resulting filter. Chebyshev filters of the first and of the second kind, and also Legendre and Butterworth filters are shown to be special cases of these allpole recursive digital filters. Closed form equations for the computation of the filter coefficients are provided. The design technique is illustrated with examples.

  17. Exponential operators, generalized polynomials and evolution problems

    International Nuclear Information System (INIS)

    Dattoli, G.; Mancho, A.M.; Quattromini, M.; Torre, A.

    2001-01-01

    The operator (d/dx) χ d/dx plays a central role in the theory of operational calculus. Its exponential form is crucial in problems relevant to solutions of Fokker-Planck and Schroedinger equations. We explore the formal properties of the evolution operators associated to (d/dx) χ d/dx, discuss its link to special forms of Laguerre polynomials and Laguerre-based functions. The obtained results are finally applied to specific problems concerning the solution of Fokker-Planck equations relevant to the beam lifetime in storage rings

  18. Orthogonal Polynomials and Special Functions

    CERN Document Server

    Assche, Walter

    2003-01-01

    The set of lectures from the Summer School held in Leuven in 2002 provide an up-to-date account of recent developments in orthogonal polynomials and special functions, in particular for algorithms for computer algebra packages, 3nj-symbols in representation theory of Lie groups, enumeration, multivariable special functions and Dunkl operators, asymptotics via the Riemann-Hilbert method, exponential asymptotics and the Stokes phenomenon. The volume aims at graduate students and post-docs working in the field of orthogonal polynomials and special functions, and in related fields interacting with orthogonal polynomials, such as combinatorics, computer algebra, asymptotics, representation theory, harmonic analysis, differential equations, physics. The lectures are self-contained requiring only a basic knowledge of analysis and algebra, and each includes many exercises.

  19. Plain Polynomial Arithmetic on GPU

    International Nuclear Information System (INIS)

    Haque, Sardar Anisul; Maza, Marc Moreno

    2012-01-01

    As for serial code on CPUs, parallel code on GPUs for dense polynomial arithmetic relies on a combination of asymptotically fast and plain algorithms. Those are employed for data of large and small size, respectively. Parallelizing both types of algorithms is required in order to achieve peak performances. In this paper, we show that the plain dense polynomial multiplication can be efficiently parallelized on GPUs. Remarkably, it outperforms (highly optimized) FFT-based multiplication up to degree 2 12 while on CPU the same threshold is usually at 2 6 . We also report on a GPU implementation of the Euclidean Algorithm which is both work-efficient and runs in linear time for input polynomials up to degree 2 18 thus showing the performance of the GCD algorithm based on systolic arrays.

  20. Orthogonal Polynomials and their Applications

    CERN Document Server

    Dehesa, Jesús; Marcellan, Francisco; Francia, José; Vinuesa, Jaime

    1988-01-01

    The Segovia meeting set out to stimulate an intensive exchange of ideas between experts in the area of orthogonal polynomials and its applications, to present recent research results and to reinforce the scientific and human relations among the increasingly international community working in orthogonal polynomials. This volume contains original research papers as well as survey papers about fundamental questions in the field (Nevai, Rakhmanov & López) and its relationship with other fields such as group theory (Koornwinder), Padé approximation (Brezinski), differential equations (Krall, Littlejohn) and numerical methods (Rivlin).

  1. Recursive formula to compute Zernike radial polynomials.

    Science.gov (United States)

    Honarvar Shakibaei, Barmak; Paramesran, Raveendran

    2013-07-15

    In optics, Zernike polynomials are widely used in testing, wavefront sensing, and aberration theory. This unique set of radial polynomials is orthogonal over the unit circle and finite on its boundary. This Letter presents a recursive formula to compute Zernike radial polynomials using a relationship between radial polynomials and Chebyshev polynomials of the second kind. Unlike the previous algorithms, the derived recurrence relation depends neither on the degree nor on the azimuthal order of the radial polynomials. This leads to a reduction in the computational complexity.

  2. Bilateral generating functions for a new class of generalized Legendre polynominals

    Directory of Open Access Journals (Sweden)

    A. N. Srivastava

    1980-01-01

    Full Text Available Recently Chatterjea (1 has proved a theorem to deduce a bilateral generating function for the Ultraspherical polynomials. In the present paper an attempt has been made to give a general version of Chatterjea's theorem. Finally, the theorem has been specialized to obtain a bilateral generating function for a class of polynomials {Pn(x;α,β} introduced by Bhattacharjya (2.

  3. Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

    Science.gov (United States)

    Díaz Mendoza, C.; Orive, R.; Pijeira Cabrera, H.

    2008-10-01

    We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e-[phi](x), giving a unified treatment for the so-called Freud (i.e., when [phi] has polynomial growth at infinity) and Erdös (when [phi] grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.

  4. Global Polynomial Kernel Hazard Estimation

    DEFF Research Database (Denmark)

    Hiabu, Munir; Miranda, Maria Dolores Martínez; Nielsen, Jens Perch

    2015-01-01

    This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically redu...

  5. Dirichlet polynomials, majorization, and trumping

    International Nuclear Information System (INIS)

    Pereira, Rajesh; Plosker, Sarah

    2013-01-01

    Majorization and trumping are two partial orders which have proved useful in quantum information theory. We show some relations between these two partial orders and generalized Dirichlet polynomials, Mellin transforms, and completely monotone functions. These relations are used to prove a succinct generalization of Turgut’s characterization of trumping. (paper)

  6. Two polynomial division inequalities in

    Directory of Open Access Journals (Sweden)

    Goetgheluck P

    1998-01-01

    Full Text Available This paper is a first attempt to give numerical values for constants and , in classical estimates and where is an algebraic polynomial of degree at most and denotes the -metric on . The basic tools are Markov and Bernstein inequalities.

  7. Aspects of the Tutte polynomial

    DEFF Research Database (Denmark)

    Ok, Seongmin

    This thesis studies various aspects of the Tutte polynomial, especially focusing on the Merino-Welsh conjecture. We write T(G;x,y) for the Tutte polynomial of a graph G with variables x and y. In 1999, Merino and Welsh conjectured that if G is a loopless 2-connected graph, then T(G;1,1) ≤ max{T(G;2......,0), T(G;0,2)}. The three numbers, T(G;1,1), T(G;2,0) and T(G;0,2) are respectively the numbers of spanning trees, acyclic orientations and totally cyclic orientations of G. First, I extend Negami's splitting formula to the multivariate Tutte polynomial. Using the splitting formula, Thomassen and I found......-Welsh conjecture has a stronger version claiming that the Tutte polynomial is convex on the line segment between (2,0) and (0,2) for loopless 2-connected graphs. Chavez-Lomeli et al. proved that this holds for coloopless paving matroids, and I provide a shorter proof of their theorem. I also prove it for minimally...

  8. Sheffer and Non-Sheffer Polynomial Families

    Directory of Open Access Journals (Sweden)

    G. Dattoli

    2012-01-01

    Full Text Available By using the integral transform method, we introduce some non-Sheffer polynomial sets. Furthermore, we show how to compute the connection coefficients for particular expressions of Appell polynomials.

  9. Tables of properties of airfoil polynomials

    Science.gov (United States)

    Desmarais, Robert N.; Bland, Samuel R.

    1995-01-01

    This monograph provides an extensive list of formulas for airfoil polynomials. These polynomials provide convenient expansion functions for the description of the downwash and pressure distributions of linear theory for airfoils in both steady and unsteady subsonic flow.

  10. Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems

    Science.gov (United States)

    Chen, Zhijie; Kuo, Ting-Jung; Lin, Chang-Shou; Takemura, Kouichi

    2018-04-01

    We study the spectral polynomial of the Treibich-Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.

  11. Random walk polynomials and random walk measures

    NARCIS (Netherlands)

    van Doorn, Erik A.; Schrijner, Pauline

    1993-01-01

    Random walk polynomials and random walk measures play a prominent role in the analysis of a class of Markov chains called random walks. Without any reference to random walks, however, a random walk polynomial sequence can be defined (and will be defined in this paper) as a polynomial sequence{Pn(x)}

  12. Positive trigonometric polynomials and signal processing applications

    CERN Document Server

    Dumitrescu, Bogdan

    2007-01-01

    Presents the results on positive trigonometric polynomials within a unitary framework; the theoretical results obtained partly from the general theory of real polynomials, partly from self-sustained developments. This book provides information on the theory of sum-of-squares trigonometric polynomials in two parts: theory and applications.

  13. Associated polynomials and birth-death processes

    NARCIS (Netherlands)

    van Doorn, Erik A.

    2001-01-01

    We consider sequences of orthogonal polynomials with positive zeros, and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials, with a view to

  14. Sibling curves of polynomials | Wiggins | Quaestiones Mathematicae

    African Journals Online (AJOL)

    Sibling curves were demonstrated in papers [2, 3] as a novel way to visualize the zeros of complex valued functions. In this paper, we continue the work done in those papers by focusing solely on polynomials. We proceed to prove that the number of sibling curves of a polynomial is the degree of the polynomial. Keywords: ...

  15. Restricted Schur polynomials and finite N counting

    International Nuclear Information System (INIS)

    Collins, Storm

    2009-01-01

    Restricted Schur polynomials have been posited as orthonormal operators for the change of basis from N=4 SYM to type IIB string theory. In this paper we briefly expound the relationship between the restricted Schur polynomials and the operators forwarded by Brown, Heslop, and Ramgoolam. We then briefly examine the finite N counting of the restricted Schur polynomials.

  16. On polynomial and multiplicative radicals | Tumurbat | Quaestiones ...

    African Journals Online (AJOL)

    We show that polynomial and multiplicative radicals in [1] are special cases of radicals defined by means of elements. We scrutinize the way of defining a radical γG by a subset G of polynomials in noncommuting indeterminates. Defining polynomial radicals, Drazin and Roberts [1] required that the set G be closed

  17. Hermite polynomials in asymptotic representations of generalized Bernoulli,Euler, Bessel and Buchholz polynomials

    NARCIS (Netherlands)

    J.L. López; N.M. Temme (Nico)

    1999-01-01

    textabstractThis is a second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli,

  18. Special polynomials associated with rational solutions of some hierarchies

    NARCIS (Netherlands)

    Kudryashov, Nikolai A.

    2009-01-01

    New special polynomials associated with rational solutions of the Painleve hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential-difference hierarchies to find special polynomials are presented. These formulae allow us to search special polynomials

  19. Chemical Equilibrium and Polynomial Equations: Beware of Roots.

    Science.gov (United States)

    Smith, William R.; Missen, Ronald W.

    1989-01-01

    Describes two easily applied mathematical theorems, Budan's rule and Rolle's theorem, that in addition to Descartes's rule of signs and intermediate-value theorem, are useful in chemical equilibrium. Provides examples that illustrate the use of all four theorems. Discusses limitations of the polynomial equation representation of chemical…

  20. Fitting by Orthonormal Polynomials of Silver Nanoparticles Spectroscopic Data

    Science.gov (United States)

    Bogdanova, Nina; Koleva, Mihaela

    2018-02-01

    Our original Orthonormal Polynomial Expansion Method (OPEM) in one-dimensional version is applied for first time to describe the silver nanoparticles (NPs) spectroscopic data. The weights for approximation include experimental errors in variables. In this way we construct orthonormal polynomial expansion for approximating the curve on a non equidistant point grid. The corridors of given data and criteria define the optimal behavior of searched curve. The most important subinterval of spectra data is investigated, where the minimum (surface plasmon resonance absorption) is looking for. This study describes the Ag nanoparticles produced by laser approach in a ZnO medium forming a AgNPs/ZnO nanocomposite heterostructure.

  1. Polynomial Mappings with Small Degree

    Directory of Open Access Journals (Sweden)

    Jelonek Zbigniew

    2015-06-01

    Full Text Available Let Xn be an affine variety of dimension n and Yn be a quasi-projective variety of the same dimension. We prove that for a quasi-finite polynomial mapping ƒ : Xn → Yn, every non-empty component of the set Yn\\ ƒ (Xn is closed and it has dimension greater or equal to n μ(ƒ, where μ(ƒ is a geometric degree of ƒ. Moreover, we prove that generally, if ƒ : Xn → Yn is any polynomial mapping, then either every non-empty component of the set is of dimension ≥ n μ(ƒ or ƒ contracts a subvariety of dimension ≥ n μ(ƒ + 1.

  2. BSDEs with polynomial growth generators

    Directory of Open Access Journals (Sweden)

    Philippe Briand

    2000-01-01

    Full Text Available In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

  3. Nonlinear Legendre Spectral Finite Elements for Wind Turbine Blade Dynamics: Preprint

    Energy Technology Data Exchange (ETDEWEB)

    Wang, Q.; Sprague, M. A.; Jonkman, J.; Johnson, N.

    2014-01-01

    This paper presents a numerical implementation and examination of new wind turbine blade finite element model based on Geometrically Exact Beam Theory (GEBT) and a high-order spectral finite element method. The displacement-based GEBT is presented, which includes the coupling effects that exist in composite structures and geometric nonlinearity. Legendre spectral finite elements (LSFEs) are high-order finite elements with nodes located at the Gauss-Legendre-Lobatto points. LSFEs can be an order of magnitude more efficient that low-order finite elements for a given accuracy level. Interpolation of the three-dimensional rotation, a major technical barrier in large-deformation simulation, is discussed in the context of LSFEs. It is shown, by numerical example, that the high-order LSFEs, where weak forms are evaluated with nodal quadrature, do not suffer from a drawback that exists in low-order finite elements where the tangent-stiffness matrix is calculated at the Gauss points. Finally, the new LSFE code is implemented in the new FAST Modularization Framework for dynamic simulation of highly flexible composite-material wind turbine blades. The framework allows for fully interactive simulations of turbine blades in operating conditions. Numerical examples showing validation and LSFE performance will be provided in the final paper.

  4. Poly-Cauchy and Peters mixed-type polynomials

    OpenAIRE

    Kim, Dae San; Kim, Taekyun

    2013-01-01

    The Peters polynomials are a generalization of Boole polynomials. In this paper, we consider Peters and poly-Cauchy mixed type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities of those polynomials associated with special polynomials.

  5. Evolutionary Design of Polynomial Controller

    OpenAIRE

    R. Matousek; S. Lang; P. Minar; P. Pivonka

    2011-01-01

    In the control theory one attempts to find a controller that provides the best possible performance with respect to some given measures of performance. There are many sorts of controllers e.g. a typical PID controller, LQR controller, Fuzzy controller etc. In the paper will be introduced polynomial controller with novel tuning method which is based on the special pole placement encoding scheme and optimization by Genetic Algorithms (GA). The examples will show the perform...

  6. Space complexity in polynomial calculus

    Czech Academy of Sciences Publication Activity Database

    Filmus, Y.; Lauria, M.; Nordström, J.; Ron-Zewi, N.; Thapen, Neil

    2015-01-01

    Roč. 44, č. 4 (2015), s. 1119-1153 ISSN 0097-5397 R&D Projects: GA AV ČR IAA100190902; GA ČR GBP202/12/G061 Institutional support: RVO:67985840 Keywords : proof complexity * polynomial calculus * lower bounds Subject RIV: BA - General Mathematics Impact factor: 0.841, year: 2015 http://epubs.siam.org/doi/10.1137/120895950

  7. Codimensions of generalized polynomial identities

    International Nuclear Information System (INIS)

    Gordienko, Aleksei S

    2010-01-01

    It is proved that for every finite-dimensional associative algebra A over a field of characteristic zero there are numbers C element of Q + and t element of Z + such that gc n (A)∼Cn t d n as n→∞, where d=PI exp(A) element of Z + . Thus, Amitsur's and Regev's conjectures hold for the codimensions gc n (A) of the generalized polynomial identities. Bibliography: 6 titles.

  8. Tabulating knot polynomials for arborescent knots

    International Nuclear Information System (INIS)

    Mironov, A; Morozov, A; Morozov, A; Sleptsov, A; Ramadevi, P; Singh, Vivek Kumar

    2017-01-01

    Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys . B 899 395–413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials. (paper)

  9. Tabulating knot polynomials for arborescent knots

    Science.gov (United States)

    Mironov, A.; Morozov, A.; Morozov, A.; Ramadevi, P.; Singh, Vivek Kumar; Sleptsov, A.

    2017-02-01

    Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys. B 899 395-413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials.

  10. On Chebyshev Polynomials of Matrices

    Czech Academy of Sciences Publication Activity Database

    Faber, V.; Liesen, J.; Tichý, Petr

    2010-01-01

    Roč. 31, č. 4 (2010), s. 2205-2221 ISSN 0895-4798 R&D Projects: GA AV ČR IAA100300802 Grant - others:GA AV ČR(CZ) M100300901 Institutional research plan: CEZ:AV0Z10300504 Source of funding: I - inštitucionálna podpora na rozvoj VO Keywords : matrix approximation problems * Chebyshev polynomials * complex approximation theory * Krylov subspace methods * Arnoldi's method Subject RIV: BA - General Mathematics Impact factor: 1.725, year: 2010

  11. On classical orthogonal polynomials and differential operators

    International Nuclear Information System (INIS)

    Miranian, L

    2005-01-01

    It is well known that the four families of classical orthogonal polynomials (Jacobi, Bessel, Hermite and Laguerre) each satisfy an equation FP n (x) = λ n P n (x), n ≥ 0, for an appropriate second-order differential operator F. In this paper it is shown that any linear differential operator U which has the Jacobi, Bessel, Hermite or Laguerre polynomials as eigenfunctions has to be a polynomial with constant coefficients in the classical second-order operator F

  12. Model selection for univariable fractional polynomials.

    Science.gov (United States)

    Royston, Patrick

    2017-07-01

    Since Royston and Altman's 1994 publication ( Journal of the Royal Statistical Society, Series C 43: 429-467), fractional polynomials have steadily gained popularity as a tool for flexible parametric modeling of regression relationships. In this article, I present fp_select, a postestimation tool for fp that allows the user to select a parsimonious fractional polynomial model according to a closed test procedure called the fractional polynomial selection procedure or function selection procedure. I also give a brief introduction to fractional polynomial models and provide examples of using fp and fp_select to select such models with real data.

  13. Acceleration of computation of φ-polynomials.

    Science.gov (United States)

    Kaya, Ilhan; Rolland, Jannick

    2013-11-18

    The benefits of making an effective use of impressive computational power offered by multi-core platforms are investigated for the computation of φ-polynomials used in the description of freeform surfaces. Specifically, we devise parallel algorithms based upon the recurrence relations of both Zernike polynomials and gradient orthogonal Q-polynomials and implement these parallel algorithms on Graphical Processing Units (GPUs) respectively. The results show that more than an order of magnitude improvement is achieved in computational time over a sequential implementation if these recurrence-based parallel algorithms are adopted in the computation of the φ-polynomials.

  14. The q-Laguerre matrix polynomials.

    Science.gov (United States)

    Salem, Ahmed

    2016-01-01

    The Laguerre polynomials have been extended to Laguerre matrix polynomials by means of studying certain second-order matrix differential equation. In this paper, certain second-order matrix q-difference equation is investigated and solved. Its solution gives a generalized of the q-Laguerre polynomials in matrix variable. Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.

  15. Control to Facet for Polynomial Systems

    DEFF Research Database (Denmark)

    Sloth, Christoffer; Wisniewski, Rafael

    2014-01-01

    This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up...... for the controller design are solved by searching for polynomials in Bernstein form. This allows the controller design problem to be formulated as a linear programming problem. Examples are provided that demonstrate the efficiency of the method for designing controls for polynomial systems....

  16. Orthonormal polynomials in wavefront analysis: analytical solution.

    Science.gov (United States)

    Mahajan, Virendra N; Dai, Guang-ming

    2007-09-01

    Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.

  17. On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials

    Directory of Open Access Journals (Sweden)

    Ryoo CS

    2010-01-01

    Full Text Available The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.

  18. Mirror symmetry, toric branes and topological string amplitudes as polynomials

    International Nuclear Information System (INIS)

    Alim, Murad

    2009-01-01

    The central theme of this thesis is the extension and application of mirror symmetry of topological string theory. The contribution of this work on the mathematical side is given by interpreting the calculated partition functions as generating functions for mathematical invariants which are extracted in various examples. Furthermore the extension of the variation of the vacuum bundle to include D-branes on compact geometries is studied. Based on previous work for non-compact geometries a system of differential equations is derived which allows to extend the mirror map to the deformation spaces of the D-Branes. Furthermore, these equations allow the computation of the full quantum corrected superpotentials which are induced by the D-branes. Based on the holomorphic anomaly equation, which describes the background dependence of topological string theory relating recursively loop amplitudes, this work generalizes a polynomial construction of the loop amplitudes, which was found for manifolds with a one dimensional space of deformations, to arbitrary target manifolds with arbitrary dimension of the deformation space. The polynomial generators are determined and it is proven that the higher loop amplitudes are polynomials of a certain degree in the generators. Furthermore, the polynomial construction is generalized to solve the extension of the holomorphic anomaly equation to D-branes without deformation space. This method is applied to calculate higher loop amplitudes in numerous examples and the mathematical invariants are extracted. (orig.)

