Lorber, A.A.; Carey, G.F.; Bova, S.W.; Harle, C.H.
1996-12-31
The connection between the solution of linear systems of equations by iterative methods and explicit time stepping techniques is used to accelerate to steady state the solution of ODE systems arising from discretized PDEs which may involve either physical or artificial transient terms. Specifically, a class of Runge-Kutta (RK) time integration schemes with extended stability domains has been used to develop recursion formulas which lead to accelerated iterative performance. The coefficients for the RK schemes are chosen based on the theory of Chebyshev iteration polynomials in conjunction with a local linear stability analysis. We refer to these schemes as Chebyshev Parameterized Runge Kutta (CPRK) methods. CPRK methods of one to four stages are derived as functions of the parameters which describe an ellipse {Epsilon} which the stability domain of the methods is known to contain. Of particular interest are two-stage, first-order CPRK and four-stage, first-order methods. It is found that the former method can be identified with any two-stage RK method through the correct choice of parameters. The latter method is found to have a wide range of stability domains, with a maximum extension of 32 along the real axis. Recursion performance results are presented below for a model linear convection-diffusion problem as well as non-linear fluid flow problems discretized by both finite-difference and finite-element methods.
Hermite base Bernoulli type polynomials on the umbral algebra
NASA Astrophysics Data System (ADS)
Dere, R.; Simsek, Y.
2015-01-01
The aim of this paper is to construct new generating functions for Hermite base Bernoulli type polynomials, which generalize not only the Milne-Thomson polynomials but also the two-variable Hermite polynomials. We also modify the Milne-Thomson polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. Moreover, by applying the umbral algebra to these generating functions, we derive new identities for the Bernoulli polynomials of higher order, the Hermite polynomials and numbers of higher order, and the Stirling numbers of the second kind.
Affine and deformable registration based on polynomial expansion.
Farnebäck, Gunnar; Westin, Carl-Fredrik
2006-01-01
This paper presents a registration framework based on the polynomial expansion transform. The idea of polynomial expansion is that the image is locally approximated by polynomials at each pixel. Starting with observations of how the coefficients of ideal linear and quadratic polynomials change under translation and affine transformation, algorithms are developed to estimate translation and compute affine and deformable registration between a fixed and a moving image, from the polynomial expansion coefficients. All algorithms can be used for signals of any dimensionality. The algorithms are evaluated on medical data. PMID:17354971
An error embedded method based on generalized Chebyshev polynomials
NASA Astrophysics Data System (ADS)
Kim, Philsu; Kim, Junghan; Jung, WonKyu; Bu, Sunyoung
2016-02-01
In this paper, we develop an error embedded method based on generalized Chebyshev polynomials for solving stiff initial value problems. The solution and the error at each integration step are calculated by generalized Chebyshev polynomials of two consecutive degrees having overlapping zeros, which enables us to minimize overall computational costs. Further the errors at each integration step are embedded in the algorithm itself. In terms of concrete convergence and stability analysis, the constructed algorithm turns out to have the 6th order convergence and an almost L-stability. We assess the proposed method with several numerical results, showing that it uses larger time step sizes and is numerically more efficient.
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
NASA Astrophysics Data System (ADS)
Ahlfeld, R.; Belkouchi, B.; Montomoli, F.
2016-09-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10
Cryptanalysis of Multiplicative Coupled Cryptosystems Based on the Chebyshev Polynomials
NASA Astrophysics Data System (ADS)
Shakiba, Ali; Hooshmandasl, Mohammad Reza; Meybodi, Mohsen Alambardar
2016-06-01
In this work, we propose a class of public-key cryptosystems called multiplicative coupled cryptosystem, or MCC for short, as well as discuss its security within three different models. Moreover, we discuss a chaotic instance of MCC based on the first and the second types of Chebyshev polynomials over real numbers for these three security models. To avoid round-off errors in floating point arithmetic as well as to enhance the security of the chaotic instance discussed, the Chebyshev polynomials of the first and the second types over a finite field are employed. We also consider the efficiency of the proposed MCCs. The discussions throughout the paper are supported by practical examples.
Fast complex memory polynomial-based adaptive digital predistorter
NASA Astrophysics Data System (ADS)
Singh Sappal, Amandeep; Singh Patterh, Manjeet; Sharma, Sanjay
2011-07-01
Today's 3G wireless systems require both high linearity and high power amplifier (PA) efficiency. The high peak-to-average ratios of the digital modulation schemes used in 3G wireless systems require that the RF PA maintain high linearity over a large range while maintaining this high efficiency; these two requirements are often at odds with each other with many of the traditional amplifier architectures. In this article, a fast and easy-to-implement adaptive digital predistorter has been presented for Wideband Code Division Multiplexed signals using complex memory polynomial work function. The proposed algorithm has been implemented to test a Motorola LDMOSFET PA. The proposed technique also takes care of the memory effects of the PA, which have been ignored in many proposed techniques in the literature. The results show that the new complex memory polynomial-based adaptive digital predistorter has better linearisation performance than conventional predistortion techniques.
Enhanced Access Polynomial Based Self-healing Key Distribution
NASA Astrophysics Data System (ADS)
Dutta, Ratna; Mukhopadhyay, Sourav; Dowling, Tom
A fundamental concern of any secure group communication system is that of key management. Wireless environments create new key management problems and requirements to solve these problems. One such core requirement in these emerging networks is that of self-healing. In systems where users can be offline and miss updates self healing allows a user to recover lost keys and get back into the secure communication without putting extra burden on the group manager. Clearly self healing must be only available to authorized users and this creates more challenges in that we must ensure unauthorized or revoked users cannot, themselves or by means of collusion, avail of self healing. To this end we enhance the one-way key chain based self-healing key distribution of Dutta et al. by introducing a collusion resistance property between the revoked users and the newly joined users. Our scheme is based on the concept of access polynomials. These can be loosely thought of as white lists of authorized users as opposed to the more widely used revocation polynomials or black lists of revoked users. We also allow each user a pre-arranged life cycle distributed by the group manager. Our scheme provides better efficiency in terms of storage, and the communication and computation costs do not increase as the number of sessions grows as compared to most current schemes. We analyze our scheme in an appropriate security model and prove that the proposed scheme is computationally secure and not only achieving forward and backward secrecy, but also resisting collusion between the new joined users and the revoked users. Unlike most existing schemes the new scheme allows temporary revocation. Also unlike existing schemes, our construction does not collapse if the number of revoked users crosses a threshold value. This feature increases resilience against revocation based denial of service (DOS) attacks and thus improves availability of communication channel.
Tracking control of piezoelectric actuators using a polynomial-based hysteresis model
NASA Astrophysics Data System (ADS)
Gan, Jinqiang; Zhang, Xianmin; Wu, Heng
2016-06-01
A polynomial-based hysteresis model that describes hysteresis behavior in piezoelectric actuators is presented. The polynomial-based model is validated by comparing with the classic Prandtl-Ishlinskii model. Taking the advantages of the proposed model into consideration, inverse control using the polynomial-based model is proposed. To achieve better tracking performance, a hybrid control combining the developed inverse control and a proportional-integral-differential feedback loop is then proposed. To demonstrate the effectiveness of the proposed tracking controls, several comparative experiments of the polynomial-based model and Prandtl-Ishlinskii model are conducted. The experimental results show that inverse control and hybrid control using the polynomial-based model in trajectory-tracking applications are effective and meaningful.
Gabor-based kernel PCA with fractional power polynomial models for face recognition.
Liu, Chengjun
2004-05-01
This paper presents a novel Gabor-based kernel Principal Component Analysis (PCA) method by integrating the Gabor wavelet representation of face images and the kernel PCA method for face recognition. Gabor wavelets first derive desirable facial features characterized by spatial frequency, spatial locality, and orientation selectivity to cope with the variations due to illumination and facial expression changes. The kernel PCA method is then extended to include fractional power polynomial models for enhanced face recognition performance. A fractional power polynomial, however, does not necessarily define a kernel function, as it might not define a positive semidefinite Gram matrix. Note that the sigmoid kernels, one of the three classes of widely used kernel functions (polynomial kernels, Gaussian kernels, and sigmoid kernels), do not actually define a positive semidefinite Gram matrix either. Nevertheless, the sigmoid kernels have been successfully used in practice, such as in building support vector machines. In order to derive real kernel PCA features, we apply only those kernel PCA eigenvectors that are associated with positive eigenvalues. The feasibility of the Gabor-based kernel PCA method with fractional power polynomial models has been successfully tested on both frontal and pose-angled face recognition, using two data sets from the FERET database and the CMU PIE database, respectively. The FERET data set contains 600 frontal face images of 200 subjects, while the PIE data set consists of 680 images across five poses (left and right profiles, left and right half profiles, and frontal view) with two different facial expressions (neutral and smiling) of 68 subjects. The effectiveness of the Gabor-based kernel PCA method with fractional power polynomial models is shown in terms of both absolute performance indices and comparative performance against the PCA method, the kernel PCA method with polynomial kernels, the kernel PCA method with fractional power
NASA Astrophysics Data System (ADS)
Liang, Xie; Min, Xu; Bin, Zhang; Zihua, Qiu
2015-03-01
To solve hyperbolic conservation laws, a new method is developed based on the spectral difference (SD) algorithm. The new scheme adopts hierarchical polynomials to represent the solution in each cell instead of Lagrange interpolation polynomials used by the original one. The degrees of freedom (DOFs) of the present scheme are the coefficients of these polynomials, which do not represent the states at the solution points like the original method. Therefore, the solution points defined in the original SD scheme are discarded, while the flux points are preserved to construct a Lagrange interpolation polynomial to approximate flux function in each cell. To update the DOFs, differential operators are applied to the governing equation as well as the Lagrange interpolation polynomial of flux function to evaluate first and higher order derivatives of both solution and flux at the centroid of the cell. The stability property of the current scheme is proved to be the same as the original SD method when the same solution space is adopted. One dimensional methods are always stable by the use of zeros of Legendre polynomials as inner flux points. For two dimensional problems, the introduction of Raviart-Thomas spaces for the interpolation of flux function proves stable schemes for triangles. Accuracy studies are performed with one- and two-dimensional problems. p-Multigrid algorithm is implemented with orthogonal hierarchical bases. The results verify the high efficiency and low memory requirements of implementation of p-multigrid algorithm with the proposed scheme.
An Accurate Projector Calibration Method Based on Polynomial Distortion Representation
Liu, Miao; Sun, Changku; Huang, Shujun; Zhang, Zonghua
2015-01-01
In structure light measurement systems or 3D printing systems, the errors caused by optical distortion of a digital projector always affect the precision performance and cannot be ignored. Existing methods to calibrate the projection distortion rely on calibration plate and photogrammetry, so the calibration performance is largely affected by the quality of the plate and the imaging system. This paper proposes a new projector calibration approach that makes use of photodiodes to directly detect the light emitted from a digital projector. By analyzing the output sequence of the photoelectric module, the pixel coordinates can be accurately obtained by the curve fitting method. A polynomial distortion representation is employed to reduce the residuals of the traditional distortion representation model. Experimental results and performance evaluation show that the proposed calibration method is able to avoid most of the disadvantages in traditional methods and achieves a higher accuracy. This proposed method is also practically applicable to evaluate the geometric optical performance of other optical projection system. PMID:26492247
Online segmentation of time series based on polynomial least-squares approximations.
Fuchs, Erich; Gruber, Thiemo; Nitschke, Jiri; Sick, Bernhard
2010-12-01
The paper presents SwiftSeg, a novel technique for online time series segmentation and piecewise polynomial representation. The segmentation approach is based on a least-squares approximation of time series in sliding and/or growing time windows utilizing a basis of orthogonal polynomials. This allows the definition of fast update steps for the approximating polynomial, where the computational effort depends only on the degree of the approximating polynomial and not on the length of the time window. The coefficients of the orthogonal expansion of the approximating polynomial-obtained by means of the update steps-can be interpreted as optimal (in the least-squares sense) estimators for average, slope, curvature, change of curvature, etc., of the signal in the time window considered. These coefficients, as well as the approximation error, may be used in a very intuitive way to define segmentation criteria. The properties of SwiftSeg are evaluated by means of some artificial and real benchmark time series. It is compared to three different offline and online techniques to assess its accuracy and runtime. It is shown that SwiftSeg-which is suitable for many data streaming applications-offers high accuracy at very low computational costs. PMID:20975120
Aspherical surface profile fitting based on the relationship between polynomial and inner products
NASA Astrophysics Data System (ADS)
Cheng, Xuemin; Yang, Yikang; Hao, Qun
2016-01-01
High-precision aspherical polynomial fitting is essential to image quality evaluation in optical design and optimization. However, conventional fitting methods cannot reach optimal fitting precision and may somehow induce numerical ill-conditioning, such as excessively high coefficients. For this reason, a projection from polynomial equations to vector space was here proposed such that polynomial solutions could be obtained based on matrix and vector operation, so avoiding the problem of excessive coefficients. The Newton-Raphson iteration method was used to search for optimal fitting of the spherical surface. The profile fitting test showed that the proposed approach was able to obtain results with high precision and small value, which solved the numerical ill-conditioning phenomenon effectively.
NASA Astrophysics Data System (ADS)
Wang, Zhengzi
2015-08-01
The influence of ambient temperature is a big challenge to robust infrared face recognition. This paper proposes a new ambient temperature normalization algorithm to improve the performance of infrared face recognition under variable ambient temperatures. Based on statistical regression theory, a second order polynomial model is learned to describe the ambient temperature's impact on infrared face image. Then, infrared image was normalized to reference ambient temperature by the second order polynomial model. Finally, this normalization method is applied to infrared face recognition to verify its efficiency. The experiments demonstrate that the proposed temperature normalization method is feasible and can significantly improve the robustness of infrared face recognition.
A comparison of high-order polynomial and wave-based methods for Helmholtz problems
NASA Astrophysics Data System (ADS)
Lieu, Alice; Gabard, Gwénaël; Bériot, Hadrien
2016-09-01
The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.
Coherent orthogonal polynomials
Celeghini, E.; Olmo, M.A. del
2013-08-15
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{sup 2} functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L{sup 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
ISAR Imaging of Maneuvering Targets Based on the Modified Discrete Polynomial-Phase Transform.
Wang, Yong; Abdelkader, Ali Cherif; Zhao, Bin; Wang, Jinxiang
2015-01-01
Inverse synthetic aperture radar (ISAR) imaging of a maneuvering target is a challenging task in the field of radar signal processing. The azimuth echo can be characterized as a multi-component polynomial phase signal (PPS) after the translational compensation, and the high quality ISAR images can be obtained by the parameters estimation of it combined with the Range-Instantaneous-Doppler (RID) technique. In this paper, a novel parameters estimation algorithm of the multi-component PPS with order three (cubic phase signal-CPS) based on the modified discrete polynomial-phase transform (MDPT) is proposed, and the corresponding new ISAR imaging algorithm is presented consequently. This algorithm is efficient and accurate to generate a focused ISAR image, and the results of real data demonstrate the effectiveness of it. PMID:26404299
ISAR Imaging of Maneuvering Targets Based on the Modified Discrete Polynomial-Phase Transform
Wang, Yong; Abdelkader, Ali Cherif; Zhao, Bin; Wang, Jinxiang
2015-01-01
Inverse synthetic aperture radar (ISAR) imaging of a maneuvering target is a challenging task in the field of radar signal processing. The azimuth echo can be characterized as a multi-component polynomial phase signal (PPS) after the translational compensation, and the high quality ISAR images can be obtained by the parameters estimation of it combined with the Range-Instantaneous-Doppler (RID) technique. In this paper, a novel parameters estimation algorithm of the multi-component PPS with order three (cubic phase signal-CPS) based on the modified discrete polynomial-phase transform (MDPT) is proposed, and the corresponding new ISAR imaging algorithm is presented consequently. This algorithm is efficient and accurate to generate a focused ISAR image, and the results of real data demonstrate the effectiveness of it. PMID:26404299
A weighted polynomial based material decomposition method for spectral x-ray CT imaging.
Wu, Dufan; Zhang, Li; Zhu, Xiaohua; Xu, Xiaofei; Wang, Sen
2016-05-21
Currently in photon counting based spectral x-ray computed tomography (CT) imaging, pre-reconstruction basis materials decomposition is an effective way to reconstruct densities of various materials. The iterative maximum-likelihood method requires precise spectrum information and is time-costly. In this paper, a novel non-iterative decomposition method based on polynomials is proposed for spectral CT, whose aim was to optimize the noise performance when there is more energy bins than the number of basis materials. Several subsets were taken from all the energy bins and conventional polynomials were established for each of them. The decomposition results from each polynomial were summed with pre-calculated weighting factors, which were designed to minimize the overall noises. Numerical studies showed that the decomposition noise of the proposed method was close to the Cramer-Rao lower bound under Poisson noises. Furthermore, experiments were carried out with an XCounter Filte X1 photon counting detector for two-material decomposition and three-material decomposition for validation. PMID:27082291
A weighted polynomial based material decomposition method for spectral x-ray CT imaging
NASA Astrophysics Data System (ADS)
Wu, Dufan; Zhang, Li; Zhu, Xiaohua; Xu, Xiaofei; Wang, Sen
2016-05-01
Currently in photon counting based spectral x-ray computed tomography (CT) imaging, pre-reconstruction basis materials decomposition is an effective way to reconstruct densities of various materials. The iterative maximum-likelihood method requires precise spectrum information and is time-costly. In this paper, a novel non-iterative decomposition method based on polynomials is proposed for spectral CT, whose aim was to optimize the noise performance when there is more energy bins than the number of basis materials. Several subsets were taken from all the energy bins and conventional polynomials were established for each of them. The decomposition results from each polynomial were summed with pre-calculated weighting factors, which were designed to minimize the overall noises. Numerical studies showed that the decomposition noise of the proposed method was close to the Cramer–Rao lower bound under Poisson noises. Furthermore, experiments were carried out with an XCounter Filte X1 photon counting detector for two-material decomposition and three-material decomposition for validation.
Novel Image Encryption Scheme Based on Chebyshev Polynomial and Duffing Map
2014-01-01
We present a novel image encryption algorithm using Chebyshev polynomial based on permutation and substitution and Duffing map based on substitution. Comprehensive security analysis has been performed on the designed scheme using key space analysis, visual testing, histogram analysis, information entropy calculation, correlation coefficient analysis, differential analysis, key sensitivity test, and speed test. The study demonstrates that the proposed image encryption algorithm shows advantages of more than 10113 key space and desirable level of security based on the good statistical results and theoretical arguments. PMID:25143970
Discrimination Power of Polynomial-Based Descriptors for Graphs by Using Functional Matrices
Dehmer, Matthias; Emmert-Streib, Frank; Shi, Yongtang; Stefu, Monica; Tripathi, Shailesh
2015-01-01
In this paper, we study the discrimination power of graph measures that are based on graph-theoretical matrices. The paper generalizes the work of [M. Dehmer, M. Moosbrugger. Y. Shi, Encoding structural information uniquely with polynomial-based descriptors by employing the Randić matrix, Applied Mathematics and Computation, 268(2015), 164–168]. We demonstrate that by using the new functional matrix approach, exhaustively generated graphs can be discriminated more uniquely than shown in the mentioned previous work. PMID:26479495
Discrimination Power of Polynomial-Based Descriptors for Graphs by Using Functional Matrices.
Dehmer, Matthias; Emmert-Streib, Frank; Shi, Yongtang; Stefu, Monica; Tripathi, Shailesh
2015-01-01
In this paper, we study the discrimination power of graph measures that are based on graph-theoretical matrices. The paper generalizes the work of [M. Dehmer, M. Moosbrugger. Y. Shi, Encoding structural information uniquely with polynomial-based descriptors by employing the Randić matrix, Applied Mathematics and Computation, 268(2015), 164-168]. We demonstrate that by using the new functional matrix approach, exhaustively generated graphs can be discriminated more uniquely than shown in the mentioned previous work. PMID:26479495
Spline-based high-accuracy piecewise-polynomial phase-to-sinusoid amplitude converters.
Petrinović, Davor; Brezović, Marko
2011-04-01
We propose a method for direct digital frequency synthesis (DDS) using a cubic spline piecewise-polynomial model for a phase-to-sinusoid amplitude converter (PSAC). This method offers maximum smoothness of the output signal. Closed-form expressions for the cubic polynomial coefficients are derived in the spectral domain and the performance analysis of the model is given in the time and frequency domains. We derive the closed-form performance bounds of such DDS using conventional metrics: rms and maximum absolute errors (MAE) and maximum spurious free dynamic range (SFDR) measured in the discrete time domain. The main advantages of the proposed PSAC are its simplicity, analytical tractability, and inherent numerical stability for high table resolutions. Detailed guidelines for a fixed-point implementation are given, based on the algebraic analysis of all quantization effects. The results are verified on 81 PSAC configurations with the output resolutions from 5 to 41 bits by using a bit-exact simulation. The VHDL implementation of a high-accuracy DDS based on the proposed PSAC with 28-bit input phase word and 32-bit output value achieves SFDR of its digital output signal between 180 and 207 dB, with a signal-to-noise ratio of 192 dB. Its implementation requires only one 18 kB block RAM and three 18-bit embedded multipliers in a typical field-programmable gate array (FPGA) device. PMID:21507749
ERIC Educational Resources Information Center
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…
Regression-based adaptive sparse polynomial dimensional decomposition for sensitivity analysis
NASA Astrophysics Data System (ADS)
Tang, Kunkun; Congedo, Pietro; Abgrall, Remi
2014-11-01
Polynomial dimensional decomposition (PDD) is employed in this work for global sensitivity analysis and uncertainty quantification of stochastic systems subject to a large number of random input variables. Due to the intimate structure between PDD and Analysis-of-Variance, PDD is able to provide simpler and more direct evaluation of the Sobol' sensitivity indices, when compared to polynomial chaos (PC). Unfortunately, the number of PDD terms grows exponentially with respect to the size of the input random vector, which makes the computational cost of the standard method unaffordable for real engineering applications. In order to address this problem of curse of dimensionality, this work proposes a variance-based adaptive strategy aiming to build a cheap meta-model by sparse-PDD with PDD coefficients computed by regression. During this adaptive procedure, the model representation by PDD only contains few terms, so that the cost to resolve repeatedly the linear system of the least-square regression problem is negligible. The size of the final sparse-PDD representation is much smaller than the full PDD, since only significant terms are eventually retained. Consequently, a much less number of calls to the deterministic model is required to compute the final PDD coefficients.
NASA Astrophysics Data System (ADS)
Vittaldev, V.; Linares, R.; Godinez, H. C.; Koller, J.; Russell, R. P.
