A phenomenological biological dose model for proton therapy based on linear energy transfer spectra.

PubMed

Rørvik, Eivind; Thörnqvist, Sara; Stokkevåg, Camilla H; Dahle, Tordis J; Fjaera, Lars Fredrik; Ytre-Hauge, Kristian S

2017-06-01

The relative biological effectiveness (RBE) of protons varies with the radiation quality, quantified by the linear energy transfer (LET). Most phenomenological models employ a linear dependency of the dose-averaged LET (LET d ) to calculate the biological dose. However, several experiments have indicated a possible non-linear trend. Our aim was to investigate if biological dose models including non-linear LET dependencies should be considered, by introducing a LET spectrum based dose model. The RBE-LET relationship was investigated by fitting of polynomials from 1st to 5th degree to a database of 85 data points from aerobic in vitro experiments. We included both unweighted and weighted regression, the latter taking into account experimental uncertainties. Statistical testing was performed to decide whether higher degree polynomials provided better fits to the data as compared to lower degrees. The newly developed models were compared to three published LET d based models for a simulated spread out Bragg peak (SOBP) scenario. The statistical analysis of the weighted regression analysis favored a non-linear RBE-LET relationship, with the quartic polynomial found to best represent the experimental data (P = 0.010). The results of the unweighted regression analysis were on the borderline of statistical significance for non-linear functions (P = 0.053), and with the current database a linear dependency could not be rejected. For the SOBP scenario, the weighted non-linear model estimated a similar mean RBE value (1.14) compared to the three established models (1.13-1.17). The unweighted model calculated a considerably higher RBE value (1.22). The analysis indicated that non-linear models could give a better representation of the RBE-LET relationship. However, this is not decisive, as inclusion of the experimental uncertainties in the regression analysis had a significant impact on the determination and ranking of the models. As differences between the models were observed for the SOBP scenario, both non-linear LET spectrum- and linear LET d based models should be further evaluated in clinically realistic scenarios. © 2017 American Association of Physicists in Medicine.

Degenerate r-Stirling Numbers and r-Bell Polynomials

NASA Astrophysics Data System (ADS)

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

2018-01-01

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

Measurement of pediatric regional cerebral blood flow from 6 months to 15 years of age in a clinical population.

PubMed

Carsin-Vu, Aline; Corouge, Isabelle; Commowick, Olivier; Bouzillé, Guillaume; Barillot, Christian; Ferré, Jean-Christophe; Proisy, Maia

2018-04-01

To investigate changes in cerebral blood flow (CBF) in gray matter (GM) between 6 months and 15 years of age and to provide CBF values for the brain, GM, white matter (WM), hemispheres and lobes. Between 2013 and 2016, we retrospectively included all clinical MRI examinations with arterial spin labeling (ASL). We excluded subjects with a condition potentially affecting brain perfusion. For each subject, mean values of CBF in the brain, GM, WM, hemispheres and lobes were calculated. GM CBF was fitted using linear, quadratic and cubic polynomial regression against age. Regression models were compared with Akaike's information criterion (AIC), and Likelihood Ratio tests. 84 children were included (44 females/40 males). Mean CBF values were 64.2 ± 13.8 mL/100 g/min in GM, and 29.3 ± 10.0 mL/100 g/min in WM. The best-fit model of brain perfusion was the cubic polynomial function (AIC = 672.7, versus respectively AIC = 673.9 and AIC = 674.1 with the linear negative function and the quadratic polynomial function). A statistically significant difference between the tested models demonstrating the superiority of the quadratic (p = 0.18) or cubic polynomial model (p = 0.06), over the negative linear regression model was not found. No effect of general anesthesia (p = 0.34) or of gender (p = 0.16) was found. we provided values for ASL CBF in the brain, GM, WM, hemispheres, and lobes over a wide pediatric age range, approximately showing inverted U-shaped changes in GM perfusion over the course of childhood. Copyright © 2018 Elsevier B.V. All rights reserved.

Orthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomials.

PubMed

Mafusire, Cosmas; Krüger, Tjaart P J

2018-06-01

The concept of orthonormal vector circle polynomials is revisited by deriving a set from the Cartesian gradient of Zernike polynomials in a unit circle using a matrix-based approach. The heart of this model is a closed-form matrix equation of the gradient of Zernike circle polynomials expressed as a linear combination of lower-order Zernike circle polynomials related through a gradient matrix. This is a sparse matrix whose elements are two-dimensional standard basis transverse Euclidean vectors. Using the outer product form of the Cholesky decomposition, the gradient matrix is used to calculate a new matrix, which we used to express the Cartesian gradient of the Zernike circle polynomials as a linear combination of orthonormal vector circle polynomials. Since this new matrix is singular, the orthonormal vector polynomials are recovered by reducing the matrix to its row echelon form using the Gauss-Jordan elimination method. We extend the model to derive orthonormal vector general polynomials, which are orthonormal in a general pupil by performing a similarity transformation on the gradient matrix to give its equivalent in the general pupil. The outer form of the Gram-Schmidt procedure and the Gauss-Jordan elimination method are then applied to the general pupil to generate the orthonormal vector general polynomials from the gradient of the orthonormal Zernike-based polynomials. The performance of the model is demonstrated with a simulated wavefront in a square pupil inscribed in a unit circle.

Computing Gröbner Bases within Linear Algebra

NASA Astrophysics Data System (ADS)

Suzuki, Akira

In this paper, we present an alternative algorithm to compute Gröbner bases, which is based on computations on sparse linear algebra. Both of S-polynomial computations and monomial reductions are computed in linear algebra simultaneously in this algorithm. So it can be implemented to any computational system which can handle linear algebra. For a given ideal in a polynomial ring, it calculates a Gröbner basis along with the corresponding term order appropriately.

A Linear Algebraic Approach to Teaching Interpolation

ERIC Educational Resources Information Center

Tassa, Tamir

2007-01-01

A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…

On Partial Fraction Decompositions by Repeated Polynomial Divisions

ERIC Educational Resources Information Center

Man, Yiu-Kwong

2017-01-01

We present a method for finding partial fraction decompositions of rational functions with linear or quadratic factors in the denominators by means of repeated polynomial divisions. This method does not involve differentiation or solving linear equations for obtaining the unknown partial fraction coefficients, which is very suitable for either…

On the Numerical Formulation of Parametric Linear Fractional Transformation (LFT) Uncertainty Models for Multivariate Matrix Polynomial Problems

NASA Technical Reports Server (NTRS)

Belcastro, Christine M.

1998-01-01

Robust control system analysis and design is based on an uncertainty description, called a linear fractional transformation (LFT), which separates the uncertain (or varying) part of the system from the nominal system. These models are also useful in the design of gain-scheduled control systems based on Linear Parameter Varying (LPV) methods. Low-order LFT models are difficult to form for problems involving nonlinear parameter variations. This paper presents a numerical computational method for constructing and LFT model for a given LPV model. The method is developed for multivariate polynomial problems, and uses simple matrix computations to obtain an exact low-order LFT representation of the given LPV system without the use of model reduction. Although the method is developed for multivariate polynomial problems, multivariate rational problems can also be solved using this method by reformulating the rational problem into a polynomial form.