  19. Weighted Polynomial Approximation for Automated Detection of Inspiratory Flow Limitation

    Directory of Open Access Journals (Sweden)

    Sheng-Cheng Huang

    2017-01-01

    Full Text Available Inspiratory flow limitation (IFL is a critical symptom of sleep breathing disorders. A characteristic flattened flow-time curve indicates the presence of highest resistance flow limitation. This study involved investigating a real-time algorithm for detecting IFL during sleep. Three categories of inspiratory flow shape were collected from previous studies for use as a development set. Of these, 16 cases were labeled as non-IFL and 78 as IFL which were further categorized into minor level (20 cases and severe level (58 cases of obstruction. In this study, algorithms using polynomial functions were proposed for extracting the features of IFL. Methods using first- to third-order polynomial approximations were applied to calculate the fitting curve to obtain the mean absolute error. The proposed algorithm is described by the weighted third-order (w.3rd-order polynomial function. For validation, a total of 1,093 inspiratory breaths were acquired as a test set. The accuracy levels of the classifications produced by the presented feature detection methods were analyzed, and the performance levels were compared using a misclassification cobweb. According to the results, the algorithm using the w.3rd-order polynomial approximation achieved an accuracy of 94.14% for IFL classification. We concluded that this algorithm achieved effective automatic IFL detection during sleep.

  20. Polynomial fitting of tight-binding method in carbon

    Science.gov (United States)

    Haa, Wai Kang; Yeak, Su Hoe

    2017-04-01

    Carbon is very unique in among the elements and its ability to form strong chemical bonds with a variety number such as two carbons (graphene) and four carbons (diamond). This combination of strong bonds with tight mass and high melting point makes them technologically and scientifically important in nanoscience development. Tight-binding model (TB) is one of the semi-empirical approximations used in quantum mechanical world which is restricted to the Linear Combinations of Localized Atomic Orbitals (LCAO). Currently, there are many approaches in tight-binding calculation. In this paper, we have reproduced a polynomial scaling function by fitting to the TB model. The model is then applied into carbon molecules and obtained the energy bands of the system. The elements of the overlap Hamiltonian matrix in the model will be depending on the parameter of the polynomials. Our purpose is to find out a set of parameters in the polynomial which were commonly fit to an independently calculated band structure. We used minimization approach to calculate the polynomial coefficients which involves differentiation of eigenvalues in the eigensystem. The algorithm of fitting the parameters is carried out in FORTRAN.

  1. Mirror symmetry, toric branes and topological string amplitudes as polynomials

    Energy Technology Data Exchange (ETDEWEB)

    Alim, Murad

    2009-07-13

    The central theme of this thesis is the extension and application of mirror symmetry of topological string theory. The contribution of this work on the mathematical side is given by interpreting the calculated partition functions as generating functions for mathematical invariants which are extracted in various examples. Furthermore the extension of the variation of the vacuum bundle to include D-branes on compact geometries is studied. Based on previous work for non-compact geometries a system of differential equations is derived which allows to extend the mirror map to the deformation spaces of the D-Branes. Furthermore, these equations allow the computation of the full quantum corrected superpotentials which are induced by the D-branes. Based on the holomorphic anomaly equation, which describes the background dependence of topological string theory relating recursively loop amplitudes, this work generalizes a polynomial construction of the loop amplitudes, which was found for manifolds with a one dimensional space of deformations, to arbitrary target manifolds with arbitrary dimension of the deformation space. The polynomial generators are determined and it is proven that the higher loop amplitudes are polynomials of a certain degree in the generators. Furthermore, the polynomial construction is generalized to solve the extension of the holomorphic anomaly equation to D-branes without deformation space. This method is applied to calculate higher loop amplitudes in numerous examples and the mathematical invariants are extracted. (orig.)

  2. Network meta-analysis of longitudinal data using fractional polynomials.

    Science.gov (United States)

    Jansen, J P; Vieira, M C; Cope, S

    2015-07-10

    Network meta-analysis of randomized controlled trials (RCTs) are often based on one treatment effect measure per study. However, many studies report data at multiple time points. Furthermore, not all studies measure the outcomes at the same time points. As an alternative to a network meta-analysis based on a synthesis of the results at one time point, a network meta-analysis method is presented that allows for the simultaneous analysis of outcomes at multiple time points. The development of outcomes over time of interventions compared in an RCT is modeled with fractional polynomials, and the differences between the parameters of these polynomials within a trial are synthesized across studies with a Bayesian network meta-analysis. The proposed models are illustrated with an analysis of RCTs evaluating interventions for osteoarthritis of the knee. Fixed and random effects second order fractional polynomials were applied to the case study. Network meta-analysis with models that represent the treatment effects in terms of several parameters using fractional polynomials can be considered a useful addition to models for network meta-analysis of repeated measures previously proposed. When RCTs report treatment effects at multiple follow-up times, these models can be used to synthesize the results even if reporting times differ across the studies. Copyright © 2015 John Wiley & Sons, Ltd.

  3. Firefly Algorithm for Polynomial Bézier Surface Parameterization

    Directory of Open Access Journals (Sweden)

    Akemi Gálvez

    2013-01-01

    reality, medical imaging, computer graphics, computer animation, and many others. Very often, the preferred approximating surface is polynomial, usually described in parametric form. This leads to the problem of determining suitable parametric values for the data points, the so-called surface parameterization. In real-world settings, data points are generally irregularly sampled and subjected to measurement noise, leading to a very difficult nonlinear continuous optimization problem, unsolvable with standard optimization techniques. This paper solves the parameterization problem for polynomial Bézier surfaces by applying the firefly algorithm, a powerful nature-inspired metaheuristic algorithm introduced recently to address difficult optimization problems. The method has been successfully applied to some illustrative examples of open and closed surfaces, including shapes with singularities. Our results show that the method performs very well, being able to yield the best approximating surface with a high degree of accuracy.

  4. Polynomial solutions of nonlinear integral equations

    Energy Technology Data Exchange (ETDEWEB)

    Dominici, Diego [Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr. Suite 9, New Paltz, NY 12561-2443 (United States)], E-mail: dominicd@newpaltz.edu

    2009-05-22

    We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.

  5. Laguerre polynomials by a harmonic oscillator

    Science.gov (United States)

    Baykal, Melek; Baykal, Ahmet

    2014-09-01

    The study of an isotropic harmonic oscillator, using the factorization method given in Ohanian's textbook on quantum mechanics, is refined and some collateral extensions of the method related to the ladder operators and the associated Laguerre polynomials are presented. In particular, some analytical properties of the associated Laguerre polynomials are derived using the ladder operators.

  6. Laguerre polynomials by a harmonic oscillator

    International Nuclear Information System (INIS)

    Baykal, Melek; Baykal, Ahmet

    2014-01-01

    The study of an isotropic harmonic oscillator, using the factorization method given in Ohanian's textbook on quantum mechanics, is refined and some collateral extensions of the method related to the ladder operators and the associated Laguerre polynomials are presented. In particular, some analytical properties of the associated Laguerre polynomials are derived using the ladder operators. (paper)

  7. Integral inequalities for self-reciprocal polynomials

    Indian Academy of Sciences (India)

    that is, the coefficients satisfy ak = an−k for k = 0, 1,...,n. We denote the class of self-reciprocal polynomials of order n by Pn. The properties of self-reciprocal polynomials have been studied by several mathemati- cians. We refer to [1–5] and the references given therein. Here, we are concerned with the following interesting ...

  8. On Generalisation of Polynomials in Complex Plane

    Directory of Open Access Journals (Sweden)

    Maslina Darus

    2010-01-01

    Full Text Available The generalised Bell and Laguerre polynomials of fractional-order in complex z-plane are defined. Some properties are studied. Moreover, we proved that these polynomials are univalent solutions for second order differential equations. Also, the Laguerre-type of some special functions are introduced.

  9. Sibling curves of quadratic polynomials | Wiggins | Quaestiones ...

    African Journals Online (AJOL)

    Sibling curves were demonstrated in [1, 2] as a novel way to visualize the zeroes of real valued functions. In [3] it was shown that a polynomial of degree n has n sibling curves. This paper focuses on the algebraic and geometric properites of the sibling curves of real and complex quadratic polynomials. Key words: Quadratic ...

  10. Polynomial solutions of nonlinear integral equations

    International Nuclear Information System (INIS)

    Dominici, Diego

    2009-01-01

    We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials

  11. Exceptional polynomials and SUSY quantum mechanics

    Indian Academy of Sciences (India)

    We show that for the quantum mechanical problem which admit classical Laguerre/Jacobi polynomials as solutions for the Schrödinger equations (SE), will also admit exceptional Laguerre/Jacobi polynomials as solutions having the same eigenvalues but with the ground state missing after a modification of the potential.

  12. Topological string partition functions as polynomials

    International Nuclear Information System (INIS)

    Yamaguchi, Satoshi; Yau Shingtung

    2004-01-01

    We investigate the structure of the higher genus topological string amplitudes on the quintic hypersurface. It is shown that the partition functions of the higher genus than one can be expressed as polynomials of five generators. We also compute the explicit polynomial forms of the partition functions for genus 2, 3, and 4. Moreover, some coefficients are written down for all genus. (author)

  13. The Bessel polynomials and their differential operators

    International Nuclear Information System (INIS)

    Onyango Otieno, V.P.

    1987-10-01

    Differential operators associated with the ordinary and the generalized Bessel polynomials are defined. In each case the commutator bracket is constructed and shows that the differential operators associated with the Bessel polynomials and their generalized form are not commutative. Some applications of these operators to linear differential equations are also discussed. (author). 4 refs

  14. Large degree asymptotics of generalized Bessel polynomials

    NARCIS (Netherlands)

    J.L. López; N.M. Temme (Nico)

    2011-01-01

    textabstractAsymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the

  15. Block orthogonal polynomials: I. Definitions and properties

    International Nuclear Information System (INIS)

    Normand, Jean-Marie

    2007-01-01

    Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg-Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace E i of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean inner product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct inner products. An explicit two step Gram-Schmidt orthogonalization (G-SO) process to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of E i equal to the subspace of polynomials of degree less than i. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers

  16. The Medusa Algorithm for Polynomial Matings

    DEFF Research Database (Denmark)

    Boyd, Suzanne Hruska; Henriksen, Christian

    2012-01-01

    The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image...

  17. On the zeros of a polynomial

    Indian Academy of Sciences (India)

    For a polynomial of degree , we have obtained an upper bound involving coefficients of the polynomial, for moduli of its zeros of smallest moduli, and then a refinement of the well-known Eneström–Kakeya theorem (under certain conditions).

  18. Polynomial approximation approach to transient heat conduction ...

    African Journals Online (AJOL)

    This work reports polynomial approximation approach to transient heat conduction in a long slab, long cylinder and sphere with linear internal heat generation. It has been shown that the polynomial approximation method is able to calculate average temperature as a function of time for higher value of Biot numbers.

  19. Orthonormal polynomials in wavefront analysis: error analysis.

    Science.gov (United States)

    Dai, Guang-Ming; Mahajan, Virendra N

    2008-07-01

    Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

  20. Connections between the matching and chromatic polynomials

    Directory of Open Access Journals (Sweden)

    E. J. Farrell

    1992-01-01

    Full Text Available The main results established are (i a connection between the matching and chromatic polynomials and (ii a formula for the matching polynomial of a general complement of a subgraph of a graph. Some deductions on matching and chromatic equivalence and uniqueness are made.

  1. Generating polynomials using function composition | Aichinger ...

    African Journals Online (AJOL)

    We characterize the integral polynomials that can be obtained from x and x2 (or from 1, x, and x3) by using +;-, and function composition. Keywords: Near-rings, integer polynomials, subnear-ring membership problem. Quaestiones Mathematicae 36(2013), 39-46 ...

  2. Exceptional polynomials and SUSY quantum mechanics

    Indian Academy of Sciences (India)

    Abstract. We show that for the quantum mechanical problem which admit classical Laguerre/. Jacobi polynomials as solutions for the Schrödinger equations (SE), will also admit exceptional. Laguerre/Jacobi polynomials as solutions having the same eigenvalues but with the ground state missing after a modification of the ...

  3. Application of polynomial preconditioners to conservation laws

    NARCIS (Netherlands)

    Geurts, Bernardus J.; van Buuren, R.; Lu, H.

    2000-01-01

    Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of

  4. Polynomials formalism of quantum numbers

    International Nuclear Information System (INIS)

    Kazakov, K.V.

    2005-01-01

    Theoretical aspects of the recently suggested perturbation formalism based on the method of quantum number polynomials are considered in the context of the general anharmonicity problem. Using a biatomic molecule by way of example, it is demonstrated how the theory can be extrapolated to the case of vibrational-rotational interactions. As a result, an exact expression for the first coefficient of the Herman-Wallis factor is derived. In addition, the basic notions of the formalism are phenomenologically generalized and expanded to the problem of spin interaction. The concept of magneto-optical anharmonicity is introduced. As a consequence, an exact analogy is drawn with the well-known electro-optical theory of molecules, and a nonlinear dependence of the magnetic dipole moment of the system on the spin and wave variables is established [ru

  5. Polynomial weights and code constructions

    DEFF Research Database (Denmark)

    Massey, J; Costello, D; Justesen, Jørn

    1973-01-01

    polynomial included. This fundamental property is then used as the key to a variety of code constructions including 1) a simplified derivation of the binary Reed-Muller codes and, for any primepgreater than 2, a new extensive class ofp-ary "Reed-Muller codes," 2) a new class of "repeated-root" cyclic codes...... that are subcodes of the binary Reed-Muller codes and can be very simply instrumented, 3) a new class of constacyclic codes that are subcodes of thep-ary "Reed-Muller codes," 4) two new classes of binary convolutional codes with large "free distance" derived from known binary cyclic codes, 5) two new classes...... of long constraint length binary convolutional codes derived from2^r-ary Reed-Solomon codes, and 6) a new class ofq-ary "repeated-root" constacyclic codes with an algebraic decoding algorithm....

  6. Differential expansion for link polynomials

    Science.gov (United States)

    Bai, C.; Jiang, J.; Liang, J.; Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A.

    2018-03-01

    The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the 6j-symbols, at least, for the simplest triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered.

  7. Resonance – Journal of Science Education | Indian Academy of ...

    Indian Academy of Sciences (India)

    Keywords. Weierstrass approximation; Fejer's theorem; Fejer kernel; polynomial approximation; Hausdorff theorem; moments; Bernstein polynomial; Gaussian quadrature; Legendre polynomial; Fourier series.

  8. Matrix product formula for Macdonald polynomials

    International Nuclear Information System (INIS)

    Cantini, Luigi; Gier, Jan de; Michael Wheeler

    2015-01-01

    We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik–Zamolodchikov equations, which arise by considering representations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1. (paper)

  9. Resolving capacity of the circular Zernike polynomials.

    Science.gov (United States)

    Svechnikov, M V; Chkhalo, N I; Toropov, M N; Salashchenko, N N

    2015-06-01

    Circular Zernike polynomials are often used for approximation and analysis of optical surfaces. In this paper, we analyse their lateral resolving capacity, illustrating the effects of a lack of approximation by a finite set of polynomials and answering the following questions: What is the minimum number of polynomials that is necessary to describe a local deformation of a certain size? What is the relationship between the number of approximating polynomials and the spatial spectrum of the approximation? What is the connection between the mean-square error of approximation and the number of polynomials? The main results of this work are the formulas for calculating the error of fitting the relief and the connection between the width of the spatial spectrum and the order of approximation.

  10. A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems

    International Nuclear Information System (INIS)

    Aguirre-Hernández, B.; Campos-Cantón, E.; López-Renteria, J.A.; Díaz González, E.C.

    2015-01-01

    In this paper, we consider characteristic polynomials of n-dimensional systems that determine a segment of polynomials. One parameter is used to characterize this segment of polynomials in order to determine the maximal interval of dissipativity and unstability. Then we apply this result to the generation of a family of attractors based on a class of unstable dissipative systems (UDS) of type affine linear systems. This class of systems is comprised of switched linear systems yielding strange attractors. A family of these chaotic switched systems is determined by the maximal interval of perturbation of the matrix that governs the dynamics for still having scroll attractors

  11. Bernoulli numbers and polynomials from a more general point of view

    Energy Technology Data Exchange (ETDEWEB)

    Dattoli, G. [ENEA, Centro Ricerche Frascati, Frascati, RM(Italy). Div. Fisica Applicata; Cesarano, C. [Ulm Univ., Ulm (Germany). Dept. of Mathematics; Lonzellutta, S. [ENEA, Centro Ricerche E. Clementel, Bologna (Italy). Div. Fisica Applicata

    2000-07-01

    In this work it is applied the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials. [Italian] Si applica il metodo della funzione generatrice per introdurre nuove forme di numeri e polinomi di Bernoulli che vengono utilizzati per sviluppare e per calcolare somme parziali che coinvolgono polinomi a piu' indici ed a piu' variabili. Si sviluppano considerazioni analoghe per i polinomi ed i numeri di Eulero.

  12. Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations

    International Nuclear Information System (INIS)

    Sasaki, Ryu; Tsujimoto, Satoshi; Zhedanov, Alexei

    2010-01-01

    A simple derivation is presented of the four families of infinitely many shape-invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. The Darboux-Crum transformations are applied to connect the well-known shape-invariant Hamiltonians of the radial oscillator and the Darboux-Poeschl-Teller potential to the shape-invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of the generalized Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.

  13. Chebyshev recursion methods: Kernel polynomials and maximum entropy

    Energy Technology Data Exchange (ETDEWEB)

    Silver, R.N.; Roeder, H.; Voter, A.F.; Kress, J.D. [Los Alamos National Lab., NM (United States). Theoretical Div.

    1995-10-01

    The authors describe two Chebyshev recursion methods for calculations with very large sparse Hamiltonians, the kernel polynomial method (KPM) and the maximum entropy method (MEM). They are especially applicable to physical properties involving large numbers of eigenstates, which include densities of states, spectral functions, thermodynamics, total energies, as well as forces for molecular dynamics and Monte Carlo simulations. The authors apply Chebyshev methods to the electronic structure of Si, the thermodynamics of Heisenberg antiferromagnets, and a polaron problem.

  14. Transfer matrix computation of generalized critical polynomials in percolation

    Science.gov (United States)

    Scullard, Christian R.; Lykke Jacobsen, Jesper

    2012-12-01

    Percolation thresholds have recently been studied by means of a graph polynomial PB(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of PB(p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially PB(p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of PB(p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162 and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds pc(4, 82) = 0.676 803 329…, pc(kagome) = 0.524 404 998…, pc(3, 122) = 0.740 420 798…, comparable to the best simulation results. We also show that the alternative definition of PB(p) can be applied to study site percolation problems. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

  15. Transfer matrix computation of generalized critical polynomials in percolation

    International Nuclear Information System (INIS)

    Scullard, Christian R; Jacobsen, Jesper Lykke

    2012-01-01

    Percolation thresholds have recently been studied by means of a graph polynomial P B (p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of P B (p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially P B (p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of P B (p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 8 2 ), kagome, and (3, 12 2 ) lattices for bases of up to respectively 96, 162 and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds p c (4, 8 2 ) = 0.676 803 329…, p c (kagome) = 0.524 404 998…, p c (3, 12 2 ) = 0.740 420 798…, comparable to the best simulation results. We also show that the alternative definition of P B (p) can be applied to study site percolation problems. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday. (paper)

  16. Recurrence relations on the generelizad poly-Genocchi polynomials

    Science.gov (United States)

    Kurt, Burak; Bilgic, Secil

    2017-07-01

    In this work, we introduce and investigate poly-Genocchi numbers and polynomials. We give recurrence relations for the poly-Genocchi numbers. Also, we prove a relation between multi-poly-Genocchi polynomial and multi-poly-Bernoulli polynomial.

  17. Orthonormal polynomial approximation of water drop evaporation data with errors in two variables

    International Nuclear Information System (INIS)

    Bogdanova, N.; Todorov, S.