2013-12-01
Recent events in space, including the collision of Russia's Cosmos 2251 satellite with Iridium 33 and China's Feng Yun 1C anti-satellite demonstration, have stressed the capabilities of the Space Surveillance Network and its ability to provide accurate and actionable impact probability estimates. In particular low-Earth orbiting satellites are heavily influenced by upper atmospheric density, due to drag, which is very difficult to model accurately. This work focuses on the generalized Polynomial Chaos (gPC) technique for Uncertainty Quantification (UQ) in physics-based atmospheric models. The advantage of the gPC approach is that it can efficiently model non-Gaussian probability distribution functions (pdfs). The gPC approach is used to create a polynomial chaos in F10.7, AP, and solar wind parameters; this chaos is used to perform UQ on future atmospheric conditions. A number of physics-based models are used as test cases, including GITM and TIE-GCM, and the gPC is shown to have good performance in modeling non-Gaussian pdfs. Los Alamos National Laboratory (LANL) has established a research effort, called IMPACT (Integrated Modeling of Perturbations in Atmospheres for Conjunction Tracking), to improve impact assessment via improved physics-based modeling. A number of atmospheric models exist which can be classified as either empirical or physics-based. Physics-based models can be used to provide a forward prediction which is required for accurate collision assessments. As part of this effort, accurate and consistent UQ is required for the atmospheric models used. One of the primary sources of uncertainty is input parameter uncertainty. These input parameters, which include F10.7, AP, and solar wind parameters, are measured constantly. In turn, these measurements are used to provide a prediction for future parameter values. Therefore, the uncertainty of the atmospheric model forecast, due to potential error in the input parameters, must be correctly characterized to
Solving fuzzy polynomial equation and the dual fuzzy polynomial equation using the ranking method
NASA Astrophysics Data System (ADS)
Rahman, Nurhakimah Ab.; Abdullah, Lazim
2014-06-01
Fuzzy polynomials with trapezoidal and triangular fuzzy numbers have attracted interest among some researchers. Many studies have been done by researchers to obtain real roots of fuzzy polynomials. As a result, there are many numerical methods involved in obtaining the real roots of fuzzy polynomials. In this study, we will present the solution to the fuzzy polynomial equation and dual fuzzy polynomial equation using the ranking method of fuzzy numbers and subsequently transforming fuzzy polynomials to crisp polynomials. This transformation is performed using the ranking method based on three parameters, namely Value, Ambiguity and Fuzziness. Finally, we illustrate our approach with two numerical examples for fuzzy polynomial equation and dual fuzzy polynomial equation.
Azimipour, Mehdi; Atry, Farid; Pashaie, Ramin
2016-04-10
Digital optical phase conjugation (DOPC) has proven to be a promising technique in deep tissue fluorescence imaging. Nonetheless, DOPC optical setups require precise alignment of all optical components to accurately read the wavefront of scattered light in a turbid medium and playback the conjugated beam toward the sample. Minor misalignments and possible imperfections in the arrangement or the structure of the optical components significantly reduce the performance of the method. In this paper, a calibration procedure based on orthogonal rectangular polynomials is introduced to compensate major imperfections including the optical aberration in the wavefront of the reference beam and the substrate curvature of the spatial light modulator without adding extra optical components to the original setup. The proposed algorithm also provides a systematic calibration procedure for mechanical fine tuning of DOPC systems. It is shown experimentally that the proposed calibration process improves the peak-to-background ratio when focusing light after passing through a highly scattering medium. PMID:27139849
Krishnamoorthi, R; Anna Poorani, G
2016-01-01
Iris normalization is an important stage in any iris biometric, as it has a propensity to trim down the consequences of iris distortion. To indemnify the variation in size of the iris owing to the action of stretching or enlarging the pupil in iris acquisition process and camera to eyeball distance, two normalization schemes has been proposed in this work. In the first method, the iris region of interest is normalized by converting the iris into the variable size rectangular model in order to avoid the under samples near the limbus border. In the second method, the iris region of interest is normalized by converting the iris region into a fixed size rectangular model in order to avoid the dimensional discrepancies between the eye images. The performance of the proposed normalization methods is evaluated with orthogonal polynomials based iris recognition in terms of FAR, FRR, GAR, CRR and EER. PMID:27066376
NASA Astrophysics Data System (ADS)
Lee, Joohwi; Kim, Sun Hyung; Oguz, Ipek; Styner, Martin
2016-03-01
The cortical thickness of the mammalian brain is an important morphological characteristic that can be used to investigate and observe the brain's developmental changes that might be caused by biologically toxic substances such as ethanol or cocaine. Although various cortical thickness analysis methods have been proposed that are applicable for human brain and have developed into well-validated open-source software packages, cortical thickness analysis methods for rodent brains have not yet become as robust and accurate as those designed for human brains. Based on a previously proposed cortical thickness measurement pipeline for rodent brain analysis,1 we present an enhanced cortical thickness pipeline in terms of accuracy and anatomical consistency. First, we propose a Lagrangian-based computational approach in the thickness measurement step in order to minimize local truncation error using the fourth-order Runge-Kutta method. Second, by constructing a line object for each streamline of the thickness measurement, we can visualize the way the thickness is measured and achieve sub-voxel accuracy by performing geometric post-processing. Last, with emphasis on the importance of an anatomically consistent partial differential equation (PDE) boundary map, we propose an automatic PDE boundary map generation algorithm that is specific to rodent brain anatomy, which does not require manual labeling. The results show that the proposed cortical thickness pipeline can produce statistically significant regions that are not observed in the previous cortical thickness analysis pipeline.
Lee, Joohwi; Kim, Sun Hyung; Oguz, Ipek; Styner, Martin
2016-01-01
The cortical thickness of the mammalian brain is an important morphological characteristic that can be used to investigate and observe the brain’s developmental changes that might be caused by biologically toxic substances such as ethanol or cocaine. Although various cortical thickness analysis methods have been proposed that are applicable for human brain and have developed into well-validated open-source software packages, cortical thickness analysis methods for rodent brains have not yet become as robust and accurate as those designed for human brains. Based on a previously proposed cortical thickness measurement pipeline for rodent brain analysis,1 we present an enhanced cortical thickness pipeline in terms of accuracy and anatomical consistency. First, we propose a Lagrangian-based computational approach in the thickness measurement step in order to minimize local truncation error using the fourth-order Runge-Kutta method. Second, by constructing a line object for each streamline of the thickness measurement, we can visualize the way the thickness is measured and achieve sub-voxel accuracy by performing geometric post-processing. Last, with emphasis on the importance of an anatomically consistent partial differential equation (PDE) boundary map, we propose an automatic PDE boundary map generation algorithm that is specific to rodent brain anatomy, which does not require manual labeling. The results show that the proposed cortical thickness pipeline can produce statistically significant regions that are not observed in the the previous cortical thickness analysis pipeline. PMID:27065047
Phase demodulation method from a single fringe pattern based on correlation with a polynomial form.
Robin, Eric; Valle, Valéry; Brémand, Fabrice
2005-12-01
The method presented extracts the demodulated phase from only one fringe pattern. Locally, this method approaches the fringe pattern morphology with the help of a mathematical model. The degree of similarity between the mathematical model and the real fringe is estimated by minimizing a correlation function. To use an optimization process, we have chosen a polynomial form such as a mathematical model. However, the use of a polynomial form induces an identification procedure with the purpose of retrieving the demodulated phase. This method, polynomial modulated phase correlation, is tested on several examples. Its performance, in terms of speed and precision, is presented on very noised fringe patterns. PMID:16353793
Phase demodulation method from a single fringe pattern based on correlation with a polynomial form
Robin, Eric; Valle, Valery; Bremand, Fabrice
2005-12-01
The method presented extracts the demodulated phase from only one fringe pattern. Locally, this method approaches the fringe pattern morphology with the help of a mathematical model. The degree of similarity between the mathematical model and the real fringe is estimated by minimizing a correlation function. To use an optimization process, we have chosen a polynomial form such as a mathematical model. However, the use of a polynomial form induces an identification procedure with the purpose of retrieving the demodulated phase. This method, polynomial modulated phase correlation, is tested on several examples. Its performance, in terms of speed and precision, is presented on very noised fringe patterns.
Optimization of LED-based non-imaging optics with orthogonal polynomial shapes
NASA Astrophysics Data System (ADS)
Brick, Peter; Wiesmann, Christopher
2012-10-01
Starting with a seminal paper by Forbes [1], orthogonal polynomials have received considerable interest as descriptors of lens shapes for imaging optics. However, there is little information on the application of orthogonal polynomials in the field of non-imaging optics. Here, we consider fundamental cases related to LED primary and secondary optics. To make it most realistic, we avoid many of the simplifications of non-imaging theory and consider the full complexity of LED optics. In this framework, the benefits of orthogonal polynomial surface description for LED optics are evaluated in comparison to a surface description by widely used monomials.
NASA Astrophysics Data System (ADS)
Papila, Nilay Uzgoren
Turbine performance directly affects engine specific impulse, thrust-to-weight ratio, and cost in a rocket propulsion system. This dissertation focuses on methodology and application of employing optimization techniques, with the neural network (NN) and polynomial-based response surface method (RSM), for supersonic turbine optimization. The research is relevant to NASA's reusable launching vehicle initiatives. It is demonstrated that accuracy of the response surface (RS) approximations can be improved with combined utilization of the NN and polynomial techniques, and higher emphases on data in regions of interests. The design of experiment methodology is critical while performing optimization in efficient and effective manners. In physical applications, both preliminary design and detailed shape design optimization are investigated. For preliminary design level, single-, two-, and three-stage turbines are considered with the number of design variables increasing from six to 11 and then to 15, in accordance with the number of stages. A major goal of the preliminary optimization effort is to balance the desire of maximizing aerodynamic performance and minimizing weight. To ascertain required predictive capability of the RSM, a two-level domain refinement approach (windowing) has been adopted. The accuracy of the predicted optimal design points based on this strategy is shown to be satisfactory. The results indicate that the two-stage turbine is the optimum configuration with the higher efficiency corresponding to smaller weights. It is demonstrated that the criteria for selecting the database exhibit significant impact on the efficiency and effectiveness of the construction of the response surface. Based on the optimized preliminary design outcome, shape optimization is performed for vanes and blades of a two-stage supersonic turbine, involving O(10) design variables. It is demonstrated that a major merit of the RS-based optimization approach is that it enables one
Cloning and characterization of a SnRK2 gene from Jatropha curcas L.
Chun, J; Li, F-S; Ma, Y; Wang, S-H; Chen, F
2014-01-01
Although the SnRK2 class of Ser/Thr protein kinases is critical for signal transduction and abiotic stress resistance in plants, there have been no studies to examine SnRK2 in Jatropha curcas L. In the present study, JcSnRK2 was cloned from J. curcas using the rapid amplification of cDNA end technique and characterized. The JcSnRK2 genomic sequence is 2578 base pairs (bp), includes 10 exons and 9 introns, and the 1017-bp open reading frame encodes 338 amino acids. JcSnRK2 was transcribed in all examined tissues, with the highest transcription rate observed in the roots, followed by the leaves and stalks; the lowest rate was observed in flowers and seeds. JcSnRK2 expression increased following abscisic acid treatment, salinity, and drought stress. During a 48-h stress period, the expression of JcSnRK2 showed 2 peaks and periodic up- and downregulation. JcSnRK2 was rapidly activated within 1 h under salt and drought stress, but not under cold stress. Because of the gene sequence and expression similarity of JcSnRK2 to AtSnRK2.8, primarily in the roots, an eukaryotic expression vector containing the JcSnRK2 gene (pBI121-JcSnRK2) was constructed and introduced to the Arabidopsis AtSnRK2.8 mutant snf2.8. JcSnRK2-overexpressing plants exhibited higher salt and drought tolerance, further demonstrating the function of JcSnRK2 in the osmotic stress response. J. curcas is highly resistant to extreme salt and drought conditions and JcSnRK2 was found to be activated under these conditions. Thus, JcSnRK2 is potential candidate for improving crop tolerance to salt and drought stress. PMID:25526217
NASA Astrophysics Data System (ADS)
Erdogan, Eren; Onur Karslioglu, Mahmut; Durmaz, Murat; Aghakarimi, Armin
2014-05-01
In this study, particle filter (PF) which is mainly based on the Monte Carlo simulation technique has been carried out for polynomial modeling of the local ionospheric conditions above the selected ground based stations. Less sensitivity to the errors caused by linearization of models and the effect of unknown or unmodeled components in the system model is one of the advantages of the particle filter as compared to the Kalman filter which is commonly used as a recursive filtering method in VTEC modeling. Besides, probability distribution of the system models is not necessarily required to be Gaussian. In this work third order polynomial function has been incorporated into the particle filter implementation to represent the local VTEC distribution. Coefficients of the polynomial model presenting the ionospheric parameters and the receiver inter frequency biases are the unknowns forming the state vector which has been estimated epoch-wise for each ground station. To consider the time varying characteristics of the regional VTEC distribution, dynamics of the state vector parameters changing permanently have been modeled using the first order Gauss-Markov process. In the processing of the particle filtering, multi-variety probability distribution of the state vector through the time has been approximated by means of randomly selected samples and their associated weights. A known drawback of the particle filtering is that the increasing number of the state vector parameters results in an inefficient filter performance and requires more samples to represent the probability distribution of the state vector. Considering the total number of unknown parameters for all ground stations, estimation of these parameters which were inserted into a single state vector has caused the particle filter to produce inefficient results. To solve this problem, the PF implementation has been carried out separately for each ground station at current time epochs. After estimation of unknown
Polynomial fitting-based shape matching algorithm for multi-sensors remote sensing images
NASA Astrophysics Data System (ADS)
Gu, Yujie; Ren, Kan; Wang, Pengcheng; Gu, Guohua
2016-05-01
According to the characteristics of multi-sensors remote sensing images, a new registration algorithm based on shape contour feature is proposed. Firstly, the edge features of remote sensing images are extracted by Canny operator, and the edge of the main contour is retained. According to the characteristics of the contour pixels, a new feature extraction algorithm based on polynomial fitting is proposed and it is used to determine the principal directions of the feature points. On this basis, we improved the shape context descriptor and completed coarse registration by minimizing the matching cost between the feature points. The shape context has been found to be robust in Simple object registration, and in this paper, it is applied to remote sensing image registration after improving the circular template with rotation invariance. Finally, the fine registration is accomplished by the RANSAC algorithm. Experiments show that this algorithm can realize the automatic registration of multi-sensors remote sensing images with high accuracy, robustness and applicability.
Lauricella, Marta Alicia; Maidana, Cristina Graciela; Frias, Victoria Fragueiro; Romagosa, Carlo M; Negri, Vanesa; Benedetti, Ruben; Sinagra, Angel J; Luna, Concepcion; Tartaglino, Lilian; Laucella, Susana; Reed, Steven G; Riarte, Adelina R
2016-07-01
Direct observation of Leishmania parasites in tissue aspirates has shown low sensitivity for the detection of canine visceral leishmaniasis (VL). Therefore in the last quarter century immunoenzymatic tests have been developed to improve diagnosis. The purpose of this study was to develop a fast recombinant K28 antigen, naked-eye qualitative enzyme-linked immunosorbent assay (VL Ql-ELISA) and a quantitative version (VL Qt-ELISA), and to display it in a kit format, whose cutoff value (0.156) was selected as the most adequate one to differentiate reactive from nonreactive samples. Considering 167 cases and 300 controls, sensitivity was 91% for both assays and specificity was 100% and 98.7% in Ql-ELISA and Qt-ELISA, respectively. Positive predictive values were 100% and 97.4% for Ql-ELISA and Qt-ELISA, respectively, and negative predictive values were 95.2% for both ELISAs. Reagent stability, reliability studies, including periodic repetitions and retest of samples, cutoff selection, and comparison of rK28 ELISAs with rK39 immunochromatographic test, were the international criteria that supported the quality in both kits. The performance of both ELISA kits in this work confirmed their validity and emphasized their usefulness for low-to-medium complexity laboratories. PMID:27162270
Blind phone segmentation based on spectral change detection using Legendre polynomial approximation.
Hoang, Dac-Thang; Wang, Hsiao-Chuan
2015-02-01
Phone segmentation involves partitioning a continuous speech signal into discrete phone units. In this paper, a method for automatic phone segmentation without prior knowledge of speech content is proposed. The signal spectrum was represented by band-energies. A segment of the band-energy curve was approximated using Legendre polynomial expansion, allowing Legendre polynomial coefficients to describe the properties of the segment. The spectral changes, which imply phone boundaries in the speech signal, were then detected by monitoring the variations of Legendre polynomial coefficients. A two-step algorithm for detecting phone boundaries was derived. The first step was to detect phone boundaries using first-order and second-order coefficients of the Legendre polynomial approximation. The second step was to locate slow spectral changes in the regions of concatenated voiced phones using zero-order coefficients of the Legendre polynomial approximation. This enabled the phone boundaries missed during the first step to be recovered. An evaluation using the TIMIT corpus indicated that the proposed method is comparable to or more accurate than previous methods. PMID:25698014
Papadopoulos, Anthony
2009-01-01
The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined. PMID:19333397
NASA Technical Reports Server (NTRS)
Narkawicz, Anthony J.; Munoz, Cesar A.
2014-01-01
Sturm's Theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semiopen interval. This paper presents a formalization of this theorem in the PVS theorem prover, as well as a decision procedure that checks whether a polynomial is always positive, nonnegative, nonzero, negative, or nonpositive on any input interval. The soundness and completeness of the decision procedure is proven in PVS. The procedure and its correctness properties enable the implementation of a PVS strategy for automatically proving existential and universal univariate polynomial inequalities. Since the decision procedure is formally verified in PVS, the soundness of the strategy depends solely on the internal logic of PVS rather than on an external oracle. The procedure itself uses a combination of Sturm's Theorem, an interval bisection procedure, and the fact that a polynomial with exactly one root in a bounded interval is always nonnegative on that interval if and only if it is nonnegative at both endpoints.
Orthogonal Bases of Hermitean Monogenic Polynomials: An Explicit Construction in Complex Dimension 2
NASA Astrophysics Data System (ADS)
Brackx, F.; De Schepper, H.; Lávička, R.; Souček, V.
2010-09-01
In this contribution we construct an orthogonal basis of Hermitean monogenic polynomials for the specific case of two complex variables. The approach combines group representation theory, see [5], with a Fischer decomposition for the kernels of each of the considered Dirac operators, see [4], and a Cauchy-Kovalevskaya extension principle, see [3].
Comparison of polynomial approximations to speed up planewave-based quantum Monte Carlo calculations
NASA Astrophysics Data System (ADS)
Parker, William D.; Umrigar, C. J.; Alfè, Dario; Petruzielo, F. R.; Hennig, Richard G.; Wilkins, John W.
2015-04-01
The computational cost of quantum Monte Carlo (QMC) calculations of realistic periodic systems depends strongly on the method of storing and evaluating the many-particle wave function. Previous work by Williamson et al. (2001) [35] and Alfè and Gillan, (2004) [36] has demonstrated the reduction of the O (N3) cost of evaluating the Slater determinant with planewaves to O (N2) using localized basis functions. We compare four polynomial approximations as basis functions - interpolating Lagrange polynomials, interpolating piecewise-polynomial-form (pp-) splines, and basis-form (B-) splines (interpolating and smoothing). All these basis functions provide a similar speedup relative to the planewave basis. The pp-splines have eight times the memory requirement of the other methods. To test the accuracy of the basis functions, we apply them to the ground state structures of Si, Al, and MgO. The polynomial approximations differ in accuracy most strongly for MgO, and smoothing B-splines most closely reproduce the planewave value for of the variational Monte Carlo energy. Using separate approximations for the Laplacian of the orbitals increases the accuracy sufficiently to justify the increased memory requirement, making smoothing B-splines, with separate approximation for the Laplacian, the preferred choice for approximating planewave-represented orbitals in QMC calculations.
Spreading lengths of Hermite polynomials
NASA Astrophysics Data System (ADS)
Sánchez-Moreno, P.; Dehesa, J. S.; Manzano, D.; Yáñez, R. J.
2010-03-01
The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted Lq-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation is computationally proved. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments, mentioned previously, are posed.
Vector quantizer based on brightness maps for image compression with the polynomial transform
NASA Astrophysics Data System (ADS)
Escalante-Ramirez, Boris; Moreno-Gutierrez, Mauricio; Silvan-Cardenas, Jose L.
2002-11-01
We present a vector quantization scheme acting on brightness fields based on distance/distortion criteria correspondent with psycho-visual aspects. These criteria quantify sensorial distortion between vectors that represent either portions of a digital image or alternatively, coefficients of a transform-based coding system. In the latter case, we use an image representation model, namely the Hermite transform, that is based on some of the main perceptual characteristics of the human vision system (HVS) and in their response to light stimulus. Energy coding in the brightness domain, determination of local structure, code-book training and local orientation analysis are all obtained by means of the Hermite transform. This paper, for thematic reasons, is divided in four sections. The first one will shortly highlight the importance of having newer and better compression algorithms. This section will also serve to explain briefly the most relevant characteristics of the HVS, advantages and disadvantages related with the behavior of our vision in front of ocular stimulus. The second section shall go through a quick review of vector quantization techniques, focusing their performance on image treatment, as a preview for the image vector quantizer compressor actually constructed in section 5. Third chapter was chosen to concentrate the most important data gathered on brightness models. The building of this so-called brightness maps (quantification of the human perception on the visible objects reflectance), in a bi-dimensional model, will be addressed here. The Hermite transform, a special case of polynomial transforms, and its usefulness, will be treated, in an applicable discrete form, in the fourth chapter. As we have learned from previous works 1, Hermite transform has showed to be a useful and practical solution to efficiently code the energy within an image block, deciding which kind of quantization is to be used upon them (whether scalar or vector). It will also be
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. PMID:26367882
Complexity and Performance Results for Non FFT-Based Univariate Polynomial Multiplication
NASA Astrophysics Data System (ADS)
Chowdhury, Muhammad F. I.; Maza, Marc Moreno; Pan, Wei; Schost, Eric
2011-11-01
Today's parallel hardware architectures and computer memory hierarchies enforce revisiting fundamental algorithms which were often designed with algebraic complexity as the main complexity measure and with sequential running time as the main performance counter. This study is devoted to two algorithms of univariate polynomial multiplication; that are independent of the coefficient ring: the plain and the Toom-Cook univariate multiplications. We analyze their cache complexity and report on their parallel implementations in Cilk++ [1].
NASA Astrophysics Data System (ADS)
Withers, Christopher S.; Nadarajah, Saralees
2016-07-01
A new class of polynomials pn(x) known as β-reciprocal polynomials is defined. Given a parameter ? that is not a root of -1, we show that the only β-reciprocal polynomials are pn(x) ≡ xn. When β is a root of -1, other polynomials are possible. For example, the Hermite polynomials are i-reciprocal, ?.
Differential Activation of the Wheat SnRK2 Family by Abiotic Stresses
Zhang, Hongying; Li, Weiyu; Mao, Xinguo; Jing, Ruilian; Jia, Hongfang
2016-01-01
Plant responses to stress occur via abscisic acid (ABA) dependent or independent pathways. Sucrose non-fermenting1-related protein kinase 2 (SnRK2) play a key role in plant stress signal transduction pathways. It is known that some SnRK2 members are positive regulators of ABA signal transduction through interaction with group A type 2C protein phosphatases (PP2Cs). Here, 10 SnRK2s were isolated from wheat. Based on phylogenetic analysis using kinase domains or the C-terminus, the 10 SnRK2s were divided into three subclasses. Expression pattern analysis revealed that all TaSnRK2s were involved in the responses to PEG, NaCl, and cold stress. TaSnRK2s in subclass III were strongly induced by ABA. Subclass II TaSnRK2s responded weakly to ABA, whereas TaSnRK2s in subclass I were not activated by ABA treatment. Motif scanning in the C-terminus indicated that motifs 4 and 5 in the C-terminus were unique to subclass III. We further demonstrate the physical and functional interaction between TaSnRK2s and a typical group A PP2C (TaABI1) using Y2H and BiFC assays. The results showed that TaABI1 interacted physically with subclass III TaSnRK2s, while having no interaction with subclasses I and II TaSnRK2s. Together, these findings indicated that subclass III TaSnRK2s were involved in ABA regulated stress responses, whereas subclasses I and II TaSnRK2s responded to various abiotic stressors in an ABA-independent manner. PMID:27066054
A ROM-less direct digital frequency synthesizer based on hybrid polynomial approximation.