Local polynomial estimation of heteroscedasticity in a multivariate linear regression model and its applications in economics.

PubMed

Su, Liyun; Zhao, Yanyong; Yan, Tianshun; Li, Fenglan

2012-01-01

Multivariate local polynomial fitting is applied to the multivariate linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improving the traditional two-stage method. Due to non-parametric technique of local polynomial estimation, it is unnecessary to know the form of heteroscedastic function. Therefore, we can improve the estimation precision, when the heteroscedastic function is unknown. Furthermore, we verify that the regression coefficients is asymptotic normal based on numerical simulations and normal Q-Q plots of residuals. Finally, the simulation results and the local polynomial estimation of real data indicate that our approach is surely effective in finite-sample situations.

Robustness Analysis of Integrated LPV-FDI Filters and LTI-FTC System for a Transport Aircraft

NASA Technical Reports Server (NTRS)

Khong, Thuan H.; Shin, Jong-Yeob

2007-01-01

This paper proposes an analysis framework for robustness analysis of a nonlinear dynamics system that can be represented by a polynomial linear parameter varying (PLPV) system with constant bounded uncertainty. The proposed analysis framework contains three key tools: 1) a function substitution method which can convert a nonlinear system in polynomial form into a PLPV system, 2) a matrix-based linear fractional transformation (LFT) modeling approach, which can convert a PLPV system into an LFT system with the delta block that includes key uncertainty and scheduling parameters, 3) micro-analysis, which is a well known robust analysis tool for linear systems. The proposed analysis framework is applied to evaluating the performance of the LPV-fault detection and isolation (FDI) filters of the closed-loop system of a transport aircraft in the presence of unmodeled actuator dynamics and sensor gain uncertainty. The robustness analysis results are compared with nonlinear time simulations.

Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Huan, Xun; Safta, Cosmin; Sargsyan, Khachik

Compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quanti cation analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several com- pressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers of l1 ls, SpaRSA, CGIST, FPC AS, and ADMM, we develop techniques to mitigate over tting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendationsmore » on parameter settings for these tech- niques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-cross flow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy and computational tradeoffs between polynomial bases of different degrees, and practi- cability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.« less

Polynomial approximation of functions of matrices and its application to the solution of a general system of linear equations

NASA Technical Reports Server (NTRS)

Tal-Ezer, Hillel

1987-01-01

During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described.

Optimal control and Galois theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zelikin, M I; Kiselev, D D; Lokutsievskiy, L V

2013-11-30

An important role is played in the solution of a class of optimal control problems by a certain special polynomial of degree 2(n−1) with integer coefficients. The linear independence of a family of k roots of this polynomial over the field Q implies the existence of a solution of the original problem with optimal control in the form of an irrational winding of a k-dimensional Clifford torus, which is passed in finite time. In the paper, we prove that for n≤15 one can take an arbitrary positive integer not exceeding [n/2] for k. The apparatus developed in the paper is applied to the systems ofmore » Chebyshev-Hermite polynomials and generalized Chebyshev-Laguerre polynomials. It is proved that for such polynomials of degree 2m every subsystem of [(m+1)/2] roots with pairwise distinct squares is linearly independent over the field Q. Bibliography: 11 titles.« less

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

PubMed

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

2014-03-01

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

Semiparametric methods for estimation of a nonlinear exposure-outcome relationship using instrumental variables with application to Mendelian randomization.

PubMed

Staley, James R; Burgess, Stephen

2017-05-01

Mendelian randomization, the use of genetic variants as instrumental variables (IV), can test for and estimate the causal effect of an exposure on an outcome. Most IV methods assume that the function relating the exposure to the expected value of the outcome (the exposure-outcome relationship) is linear. However, in practice, this assumption may not hold. Indeed, often the primary question of interest is to assess the shape of this relationship. We present two novel IV methods for investigating the shape of the exposure-outcome relationship: a fractional polynomial method and a piecewise linear method. We divide the population into strata using the exposure distribution, and estimate a causal effect, referred to as a localized average causal effect (LACE), in each stratum of population. The fractional polynomial method performs metaregression on these LACE estimates. The piecewise linear method estimates a continuous piecewise linear function, the gradient of which is the LACE estimate in each stratum. Both methods were demonstrated in a simulation study to estimate the true exposure-outcome relationship well, particularly when the relationship was a fractional polynomial (for the fractional polynomial method) or was piecewise linear (for the piecewise linear method). The methods were used to investigate the shape of relationship of body mass index with systolic blood pressure and diastolic blood pressure. © 2017 The Authors Genetic Epidemiology Published by Wiley Periodicals, Inc.

Semiparametric methods for estimation of a nonlinear exposure‐outcome relationship using instrumental variables with application to Mendelian randomization

PubMed Central

Staley, James R.

2017-01-01

ABSTRACT Mendelian randomization, the use of genetic variants as instrumental variables (IV), can test for and estimate the causal effect of an exposure on an outcome. Most IV methods assume that the function relating the exposure to the expected value of the outcome (the exposure‐outcome relationship) is linear. However, in practice, this assumption may not hold. Indeed, often the primary question of interest is to assess the shape of this relationship. We present two novel IV methods for investigating the shape of the exposure‐outcome relationship: a fractional polynomial method and a piecewise linear method. We divide the population into strata using the exposure distribution, and estimate a causal effect, referred to as a localized average causal effect (LACE), in each stratum of population. The fractional polynomial method performs metaregression on these LACE estimates. The piecewise linear method estimates a continuous piecewise linear function, the gradient of which is the LACE estimate in each stratum. Both methods were demonstrated in a simulation study to estimate the true exposure‐outcome relationship well, particularly when the relationship was a fractional polynomial (for the fractional polynomial method) or was piecewise linear (for the piecewise linear method). The methods were used to investigate the shape of relationship of body mass index with systolic blood pressure and diastolic blood pressure. PMID:28317167

Reliability of the Load-Velocity Relationship Obtained Through Linear and Polynomial Regression Models to Predict the One-Repetition Maximum Load.