    2011-01-01

    The investigation for fitting drop water evaporation data as a result of original microscope observations is presented. Our approximation algorithm with construction of orthonormal polynomials (orthonormal polynomial expansion method, OPEM) is applied to data with uncertainties in both independent and dependent variables. For this purpose our numerical method is developed here to include both errors. We also review its principles and analyze the orthonormal and 'usual' expansions of the approximating function

  18. Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials

    Energy Technology Data Exchange (ETDEWEB)

    Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu [Department of Mathematics and Actuarial Science, Indiana University Northwest, 3400 Broadway, Gary IN 46408 (United States); Department of Physics, Indiana University Northwest, 3400 Broadway, Gary IN 46408 (United States); Roy, Pinaki, E-mail: pinaki@isical.ac.in [Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108 (India)

    2017-03-15

    We construct energy-dependent potentials for which the Schrödinger equations admit solutions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.

  19. Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials

    International Nuclear Information System (INIS)

    Schulze-Halberg, Axel; Roy, Pinaki

    2017-01-01

    We construct energy-dependent potentials for which the Schrödinger equations admit solutions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.

  20. Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

    Directory of Open Access Journals (Sweden)

    Bolandtalat A.

    2016-01-01

    Full Text Available In this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

  1. Exact multi-restricted Schur polynomial correlators

    International Nuclear Information System (INIS)

    Bhattacharyya, Rajsekhar; Koch, Robert de Mello; Stephanou, Michael

    2008-01-01

    We derive a product rule satisfied by restricted Schur polynomials. We focus mostly on the case that the restricted Schur polynomial is built using two matrices, although our analysis easily extends to more than two matrices. This product rule allows us to compute exact multi-point correlation functions of restricted Schur polynomials, in the free field theory limit. As an example of the use of our formulas, we compute two point functions of certain single trace operators built using two matrices and three point functions of certain restricted Schur polynomials, exactly, in the free field theory limit. Our results suggest that gravitons become strongly coupled at sufficiently high energy, while the restricted Schur polynomials for totally antisymmetric representations remain weakly interacting at these energies. This is in perfect accord with the half-BPS (single matrix) results of hep-th/0512312. Finally, by studying the interaction of two restricted Schur polynomials we suggest a physical interpretation for the labels of the restricted Schur polynomial: the composite operator χ R,(r n ,r m ) (Z,X) is constructed from the half BPS 'partons' χ r n (Z) and χ r m (X).

  2. Polynomial realization of the Uq (sl(3)) Gel'fand-(Weyl)-Zetlin basis

    International Nuclear Information System (INIS)

    Dobrev, V.K.; Truini, P.

    1996-01-01

    We give an explicit realization of the U ≡ U q (sl(3)) Gel'fand-(Weyl)-Zetlin (GWZ) basis as polynomial functions in three variables. This realization is obtained in two complementary ways. First we establish a 1-to-1 correspondence between the abstract GWZ basis and explicit polynomials in the quantum subgroup U + of the raising generators. We then use an explicit construction of arbitrary lowest weight (holomorphic) representations of U in terms of three variables on which the generators of U are realized as q-difference operators. Applying the GWZ corresponding polynomials in this realization to the lowest weight vector (the function 1) produces one realization of our GWZ basis. Another realization of the GWZ polynomial basis is found by the explicit diagonalization of the operators of isospin I-circumflex 2 , third component of isospin I-circumflex z , and hypercharge Y-circumflex, in the same realization as q-difference operators. The result is that the eigenvectors can be written in terms of q-hypergeometric polynomials in our three variables. Finally we construct an explicit scalar product (adapting the Shapovalov form to our setting). Using it we prove the orthogonality of our GWZ polynomials for which we use both realizations. This provides a polynomial construction for the orthonormal GWZ basis. We work for generic q, leaving the root of unity case for a following paper. It seems that our results are new also in the classical situation (q=1). (author). 20 refs

  3. Applying the expansion method in hierarchical functions to the solution of Navier-Stokes equations for incompressible fluids

    International Nuclear Information System (INIS)

    Sabundjian, Gaiane

    1999-01-01

    This work presents a novel numeric method, based on the finite element method, applied for the solution of the Navier-Stokes equations for incompressible fluids in two dimensions in laminar flow. The method is based on the expansion of the variables in almost hierarchical functions. The used expansion functions are based on Legendre polynomials, adjusted in the rectangular elements in a such a way that corner, side and area functions are defined. The order of the expansion functions associated with the sides and with the area of the elements can be adjusted to the necessary or desired degree. This novel numeric method is denominated by Hierarchical Expansion Method. In order to validate the proposed numeric method three well-known problems of the literature in two dimensions are analyzed. The results show the method capacity in supplying precise results. From the results obtained in this thesis it is possible to conclude that the hierarchical expansion method can be applied successfully for the solution of fluid dynamic problems that involve incompressible fluids. (author)

  4. Scalable explicit implementation of anisotropic diffusion with Runge-Kutta-Legendre super-time stepping

    Science.gov (United States)

    Vaidya, Bhargav; Prasad, Deovrat; Mignone, Andrea; Sharma, Prateek; Rickler, Luca

    2017-12-01

    An important ingredient in numerical modelling of high temperature magnetized astrophysical plasmas is the anisotropic transport of heat along magnetic field lines from higher to lower temperatures. Magnetohydrodynamics typically involves solving the hyperbolic set of conservation equations along with the induction equation. Incorporating anisotropic thermal conduction requires to also treat parabolic terms arising from the diffusion operator. An explicit treatment of parabolic terms will considerably reduce the simulation time step due to its dependence on the square of the grid resolution (Δx) for stability. Although an implicit scheme relaxes the constraint on stability, it is difficult to distribute efficiently on a parallel architecture. Treating parabolic terms with accelerated super-time-stepping (STS) methods has been discussed in literature, but these methods suffer from poor accuracy (first order in time) and also have difficult-to-choose tuneable stability parameters. In this work, we highlight a second-order (in time) Runge-Kutta-Legendre (RKL) scheme (first described by Meyer, Balsara & Aslam 2012) that is robust, fast and accurate in treating parabolic terms alongside the hyperbolic conversation laws. We demonstrate its superiority over the first-order STS schemes with standard tests and astrophysical applications. We also show that explicit conduction is particularly robust in handling saturated thermal conduction. Parallel scaling of explicit conduction using RKL scheme is demonstrated up to more than 104 processors.

  5. Neck curve polynomials in neck rupture model

    International Nuclear Information System (INIS)

    Kurniadi, Rizal; Perkasa, Yudha S.; Waris, Abdul

    2012-01-01

    The Neck Rupture Model is a model that explains the scission process which has smallest radius in liquid drop at certain position. Old fashion of rupture position is determined randomly so that has been called as Random Neck Rupture Model (RNRM). The neck curve polynomials have been employed in the Neck Rupture Model for calculation the fission yield of neutron induced fission reaction of 280 X 90 with changing of order of polynomials as well as temperature. The neck curve polynomials approximation shows the important effects in shaping of fission yield curve.

  6. More on rotations as spin matrix polynomials

    International Nuclear Information System (INIS)

    Curtright, Thomas L.

    2015-01-01

    Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined

  7. The Translated Dowling Polynomials and Numbers.

    Science.gov (United States)

    Mangontarum, Mahid M; Macodi-Ringia, Amila P; Abdulcarim, Normalah S

    2014-01-01

    More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.

  8. Small oscillations, Sturm sequences, and orthogonal polynomials

    International Nuclear Information System (INIS)

    Baake, M.

    1986-07-01

    The relation between small oscillations of one-dimensional mechanical systems and the theory of orthogonal polynomials is investigated. It is shown how the polynomials provide a natural tool to determine the eigenfrequencies and eigencoordinates completely, where the existence of a certain two-termed recurrence formula is essential. Physical and mathematical statements are formulated in terms of the recursion coefficients which can directly be obtained from the corresponding secular equation. Several known as well as new results on Sturm sequences and orthogonal polynomials are presented with respect to the treatment of small oscillations. (orig.)

  9. Wick polynomials and time-evolution of cumulants

    Science.gov (United States)

    Lukkarinen, Jani; Marcozzi, Matteo

    2016-08-01

    We show how Wick polynomials of random variables can be defined combinatorially as the unique choice, which removes all "internal contractions" from the related cumulant expansions, also in a non-Gaussian case. We discuss how an expansion in terms of the Wick polynomials can be used for derivation of a hierarchy of equations for the time-evolution of cumulants. These methods are then applied to simplify the formal derivation of the Boltzmann-Peierls equation in the kinetic scaling limit of the discrete nonlinear Schödinger equation (DNLS) with suitable random initial data. We also present a reformulation of the standard perturbation expansion using cumulants, which could simplify the problem of a rigorous derivation of the Boltzmann-Peierls equation by separating the analysis of the solutions to the Boltzmann-Peierls equation from the analysis of the corrections. This latter scheme is general and not tied to the DNLS evolution equations.

  10. Orthonormal curvature polynomials over a unit circle: basis set derived from curvatures of Zernike polynomials.

    Science.gov (United States)

    Zhao, Chunyu; Burge, James H

    2013-12-16

    Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.

  11. Adapted polynomial chaos expansion for failure detection

    International Nuclear Information System (INIS)

    Paffrath, M.; Wever, U.

    2007-01-01

    In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator-prey model and a chemical reaction problem

  12. Transversals of Complex Polynomial Vector Fields

    DEFF Research Database (Denmark)

    Dias, Kealey

    Vector fields in the complex plane are defined by assigning the vector determined by the value P(z) to each point z in the complex plane, where P is a polynomial of one complex variable. We consider special families of so-called rotated vector fields that are determined by a polynomial multiplied...... by rotational constants. Transversals are a certain class of curves for such a family of vector fields that represent the bifurcation states for this family of vector fields. More specifically, transversals are curves that coincide with a homoclinic separatrix for some rotation of the vector field. Given...... a concrete polynomial, it seems to take quite a bit of work to prove that it is generic, i.e. structurally stable. This has been done for a special class of degree d polynomial vector fields having simple equilibrium points at the d roots of unity, d odd. In proving that such vector fields are generic...

  13. Twisted Polynomials and Forgery Attacks on GCM

    DEFF Research Database (Denmark)

    Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey

    2015-01-01

    nonce misuse resistance, such as POET. The algebraic structure of polynomial hashing has given rise to security concerns: At CRYPTO 2008, Handschuh and Preneel describe key recovery attacks, and at FSE 2013, Procter and Cid provide a comprehensive framework for forgery attacks. Both approaches rely...... heavily on the ability to construct forgery polynomials having disjoint sets of roots, with many roots (“weak keys”) each. Constructing such polynomials beyond naïve approaches is crucial for these attacks, but still an open problem. In this paper, we comprehensively address this issue. We propose to use...... in an improved key recovery algorithm. As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which...

  14. Some results for extended Jacobi polynomials

    International Nuclear Information System (INIS)

    Khan, I.A.

    1991-06-01

    In this paper, some contiguous function relations and generating functions have been established for extended Jacobi polynomials. The results obtained here, are of very general nature. (author). 5 refs

  15. Schur Stability Regions for Complex Quadratic Polynomials

    Science.gov (United States)

    Cheng, Sui Sun; Huang, Shao Yuan

    2010-01-01

    Given a quadratic polynomial with complex coefficients, necessary and sufficient conditions are found in terms of the coefficients such that all its roots have absolute values less than 1. (Contains 3 figures.)

  16. Some generating functions of Laguerre polynomials

    Directory of Open Access Journals (Sweden)

    Bidyut Kumar Guha Thakurta

    1987-01-01

    Full Text Available In this note a class of interesting generating relation, which is stated in the form of theorem, involving Laguerre polynomials is derived. Some applications of the theorem are also given here.

  17. On linear equations with general polynomial solutions

    Science.gov (United States)

    Laradji, A.

    2018-04-01

    We provide necessary and sufficient conditions for which an nth-order linear differential equation has a general polynomial solution. We also give necessary conditions that can directly be ascertained from the coefficient functions of the equation.

  18. Handbook on semidefinite, conic and polynomial optimization

    CERN Document Server

    Anjos, Miguel F

    2012-01-01

    This book offers the reader a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization and polynomial optimization. It covers theory, algorithms, software and applications.

  19. Solving Bivariate Polynomial Systems on a GPU

    International Nuclear Information System (INIS)

    Moreno Maza, Marc; Pan Wei

    2012-01-01

    We present a CUDA implementation of dense multivariate polynomial arithmetic based on Fast Fourier Transforms over finite fields. Our core routine computes on the device (GPU) the subresultant chain of two polynomials with respect to a given variable. This subresultant chain is encoded by values on a FFT grid and is manipulated from the host (CPU) in higher-level procedures. We have realized a bivariate polynomial system solver supported by our GPU code. Our experimental results (including detailed profiling information and benchmarks against a serial polynomial system solver implementing the same algorithm) demonstrate that our strategy is well suited for GPU implementation and provides large speedup factors with respect to pure CPU code.

  20. Polynomial analysis of ambulatory blood pressure measurements

    NARCIS (Netherlands)

    Zwinderman, A. H.; Cleophas, T. A.; Cleophas, T. J.; van der Wall, E. E.

    2001-01-01

    In normotensive subjects blood pressures follow a circadian rhythm. A circadian rhythm in hypertensive patients is less well established, and may be clinically important, particularly with rigorous treatments of daytime blood pressures. Polynomial analysis of ambulatory blood pressure monitoring

  1. vs. a polynomial chaos-based MCMC

    KAUST Repository

    Siripatana, Adil

    2014-08-01

    Bayesian Inference of Manning\\'s n coefficient in a Storm Surge Model Framework: comparison between Kalman lter and polynomial based method Adil Siripatana Conventional coastal ocean models solve the shallow water equations, which describe the conservation of mass and momentum when the horizontal length scale is much greater than the vertical length scale. In this case vertical pressure gradients in the momentum equations are nearly hydrostatic. The outputs of coastal ocean models are thus sensitive to the bottom stress terms de ned through the formulation of Manning\\'s n coefficients. This thesis considers the Bayesian inference problem of the Manning\\'s n coefficient in the context of storm surge based on the coastal ocean ADCIRC model. In the first part of the thesis, we apply an ensemble-based Kalman filter, the singular evolutive interpolated Kalman (SEIK) filter to estimate both a constant Manning\\'s n coefficient and a 2-D parameterized Manning\\'s coefficient on one ideal and one of more realistic domain using observation system simulation experiments (OSSEs). We study the sensitivity of the system to the ensemble size. we also access the benefits from using an in ation factor on the filter performance. To study the limitation of the Guassian restricted assumption on the SEIK lter, 5 we also implemented in the second part of this thesis a Markov Chain Monte Carlo (MCMC) method based on a Generalized Polynomial chaos (gPc) approach for the estimation of the 1-D and 2-D Mannning\\'s n coe cient. The gPc is used to build a surrogate model that imitate the ADCIRC model in order to make the computational cost of implementing the MCMC with the ADCIRC model reasonable. We evaluate the performance of the MCMC-gPc approach and study its robustness to di erent OSSEs scenario. we also compare its estimates with those resulting from SEIK in term of parameter estimates and full distributions. we present a full analysis of the solution of these two methods, of the

  2. Rotation of 2D orthogonal polynomials

    Czech Academy of Sciences Publication Activity Database

    Yang, B.; Flusser, Jan; Kautský, J.

    2018-01-01

    Roč. 102, č. 1 (2018), s. 44-49 ISSN 0167-8655 R&D Projects: GA ČR GA15-16928S Institutional support: RVO:67985556 Keywords : Rotation invariants * Orthogonal polynomials * Recurrent relation * Hermite-like polynomials * Hermite moments Subject RIV: JD - Computer Applications, Robotics Impact factor: 1.995, year: 2016 http://library.utia.cas.cz/separaty/2017/ZOI/flusser-0483250.pdf

  3. A Polynomial Estimate of Railway Line Delay

    DEFF Research Database (Denmark)

    Cerreto, Fabrizio; Harrod, Steven; Nielsen, Otto Anker

    2017-01-01

    by numerical analysis with simulation. This paper proposes an analytical estimate of aggregate delay with a polynomial form. The function returns the aggregate delay of a railway line resulting from an initial, primary, delay. Analysis of the function demonstrates that there should be a balance between the two...... remedial measures, supplement and buffer. Numerical analysis of a Copenhagen Sbane line shows that the polynomial function is valid even when theoretical assumptions are violated....

  4. About the solvability of matrix polynomial equations

    OpenAIRE

    Netzer, Tim; Thom, Andreas

    2016-01-01

    We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd degree with non-degenerate leading form can be solved in self-adjoint matrices. We also study equations of even degree and equations in many variables.

  5. Integral inequalities for self-reciprocal polynomials

    Indian Academy of Sciences (India)

    Integral inequalities for self-reciprocal polynomials. HORST ALZER. Morsbacher Str. 10, D-51545 Waldbröl, Germany. E-mail: H.Alzer@gmx.de. MS received 14 November 2007. Abstract. Let n ≥ 1 be an integer and let Pn be the class of polynomials P of degree at most n satisfying znP(1/z) = P (z) for all z ∈ C. Moreover, ...

  6. Two polynomial representations of experimental design

    OpenAIRE

    Notari, Roberto; Riccomagno, Eva; Rogantin, Maria-Piera

    2007-01-01

    In the context of algebraic statistics an experimental design is described by a set of polynomials called the design ideal. This, in turn, is generated by finite sets of polynomials. Two types of generating sets are mostly used in the literature: Groebner bases and indicator functions. We briefly describe them both, how they are used in the analysis and planning of a design and how to switch between them. Examples include fractions of full factorial designs and designs for mixture experiments.

  7. Local fibred right adjoints are polynomial

    DEFF Research Database (Denmark)

    Kock, Anders; Kock, Joachim

    2013-01-01

    For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense......For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense...

  8. Fast polynomial multiplication on a GPU

    International Nuclear Information System (INIS)

    Maza, Marc Moreno; Pan Wei

    2010-01-01

    We present CUDA implementations of Fast Fourier Transforms over finite fields. This allows us to develop GPU support for dense univariate polynomial multiplication leading to speedup factors in the range 21 - 37 with respect to the best serial C-code available to us, for our largest input data sets. Since dense univariate polynomial multiplication is a core routine in symbolic computation, this is promising result for the integration of GPU support into computer algebra systems.

  9. Generating the patterns of variation with GeoGebra: the case of polynomial approximations

    Science.gov (United States)

    Attorps, Iiris; Björk, Kjell; Radic, Mirko

    2016-01-01

    In this paper, we report a teaching experiment regarding the theory of polynomial approximations at the university mathematics teaching in Sweden. The experiment was designed by applying Variation theory and by using the free dynamic mathematics software GeoGebra. The aim of this study was to investigate if the technology-assisted teaching of Taylor polynomials compared with traditional way of work at the university level can support the teaching and learning of mathematical concepts and ideas. An engineering student group (n = 19) was taught Taylor polynomials with the assistance of GeoGebra while a control group (n = 18) was taught in a traditional way. The data were gathered by video recording of the lectures, by doing a post-test concerning Taylor polynomials in both groups and by giving one question regarding Taylor polynomials at the final exam for the course in Real Analysis in one variable. In the analysis of the lectures, we found Variation theory combined with GeoGebra to be a potentially powerful tool for revealing some critical aspects of Taylor Polynomials. Furthermore, the research results indicated that applying Variation theory, when planning the technology-assisted teaching, supported and enriched students' learning opportunities in the study group compared with the control group.

  10. Vector-Valued Jack Polynomials from Scratch

    Directory of Open Access Journals (Sweden)

    Jean-Gabriel Luque

    2011-03-01

    Full Text Available Vector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of S_N and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N and studied by one of the authors (C. Dunkl in the specialization r=p=1 (i.e. for the symmetric group. By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.

  11. Strong asymptotics for extremal polynomials associated with weights on ℝ

    CERN Document Server

    Lubinsky, Doron S

    1988-01-01

    0. The results are consequences of a strengthened form of the following assertion: Given 0 1. Auxiliary results include inequalities for weighted polynomials, and zeros of extremal polynomials. The monograph is fairly self-contained, with proofs involving elementary complex analysis, and the theory of orthogonal and extremal polynomials. It should be of interest to research workers in approximation theory and orthogonal polynomials.

  12. Zernike olivary polynomials for applications with olivary pupils.

    Science.gov (United States)

    Zheng, Yi; Sun, Shanshan; Li, Ying

    2016-04-20

    Orthonormal polynomials have been extensively applied in optical image systems. One important optical pupil, which is widely processed in lateral shearing interferometers (LSI) and subaperture stitch tests (SST), is the overlap region of two circular wavefronts that are displaced from each other. We call it an olivary pupil. In this paper, the normalized process of an olivary pupil in a unit circle is first presented. Then, using a nonrecursive matrix method, Zernike olivary polynomials (ZOPs) are obtained. Previously, Zernike elliptical polynomials (ZEPs) have been considered as an approximation over an olivary pupil. We compare ZOPs with their ZEPs counterparts. Results show that they share the same components but are in different proportions. For some low-order aberrations such as defocus, coma, and spherical, the differences are considerable and may lead to deviations. Using a least-squares method to fit coefficient curves, we present a power-series expansion form for the first 15 ZOPs, which can be used conveniently with less than 0.1% error. The applications of ZOP are demonstrated in wavefront decomposition, LSI interferogram reconstruction, and SST overlap domain evaluation.