Omran, Qahtan Khalaf; Islam, Mohammad Tariqul; Misran, Norbahiah; Faruque, Mohammad Rashed Iqbal
2014-01-01
In this paper, a novel design approach for a phase to sinusoid amplitude converter (PSAC) has been investigated. Two segments have been used to approximate the first sine quadrant. A first linear segment is used to fit the region near the zero point, while a second fourth-order parabolic segment is used to approximate the rest of the sine curve. The phase sample, where the polynomial changed, was chosen in such a way as to achieve the maximum spurious free dynamic range (SFDR). The invented direct digital frequency synthesizer (DDFS) has been encoded in VHDL and post simulation was carried out. The synthesized architecture exhibits a promising result of 90 dBc SFDR. The targeted structure is expected to show advantages for perceptible reduction of hardware resources and power consumption as well as high clock speeds. PMID:24892092
NASA Astrophysics Data System (ADS)
Suwansukho, Kajpanya; Sumriddetchkajorn, Sarun; Buranasiri, Prathan
2013-06-01
We previously showed that a combination of image thresholding, chain coding, elliptic Fourier descriptors, and artificial neural network analysis provided a low false acceptance rate (FAR) and a false rejection rate (FRR) of 11.0% and 19.0%, respectively, in identify Thai jasmine rice from three unwanted rice varieties. In this work, we highlight that only a polynomial function fitting on the determined chain code and the neural network analysis are highly sufficient in obtaining a very low FAR of < 3.0% and a very low 0.3% FRR for the separation of Thai jasmine rice from Chainat 1 (CNT1), Prathumtani 1 (PTT1), and Hom-Pitsanulok (HPSL) rice varieties. With this proposed approach, the analytical time is tremendously suppressed from 4,250 seconds down to 2 seconds, implying extremely high potential in practical deployment.
A ROM-Less Direct Digital Frequency Synthesizer Based on Hybrid Polynomial Approximation
Omran, Qahtan Khalaf; Islam, Mohammad Tariqul; Misran, Norbahiah; Faruque, Mohammad Rashed Iqbal
2014-01-01
In this paper, a novel design approach for a phase to sinusoid amplitude converter (PSAC) has been investigated. Two segments have been used to approximate the first sine quadrant. A first linear segment is used to fit the region near the zero point, while a second fourth-order parabolic segment is used to approximate the rest of the sine curve. The phase sample, where the polynomial changed, was chosen in such a way as to achieve the maximum spurious free dynamic range (SFDR). The invented direct digital frequency synthesizer (DDFS) has been encoded in VHDL and post simulation was carried out. The synthesized architecture exhibits a promising result of 90 dBc SFDR. The targeted structure is expected to show advantages for perceptible reduction of hardware resources and power consumption as well as high clock speeds. PMID:24892092
Accelerated Hazards Model based on Parametric Families Generalized with Bernstein Polynomials
Chen, Yuhui; Hanson, Timothy; Zhang, Jiajia
2015-01-01
Summary A transformed Bernstein polynomial that is centered at standard parametric families, such as Weibull or log-logistic, is proposed for use in the accelerated hazards model. This class provides a convenient way towards creating a Bayesian non-parametric prior for smooth densities, blending the merits of parametric and non-parametric methods, that is amenable to standard estimation approaches. For example optimization methods in SAS or R can yield the posterior mode and asymptotic covariance matrix. This novel nonparametric prior is employed in the accelerated hazards model, which is further generalized to time-dependent covariates. The proposed approach fares considerably better than previous approaches in simulations; data on the effectiveness of biodegradable carmustine polymers on recurrent brain malignant gliomas is investigated. PMID:24261450
NASA Astrophysics Data System (ADS)
Wang, Y. P.; Lu, Z. P.; Sun, D. S.; Wang, N.
2016-01-01
In order to better express the characteristics of satellite clock bias (SCB) and improve SCB prediction precision, this paper proposed a new SCB prediction model which can take physical characteristics of space-borne atomic clock, the cyclic variation, and random part of SCB into consideration. First, the new model employs a quadratic polynomial model with periodic items to fit and extract the trend term and cyclic term of SCB; then based on the characteristics of fitting residuals, a time series ARIMA ~(Auto-Regressive Integrated Moving Average) model is used to model the residuals; eventually, the results from the two models are combined to obtain final SCB prediction values. At last, this paper uses precise SCB data from IGS (International GNSS Service) to conduct prediction tests, and the results show that the proposed model is effective and has better prediction performance compared with the quadratic polynomial model, grey model, and ARIMA model. In addition, the new method can also overcome the insufficiency of the ARIMA model in model recognition and order determination.
NASA Astrophysics Data System (ADS)
Bagchi, B.; Grandati, Y.; Quesne, C.
2015-06-01
The possibility for the Jacobi equation to admit, in some cases, general solutions that are polynomials has been recently highlighted by Calogero and Yi, who termed them para-Jacobi polynomials. Such polynomials are used here to build seed functions of a Darboux-Bäcklund transformation for the trigonometric Darboux-Pöschl-Teller potential. As a result, one-step regular rational extensions of the latter depending both on an integer index n and on a continuously varying parameter λ are constructed. For each n value, the eigenstates of these extended potentials are associated with a novel family of λ-dependent polynomials, which are orthogonal on [-1,1].
Silbar, R.R.
1998-09-28
WhistleSoft, Inc., proposed to convert a successful pedagogical experiment into multimedia software, making it accessible to a much broader audience. A colleague, Richard J. Jacob, has been teaching a workshop course in mathematical methods at Arizona State University (ASU) for lower undergraduate science majors. Students work at their own pace through paper-based tutorials containing many exercises, either with pencil and paper or with computer tools such as spreadsheets. These tutorial modules cry out for conversion into an interactive computer-based tutorial course that is suitable both for the classroom and for self-paced, independent learning. WhistleSoft has made a prototype of one such module, Legendre Polynomials, under Subcontract (No F97440018-35) with the Los Alamos Laboratory`s Technology Commercialization Office for demonstration and marketing purposes.
Polynomial regression calculation of the Earth's position based on millisecond pulsar timing
NASA Astrophysics Data System (ADS)
Tian, Feng; Tang, Zheng-Hong; Yan, Qing-Zeng; Yu, Yong
2012-02-01
Prior to achieving high precision navigation of a spacecraft using X-ray observations, a pulsar rotation model must be built and analysis of the precise position of the Earth should be performed using ground pulsar timing observations. We can simulate time-of-arrival ground observation data close to actual observed values before using pulsar timing observation data. Considering the correlation between the Earth's position and its short arc section of an orbit, we use polynomial regression to build the correlation. Regression coefficients can be calculated using the least square method, and a coordinate component series can also be obtained; that is, we can calculate Earth's position in the Barycentric Celestial Reference System according to pulse arrival time data and a precise pulsar rotation model. In order to set appropriate parameters before the actual timing observations for Earth positioning, we can calculate the influence of the spatial distribution of pulsars on errors in the positioning result and the influence of error source variation on positioning by simulation. It is significant that the threshold values of the observation and systematic errors can be established before an actual observation occurs; namely, we can determine the observation mode with small errors and reject the observed data with big errors, thus improving the positioning result.
Ultrafast laser spatial beam shaping based on Zernike polynomials for surface processing.
Houzet, J; Faure, N; Larochette, M; Brulez, A-C; Benayoun, S; Mauclair, C
2016-03-21
In femtosecond laser machining, spatial beam shaping can be achieved with wavefront modulators. The wavefront modulator displays a pre-calculated phase mask that modulates the laser wavefront to generate a target intensity distribution in the processing plane. Due to the non-perfect optical response of wavefront modulators, the experimental distribution may significantly differ from the target, especially for continuous shapes. We propose an alternative phase mask calculation method that can be adapted to the phase modulator optical performance. From an adjustable number of Zernike polynomials according to this performance, a least square fitting algorithm numerically determines their coefficients to obtain the desired wavefront modulation. We illustrate the technique with an optically addressed liquid-crystal light valve to produce continuous intensity distributions matching a desired ablation profile, without the need of a wavefront sensor. The projection of the experimental laser distribution shows a 5% RMS error compared to the calculated one. Ablation of steel is achieved following user-defined micro-dimples and micro-grooves targets on mold surfaces. The profiles of the microgrooves and the injected polycarbonate closely match the target (RMS below 4%). PMID:27136844
Karagiannis, Georgios Lin, Guang
2014-02-15
Generalized polynomial chaos (gPC) expansions allow us to represent the solution of a stochastic system using a series of polynomial chaos basis functions. The number of gPC terms increases dramatically as the dimension of the random input variables increases. When the number of the gPC terms is larger than that of the available samples, a scenario that often occurs when the corresponding deterministic solver is computationally expensive, evaluation of the gPC expansion can be inaccurate due to over-fitting. We propose a fully Bayesian approach that allows for global recovery of the stochastic solutions, in both spatial and random domains, by coupling Bayesian model uncertainty and regularization regression methods. It allows the evaluation of the PC coefficients on a grid of spatial points, via (1) the Bayesian model average (BMA) or (2) the median probability model, and their construction as spatial functions on the spatial domain via spline interpolation. The former accounts for the model uncertainty and provides Bayes-optimal predictions; while the latter provides a sparse representation of the stochastic solutions by evaluating the expansion on a subset of dominating gPC bases. Moreover, the proposed methods quantify the importance of the gPC bases in the probabilistic sense through inclusion probabilities. We design a Markov chain Monte Carlo (MCMC) sampler that evaluates all the unknown quantities without the need of ad-hoc techniques. The proposed methods are suitable for, but not restricted to, problems whose stochastic solutions are sparse in the stochastic space with respect to the gPC bases while the deterministic solver involved is expensive. We demonstrate the accuracy and performance of the proposed methods and make comparisons with other approaches on solving elliptic SPDEs with 1-, 14- and 40-random dimensions.
Karagiannis, Georgios; Lin, Guang
2014-02-15
Generalized polynomial chaos (gPC) expansions allow the representation of the solution of a stochastic system as a series of polynomial terms. The number of gPC terms increases dramatically with the dimension of the random input variables. When the number of the gPC terms is larger than that of the available samples, a scenario that often occurs if the evaluations of the system are expensive, the evaluation of the gPC expansion can be inaccurate due to over-fitting. We propose a fully Bayesian approach that allows for global recovery of the stochastic solution, both in spacial and random domains, by coupling Bayesian model uncertainty and regularization regression methods. It allows the evaluation of the PC coefficients on a grid of spacial points via (1) Bayesian model average or (2) medial probability model, and their construction as functions on the spacial domain via spline interpolation. The former accounts the model uncertainty and provides Bayes-optimal predictions; while the latter, additionally, provides a sparse representation of the solution by evaluating the expansion on a subset of dominating gPC bases when represented as a gPC expansion. Moreover, the method quantifies the importance of the gPC bases through inclusion probabilities. We design an MCMC sampler that evaluates all the unknown quantities without the need of ad-hoc techniques. The proposed method is suitable for, but not restricted to, problems whose stochastic solution is sparse at the stochastic level with respect to the gPC bases while the deterministic solver involved is expensive. We demonstrate the good performance of the proposed method and make comparisons with others on 1D, 14D and 40D in random space elliptic stochastic partial differential equations.
Generalized Polynomial Chaos Based Uncertainty Quantification for Planning MRgLITT Procedures
Fahrenholtz, S.; Stafford, R. J.; Maier, F.; Hazle, J. D.; Fuentes, D.
2014-01-01
Purpose A generalized polynomial chaos (gPC) method is used to incorporate constitutive parameter uncertainties within the Pennes representation of bioheat transfer phenomena. The stochastic temperature predictions of the mathematical model are critically evaluated against MR thermometry data for planning MR-guided Laser Induced Thermal Therapies (MRgLITT). Methods Pennes bioheat transfer model coupled with a diffusion theory approximation of laser tissue interaction was implemented as the underlying deterministic kernel. A probabilistic sensitivity study was used to identify parameters that provide the most variance in temperature output. Confidence intervals of the temperature predictions are compared to MR temperature imaging (MRTI) obtained during phantom and in vivo canine (n=4) MRgLITT experiments. The gPC predictions were quantitatively compared to MRTI data using probabilistic linear and temporal profiles as well as 2-D 60 °C isotherms. Results Within the range of physically meaningful constitutive values relevant to the ablative temperature regime of MRgLITT, the sensitivity study indicated that the optical parameters, particularly the anisotropy factor, created the most variance in the stochastic model's output temperature prediction. Further, within the statistical sense considered, a nonlinear model of the temperature and damage dependent perfusion, absorption, and scattering is captured within the confidence intervals of the linear gPC method. Multivariate stochastic model predictions using parameters with the dominant sensitivities show good agreement with experimental MRTI data. Conclusions Given parameter uncertainties and mathematical modeling approximations of the Pennes bioheat model, the statistical framework demonstrates conservative estimates of the therapeutic heating and has potential for use as a computational prediction tool for thermal therapy planning. PMID:23692295
A generalized polynomial chaos based ensemble Kalman filter with high accuracy
Li Jia; Xiu Dongbin
2009-08-20
As one of the most adopted sequential data assimilation methods in many areas, especially those involving complex nonlinear dynamics, the ensemble Kalman filter (EnKF) has been under extensive investigation regarding its properties and efficiency. Compared to other variants of the Kalman filter (KF), EnKF is straightforward to implement, as it employs random ensembles to represent solution states. This, however, introduces sampling errors that affect the accuracy of EnKF in a negative manner. Though sampling errors can be easily reduced by using a large number of samples, in practice this is undesirable as each ensemble member is a solution of the system of state equations and can be time consuming to compute for large-scale problems. In this paper we present an efficient EnKF implementation via generalized polynomial chaos (gPC) expansion. The key ingredients of the proposed approach involve (1) solving the system of stochastic state equations via the gPC methodology to gain efficiency; and (2) sampling the gPC approximation of the stochastic solution with an arbitrarily large number of samples, at virtually no additional computational cost, to drastically reduce the sampling errors. The resulting algorithm thus achieves a high accuracy at reduced computational cost, compared to the classical implementations of EnKF. Numerical examples are provided to verify the convergence property and accuracy improvement of the new algorithm. We also prove that for linear systems with Gaussian noise, the first-order gPC Kalman filter method is equivalent to the exact Kalman filter.
Chaos, Fractals, and Polynomials.
ERIC Educational Resources Information Center
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)
Factoring Polynomials and Fibonacci.
ERIC Educational Resources Information Center
Schwartzman, Steven
1986-01-01
Discusses the factoring of polynomials and Fibonacci numbers, offering several challenges teachers can give students. For example, they can give students a polynomial containing large numbers and challenge them to factor it. (JN)
NASA Astrophysics Data System (ADS)
Ammari, Amara; Karoui, Abderrazek
2012-05-01
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.
Interval polynomial positivity
NASA Technical Reports Server (NTRS)
Bose, N. K.; Kim, K. D.
1989-01-01
It is shown that a univariate interval polynomial is globally positive if and only if two extreme polynomials are globally positive. It is shown that the global positivity property of a bivariate interval polynomial is completely determined by four extreme bivariate polynomials. The cardinality of the determining set for k-variate interval polynomials is 2k. One of many possible generalizations, where vertex implication for global positivity holds, is made by considering the parameter space to be the set dual of a boxed domain.
Shityakov, Sergey; Förster, Carola
2014-01-01
P-glycoprotein (P-gp) is an ATP (adenosine triphosphate)-binding cassette transporter that causes multidrug resistance of various chemotherapeutic substances by active efflux from mammalian cells. P-gp plays a pivotal role in limiting drug absorption and distribution in different organs, including the intestines and brain. Thus, the prediction of P-gp–drug interactions is of vital importance in assessing drug pharmacokinetic and pharmacodynamic properties. To find the strongest P-gp blockers, we performed an in silico structure-based screening of P-gp inhibitor library (1,300 molecules) by the gradient optimization method, using polynomial empirical scoring (POLSCORE) functions. We report a strong correlation (r2=0.80, F=16.27, n=6, P<0.0157) of inhibition constants (Kiexp or pKiexp; experimental Ki or negative decimal logarithm of Kiexp) converted from experimental IC50 (half maximal inhibitory concentration) values with POLSCORE-predicted constants (KiPOLSCORE or pKiPOLSCORE), using a linear regression fitting technique. The hydrophobic interactions between P-gp and selected drug substances were detected as the main forces responsible for the inhibition effect. The results showed that this scoring technique might be useful in the virtual screening and filtering of databases of drug-like compounds at the early stage of drug development processes. PMID:24711707
Shityakov, Sergey; Förster, Carola
2014-01-01
P-glycoprotein (P-gp) is an ATP (adenosine triphosphate)-binding cassette transporter that causes multidrug resistance of various chemotherapeutic substances by active efflux from mammalian cells. P-gp plays a pivotal role in limiting drug absorption and distribution in different organs, including the intestines and brain. Thus, the prediction of P-gp-drug interactions is of vital importance in assessing drug pharmacokinetic and pharmacodynamic properties. To find the strongest P-gp blockers, we performed an in silico structure-based screening of P-gp inhibitor library (1,300 molecules) by the gradient optimization method, using polynomial empirical scoring (POLSCORE) functions. We report a strong correlation (r (2)=0.80, F=16.27, n=6, P<0.0157) of inhibition constants (Kiexp or pKiexp; experimental Ki or negative decimal logarithm of Kiexp) converted from experimental IC50 (half maximal inhibitory concentration) values with POLSCORE-predicted constants (KiPOLSCORE or pKiPOLSCORE), using a linear regression fitting technique. The hydrophobic interactions between P-gp and selected drug substances were detected as the main forces responsible for the inhibition effect. The results showed that this scoring technique might be useful in the virtual screening and filtering of databases of drug-like compounds at the early stage of drug development processes. PMID:24711707
POLYNOMIAL-BASED DISAGGREGATION OF HOURLY RAINFALL FOR CONTINUOUS HYDROLOGIC SIMULATION
Hydrologic modeling of urban watersheds for designs and analyses of stormwater conveyance facilities can be performed in either an event-based or continuous fashion. Continuous simulation requires, among other things, the use of a time series of rainfall amounts. However, for urb...
NASA Technical Reports Server (NTRS)
Wood, C. A.
1974-01-01
For polynomials of higher degree, iterative numerical methods must be used. Four iterative methods are presented for approximating the zeros of a polynomial using a digital computer. Newton's method and Muller's method are two well known iterative methods which are presented. They extract the zeros of a polynomial by generating a sequence of approximations converging to each zero. However, both of these methods are very unstable when used on a polynomial which has multiple zeros. That is, either they fail to converge to some or all of the zeros, or they converge to very bad approximations of the polynomial's zeros. This material introduces two new methods, the greatest common divisor (G.C.D.) method and the repeated greatest common divisor (repeated G.C.D.) method, which are superior methods for numerically approximating the zeros of a polynomial having multiple zeros. These methods were programmed in FORTRAN 4 and comparisons in time and accuracy are given.
Polynomial Algorithms for Item Matching.
ERIC Educational Resources Information Center
Armstrong, Ronald D.; Jones, Douglas H.
1992-01-01
Polynomial algorithms are presented that are used to solve selected problems in test theory, and computational results from sample problems with several hundred decision variables are provided that demonstrate the benefits of these algorithms. The algorithms are based on optimization theory in networks (graphs). (SLD)
Entanglement conditions and polynomial identities
Shchukin, E.
2011-11-15
We develop a rather general approach to entanglement characterization based on convexity properties and polynomial identities. This approach is applied to obtain simple and efficient entanglement conditions that work equally well in both discrete as well as continuous-variable environments. Examples of violations of our conditions are presented.
Fink, Wolfgang; Micol, Daniel
2006-01-01
We describe a computer eye model that allows for aspheric surfaces and a three-dimensional computer-based ray-tracing technique to simulate optical properties of the human eye and visual perception under various eye defects. Eye surfaces, such as the cornea, eye lens, and retina, are modeled or approximated by a set of Zernike polynomials that are fitted to input data for the respective surfaces. A ray-tracing procedure propagates light rays using Snell's law of refraction from an input object (e.g., digital image) through the eye under investigation (i.e., eye with defects to be modeled) to form a retinal image that is upside down and left-right inverted. To obtain a first-order realistic visual perception without having to model or simulate the retina and the visual cortex, this retinal image is then back-propagated through an emmetropic eye (e.g., Gullstrand exact schematic eye model with no additional eye defects) to an output screen of the same dimensions and at the same distance from the eye as the input object. Visual perception under instances of emmetropia, regular astigmatism, irregular astigmatism, and (central symmetric) keratoconus is simulated and depicted. In addition to still images, the computer ray-tracing tool presented here (simEye) permits the production of animated movies. These developments may have scientific and educational value. This tool may facilitate the education and training of both the public, for example, patients before undergoing eye surgery, and those in the medical field, such as students and professionals. Moreover, simEye may be used as a scientific research tool to investigate optical lens systems in general and the visual perception under a variety of eye conditions and surgical procedures such as cataract surgery and laser assisted in situ keratomileusis (LASIK) in particular. PMID:17092160
Polynomial Graphs and Symmetry
ERIC Educational Resources Information Center
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…
NASA Astrophysics Data System (ADS)
Bogner, Christian; Weinzierl, Stefan
The integrand of any multiloop integral is characterized after Feynman parametrization by two polynomials. In this review we summarize the properties of these polynomials. Topics covered in this paper include among others: spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.
NASA Astrophysics Data System (ADS)
Duan, Lei; Hui, Mei; Deng, Jiayuan; Gong, Cheng; Zhao, Yuejin
2012-11-01
Annular sub-aperture stitching method was developed for testing large-aperture aspheric surfaces without using of any compensating element for measurement. It is necessary to correct measurement of aspheric optical aberrations and create mathematical description to describe wave-front aberrations. Zernike polynomials are suitable to describe wave aberration functions and data fitting of experimental measurements for the annular sub-aperture stitching system. This paper uses Zernike polynomials to describe the wave-front aberrations of full wave-front and reconstructed wave-front by annular sub-aperture stitching algorithm. At the same time, the imaging quality of the aspheric optical element can be contrasted. The stitching result shows good agreement with the full aperture result.
NASA Astrophysics Data System (ADS)
Mironov, A.; Mkrtchyan, R.; Morozov, A.
2016-02-01
We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, respectively and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.
Efficient Multiplication of Polynomials on Graphics Hardware
NASA Astrophysics Data System (ADS)
Emeliyanenko, Pavel
We present the algorithm to multiply univariate polynomials with integer coefficients efficiently using the Number Theoretic transform (NTT) on Graphics Processing Units (GPU). The same approach can be used to multiply large integers encoded as polynomials. Our algorithm exploits fused multiply-add capabilities of the graphics hardware. NTT multiplications are executed in parallel for a set of distinct primes followed by reconstruction using the Chinese Remainder theorem (CRT) on the GPU. Our benchmarking experiences show the NTT multiplication performance up to 77 GMul/s. We compared our approach with CPU-based implementations of polynomial and large integer multiplication provided by NTL and GMP libraries.