PubMed

Pestaña-Melero, Francisco Luis; Haff, G Gregory; Rojas, Francisco Javier; Pérez-Castilla, Alejandro; García-Ramos, Amador

2017-12-18

This study aimed to compare the between-session reliability of the load-velocity relationship between (1) linear vs. polynomial regression models, (2) concentric-only vs. eccentric-concentric bench press variants, as well as (3) the within-participants vs. the between-participants variability of the velocity attained at each percentage of the one-repetition maximum (%1RM). The load-velocity relationship of 30 men (age: 21.2±3.8 y; height: 1.78±0.07 m, body mass: 72.3±7.3 kg; bench press 1RM: 78.8±13.2 kg) were evaluated by means of linear and polynomial regression models in the concentric-only and eccentric-concentric bench press variants in a Smith Machine. Two sessions were performed with each bench press variant. The main findings were: (1) first-order-polynomials (CV: 4.39%-4.70%) provided the load-velocity relationship with higher reliability than second-order-polynomials (CV: 4.68%-5.04%); (2) the reliability of the load-velocity relationship did not differ between the concentric-only and eccentric-concentric bench press variants; (3) the within-participants variability of the velocity attained at each %1RM was markedly lower than the between-participants variability. Taken together, these results highlight that, regardless of the bench press variant considered, the individual determination of the load-velocity relationship by a linear regression model could be recommended to monitor and prescribe the relative load in the Smith machine bench press exercise.

Gauss Elimination: Workhorse of Linear Algebra.

DTIC Science & Technology

1995-08-05

linear algebra computation for solving systems, computing determinants and determining the rank of matrix. All of these are discussed in varying contexts. These include different arithmetic or algebraic setting such as integer arithmetic or polynomial rings as well as conventional real (floating-point) arithmetic. These have effects on both accuracy and complexity analyses of the algorithm. These, too, are covered here. The impact of modern parallel computer architecture on GE is also

Random regression models using different functions to model milk flow in dairy cows.

PubMed

Laureano, M M M; Bignardi, A B; El Faro, L; Cardoso, V L; Tonhati, H; Albuquerque, L G

2014-09-12

We analyzed 75,555 test-day milk flow records from 2175 primiparous Holstein cows that calved between 1997 and 2005. Milk flow was obtained by dividing the mean milk yield (kg) of the 3 daily milking by the total milking time (min) and was expressed as kg/min. Milk flow was grouped into 43 weekly classes. The analyses were performed using a single-trait Random Regression Models that included direct additive genetic, permanent environmental, and residual random effects. In addition, the contemporary group and linear and quadratic effects of cow age at calving were included as fixed effects. Fourth-order orthogonal Legendre polynomial of days in milk was used to model the mean trend in milk flow. The additive genetic and permanent environmental covariance functions were estimated using random regression Legendre polynomials and B-spline functions of days in milk. The model using a third-order Legendre polynomial for additive genetic effects and a sixth-order polynomial for permanent environmental effects, which contained 7 residual classes, proved to be the most adequate to describe variations in milk flow, and was also the most parsimonious. The heritability in milk flow estimated by the most parsimonious model was of moderate to high magnitude.

Algebraic solutions of shape-invariant position-dependent effective mass systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Amir, Naila, E-mail: naila.amir@live.com, E-mail: naila.amir@seecs.edu.pk; Iqbal, Shahid, E-mail: sic80@hotmail.com, E-mail: siqbal@sns.nust.edu.pk

2016-06-15

Keeping in view the ordering ambiguity that arises due to the presence of position-dependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with position-dependent effective mass is discussed. We quantize the Hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass, initially proposed by von Roos and Lévy-Leblond. The algebraic method, used to obtain the solutions, is based on the concepts of supersymmetric quantum mechanics and shape invariance. In order to exemplify the general formalism a class ofmore » non-linear oscillators has been considered. This class includes the particular example of a one-dimensional oscillator with different position-dependent effective mass profiles. Explicit expressions for the eigenenergies and eigenfunctions in terms of generalized Hermite polynomials are presented. Moreover, properties of these modified Hermite polynomials, like existence of generating function and recurrence relations among the polynomials have also been studied. Furthermore, it has been shown that in the harmonic limit, all the results for the linear harmonic oscillator are recovered.« less

Why High-Order Polynomials Should Not Be Used in Regression Discontinuity Designs. NBER Working Paper No. 20405

ERIC Educational Resources Information Center

Gelman, Andrew; Imbens, Guido

2014-01-01

It is common in regression discontinuity analysis to control for high order (third, fourth, or higher) polynomials of the forcing variable. We argue that estimators for causal effects based on such methods can be misleading, and we recommend researchers do not use them, and instead use estimators based on local linear or quadratic polynomials or…

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.

Phase unwrapping algorithm using polynomial phase approximation and linear Kalman filter.

PubMed

Kulkarni, Rishikesh; Rastogi, Pramod

2018-02-01

A noise-robust phase unwrapping algorithm is proposed based on state space analysis and polynomial phase approximation using wrapped phase measurement. The true phase is approximated as a two-dimensional first order polynomial function within a small sized window around each pixel. The estimates of polynomial coefficients provide the measurement of phase and local fringe frequencies. A state space representation of spatial phase evolution and the wrapped phase measurement is considered with the state vector consisting of polynomial coefficients as its elements. Instead of using the traditional nonlinear Kalman filter for the purpose of state estimation, we propose to use the linear Kalman filter operating directly with the wrapped phase measurement. The adaptive window width is selected at each pixel based on the local fringe density to strike a balance between the computation time and the noise robustness. In order to retrieve the unwrapped phase, either a line-scanning approach or a quality guided strategy of pixel selection is used depending on the underlying continuous or discontinuous phase distribution, respectively. Simulation and experimental results are provided to demonstrate the applicability of the proposed method.

On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials

NASA Astrophysics Data System (ADS)

Doha, E. H.

2003-05-01

A formula expressing the Laguerre coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Laguerre polynomials of any degree and for any order as a linear combination of suitable Laguerre polynomials is deduced. A formula for the Laguerre coefficients of the moments of one single Laguerre polynomial of certain degree is given. Formulae for the Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Laguerre coefficients are also obtained. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi-Laguerre and Hermite-Laguerre polynomials is described. An explicit formula for these coefficients between Jacobi and Laguerre polynomials is given, of which the ultra-spherical polynomials of the first and second kinds and Legendre polynomials are important special cases. An analytical formula for the connection coefficients between Hermite and Laguerre polynomials is also obtained.

Geometric analysis and restitution of digital multispectral scanner data arrays

NASA Technical Reports Server (NTRS)

Baker, J. R.; Mikhail, E. M.

1975-01-01

An investigation was conducted to define causes of geometric defects within digital multispectral scanner (MSS) data arrays, to analyze the resulting geometric errors, and to investigate restitution methods to correct or reduce these errors. Geometric transformation relationships for scanned data, from which collinearity equations may be derived, served as the basis of parametric methods of analysis and restitution of MSS digital data arrays. The linearization of these collinearity equations is presented. Algorithms considered for use in analysis and restitution included the MSS collinearity equations, piecewise polynomials based on linearized collinearity equations, and nonparametric algorithms. A proposed system for geometric analysis and restitution of MSS digital data arrays was used to evaluate these algorithms, utilizing actual MSS data arrays. It was shown that collinearity equations and nonparametric algorithms both yield acceptable results, but nonparametric algorithms possess definite advantages in computational efficiency. Piecewise polynomials were found to yield inferior results.