  13. On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

    Directory of Open Access Journals (Sweden)

    Paul Barry

    2013-01-01

    Full Text Available The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.

  14. On the connection coefficients of the Chebyshev-Boubaker polynomials.

    Science.gov (United States)

    Barry, Paul

    2013-01-01

    The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.

  15. Beta-integrals and finite orthogonal systems of Wilson polynomials

    International Nuclear Information System (INIS)

    Neretin, Yu A

    2002-01-01

    The integral is calculated and the system of orthogonal polynomials with weight equal to the corresponding integrand is constructed. This weight decreases polynomially, therefore only finitely many of its moments converge. As a result the system of orthogonal polynomials is finite. Systems of orthogonal polynomials related to 5 H 5 -Dougall's formula and the Askey integral is also constructed. All the three systems consist of Wilson polynomials outside the domain of positiveness of the usual weight

  16. Special polynomials associated with rational solutions of some hierarchies

    International Nuclear Information System (INIS)

    Kudryashov, Nikolai A.

    2009-01-01

    New special polynomials associated with rational solutions of the Painleve hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential-difference hierarchies to find special polynomials are presented. These formulae allow us to search special polynomials associated with the hierarchies. It is shown that rational solutions of the Caudrey-Dodd-Gibbon, the Kaup-Kupershmidt and the modified hierarchy for these ones can be obtained using new special polynomials.

  17. New polynomial-based molecular descriptors with low degeneracy.

    Directory of Open Access Journals (Sweden)

    Matthias Dehmer

    Full Text Available In this paper, we introduce a novel graph polynomial called the 'information polynomial' of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform a numerical study using real chemical databases. We obtain that the novel descriptors do have a high discrimination power.

  18. Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials

    KAUST Repository

    Ait-Haddou, Rachid

    2015-06-07

    We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials. © 2015 Elsevier Inc. All rights reserved.

  19. Extending a Property of Cubic Polynomials to Higher-Degree Polynomials

    Science.gov (United States)

    Miller, David A.; Moseley, James

    2012-01-01

    In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…

  20. Real-time task recognition in cataract surgery videos using adaptive spatiotemporal polynomials.

    Science.gov (United States)

    Quellec, Gwénolé; Lamard, Mathieu; Cochener, Béatrice; Cazuguel, Guy

    2015-04-01

    This paper introduces a new algorithm for recognizing surgical tasks in real-time in a video stream. The goal is to communicate information to the surgeon in due time during a video-monitored surgery. The proposed algorithm is applied to cataract surgery, which is the most common eye surgery. To compensate for eye motion and zoom level variations, cataract surgery videos are first normalized. Then, the motion content of short video subsequences is characterized with spatiotemporal polynomials: a multiscale motion characterization based on adaptive spatiotemporal polynomials is presented. The proposed solution is particularly suited to characterize deformable moving objects with fuzzy borders, which are typically found in surgical videos. Given a target surgical task, the system is trained to identify which spatiotemporal polynomials are usually extracted from videos when and only when this task is being performed. These key spatiotemporal polynomials are then searched in new videos to recognize the target surgical task. For improved performances, the system jointly adapts the spatiotemporal polynomial basis and identifies the key spatiotemporal polynomials using the multiple-instance learning paradigm. The proposed system runs in real-time and outperforms the previous solution from our group, both for surgical task recognition ( Az = 0.851 on average, as opposed to Az = 0.794 previously) and for the joint segmentation and recognition of surgical tasks ( Az = 0.856 on average, as opposed to Az = 0.832 previously).

  1. Chemical Reaction Networks for Computing Polynomials.

    Science.gov (United States)

    Salehi, Sayed Ahmad; Parhi, Keshab K; Riedel, Marc D

    2017-01-20

    Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: a type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.

  2. Multilevel weighted least squares polynomial approximation

    KAUST Repository

    Haji-Ali, Abdul-Lateef

    2017-06-30

    Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

  3. Quantum chaotic dynamics and random polynomials

    International Nuclear Information System (INIS)

    Bogomolny, E.; Bohigas, O.; Leboeuf, P.

    1995-11-01

    The distribution of roots of polynomials of high degree with random coefficients is investigated which, among others, appear naturally in the context of 'quantum chaotic dynamics'. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, the particular case of self-inverse random polynomials is studied, and it is shown that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials. (author)

  4. Markov and Bernstein type inequalities for polynomials

    Directory of Open Access Journals (Sweden)

    Mohapatra RN

    1999-01-01

    Full Text Available In an answer to a question raised by chemist Mendeleev, A. Markov proved that if is a real polynomial of degree , then The above inequality which is known as Markov's Inequality is best possible and becomes equality for the Chebyshev polynomial . Few years later, Serge Bernstein needed the analogue of this result for the unit disk in the complex plane instead of the interval and the following is known as Bernstein's Inequality. If is a polynomial of degree then This inequality is also best possible and is attained for , being a complex number. The above two inequalities have been the starting point of a considerable literature in Mathematics and in this article we discuss some of the research centered around these inequalities.

  5. Dominating Sets and Domination Polynomials of Paths

    Directory of Open Access Journals (Sweden)

    Saeid Alikhani

    2009-01-01

    Full Text Available Let G=(V,E be a simple graph. A set S⊆V is a dominating set of G, if every vertex in V\\S is adjacent to at least one vertex in S. Let 𝒫ni be the family of all dominating sets of a path Pn with cardinality i, and let d(Pn,j=|𝒫nj|. In this paper, we construct 𝒫ni, and obtain a recursive formula for d(Pn,i. Using this recursive formula, we consider the polynomial D(Pn,x=∑i=⌈n/3⌉nd(Pn,ixi, which we call domination polynomial of paths and obtain some properties of this polynomial.

  6. Minimal residual method stronger than polynomial preconditioning

    Energy Technology Data Exchange (ETDEWEB)

    Faber, V.; Joubert, W.; Knill, E. [Los Alamos National Lab., NM (United States)] [and others

    1994-12-31

    Two popular methods for solving symmetric and nonsymmetric systems of equations are the minimal residual method, implemented by algorithms such as GMRES, and polynomial preconditioning methods. In this study results are given on the convergence rates of these methods for various classes of matrices. It is shown that for some matrices, such as normal matrices, the convergence rates for GMRES and for the optimal polynomial preconditioning are the same, and for other matrices such as the upper triangular Toeplitz matrices, it is at least assured that if one method converges then the other must converge. On the other hand, it is shown that matrices exist for which restarted GMRES always converges but any polynomial preconditioning of corresponding degree makes no progress toward the solution for some initial error. The implications of these results for these and other iterative methods are discussed.

  7. Some Inequalities for the Derivative of Polynomials

    Directory of Open Access Journals (Sweden)

    Sunil Hans

    2014-01-01

    Full Text Available If pz=∑υ=0ncυzυ is a polynomial of degree n, having no zeros in z<1, then Aziz (1989 proved maxz=1p′z≤n/2Mα2+Mα+π21/2, where Mα=max1≤k≤npeiα+2kπ/n. In this paper, we consider a class of polynomial Pnμ of degree n, defined as pz=a0+∑υ=μnaυzυ and present certain generalizations of above inequality and some other well-known results.

  8. The chromatic polynomial and list colorings

    DEFF Research Database (Denmark)

    Thomassen, Carsten

    2009-01-01

    We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph.......We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph....

  9. Some Relationships between the Analogs of Euler Numbers and Polynomials

    Directory of Open Access Journals (Sweden)

    Kim T

    2007-01-01

    Full Text Available We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet -Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted -Euler polynomials. Finally, we give a table for the solutions of the twisted -Euler polynomials.

  10. Some Relationships between the Analogs of Euler Numbers and Polynomials

    Directory of Open Access Journals (Sweden)

    C. S. Ryoo

    2007-10-01

    Full Text Available We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet l-Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted q-Euler polynomials. Finally, we give a table for the solutions of the twisted q-Euler polynomials.

  11. Equivalences of the multi-indexed orthogonal polynomials

    International Nuclear Information System (INIS)

    Odake, Satoru

    2014-01-01

    Multi-indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape-invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion. Multi-indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types, two different index sets may give equivalent multi-indexed orthogonal polynomials. We clarify these equivalences. Multi-indexed orthogonal polynomials with both type I and II indices are proportional to those of type I indices only (or type II indices only) with shifted parameters

  12. Polynomial regression analysis and significance test of the regression function

    International Nuclear Information System (INIS)

    Gao Zhengming; Zhao Juan; He Shengping

    2012-01-01

    In order to analyze the decay heating power of a certain radioactive isotope per kilogram with polynomial regression method, the paper firstly demonstrated the broad usage of polynomial function and deduced its parameters with ordinary least squares estimate. Then significance test method of polynomial regression function is derived considering the similarity between the polynomial regression model and the multivariable linear regression model. Finally, polynomial regression analysis and significance test of the polynomial function are done to the decay heating power of the iso tope per kilogram in accord with the authors' real work. (authors)

  13. Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials

    International Nuclear Information System (INIS)

    Nesterenko, M; Patera, J; Szajewska, M; Tereszkiewicz, A

    2010-01-01

    Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank ≤3 are explicitly studied. We derive the polynomials of simple Lie groups B 3 and C 3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.

  14. Airy polynomials, three-variable Hermite polynomials and the paraxial wave equation

    International Nuclear Information System (INIS)

    Torre, A

    2012-01-01

    Within the context of an investigation of the potential of functions, which directly or indirectly relate to the Airy functions, to yield solutions of the 2D paraxial wave equation, we consider here the Airy polynomials. We see that appropriate Gaussian-modulated versions of such polynomials, which could in a sense parallel the Hermite–Gaussian wavefunctions of both elegant and standard type, yield, when assumed as initial functions, solutions of the 2D PWE shaping in the form of a complex Gaussian modulation of three-variable Hermite polynomials. A basic characterization of these wavefunctions is provided, having as a touchstone the Hermite–Gaussian wavefunctions. (paper)

  15. A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions

    KAUST Repository

    Butler, T.

    2011-01-01

    We develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear stochastic differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. We present numerical examples showing that there is excellent agreement between these methods. © 2011 Society for Industrial and Applied Mathematics.

  16. Solution of Radiative Transfer Equation with a Continuous and Stochastic Varying Refractive Index by Legendre Transform Method

    Directory of Open Access Journals (Sweden)

    M. Gantri

    2014-01-01

    Full Text Available The present paper gives a new computational framework within which radiative transfer in a varying refractive index biological tissue can be studied. In our previous works, Legendre transform was used as an innovative view to handle the angular derivative terms in the case of uniform refractive index spherical medium. In biomedical optics, our analysis can be considered as a forward problem solution in a diffuse optical tomography imaging scheme. We consider a rectangular biological tissue-like domain with spatially varying refractive index submitted to a near infrared continuous light source. Interaction of radiation with the biological material into the medium is handled by a radiative transfer model. In the studied situation, the model displays two angular redistribution terms that are treated with Legendre integral transform. The model is used to study a possible detection of abnormalities in a general biological tissue. The effect of the embedded nonhomogeneous objects on the transmitted signal is studied. Particularly, detection of targets of localized heterogeneous inclusions within the tissue is discussed. Results show that models accounting for variation of refractive index can yield useful predictions about the target and the location of abnormal inclusions within the tissue.

  17. A summation formula over the zeros of a combination of the associated Legendre functions with a physical application

    International Nuclear Information System (INIS)

    Saharian, A A

    2009-01-01

    By using the generalized Abel-Plana formula, we derive a summation formula for the series over the zeros of a combination of the associated Legendre functions with respect to the degree. The summation formula for the series over the zeros of the combination of the Bessel functions, previously discussed in the literature, is obtained as a limiting case. As an application we evaluate the Wightman function for a scalar field with a general curvature coupling parameter in the region between concentric spherical shells on a background of constant negative curvature space. For the Dirichlet boundary conditions the corresponding mode-sum contains the series over the zeros of the combination of the associated Legendre functions. The application of the summation formula allows us to present the Wightman function in the form of the sum of two integrals. The first one corresponds to the Wightman function for the geometry of a single spherical shell and the second one is induced by the presence of the second shell. The boundary-induced part in the vacuum expectation value of the field squared is investigated. For points away from the boundaries the corresponding renormalization procedure is reduced to that for the boundary-free part.

  18. Inequalities for a Polynomial and its Derivative

    Indian Academy of Sciences (India)

    Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 110; Issue 2. Inequalities for a Polynomial and its Derivative. V K Jain. Volume 110 Issue 2 May 2000 pp 137- ...

  19. Integral Inequalities for Self-Reciprocal Polynomials

    Indian Academy of Sciences (India)

    Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 120; Issue 2. Integral Inequalities for Self-Reciprocal Polynomials. Horst Alzer. Volume 120 Issue 2 April 2010 ...

  20. Integral inequalities for self-reciprocal polynomials

    Indian Academy of Sciences (India)

    Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 120; Issue 2. Integral Inequalities for Self-Reciprocal Polynomials. Horst Alzer. Volume 120 Issue 2 April 2010 ...

  1. Polynomial stabilization of some dissipative hyperbolic systems

    Czech Academy of Sciences Publication Activity Database

    Ammari, K.; Feireisl, Eduard; Nicaise, S.

    2014-01-01

    Roč. 34, č. 11 (2014), s. 4371-4388 ISSN 1078-0947 R&D Projects: GA ČR GA201/09/0917 Institutional support: RVO:67985840 Keywords : exponential stability * polynomial stability * observability inequality Subject RIV: BA - General Mathematics Impact factor: 0.826, year: 2014 http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=9924

  2. Bernoulli Polynomials, Fourier Series and Zeta Numbers

    DEFF Research Database (Denmark)

    Scheufens, Ernst E

    2013-01-01

    Fourier series for Bernoulli polynomials are used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent...

  3. Euler Polynomials, Fourier Series and Zeta Numbers

    DEFF Research Database (Denmark)

    Scheufens, Ernst E

    2012-01-01

    Fourier series for Euler polynomials is used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent series....

  4. D-stability of polynomial matrices

    Czech Academy of Sciences Publication Activity Database

    Henrion, Didier; Bachelier, O.; Šebek, M.

    2001-01-01

    Roč. 74, č. 8 (2001), s. 845-856 ISSN 0020-7179 R&D Projects: GA AV ČR KSK1019101 Institutional research plan: AV0Z1075907 Keywords : polynomial methods * robust control * linear matrix inequalities Subject RIV: BC - Control Systems Theory Impact factor: 0.736, year: 2001

  5. Odd Degree Polynomials on Real Banach Spaces

    Czech Academy of Sciences Publication Activity Database

    Aron, R. M.; Hájek, Petr Pavel

    2007-01-01

    Roč. 11, č. 1 (2007), s. 143-153 ISSN 1385-1292 R&D Projects: GA AV ČR IAA100190502; GA ČR GA201/04/0090 Institutional research plan: CEZ:AV0Z10190503 Keywords : odd degree polynomials * zero sets Subject RIV: BA - General Mathematics Impact factor: 0.356, year: 2007

  6. Quantum Hilbert matrices and orthogonal polynomials

    DEFF Research Database (Denmark)

    Andersen, Jørgen Ellegaard; Berg, Christian

    2009-01-01

    Using the notion of quantum integers associated with a complex number q≠0 , we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q -Jacobi polynomials when |q|... of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix....

  7. Characteristic polynomials of linear polyacenes and their ...

    Indian Academy of Sciences (India)

    Administrator

    Abstract. Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal's triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP's.

  8. Polynomial computation of Hankel singular values

    NARCIS (Netherlands)

    Kwakernaak, H.

    1992-01-01

    A revised and improved version of a polynomial algorithm is presented. It was published by N.J. Young (1990) for the computation of the singular values and vectors of the Hankel operator defined by a linear time-invariant system with a rotational transfer matrix. Tentative numerical experiments

  9. Characteristic polynomials of linear polyacenes and their ...

    Indian Academy of Sciences (India)

    Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal's triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP's. Thus it ...

  10. Coherent states for polynomial su(2) algebra

    International Nuclear Information System (INIS)

    Sadiq, Muhammad; Inomata, Akira

    2007-01-01

    A class of generalized coherent states is constructed for a polynomial su(2) algebra in a group-free manner. As a special case, the coherent states for the cubic su(2) algebra are discussed. The states so constructed reduce to the usual SU(2) coherent states in the linear limit

  11. On zeros of certain Dirichlet polynomials

    OpenAIRE

    BLLACA, KAJTAZ H.

    2015-01-01

    In this article we establish the zero-free region of certain Dirichlet polynomials $L_{F, X}$ arising in approximate functional equation for functions in the Selberg class and we prove an asymptotic formula for the number of zeros of $L_{F, X}$.

  12. Exceptional polynomials and SUSY quantum mechanics

    Indian Academy of Sciences (India)

    Exceptional polynomials and SUSY quantum mechanics. K V S SHIV CHAITANYA1,∗, S SREE RANJANI2,. PRASANTA K PANIGRAHI3, R RADHAKRISHNAN4 and V SRINIVASAN4. 1BITS Pilani, Hyderabad Campus, Jawahar Nagar, Shameerpet Mandal, Hyderabad 500 078, India. 2Faculty of Science and Technology, ...

  13. Deformation Prediction Using Linear Polynomial Functions ...

    African Journals Online (AJOL)

    The predictions are compared with measured data reported in literature and the results are discussed. The computational aspects of implementation of the model are also discussed briefly. Keywords: Linear Polynomial, Structural Deformation, Prediction Journal of the Nigerian Association of Mathematical Physics, Volume ...

  14. Subintegrality, invertible modules and Laurent polynomial extensions

    Indian Academy of Sciences (India)

    Indian Acad. Sci. (Math. Sci.) Vol. 125, No. 2, May 2015, pp. 149–160. c Indian Academy of Sciences. Subintegrality, invertible modules and Laurent polynomial extensions. VIVEK SADHU. Department of Mathematics ...... comments which have improved the exposition. Further, he would like to thank CSIR,. India for financial ...

  15. Information-theoretic lengths of Jacobi polynomials

    Energy Technology Data Exchange (ETDEWEB)

    Guerrero, A; Dehesa, J S [Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Granada, Granada (Spain); Sanchez-Moreno, P, E-mail: agmartinez@ugr.e, E-mail: pablos@ugr.e, E-mail: dehesa@ugr.e [Instituto ' Carlos I' de Fisica Teorica y Computacional, Universidad de Granada, Granada (Spain)

    2010-07-30

    The information-theoretic lengths of the Jacobi polynomials P{sup ({alpha}, {beta})}{sub n}(x), which are information-theoretic measures (Renyi, Shannon and Fisher) of their associated Rakhmanov probability density, are investigated. They quantify the spreading of the polynomials along the orthogonality interval [- 1, 1] in a complementary but different way as the root-mean-square or standard deviation because, contrary to this measure, they do not refer to any specific point of the interval. The explicit expressions of the Fisher length are given. The Renyi lengths are found by the use of the combinatorial multivariable Bell polynomials in terms of the polynomial degree n and the parameters ({alpha}, {beta}). The Shannon length, which cannot be exactly calculated because of its logarithmic functional form, is bounded from below by using sharp upper bounds to general densities on [- 1, +1] given in terms of various expectation values; moreover, its asymptotics is also pointed out. Finally, several computational issues relative to these three quantities are carefully analyzed.

  16. Error Minimization of Polynomial Approximation of Delta

    Indian Academy of Sciences (India)

    2016-01-27

    Jan 27, 2016 ... The difference between Universal time (UT) and Dynamical time (TD), known as Delta ( ) is tabulated for the first day of each year in the Astronomical Almanac. During the last four centuries it is found that there are large differences between its values for two consecutive years. Polynomial ...

  17. On Arithmetic-Geometric-Mean Polynomials

    Science.gov (United States)

    Griffiths, Martin; MacHale, Des

    2017-01-01

    We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…

  18. On the zeros of the classical polynomials

    International Nuclear Information System (INIS)

    Calogero, F.