Shan, Peng; Peng, Silong; Zhao, Yuhui; Tang, Liang
2016-03-01
An analysis of binary mixtures of hydroxyl compound by Attenuated Total Reflection Fourier transform infrared spectroscopy (ATR FT-IR) and classical least squares (CLS) yield large model error due to the presence of unmodeled components such as H-bonded components. To accommodate these spectral variations, polynomial-based least squares (LSP) and polynomial-based total least squares (TLSP) are proposed to capture the nonlinear absorbance-concentration relationship. LSP is based on assuming that only absorbance noise exists; while TLSP takes both absorbance noise and concentration noise into consideration. In addition, based on different solving strategy, two optimization algorithms (limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm and Levenberg-Marquardt (LM) algorithm) are combined with TLSP and then two different TLSP versions (termed as TLSP-LBFGS and TLSP-LM) are formed. The optimum order of each nonlinear model is determined by cross-validation. Comparison and analyses of the four models are made from two aspects: absorbance prediction and concentration prediction. The results for water-ethanol solution and ethanol-ethyl lactate solution show that LSP, TLSP-LBFGS, and TLSP-LM can, for both absorbance prediction and concentration prediction, obtain smaller root mean square error of prediction than CLS. Additionally, they can also greatly enhance the accuracy of estimated pure component spectra. However, from the view of concentration prediction, the Wilcoxon signed rank test shows that there is no statistically significant difference between each nonlinear model and CLS. PMID:26810185
Muralidhar, K Raja; Komanduri, K
2014-06-01
Purpose: The objective of this work is to present a mechanism for calculating inflection points on profiles at various depths and field sizes and also a significant study on the percentage of doses at the inflection points for various field sizes and depths for 6XFFF and 10XFFF energy profiles. Methods: Graphical representation was done on Percentage of dose versus Inflection points. Also using the polynomial function, the authors formulated equations for calculating spot-on inflection point on the profiles for 6X FFF and 10X FFF energies for all field sizes and at various depths. Results: In a flattening filter free radiation beam which is not like in Flattened beams, the dose at inflection point of the profile decreases as field size increases for 10XFFF. Whereas in 6XFFF, the dose at the inflection point initially increases up to 10x10cm2 and then decreases. The polynomial function was fitted for both FFF beams for all field sizes and depths. For small fields less than 5x5 cm2 the inflection point and FWHM are almost same and hence analysis can be done just like in FF beams. A change in 10% of dose can change the field width by 1mm. Conclusion: The present study, Derivative of equations based on the polynomial equation to define inflection point concept is precise and accurate way to derive the inflection point dose on any FFF beam profile at any depth with less than 1% accuracy. Corrections can be done in future studies based on the multiple number of machine data. Also a brief study was done to evaluate the inflection point positions with respect to dose in FFF energies for various field sizes and depths for 6XFFF and 10XFFF energy profiles.
Distortion theorems for polynomials on a circle
Dubinin, V N
2000-12-31
Inequalities for the derivatives with respect to {phi}=arg z the functions ReP(z), |P(z)|{sup 2} and arg P(z) are established for an algebraic polynomial P(z) at points on the circle |z|=1. These estimates depend, in particular, on the constant term and the leading coefficient of the polynomial P(z) and improve the classical Bernstein and Turan inequalities. The method of proof is based on the techniques of generalized reduced moduli.
Tutte Polynomial of Scale-Free Networks
NASA Astrophysics Data System (ADS)
Chen, Hanlin; Deng, Hanyuan
2016-05-01
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics. The computation of this invariant for a graph is NP-hard in general. In this paper, we focus on two iteratively growing scale-free networks, which are ubiquitous in real-life systems. Based on their self-similar structures, we mainly obtain recursive formulas for the Tutte polynomials of two scale-free networks (lattices), one is fractal and "large world", while the other is non-fractal but possess the small-world property. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at ( x, y)-plane, such as, the number of spanning trees, the number of acyclic orientations, etc.
Polynomials with small Mahler measure
NASA Astrophysics Data System (ADS)
Mossinghoff, M. J.
1998-10-01
We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than 1.3, test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large as 181. We find a new limit point of Mahler measures near 1.309, four new Salem numbers less than 1.3, and many new polynomials with small Mahler measure. None has measure smaller than that of Lehmer's degree 10 polynomial.
Chen, Huifang; Xie, Lei
2014-01-01
Self-healing group key distribution (SGKD) aims to deal with the key distribution problem over an unreliable wireless network. In this paper, we investigate the SGKD issue in resource-constrained wireless networks. We propose two improved SGKD schemes using the one-way hash chain (OHC) and the revocation polynomial (RP), the OHC&RP-SGKD schemes. In the proposed OHC&RP-SGKD schemes, by introducing the unique session identifier and binding the joining time with the capability of recovering previous session keys, the problem of the collusion attack between revoked users and new joined users in existing hash chain-based SGKD schemes is resolved. Moreover, novel methods for utilizing the one-way hash chain and constructing the personal secret, the revocation polynomial and the key updating broadcast packet are presented. Hence, the proposed OHC&RP-SGKD schemes eliminate the limitation of the maximum allowed number of revoked users on the maximum allowed number of sessions, increase the maximum allowed number of revoked/colluding users, and reduce the redundancy in the key updating broadcast packet. Performance analysis and simulation results show that the proposed OHC&RP-SGKD schemes are practical for resource-constrained wireless networks in bad environments, where a strong collusion attack resistance is required and many users could be revoked. PMID:25529204
Calculators and Polynomial Evaluation.
ERIC Educational Resources Information Center
Weaver, J. F.
The intent of this paper is to suggest and illustrate how electronic hand-held calculators, especially non-programmable ones with limited data-storage capacity, can be used to advantage by students in one particular aspect of work with polynomial functions. The basic mathematical background upon which calculator application is built is summarized.…
ERIC Educational Resources Information Center
Gordon, Sheldon P.
1992-01-01
Demonstrates how the uniqueness and anonymity of a student's Social Security number can be utilized to create individualized polynomial equations that students can investigate using computers or graphing calculators. Students write reports of their efforts to find and classify all real roots of their equation. (MDH)
On polynomial preconditioning for indefinite Hermitian matrices
NASA Technical Reports Server (NTRS)
Freund, Roland W.
1989-01-01
The minimal residual method is studied combined with polynomial preconditioning for solving large linear systems (Ax = b) with indefinite Hermitian coefficient matrices (A). The standard approach for choosing the polynomial preconditioners leads to preconditioned systems which are positive definite. Here, a different strategy is studied which leaves the preconditioned coefficient matrix indefinite. More precisely, the polynomial preconditioner is designed to cluster the positive, resp. negative eigenvalues of A around 1, resp. around some negative constant. In particular, it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two parameter family of Chebyshev approximation problems. Some basic results are established for these approximation problems and a Remez type algorithm is sketched for their numerical solution. The problem of selecting the parameters such that the resulting indefinite polynomial preconditioners speeds up the convergence of minimal residual method optimally is also addressed. An approach is proposed based on the concept of asymptotic convergence factors. Finally, some numerical examples of indefinite polynomial preconditioners are given.
Interpolation and Polynomial Curve Fitting
ERIC Educational Resources Information Center
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…
Determinants and Polynomial Root Structure
ERIC Educational Resources Information Center
De Pillis, L. G.
2005-01-01
A little known property of determinants is developed in a manner accessible to beginning undergraduates in linear algebra. Using the language of matrix theory, a classical result by Sylvester that describes when two polynomials have a common root is recaptured. Among results concerning the structure of polynomial roots, polynomials with pairs of…
Interpolation algorithm of Leverrier?Faddev type for polynomial matrices
NASA Astrophysics Data System (ADS)
Petkovic, Marko; Stanimirovic, Predrag
2006-07-01
We investigated an interpolation algorithm for computing outer inverses of a given polynomial matrix, based on the Leverrier?Faddeev method. This algorithm is a continuation of the finite algorithm for computing generalized inverses of a given polynomial matrix, introduced in [11]. Also, a method for estimating the degrees of polynomial matrices arising from the Leverrier?Faddeev algorithm is given as the improvement of the interpolation algorithm. Based on similar idea, we introduced methods for computing rank and index of polynomial matrix. All algorithms are implemented in the symbolic programming language MATHEMATICA , and tested on several different classes of test examples.
NASA Astrophysics Data System (ADS)
Wong-Loya, J. A.; Andaverde, J.; Santoyo, E.
2012-12-01
A new practical method based on rational polynomial (RP) functions to estimate the static formation temperatures (SFT) in geothermal and petroleum boreholes is described. Thermal recovery processes involved during borehole drilling and completion operations were represented by mathematical asymptotic trends. Measurements of bottom-hole temperature and shut-in times (at least three or more) have been used both to obtain a mathematical function that describes the thermal recovery process of drilled boreholes, and to estimate the SFT. Using build-up temperature logs, the SFT have been reliably estimated with precision and accuracy. With these results, it was successfully demonstrated that the new RP method provides a practical tool for the reliable prediction of SFT in geothermal and petroleum boreholes.
Some discrete multiple orthogonal polynomials
NASA Astrophysics Data System (ADS)
Arvesú, J.; Coussement, J.; van Assche, W.
2003-04-01
In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317-347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r+1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r=2.
Independence polynomial and matching polynomial of the Koch network
NASA Astrophysics Data System (ADS)
Liao, Yunhua; Xie, Xiaoliang
2015-11-01
The lattice gas model and the monomer-dimer model are two classical models in statistical mechanics. It is well known that the partition functions of these two models are associated with the independence polynomial and the matching polynomial in graph theory, respectively. Both polynomials have been shown to belong to the “#P-complete” class, which indicate the problems are computationally “intractable”. We consider these two polynomials of the Koch networks which are scale-free with small-world effects. Explicit recurrences are derived, and explicit formulae are presented for the number of independent sets of a certain type.
Orthogonal polynomials and tolerancing
NASA Astrophysics Data System (ADS)
Rogers, John R.
2011-10-01
Previous papers have established the inadvisability of applying tolerances directly to power-series aspheric coefficients. The basic reason is that the individual terms are far from orthogonal. Zernike surfaces and the new Forbes surface types have certain orthogonality properties over the circle described by the "normalization radius." However, at surfaces away from the stop, the optical beam is smaller than the surface, and the polynomials are not orthogonal over the area sampled by the beam. In this paper, we investigate the breakdown of orthogonality as the surface moves away from the aperture stop, and the implications of this to tolerancing.
NASA Astrophysics Data System (ADS)
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.
NASA Astrophysics Data System (ADS)
Chaves, Rafael
2016-01-01
It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the mathematical theory of causality, allowing for the representation of arbitrary causal structures and providing a rigorous tool to reason about probabilistic causation. Indeed, Bell's theorem concerns a very particular kind of causal structure and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalizations involving more complex Bell scenarios. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a new, general, and conceptually clear method for the derivation of polynomial Bell inequalities in a wide class of scenarios. We also show how our construction can be used to allow for relaxations of causal constraints and naturally gives rise to a notion of nonsignaling in generalized Bell networks.
NASA Astrophysics Data System (ADS)
Zhang, Xu
This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange attractors, and shows that some maps are chaotic in the sense of Li-Yorke or Devaney. This type of maps includes both the Logistic map and the Hénon map. For some diffeomorphisms with the expansion dimension equal to one or two in three-dimensional spaces, the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets on which the systems are topologically conjugate to the two-sided fullshift on finite alphabet are obtained; for some expanding maps, the chaotic region is analyzed by using the coupled-expansion theory and the Brouwer degree theory. For three types of higher-dimensional polynomial maps with degree two, the conditions under which there are Smale horseshoes and uniformly hyperbolic invariant sets are given, and the topological conjugacy between the maps on the invariant sets and the two-sided fullshift on finite alphabet is obtained. Some interesting maps with chaotic attractors and positive Lyapunov exponents in three-dimensional spaces are found by using computer simulations. In the end, two examples are provided to illustrate the theoretical results.
Molecular Insights into the Enigmatic Metabolic Regulator, SnRK1.
Emanuelle, Shane; Doblin, Monika S; Stapleton, David I; Bacic, Antony; Gooley, Paul R
2016-04-01
Sucrose non-fermenting-1 (SNF1)-related kinase 1 (SnRK1) lies at the heart of metabolic homeostasis in plants and is crucial for normal development and response to stress. Evolutionarily related to SNF1 in yeast and AMP-activated kinase (AMPK) in mammals, SnRK1 acts protectively to maintain homeostasis in the face of fluctuations in energy status. Despite a conserved function, the structure and regulation of the plant kinase differ considerably from its relatively well-understood opisthokont orthologues. In this review, we highlight the known plant-specific modes of regulation involving SnRK1 together with new insights based on a 3D molecular model of the kinase. We also summarise how these differences from other orthologues may be specific adaptations to plant metabolism, and offer insights into possible avenues of future inquiry into this enigmatic enzyme. PMID:26642889
Satellite Orbital Interpolation using Tchebychev Polynomials
NASA Astrophysics Data System (ADS)
Richard, Jean-Yves; Deleflie, Florent; Edorh, Sémého
2014-05-01
A satellite or artificial probe orbit is made of time series of orbital elements such as state vectors (position and velocities, keplerian orbital elements) given at regular or irregular time intervals. These time series are fitted to observations, so that differences between observations (distance, radial velocity) and the theoretical quantity be minimal, according to a statistical criterion, mostly based on the least-squared algorithm. These computations are carried out using dedicated software, such as the GINS used by GRGS, mainly at CNES Toulouse and Paris Observatory. From an operational point of view, time series of orbital elements are 7-day long. Depending on the dynamical configurations, more generally, they can typically vary from a couple of days to some weeks. One of the fundamental parameters to be adjusted is the initial state vector. This can lead to time gaps, at the level of a few dozen of centimetres between the last point of a time series to the first one of the following data set. The objective of this presentation consists in the improvement of an interpolation method freed itself of such possible "discontinuities" resulting between satellite's orbit arcs when a new initial bulletin is adjusted. We compare solutions of different Satellite Laser Ranging using interpolation methods such as Lagrange polynomial, spline cubic, Tchebychev orthogonal polynomial and cubic Hermite polynomial. These polynomial coefficients are used to reconstruct and interpolate the satellite orbits without time gaps and discontinuities and requiring a weak memory size. In this approach, we have tested the orbital reconstruction using Tchebychev polynomial coefficients for the LAGEOS and Starlette satellites. In this presentation, it is showed that Tchebychev's polynomial interpolation can achieve accuracy in the orbit reconstruction at the sub-centimetre level and allowing a gain of a factor 5 of memory size of the satellite orbit with respect to the Cartesian
On a Perplexing Polynomial Puzzle
ERIC Educational Resources Information Center
Richmond, Bettina
2010-01-01
It seems rather surprising that any given polynomial p(x) with nonnegative integer coefficients can be determined by just the two values p(1) and p(a), where a is any integer greater than p(1). This result has become known as the "perplexing polynomial puzzle." Here, we address the natural question of what might be required to determine a…
Graphical Solution of Polynomial Equations
ERIC Educational Resources Information Center
Grishin, Anatole
2009-01-01
Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…
Pollok, Jill R; Johnson, Charles S; Eisenback, J D; Reed, T David
2016-06-01
Most commercial tobacco cultivars possess the Rk1 resistance gene to races 1 and 3 of Meloidogyne incognita and race 1 of Meloidogyne arenaria, which has caused a shift in population prevalence in Virginia tobacco fields toward other species and races. A number of cultivars now also possess the Rk2 gene for root-knot resistance. Experiments were conducted in 2013 to 2014 to examine whether possessing both Rk1 and Rk2 increases resistance to a variant of M. incognita race 3 compared to either gene alone. Greenhouse trials were arranged in a completely randomized design with Coker 371-Gold (C371G; susceptible), NC 95 and SC 72 (Rk1Rk1), T-15-1-1 (Rk2Rk2), and STNCB-2-28 and NOD 8 (Rk1Rk1 and Rk2Rk2). Each plant was inoculated with 5,000 root-knot nematode eggs; data were collected 60 d postinoculation. Percent galling and numbers of egg masses and eggs were counted, the latter being used to calculate the reproductive index on each host. Despite variability, entries with both Rk1 and Rk2 conferred greater resistance to a variant of M. incognita race 3 than plants with Rk1 or Rk2 alone. Entries with Rk1 alone were successful in reducing root galling and nematode reproduction compared to the susceptible control. Entry T-15-1-1 did not reduce galling compared to the susceptible control but often suppressed reproduction. PMID:27418700
Pollok, Jill R.; Johnson, Charles S.; Eisenback, J. D.; Reed, T. David
2016-01-01
Most commercial tobacco cultivars possess the Rk1 resistance gene to races 1 and 3 of Meloidogyne incognita and race 1 of Meloidogyne arenaria, which has caused a shift in population prevalence in Virginia tobacco fields toward other species and races. A number of cultivars now also possess the Rk2 gene for root-knot resistance. Experiments were conducted in 2013 to 2014 to examine whether possessing both Rk1 and Rk2 increases resistance to a variant of M. incognita race 3 compared to either gene alone. Greenhouse trials were arranged in a completely randomized design with Coker 371-Gold (C371G; susceptible), NC 95 and SC 72 (Rk1Rk1), T-15-1-1 (Rk2Rk2), and STNCB-2-28 and NOD 8 (Rk1Rk1 and Rk2Rk2). Each plant was inoculated with 5,000 root-knot nematode eggs; data were collected 60 d postinoculation. Percent galling and numbers of egg masses and eggs were counted, the latter being used to calculate the reproductive index on each host. Despite variability, entries with both Rk1 and Rk2 conferred greater resistance to a variant of M. incognita race 3 than plants with Rk1 or Rk2 alone. Entries with Rk1 alone were successful in reducing root galling and nematode reproduction compared to the susceptible control. Entry T-15-1-1 did not reduce galling compared to the susceptible control but often suppressed reproduction. PMID:27418700
NASA Astrophysics Data System (ADS)
Wong-Loya, J. A.; Santoyo, E.; Andaverde, J. A.; Quiroz-Ruiz, A.
2015-12-01
A Web-Based Computer System (RPM-WEBBSYS) has been developed for the application of the Rational Polynomial Method (RPM) to estimate static formation temperatures (SFT) of geothermal and petroleum wells. The system is also capable to reproduce the full thermal recovery processes occurred during the well completion. RPM-WEBBSYS has been programmed using advances of the information technology to perform more efficiently computations of SFT. RPM-WEBBSYS may be friendly and rapidly executed by using any computing device (e.g., personal computers and portable computing devices such as tablets or smartphones) with Internet access and a web browser. The computer system was validated using bottomhole temperature (BHT) measurements logged in a synthetic heat transfer experiment, where a good matching between predicted and true SFT was achieved. RPM-WEBBSYS was finally applied to BHT logs collected from well drilling and shut-in operations, where the typical problems of the under- and over-estimation of the SFT (exhibited by most of the existing analytical methods) were effectively corrected.
Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials.
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. PMID:25791164
NASA Astrophysics Data System (ADS)
Oh, Jaehong; Lee, Changno
2015-02-01
As the need for efficient methods to accurately update and refine geospatial satellite image databases is increasing, we have proposed the use of 3-dimensional digital maps for the fully-automated RPCs bias compensation of high resolution satellite imagery. The basic idea is that the map features are scaled and aligned to the image features, except for the local shift, through the RPCs-based image projection, and then the shifts are automatically determined over the entire image space by template-based edge matching of the heterogeneous data set. This enables modeling of RPCs bias compensation parameters for accurate georeferencing. The map features are selected based on four suggested rules. Experiments were carried out for three Kompsat-2 images and stereo IKONOS images with 1:5000 scale Korean national topographic maps. Image matching performance is discussed with justification of the parameter selection, and the georeferencing accuracy is analyzed. The experimental results showed the automated approach can achieve one-pixel level of georeferencing accuracy, enabling economical hybrid map creation as well as large scale map updates.
Thermodynamic characterization of networks using graph polynomials
NASA Astrophysics Data System (ADS)
Ye, Cheng; Comin, César H.; Peron, Thomas K. DM.; Silva, Filipi N.; Rodrigues, Francisco A.; Costa, Luciano da F.; Torsello, Andrea; Hancock, Edwin R.
2015-09-01
In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.
Hadamard Factorization of Stable Polynomials
NASA Astrophysics Data System (ADS)
Loredo-Villalobos, Carlos Arturo; Aguirre-Hernández, Baltazar
2011-11-01
The stable (Hurwitz) polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p,q ∈ R[x]:p(x) = anxn+an-1xn-1+...+a1x+a0q(x) = bmx m+bm-1xm-1+...+b1x+b0the Hadamard product (p × q) is defined as (p×q)(x) = akbkxk+ak-1bk-1xk-1+...+a1b1x+a0b0where k = min(m,n). Some results (see [16]) shows that if p,q ∈R[x] are stable polynomials then (p×q) is stable, also, i.e. the Hadamard product is closed; however, the reciprocal is not always true, that is, not all stable polynomial has a factorization into two stable polynomials the same degree n, if n> 4 (see [15]).In this work we will give some conditions to Hadamard factorization existence for stable polynomials.
Orthogonal polynomials and deformed oscillators
NASA Astrophysics Data System (ADS)
Borzov, V. V.; Damaskinsky, E. V.
2015-10-01
In the example of the Fibonacci oscillator, we discuss the construction of oscillator-like systems associated with orthogonal polynomials. We also consider the question of the dimensions of the corresponding Lie algebras.
NASA Astrophysics Data System (ADS)
Rochoux, M. C.; Ricci, S.; Lucor, D.; Cuenot, B.; Trouvé, A.
2014-05-01
This paper is the first part in a series of two articles and presents a data-driven wildfire simulator for forecasting wildfire spread scenarios, at a reduced computational cost that is consistent with operational systems. The prototype simulator features the following components: a level-set-based fire propagation solver FIREFLY that adopts a regional-scale modeling viewpoint, treats wildfires as surface propagating fronts, and uses a description of the local rate of fire spread (ROS) as a function of environmental conditions based on Rothermel's model; a series of airborne-like observations of the fire front positions; and a data assimilation algorithm based on an ensemble Kalman filter (EnKF) for parameter estimation. This stochastic algorithm partly accounts for the non-linearities between the input parameters of the semi-empirical ROS model and the fire front position, and is sequentially applied to provide a spatially-uniform correction to wind and biomass fuel parameters as observations become available. A wildfire spread simulator combined with an ensemble-based data assimilation algorithm is therefore a promising approach to reduce uncertainties in the forecast position of the fire front and to introduce a paradigm-shift in the wildfire emergency response. In order to reduce the computational cost of the EnKF algorithm, a surrogate model based on a polynomial chaos (PC) expansion is used in place of the forward model FIREFLY in the resulting hybrid PC-EnKF algorithm. The performance of EnKF and PC-EnKF is assessed on synthetically-generated simple configurations of fire spread to provide valuable information and insight on the benefits of the PC-EnKF approach as well as on a controlled grassland fire experiment. The results indicate that the proposed PC-EnKF algorithm features similar performance to the standard EnKF algorithm, but at a much reduced computational cost. In particular, the re-analysis and forecast skills of data assimilation strongly relate
NASA Astrophysics Data System (ADS)
Rochoux, M. C.; Ricci, S.; Lucor, D.; Cuenot, B.; Trouvé, A.