Factorizing the factorization - a spectral-element solver for elliptic equations with linear operation count

NASA Astrophysics Data System (ADS)

Huismann, Immo; Stiller, Jörg; Fröhlich, Jochen

2017-10-01

The paper proposes a novel factorization technique for static condensation of a spectral-element discretization matrix that yields a linear operation count of just 13N multiplications for the residual evaluation, where N is the total number of unknowns. In comparison to previous work it saves a factor larger than 3 and outpaces unfactored variants for all polynomial degrees. Using the new technique as a building block for a preconditioned conjugate gradient method yields linear scaling of the runtime with N which is demonstrated for polynomial degrees from 2 to 32. This makes the spectral-element method cost effective even for low polynomial degrees. Moreover, the dependence of the iterative solution on the element aspect ratio is addressed, showing only a slight increase in the number of iterations for aspect ratios up to 128. Hence, the solver is very robust for practical applications.

Higher-order Fourier analysis over finite fields and applications

NASA Astrophysics Data System (ADS)

Hatami, Pooya

Higher-order Fourier analysis is a powerful tool in the study of problems in additive and extremal combinatorics, for instance the study of arithmetic progressions in primes, where the traditional Fourier analysis comes short. In recent years, higher-order Fourier analysis has found multiple applications in computer science in fields such as property testing and coding theory. In this thesis, we develop new tools within this theory with several new applications such as a characterization theorem in algebraic property testing. One of our main contributions is a strong near-equidistribution result for regular collections of polynomials. The densities of small linear structures in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate the average, it suffices to know the joint distribution of the polynomials applied to the linear forms. We prove a near-equidistribution theorem that describes these distributions for the group F(n/p) when p is a fixed prime. This fundamental fact was previously known only under various extra assumptions about the linear forms or the field size. We use this near-equidistribution theorem to settle a conjecture of Gowers and Wolf on the true complexity of systems of linear forms. Our next application is towards a characterization of testable algebraic properties. We prove that every locally characterized affine-invariant property of functions f : F(n/p) → R with n∈ N, is testable. In fact, we prove that any such property P is proximity-obliviously testable. More generally, we show that any affine-invariant property that is closed under subspace restrictions and has "bounded complexity" is testable. We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. We discuss several notions of regularity which allow us to deduce algorithmic versions of various regularity lemmas for polynomials by Green and Tao and by Kaufman and Lovett. We show that our algorithmic regularity lemmas for polynomials imply algorithmic versions of several results relying on regularity, such as decoding Reed-Muller codes beyond the list decoding radius (for certain structured errors), and prescribed polynomial decompositions. Finally, motivated by the definition of Gowers norms, we investigate norms defined by different systems of linear forms. We give necessary conditions on the structure of systems of linear forms that define norms. We prove that such norms can be one of only two types, and assuming that |F p| is sufficiently large, they essentially are equivalent to either a Gowers norm or Lp norms.

Covariance functions for body weight from birth to maturity in Nellore cows.

PubMed

Boligon, A A; Mercadante, M E Z; Forni, S; Lôbo, R B; Albuquerque, L G

2010-03-01

The objective of this study was to estimate (co)variance functions using random regression models on Legendre polynomials for the analysis of repeated measures of BW from birth to adult age. A total of 82,064 records from 8,145 females were analyzed. Different models were compared. The models included additive direct and maternal effects, and animal and maternal permanent environmental effects as random terms. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of animal age (cubic regression) were considered as random covariables. Eight models with polynomials of third to sixth order were used to describe additive direct and maternal effects, and animal and maternal permanent environmental effects. Residual effects were modeled using 1 (i.e., assuming homogeneity of variances across all ages) or 5 age classes. The model with 5 classes was the best to describe the trajectory of residuals along the growth curve. The model including fourth- and sixth-order polynomials for additive direct and animal permanent environmental effects, respectively, and third-order polynomials for maternal genetic and maternal permanent environmental effects were the best. Estimates of (co)variance obtained with the multi-trait and random regression models were similar. Direct heritability estimates obtained with the random regression models followed a trend similar to that obtained with the multi-trait model. The largest estimates of maternal heritability were those of BW taken close to 240 d of age. In general, estimates of correlation between BW from birth to 8 yr of age decreased with increasing distance between ages.

Virasoro constraints and polynomial recursion for the linear Hodge integrals

NASA Astrophysics Data System (ADS)

Guo, Shuai; Wang, Gehao

2017-04-01

The Hodge tau-function is a generating function for the linear Hodge integrals. It is also a tau-function of the KP hierarchy. In this paper, we first present the Virasoro constraints for the Hodge tau-function in the explicit form of the Virasoro equations. The expression of our Virasoro constraints is simply a linear combination of the Virasoro operators, where the coefficients are restored from a power series for the Lambert W function. Then, using this result, we deduce a simple version of the Virasoro constraints for the linear Hodge partition function, where the coefficients are restored from the Gamma function. Finally, we establish the equivalence relation between the Virasoro constraints and polynomial recursion formula for the linear Hodge integrals.

Improved multivariate polynomial factoring algorithm

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wang, P.S.

1978-10-01

A new algorithm for factoring multivariate polynomials over the integers based on an algorithm by Wang and Rothschild is described. The new algorithm has improved strategies for dealing with the known problems of the original algorithm, namely, the leading coefficient problem, the bad-zero problem and the occurrence of extraneous factors. It has an algorithm for correctly predetermining leading coefficients of the factors. A new and efficient p-adic algorithm named EEZ is described. Bascially it is a linearly convergent variable-by-variable parallel construction. The improved algorithm is generally faster and requires less store then the original algorithm. Machine examples with comparative timingmore » are included.« less

Maximum Marginal Likelihood Estimation of a Monotonic Polynomial Generalized Partial Credit Model with Applications to Multiple Group Analysis.

PubMed

Falk, Carl F; Cai, Li

2016-06-01

We present a semi-parametric approach to estimating item response functions (IRF) useful when the true IRF does not strictly follow commonly used functions. Our approach replaces the linear predictor of the generalized partial credit model with a monotonic polynomial. The model includes the regular generalized partial credit model at the lowest order polynomial. Our approach extends Liang's (A semi-parametric approach to estimate IRFs, Unpublished doctoral dissertation, 2007) method for dichotomous item responses to the case of polytomous data. Furthermore, item parameter estimation is implemented with maximum marginal likelihood using the Bock-Aitkin EM algorithm, thereby facilitating multiple group analyses useful in operational settings. Our approach is demonstrated on both educational and psychological data. We present simulation results comparing our approach to more standard IRF estimation approaches and other non-parametric and semi-parametric alternatives.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lue Xing; Sun Kun; Wang Pan