    1977-01-01

    Some theorems and conjectures concerning the zeros of the classical polynomials (of Hermite, Laguerre and Jacobi) are announced. The theorems, whose proof is not reported, have been obtained from the relationship between the motion of the zeros of special solutions of simple linear partial differential equations (in x and t), and one-dimensional many-body problems. (author)

  19. Hermite polynomials and quasi-classical asymptotics

    Czech Academy of Sciences Publication Activity Database

    Ali, S. T.; Engliš, Miroslav

    2014-01-01

    Roč. 55, č. 4 (2014), 042102 ISSN 0022-2488 R&D Projects: GA ČR GA201/09/0473 Institutional support: RVO:67985840 Keywords : quantization * Hermite polynomials * reproducing kernel Subject RIV: BA - General Mathematics Impact factor: 1.243, year: 2014 http://scitation.aip.org/content/aip/journal/jmp/55/4/10.1063/1.4869325

  20. Polynomial Asymptotes of the Second Kind

    Science.gov (United States)

    Dobbs, David E.

    2011-01-01

    This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…

  1. The 6 Vertex Model and Schubert Polynomials

    Directory of Open Access Journals (Sweden)

    Alain Lascoux

    2007-02-01

    Full Text Available We enumerate staircases with fixed left and right columns. These objects correspond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization factor, to some Schubert polynomials.

  2. Polynomial Vector Fields in One Complex Variable

    DEFF Research Database (Denmark)

    Branner, Bodil

    In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias....

  3. Nonclassical Orthogonal Polynomials and Corresponding Quadratures

    CERN Document Server

    Fukuda, H; Alt, E O; Matveenko, A V

    2004-01-01

    We construct nonclassical orthogonal polynomials and calculate abscissas and weights of Gaussian quadrature for arbitrary weight and interval. The program is written by Mathematica and it works if moment integrals are given analytically. The result is a FORTRAN subroutine ready to utilize the quadrature.

  4. Indecomposability of polynomials via Jacobian matrix

    International Nuclear Information System (INIS)

    Cheze, G.; Najib, S.

    2007-12-01

    Uni-multivariate decomposition of polynomials is a special case of absolute factorization. Recently, thanks to the Ruppert's matrix some effective results about absolute factorization have been improved. Here we show that with a jacobian matrix we can get sharper bounds for the special case of uni-multivariate decomposition. (author)

  5. On an Inequality Concerning the Polar Derivative of a Polynomial

    Indian Academy of Sciences (India)

    Abstract. In this paper, we present a correct proof of an -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.

  6. On the Symmetric Properties for the Generalized Twisted Bernoulli Polynomials

    Directory of Open Access Journals (Sweden)

    Kim Taekyun

    2009-01-01

    Full Text Available We study the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for the -adic invariant integral.

  7. On an inequality concerning the polar derivative of a polynomial

    Indian Academy of Sciences (India)

    Abstract. In this paper, we present a correct proof of an -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.

  8. On a Family of Multivariate Modified Humbert Polynomials

    Science.gov (United States)

    Aktaş, Rabia; Erkuş-Duman, Esra

    2013-01-01

    This paper attempts to present a multivariable extension of generalized Humbert polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties, and also some special cases for these multivariable polynomials. PMID:23935411

  9. Irreducibility results for compositions of polynomials in several ...

    Indian Academy of Sciences (India)

    We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.

  10. Recurrence relations for the Cartesian derivatives of the Zernike polynomials.

    Science.gov (United States)

    Stephenson, Philip C L

    2014-04-01

    A recurrence relation for the first-order Cartesian derivatives of the Zernike polynomials is derived. This relation is used with the Clenshaw method to determine an efficient method for calculating the derivatives of any linear series of Zernike polynomials.

  11. Translational Bounds for Factorial n and the Factorial Polynomial

    Science.gov (United States)

    Mahmood, Munir; Edwards, Phillip

    2009-01-01

    During the period 1729-1826 Bernoulli, Euler, Goldbach and Legendre developed expressions for defining and evaluating "n"! and the related gamma function. Expressions related to "n"! and the gamma function are a common feature in computer science and engineering applications. In the modern computer age people live in now, two common tests to…

  12. Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials.

    Science.gov (United States)

    Khan, Waseem A; Haroon, Hiba

    2016-01-01

    In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite-Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.

  13. The generalized Yablonskii-Vorob'ev polynomials and their properties

    International Nuclear Information System (INIS)

    Kudryashov, Nikolai A.; Demina, Maria V.

    2008-01-01

    Rational solutions of the generalized second Painleve hierarchy are classified. Representation of the rational solutions in terms of special polynomials, the generalized Yablonskii-Vorob'ev polynomials, is introduced. Differential-difference relations satisfied by the polynomials are found. Hierarchies of differential equations related to the generalized second Painleve hierarchy are derived. One of these hierarchies is a sequence of differential equations satisfied by the generalized Yablonskii-Vorob'ev polynomials

  14. Weighted inequalities for generalized polynomials with doubling weights.

    Science.gov (United States)

    Joung, Haewon

    2017-01-01

    Many weighted polynomial inequalities, such as the Bernstein, Marcinkiewicz, Schur, Remez, Nikolskii inequalities, with doubling weights were proved by Mastroianni and Totik for the case [Formula: see text], and by Tamás Erdélyi for [Formula: see text]. In this paper we extend such polynomial inequalities to those for generalized trigonometric polynomials. We also prove the large sieve for generalized trigonometric polynomials with doubling weights.

  15. Systematic comparison of the use of annular and Zernike circle polynomials for annular wavefronts.

    Science.gov (United States)

    Mahajan, Virendra N; Aftab, Maham

    2010-11-20

    The theory of wavefront analysis of a noncircular wavefront is given and applied for a systematic comparison of the use of annular and Zernike circle polynomials for the analysis of an annular wavefront. It is shown that, unlike the annular coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is identically the same as that with the corresponding annular polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. The interferometer setting errors of tip, tilt, and defocus from a four-circle-polynomial expansion are the same as those from the annular-polynomial expansion. However, if these errors are obtained from, say, an 11-circle-polynomial expansion, and are removed from the aberration function, wrong polishing will result by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for an annular pupil. The various results are illustrated analytically and numerically by considering an annular Seidel aberration function.

  16. Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation

    Science.gov (United States)

    Gordon, Sheldon P.; Yang, Yajun

    2017-01-01

    This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…

  17. On the estimation of the degree of regression polynomial

    International Nuclear Information System (INIS)

    Toeroek, Cs.

    1997-01-01

    The mathematical functions most commonly used to model curvature in plots are polynomials. Generally, the higher the degree of the polynomial, the more complex is the trend that its graph can represent. We propose a new statistical-graphical approach based on the discrete projective transformation (DPT) to estimating the degree of polynomial that adequately describes the trend in the plot

  18. Interlacing of zeros of quasi-orthogonal meixner polynomials | Driver ...

    African Journals Online (AJOL)

    ... interlacing of zeros of quasi-orthogonal Meixner polynomials Mn(x;β; c) with the zeros of their nearest orthogonal counterparts Mt(x;β + k; c), l; n ∈ ℕ, k ∈ {1; 2}; is also discussed. Mathematics Subject Classication (2010): 33C45, 42C05. Key words: Discrete orthogonal polynomials, quasi-orthogonal polynomials, Meixner

  19. Commutators with idempotent values on multilinear polynomials in ...

    Indian Academy of Sciences (India)

    Multilinear polynomial; derivations; generalized polynomial identity; prime ring; right ideal. Abstract. Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R , f ( x 1 , … , x n ) a multilinear polynomial over C , ϱ a nonzero right ideal of R and m > 1 a fixed integer such that.

  20. On the Lorentz degree of a product of polynomials

    KAUST Repository

    Ait-Haddou, Rachid

    2015-01-01

    In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al. (1991) on polynomials with nonnegative coefficients.

  1. Degenerate r-Stirling Numbers and r-Bell Polynomials

    Science.gov (United States)

    Kim, T.; Yao, Y.; Kim, D. S.; Jang, G.-W.

    2018-01-01

    The purpose of this paper is to exploit umbral calculus in order to derive some properties, recurrence relations, and identities related to the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials. Especially, we will express the degenerate r-Bell polynomials as linear combinations of many well-known families of special polynomials.

  2. The quality of zero bounds for complex polynomials.

    Science.gov (United States)

    Dehmer, Matthias; Tsoy, Yury Robertovich

    2012-01-01

    In this paper, we evaluate the quality of zero bounds on the moduli of univariate complex polynomials. We select classical and recently developed bounds and evaluate their quality by using several sets of complex polynomials. As the quality of priori bounds has not been investigated thoroughly, our results can be useful to find optimal bounds to locate the zeros of complex polynomials.

  3. Lowering and raising operators for some special orthogonal polynomials

    NARCIS (Netherlands)

    Koornwinder, T.H.

    2006-01-01

    Abstract: This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated with root system BC2 a new pair of lowering and

  4. Multiple Representations and the Understanding of Taylor Polynomials

    Science.gov (United States)

    Habre, Samer

    2009-01-01

    The study of Maclaurin and Taylor polynomials entails the comprehension of various new mathematical ideas. Those polynomials are initially discussed at the college level in a calculus class and then again in a course on numerical methods. This article investigates the understanding of these polynomials by students taking a numerical methods class…

  5. Root and critical point behaviors of certain sums of polynomials

    Indian Academy of Sciences (India)

    Seon-Hong Kim

    2018-04-24

    Apr 24, 2018 ... Introduction. There is an extensive literature concerning roots of sums of polynomials. Many authors. [5–7] have written about these polynomials. Perhaps the most immediate question of sums of polynomials, A+ B = C is: 'given bounds for the roots of A and B, what bounds can be given for the roots of C?' ...

  6. Uniqueness and zeros of q-shift difference polynomials

    Indian Academy of Sciences (India)

    In this paper, we consider the zero distributions of -shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to -shift difference polynomials. We also investigate the uniqueness problem of -shift ...

  7. Strong result for real zeros of random algebraic polynomials

    Directory of Open Access Journals (Sweden)

    T. Uno

    2001-01-01

    Full Text Available An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non-identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.

  8. Generating the Patterns of Variation with GeoGebra: The Case of Polynomial Approximations

    Science.gov (United States)

    Attorps, Iiris; Björk, Kjell; Radic, Mirko

    2016-01-01

    In this paper, we report a teaching experiment regarding the theory of polynomial approximations at the university mathematics teaching in Sweden. The experiment was designed by applying Variation theory and by using the free dynamic mathematics software GeoGebra. The aim of this study was to investigate if the technology-assisted teaching of…

  9. A polynomial analytical method for one-group slab-geometry discrete ordinates heterogeneous problems

    International Nuclear Information System (INIS)

    Leal, Andre Luiz do Carmo

    2008-01-01

    In this work we evaluate polynomial approximations to obtain the transfer functions that appear in SGF auxiliary equations (Green's Functions) for monoenergetic linearly anisotropic scattering SN equations in one-dimensional Cartesian geometry. For this task we use Lagrange Polynomials in order to compare the numerical results with the ones generated by the standard SGF method applied to SN problems in heterogeneous domains. This work is a preliminary investigation of a new proposal for handling the transverse leakage terms that appear in the transverse-integrated one-dimensional SN equations when we use the SGF - exponential nodal method (SGF-ExpN) in multidimensional rectangular geometry. (author)

  10. The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

    Directory of Open Access Journals (Sweden)

    Luca Bagnato

    2016-08-01

    Full Text Available This paper carries out an investigation of the orthogonal-polynomial approach to reshaping symmetric distributions to fit in with data requirements so as to cover the multivariate case. With this objective in mind, reference is made to the class of spherical distributions, given that they provide a natural multivariate generalization of univariate even densities. After showing how to tailor a spherical distribution via orthogonal polynomials to better comply with kurtosis requirements, we provide operational conditions for the positiveness of the resulting multivariate Gram–Charlier-like expansion, together with its kurtosis range. Finally, the approach proposed here is applied to some selected spherical distributions.

  11. Adaptive method for multi-dimensional integration and selection of a base of chaos polynomials

    International Nuclear Information System (INIS)

    Crestaux, T.

    2011-01-01

    This research thesis addresses the propagation of uncertainty in numerical simulations and its processing within a probabilistic framework by a functional approach based on random variable functions. The author reports the use of the spectral method to represent random variables by development in polynomial chaos. More precisely, the author uses the method of non-intrusive projection which uses the orthogonality of Chaos Polynomials to compute the development coefficients by approximation of scalar products. The approach is applied to a cavity and to waste storage [fr

  12. Polynomial filtered HMC-an algorithm for lattice QCD with dynamical quarks

    Science.gov (United States)

    Kamleh, Waseem; Peardon, Mike

    2012-09-01

    Polynomial approximations to the inverse of the fermion matrix are used to filter the dynamics of the upper energy scales in Hybrid Monte Carlo (HMC) simulations. The use of a multiple time-scale integration scheme then allows the filtered pseudofermions to be evolved using a coarse step size. The algorithm is applied to Nf=2 flavour simulations at two different quark masses. Polynomial filtering yields a reduction in the computational expense of the molecular dynamics integration of between three- and fivefold when compared to standard HMC. Significantly, the observed speedup is better at the lower quark mass.

  13. Fundamental Frequency Estimation using Polynomial Rooting of a Subspace-Based Method

    DEFF Research Database (Denmark)

    Jensen, Jesper Rindom; Christensen, Mads Græsbøll; Jensen, Søren Holdt

    2010-01-01

    We consider the problem of estimating the fundamental frequency of periodic signals such as audio and speech. A novel estimation method based on polynomial rooting of the harmonic MUltiple SIgnal Classification (HMUSIC) is presented. By applying polynomial rooting, we obtain two significant...... improvements compared to HMUSIC. First, by using the proposed method we can obtain an estimate of the fundamental frequency without doing a grid search like in HMUSIC. This is due to that the fundamental frequency is estimated as the argument of the root lying closest to the unit circle. Second, we obtain...

  14. Non-axisymmetric Aberration Patterns from Wide-field Telescopes Using Spin-weighted Zernike Polynomials

    Science.gov (United States)

    Kent, Stephen M.

    2018-04-01

    If the optical system of a telescope is perturbed from rotational symmetry, the Zernike wavefront aberration coefficients describing that system can be expressed as a function of position in the focal plane using spin-weighted Zernike polynomials. Methodologies are presented to derive these polynomials to arbitrary order. This methodology is applied to aberration patterns produced by a misaligned Ritchey–Chrétien telescope and to distortion patterns at the focal plane of the DESI optical corrector, where it is shown to provide a more efficient description of distortion than conventional expansions.

  15. Chebyshev polynomials in the spectral Tau method and applications to Eigenvalue problems

    Science.gov (United States)

    Johnson, Duane

    1996-01-01

    Chebyshev Spectral methods have received much attention recently as a technique for the rapid solution of ordinary differential equations. This technique also works well for solving linear eigenvalue problems. Specific detail is given to the properties and algebra of chebyshev polynomials; the use of chebyshev polynomials in spectral methods; and the recurrence relationships that are developed. These formula and equations are then applied to several examples which are worked out in detail. The appendix contains an example FORTRAN program used in solving an eigenvalue problem.

  16. Identification of Super Phenix steam generator by a simple polynomial model

    International Nuclear Information System (INIS)

    Rousseau, I.

    1981-01-01

    This note suggests a method of identification for the steam generator of the Super-Phenix fast neutron power plant for simple polynomial models. This approach is justified in the selection of the adaptive control. The identification algorithms presented will be applied to multivariable input-output behaviours. The results obtained with the representation in self-regressive form and by simple polynomial models will be compared and the effect of perturbations on the output signal will be tested, in order to select a good identification algorithm for multivariable adaptive regulation [fr

  17. Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials

    Directory of Open Access Journals (Sweden)

    Claus Vogl

    2014-11-01

    Full Text Available In population genetics, parameters describing forces such as mutation, migration and drift are generally inferred from molecular data. Lately, approximate methods based on simulations and summary statistics have been widely applied for such inference, even though these methods waste information. In contrast, probabilistic methods of inference can be shown to be optimal, if their assumptions are met. In genomic regions where recombination rates are high relative to mutation rates, polymorphic nucleotide sites can be assumed to evolve independently from each other. The distribution of allele frequencies at a large number of such sites has been called “allele-frequency spectrum” or “site-frequency spectrum” (SFS. Conditional on the allelic proportions, the likelihoods of such data can be modeled as binomial. A simple model representing the evolution of allelic proportions is the biallelic mutation-drift or mutation-directional selection-drift diffusion model. With series of orthogonal polynomials, specifically Jacobi and Gegenbauer polynomials, or the related spheroidal wave function, the diffusion equations can be solved efficiently. In the neutral case, the product of the binomial likelihoods with the sum of such polynomials leads to finite series of polynomials, i.e., relatively simple equations, from which the exact likelihoods can be calculated. In this article, the use of orthogonal polynomials for inferring population genetic parameters is investigated.

  18. On rational classical orthogonal polynomials and their application for explicit computation of inverse Laplace transforms

    Directory of Open Access Journals (Sweden)

    Masjed-Jamei Mohammad

    2005-01-01

    Full Text Available From the main equation ( a x 2 +bx+c y ″ n ( x +( dx+e y ′ n ( x −n( ( n−1 a+d y n ( x =0 , n∈ ℤ + , six finite and infinite classes of orthogonal polynomials can be extracted. In this work, first we have a survey on these classes, particularly on finite classes, and their corresponding rational orthogonal polynomials, which are generated by Mobius transform x=p z −1 +q , p≠0 , q∈ℝ . Some new integral relations are also given in this section for the Jacobi, Laguerre, and Bessel orthogonal polynomials. Then we show that the rational orthogonal polynomials can be a very suitable tool to compute the inverse Laplace transform directly, with no additional calculation for finding their roots. In this way, by applying infinite and finite rational classical orthogonal polynomials, we give three basic expansions of six ones as a sample for computation of inverse Laplace transform.

  19. Polynomial expansions and transition strengths

    International Nuclear Information System (INIS)

    Draayer, J.P.

    1980-01-01

    The subject is statistical spectroscopy applied to determining strengths and strength sums of excitation processes in nuclei. The focus will be on a ds-shell isoscalar E2 study with detailed shell-model results providing the standard for comparison; similar results are available for isovector E2 and M1 and E4 transitions as well as for single-particle transfer and ν +- decay. The present study is intended to serve as a tutorial for applications where shell-model calculations are not feasible. The problem is posed and a schematic theory for strengths and sums is presented. The theory is extended to include the effect of correlations between H, the system Hamiltonian, and theta, the excitation operator. Associated with correlation measures is a geometry that can be used to anticipate the goodness of a symmetry. This is illustrated for pseudo SU(3) in the fp-shell. Some conclusions about fluctuations and collectivity that one can deduce from the statistical results for strengths are presented

  20. Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials.

    Science.gov (United States)

    Lombardo, Rosaria; Beh, Eric J; Kroonenberg, Pieter M

    2016-06-01

    The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display-the polynomial biplot.

  1. Describing Quadratic Cremer Point Polynomials by Parabolic Perturbations

    DEFF Research Database (Denmark)

    Sørensen, Dan Erik Krarup

    1996-01-01

    We describe two infinite order parabolic perturbation proceduresyielding quadratic polynomials having a Cremer fixed point. The main ideais to obtain the polynomial as the limit of repeated parabolic perturbations.The basic tool at each step is to control the behaviour of certain externalrays.Polynomials...... the existence of polynomials having an explicitlygiven external ray accumulating both at the Cremer point and at its non-periodicpreimage. We think of the Julia set as containing a "topologists double comb".In the one-sided case we prove a weaker result: the existence of polynomials havingan explicitly given...

  2. Planar Harmonic and Monogenic Polynomials of Type A

    Directory of Open Access Journals (Sweden)

    Charles F. Dunkl

    2016-10-01

    Full Text Available Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on R N . The Dunkl operators are denoted by T j for 1 ≤ j ≤ N , and the Laplacian Δ κ = ∑ j = 1 N T j 2 . This paper finds the homogeneous harmonic polynomials annihilated by all T j for j > 2 . The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.