2014-11-01
This paper is the first part in a series of two articles and presents a data-driven wildfire simulator for forecasting wildfire spread scenarios, at a reduced computational cost that is consistent with operational systems. The prototype simulator features the following components: an Eulerian front propagation solver FIREFLY that adopts a regional-scale modeling viewpoint, treats wildfires as surface propagating fronts, and uses a description of the local rate of fire spread (ROS) as a function of environmental conditions based on Rothermel's model; a series of airborne-like observations of the fire front positions; and a data assimilation (DA) algorithm based on an ensemble Kalman filter (EnKF) for parameter estimation. This stochastic algorithm partly accounts for the nonlinearities between the input parameters of the semi-empirical ROS model and the fire front position, and is sequentially applied to provide a spatially uniform correction to wind and biomass fuel parameters as observations become available. A wildfire spread simulator combined with an ensemble-based DA algorithm is therefore a promising approach to reduce uncertainties in the forecast position of the fire front and to introduce a paradigm-shift in the wildfire emergency response. In order to reduce the computational cost of the EnKF algorithm, a surrogate model based on a polynomial chaos (PC) expansion is used in place of the forward model FIREFLY in the resulting hybrid PC-EnKF algorithm. The performance of EnKF and PC-EnKF is assessed on synthetically generated simple configurations of fire spread to provide valuable information and insight on the benefits of the PC-EnKF approach, as well as on a controlled grassland fire experiment. The results indicate that the proposed PC-EnKF algorithm features similar performance to the standard EnKF algorithm, but at a much reduced computational cost. In particular, the re-analysis and forecast skills of DA strongly relate to the spatial and temporal
ERIC Educational Resources Information Center
Young, Forrest W.
A model permitting construction of algorithms for the polynomial conjoint analysis of similarities is presented. This model, which is based on concepts used in nonmetric scaling, permits one to obtain the best approximate solution. The concepts used to construct nonmetric scaling algorithms are reviewed. Finally, examples of algorithmic models for…
Constructing general partial differential equations using polynomial and neural networks.
Zjavka, Ladislav; Pedrycz, Witold
2016-01-01
Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems. PMID:26547244
Zernike polynomials for photometric characterization of LEDs
NASA Astrophysics Data System (ADS)
Velázquez, J. L.; Ferrero, A.; Pons, A.; Campos, J.; Hernanz, M. L.
2016-02-01
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.
On the formulae for the colored HOMFLY polynomials
NASA Astrophysics Data System (ADS)
Kawagoe, Kenichi
2016-08-01
We provide methods to compute the colored HOMFLY polynomials of knots and links with symmetric representations based on the linear skein theory. By using diagrammatic calculations, several formulae for the colored HOMFLY polynomials are obtained. As an application, we calculate some examples for hyperbolic knots and links, and we study a generalization of the volume conjecture by means of numerical calculations. In these examples, we observe that asymptotic behaviors of invariants seem to have relations to the volume conjecture.
Using Tutte polynomials to analyze the structure of the benzodiazepines
NASA Astrophysics Data System (ADS)
Cadavid Muñoz, Juan José
2014-05-01
Graph theory in general and Tutte polynomials in particular, are implemented for analyzing the chemical structure of the benzodiazepines. Similarity analysis are used with the Tutte polynomials for finding other molecules that are similar to the benzodiazepines and therefore that might show similar psycho-active actions for medical purpose, in order to evade the drawbacks associated to the benzodiazepines based medicine. For each type of benzodiazepines, Tutte polynomials are computed and some numeric characteristics are obtained, such as the number of spanning trees and the number of spanning forests. Computations are done using the computer algebra Maple's GraphTheory package. The obtained analytical results are of great importance in pharmaceutical engineering. As a future research line, the usage of the chemistry computational program named Spartan, will be used to extent and compare it with the obtained results from the Tutte polynomials of benzodiazepines.
Georeferencing CAMS data: Polynomial rectification and beyond
NASA Astrophysics Data System (ADS)
Yang, Xinghe
The Calibrated Airborne Multispectral Scanner (CAMS) is a sensor used in the commercial remote sensing program at NASA Stennis Space Center. In geographic applications of the CAMS data, accurate geometric rectification is essential for the analysis of the remotely sensed data and for the integration of the data into Geographic Information Systems (GIS). The commonly used rectification techniques such as the polynomial transformation and ortho rectification have been very successful in the field of remote sensing and GIS for most remote sensing data such as Landsat imagery, SPOT imagery and aerial photos. However, due to the geometric nature of the airborne line scanner which has high spatial frequency distortions, the polynomial model and the ortho rectification technique in current commercial software packages such as Erdas Imagine are not adequate for obtaining sufficient geometric accuracy. In this research, the geometric nature, especially the major distortions, of the CAMS data has been described. An analytical step-by-step geometric preprocessing has been utilized to deal with the potential high frequency distortions of the CAMS data. A generic sensor-independent photogrammetric model has been developed for the ortho-rectification of the CAMS data. Three generalized kernel classes and directional elliptical basis have been formulated into a rectification model of summation of multisurface functions, which is a significant extension to the traditional radial basis functions. The preprocessing mechanism has been fully incorporated into the polynomial, the triangle-based finite element analysis as well as the summation of multisurface functions. While the multisurface functions and the finite element analysis have the characteristics of localization, piecewise logic has been applied to the polynomial and photogrammetric methods, which can produce significant accuracy improvement over the global approach. A software module has been implemented with full
General complex polynomial root solver
NASA Astrophysics Data System (ADS)
Skowron, J.; Gould, A.
2012-12-01
This general complex polynomial root solver, implemented in Fortran and further optimized for binary microlenses, uses a new algorithm to solve polynomial equations and is 1.6-3 times faster than the ZROOTS subroutine that is commercially available from Numerical Recipes, depending on application. The largest improvement, when compared to naive solvers, comes from a fail-safe procedure that permits skipping the majority of the calculations in the great majority of cases, without risking catastrophic failure in the few cases that these are actually required.
On the minimum polynomial of supermatrices
NASA Astrophysics Data System (ADS)
Fellouris, Anargyros G.; Matiadou, Lina K.
2002-11-01
In this paper, a new selection of factors for the construction of the minimum polynomial of a supermatrix M is proposed, leading to null polynomials of M of lower degree than the degree of the corresponding polynomial obtained by using the method proposed in the work of Urrutia and Morales [1]. The case of (1 + 1) × (1 + 1) supermatrices has been completely discussed. Moreover, the main theorem concerning the construction of the minimum polynomial as a product of factors from the characteristic polynomial in the general case of (m + n) × (m + n) supermatrices is given. Finally, we prove that the minimum polynomial of a supermatrix M, in general, is not unique.
Optical homodyne tomography with polynomial series expansion
Benichi, Hugo; Furusawa, Akira
2011-09-15
We present and demonstrate a method for optical homodyne tomography based on the inverse Radon transform. Different from the usual filtered back-projection algorithm, this method uses an appropriate polynomial series to expand the Wigner function and the marginal distribution, and discretize Fourier space. We show that this technique solves most technical difficulties encountered with kernel deconvolution-based methods and reconstructs overall better and smoother Wigner functions. We also give estimators of the reconstruction errors for both methods and show improvement in noise handling properties and resilience to statistical errors.
Polynomial Beam Element Analysis Module
Energy Science and Technology Software Center (ESTSC)
2013-05-01
pBEAM (Polynomial Beam Element Analysis Module) is a finite element code for beam-like structures. The methodology uses Euler? Bernoulli beam elements with 12 degrees of freedom (3 translation and 3 rotational at each end of the element).
SymRK and the nodule vascular system
Sanchez-Lopez, Rosana; Jáuregui, David; Quinto, Carmen
2012-01-01
Symbiotic legume-rhizobia relationship leads to the formation of nitrogen-fixing nodules. Successful nodulation depends on the expression and cross-talk of a batttery of genes, among them SymRK (symbiosis receptor-like kinase), a leucine-rich repeat receptor-like kinase. SymRK is required for the rhizobia invasion of root hairs, as well as for the infection thread and symbiosome formation. Using immunolocalization and downregulation strategies we have recently provided evidence of a new function of PvSymRK in nodulation. We have found that a tight regulation of PvSymRK expression is required for the accurate development of the vascular bundle system in Phaseolus vulgaris nodules. PMID:22580688
A Summation Formula for Macdonald Polynomials
NASA Astrophysics Data System (ADS)
de Gier, Jan; Wheeler, Michael
2016-03-01
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases {t = 1} and {q = 0}, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q-Whittaker polynomials.
Nodal Statistics for the Van Vleck Polynomials
NASA Astrophysics Data System (ADS)
Bourget, Alain
The Van Vleck polynomials naturally arise from the generalized Lamé equation
Restricted Schur polynomials and finite N counting
Collins, Storm
2009-01-15
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.
Modal wavefront reconstruction over general shaped aperture by numerical orthogonal polynomials
NASA Astrophysics Data System (ADS)
Ye, Jingfei; Li, Xinhua; Gao, Zhishan; Wang, Shuai; Sun, Wenqing; Wang, Wei; Yuan, Qun
2015-03-01
In practical optical measurements, the wavefront data are recorded by pixelated imaging sensors. The closed-form analytical base polynomial will lose its orthogonality in the discrete wavefront database. For a wavefront with an irregularly shaped aperture, the corresponding analytical base polynomials are laboriously derived. The use of numerical orthogonal polynomials for reconstructing a wavefront with a general shaped aperture over the discrete data points is presented. Numerical polynomials are orthogonal over the discrete data points regardless of the boundary shape of the aperture. The performance of numerical orthogonal polynomials is confirmed by theoretical analysis and experiments. The results demonstrate the adaptability, validity, and accuracy of numerical orthogonal polynomials for estimating the wavefront over a general shaped aperture from regular boundary to an irregular boundary.
Measuring polynomial invariants of multiparty quantum states
Leifer, M.S.; Linden, N.; Winter, A.
2004-05-01
We present networks for directly estimating the polynomial invariants of multiparty quantum states under local transformations. The structure of these networks is closely related to the structure of the invariants themselves and this lends a physical interpretation to these otherwise abstract mathematical quantities. Specifically, our networks estimate the invariants under local unitary (LU) transformations and under stochastic local operations and classical communication (SLOCC). Our networks can estimate the LU invariants for multiparty states, where each party can have a Hilbert space of arbitrary dimension and the SLOCC invariants for multiqubit states. We analyze the statistical efficiency of our networks compared to methods based on estimating the state coefficients and calculating the invariants.
Quadratic-Like Dynamics of Cubic Polynomials
NASA Astrophysics Data System (ADS)
Blokh, Alexander; Oversteegen, Lex; Ptacek, Ross; Timorin, Vladlen
2016-02-01
A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.
NASA Astrophysics Data System (ADS)
Leont'ev, V. K.
2015-11-01
A pseudo-Boolean function is an arbitrary mapping of the set of binary n-tuples to the real line. Such functions are a natural generalization of classical Boolean functions and find numerous applications in various applied studies. Specifically, the Fourier transform of a Boolean function is a pseudo-Boolean function. A number of facts associated with pseudo-Boolean polynomials are presented, and their applications to well-known discrete optimization problems are described.
Solving the interval type-2 fuzzy polynomial equation using the ranking method
NASA Astrophysics Data System (ADS)
Rahman, Nurhakimah Ab.; Abdullah, Lazim
2014-07-01
Polynomial equations with trapezoidal and triangular fuzzy numbers have attracted some interest among researchers in mathematics, engineering and social sciences. There are some methods that have been developed in order to solve these equations. In this study we are interested in introducing the interval type-2 fuzzy polynomial equation and solving it using the ranking method of fuzzy numbers. The ranking method concept was firstly proposed to find real roots of fuzzy polynomial equation. Therefore, the ranking method is applied to find real roots of the interval type-2 fuzzy polynomial equation. We transform the interval type-2 fuzzy polynomial equation to a system of crisp interval type-2 fuzzy polynomial equation. This transformation is performed using the ranking method of fuzzy numbers based on three parameters, namely value, ambiguity and fuzziness. Finally, we illustrate our approach by numerical example.
On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices
NASA Technical Reports Server (NTRS)
Fischer, Bernd; Freund, Roland W.
1992-01-01
The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.
Solutions of interval type-2 fuzzy polynomials using a new ranking method
NASA Astrophysics Data System (ADS)
Rahman, Nurhakimah Ab.; Abdullah, Lazim; Ghani, Ahmad Termimi Ab.; Ahmad, Noor'Ani
2015-10-01
A few years ago, a ranking method have been introduced in the fuzzy polynomial equations. Concept of the ranking method is proposed to find actual roots of fuzzy polynomials (if exists). Fuzzy polynomials are transformed to system of crisp polynomials, performed by using ranking method based on three parameters namely, Value, Ambiguity and Fuzziness. However, it was found that solutions based on these three parameters are quite inefficient to produce answers. Therefore in this study a new ranking method have been developed with the aim to overcome the inherent weakness. The new ranking method which have four parameters are then applied in the interval type-2 fuzzy polynomials, covering the interval type-2 of fuzzy polynomial equation, dual fuzzy polynomial equations and system of fuzzy polynomials. The efficiency of the new ranking method then numerically considered in the triangular fuzzy numbers and the trapezoidal fuzzy numbers. Finally, the approximate solutions produced from the numerical examples indicate that the new ranking method successfully produced actual roots for the interval type-2 fuzzy polynomials.
Point estimation of simultaneous methods for solving polynomial equations
NASA Astrophysics Data System (ADS)
Petkovic, Miodrag S.; Petkovic, Ljiljana D.; Rancic, Lidija Z.
2007-08-01
The construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root-finding algorithm is one of the most important problems in solving nonlinear equations. Smale's "point estimation theory" from 1981 was a great advance in this topic; it treats convergence conditions and the domain of convergence in solving an equation f(z)=0 using only the information of f at the initial point z0. The study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult, even in the case of algebraic polynomials. In the light of Smale's point estimation theory, an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials. We state new, improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations: Ehrlich-Aberth's method, Ehrlich-Aberth's method with Newton's correction, Borsch-Supan's method with Weierstrass' correction and Halley-like (or Wang-Zheng) method. The introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods, but also explicitly gives their order of convergence. The stated initial conditions are of significant practical importance since they are computationally verifiable; they depend only on the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.
Properties of convergence for [omega],q-Bernstein polynomials
NASA Astrophysics Data System (ADS)
Wang, Heping
2008-04-01
In this paper, we discuss properties of the [omega],q-Bernstein polynomials introduced by S. Lewanowicz and P. Wozny in [S. Lewanowicz, P. Wozny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63-78], where f[set membership, variant]C[0,1], [omega],q>0, [omega][not equal to]1,q-1,...,q-n+1. When [omega]=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511-518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and [omega],q[set membership, variant](0,1) or (1,[infinity]), then are monotonically decreasing in n for all x[set membership, variant][0,1]. We prove that for [omega][set membership, variant](0,1), qn[set membership, variant](0,1], the sequence converges to f uniformly on [0,1] for each f[set membership, variant]C[0,1] if and only if limn-->[infinity]qn=1. For fixed [omega],q[set membership, variant](0,1), we prove that the sequence converges for each f[set membership, variant]C[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.
Orbifold E-functions of dual invertible polynomials
NASA Astrophysics Data System (ADS)
Ebeling, Wolfgang; Gusein-Zade, Sabir M.; Takahashi, Atsushi
2016-08-01
An invertible polynomial is a weighted homogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau-Ginzburg models, P. Berglund and M. Henningson considered a pair (f , G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair (f ˜ , G ˜) . We consider the so-called orbifold E-function of such a pair (f , G) which is a generating function for the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of f. We prove that the orbifold E-functions of Berglund-Henningson dual pairs coincide up to a sign depending on the number of variables and a simple change of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial.
A Formally Verified Conflict Detection Algorithm for Polynomial Trajectories
NASA Technical Reports Server (NTRS)
Narkawicz, Anthony; Munoz, Cesar
2015-01-01
In air traffic management, conflict detection algorithms are used to determine whether or not aircraft are predicted to lose horizontal and vertical separation minima within a time interval assuming a trajectory model. In the case of linear trajectories, conflict detection algorithms have been proposed that are both sound, i.e., they detect all conflicts, and complete, i.e., they do not present false alarms. In general, for arbitrary nonlinear trajectory models, it is possible to define detection algorithms that are either sound or complete, but not both. This paper considers the case of nonlinear aircraft trajectory models based on polynomial functions. In particular, it proposes a conflict detection algorithm that precisely determines whether, given a lookahead time, two aircraft flying polynomial trajectories are in conflict. That is, it has been formally verified that, assuming that the aircraft trajectories are modeled as polynomial functions, the proposed algorithm is both sound and complete.
A new Arnoldi approach for polynomial eigenproblems
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.
The q-Laguerre matrix polynomials.
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. PMID:27190749
From Jack polynomials to minimal model spectra
NASA Astrophysics Data System (ADS)
Ridout, David; Wood, Simon
2015-01-01
In this note, a deep connection between free field realizations of conformal field theories and symmetric polynomials is presented. We give a brief introduction into the necessary prerequisites of both free field realizations and symmetric polynomials, in particular Jack symmetric polynomials. Then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods. While these results on the representation theory of the minimal models are all known, this note exploits the full power of Jack polynomials to present significant simplifications of the original proofs in the literature.
Song, Xueqing; Yu, Xiang; Hori, Chiaki; Demura, Taku; Ohtani, Misato; Zhuge, Qiang
2016-01-01
Subfamily 2 of SNF1-related protein kinase (SnRK2) plays important roles in plant abiotic stress responses as a global positive regulator of abscisic acid signaling. In the genome of the model tree Populus trichocarpa, 12 SnRK2 genes have been identified, and some are upregulated by abiotic stresses. In this study, we heterologously overexpressed the PtSnRK2 genes in Arabidopsis thaliana and found that overexpression of PtSnRK2.5 and PtSnRK2.7 genes enhanced stress tolerance. In the PtSnRK2.5 and PtSnRK2.7 overexpressors, chlorophyll content, and root elongation were maintained under salt stress conditions, leading to higher survival rates under salt stress compared with those in the wild type. Transcriptomic analysis revealed that PtSnRK2.7 overexpression affected stress-related metabolic genes, including lipid metabolism and flavonoid metabolism, even under normal growth conditions. However, the stress response genes reported to be upregulated in Arabidopsis SRK2C/SnRK2.6 and wheat SnRK2.8 overexpressors were not changed by PtSnRK2.7 overexpression. Furthermore, PtSnRK2.7 overexpression widely and largely influenced the transcriptome in response to salt stress; genes related to transport activity, including anion transport-related genes, were characteristically upregulated, and a variety of metabolic genes were specifically downregulated. We also found that the salt stress response genes were greatly upregulated in the PtSnRK2.7 overexpressor. Taken together, poplar subclass 2 PtSnRK2 genes can modulate salt stress tolerance in Arabidopsis, through the activation of cellular signaling pathways in a different manner from that by herbal subclass 2 SnRK2 genes. PMID:27242819
Network meta-analysis of survival data with fractional polynomials
2011-01-01
Background Pairwise meta-analysis, indirect treatment comparisons and network meta-analysis for aggregate level survival data are often based on the reported hazard ratio, which relies on the proportional hazards assumption. This assumption is implausible when hazard functions intersect, and can have a huge impact on decisions based on comparisons of expected survival, such as cost-effectiveness analysis. Methods As an alternative to network meta-analysis of survival data in which the treatment effect is represented by the constant hazard ratio, a multi-dimensional treatment effect approach is presented. With fractional polynomials the hazard functions of interventions compared in a randomized controlled trial are modeled, and the difference between the parameters of these fractional polynomials within a trial are synthesized (and indirectly compared) across studies. Results The proposed models are illustrated with an analysis of survival data in non-small-cell lung cancer. Fixed and random effects first and second order fractional polynomials were evaluated. Conclusion (Network) meta-analysis of survival data with models where the treatment effect is represented with several parameters using fractional polynomials can be more closely fitted to the available data than meta-analysis based on the constant hazard ratio. PMID:21548941
Do r/K reproductive strategies apply to human differences?
Rushton, J P
1988-01-01
This article discusses the r/K theory of Social Biology and how it relates to humans. The symbols r and K originate in the mathematics of population biology and refer to 2 ends of a continuum in which a compensatory exchange occurs between gamete production (the r-strategy) and longevity (the K-strategy). Both across and within species, r and K strategists differ in a suite of correlated characteristics. Humans are the most K of all. K's supposedly have a longer gestation period, a higher birthweight, a more delayed sexual maturation, a lower sex drive, and a longer life. Studies providing evidence for the expected covariation among K attributes are presented. Additional evidence for r/K theory comes from the comparison of human population known to differ in gamete production. The pattern of racial differences observed to occur in sexual behavior has also been found to exist on numerous other indices of K. For instance, there are racial differences in brain size, intelligence, and maturation rate, among others. The findings suggest that, on the average, Mongoloids are more K than Caucasoids, who in turn, are more K than Negroids. Recently conducted studies have extended the data in favor of r/K theory, and further research is currently underway, including whether r/K attributes underlie individual and social class differences in health and longevity. PMID:3241997
On polynomial-time testable combinational circuits
Rao, N.S.V.; Toida, Shunichi
1994-11-01
The problems of identifying several nontrivial classes of Polynomial-Time Testable (PTT) circuits are shown to be NP-complete or harder. First, PTT classes obtained by using circuit decompositions proposed by Fujiwara and Chakradhar et al. are considered. Another type of decompositions, based on fanout-reconvergent (f-r) pairs, which also lead to PTT classes are proposed. The problems of obtaining these decompositions, and also some structurally similar general graph decompositions, are shown to be NP-complete or harder. Then, the problems of recognizing PTT classes formed by the Boolean formulae belonging to the weakly positive, weakly negative, bijunctive and affine classes (proposed by Schaefer) are shown to be NP-complete.
Polynomial chaotic inflation in supergravity
Nakayama, Kazunori; Takahashi, Fuminobu; Yanagida, Tsutomu T. E-mail: fumi@tuhep.phys.tohoku.ac.jp
2013-08-01
We present a general polynomial chaotic inflation model in supergravity, for which the predicted spectral index and tensor-to-scalar ratio can lie within the 1σ region allowed by the Planck results. Most importantly, the predicted tensor-to-scalar ratio is large enough to be probed in the on-going and future B-mode experiments. We study the inflaton dynamics and the subsequent reheating process in a couple of specific examples. The non-thermal gravitino production from the inflaton decay can be suppressed in a case with a discrete Z{sub 2} symmetry. We find that the reheating temperature can be naturally as high as O(10{sup 9−10}) GeV, sufficient for baryon asymmetry generation through (non-)thermal leptogenesis.
Fractal Trigonometric Polynomials for Restricted Range Approximation
NASA Astrophysics Data System (ADS)
Chand, A. K. B.; Navascués, M. A.; Viswanathan, P.; Katiyar, S. K.
2016-05-01
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
On the Waring problem for polynomial rings
Fröberg, Ralf; Ottaviani, Giorgio; Shapiro, Boris
2012-01-01
In this note we discuss an analog of the classical Waring problem for . Namely, we show that a general homogeneous polynomial of degree divisible by k≥2 can be represented as a sum of at most kn k-th powers of homogeneous polynomials in . Noticeably, kn coincides with the number obtained by naive dimension count. PMID:22460787
Point vortex equilibria related to Bessel polynomials
NASA Astrophysics Data System (ADS)
O'Neil, Kevin A.