In the framework of Bell-polynomial manipulations, under investigation hereby are three single-field bilinearizable equations: the (1+1)-dimensional shallow water wave model, Boiti-Leon-Manna-Pempinelli model, and (2+1)-dimensional Sawada-Kotera model. Based on the concept of scale invariance, a direct and unifying Bell-polynomial scheme is employed to achieve the Baecklund transformations and Lax pairs associated with those three soliton equations. Note that the Bell-polynomial expressions and Bell-polynomial-typed Baecklund transformations for those three soliton equations can be, respectively, cast into the bilinear equations and bilinear Baecklund transformations with symbolic computation. Consequently, it is also shown that the Bell-polynomial-typed Baecklund transformations can be linearized into the correspondingmore » Lax pairs.« less

Polynomials with Restricted Coefficients and Their Applications

DTIC Science & Technology

1987-01-01

sums of exponentials of quadratics, he reduced such ýzums to exponentials of linears (geometric sums!) by simplg multiplying by their conjugates...n, the same algebraic manipulations as before lead to rn V`-~ v ie ? --8-- el4V’ .fk ts with 𔄃 = a+(2r+l)t, A = a+(2r+2m+l)t. To estimate the right...coefficients. These random polynomials represent the deviation in frequency response of a linear , equispaced antenna array cauised by coefficient

Nodal-line dynamics via exact polynomial solutions for coherent waves traversing aberrated imaging systems.

PubMed

Paganin, David M; Beltran, Mario A; Petersen, Timothy C

2018-03-01

We obtain exact polynomial solutions for two-dimensional coherent complex scalar fields propagating through arbitrary aberrated shift-invariant linear imaging systems. These solutions are used to model nodal-line dynamics of coherent fields output by such systems.

Algebraic approach to solve ttbar dilepton equations

NASA Astrophysics Data System (ADS)

Sonnenschein, Lars

2006-01-01

The set of non-linear equations describing the Standard Model kinematics of the top quark an- tiqark production system in the dilepton decay channel has at most a four-fold ambiguity due to two not fully reconstructed neutrinos. Its most precise and robust solution is of major importance for measurements of top quark properties like the top quark mass and t t spin correlations. Simple algebraic operations allow to transform the non-linear equations into a system of two polynomial equations with two unknowns. These two polynomials of multidegree eight can in turn be an- alytically reduced to one polynomial with one unknown by means of resultants. The obtained univariate polynomial is of degree sixteen and the coefficients are free of any singularity. The number of its real solutions is determined analytically by means of Sturm’s theorem, which is as well used to isolate each real solution into a unique pairwise disjoint interval. The solutions are polished by seeking the sign change of the polynomial in a given interval through binary brack- eting. Further a new Ansatz - exploiting an accidental cancelation in the process of transforming the equations - is presented. It permits to transform the initial system of equations into two poly- nomial equations with two unknowns. These two polynomials of multidegree two can be reduced to one univariate polynomial of degree four by means of resultants. The obtained quartic equation can be solved analytically. The analytical solution has singularities which can be circumvented by the algebraic approach described above.

Evaluation of Piecewise Polynomial Equations for Two Types of Thermocouples

PubMed Central

Chen, Andrew; Chen, Chiachung

2013-01-01

Thermocouples are the most frequently used sensors for temperature measurement because of their wide applicability, long-term stability and high reliability. However, one of the major utilization problems is the linearization of the transfer relation between temperature and output voltage of thermocouples. The linear calibration equation and its modules could be improved by using regression analysis to help solve this problem. In this study, two types of thermocouple and five temperature ranges were selected to evaluate the fitting agreement of different-order polynomial equations. Two quantitative criteria, the average of the absolute error values |e|ave and the standard deviation of calibration equation estd, were used to evaluate the accuracy and precision of these calibrations equations. The optimal order of polynomial equations differed with the temperature range. The accuracy and precision of the calibration equation could be improved significantly with an adequate higher degree polynomial equation. The technique could be applied with hardware modules to serve as an intelligent sensor for temperature measurement. PMID:24351627

Matrix-Free Polynomial-Based Nonlinear Least Squares Optimized Preconditioning and its Application to Discontinuous Galerkin Discretizations of the Euler Equations

DTIC Science & Technology

2015-06-01

cient parallel code for applying the operator. Our method constructs a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show...apply the underlying operator. Such a preconditioner can be very attractive in scenarios where one has a highly efficient parallel code for applying...repeatedly solve a large system of linear equations where one has an extremely fast parallel code for applying an underlying fixed linear operator

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

PubMed

Richardson, Megan; Lambers, James V

2016-01-01

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

Frequency domain system identification methods - Matrix fraction description approach

NASA Technical Reports Server (NTRS)

Horta, Luca G.; Juang, Jer-Nan

1993-01-01

This paper presents the use of matrix fraction descriptions for least-squares curve fitting of the frequency spectra to compute two matrix polynomials. The matrix polynomials are intermediate step to obtain a linearized representation of the experimental transfer function. Two approaches are presented: first, the matrix polynomials are identified using an estimated transfer function; second, the matrix polynomials are identified directly from the cross/auto spectra of the input and output signals. A set of Markov parameters are computed from the polynomials and subsequently realization theory is used to recover a minimum order state space model. Unevenly spaced frequency response functions may be used. Results from a simple numerical example and an experiment are discussed to highlight some of the important aspect of the algorithm.

Parameterized LMI Based Diagonal Dominance Compensator Study for Polynomial Linear Parameter Varying System

NASA Astrophysics Data System (ADS)

Han, Xiaobao; Li, Huacong; Jia, Qiusheng

2017-12-01

For dynamic decoupling of polynomial linear parameter varying(PLPV) system, a robust dominance pre-compensator design method is given. The parameterized precompensator design problem is converted into an optimal problem constrained with parameterized linear matrix inequalities(PLMI) by using the conception of parameterized Lyapunov function(PLF). To solve the PLMI constrained optimal problem, the precompensator design problem is reduced into a normal convex optimization problem with normal linear matrix inequalities (LMI) constraints on a new constructed convex polyhedron. Moreover, a parameter scheduling pre-compensator is achieved, which satisfies robust performance and decoupling performances. Finally, the feasibility and validity of the robust diagonal dominance pre-compensator design method are verified by the numerical simulation on a turbofan engine PLPV model.

Short communication: Genetic variation of saturated fatty acids in Holsteins in the Walloon region of Belgium.

PubMed

Arnould, V M-R; Hammami, H; Soyeurt, H; Gengler, N

2010-09-01

Random regression test-day models using Legendre polynomials are commonly used for the estimation of genetic parameters and genetic evaluation for test-day milk production traits. However, some researchers have reported that these models present some undesirable properties such as the overestimation of variances at the edges of lactation. Describing genetic variation of saturated fatty acids expressed in milk fat might require the testing of different models. Therefore, 3 different functions were used and compared to take into account the lactation curve: (1) Legendre polynomials with the same order as currently applied for genetic model for production traits; 2) linear splines with 10 knots; and 3) linear splines with the same 10 knots reduced to 3 parameters. The criteria used were Akaike's information and Bayesian information criteria, percentage square biases, and log-likelihood function. These criteria indentified Legendre polynomials and linear splines with 10 knots reduced to 3 parameters models as the most useful. Reducing more complex models using eigenvalues seemed appealing because the resulting models are less time demanding and can reduce convergence difficulties, because convergence properties also seemed to be improved. Finally, the results showed that the reduced spline model was very similar to the Legendre polynomials model. Copyright (c) 2010 American Dairy Science Association. Published by Elsevier Inc. All rights reserved.