  3. Zernike polynomials for photometric characterization of LEDs

    International Nuclear Information System (INIS)

    Velázquez, J L; Ferrero, A; Pons, A; Campos, J; Hernanz, M L

    2016-01-01

    We propose a method based on Zernike polynomials to characterize photometric quantities and descriptors of light emitting diodes (LEDs) from measurements of the angular distribution of the luminous intensity, such as total luminous flux, BA, inhomogeneity, anisotropy, direction of the optical axis and Lambertianity of the source. The performance of this method was experimentally tested for 18 high-power LEDs from different manufacturers and with different photometric characteristics. A small set of Zernike coefficients can be used to calculate all the mentioned photometric quantities and descriptors. For applications not requiring a great accuracy such as those of lighting design, the angular distribution of the luminous intensity of most of the studied LEDs can be interpolated with only two Zernike polynomials. (paper)

  4. Orthogonal polynomials in neutron transport theory

    Energy Technology Data Exchange (ETDEWEB)

    Dehesa, J.S. (Granada Univ. (Spain). Facultad de Ciencias)

    1982-01-01

    The asymptotic average properties of zeros of the polynomials gsub(k)sup(m) (x), which play a fundamental role in neutron transport and radiative transfer theories, are investigated analytically in terms of the angular expansion coefficients wsub(k) of the scattering kernel for three wide classes of scattering models. In particular it is found that the scattering models of Eccleston-McCormick (J. Nucl. Energy.; 24:23 (1970)), Shultis et al (Nucl. Sci. Eng.; 59:53 (1976)) and Henyey-Greenstein (Astrophys. J.; 93:70 (1941)) belong in one of the above-mentioned classes, and their associated polynomials gsub(k)sup(m) (x) have the same asymptotic density of zeros.

  5. Polynomial chaos representation of databases on manifolds

    Energy Technology Data Exchange (ETDEWEB)

    Soize, C., E-mail: christian.soize@univ-paris-est.fr [Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France); Ghanem, R., E-mail: ghanem@usc.edu [University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)

    2017-04-15

    Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.

  6. Polynomial structures in one-loop amplitudes

    International Nuclear Information System (INIS)

    Britto, Ruth; Feng Bo; Yang Gang

    2008-01-01

    A general one-loop scattering amplitude may be expanded in terms of master integrals. The coefficients of the master integrals can be obtained from tree-level input in a two-step process. First, use known formulas to write the coefficients of (4-2ε)-dimensional master integrals; these formulas depend on an additional variable, u, which encodes the dimensional shift. Second, convert the u-dependent coefficients of (4-2ε)-dimensional master integrals to explicit coefficients of dimensionally shifted master integrals. This procedure requires the initial formulas for coefficients to have polynomial dependence on u. Here, we give a proof of this property in the case of massless propagators. The proof is constructive. Thus, as a byproduct, we produce different algebraic expressions for the scalar integral coefficients, in which the polynomial property is apparent. In these formulas, the box and pentagon contributions are separated explicitly.

  7. Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs

    KAUST Repository

    Chkifa, Abdellah

    2015-04-08

    Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case, the least-squares method is quasi-optimal in expectation in [A. Cohen, M A. Davenport and D. Leviatan. Found. Comput. Math. 13 (2013) 819–834] and in probability in [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Found. Comput. Math. 14 (2014) 419–456], under suitable conditions that relate the number of samples with respect to the dimension of the polynomial space. Here “quasi-optimal” means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the quasi-optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space, independently of the anisotropic shape and of the number of variables. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss “inclusion type” elliptic PDE models, and derive an exponential convergence estimate for the least-squares method. Numerical results confirm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate.

  8. A Deterministic and Polynomial Modified Perceptron Algorithm

    Directory of Open Access Journals (Sweden)

    Olof Barr

    2006-01-01

    Full Text Available We construct a modified perceptron algorithm that is deterministic, polynomial and also as fast as previous known algorithms. The algorithm runs in time O(mn3lognlog(1/ρ, where m is the number of examples, n the number of dimensions and ρ is approximately the size of the margin. We also construct a non-deterministic modified perceptron algorithm running in timeO(mn2lognlog(1/ρ.

  9. Polynomials and identities on real Banach spaces

    Czech Academy of Sciences Publication Activity Database

    Hájek, Petr Pavel; Kraus, M.

    2012-01-01

    Roč. 385, č. 2 (2012), s. 1015-1026 ISSN 0022-247X R&D Projects: GA ČR(CZ) GAP201/11/0345 Institutional research plan: CEZ:AV0Z10190503 Keywords : Polynomials on Banach spaces Subject RIV: BA - General Mathematics Impact factor: 1.050, year: 2012 http://www.sciencedirect.com/science/article/pii/S0022247X11006743

  10. Piecewise polynomial solutions to linear inverse problems

    DEFF Research Database (Denmark)

    Hansen, Per Christian; Mosegaard, K.

    1996-01-01

    We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.g.......g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly....

  11. Polynomial Variables and the Jacobian Problem

    Indian Academy of Sciences (India)

    Most readers would be familiar with the technique of shifting the origin to simplify the equation -of a locus. This process is the same as choosing new coordinate axes, ..... 1982 by H Bass, E H Connell and D Wright. Points at Infinity. Let f be a nonzero polynomial in X, Y, and let d = deg (f). Then f can be written in the form f =.

  12. Completeness of the ring of polynomials

    DEFF Research Database (Denmark)

    Thorup, Anders

    2015-01-01

    Consider the polynomial ring R:=k[X1,…,Xn]R:=k[X1,…,Xn] in n≥2n≥2 variables over an uncountable field k. We prove that R   is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we prove...

  13. Zero sets of polynomials in several variables

    Czech Academy of Sciences Publication Activity Database

    Aron, R. M.; Hájek, Petr Pavel

    2006-01-01

    Roč. 86, č. 6 (2006), s. 561-568 ISSN 0003-889X R&D Projects: GA ČR(CZ) GA201/01/1198; GA ČR(CZ) GA201/04/0090; GA AV ČR(CZ) IAA1019205 Institutional research plan: CEZ:AV0Z10190503 Keywords : polynomial * zero set Subject RIV: BA - General Mathematics Impact factor: 0.341, year: 2006

  14. Trigonometric Polynomials For Estimation Of Spectra

    Science.gov (United States)

    Greenhall, Charles A.

    1990-01-01

    Orthogonal sets of trigonometric polynomials used as suboptimal substitutes for discrete prolate-spheroidal "windows" of Thomson method of estimation of spectra. As used here, "windows" denotes weighting functions used in sampling time series to obtain their power spectra within specified frequency bands. Simplified windows designed to require less computation than do discrete prolate-spheroidal windows, albeit at price of some loss of accuracy.

  15. Detecting Prime Numbers via Roots of Polynomials

    Science.gov (United States)

    Dobbs, David E.

    2012-01-01

    It is proved that an integer n [greater than or equal] 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z[subscript n], the ring of integers modulo n, such that each element of Z[subscript n] is a root of f. This classroom note could find use in…

  16. Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes

    Directory of Open Access Journals (Sweden)

    Robert Griffiths

    2016-05-01

    Full Text Available This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous time, there are N independent and identically distributed birth and death processes each with support 0 , 1 , … . The state space in the composition process is the number of processes in the different states 0 , 1 , … . Dealing with the spectral representation requires new extensions of the multivariate Krawtchouk polynomials to orthogonal polynomials on a multinomial distribution with a countable infinity of states.

  17. From $r$-Linearized Polynomial Equations to $r^m$-Linearized Polynomial Equations

    OpenAIRE

    Fernando, Neranga; Hou, Xiang-dong

    2015-01-01

    Let $r$ be a prime power and $q=r^m$. For $0\\le i\\le m-1$, let $f_i\\in \\mathbb{F}_r[x]$ be $q$-linearized and $a_i\\in \\mathbb{F}_q$. Assume that $z\\in \\mathbb{\\bar{F}}_r$ satisfies the equation $\\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$, where $\\sum_{i=0}^{m-1}a_if_i^{r_i}\\in \\mathbb{F}_q[x]$ is an $r$-linearized polynomial. It is shown that $z$ satisfies a $q$-linearized polynomial equation with coefficients in $\\mathbb{F}_r$. This result provides an explanation for numerous permutation polynomials...

  18. A Polynomial Subset-Based Efficient Multi-Party Key Management System for Lightweight Device Networks.

    Science.gov (United States)

    Mahmood, Zahid; Ning, Huansheng; Ghafoor, AtaUllah

    2017-03-24

    Wireless Sensor Networks (WSNs) consist of lightweight devices to measure sensitive data that are highly vulnerable to security attacks due to their constrained resources. In a similar manner, the internet-based lightweight devices used in the Internet of Things (IoT) are facing severe security and privacy issues because of the direct accessibility of devices due to their connection to the internet. Complex and resource-intensive security schemes are infeasible and reduce the network lifetime. In this regard, we have explored the polynomial distribution-based key establishment schemes and identified an issue that the resultant polynomial value is either storage intensive or infeasible when large values are multiplied. It becomes more costly when these polynomials are regenerated dynamically after each node join or leave operation and whenever key is refreshed. To reduce the computation, we have proposed an Efficient Key Management (EKM) scheme for multiparty communication-based scenarios. The proposed session key management protocol is established by applying a symmetric polynomial for group members, and the group head acts as a responsible node. The polynomial generation method uses security credentials and secure hash function. Symmetric cryptographic parameters are efficient in computation, communication, and the storage required. The security justification of the proposed scheme has been completed by using Rubin logic, which guarantees that the protocol attains mutual validation and session key agreement property strongly among the participating entities. Simulation scenarios are performed using NS 2.35 to validate the results for storage, communication, latency, energy, and polynomial calculation costs during authentication, session key generation, node migration, secure joining, and leaving phases. EKM is efficient regarding storage, computation, and communication overhead and can protect WSN-based IoT infrastructure.

  19. Collective motions and non-polynomial lagrangians in Fokker-Planck dynamics

    International Nuclear Information System (INIS)

    Spina, A.; Vucetich, H.

    1986-04-01

    A method based on the ideas of collective motion is applied to Fokker-Planck dynamics. The usual diagramatic techniques used in stochastic dynamics are enlarged in order to deal with the non-polynomial Lagrangians that appear in the theory. The technique is tested and applied to the case of a self-sustained sinusoidal oscillator whose statistical behaviour is well understood: the Poincare oscillator. (author)

  20. Segmented polynomials for incidence rate estimation from prevalence data.

    Science.gov (United States)

    Mahiané, Severin Guy; Laeyendecker, Oliver

    2017-01-30

    The study considers the problem of estimating incidence of a non remissible infection (or disease) with possibly differential mortality using data from a(several) cross-sectional prevalence survey(s). Fitting segmented polynomial models is proposed to estimate the incidence as a function of age, using the maximum likelihood method. The approach allows automatic search for optimal position of knots, and model selection is performed using the Akaike information criterion. The method is applied to simulated data and to estimate HIV incidence among men in Zimbabwe using data from both the NIMH Project Accept (HPTN 043) and Zimbabwe Demographic Health Surveys (2005-2006). Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

  1. Polynomial solutions of the Monge-Ampère equation

    Energy Technology Data Exchange (ETDEWEB)

    Aminov, Yu A [B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar' kov (Ukraine)

    2014-11-30

    The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.

  2. Least squares orthogonal polynomial approximation in several independent variables

    International Nuclear Information System (INIS)

    Caprari, R.S.

    1992-06-01

    This paper begins with an exposition of a systematic technique for generating orthonormal polynomials in two independent variables by application of the Gram-Schmidt orthogonalization procedure of linear algebra. It is then demonstrated how a linear least squares approximation for experimental data or an arbitrary function can be generated from these polynomials. The least squares coefficients are computed without recourse to matrix arithmetic, which ensures both numerical stability and simplicity of implementation as a self contained numerical algorithm. The Gram-Schmidt procedure is then utilised to generate a complete set of orthogonal polynomials of fourth degree. A theory for the transformation of the polynomial representation from an arbitrary basis into the familiar sum of products form is presented, together with a specific implementation for fourth degree polynomials. Finally, the computational integrity of this algorithm is verified by reconstructing arbitrary fourth degree polynomials from their values at randomly chosen points in their domain. 13 refs., 1 tab

  3. Explicitly integrable polynomial Hamiltonians and evaluation of Lie transformations

    International Nuclear Information System (INIS)

    Shi, J.; Yan, Y.T.

    1993-01-01

    We have found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree, with which each associated polynomial Hamiltonian is integrable, and the associated Lie transformation can be evaluated exactly. An integrable polynomial factorization has thus been developed to convert a sympletic map in the form of a Dragt-Finn factorization into a product of exactly evaluable Lie transformations associated with integrable polynomials. Having a small number of factorization bases of integrable polynomials enables one to consider a factorization with the use of high-order symplectic integrators so that a symplectic map can always be evaluated with the desired accuracy. The results are significant for studying the long-term stability of beams in accelerators

  4. Comparative assessment of freeform polynomials as optical surface descriptions.

    Science.gov (United States)

    Kaya, Ilhan; Thompson, Kevin P; Rolland, Jannick P

    2012-09-24

    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.

  5. Polynomials for torus links from Chern-Simons gauge theories

    International Nuclear Information System (INIS)

    Isidro, J.M.; Labastida, J.M.F.; Ramallo, A.V.

    1993-01-01

    Invariant polynomials for torus links are obtained in the framework of the Chern-Simons topological gauge theory. The polynomials are computed as vacuum expectation values on the three-sphere of Wilson line operators representing the Verlinde algebra of the corresponding rational conformal field theory. In the case of the SU(2) gauge theory our results provide explicit expressions for the Jones polynomial as well as for the polynomials associated to the N-state (N>2) vertex models (Akutsu-Wadati polynomials). By means of the Chern-Simons coset construction, the minimal unitary models are analyzed, showing that the corresponding link invariants factorize into two SU(2) polynomials. A method to obtain skein rules from the Chern-Simons knot operators is developed. This procedure yields the eigenvalues of the braiding matrix of the corresponding conformal field theory. (orig.)

  6. Note on Generating Orthogonal Polynomials and Their Application in Solving Complicated Polynomial Regression Tasks

    Czech Academy of Sciences Publication Activity Database

    Knížek, J.; Tichý, Petr; Beránek, L.; Šindelář, Jan; Vojtěšek, B.; Bouchal, P.; Nenutil, R.; Dedík, O.

    2010-01-01

    Roč. 7, č. 10 (2010), s. 48-60 ISSN 0974-5718 Grant - others:GA MZd(CZ) NS9812; GA ČR(CZ) GAP304/10/0868 Institutional research plan: CEZ:AV0Z10300504; CEZ:AV0Z10750506 Keywords : polynomial regression * orthogonalization * numerical methods * markers * biomarkers Subject RIV: BA - General Mathematics

  7. Open Problems Related to the Hurwitz Stability of Polynomials Segments

    Directory of Open Access Journals (Sweden)

    Baltazar Aguirre-Hernández

    2018-01-01

    Full Text Available In the framework of robust stability analysis of linear systems, the development of techniques and methods that help to obtain necessary and sufficient conditions to determine stability of convex combinations of polynomials is paramount. In this paper, knowing that Hurwitz polynomials set is not a convex set, a brief overview of some results and open problems concerning the stability of the convex combinations of Hurwitz polynomials is then provided.

  8. Recurrence relations and weight functions for approximation by orthonormal polynomials

    International Nuclear Information System (INIS)

    Bisoi, A.K.

    1989-12-01

    The popular three-term recurrence relation is found to be a numerically unstable algorithm for generation of higher order orthogonal polynomials. For stability, a k-term recurrence relation for generating the k th order polynomial is suggested. Instead of assigning equal weight to data points with unspecified errors, the use of the Jacobian weight function is suggested for generating the orthogonal polynomials to effect a closer representation. (author). 6 refs, 1 tab

  9. Vector polynomials for direct analysis of circular wavefront slope data.

    Science.gov (United States)

    Mahajan, Virendra N; Acosta, Eva

    2017-10-01

    In the aberration analysis of a circular wavefront, Zernike circle polynomials are used to obtain its wave aberration coefficients. To obtain these coefficients from the wavefront slope data, we need vector functions that are orthogonal to the gradients of the Zernike polynomials, and are irrotational so as to propagate minimum uncorrelated random noise from the data to the coefficients. In this paper, we derive such vector functions, which happen to be polynomials.

  10. Root and Critical Point Behaviors of Certain Sums of Polynomials

    Indian Academy of Sciences (India)

    13

    There is an extensive literature concerning roots of sums of polynomials. Many papers and books([5], [6],. [7]) have written about these polynomials. Perhaps the most immediate question of sums of polynomials,. A + B = C, is “given bounds for the roots of A and B, what bounds can be given for the roots of C?” By. Fell [3], if ...

  11. Applications of polynomial optimization in financial risk investment

    Science.gov (United States)

    Zeng, Meilan; Fu, Hongwei

    2017-09-01

    Recently, polynomial optimization has many important applications in optimization, financial economics and eigenvalues of tensor, etc. This paper studies the applications of polynomial optimization in financial risk investment. We consider the standard mean-variance risk measurement model and the mean-variance risk measurement model with transaction costs. We use Lasserre's hierarchy of semidefinite programming (SDP) relaxations to solve the specific cases. The results show that polynomial optimization is effective for some financial optimization problems.

  12. A Fast lattice-based polynomial digital signature system for m-commerce

    Science.gov (United States)

    Wei, Xinzhou; Leung, Lin; Anshel, Michael

    2003-01-01

    The privacy and data integrity are not guaranteed in current wireless communications due to the security hole inside the Wireless Application Protocol (WAP) version 1.2 gateway. One of the remedies is to provide an end-to-end security in m-commerce by applying application level security on top of current WAP1.2. The traditional security technologies like RSA and ECC applied on enterprise's server are not practical for wireless devices because wireless devices have relatively weak computation power and limited memory compared with server. In this paper, we developed a lattice based polynomial digital signature system based on NTRU's Polynomial Authentication and Signature Scheme (PASS), which enabled the feasibility of applying high-level security on both server and wireless device sides.

  13. An Analytic Formula for the A_2 Jack Polynomials

    Directory of Open Access Journals (Sweden)

    Vladimir V. Mangazeev

    2007-01-01

    Full Text Available In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003, 451-482] on separation of variables (SoV for the $A_n$ Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995, 27-34] where the integral representations for the $A_2$ Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the $A_2$ Jack polynomials in terms of generalised hypergeometric functions.

  14. Positive trigonometric polynomials and signal processing applications

    CERN Document Server

    Dumitrescu, Bogdan

    2017-01-01

    This revised edition is made up of two parts: theory and applications. Though many of the fundamental results are still valid and used, new and revised material is woven throughout the text. As with the original book, the theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The programming environment has also evolved, and the books examples are changed accordingly. The applications section is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semi-definite programming form, ready to be solved with algorithms freely available, like those from the libraries SeDuMi, CVX and Pos3Poly. A new chapter discusses applications in super-resolution theory, where Bounded Real Lemma for trigonometric polynomials is an important tool. This revision is written to be more appealing and easier to use for new readers. < Features updated information on LMI...

  15. Simulation of aspheric tolerance with polynomial fitting

    Science.gov (United States)

    Li, Jing; Cen, Zhaofeng; Li, Xiaotong

    2018-01-01

    The shape of the aspheric lens changes caused by machining errors, resulting in a change in the optical transfer function, which affects the image quality. At present, there is no universally recognized tolerance criterion standard for aspheric surface. To study the influence of aspheric tolerances on the optical transfer function, the tolerances of polynomial fitting are allocated on the aspheric surface, and the imaging simulation is carried out by optical imaging software. Analysis is based on a set of aspheric imaging system. The error is generated in the range of a certain PV value, and expressed as a form of Zernike polynomial, which is added to the aspheric surface as a tolerance term. Through optical software analysis, the MTF of optical system can be obtained and used as the main evaluation index. Evaluate whether the effect of the added error on the MTF of the system meets the requirements of the current PV value. Change the PV value and repeat the operation until the acceptable maximum allowable PV value is obtained. According to the actual processing technology, consider the error of various shapes, such as M type, W type, random type error. The new method will provide a certain development for the actual free surface processing technology the reference value.

  16. Hermite Polynomials and the Inverse Problem for Collisionless Equilibria

    Science.gov (United States)

    Allanson, O.; Neukirch, T.; Troscheit, S.; Wilson, F.