2016-05-01
The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.
Wang, Lianzhe; Hu, Wei; Sun, Jiutong; Liang, Xiaoyu; Yang, Xiaoyue; Wei, Shuya; Wang, Xiatian; Zhou, Yi; Xiao, Qiang; Yang, Guangxiao; He, Guangyuan
2015-08-01
The sucrose non-fermenting 1 (SNF1)-related protein kinases (SnRKs) play key roles in plant signaling pathways including responses to biotic and abiotic stresses. Although SnRKs have been systematically studied in Arabidopsis and rice, there is no information concerning SnRKs in the new Poaceae model plant Brachypodium distachyon. In the present study, a total of 44 BdSnRKs were identified and classified into three subfamilies, including three members of BdSnRK1, 10 of BdSnRK2 and 31 of BdSnRK3 (CIPK) subfamilies. Phylogenetic reconstruction, chromosome distribution and synteny analyses suggested that BdSnRK family had been established before the dicot-monocot lineage parted, and had experienced rapid expansion during the process of plant evolution since then. Expression analysis of the BdSnRK2 subfamily showed that the majority of them could respond to abiotic stress and related signal molecules treatments. Protein-protein interaction and co-expression analyses of BdSnRK2s network showed that SnRK2s might be involved in biological pathway different from that of dicot model plant Arabidopsis. Expression of BdSnRK2.9 in tobacco resulted in increased tolerance to drought and salt stresses through activation of NtABF2. Taken together, comprehensive analyses of BdSnRKs would provide a basis for understanding of evolution and function of BdSnRK family. PMID:26089150
PLGA nanoparticle formulation of RK-33: an RNA helicase inhibitor against DDX3
Bol, Guus Martinus; Khan, Raheela; van Voss, Marise Rosa Heerma; Tantravedi, Saritha; Korz, Dorian
2016-01-01
Background The DDX3 helicase inhibitor RK-33 is a newly developed anticancer agent that showed promising results in preclinical research (Bol et al. EMBO Mol Med, 7(5):648–649, 2015). However, due to the physicochemical and pharmacological characteristics of RK-33, we initiated development of alternative formulations of RK-33 by preparing sustained release nanoparticles that can be administered intravenously. Methods In this study, RK-33 was encapsulated in poly(lactic-co-glycolic acid) (PLGA), one of the most well-developed biodegradable polymers, using the emulsion solvent evaporation method. Results Hydrodynamic diameter of RK-33-PLGA nanoparticles was about 245 nm with a negative charge, and RK-33-PLGA nanoparticles had a payload of 1.4 % RK-33. RK-33 was released from the PLGA nanoparticles over 7 days (90 ± 5.7 % released by day 7) and exhibited cytotoxicity to human breast carcinoma MCF-7 cells in a time-dependent manner. Moreover, RK-33-PLGA nanoparticles were well tolerated, and systemic retention of RK-33 was markedly improved in normal mice. Conclusions PLGA nanoparticles have a potential as a parenteral formulation of RK-33. PMID:26330329
Matrix product formula for Macdonald polynomials
NASA Astrophysics Data System (ADS)
Cantini, Luigi; de Gier, Jan; Wheeler, Michael
2015-09-01
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.
Tutte polynomial in functional magnetic resonance imaging
NASA Astrophysics Data System (ADS)
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.
DIFFERENTIAL CROSS SECTION ANALYSIS IN KAON PHOTOPRODUCTION USING ASSOCIATED LEGENDRE POLYNOMIALS
P. T. P. HUTAURUK, D. G. IRELAND, G. ROSNER
2009-04-01
Angular distributions of differential cross sections from the latest CLAS data sets,6 for the reaction γ + p→K+ + Λ have been analyzed using associated Legendre polynomials. This analysis is based upon theoretical calculations in Ref. 1 where all sixteen observables in kaon photoproduction can be classified into four Legendre classes. Each observable can be described by an expansion of associated Legendre polynomial functions. One of the questions to be addressed is how many associated Legendre polynomials are required to describe the data. In this preliminary analysis, we used data models with different numbers of associated Legendre polynomials. We then compared these models by calculating posterior probabilities of the models. We found that the CLAS data set needs no more than four associated Legendre polynomials to describe the differential cross section data. In addition, we also show the extracted coefficients of the best model.
An algorithm for constructing polynomial systems whose solution space characterizes quantum circuits
NASA Astrophysics Data System (ADS)
Gerdt, Vladimir P.; Severyanov, Vasily M.
2006-05-01
An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z II arising when applying the Feynman's sum-over-paths approach to quantum circuits. The matrix elements determined by a circuit can be computed by counting the number of common roots in Z II for the polynomial system associated with the circuit. To determine the number of solutions in Z II for the output polynomial system, one can use the Grobner bases method and the relevant algorithms for computing Grobner bases.
Temperature dependence of gas properties in polynomial form
NASA Astrophysics Data System (ADS)
Andrews, J. R.; Biblarz, O.
1981-01-01
Based on a least-squares polynomial approximation, a procedure is introduced for calculating existing tabular values of thermodynamic and transport properties for common gases. The specific heat at constant pressure is given for 238 gases, the thermal conductivity for 55 gases, the dynamic viscocity for 58 gases, and the second and third virial coefficients for 14 gases. At sufficiently low pressures, ideal gas behavior prevails and temperature may be used as the single independent variable. The algorithm for nested multiplication is presented, optimized for hand-held or desktop electronic calculators. Using the polynomial approximations and a suitable calculator, it is possible to duplicate existing reference source tabular values directly, obviating the need for interpolation or further reference to the tables per se. The accuracy of the calculated values can be within 0.5% of the tabular values. The polynomial coefficients are given in the International System of Units (SI). Methods are presented to calculate the temperature corresponding to a given property value. Extrapolation features of the polynomials are discussed.
More on rotations as spin matrix polynomials
Curtright, Thomas L.
2015-09-15
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.
The Translated Dowling Polynomials and Numbers
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.
Heisenberg algebra, umbral calculus and orthogonal polynomials
Dattoli, G.; Levi, D.; Winternitz, P.
2008-05-15
Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation [P,M]=1. In ordinary quantum mechanics, P is the derivative and M the coordinate operator. Here, we shall realize P as a second order differential operator and M as a first order integral one. We show that this makes it possible to solve large classes of differential and integrodifferential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing the so-called flatenned beams in laser theory.
NASA Astrophysics Data System (ADS)
Konakli, Katerina; Sudret, Bruno
2016-09-01
The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the "curse of dimensionality", namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor-product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input dimension, a
Uncertainty quantification in simulations of epidemics using polynomial chaos.
Santonja, F; Chen-Charpentier, B
2012-01-01
Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the first-and the second-order moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model. PMID:22927889
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. PMID:24514717
Adapted polynomial chaos expansion for failure detection
Paffrath, M. Wever, U.
2007-09-10
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.
Hermite polynomials and quasi-classical asymptotics
Ali, S. Twareque; Engliš, Miroslav
2014-04-15
We study an unorthodox variant of the Berezin-Toeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated quasi-classical asymptotics.
Inequalities for a polynomial and its derivative
NASA Astrophysics Data System (ADS)
Chanam, Barchand; Dewan, K. K.
2007-12-01
Let , 1[less-than-or-equals, slant][mu][less-than-or-equals, slant]n, be a polynomial of degree n such that p(z)[not equal to]0 in z
Schur Stability Regions for Complex Quadratic Polynomials
ERIC Educational Resources Information Center
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.)
Stochastic processes with orthogonal polynomial eigenfunctions
NASA Astrophysics Data System (ADS)
Griffiths, Bob
2009-12-01
Markov processes which are reversible with either Gamma, Normal, Poisson or Negative Binomial stationary distributions in the Meixner class and have orthogonal polynomial eigenfunctions are characterized as being processes subordinated to well-known diffusion processes for the Gamma and Normal, and birth and death processes for the Poisson and Negative Binomial. A characterization of Markov processes with Beta stationary distributions and Jacobi polynomial eigenvalues is also discussed.
Combinatorial and algorithm aspects of hyperbolic polynomials
Gurvits, Leonid I.
2004-01-01
Univariate polynomials with real roots appear quite often in modern combinatorics, especially in the context of integer polytopes. We discovered in this paper rather unexpected and very likely far-reaching connections between hyperbolic polynomials and many classical combinatorial and algorithmic problems. There are still several open problems. The most interesting is a hyperbolic generalization of the van der Waerden conjecture for permanents of doubly stochastic matrices.
ERIC Educational Resources Information Center
Schweizer, Karl
2006-01-01
A model with fixed relations between manifest and latent variables is presented for investigating choice reaction time data. The numbers for fixation originate from the polynomial function. Two options are considered: the component-based (1 latent variable for each component of the polynomial function) and composite-based options (1 latent…
On the cardinality of twelfth degree polynomial
NASA Astrophysics Data System (ADS)
Lasaraiya, S.; Sapar, S. H.; Johari, M. A. Mohamat
2016-06-01
Let p be a prime and f (x, y) be a polynomial in Zp[x, y]. It is defined that the exponential sums associated with f modulo a prime pα is S (f :q )= ∑ e2/π i f (x ) q for α >1 , where f (x) is in Z[x] and the sum is taken over a complete set of residues x modulo positive integer q. Previous studies has shown that estimation of S (f; pα) is depends on the cardinality of the set of solutions to congruence equation associated with the polynomial. In order to estimate the cardinality, we need to have the value of p-adic sizes of common zeros of partial derivative polynomials associated with polynomial. Hence, p-adic method and newton polyhedron technique will be applied to this approach. After that, indicator diagram will be constructed and analyzed. The cardinality will in turn be used to estimate the exponential sums of the polynomials. This paper concentrates on the cardinality of the set of solutions to congruence equation associated with polynomial in the form of f (x, y) = ax12 + bx11y + cx10y2 + sx + ty + k.
Frameworks for Logically Classifying Polynomial-Time Optimisation Problems
NASA Astrophysics Data System (ADS)
Gate, James; Stewart, Iain A.
We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems.
Beta-integrals and finite orthogonal systems of Wilson polynomials
Neretin, Yu A
2002-08-31
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 {sub 5}H{sub 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.
Quasi-kernel polynomials and convergence results for quasi-minimal residual iterations
NASA Technical Reports Server (NTRS)
Freund, Roland W.
1992-01-01
Recently, Freund and Nachtigal have proposed a novel polynominal-based iteration, the quasi-minimal residual algorithm (QMR), for solving general nonsingular non-Hermitian linear systems. Motivated by the QMR method, we have introduced the general concept of quasi-kernel polynomials, and we have shown that the QMR algorithm is based on a particular instance of quasi-kernel polynomials. In this paper, we continue our study of quasi-kernel polynomials. In particular, we derive bounds for the norms of quasi-kernel polynomials. These results are then applied to obtain convergence theorems both for the QMR method and for a transpose-free variant of QMR, the TFQMR algorithm.
The Rational Polynomial Coefficients Modification Using Digital Elevation Models
NASA Astrophysics Data System (ADS)
Alidoost, F.; Azizi, A.; Arefi, H.
2015-12-01
The high-resolution satellite imageries (HRSI) are as primary dataset for different applications such as DEM generation, 3D city mapping, change detection, monitoring, and deformation detection. The geo-location information of HRSI are stored in metadata called Rational Polynomial Coefficients (RPCs). There are many methods to improve and modify the RPCs in order to have a precise mapping. In this paper, an automatic approach is presented for the RPC modification using global Digital Elevation Models. The main steps of this approach are: relative digital elevation model generation, shift parameters calculation, sparse point cloud generation and shift correction, and rational polynomial fitting. Using some ground control points, the accuracy of the proposed method is evaluated based on statistical descriptors in which the results show that the geo-location accuracy of HRSI can be improved without using Ground Control Points (GCPs).
Extending a Property of Cubic Polynomials to Higher-Degree Polynomials
ERIC Educational Resources Information Center
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…
Recognition of Arabic Sign Language Alphabet Using Polynomial Classifiers
NASA Astrophysics Data System (ADS)
Assaleh, Khaled; Al-Rousan, M.
2005-12-01
Building an accurate automatic sign language recognition system is of great importance in facilitating efficient communication with deaf people. In this paper, we propose the use of polynomial classifiers as a classification engine for the recognition of Arabic sign language (ArSL) alphabet. Polynomial classifiers have several advantages over other classifiers in that they do not require iterative training, and that they are highly computationally scalable with the number of classes. Based on polynomial classifiers, we have built an ArSL system and measured its performance using real ArSL data collected from deaf people. We show that the proposed system provides superior recognition results when compared with previously published results using ANFIS-based classification on the same dataset and feature extraction methodology. The comparison is shown in terms of the number of misclassified test patterns. The reduction in the rate of misclassified patterns was very significant. In particular, we have achieved a 36% reduction of misclassifications on the training data and 57% on the test data.
Time-dependent generalized polynomial chaos
Gerritsma, Marc; Steen, Jan-Bart van der; Vos, Peter; Karniadakis, George
2010-11-01
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.
Fitting parametrized polynomials with scattered surface data.
van Ruijven, L J; Beek, M; van Eijden, T M
1999-07-01
Currently used joint-surface models require the measurements to be structured according to a grid. With the currently available tracking devices a large quantity of unstructured surface points can be measured in a relatively short time. In this paper a method is presented to fit polynomial functions to three-dimensional unstructured data points. To test the method spherical, cylindrical, parabolic, hyperbolic, exponential, logarithmic, and sellar surfaces with different undulations were used. The resulting polynomials were compared with the original shapes. The results show that even complex joint surfaces can be modelled with polynomial functions. In addition, the influence of noise and the number of data points was also analyzed. From a surface (diam: 20 mm) which is measured with a precision of 0.2 mm a model can be constructed with a precision of 0.02 mm. PMID:10400359
Torus Knot Polynomials and Susy Wilson Loops
NASA Astrophysics Data System (ADS)
Giasemidis, Georgios; Tierz, Miguel
2014-12-01
We give, using an explicit expression obtained in (Jones V, Ann Math 126:335, 1987), a basic hypergeometric representation of the HOMFLY polynomial of ( n, m) torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result, the symmetry and the leading polynomial at large N are explicit. We show the latter to be the Wilson loop of 2d Yang-Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang-Mills theory, which is known to give averages of Wilson loops in = 4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones-Rosso representation in terms of q-harmonic oscillators.
Chebyshev Polynomials Are Not Always Optimal
NASA Technical Reports Server (NTRS)
Fischer, B.; Freund, E.
1989-01-01
The authors are concerned with the problem of finding among all polynomials of degree at most n and normalized to be 1 at c the one with minimal uniform norm on Epsilon. Here, Epsilon is a given ellipse with both foci on the real axis and c is a given real point not contained in Epsilon. Problems of this type arise in certain iterative matrix computations, and, in this context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems. In this note, the authors show that this is not true in general. Moreover, the authors derive sufficient conditions which guarantee that Chebyshev polynomials are optimal. Also, some numerical examples are presented.
Minimal residual method stronger than polynomial preconditioning
Faber, V.; Joubert, W.; Knill, E.
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.
Bispectrality of the Complementary Bannai-Ito Polynomials
NASA Astrophysics Data System (ADS)
Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei
2013-03-01
A one-parameter family of operators that have the complementary Bannai-Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai-Ito polynomials and also correspond to a q→"1 limit of the Askey-Wilson polynomials. The eigenvalue equations for the CBI polynomials are found to involve second order Dunkl shift operators with reflections and exhibit quadratic spectra. The algebra associated to the CBI polynomials is given and seen to be a deformation of the Askey-Wilson algebra with an involution. The relation between the CBI polynomials and the recently discovered dual "1 Hahn and para-Krawtchouk polynomials, as well as their relation with the symmetric Hahn polynomials, is also discussed.
Sun, Liang; Wang, Yan-Ping; Chen, Pei; Ren, Jie; Ji, Kai; Li, Qian; Li, Ping; Dai, Sheng-Jie; Leng, Ping
2011-01-01
In order to characterize the potential transcriptional regulation of core components of abscisic acid (ABA) signal transduction in tomato fruit development and drought stress, eight SlPYL (ABA receptor), seven SlPP2C (type 2C protein phosphatase), and eight SlSnRK2 (subfamily 2 of SNF1-related kinases) full-length cDNA sequences were isolated from the tomato nucleotide database of NCBI GenBank. All SlPYL, SlPP2C, and SlSnRK2 genes obtained are homologous to Arabidopsis AtPYL, AtPP2C, and AtSnRK2 genes, respectively. Based on phylogenetic analysis, SlPYLs and SlSnRK2s were clustered into three subfamilies/subclasses, and all SlPP2Cs belonged to PP2C group A. Within the SlPYL gene family, SlPYL1, SlPYL2, SlPYL3, and SlPYL6 were the major genes involved in the regulation of fruit development. Among them, SlPYL1 and SlPYL2 were expressed at high levels throughout the process of fruit development and ripening; SlPYL3 was strongly expressed at the immature green (IM) and mature green (MG) stages, while SlPYL6 was expressed strongly at the IM and red ripe (RR) stages. Within the SlPP2C gene family, the expression of SlPP2C, SlPP2C3, and SlPP2C4 increased after the MG stage; SlPP2C1 and SlPP2C5 peaked at the B3 stage, while SlPP2C2 and SlPP2C6 changed little during fruit development. Within the SlSnRK2 gene family, the expression of SlSnRK2.2, SlSnRK2.3, SlSnRK2.4, and SlSnRK2C was higher than that of other members during fruit development. Additionally, most SlPYL genes were down-regulated, while most SlPP2C and SlSnRK2 genes were up-regulated by dehydration in tomato leaf. PMID:21873532
On the derivatives of unimodular polynomials
NASA Astrophysics Data System (ADS)
Nevai, P.; Erdélyi, T.
2016-04-01
Let D be the open unit disk of the complex plane; its boundary, the unit circle of the complex plane, is denoted by \\partial D. Let \\mathscr P_n^c denote the set of all algebraic polynomials of degree at most n with complex coefficients. For λ ≥ 0, let {\\mathscr K}_n^λ \\stackrel{{def}}{=} \\biggl\\{P_n: P_n(z) = \\sumk=0^n{ak k^λ z^k}, ak \\in { C}, |a_k| = 1 \\biggr\\} \\subset {\\mathscr P}_n^c.The class \\mathscr K_n^0 is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (\\varepsilon_n) of positive numbers tending to 0, we say that a sequence (P_n) of polynomials P_n\\in\\mathscr K_n^λ is \\{λ, (\\varepsilon_n)\\}-ultraflat if \\displaystyle (1-\\varepsilon_n)\\frac{nλ+1/2}{\\sqrt{2λ+1}}≤\\ve......a +1/2}}{\\sqrt{2λ +1}},\\qquad z \\in \\partial D,\\quad n\\in N_0.Although we do not know, in general, whether or not \\{λ, (\\varepsilon_n)\\}-ultraflat sequences of polynomials P_n\\in\\mathscr K_n^λ exist for each fixed λ>0, we make an effort to prove various interesting properties of them. These allow us to conclude that there are no sequences (P_n) of either conjugate, or plain, or skew reciprocal unimodular polynomials P_n\\in\\mathscr K_n^0 such that (Q_n) with Q_n(z)\\stackrel{{def}}{=} zP_n'(z)+1 is a \\{1,(\\varepsilon_n)\\}-ultraflat sequence of polynomials.Bibliography: 18 titles.
High degree interpolation polynomial in Newton form
NASA Technical Reports Server (NTRS)
Tal-Ezer, Hillel
1988-01-01
Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4.
On the Waring problem for polynomial rings.
Fröberg, Ralf; Ottaviani, Giorgio; Shapiro, Boris
2012-04-10
In this note we discuss an analog of the classical Waring problem for C[x0,x1,...,x(n)]. Namely, we show that a general homogeneous polynomial p ∈ C[x0,x1,...,x(n)] of degree divisible by k≥2 can be represented as a sum of at most k(n) k-th powers of homogeneous polynomials in C[x0,x1,...,x(n)]. Noticeably, k(n) coincides with the number obtained by naive dimension count. PMID:22460787
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Perturbations around the zeros of classical orthogonal polynomials
NASA Astrophysics Data System (ADS)
Sasaki, Ryu
2015-04-01
Starting from degree N solutions of a time dependent Schrödinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree ( 0 , 1 , … , N - 1 ) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
A transfectant RK13 cell line permissive to classical caprine scrapie prion propagation.
Dassanayake, Rohana P; Zhuang, Dongyue; Truscott, Thomas C; Madsen-Bouterse, Sally A; O'Rourke, Katherine I; Schneider, David A
2016-03-01
To assess scrapie infectivity associated with caprine-origin tissues, bioassay can be performed using kids, lambs or transgenic mice expressing caprine or ovine prion (PRNP) alleles, but the incubation periods are fairly long. Although several classical ovine scrapie prion permissive cell lines with the ability to detect brain-derived scrapie prion have been available, no classical caprine scrapie permissive cell line is currently available. Therefore, the aims of this study were to generate a rabbit kidney epithelial cell line (RK13) stably expressing caprine wild-type PRNP (cpRK13) and then to assess permissiveness of cpRK13 cells to classical caprine scrapie prion propagation. The cpRK13 and plasmid control RK13 (pcRK13) cells were incubated with brain-derived classical caprine scrapie inocula prepared from goats or ovinized transgenic mice (Tg338, express ovine VRQ allele) infected with caprine scrapie. Significant PrP(Sc) accumulation, which is indicative of scrapie prion propagation, was detected by TSE ELISA and immunohistochemistry in cpRK13 cells inoculated with classical caprine scrapie inocula. Western blot analysis revealed the typical proteinase K-resistant 3 PrP(res) isoforms in the caprine scrapie prion inoculated cpRK13 cell lysate. Importantly, PrP(Sc) accumulation was not detected in similarly inoculated pcRK13 cells, whether by TSE ELISA, immunohistochemistry, or western blot. These findings suggest that caprine scrapie prions can be propagated in cpRK13 cells, thus this cell line may be a useful tool for the assessment of classical caprine prions in the brain tissues of goats. PMID:27216989
Simultaneous explanation of the RK and R (D (*)) puzzles
NASA Astrophysics Data System (ADS)
Bhattacharya, Bhubanjyoti; Datta, Alakabha; London, David; Shivashankara, Shanmuka
2015-03-01
At present, there are several hints of lepton flavor non-universality. The LHCb Collaboration has measured RK ≡ B (B+ →K+μ+μ-) / B (B+ →K+e+e-), and the BaBar Collaboration has measured R (D (*)) ≡ B (B bar →D (*) +τ-νbarτ) / B (B bar →D (*) +ℓ-νbarℓ) (ℓ = e , μ). In all cases, the experimental results differ from the standard model predictions by 2- 3 σ. Recently, an explanation of the RK puzzle was proposed in which new physics (NP) generates a neutral-current operator involving only third-generation particles. Now, assuming the scale of NP is much larger than the weak scale, this NP operator must be made invariant under the full SU (3)C × SU (2)L × U(1)Y gauge group. In this Letter, we note that, when this is done, a new charged-current operator can appear, and this can explain the R (D (*)) puzzle. A more precise measurement of the double ratio R (D) / R (D*) can rule out this model.