Segmented Polynomial Models in Quasi-Experimental Research.

ERIC Educational Resources Information Center

Wasik, John L.

1981-01-01

The use of segmented polynomial models is explained. Examples of design matrices of dummy variables are given for the least squares analyses of time series and discontinuity quasi-experimental research designs. Linear combinations of dummy variable vectors appear to provide tests of effects in the two quasi-experimental designs. (Author/BW)

Computing Galois Groups of Eisenstein Polynomials Over P-adic Fields

NASA Astrophysics Data System (ADS)

Milstead, Jonathan

The most efficient algorithms for computing Galois groups of polynomials over global fields are based on Stauduhar's relative resolvent method. These methods are not directly generalizable to the local field case, since they require a field that contains the global field in which all roots of the polynomial can be approximated. We present splitting field-independent methods for computing the Galois group of an Eisenstein polynomial over a p-adic field. Our approach is to combine information from different disciplines. We primarily, make use of the ramification polygon of the polynomial, which is the Newton polygon of a related polynomial. This allows us to quickly calculate several invariants that serve to reduce the number of possible Galois groups. Algorithms by Greve and Pauli very efficiently return the Galois group of polynomials where the ramification polygon consists of one segment as well as information about the subfields of the stem field. Second, we look at the factorization of linear absolute resolvents to further narrow the pool of possible groups.

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.

Kirchhoff index of linear hexagonal chains

NASA Astrophysics Data System (ADS)

Yang, Yujun; Zhang, Heping

The resistance distance rij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain Ln consists of the Laplacian spectrum of path P2n+1 and eigenvalues of a symmetric tridiagonal matrix of order 2n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed-form formula for Kirchhoff index of Ln is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of Ln is approximately to one half of its Wiener index. Finally, we show that holds for all graphs G in a class of graphs including Ln.0

a Comparison Study of Different Kernel Functions for Svm-Based Classification of Multi-Temporal Polarimetry SAR Data

NASA Astrophysics Data System (ADS)

Yekkehkhany, B.; Safari, A.; Homayouni, S.; Hasanlou, M.

2014-10-01

In this paper, a framework is developed based on Support Vector Machines (SVM) for crop classification using polarimetric features extracted from multi-temporal Synthetic Aperture Radar (SAR) imageries. The multi-temporal integration of data not only improves the overall retrieval accuracy but also provides more reliable estimates with respect to single-date data. Several kernel functions are employed and compared in this study for mapping the input space to higher Hilbert dimension space. These kernel functions include linear, polynomials and Radial Based Function (RBF). The method is applied to several UAVSAR L-band SAR images acquired over an agricultural area near Winnipeg, Manitoba, Canada. In this research, the temporal alpha features of H/A/α decomposition method are used in classification. The experimental tests show an SVM classifier with RBF kernel for three dates of data increases the Overall Accuracy (OA) to up to 3% in comparison to using linear kernel function, and up to 1% in comparison to a 3rd degree polynomial kernel function.

Linear decomposition approach for a class of nonconvex programming problems.

PubMed

Shen, Peiping; Wang, Chunfeng

2017-01-01

This paper presents a linear decomposition approach for a class of nonconvex programming problems by dividing the input space into polynomially many grids. It shows that under certain assumptions the original problem can be transformed and decomposed into a polynomial number of equivalent linear programming subproblems. Based on solving a series of liner programming subproblems corresponding to those grid points we can obtain the near-optimal solution of the original problem. Compared to existing results in the literature, the proposed algorithm does not require the assumptions of quasi-concavity and differentiability of the objective function, and it differs significantly giving an interesting approach to solving the problem with a reduced running time.

Re-entry vehicle shape for enhanced performance

NASA Technical Reports Server (NTRS)

Brown, James L. (Inventor); Garcia, Joseph A. (Inventor); Prabhu, Dinesh K. (Inventor)

2008-01-01

A convex shell structure for enhanced aerodynamic performance and/or reduced heat transfer requirements for a space vehicle that re-enters an atmosphere. The structure has a fore-body, an aft-body, a longitudinal axis and a transverse cross sectional shape, projected on a plane containing the longitudinal axis, that includes: first and second linear segments, smoothly joined at a first end of each the first and second linear segments to an end of a third linear segment by respective first and second curvilinear segments; and a fourth linear segment, joined to a second end of each of the first and second segments by curvilinear segments, including first and second ellipses having unequal ellipse parameters. The cross sectional shape is non-symmetric about the longitudinal axis. The fourth linear segment can be replaced by a sum of one or more polynomials, trigonometric functions or other functions satisfying certain constraints.

Local polynomial chaos expansion for linear differential equations with high dimensional random inputs

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chen, Yi; Jakeman, John; Gittelson, Claude

2015-01-08

In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. Furthermore, the local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained frommore » the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In our paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.« less

Explaining Support Vector Machines: A Color Based Nomogram

PubMed Central

Van Belle, Vanya; Van Calster, Ben; Van Huffel, Sabine; Suykens, Johan A. K.; Lisboa, Paulo

2016-01-01

Problem setting Support vector machines (SVMs) are very popular tools for classification, regression and other problems. Due to the large choice of kernels they can be applied with, a large variety of data can be analysed using these tools. Machine learning thanks its popularity to the good performance of the resulting models. However, interpreting the models is far from obvious, especially when non-linear kernels are used. Hence, the methods are used as black boxes. As a consequence, the use of SVMs is less supported in areas where interpretability is important and where people are held responsible for the decisions made by models. Objective In this work, we investigate whether SVMs using linear, polynomial and RBF kernels can be explained such that interpretations for model-based decisions can be provided. We further indicate when SVMs can be explained and in which situations interpretation of SVMs is (hitherto) not possible. Here, explainability is defined as the ability to produce the final decision based on a sum of contributions which depend on one single or at most two input variables. Results Our experiments on simulated and real-life data show that explainability of an SVM depends on the chosen parameter values (degree of polynomial kernel, width of RBF kernel and regularization constant). When several combinations of parameter values yield the same cross-validation performance, combinations with a lower polynomial degree or a larger kernel width have a higher chance of being explainable. Conclusions This work summarizes SVM classifiers obtained with linear, polynomial and RBF kernels in a single plot. Linear and polynomial kernels up to the second degree are represented exactly. For other kernels an indication of the reliability of the approximation is presented. The complete methodology is available as an R package and two apps and a movie are provided to illustrate the possibilities offered by the method. PMID:27723811

Vector-valued Jack polynomials and wavefunctions on the torus

NASA Astrophysics Data System (ADS)

Dunkl, Charles F.