    2017-12-01

    It is long established that Hermite polynomial expansions in either velocity or momentum space can elegantly encode the non-Maxwellian velocity-space structure of a collisionless plasma distribution function (DF). In particular, Hermite polynomials in the canonical momenta naturally arise in the consideration of the 'inverse problem in collisionless equilibria' (IPCE): "for a given macroscopic/fluid equilibrium, what are the self-consistent Vlasov-Maxwell equilibrium DFs?". This question is of particular interest for the equilibrium and stability properties of a given macroscopic configuration, e.g. a current sheet. It can be relatively straightforward to construct a formal solution to IPCE by a Hermite expansion method, but several important questions remain regarding the use of this method. We present recent work that considers the necessary conditions of non-negativity, convergence, and the existence of all moments of an equilibrium DF solution found for IPCE. We also establish meaningful analogies between the equations that link the microscopic and macrosopic descriptions of the Vlasov-Maxwell equilibrium, and those that solve the initial value problem for the heat equation. In the language of the heat equation, IPCE poses the pressure tensor as the 'present' heat distribution over an infinite domain, and the non-Maxwellian features of the DF as the 'past' distribution. We find sufficient conditions for the convergence of the Hermite series representation of the DF, and prove that the non-negativity of the DF can be dependent on the magnetisation of the plasma. For DFs that decay at least as quickly as exp(-v^2/4), we show non-negativity is guaranteed for at least a finite range of magnetisation values, as parameterised by the ratio of the Larmor radius to the gradient length scale. 1. O. Allanson, T. Neukirch, S. Troscheit & F. Wilson: From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials, Journal of Plasma Physics, 82

  17. Supersymmetric quantum mechanics: Engineered hierarchies of integrable potentials and related orthogonal polynomials

    Energy Technology Data Exchange (ETDEWEB)

    Balondo Iyela, Daddy [International Chair in Mathematical Physics and Applications (ICMPA–UNESCO Chair), University of Abomey–Calavi, 072 B. P. 50 Cotonou, Republic of Benin (Benin); Centre for Cosmology, Particle Physics and Phenomenology (CP3), Institut de Recherche en Mathématique et Physique (IRMP), Université catholique de Louvain U.C.L., 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve (Belgium); Département de Physique, Université de Kinshasa (UNIKIN), B.P. 190 Kinshasa XI, Democratic Republic of Congo (Congo, The Democratic Republic of the); Govaerts, Jan [International Chair in Mathematical Physics and Applications (ICMPA–UNESCO Chair), University of Abomey–Calavi, 072 B. P. 50 Cotonou, Republic of Benin (Benin); Centre for Cosmology, Particle Physics and Phenomenology (CP3), Institut de Recherche en Mathématique et Physique (IRMP), Université catholique de Louvain U.C.L., 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve (Belgium); Hounkonnou, M. Norbert [International Chair in Mathematical Physics and Applications (ICMPA–UNESCO Chair), University of Abomey–Calavi, 072 B. P. 50 Cotonou, Republic of Benin (Benin)

    2013-09-15

    Within the context of supersymmetric quantum mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N. By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy which should allow for its resolution. Specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated, while the methods of supersymmetric quantum mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper, these ideas are presented and solved explicitly for the cases N= 1 and N= 2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. In the context of dressing chains and deformed polynomial Heisenberg algebras, some partial results for N⩾ 3 also exist in the literature, which should be relevant to a complete study of the N⩾ 3 general periodic hierarchies.

  18. Mobile device camera design with Q-type polynomials to achieve higher production yield.

    Science.gov (United States)

    Ma, Bin; Sharma, Katelynn; Thompson, Kevin P; Rolland, Jannick P

    2013-07-29

    The camera lenses that are built into the current generation of mobile devices are extremely stressed by the excessively tight packaging requirements, particularly the length. As a result, the aspheric departures and slopes on the lens surfaces, when designed with conventional power series based aspheres, are well beyond those encountered in most optical systems. When the as-manufactured performance is considered, the excessive aspheric slopes result in unusually high sensitivity to tilt and decenter and even despace resulting in unusually low manufacturing yield. Q(bfs) polynomials, a new formulation for nonspherical optical surfaces introduced by Forbes, not only build on orthogonal polynomials, but their unique normalization provides direct access to the RMS slope of the aspheric departure during optimization. Using surface shapes with this description in optimization results in equivalent performance with reduced alignment sensitivity and higher yield. As an additional approach to increasing yield, mechanically imposed external pivot points, introduced by Bottema, can be used as a design technique to further reduce alignment sensitivity and increase yield. In this paper, the Q-type polynomials and external pivot points were applied to a mobile device camera lens designed using an active RMS slope constraint that was then compared to a design developed using conventional power series surface descriptions. Results show that slope constrained Q-type polynomial description together with external pivot points lead directly to solutions with significantly higher manufacturing yield.

  19. Fast track segment finding in the Monitored Drift Tubes (MDT) of the ATLAS Muon Spectrometer using a Legendre transform algorithm

    CERN Document Server

    Ntekas, Konstantinos; The ATLAS collaboration

    2018-01-01

    Many of the physics goals of ATLAS in the High Luminosity LHC era, including precision studies of the Higgs boson, require an unprescaled single muon trigger with a 20 GeV threshold. The selectivity of the current ATLAS first-level muon trigger is limited by the moderate spatial resolution of the muon trigger chambers. By incorporating the precise tracking of the MDT, the muon transverse momentum can be measured with an accuracy close to that of the offline reconstruction at the trigger level, sharpening the trigger turn-on curves and reducing the single muon trigger rate. A novel algorithm is proposed which reconstructs segments from MDT hits in an FPGA and find tracks within the tight latency constraints of the ATLAS first-level muon trigger. The algorithm represents MDT drift circles as curves in the Legendre space and returns one or more segment lines tangent to the maximum possible number of drift circles.  This algorithm is implemented without the need of resource and time consuming hit position calcul...

  20. Two-Stage Method Based on Local Polynomial Fitting for a Linear Heteroscedastic Regression Model and Its Application in Economics

    Directory of Open Access Journals (Sweden)

    Liyun Su

    2012-01-01

    Full Text Available We introduce the extension of local polynomial fitting to the linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improving the traditional two-stage method. Due to nonparametric technique of local polynomial estimation, we do not need to know the heteroscedastic function. Therefore, we can improve the estimation precision, when the heteroscedastic function is unknown. Furthermore, we focus on comparison of parameters and reach an optimal fitting. Besides, we verify the asymptotic normality of parameters based on numerical simulations. Finally, this approach is applied to a case of economics, and it indicates that our method is surely effective in finite-sample situations.

  1. On Continued Fraction Expansion of Real Roots of Polynomial Systems

    DEFF Research Database (Denmark)

    Mantzaflaris, Angelos; Mourrain, Bernard; Tsigaridas, Elias

    2011-01-01

    We elaborate on a correspondence between the coefficients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions that use only integer arithmetic (in contrast to the Bernstein basis) and are...

  2. On polynomial selection for the general number field sieve

    Science.gov (United States)

    Kleinjung, Thorsten

    2006-12-01

    The general number field sieve (GNFS) is the asymptotically fastest algorithm for factoring large integers. Its runtime depends on a good choice of a polynomial pair. In this article we present an improvement of the polynomial selection method of Montgomery and Murphy which has been used in recent GNFS records.

  3. Optimal Approximation of Biquartic Polynomials by Bicubic Splines

    Directory of Open Access Journals (Sweden)

    Kačala Viliam

    2018-01-01

    The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.

  4. Classification of complex polynomial vector fields in one complex variable

    DEFF Research Database (Denmark)

    Branner, Bodil; Dias, Kealey

    2010-01-01

    This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing the top...

  5. A Combinatorial Proof of a Result on Generalized Lucas Polynomials

    Directory of Open Access Journals (Sweden)

    Laugier Alexandre

    2016-09-01

    Full Text Available We give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation 〈n〉 = s〈n-1〉+t〈n-2〉 for n ≥ 2. The initial values are 〈0〉 = 2; 〈1〉= s, respectively.

  6. Orthonormal polynomials for elliptical wavefronts with an arbitrary orientation.

    Science.gov (United States)

    Díaz, José A; Navarro, Rafael

    2014-04-01

    We generalize the analytical form of the orthonormal elliptical polynomials for any arbitrary aspect ratio to arbitrary orientation and give expression for them up to the 4th order. The utility of the polynomials is demonstrated by obtaining the expansion up to the 8th order in two examples of an off-axis wavefront exiting from an optical system with a vignetted pupil.

  7. Animating Nested Taylor Polynomials to Approximate a Function

    Science.gov (United States)

    Mazzone, Eric F.; Piper, Bruce R.

    2010-01-01

    The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or…

  8. On peculiar properties of generating functions of some orthogonal polynomials

    International Nuclear Information System (INIS)

    Szabłowski, Paweł J

    2012-01-01

    We prove that for |x| ⩽ |t| ≥( t i )/(q) i h n+i ( x|q) =h n (x|t,q) Σ i≥0 (t i )/(q) i h i (x|q), where h n (x|q) and h n (x|t, q) are respectively the so-called q-Hermite and the big q-Hermite polynomials, and (q) n denotes the so-called q-Pochhammer symbol. We prove similar equalities involving big q-Hermite and Al-Salam–Chihara polynomials, and Al-Salam–Chihara and the so-called continuous dual q-Hahn polynomials. Moreover, we are able to relate in this way some other ‘ordinary’ orthogonal polynomials such as, e.g., Hermite, Chebyshev or Laguerre. These equalities give a new interpretation of the polynomials involved and moreover can give rise to a simple method of generating more and more general (i.e. involving more and more parameters) families of orthogonal polynomials. We pose some conjectures concerning Askey–Wilson polynomials and their possible generalizations. We prove that these conjectures are true for the cases q = 1 (classical case) and q = 0 (free case), thus paving the way to generalization of Askey–Wilson polynomials at least in these two cases. (paper)

  9. On an inequality concerning the polar derivative of a polynomial

    Indian Academy of Sciences (India)

    We also extend Zygmund's inequality to the polar derivative of a polynomial. Keywords. Zygmund's inequality; polar derivative; Lp-norm inequalities. 1. Introduction and statement of results. Let P (z) be a polynomial of degreen and let P (z) be its derivative. Then according to the famous result known as Bernstein's inequality ...

  10. Mathematical Use Of Polynomials Of Different End Periods Of ...

    African Journals Online (AJOL)

    This paper focused on how polynomials of different end period of random numbers can be used in the application of encryption and decryption of a message. Eight steps were used in generating information on how polynomials of different end periods of random numbers in the application of encryption and decryption of a ...

  11. Generalized Freud's equation and level densities with polynomial

    Indian Academy of Sciences (India)

    Home; Journals; Pramana – Journal of Physics; Volume 81; Issue 2. Generalized Freud's equation and level densities with polynomial potential. Akshat Boobna Saugata Ghosh. Research Articles Volume 81 ... Keywords. Orthogonal polynomial; Freud's equation; Dyson–Mehta method; methods of resolvents; level density.

  12. Zeros and uniqueness of Q-difference polynomials of meromorphic ...

    Indian Academy of Sciences (India)

    In this paper, we investigate the value distribution of -difference polynomials of meromorphic function of finite logarithmic order, and study the zero distribution of difference-differential polynomials [ f n ( z ) f ( q z + c ) ] ( k ) and [ f n ( z ) ( f ( q z + c ) − f ( z ) ) ] ( k ) , where () is a transcendental function of zero order.

  13. Conditional Density Approximations with Mixtures of Polynomials

    DEFF Research Database (Denmark)

    Varando, Gherardo; López-Cruz, Pedro L.; Nielsen, Thomas Dyhre

    2015-01-01

    Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce...... is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated...... basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method....

  14. Kinetic term anarchy for polynomial chaotic inflation

    Science.gov (United States)

    Nakayama, Kazunori; Takahashi, Fuminobu; Yanagida, Tsutomu T.

    2014-09-01

    We argue that there may arise a relatively flat inflaton potential over super-Planckian field values with an approximate shift symmetry, if the coefficients of the kinetic terms for many singlet scalars are subject to a certain random distribution. The inflation takes place along the flat direction with a super-Planckian length, whereas the other light directions can be stabilized by the Hubble-induced mass. The inflaton potential generically contains various shift-symmetry breaking terms, leading to a possibly large deviation of the predicted values of the spectral index and tensor-to-scalar ratio from those of the simple quadratic chaotic inflation. We revisit a polynomial chaotic inflation in supergravity as such.

  15. Dynamic normal forms and dynamic characteristic polynomial

    DEFF Research Database (Denmark)

    Frandsen, Gudmund Skovbjerg; Sankowski, Piotr

    2011-01-01

    We present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix. In the generic symmetric case, our algorithm supports rank-one updates in O(n2logn) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n2klogn......) randomized time, where k is the number of invariant factors of the matrix. The algorithm is based on the first dynamic algorithm for computing normal forms of a matrix such as the Frobenius normal form or the tridiagonal symmetric form. The algorithm can be extended to solve the matrix eigenproblem...... with relative error 2−b in additional O(nlog2nlogb) time. Furthermore, it can be used to dynamically maintain the singular value decomposition (SVD) of a generic matrix. Together with the algorithm, the hardness of the problem is studied. For the symmetric case, we present an Ω(n2) lower bound for rank...

  16. On a novel iterative method to compute polynomial approximations to Bessel functions of the first kind and its connection to the solution of fractional diffusion/diffusion-wave problems

    International Nuclear Information System (INIS)

    Yuste, Santos Bravo; Abad, Enrique

    2011-01-01

    We present an iterative method to obtain approximations to Bessel functions of the first kind J p (x) (p > -1) via the repeated application of an integral operator to an initial seed function f 0 (x). The class of seed functions f 0 (x) leading to sets of increasingly accurate approximations f n (x) is considerably large and includes any polynomial. When the operator is applied once to a polynomial of degree s, it yields a polynomial of degree s + 2, and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree. We focus on the set of polynomial approximations generated from the seed function f 0 (x) = 1. This set of polynomials is useful not only for the computation of J p (x) but also from a physical point of view, as it describes the long-time decay modes of certain fractional diffusion and diffusion-wave problems.

  17. Orthogonal polynomials derived from the tridiagonal representation approach

    Science.gov (United States)

    Alhaidari, A. D.

    2018-01-01

    The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials whose properties give the structure and dynamics of the corresponding physical system. For a certain range of parameters, one of these polynomials has a mix of continuous and discrete spectra making it suitable for describing physical systems with both scattering and bound states. In this work, we define these polynomials by their recursion relations and highlight some of their properties using numerical means. Due to the prime significance of these polynomials in physics, we hope that our short expose will encourage experts in the field of orthogonal polynomials to study them and derive their properties (weight functions, generating functions, asymptotics, orthogonality relations, zeros, etc.) analytically.

  18. Higher order branching of periodic orbits from polynomial isochrones

    Directory of Open Access Journals (Sweden)

    B. Toni

    1999-09-01

    Full Text Available We discuss the higher order local bifurcations of limit cycles from polynomial isochrones (linearizable centers when the linearizing transformation is explicitly known and yields a polynomial perturbation one-form. Using a method based on the relative cohomology decomposition of polynomial one-forms complemented with a step reduction process, we give an explicit formula for the overall upper bound of branch points of limit cycles in an arbitrary $n$ degree polynomial perturbation of the linear isochrone, and provide an algorithmic procedure to compute the upper bound at successive orders. We derive a complete analysis of the nonlinear cubic Hamiltonian isochrone and show that at most nine branch points of limit cycles can bifurcate in a cubic polynomial perturbation. Moreover, perturbations with exactly two, three, four, six, and nine local families of limit cycles may be constructed.

  19. Structured matrix based methods for approximate polynomial GCD

    CERN Document Server

    Boito, Paola

    2011-01-01

    Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree. .

  20. Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians

    International Nuclear Information System (INIS)

    Ndayiragije, F; Van Assche, W

    2013-01-01

    Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to r > 1 different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of the first kind), we construct r > 1 non-Hermitian oscillator Hamiltonians in r dimensions which are simultaneously diagonalizable and for which the common eigenstates are expressed in terms of multiple Meixner polynomials of the second kind. (paper)

  1. Jack polynomial fractional quantum Hall states and their generalizations

    International Nuclear Information System (INIS)

    Baratta, Wendy; Forrester, Peter J.

    2011-01-01

    In the study of fractional quantum Hall states, a certain clustering condition involving up to four integers has been identified. We give a simple proof that particular Jack polynomials with α=-(r-1)/(k+1), (r-1) and (k+1) relatively prime, and with partition given in terms of its frequencies by [n 0 0 (r-1)s k0 r-1 k0 r-1 k...0 r-1 m] satisfy this clustering condition. Our proof makes essential use of the fact that these Jack polynomials are translationally invariant. We also consider nonsymmetric Jack polynomials, symmetric and nonsymmetric generalized Hermite and Laguerre polynomials, and Macdonald polynomials from the viewpoint of the clustering.

  2. A note on some identities of derangement polynomials.

    Science.gov (United States)

    Kim, Taekyun; Kim, Dae San; Jang, Gwan-Woo; Kwon, Jongkyum

    2018-01-01

    The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255-258, 1978, Clarke and Sved in Math. Mag. 66(5):299-303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1-11 2018. A derangement is a permutation that has no fixed points, and the derangement number [Formula: see text] is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r -derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.

  3. Linear and evolutionary polynomial regression models to forecast coastal dynamics: Comparison and reliability assessment

    Science.gov (United States)

    Bruno, Delia Evelina; Barca, Emanuele; Goncalves, Rodrigo Mikosz; de Araujo Queiroz, Heithor Alexandre; Berardi, Luigi; Passarella, Giuseppe

    2018-01-01

    In this paper, the Evolutionary Polynomial Regression data modelling strategy has been applied to study small scale, short-term coastal morphodynamics, given its capability for treating a wide database of known information, non-linearly. Simple linear and multilinear regression models were also applied to achieve a balance between the computational load and reliability of estimations of the three models. In fact, even though it is easy to imagine that the more complex the model, the more the prediction improves, sometimes a "slight" worsening of estimations can be accepted in exchange for the time saved in data organization and computational load. The models' outcomes were validated through a detailed statistical, error analysis, which revealed a slightly better estimation of the polynomial model with respect to the multilinear model, as expected. On the other hand, even though the data organization was identical for the two models, the multilinear one required a simpler simulation setting and a faster run time. Finally, the most reliable evolutionary polynomial regression model was used in order to make some conjecture about the uncertainty increase with the extension of extrapolation time of the estimation. The overlapping rate between the confidence band of the mean of the known coast position and the prediction band of the estimated position can be a good index of the weakness in producing reliable estimations when the extrapolation time increases too much. The proposed models and tests have been applied to a coastal sector located nearby Torre Colimena in the Apulia region, south Italy.

  4. Fulltext PDF

    Indian Academy of Sciences (India)

    IAS Admin

    experiences and viewpoints on matters related to teaching and learning science. Gram–Schmidt Orthogonalization and. Legendre Polynomials. Keywords. Orthonormal subsets, moment matrix, orthogonal polynomial sequence, Legendre polynomi- als. Chanchal Kumar. Indian Institute of Science. Education and Research.

  5. Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format

    KAUST Repository

    Dolgov, Sergey

    2015-11-03

    We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its postprocessing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.

  6. Analysis of the performance of a H-Darrieus rotor under uncertainty using Polynomial Chaos Expansion

    International Nuclear Information System (INIS)

    Daróczy, László; Janiga, Gábor; Thévenin, Dominique

    2016-01-01

    Due to the growing importance of wind energy, improving the efficiency of energy conversion is essential. Horizontal Axis Wind Turbines are the most well-spread, but H-Darrieus turbines are becoming popular as well due to their simple design and easier integration. Due to the high efficiency of existing wind turbines, further improvements require numerical optimization. One important aspect is to find a better configuration that is also robust, i.e., a configuration that retains its performance under uncertainties. For this purpose, forward uncertainty propagation has to be applied. In the present work, an Uncertainty Quantification (UQ) method, Polynomial Chaos Expansion, is applied to transient, turbulent flow simulations of a variable-speed H-Darrieus turbine, taking into account uncertainty in the preset pitch angle and in the angular velocity. The resulting uncertainty of the performance coefficient and of the quasi-periodic torque curve are quantified. In the presence of stall the instantaneous torque coefficients tend to show asymmetric distributions, meaning that error bars cannot be correctly reconstructed using only mean value and standard deviation. The expected performance was always found to be smaller than in computations without UQ techniques, corresponding to up to 10% of relative losses for λ = 2.5. - Highlights: • Uncertainty Quantification/Polynomial Chaos Expansion successfully applied to H-rotor. • Accounting simultaneously for uncertainty in pitch angle and angular velocity. • Performance coefficient decreases by up to 10% when accounting for uncertainty. • For low tip-speed-ratio, high-order polynomials are needed. • Polynomial order 4 is sufficient to reconstruct distribution at higher TSR.

  7. Generalized Freud's equation and level densities with polynomial potential

    Science.gov (United States)

    Boobna, Akshat; Ghosh, Saugata

    2013-08-01

    We study orthogonal polynomials with weight $\\exp[-NV(x)]$, where $V(x)=\\sum_{k=1}^{d}a_{2k}x^{2k}/2k$ is a polynomial of order 2d. We derive the generalised Freud's equations for $d=3$, 4 and 5 and using this obtain $R_{\\mu}=h_{\\mu}/h_{\\mu -1}$, where $h_{\\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\\mu}$, are obtained using Freud's equation and using this, explicit results of level densities as $N\\rightarrow\\infty$ are derived.