Information-theoretic lengths of Jacobi polynomials
NASA Astrophysics Data System (ADS)
Guerrero, A.; Sánchez-Moreno, P.; Dehesa, J. S.
2010-07-01
The information-theoretic lengths of the Jacobi polynomials P(α, β)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 (α, β). 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.
Piecewise Polynomial Representations of Genomic Tracks
Tarabichi, Maxime; Detours, Vincent; Konopka, Tomasz
2012-01-01
Genomic data from micro-array and sequencing projects consist of associations of measured values to chromosomal coordinates. These associations can be thought of as functions in one dimension and can thus be stored, analyzed, and interpreted as piecewise-polynomial curves. We present a general framework for building piecewise polynomial representations of genome-scale signals and illustrate some of its applications via examples. We show that piecewise constant segmentation, a typical step in copy-number analyses, can be carried out within this framework for both array and (DNA) sequencing data offering advantages over existing methods in each case. Higher-order polynomial curves can be used, for example, to detect trends and/or discontinuities in transcription levels from RNA-seq data. We give a concrete application of piecewise linear functions to diagnose and quantify alignment quality at exon borders (splice sites). Our software (source and object code) for building piecewise polynomial models is available at http://sourceforge.net/projects/locsmoc/. PMID:23166601
Polynomials Generated by the Fibonacci Sequence
NASA Astrophysics Data System (ADS)
Garth, David; Mills, Donald; Mitchell, Patrick
2007-06-01
The Fibonacci sequence's initial terms are F_0=0 and F_1=1, with F_n=F_{n-1}+F_{n-2} for n>=2. We define the polynomial sequence p by setting p_0(x)=1 and p_{n}(x)=x*p_{n-1}(x)+F_{n+1} for n>=1, with p_{n}(x)= sum_{k=0}^{n} F_{k+1}x^{n-k}. We call p_n(x) the Fibonacci-coefficient polynomial (FCP) of order n. The FCP sequence is distinct from the well-known Fibonacci polynomial sequence. We answer several questions regarding these polynomials. Specifically, we show that each even-degree FCP has no real zeros, while each odd-degree FCP has a unique, and (for degree at least 3) irrational, real zero. Further, we show that this sequence of unique real zeros converges monotonically to the negative of the golden ratio. Using Rouche's theorem, we prove that the zeros of the FCP's approach the golden ratio in modulus. We also prove a general result that gives the Mahler measures of an infinite subsequence of the FCP sequence whose coefficients are reduced modulo an integer m>=2. We then apply this to the case that m=L_n, the nth Lucas number, showing that the Mahler measure of the subsequence is phi^{n-1}, where phi=(1+sqrt 5)/2.
On solvable Dirac equation with polynomial potentials
Stachowiak, Tomasz
2011-01-15
One-dimensional Dirac equation is analyzed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the equation in question is not solvable.
A recursive algorithm for Zernike polynomials
NASA Technical Reports Server (NTRS)
Davenport, J. W.
1982-01-01
The analysis of a function defined on a rotationally symmetric system, with either a circular or annular pupil is discussed. In order to numerically analyze such systems it is typical to expand the given function in terms of a class of orthogonal polynomials. Because of their particular properties, the Zernike polynomials are especially suited for numerical calculations. Developed is a recursive algorithm that can be used to generate the Zernike polynomials up to a given order. The algorithm is recursively defined over J where R(J,N) is the Zernike polynomial of degree N obtained by orthogonalizing the sequence R(J), R(J+2), ..., R(J+2N) over (epsilon, 1). The terms in the preceding row - the (J-1) row - up to the N+1 term is needed for generating the (J,N)th term. Thus, the algorith generates an upper left-triangular table. This algorithm was placed in the computer with the necessary support program also included.
Piecewise polynomial representations of genomic tracks.
Tarabichi, Maxime; Detours, Vincent; Konopka, Tomasz
2012-01-01
Genomic data from micro-array and sequencing projects consist of associations of measured values to chromosomal coordinates. These associations can be thought of as functions in one dimension and can thus be stored, analyzed, and interpreted as piecewise-polynomial curves. We present a general framework for building piecewise polynomial representations of genome-scale signals and illustrate some of its applications via examples. We show that piecewise constant segmentation, a typical step in copy-number analyses, can be carried out within this framework for both array and (DNA) sequencing data offering advantages over existing methods in each case. Higher-order polynomial curves can be used, for example, to detect trends and/or discontinuities in transcription levels from RNA-seq data. We give a concrete application of piecewise linear functions to diagnose and quantify alignment quality at exon borders (splice sites). Our software (source and object code) for building piecewise polynomial models is available at http://sourceforge.net/projects/locsmoc/. PMID:23166601
Polynomial preconditioning for conjugate gradient methods
Ashby, S.F.
1987-12-01
The solution of a linear system of equations, Ax = b, arises in many scientific applications. If A is large and sparse, an iterative method is required. When A is hermitian positive definite (hpd), the conjugate gradient method of Hestenes and Stiefel is popular. When A is hermitian indefinite (hid), the conjugate residual method may be used. If A is ill-conditioned, these methods may converge slowly, in which case a preconditioner is needed. In this thesis we examine the use of polynomial preconditioning in CG methods for both hermitian positive definite and indefinite matrices. Such preconditioners are easy to employ and well-suited to vector and/or parallel architectures. We first show that any CG method is characterized by three matrices: an hpd inner product matrix B, a preconditioning matrix C, and the hermitian matrix A. The resulting method, CG(B,C,A), minimizes the B-norm of the error over a Krylov subspace. We next exploit the versatility of polynomial preconditioners to design several new CG methods. To obtain an optimum preconditioner, we solve a constrained minimax approximation problem. The preconditioning polynomial, C(lambda), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix, p/sub m/(A). An adaptive procedure for dynamically determining the optimum preconditioner is also discussed. Finally, in a variety of numerical experiments, conducted on a Cray X-MP/48, we demonstrate the effectiveness of polynomial preconditioning. 66 ref., 19 figs., 39 tabs.
Polynomial Asymptotes of the Second Kind
ERIC Educational Resources Information Center
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…
Optimization of Cubic Polynomial Functions without Calculus
ERIC Educational Resources Information Center
Taylor, Ronald D., Jr.; Hansen, Ryan
2008-01-01
In algebra and precalculus courses, students are often asked to find extreme values of polynomial functions in the context of solving an applied problem; but without the notion of derivative, something is lost. Either the functions are reduced to quadratics, since students know the formula for the vertex of a parabola, or solutions are…
NASA Astrophysics Data System (ADS)
Recchioni, Maria Cristina
2001-12-01
This paper investigates the application of the method introduced by L. Pasquini (1989) for simultaneously approaching the zeros of polynomial solutions to a class of second-order linear homogeneous ordinary differential equations with polynomial coefficients to a particular case in which these polynomial solutions have zeros symmetrically arranged with respect to the origin. The method is based on a family of nonlinear equations which is associated with a given class of differential equations. The roots of the nonlinear equations are related to the roots of the polynomial solutions of differential equations considered. Newton's method is applied to find the roots of these nonlinear equations. In (Pasquini, 1994) the nonsingularity of the roots of these nonlinear equations is studied. In this paper, following the lines in (Pasquini, 1994), the nonsingularity of the roots of these nonlinear equations is studied. More favourable results than the ones in (Pasquini, 1994) are proven in the particular case of polynomial solutions with symmetrical zeros. The method is applied to approximate the roots of Hermite-Sobolev type polynomials and Freud polynomials. A lower bound for the smallest positive root of Hermite-Sobolev type polynomials is given via the nonlinear equation. The quadratic convergence of the method is proven. A comparison with a classical method that uses the Jacobi matrices is carried out. We show that the algorithm derived by the proposed method is sometimes preferable to the classical QR type algorithms for computing the eigenvalues of the Jacobi matrices even if these matrices are real and symmetric.
The ratio monotonicity of the Boros-Moll polynomials
NASA Astrophysics Data System (ADS)
Chen, William Y. C.; Xia, Ernest X. W.
2009-12-01
In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a conjecture which is stronger than Moll's conjecture on the infty -log-concavity.
Herman's Condition and Siegel Disks of Bi-Critical Polynomials
NASA Astrophysics Data System (ADS)
Chéritat, Arnaud; Roesch, Pascale
2016-06-01
We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's condition) on the rotation number, must have a critical point on their boundaries.
The New Polynomial Invariants of Knots and Links.
ERIC Educational Resources Information Center
Lickorish, W. B. R.; Millett, K. C.
1988-01-01
Knot theory has been inspirational to algebraic and geometric topology. The principal problem has been to ascertain whether two links are equivalent. New methods have been discovered which are effective and simple. Considered are background information; the oriented polynomial; the Jones polynomial; the semioriented polynomial; and calculations,…
The Gibbs Phenomenon for Series of Orthogonal Polynomials
ERIC Educational Resources Information Center
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…
An acoustical interpretation of the zeroes of ultraspherical polynomials
NASA Astrophysics Data System (ADS)
Le Vey, Georges
2016-06-01
In 1887, T.J. Stieltjes gave an electrostatical interpretation of the zeroes of Jacobi polynomials. This was extended later to Laguerre and Hermite polynomials by G. Szegö. An analogous interpretation is given here for ultraspherical polynomials in terms of piecewise cylindrical acoustical resonators. xml:lang="fr"
Inverse of polynomial matrices in the irreducible form
NASA Technical Reports Server (NTRS)
Chang, Fan R.; Shieh, Leang S.; Mcinnis, Bayliss C.
1987-01-01
An algorithm is developed for finding the inverse of polynomial matrices in the irreducible form. The computational method involves the use of the left (right) matrix division method and the determination of linearly dependent vectors of the remainders. The obtained transfer function matrix has no nontrivial common factor between the elements of the numerator polynomial matrix and the denominator polynomial.
Global Monte Carlo Simulation with High Order Polynomial Expansions
William R. Martin; James Paul Holloway; Kaushik Banerjee; Jesse Cheatham; Jeremy Conlin
2007-12-13
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
Polynomial search and global modeling: Two algorithms for modeling chaos.
Mangiarotti, S; Coudret, R; Drapeau, L; Jarlan, L
2012-10-01
Global modeling aims to build mathematical models of concise description. Polynomial Model Search (PoMoS) and Global Modeling (GloMo) are two complementary algorithms (freely downloadable at the following address: http://www.cesbio.ups-tlse.fr/us/pomos_et_glomo.html) designed for the modeling of observed dynamical systems based on a small set of time series. Models considered in these algorithms are based on ordinary differential equations built on a polynomial formulation. More specifically, PoMoS aims at finding polynomial formulations from a given set of 1 to N time series, whereas GloMo is designed for single time series and aims to identify the parameters for a selected structure. GloMo also provides basic features to visualize integrated trajectories and to characterize their structure when it is simple enough: One allows for drawing the first return map for a chosen Poincaré section in the reconstructed space; another one computes the Lyapunov exponent along the trajectory. In the present paper, global modeling from single time series is considered. A description of the algorithms is given and three examples are provided. The first example is based on the three variables of the Rössler attractor. The second one comes from an experimental analysis of the copper electrodissolution in phosphoric acid for which a less parsimonious global model was obtained in a previous study. The third example is an exploratory case and concerns the cycle of rainfed wheat under semiarid climatic conditions as observed through a vegetation index derived from a spatial sensor. PMID:23214661
Possible quantum algorithms for the Bollobas-Riordan-Tutte polynomial of a ribbon graph
NASA Astrophysics Data System (ADS)
Vélez, Mario; Ospina, Juan
2008-04-01
Three possible quantum algorithms, for the computation of the Bollobás-Riordan-Tutte polynomial of a given ribbon graph, are presented and discussed. The first possible algorithm is based on the spanning quasi-trees expansion for generalized Tutte polynomials of generalized graphs and on a quantum version of the Binary Decision Diagram (BDD) for quasi-trees . The second possible algorithm is based on the relation between the Kauffman bracket and the Tutte polynomial; and with an application of the recently introduced Aharonov-Arad-Eban-Landau quantum algorithm. The third possible algorithm is based on the relation between the HOMFLY polynomial and the Tutte polynomial; and with an application of the Wocjan-Yard quantum algorithm. It is claimed that these possible algorithms may be more efficient that the best known classical algorithms. These three algorithms may have interesting applications in computer science at general or in computational biology and bio-informatics in particular. A line for future research based on the categorification project is mentioned.
New optimal polynomial theory for NN-scattering
Rijken, T A; Signell, P
1980-01-01
A new optimal polynomial theory for nucleon-nucleon scattering is presented. For the first time in nucleon-nucleon scattering, the derivative amplitudes originally introduced by Fubini, Furlan, and Rosetti are applied. Based on the properties of these amplitudes we introduce K-matrix functions which have suitable analyticity properties as functions of cos theta, where theta is the center of mass scattering angle. The K-matrix functions enable introduction of a new set of functions for which the optimal mapping techniques of Cutkosky, Deo and Ciulli can be applied. Results are shown for proton-proton phase shift analyses at 210 and 330 MeV.
Cho, Hsing-Yi; Wen, Tuan-Nan; Wang, Ying-Tsui; Shih, Ming-Che
2016-04-01
SNF1 RELATED PROTEIN KINASE 1 (SnRK1) is proposed to be a central integrator of the plant stress and energy starvation signalling pathways. We observed that the Arabidopsis SnRK1.1 dominant negative mutant (SnRK1.1 (K48M) ) had lower tolerance to submergence than the wild type, suggesting that SnRK1.1-dependent phosphorylation of target proteins is important in signalling pathways triggered by submergence. We conducted quantitative phosphoproteomics and found that the phosphorylation levels of 57 proteins increased and the levels of 27 proteins decreased in Col-0 within 0.5-3h of submergence. Among the 57 proteins with increased phosphorylation in Col-0, 38 did not show increased phosphorylation levels in SnRK1.1 (K48M) under submergence. These proteins are involved mainly in sugar and protein synthesis. In particular, the phosphorylation of MPK6, which is involved in regulating ROS responses under abiotic stresses, was disrupted in the SnRK1.1 (K48M) mutant. In addition, PTP1, a negative regulator of MPK6 activity that directly dephosphorylates MPK6, was also regulated by SnRK1.1. We also showed that energy conservation was disrupted in SnRK1.1 (K48M) , mpk6, and PTP1 (S7AS8A) under submergence. These results reveal insights into the function of SnRK1 and the downstream signalling factors related to submergence. PMID:27029354
Cho, Hsing-Yi; Wen, Tuan-Nan; Wang, Ying-Tsui; Shih, Ming-Che
2016-01-01
SNF1 RELATED PROTEIN KINASE 1 (SnRK1) is proposed to be a central integrator of the plant stress and energy starvation signalling pathways. We observed that the Arabidopsis SnRK1.1 dominant negative mutant (SnRK1.1 K48M) had lower tolerance to submergence than the wild type, suggesting that SnRK1.1-dependent phosphorylation of target proteins is important in signalling pathways triggered by submergence. We conducted quantitative phosphoproteomics and found that the phosphorylation levels of 57 proteins increased and the levels of 27 proteins decreased in Col-0 within 0.5–3h of submergence. Among the 57 proteins with increased phosphorylation in Col-0, 38 did not show increased phosphorylation levels in SnRK1.1 K48M under submergence. These proteins are involved mainly in sugar and protein synthesis. In particular, the phosphorylation of MPK6, which is involved in regulating ROS responses under abiotic stresses, was disrupted in the SnRK1.1 K48M mutant. In addition, PTP1, a negative regulator of MPK6 activity that directly dephosphorylates MPK6, was also regulated by SnRK1.1. We also showed that energy conservation was disrupted in SnRK1.1 K48M, mpk6, and PTP1 S7AS8A under submergence. These results reveal insights into the function of SnRK1 and the downstream signalling factors related to submergence. PMID:27029354
NASA Astrophysics Data System (ADS)
Sakarya, Ufuk; Hakkı Demirhan, İsmail; Seda Deveci, Hüsne; Teke, Mustafa; Demirkesen, Can; Küpçü, Ramazan; Feray Öztoprak, A.; Efendioğlu, Mehmet; Fehmi Şimşek, F.; Berke, Erdinç; Zübeyde Gürbüz, Sevgi
2016-06-01
TÜBİTAK UZAY has conducted a research study on the use of space-based satellite resources for several aspects of agriculture. Especially, there are two precision agriculture related projects: HASSAS (Widespread application of sustainable precision agriculture practices in Southeastern Anatolia Project Region (GAP) Project) and AKTAR (Smart Agriculture Feasibility Project). The HASSAS project aims to study development of precision agriculture practice in GAP region. Multi-spectral satellite imagery and aerial hyperspectral data along with ground measurements was collected to analyze data in an information system. AKTAR aims to develop models for irrigation, fertilization and spectral signatures of crops in Inner Anatolia. By the end of the project precision agriculture practices to control irrigation, fertilization, pesticide and estimation of crop yield will be developed. Analyzing the phenology of crops using NDVI is critical for the projects. For this reason, absolute radiometric calibration of the Red and NIR bands in space-based satellite sensors is an important issue. The Göktürk-2 satellite is an earth observation satellite which was designed and built in Turkey and was launched in 2012. The Göktürk-2 satellite sensor has a resolution 2.5 meters in panchromatic and 5 meters in R/G/B/NIR bands. The absolute radiometric calibration of the Göktürk-2 satellite sensor was performed via the ground-based measurements - spectra-radiometer, sun photometer, and meteorological station- in Tuz Gölü cal/val site in 2015. In this paper, the first ground-based absolute radiometric calibration results of the Göktürk-2 satellite sensor using Tuz Gölü is demonstrated. The absolute radiometric calibration results of this paper are compared with the published cross-calibration results of the Göktürk-2 satellite sensor utilizing Landsat 8 imagery. According to the experimental comparison results, the Göktürk-2 satellite sensor coefficients for red and NIR bands
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
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{sub 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.
Cabling procedure for the colored HOMFLY polynomials
NASA Astrophysics Data System (ADS)
Anokhina, A. S.; Morozov, A. A.
2014-02-01
We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and -matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and -matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦ Q¦ m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦ Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental -matrices and clarifying some conjectures formulated in previous papers.
The basic function scheme of polynomial type
WU, Wang-yi; Lin, Guang
2009-12-01
A new numerical method---Basic Function Method is proposed. This method can directly discrete differential operator on unstructured grids. By using the expansion of basic function to approach the exact function, the central and upwind schemes of derivative are constructed. By using the second-order polynomial as basic function and applying the technique of flux splitting method and the combination of central and upwind schemes to suppress the non-physical fluctuation near the shock wave, the second-order basic function scheme of polynomial type for solving inviscid compressible flow numerically is constructed in this paper. Several numerical results of many typical examples for two dimensional inviscid compressible transonic and supersonic steady flow illustrate that it is a new scheme with high accuracy and high resolution for shock wave. Especially, combining with the adaptive remeshing technique, the satisfactory results can be obtained by these schemes.
On computing factors of cyclotomic polynomials
NASA Astrophysics Data System (ADS)
Brent, Richard P.
1993-07-01
For odd square-free n > 1 the cyclotomic polynomial {Φ_n}(x) satisfies the identity of Gauss, 4{Φ_n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2. A similar identity of Aurifeuille, Le Lasseur, and Lucas is {Φ_n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2 or, in the case that n is even and square-free, ± {Φ_{n/2}}( - {x^2}) = C_n^2 - nxD_n^2. Here, {A_n}(x), ldots ,{D_n}(x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O({n^2}) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for {A_n}(x), ldots ,{D_n}(x) , and illustrate the application to integer factorization with some numerical examples.
Fast and practical parallel polynomial interpolation
Egecioglu, O.; Gallopoulos, E.; Koc, C.K.
1987-01-01
We present fast and practical parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms make use of fast parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. For n + 1 given input pairs the proposed interpolation algorithm requires 2 (log (n + 1)) + 2 parallel arithmetic steps and circuit size O(n/sup 2/). The algorithms are numerically stable and their floating-point implementation results in error accumulation similar to that of the widely used serial algorithms. This is in contrast to other fast serial and parallel interpolation algorithms which are subject to much larger roundoff. We demonstrate that in a distributed memory environment context, a cube connected system is very suitable for the algorithms' implementation, exhibiting very small communication cost. As further advantages we note that our techniques do not require equidistant points, preconditioning, or use of the Fast Fourier Transform. 21 refs., 4 figs.
Polynomial Operators on Classes of Regular Languages
NASA Astrophysics Data System (ADS)
Klíma, Ondřej; Polák, Libor
We assign to each positive variety mathcal V and each natural number k the class of all (positive) Boolean combinations of the restricted polynomials, i.e. the languages of the form L_0a_1 L_1a_2dots a_ell L_ell, text{ where } ell≤ k, a 1,...,a ℓ are letters and L 0,...,L ℓ are languages from the variety mathcal V. For this polynomial operator we give a certain algebraic counterpart which works with identities satisfied by syntactic (ordered) monoids of languages considered. We also characterize the property that a variety of languages is generated by a finite number of languages. We apply our constructions to particular examples of varieties of languages which are crucial for a certain famous open problem concerning concatenation hierarchies.
Scalar Field Theories with Polynomial Shift Symmetries
NASA Astrophysics Data System (ADS)
Griffin, Tom; Grosvenor, Kevin T.; Hořava, Petr; Yan, Ziqi
2015-12-01
We continue our study of naturalness in nonrelativistic QFTs of the Lifshitz type, focusing on scalar fields that can play the role of Nambu-Goldstone (NG) modes associated with spontaneous symmetry breaking. Such systems allow for an extension of the constant shift symmetry to a shift by a polynomial of degree P in spatial coordinates. These "polynomial shift symmetries" in turn protect the technical naturalness of modes with a higher-order dispersion relation, and lead to a refinement of the proposed classification of infrared Gaussian fixed points available to describe NG modes in nonrelativistic theories. Generic interactions in such theories break the polynomial shift symmetry explicitly to the constant shift. It is thus natural to ask: Given a Gaussian fixed point with polynomial shift symmetry of degree P, what are the lowest-dimension operators that preserve this symmetry, and deform the theory into a self-interacting scalar field theory with the shift symmetry of degree P? To answer this (essentially cohomological) question, we develop a new graph-theoretical technique, and use it to prove several classification theorems. First, in the special case of P = 1 (essentially equivalent to Galileons), we reproduce the known Galileon N-point invariants, and find their novel interpretation in terms of graph theory, as an equal-weight sum over all labeled trees with N vertices. Then we extend the classification to P > 1 and find a whole host of new invariants, including those that represent the most relevant (or least irrelevant) deformations of the corresponding Gaussian fixed points, and we study their uniqueness.
Detecting Prime Numbers via Roots of Polynomials
ERIC Educational Resources Information Center
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…
Detecting prime numbers via roots of polynomials
NASA Astrophysics Data System (ADS)
Dobbs, David E.
2012-04-01
It is proved that an integer n ≥ 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 n , the ring of integers modulo n, such that each element of Z n is a root of f. This classroom note could find use in any introductory course on abstract algebra or elementary number theory.
Vortex knot cascade in polynomial skein relations
NASA Astrophysics Data System (ADS)
Ricca, Renzo L.