2017-06-01

The Hamiltonian of the quantum Calogero-Sutherland model of N identical particles on the circle with 1/r 2 interactions has eigenfunctions consisting of Jack polynomials times the base state. By use of the generalized Jack polynomials taking values in modules of the symmetric group and the matrix solution of a system of linear differential equations one constructs novel eigenfunctions of the Hamiltonian. Like the usual wavefunctions each eigenfunction determines a symmetric probability density on the N-torus. The construction applies to any irreducible representation of the symmetric group. The methods depend on the theory of generalized Jack polynomials due to Griffeth, and the Yang-Baxter graph approach of Luque and the author.

Free torsional vibrations of tapered cantilever I-beams

NASA Astrophysics Data System (ADS)

Rao, C. Kameswara; Mirza, S.

1988-08-01

Torsional vibration characteristics of linearly tapered cantilever I-beams have been studied by using the Galerkin finite element method. A third degree polynomial is assumed for the angle of twist. The analysis presented is valid for long beams and includes the effect of warping. The individual as well as combined effects of linear tapers in the width of the flanges and the depth of the web on the torsional vibration of cantilever I-beams are investigated. Numerical results generated for various values of taper ratios are presented in graphical form.

Bin Packing, Number Balancing, and Rescaling Linear Programs

NASA Astrophysics Data System (ADS)

Hoberg, Rebecca

This thesis deals with several important algorithmic questions using techniques from diverse areas including discrepancy theory, machine learning and lattice theory. In Chapter 2, we construct an improved approximation algorithm for a classical NP-complete problem, the bin packing problem. In this problem, the goal is to pack items of sizes si ∈ [0,1] into as few bins as possible, where a set of items fits into a bin provided the sum of the item sizes is at most one. We give a polynomial-time rounding scheme for a standard linear programming relaxation of the problem, yielding a packing that uses at most OPT + O(log OPT) bins. This makes progress towards one of the "10 open problems in approximation algorithms" stated in the book of Shmoys and Williamson. In fact, based on related combinatorial lower bounds, Rothvoss conjectures that theta(logOPT) may be a tight bound on the additive integrality gap of this LP relaxation. In Chapter 3, we give a new polynomial-time algorithm for linear programming. Our algorithm is based on the multiplicative weights update (MWU) method, which is a general framework that is currently of great interest in theoretical computer science. An algorithm for linear programming based on MWU was known previously, but was not polynomial time--we remedy this by alternating between a MWU phase and a rescaling phase. The rescaling methods we introduce improve upon previous methods by reducing the number of iterations needed until one can rescale, and they can be used for any algorithm with a similar rescaling structure. Finally, we note that the MWU phase of the algorithm has a simple interpretation as gradient descent of a particular potential function, and we show we can speed up this phase by walking in a direction that decreases both the potential function and its gradient. In Chapter 4, we show that an approximate oracle for Minkowski's Theorem gives an approximate oracle for the number balancing problem, and conversely. Number balancing is the problem of minimizing | 〈a,x〉 | over x ∈ {-1,0,1}n \\ { 0}, given a ∈ [0,1]n. While an application of the pigeonhole principle shows that there always exists x with | 〈a,x〉| ≤ O(√ n/2n), the best known algorithm only guarantees |〈a,x〉| ≤ 2-ntheta(log n). We show that an oracle for Minkowski's Theorem with approximation factor rho would give an algorithm for NBP that guarantees | 〈a,x〉 | ≤ 2-ntheta(1/rho). In particular, this would beat the bound of Karmarkar and Karp provided rho ≤ O(logn/loglogn). In the other direction, we prove that any polynomial time algorithm for NBP that guarantees a solution of difference at most 2√n/2 n would give a polynomial approximation for Minkowski as well as a polynomial factor approximation algorithm for the Shortest Vector Problem.

Simulation of Shallow Water Jets with a Unified Element-based Continuous/Discontinuous Galerkin Model with Grid Flexibility on the Sphere

DTIC Science & Technology

2013-01-01

is the derivative of the N th-order Legendre polynomial . Given these definitions, the one-dimensional Lagrange polynomials hi(ξ) are hi(ξ) = − 1 N(N...2. Detail of one interface patch in the northern hemisphere. The high-order Legendre -Gauss-Lobatto (LGL) points are added to the linear grid by...smaller ones by a Lagrange polynomial of order nI . The number of quadrilateral elements and grid points of the final grid are then given by Np = 6(N

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

NASA Technical Reports Server (NTRS)

Johnson, Duane

1996-01-01

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

Consensus seeking in a network of discrete-time linear agents with communication noises

NASA Astrophysics Data System (ADS)

Wang, Yunpeng; Cheng, Long; Hou, Zeng-Guang; Tan, Min; Zhou, Chao; Wang, Ming

2015-07-01

This paper studies the mean square consensus of discrete-time linear time-invariant multi-agent systems with communication noises. A distributed consensus protocol, which is composed of the agent's own state feedback and the relative states between the agent and its neighbours, is proposed. A time-varying consensus gain a[k] is applied to attenuate the effect of noises which inherits in the inaccurate measurement of relative states with neighbours. A polynomial, namely 'parameter polynomial', is constructed. And its coefficients are the parameters in the feedback gain vector of the proposed protocol. It turns out that the parameter polynomial plays an important role in guaranteeing the consensus of linear multi-agent systems. By the proposed protocol, necessary and sufficient conditions for mean square consensus are presented under different topology conditions: (1) if the communication topology graph has a spanning tree and every node in the graph has at least one parent node, then the mean square consensus can be achieved if and only if ∑∞k = 0a[k] = ∞, ∑∞k = 0a2[k] < ∞ and all roots of the parameter polynomial are in the unit circle; (2) if the communication topology graph has a spanning tree and there exits one node without any parent node (the leader-follower case), then the mean square consensus can be achieved if and only if ∑∞k = 0a[k] = ∞, limk → ∞a[k] = 0 and all roots of the parameter polynomial are in the unit circle; (3) if the communication topology graph does not have a spanning tree, then the mean square consensus can never be achieved. Finally, one simulation example on the multiple aircrafts system is provided to validate the theoretical analysis.

Local invariants vanishing on stationary horizons: a diagnostic for locating black holes.

PubMed

Page, Don N; Shoom, Andrey A

2015-04-10

Inspired by the example of Abdelqader and Lake for the Kerr metric, we construct local scalar polynomial curvature invariants that vanish on the horizon of any stationary black hole: the squared norms of the wedge products of n linearly independent gradients of scalar polynomial curvature invariants, where n is the local cohomogeneity of the spacetime.

Enhancing sparsity of Hermite polynomial expansions by iterative rotations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yang, Xiu; Lei, Huan; Baker, Nathan A.