  8. q-Bernoulli numbers and q-Bernoulli polynomials revisited

    Directory of Open Access Journals (Sweden)

    Kim Taekyun

    2011-01-01

    Full Text Available Abstract This paper performs a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al. (Adv Differ Equ, Article ID 951764, 9, 2010, some incorrect properties are revised. It is point out that the generating function for the q-Bernoulli numbers and polynomials is unreasonable. By using the theorem of Kim (Kyushu J Math 48, 73-86, 1994 (see Equation 9, some new generating functions for the q-Bernoulli numbers and polynomials are shown. Mathematics Subject Classification (2000 11B68, 11S40, 11S80

  9. Coherent states and Hermite polynomials on quaternionic Hilbert spaces

    Energy Technology Data Exchange (ETDEWEB)

    Thirulogasanthar, K; Krzyzak, A [Department of Computer Science and Software Engineering, Concordia University, 1455 de Maisonneuve Blvd. W, Montreal, Quebec H3G 1M8 (Canada); Honnouvo, G, E-mail: santhar@cs.concordia.c, E-mail: krzyzak@cs.concordia.c, E-mail: g-honnouvo@yahoo.f [Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 2K6 (Canada)

    2010-09-24

    Coherent states, similar to the canonical coherent states of the harmonic oscillator, labeled by quaternions are established on the right and left quaternionic Hilbert spaces. On the left quaternionic Hilbert space reproducing kernels are established. As was claimed by Adler and Millard (J. Math. Phys. 1997 38 2117-26) it is proved that these coherent states cannot be realized as a group action on a quaternionic Hilbert space. As an extension of the complex Hermite polynomials, quaternionic Hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions.

  10. Local polynomial Whittle estimation covering non-stationary fractional processes

    DEFF Research Database (Denmark)

    Nielsen, Frank

    This paper extends the local polynomial Whittle estimator of Andrews & Sun (2004) to fractionally integrated processes covering stationary and non-stationary regions. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the local polynomial Whittle estimator...... to the non-stationary region. By approximating the short-run component of the spectrum by a polynomial, instead of a constant, in a shrinking neighborhood of zero we alleviate some of the bias that the classical local Whittle estimators is prone to. This bias reduction comes at a cost as the variance is in...

  11. Chromatic Polynomials Of Some (m,l-Hyperwheels

    Directory of Open Access Journals (Sweden)

    Julian A. Allagan

    2014-03-01

    Full Text Available In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete non-uniform $(m,l-$hyperwheels and non-uniform $(m,l-$hyperfans. These hypergraphs, constructed through a ``join" graph operation, are some generalizations of the well-known wheel and fan graphs, respectively. Further, we revisit some results concerning these graphs and present their chromatic polynomials in a standard form that involves the Stirling numbers of the second kind.

  12. Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

    Science.gov (United States)

    Horozov, Emil

    2016-05-01

    We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.

  13. Learning Read-constant Polynomials of Constant Degree modulo Composites

    DEFF Research Database (Denmark)

    Chattopadhyay, Arkadev; Gavaldá, Richard; Hansen, Kristoffer Arnsfelt

    2011-01-01

    Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class \\textACC0ACC0. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class...... is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable...

  14. Asymptotic distribution of zeros of polynomials satisfying difference equations

    Science.gov (United States)

    Krasovsky, I. V.

    2003-01-01

    We propose a way to find the asymptotic distribution of zeros of orthogonal polynomials pn(x) satisfying a difference equation of the formB(x)pn(x+[delta])-C(x,n)pn(x)+D(x)pn(x-[delta])=0.We calculate the asymptotic distribution of zeros and asymptotics of extreme zeros of the Meixner and Meixner-Pollaczek polynomials. The distribution of zeros of Meixner polynomials shows some delicate features. We indicate the relation of our technique to the approach based on the Nevai-Dehesa-Ullman distribution.

  15. Orthogonal polynomials describing polarization aberration for rotationally symmetric optical systems.

    Science.gov (United States)

    Xu, Xiangru; Huang, Wei; Xu, Mingfei

    2015-10-19

    Optical lithography has approached a regime of high numerical aperture and wide field, where the impact of polarization aberration on imaging quality turns to be serious. Most of the existing studies focused on the distribution rule of polarization aberration on the pupil, and little attention had been paid to the field. In this paper, a new orthonormal set of polynomials is established to describe the polarization aberration of rotationally symmetric optical systems. The polynomials can simultaneously reveal the distribution rules of polarization aberration on the exit pupil and the field. Two examples are given to verify the polynomials.

  16. Robust stability of fractional order polynomials with complicated uncertainty structure.

    Science.gov (United States)

    Matušů, Radek; Şenol, Bilal; Pekař, Libor

    2017-01-01

    The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition.

  17. On an Ordering-Dependent Generalization of the Tutte Polynomial

    Science.gov (United States)

    Geloun, Joseph Ben; Caravelli, Francesco

    2017-09-01

    A generalization of the Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Itô formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction-deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte-Fortuin-Kasteleyn polynomial is recovered.

  18. A growth curve model with fractional polynomials for analysing incomplete time-course data in microarray gene expression studies

    DEFF Research Database (Denmark)

    Tan, Qihua; Thomassen, Mads; Hjelmborg, Jacob V B

    2011-01-01

    -course pattern in a gene by gene manner. We introduce a growth curve model with fractional polynomials to automatically capture the various time-dependent expression patterns and meanwhile efficiently handle missing values due to incomplete observations. For each gene, our procedure compares the performances...... among fractional polynomial models with power terms from a set of fixed values that offer a wide range of curve shapes and suggests a best fitting model. After a limited simulation study, the model has been applied to our human in vivo irritated epidermis data with missing observations to investigate...... time-dependent transcriptional responses to a chemical irritant. Our method was able to identify the various nonlinear time-course expression trajectories. The integration of growth curves with fractional polynomials provides a flexible way to model different time-course patterns together with model...

  19. MAPPING LANDSLIDES IN LUNAR IMPACT CRATERS USING CHEBYSHEV POLYNOMIALS AND DEM’S

    Directory of Open Access Journals (Sweden)

    V. Yordanov

    2016-06-01

    Full Text Available Geological slope failure processes have been observed on the Moon surface for decades, nevertheless a detailed and exhaustive lunar landslide inventory has not been produced yet. For a preliminary survey, WAC images and DEM maps from LROC at 100 m/pixels have been exploited in combination with the criteria applied by Brunetti et al. (2015 to detect the landslides. These criteria are based on the visual analysis of optical images to recognize mass wasting features. In the literature, Chebyshev polynomials have been applied to interpolate crater cross-sections in order to obtain a parametric characterization useful for classification into different morphological shapes. Here a new implementation of Chebyshev polynomial approximation is proposed, taking into account some statistical testing of the results obtained during Least-squares estimation. The presence of landslides in lunar craters is then investigated by analyzing the absolute values off odd coefficients of estimated Chebyshev polynomials. A case study on the Cassini A crater has demonstrated the key-points of the proposed methodology and outlined the required future development to carry out.

  20. Prediction of zeolite-cement-sand unconfined compressive strength using polynomial neural network

    Science.gov (United States)

    MolaAbasi, H.; Shooshpasha, I.

    2016-04-01

    The improvement of local soils with cement and zeolite can provide great benefits, including strengthening slopes in slope stability problems, stabilizing problematic soils and preventing soil liquefaction. Recently, dosage methodologies are being developed for improved soils based on a rational criterion as it exists in concrete technology. There are numerous earlier studies showing the possibility of relating Unconfined Compressive Strength (UCS) and Cemented sand (CS) parameters (voids/cement ratio) as a power function fits. Taking into account the fact that the existing equations are incapable of estimating UCS for zeolite cemented sand mixture (ZCS) well, artificial intelligence methods are used for forecasting them. Polynomial-type neural network is applied to estimate the UCS from more simply determined index properties such as zeolite and cement content, porosity as well as curing time. In order to assess the merits of the proposed approach, a total number of 216 unconfined compressive tests have been done. A comparison is carried out between the experimentally measured UCS with the predictions in order to evaluate the performance of the current method. The results demonstrate that generalized polynomial-type neural network has a great ability for prediction of the UCS. At the end sensitivity analysis of the polynomial model is applied to study the influence of input parameters on model output. The sensitivity analysis reveals that cement and zeolite content have significant influence on predicting UCS.

  1. Variational Iteration Method for Fifth-Order Boundary Value Problems Using He's Polynomials

    Directory of Open Access Journals (Sweden)

    Muhammad Aslam Noor

    2008-01-01

    Full Text Available We apply the variational iteration method using He's polynomials (VIMHP for solving the fifth-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods and is mainly due to Ghorbani (2007. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

  2. Polynomial Chaos Surrogates for Bayesian Inference

    KAUST Repository

    Le Maitre, Olivier

    2016-01-06

    The Bayesian inference is a popular probabilistic method to solve inverse problems, such as the identification of field parameter in a PDE model. The inference rely on the Bayes rule to update the prior density of the sought field, from observations, and derive its posterior distribution. In most cases the posterior distribution has no explicit form and has to be sampled, for instance using a Markov-Chain Monte Carlo method. In practice the prior field parameter is decomposed and truncated (e.g. by means of Karhunen- Lo´eve decomposition) to recast the inference problem into the inference of a finite number of coordinates. Although proved effective in many situations, the Bayesian inference as sketched above faces several difficulties requiring improvements. First, sampling the posterior can be a extremely costly task as it requires multiple resolutions of the PDE model for different values of the field parameter. Second, when the observations are not very much informative, the inferred parameter field can highly depends on its prior which can be somehow arbitrary. These issues have motivated the introduction of reduced modeling or surrogates for the (approximate) determination of the parametrized PDE solution and hyperparameters in the description of the prior field. Our contribution focuses on recent developments in these two directions: the acceleration of the posterior sampling by means of Polynomial Chaos expansions and the efficient treatment of parametrized covariance functions for the prior field. We also discuss the possibility of making such approach adaptive to further improve its efficiency.

  3. Dynamics of polynomial Chaplygin gas warm inflation

    Energy Technology Data Exchange (ETDEWEB)

    Jawad, Abdul [COMSATS Institute of Information Technology, Department of Mathematics, Lahore (Pakistan); Chaudhary, Shahid [Sharif College of Engineering and Technology, Department of Mathematics, Lahore (Pakistan); Videla, Nelson [Pontificia Universidad Catolica de Valparaiso, Instituto de Fisica, Valparaiso (Chile)

    2017-11-15

    In the present work, we study the consequences of a recently proposed polynomial inflationary potential in the context of the generalized, modified, and generalized cosmic Chaplygin gas models. In addition, we consider dissipative effects by coupling the inflation field to radiation, i.e., the inflationary dynamics is studied in the warm inflation scenario. We take into account a general parametrization of the dissipative coefficient Γ for describing the decay of the inflaton field into radiation. By studying the background and perturbative dynamics in the weak and strong dissipative regimes of warm inflation separately for the positive and negative quadratic and quartic potentials, we obtain expressions for the most relevant inflationary observables as the scalar power spectrum, the scalar spectral, and the tensor-to-scalar ratio. We construct the trajectories in the n{sub s}-r plane for several expressions of the dissipative coefficient and compare with the two-dimensional marginalized contours for (n{sub s}, r) from the latest Planck data. We find that our results are in agreement with WMAP9 and Planck 2015 data. (orig.)

  4. A summation procedure for expansions in orthogonal polynomials

    International Nuclear Information System (INIS)

    Garibotti, C.R.; Grinstein, F.F.

    1977-01-01

    Approximants to functions defined by formal series expansions in orthogonal polynomials are introduced. They are shown to be convergent even out of the elliptical domain where the original expansion converges

  5. Force prediction in cold rolling mills by polynomial methods

    Directory of Open Access Journals (Sweden)

    Nicu ROMAN

    2007-12-01

    Full Text Available A method for steel and aluminium strip thickness control is provided including a new technique for predictive rolling force estimation method by statistic model based on polynomial techniques.

  6. on the performance of Autoregressive Moving Average Polynomial

    African Journals Online (AJOL)

    Timothy Ademakinwa

    Using numerical example, DL, PDL, ARPDL and. ARMAPDL models were fitted. Autoregressive Moving Average Polynomial Distributed Lag Model. (ARMAPDL) model performed better than the other models. Keywords: Distributed Lag Model, Selection Criterion, Parameter Estimation, Residual Variance. ABSTRACT. 247.

  7. Almost normed spaces of functions with polynomial asymptotic behaviour

    International Nuclear Information System (INIS)

    Kudryavtsev, L D

    2003-01-01

    We consider a linear space of functions asymptotically approaching polynomials of degree not higher than a fixed one as the independent variable approaches infinity. Such a space cannot be normed if the functions in it possess a certain smoothness. For this reason the concept of almost normed space is introduced and the spaces in question, namely, spaces of functions asymptotically or strongly asymptotically approaching polynomials, are shown to be almost normed. The completeness of these spaces in the metric generated by their almost norm is also proved, the connection between the asymptotic approach and the strong asymptotic approach of functions to polynomials is studied, and a new (and shorter) proof of the criterion for the asymptotic approach of functions to polynomials is presented

  8. Guts of surfaces and the colored Jones polynomial

    CERN Document Server

    Futer, David; Purcell, Jessica

    2013-01-01

    This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber for the knot; in particular, the surface is a fiber if and only if a particular coefficient vanishes. We also relate hyperbolic volume to colored Jones polynomials. Our method is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses, we show that these surfaces are essential, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to relate the pieces of the JSJ decomposition of the  complement to the combinatorics of certain surface spines (state graphs). Since state graphs have p...

  9. A general polynomial solution to convection–dispersion equation ...

    Indian Academy of Sciences (India)

    Jiao Wang

    s12040-017-0820-4. A general polynomial solution to convection–dispersion equation using ... water pollution of groundwater, numerical models are increasingly used in .... to convective transport by water flow is negligi- ble. Equation (4) is ...

  10. An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem

    OpenAIRE

    Ephremidze, Lasha

    2010-01-01

    A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.

  11. Skew-orthogonal polynomials and random matrix theory

    CERN Document Server

    Ghosh, Saugata

    2009-01-01

    Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward. In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, depend on weight functions. After deriving reduced expressions, called the generalized Christoffel-Darboux formulas (GCD), he obtains universal correlation functions and non-universal level densities for a wide class of random matrix ensembles using the GCD. The author also shows that once questions about higher order effects are considered (questions that are relevant in different branches of physics and mathematics) the ...

  12. Design of robust differential microphone arrays with orthogonal polynomials.

    Science.gov (United States)

    Pan, Chao; Benesty, Jacob; Chen, Jingdong

    2015-08-01

    Differential microphone arrays have the potential to be widely deployed in hands-free communication systems thanks to their frequency-invariant beampatterns, high directivity factors, and small apertures. Traditionally, they are designed and implemented in a multistage way with uniform linear geometries. This paper presents an approach to the design of differential microphone arrays with orthogonal polynomials, more specifically with Jacobi polynomials. It first shows how to express the beampatterns as a function of orthogonal polynomials. Then several differential beamformers are derived and their performance depends on the parameters of the Jacobi polynomials. Simulations show the great flexibility of the proposed method in terms of designing any order differential microphone arrays with different beampatterns and controlling white noise gain.

  13. Orthogonal sets of data windows constructed from trigonometric polynomials

    Science.gov (United States)

    Greenhall, Charles A.

    1990-01-01

    Suboptimal, easily computable substitutes for the discrete prolate-spheroidal windows used by Thomson for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated.

  14. Cubic Polynomials with Rational Roots and Critical Points

    Science.gov (United States)

    Gupta, Shiv K.; Szymanski, Waclaw

    2010-01-01

    If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.

  15. Fitting discrete aspherical surface sag data using orthonormal polynomials.

    Science.gov (United States)

    Hilbig, David; Ceyhan, Ufuk; Henning, Thomas; Fleischmann, Friedrich; Knipp, Dietmar

    2015-08-24

    Characterizing real-life optical surfaces usually involves finding the best-fit of an appropriate surface model to a set of discrete measurement data. This process can be greatly simplified by choosing orthonormal polynomials for the surface description. In case of rotationally symmetric aspherical surfaces, new sets of orthogonal polynomials were introduced by Forbes to replace the numerical unstable standard description. From these, for the application of surface retrieval using experimental ray tracing, the sag orthogonal Q(con)-polynomials are of particular interest. However, these are by definition orthogonal over continuous data and may not be orthogonal for discrete data. In this case, the simplified solution is not valid. Hence, a Gram-Schmidt orthonormalization of these polynomials over the discrete data set is proposed to solve this problem. The resulting difference will be presented by a performance analysis and comparison to the direct matrix inversion method.

  16. Global Sensitivity Analysis for multivariate output using Polynomial Chaos Expansion

    International Nuclear Information System (INIS)

    Garcia-Cabrejo, Oscar; Valocchi, Albert

    2014-01-01

    Many mathematical and computational models used in engineering produce multivariate output that shows some degree of correlation. However, conventional approaches to Global Sensitivity Analysis (GSA) assume that the output variable is scalar. These approaches are applied on each output variable leading to a large number of sensitivity indices that shows a high degree of redundancy making the interpretation of the results difficult. Two approaches have been proposed for GSA in the case of multivariate output: output decomposition approach [9] and covariance decomposition approach [14] but they are computationally intensive for most practical problems. In this paper, Polynomial Chaos Expansion (PCE) is used for an efficient GSA with multivariate output. The results indicate that PCE allows efficient estimation of the covariance matrix and GSA on the coefficients in the approach defined by Campbell et al. [9], and the development of analytical expressions for the multivariate sensitivity indices defined by Gamboa et al. [14]. - Highlights: • PCE increases computational efficiency in 2 approaches of GSA of multivariate output. • Efficient estimation of covariance matrix of output from coefficients of PCE. • Efficient GSA on coefficients of orthogonal decomposition of the output using PCE. • Analytical expressions of multivariate sensitivity indices from coefficients of PCE

  17. Least squares polynomial chaos expansion: A review of sampling strategies

    Science.gov (United States)

    Hadigol, Mohammad; Doostan, Alireza

    2018-04-01

    As non-institutive polynomial chaos expansion (PCE) techniques have gained growing popularity among researchers, we here provide a comprehensive review of major sampling strategies for the least squares based PCE. Traditional sampling methods, such as Monte Carlo, Latin hypercube, quasi-Monte Carlo, optimal design of experiments (ODE), Gaussian quadratures, as well as more recent techniques, such as coherence-optimal and randomized quadratures are discussed. We also propose a hybrid sampling method, dubbed alphabetic-coherence-optimal, that employs the so-called alphabetic optimality criteria used in the context of ODE in conjunction with coherence-optimal samples. A comparison between the empirical performance of the selected sampling methods applied to three numerical examples, including high-order PCE's, high-dimensional problems, and low oversampling ratios, is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand. We observed that the alphabetic-coherence-optimal technique outperforms other sampling methods, specially when high-order ODE are employed and/or the oversampling ratio is low.

  18. Differential cross section of the reaction γ + p -> π+ + n at intermediate angles in the γ-energy range from 0.3 to 2.0 GeV and parametrization by expansion in Legendre-Polynomials

    International Nuclear Information System (INIS)

    Durwen, E.J.

    1980-04-01

    The differential cross section of the reaction γp->π + n was increased in 6 excitation curves at pion laboratory angles from thetasub(lab)sup(π) = 35 0 to 85 0 in 10 0 -steps. The γ-energy range extended from 0.3 GeV at thetasub(lab) = 35 0 to an angle-dependent maximum value which lied between 0.77 GeV at thetasub(lab)sup(π) = 35 0 and 1.94 GeV thetasub(lab)sup(π) = 85 0 . The 705 measuring points are part of a comprehensive measuring program of this laboratory which has the aim of the establishment of a complete, consistent high precision data set for the differential cross sections of the π + photoproduction in the resonance region. (orig./HSI) [de

  19. Irreducibility results for compositions of polynomials in several ...

    Indian Academy of Sciences (India)

    R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22

    such that the corresponding polynomial f (X, Y ) is reducible over Q(X). Indeed, one may choose for instance a0(X), a1(X), . . . , am−2(X) to be any polynomials with coefficients in Q, with a0(X) = 0, of degrees less than or equal to d, and define am−1(X) by the equality am−1(X) = −Xd − 5X − 5 −. ∑. 0≤i≤m−2 ai(X).

  20. Commutators with idempotent values on multilinear polynomials in ...

    Indian Academy of Sciences (India)

    ring of R and C = Z(Q), the center of Q, which is called the extended centroid of R. We also make frequent use of the theory of generalized polynomial identities and differential identities. In particular, we need to recall that when R is a prime ring and I an ideal of R, then R, I and Q satisfy the same generalized polynomial ...