2016-06-01
The process of vortex cascade through continuous reduction of topological complexity by stepwise unlinking, that has been observed experimentally in the production of vortex knots (Kleckner & Irvine, 2013), is shown to be reproduced in the branching of the skein relations of knot polynomials (Liu & Ricca, 2015) used to identify topological complexity of vortex systems. This observation can be usefully exploited for predictions of energy-complexity estimates for fluid flows.
Trigonometric Polynomials For Estimation Of Spectra
NASA Technical Reports Server (NTRS)
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.
Polynomial approximation of functions in Sobolev spaces
NASA Technical Reports Server (NTRS)
Dupont, T.; Scott, R.
1980-01-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
Eye aberration analysis with Zernike polynomials
NASA Astrophysics Data System (ADS)
Molebny, Vasyl V.; Chyzh, Igor H.; Sokurenko, Vyacheslav M.; Pallikaris, Ioannis G.; Naoumidis, Leonidas P.
1998-06-01
New horizons for accurate photorefractive sight correction, afforded by novel flying spot technologies, require adequate measurements of photorefractive properties of an eye. Proposed techniques of eye refraction mapping present results of measurements for finite number of points of eye aperture, requiring to approximate these data by 3D surface. A technique of wave front approximation with Zernike polynomials is described, using optimization of the number of polynomial coefficients. Criterion of optimization is the nearest proximity of the resulted continuous surface to the values calculated for given discrete points. Methodology includes statistical evaluation of minimal root mean square deviation (RMSD) of transverse aberrations, in particular, varying consecutively the values of maximal coefficient indices of Zernike polynomials, recalculating the coefficients, and computing the value of RMSD. Optimization is finished at minimal value of RMSD. Formulas are given for computing ametropia, size of the spot of light on retina, caused by spherical aberration, coma, and astigmatism. Results are illustrated by experimental data, that could be of interest for other applications, where detailed evaluation of eye parameters is needed.
Nested Canalyzing, Unate Cascade, and Polynomial Functions.
Jarrah, Abdul Salam; Raposa, Blessilda; Laubenbacher, Reinhard
2007-09-15
This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions. PMID:18437250
Role of discriminantly separable polynomials in integrable dynamical systems
NASA Astrophysics Data System (ADS)
Dragović, Vladimir; Kukić, Katarina
2014-11-01
Discriminantly separable polynomials of degree two in each of the three variables are considered. Those polynomials are by definition polynomials which discriminants are factorized as the products of the polynomials in one variable. Motivating example for introducing such polynomials is the famous Kowalevski top. Motivated by the role of such polynomials in the Kowalevski top, we generalize Kowalevski's integration procedure on a whole class of systems basically obtained by replacing so called the Kowalevski's fundamental equation by some other instance of the discriminantly separable polynomial. We present also the role of the discriminantly separable polynomils in twowell-known examples: the case of Kirchhoff elasticae and the Sokolov's case of a rigid body in an ideal fluid.
Polynomials and Neural Networks for Gas Turbine Monitoring: a Comparative Study
NASA Astrophysics Data System (ADS)
Loboda, Igor; Feldshteyn, Yakov
2011-09-01
Gas turbine health monitoring includes the common stages of problem detection, fault identification, and prognostics. To extract useful diagnostic information from raw recorded data, these stages require a preliminary operation of computing differences between measurements and an engine baseline, which is a function of engine operating conditions. These deviations of measured values from the baseline data can be good indicators of engine health. However, their quality and the success of all diagnostic stages strongly depend on the adequacy of the baseline model employed and, in particular, on the mathematical techniques applied to create it. To create a baseline model, we have applied polynomials and the least squares method for computing the coefficients over a long period of time. Methods were proposed to enhance such a polynomial-based model. The resulting accuracy was sufficient for reliable monitoring of gas turbine deterioration effects. The polynomials previously investigated thus far are used in the present study as a standard for evaluating artificial neural networks, a very popular technique in gas turbine diagnostics. The focus of this comparative study is to verify whether the use of networks results in a better description of the engine baseline. Extensive field data for two different industrial gas turbines were used to compare these two techniques under various conditions. The deviations were computed for all available data, and the quality of the resulting deviation plots was compared visually. The mean error of the baseline model was used as an additional criterion for comparing the techniques. To find the best network configurations, many network variations were realised and compared with the polynomials. Although the neural networks studied were found to be close to the polynomials in accuracy, they did not exceed the polynomials in any variation. In this way, it seems that polynomials can be successfully used for engine monitoring, at least for
Long-time uncertainty propagation using generalized polynomial chaos and flow map composition
Luchtenburg, Dirk M.; Brunton, Steven L.; Rowley, Clarence W.
2014-10-01
We present an efficient and accurate method for long-time uncertainty propagation in dynamical systems. Uncertain initial conditions and parameters are both addressed. The method approximates the intermediate short-time flow maps by spectral polynomial bases, as in the generalized polynomial chaos (gPC) method, and uses flow map composition to construct the long-time flow map. In contrast to the gPC method, this approach has spectral error convergence for both short and long integration times. The short-time flow map is characterized by small stretching and folding of the associated trajectories and hence can be well represented by a relatively low-degree basis. The composition of these low-degree polynomial bases then accurately describes the uncertainty behavior for long integration times. The key to the method is that the degree of the resulting polynomial approximation increases exponentially in the number of time intervals, while the number of polynomial coefficients either remains constant (for an autonomous system) or increases linearly in the number of time intervals (for a non-autonomous system). The findings are illustrated on several numerical examples including a nonlinear ordinary differential equation (ODE) with an uncertain initial condition, a linear ODE with an uncertain model parameter, and a two-dimensional, non-autonomous double gyre flow.
Molecular Mimicry Regulates ABA Signaling by SnRK2 Kinases and PP2C Phosphatases
Soon, Fen-Fen; Ng, Ley-Moy; Zhou, X. Edward; West, Graham M.; Kovach, Amanda; Tan, M.H. Eileen; Suino-Powell, Kelly M.; He, Yuanzheng; Xu, Yong; Chalmers, Michael J.; Brunzelle, Joseph S.; Zhang, Huiming; Yang, Huaiyu; Jiang, Hualiang; Li, Jun; Yong, Eu-Leong; Cutler, Sean; Zhu, Jian-Kang; Griffin, Patrick R.; Melcher, Karsten; Xu, H. Eric
2014-10-02
Abscisic acid (ABA) is an essential hormone for plants to survive environmental stresses. At the center of the ABA signaling network is a subfamily of type 2C protein phosphatases (PP2Cs), which form exclusive interactions with ABA receptors and subfamily 2 Snfl-related kinase (SnRK2s). Here, we report a SnRK2-PP2C complex structure, which reveals marked similarity in PP2C recognition by SnRK2 and ABA receptors. In the complex, the kinase activation loop docks into the active site of PP2C, while the conserved ABA-sensing tryptophan of PP2C inserts into the kinase catalytic cleft, thus mimicking receptor-PP2C interactions. These structural results provide a simple mechanism that directly couples ABA binding to SnRK2 kinase activation and highlight a new paradigm of kinase-phosphatase regulation through mutual packing of their catalytic sites.
Combining fractional polynomial model building with multiple imputation
Morris, Tim P.; White, Ian R.; Carpenter, James R.; Stanworth, Simon J.; Royston, Patrick
2016-01-01
Multivariable fractional polynomial (MFP) models are commonly used in medical research. The datasets in which MFP models are applied often contain covariates with missing values. To handle the missing values, we describe methods for combining multiple imputation with MFP modelling, considering in turn three issues: first, how to impute so that the imputation model does not favour certain fractional polynomial (FP) models over others; second, how to estimate the FP exponents in multiply imputed data; and third, how to choose between models of differing complexity. Two imputation methods are outlined for different settings. For model selection, methods based on Wald-type statistics and weighted likelihood-ratio tests are proposed and evaluated in simulation studies. The Wald-based method is very slightly better at estimating FP exponents. Type I error rates are very similar for both methods, although slightly less well controlled than analysis of complete records; however, there is potential for substantial gains in power over the analysis of complete records. We illustrate the two methods in a dataset from five trauma registries for which a prognostic model has previously been published, contrasting the selected models with that obtained by analysing the complete records only. PMID:26095614
Uncertainty Quantification for Polynomial Systems via Bernstein Expansions
NASA Technical Reports Server (NTRS)
Crespo, Luis G.; Kenny, Sean P.; Giesy, Daniel P.
2012-01-01
This paper presents a unifying framework to uncertainty quantification for systems having polynomial response metrics that depend on both aleatory and epistemic uncertainties. The approach proposed, which is based on the Bernstein expansions of polynomials, enables bounding the range of moments and failure probabilities of response metrics as well as finding supersets of the extreme epistemic realizations where the limits of such ranges occur. These bounds and supersets, whose analytical structure renders them free of approximation error, can be made arbitrarily tight with additional computational effort. Furthermore, this framework enables determining the importance of particular uncertain parameters according to the extent to which they affect the first two moments of response metrics and failure probabilities. This analysis enables determining the parameters that should be considered uncertain as well as those that can be assumed to be constants without incurring significant error. The analytical nature of the approach eliminates the numerical error that characterizes the sampling-based techniques commonly used to propagate aleatory uncertainties as well as the possibility of under predicting the range of the statistic of interest that may result from searching for the best- and worstcase epistemic values via nonlinear optimization or sampling.
Combining fractional polynomial model building with multiple imputation.
Morris, Tim P; White, Ian R; Carpenter, James R; Stanworth, Simon J; Royston, Patrick
2015-11-10
Multivariable fractional polynomial (MFP) models are commonly used in medical research. The datasets in which MFP models are applied often contain covariates with missing values. To handle the missing values, we describe methods for combining multiple imputation with MFP modelling, considering in turn three issues: first, how to impute so that the imputation model does not favour certain fractional polynomial (FP) models over others; second, how to estimate the FP exponents in multiply imputed data; and third, how to choose between models of differing complexity. Two imputation methods are outlined for different settings. For model selection, methods based on Wald-type statistics and weighted likelihood-ratio tests are proposed and evaluated in simulation studies. The Wald-based method is very slightly better at estimating FP exponents. Type I error rates are very similar for both methods, although slightly less well controlled than analysis of complete records; however, there is potential for substantial gains in power over the analysis of complete records. We illustrate the two methods in a dataset from five trauma registries for which a prognostic model has previously been published, contrasting the selected models with that obtained by analysing the complete records only. PMID:26095614
Replacement of a Björk-Shiley Delrin Aortic Valve Still Functioning after 25 Years
Badak, M. Ismail; Ozkisacik, Erdem Ali; Boga, Mehmet; Gurcun, Ugur; Discigil, Berent
2004-01-01
We report the case of a patient who had undergone implantation of a Björk-Shiley Delrin valve in the aortic position 25 years earlier and who now presented with severe mitral stenosis. The patient underwent mitral valve replacement and aortic valve re-replacement. We review the justification for prophylactic replacement of Björk-Shiley Delrin heart valves. PMID:15562853
NASA Astrophysics Data System (ADS)
Petronilho, J.
2007-08-01
It is well-known that the classical orthogonal polynomials of Jacobi, Bessel, Laguerre and Hermite are solutions of a Sturm-Liouville problem of the type where [sigma] and [tau] are polynomials such that deg[sigma][less-than-or-equals, slant]2 and deg[tau]=1, and [lambda]n is a constant independent of x. Recently, based on the hypergeometric character of the solutions of this differential equation, W. Koepf and M. Masjed-Jamei [A generic formula for the values at the boundary points of monic classical orthogonal polynomials, J. Comput. Appl. Math. 191 (2006) 98-105] found a generic formula, only in terms of the coefficients of [sigma] and [tau], for the values of the classical orthogonal polynomials at the singular points of the above differential hypergeometric equation. In this paper, we generalize the mentioned result giving the analogous formulas for both the classical q-orthogonal polynomials (of the q-Hahn tableau) and the classical D[omega]-orthogonal polynomials. Both are special cases of the classical Hq,[omega]-orthogonal polynomials, which are solutions of the hypergeometric-type difference equation where Hq,[omega] is the difference operator introduced by Hahn, and [sigma], [tau] and [lambda]n being as above. Our approach is algebraic and it does not require hypergeometric functions.
Rk1, a ginsenoside, is a new blocker of vascular leakage acting through actin structure remodeling.
Maeng, Yong-Sun; Maharjan, Sony; Kim, Jeong-Hun; Park, Jeong-Hill; Suk Yu, Young; Kim, Young-Myoung; Kwon, Young-Guen
2013-01-01
Endothelial barrier integrity is essential for vascular homeostasis and increased vascular permeability and has been implicated in many pathological processes, including diabetic retinopathy. Here, we investigated the effect of Rk1, a ginsenoside extracted from sun ginseng, on regulation of endothelial barrier function. In human retinal endothelial cells, Rk1 strongly inhibited permeability induced by VEGF, advanced glycation end-product, thrombin, or histamine. Furthermore, Rk1 significantly reduced the vessel leakiness of retina in a diabetic mouse model. This anti-permeability activity of Rk1 is correlated with enhanced stability and positioning of tight junction proteins at the boundary between cells. Signaling experiments revealed that Rk1 induces phosphorylation of myosin light chain and cortactin, which are critical regulators for the formation of the cortical actin ring structure and endothelial barrier. These findings raise the possibility that ginsenoside Rk1 could be exploited as a novel prototype compound for the prevention of human diseases that are characterized by vascular leakage. PMID:23894330
Perturbing polynomials with all their roots on the unit circle
NASA Astrophysics Data System (ADS)
Mossinghoff, M. J.; Pinner, C. G.; Vaaler, J. D.
1998-10-01
Given a monic real polynomial with all its roots on the unit circle, we ask to what extent one can perturb its middle coefficient and still have a polynomial with all its roots on the unit circle. We show that the set of possible perturbations forms a closed interval of length at most 4, with 4 achieved only for polynomials of the form x(2n) + cx(n) + 1 with c in [-2, 2]. The problem can also be formulated in terms of perturbing the constant coefficient of a polynomial having all its roots in [-1, 1]. If we restrict to integer coefficients, then the polynomials in question are products of cyclotomics. We show that in this case there are no perturbations of length 3 that do not arise from a perturbation of length 4. We also investigate the connection between slightly perturbed products of cyclotomic polynomials and polynomials with small Mahler measure. We describe an algorithm for searching for polynomials with small Mahler measure by perturbing the middle coefficients of products of cyclotomic polynomials. We show that the complexity of this algorithm is O(C-root d), where d is the degree, and we report on the polynomials found by this algorithm through degree 64.
Momentum space orthogonal polynomial projection quantization
NASA Astrophysics Data System (ADS)
Handy, C. R.; Vrinceanu, D.; Marth, C. B.; Gupta, R.
2016-04-01
The orthogonal polynomial projection quantization (OPPQ) is an algebraic method for solving Schrödinger’s equation by representing the wave function as an expansion {{\\Psi }}(x)={\\displaystyle \\sum }n{{{Ω }}}n{P}n(x)R(x) in terms of polynomials {P}n(x) orthogonal with respect to a suitable reference function R(x), which decays asymptotically not faster than the bound state wave function. The expansion coefficients {{{Ω }}}n are obtained as linear combinations of power moments {μ }{{p}}=\\int {x}p{{\\Psi }}(x) {{d}}x. In turn, the {μ }{{p}}'s are generated by a linear recursion relation derived from Schrödinger’s equation from an initial set of low order moments. It can be readily argued that for square integrable wave functions representing physical states {{lim}}n\\to ∞ {{{Ω }}}n=0. Rapidly converging discrete energies are obtained by setting Ω coefficients to zero at arbitrarily high order. This paper introduces an extention of OPPQ in momentum space by using the representation {{Φ }}(k)={\\displaystyle \\sum }n{{{\\Xi }}}n{Q}n(k)T(k), where Q n (k) are polynomials orthogonal with respect to a suitable reference function T(k). The advantage of this new representation is that it can help solving problems for which there is no coordinate space moment equation. This is because the power moments in momentum space are the Taylor expansion coefficients, which are recursively calculated via Schrödinger’s equation. We show the convergence of this new method for the sextic anharmonic oscillator and an algebraic treatment of Gross-Pitaevskii nonlinear equation.
Wavelet approach to accelerator problems. 1: Polynomial dynamics
Fedorova, A.; Zeitlin, M.; Parsa, Z.
1997-05-01
This is the first part of a series of talks in which the authors present applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. In the general case they have the solution as a multiresolution expansion in the base of compactly supported wavelet basis. The solution is parameterized by solutions of two reduced algebraical problems, one is nonlinear and the second is some linear problem, which is obtained from one of the next wavelet constructions: Fast Wavelet Transform, Stationary Subdivision Schemes, the method of Connection Coefficients. In this paper the authors consider the problem of calculation of orbital motion in storage rings. The key point in the solution of this problem is the use of the methods of wavelet analysis, relatively novel set of mathematical methods, which gives one a possibility to work with well-localized bases in functional spaces and with the general type of operators (including pseudodifferential) in such bases.
A polynomial f(R) inflation model
Huang, Qing-Guo
2014-02-01
Motivated by the ultraviolet complete theory of quantum gravity, for example the string theory, we investigate a polynomial f(R) inflation model in detail. We calculate the spectral index and tensor-to-scalar ratio in the f(R) inflation model with the form of f(R) = R + (R{sup 2})/6M{sup 2} + (λn)/2n (R{sup n})/(3M{sup 2}){sup n-1}. Compared to Planck 2013, we find that R{sup n} term should be exponentially suppressed, i.e. |λ{sub n}|∼<10{sup −2n+2.6}.
A polynomial f(R) inflation model
Huang, Qing-Guo
2014-02-19
Motivated by the ultraviolet complete theory of quantum gravity, for example the string theory, we investigate a polynomial f(R) inflation model in detail. We calculate the spectral index and tensor-to-scalar ratio in the f(R) inflation model with the form of f(R)=R+((R{sup 2})/(6M{sup 2}))+((λ{sub n})/(2n))((R{sup n})/((3M{sup 2}){sup n−1})). Compared to Planck 2013, we find that R{sup n} term should be exponentially suppressed, i.e. |λ{sub n}|≲10{sup −2n+2.6}.
Supersymmetric Casimir energy and the anomaly polynomial
NASA Astrophysics Data System (ADS)
Bobev, Nikolay; Bullimore, Mathew; Kim, Hee-Cheol
2015-09-01
We conjecture that for superconformal field theories in even dimensions, the supersymmetric Casimir energy on a space with topology S 1 × S D-1 is equal to an equivariant integral of the anomaly polynomial. The equivariant integration is defined with respect to the Cartan subalgebra of the global symmetry algebra that commutes with a given supercharge. We test our proposal extensively by computing the supersymmetric Casimir energy for large classes of superconformal field theories, with and without known Lagrangian descriptions, in two, four and six dimensions.
A multiple-scale Pascal polynomial for 2D Stokes and inverse Cauchy-Stokes problems
NASA Astrophysics Data System (ADS)
Liu, Chein-Shan; Young, D. L.
2016-05-01
The polynomial expansion method is a useful tool for solving both the direct and inverse Stokes problems, which together with the pointwise collocation technique is easy to derive the algebraic equations for satisfying the Stokes differential equations and the specified boundary conditions. In this paper we propose two novel numerical algorithms, based on a third-first order system and a third-third order system, to solve the direct and the inverse Cauchy problems in Stokes flows by developing a multiple-scale Pascal polynomial method, of which the scales are determined a priori by the collocation points. To assess the performance through numerical experiments, we find that the multiple-scale Pascal polynomial expansion method (MSPEM) is accurate and stable against large noise.
Myers, N.J.
1994-12-31
The author gives a hybrid method for the iterative solution of linear systems of equations Ax = b, where the matrix (A) is nonsingular, sparse and nonsymmetric. As in a method developed by Starke and Varga the method begins with a number of steps of the Arnoldi method to produce some information on the location of the spectrum of A. This method then switches to an iterative method based on the Faber polynomials for an annular sector placed around these eigenvalue estimates. The Faber polynomials for an annular sector are used because, firstly an annular sector can easily be placed around any eigenvalue estimates bounded away from zero, and secondly the Faber polynomials are known analytically for an annular sector. Finally the author gives three numerical examples, two of which allow comparison with Starke and Varga`s results. The third is an example of a matrix for which many iterative methods would fall, but this method converges.
Polynomial invariants for discrimination and classification of four-qubit entanglement
Viehmann, Oliver; Eltschka, Christopher; Siewert, Jens
2011-05-15
The number of entanglement classes in stochastic local operations and classical communication (SLOCC) classifications increases with the number of qubits and is already infinite for four qubits. Criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore at the focus of intense theoretical research. We develop a general criterion for the discrimination of pure N-partite entangled states in terms of polynomial SL(d,C){sup xN} invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which entanglement types are identified through 'tangle patterns'. This scheme provides a practicable way to classify states of arbitrary multipartite systems. Moreover, the use of polynomials induces a corresponding quantification of the different types of entanglement.
Lüchow, Arne; Sturm, Alexander; Schulte, Christoph; Haghighi Mood, Kaveh
2015-02-28
Jastrow correlation factors play an important role in quantum Monte Carlo calculations. Together with an orbital based antisymmetric function, they allow the construction of highly accurate correlation wave functions. In this paper, a generic expansion of the Jastrow correlation function in terms of polynomials that satisfy both the electron exchange symmetry constraint and the cusp conditions is presented. In particular, an expansion of the three-body electron-electron-nucleus contribution in terms of cuspless homogeneous symmetric polynomials is proposed. The polynomials can be expressed in fairly arbitrary scaling function allowing a generic implementation of the Jastrow factor. It is demonstrated with a few examples that the new Jastrow factor achieves 85%–90% of the total correlation energy in a variational quantum Monte Carlo calculation and more than 90% of the diffusion Monte Carlo correlation energy.
Polynomial solutions of the Monge-Ampère equation
Aminov, Yu A
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.
Lin, Chien-Ru; Lee, Kuo-Wei; Chen, Chih-Yu; Hong, Ya-Fang; Chen, Jyh-Long; Lu, Chung-An; Chen, Ku-Ting; Ho, Tuan-Hua David; Yu, Su-May
2014-02-01
In plants, source-sink communication plays a pivotal role in crop productivity, yet the underlying regulatory mechanisms are largely unknown. The SnRK1A protein kinase and transcription factor MYBS1 regulate the sugar starvation signaling pathway during seedling growth in cereals. Here, we identified plant-specific SnRK1A-interacting negative regulators (SKINs). SKINs antagonize the function of SnRK1A, and the highly conserved GKSKSF domain is essential for SKINs to function as repressors. Overexpression of SKINs inhibits the expression of MYBS1 and hydrolases essential for mobilization of nutrient reserves in the endosperm, leading to inhibition of seedling growth. The expression of SKINs is highly inducible by drought and moderately by various stresses, which is likely related to the abscisic acid (ABA)-mediated repression of SnRK1A under stress. Overexpression of SKINs enhances ABA sensitivity for inhibition of seedling growth. ABA promotes the interaction between SnRK1A and SKINs and shifts the localization of SKINs from the nucleus to the cytoplasm, where it binds SnRK1A and prevents SnRK1A and MYBS1 from entering the nucleus. Our findings demonstrate that SnRK1A plays a key role regulating source-sink communication during seedling growth. Under abiotic stress, SKINs antagonize the function of SnRK1A, which is likely a key factor restricting seedling vigor. PMID:24569770