2016-02-01

Compressive sensing has become a powerful addition to uncertainty quantification in recent years. This paper identifies new bases for random variables through linear mappings such that the representation of the quantity of interest is more sparse with new basis functions associated with the new random variables. This sparsity increases both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. Specifically, we consider rotation- based linear mappings which are determined iteratively for Hermite polynomial expansions. We demonstrate the effectiveness of the new method with applications in solving stochastic partial differential equations and high-dimensional (O(100)) problems.

Symbolic discrete event system specification

NASA Technical Reports Server (NTRS)

Zeigler, Bernard P.; Chi, Sungdo

1992-01-01

Extending discrete event modeling formalisms to facilitate greater symbol manipulation capabilities is important to further their use in intelligent control and design of high autonomy systems. An extension to the DEVS formalism that facilitates symbolic expression of event times by extending the time base from the real numbers to the field of linear polynomials over the reals is defined. A simulation algorithm is developed to generate the branching trajectories resulting from the underlying nondeterminism. To efficiently manage symbolic constraints, a consistency checking algorithm for linear polynomial constraints based on feasibility checking algorithms borrowed from linear programming has been developed. The extended formalism offers a convenient means to conduct multiple, simultaneous explorations of model behaviors. Examples of application are given with concentration on fault model analysis.

Developing a reversible rapid coordinate transformation model for the cylindrical projection

NASA Astrophysics Data System (ADS)

Ye, Si-jing; Yan, Tai-lai; Yue, Yan-li; Lin, Wei-yan; Li, Lin; Yao, Xiao-chuang; Mu, Qin-yun; Li, Yong-qin; Zhu, De-hai

2016-04-01

Numerical models are widely used for coordinate transformations. However, in most numerical models, polynomials are generated to approximate "true" geographic coordinates or plane coordinates, and one polynomial is hard to make simultaneously appropriate for both forward and inverse transformations. As there is a transformation rule between geographic coordinates and plane coordinates, how accurate and efficient is the calculation of the coordinate transformation if we construct polynomials to approximate the transformation rule instead of "true" coordinates? In addition, is it preferable to compare models using such polynomials with traditional numerical models with even higher exponents? Focusing on cylindrical projection, this paper reports on a grid-based rapid numerical transformation model - a linear rule approximation model (LRA-model) that constructs linear polynomials to approximate the transformation rule and uses a graticule to alleviate error propagation. Our experiments on cylindrical projection transformation between the WGS 84 Geographic Coordinate System (EPSG 4326) and the WGS 84 UTM ZONE 50N Plane Coordinate System (EPSG 32650) with simulated data demonstrate that the LRA-model exhibits high efficiency, high accuracy, and high stability; is simple and easy to use for both forward and inverse transformations; and can be applied to the transformation of a large amount of data with a requirement of high calculation efficiency. Furthermore, the LRA-model exhibits advantages in terms of calculation efficiency, accuracy and stability for coordinate transformations, compared to the widely used hyperbolic transformation model.

Assessing the suitability of fractional polynomial methods in health services research: a perspective on the categorization epidemic.

PubMed

Williams, Jennifer Stewart

2011-07-01

To show how fractional polynomial methods can usefully replace the practice of arbitrarily categorizing data in epidemiology and health services research. A health service setting is used to illustrate a structured and transparent way of representing non-linear data without arbitrary grouping. When age is a regressor its effects on an outcome will be interpreted differently depending upon the placing of cutpoints or the use of a polynomial transformation. Although it is common practice, categorization comes at a cost. Information is lost, and accuracy and statistical power reduced, leading to spurious statistical interpretation of the data. The fractional polynomial method is widely supported by statistical software programs, and deserves greater attention and use.

Piecewise polynomial representations of genomic tracks.

PubMed

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/.

An algorithmic approach to solving polynomial equations associated with quantum circuits

NASA Astrophysics Data System (ADS)

Gerdt, V. P.; Zinin, M. V.

2009-12-01

In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F 2 to the canonical triangular form called lexicographical Gröbner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F 2 whose variables also take values in F 2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Gröbner bases over F 2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Gröbner bases over F 2.

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

PubMed

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

2013-01-01

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

Fitness Probability Distribution of Bit-Flip Mutation.

PubMed

Chicano, Francisco; Sutton, Andrew M; Whitley, L Darrell; Alba, Enrique

2015-01-01

Bit-flip mutation is a common mutation operator for evolutionary algorithms applied to optimize functions over binary strings. In this paper, we develop results from the theory of landscapes and Krawtchouk polynomials to exactly compute the probability distribution of fitness values of a binary string undergoing uniform bit-flip mutation. We prove that this probability distribution can be expressed as a polynomial in p, the probability of flipping each bit. We analyze these polynomials and provide closed-form expressions for an easy linear problem (Onemax), and an NP-hard problem, MAX-SAT. We also discuss a connection of the results with runtime analysis.

Polynomial approximation of Poincare maps for Hamiltonian system

NASA Technical Reports Server (NTRS)

Froeschle, Claude; Petit, Jean-Marc

1992-01-01

Different methods are proposed and tested for transforming a non-linear differential system, and more particularly a Hamiltonian one, into a map without integrating the whole orbit as in the well-known Poincare return map technique. We construct piecewise polynomial maps by coarse-graining the phase-space surface of section into parallelograms and using either only values of the Poincare maps at the vertices or also the gradient information at the nearest neighbors to define a polynomial approximation within each cell. The numerical experiments are in good agreement with both the real symplectic and Poincare maps.

The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

PubMed

Khader, M M

2013-10-01

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

Existence of entire solutions of some non-linear differential-difference equations.

PubMed

Chen, Minfeng; Gao, Zongsheng; Du, Yunfei

2017-01-01

In this paper, we investigate the admissible entire solutions of finite order of the differential-difference equations [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] are two non-zero polynomials, [Formula: see text] is a polynomial and [Formula: see text]. In addition, we investigate the non-existence of entire solutions of finite order of the differential-difference equation [Formula: see text], where [Formula: see text], [Formula: see text] are two non-constant polynomials, [Formula: see text], m , n are positive integers and satisfy [Formula: see text] except for [Formula: see text], [Formula: see text].

Constructing general partial differential equations using polynomial and neural networks.

PubMed

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. Copyright © 2015 Elsevier Ltd. All rights reserved.

Stability analysis of fuzzy parametric uncertain systems.

PubMed

Bhiwani, R J; Patre, B M

2011-10-01

In this paper, the determination of stability margin, gain and phase margin aspects of fuzzy parametric uncertain systems are dealt. The stability analysis of uncertain linear systems with coefficients described by fuzzy functions is studied. A complexity reduced technique for determining the stability margin for FPUS is proposed. The method suggested is dependent on the order of the characteristic polynomial. In order to find the stability margin of interval polynomials of order less than 5, it is not always necessary to determine and check all four Kharitonov's polynomials. It has been shown that, for determining stability margin of FPUS of order five, four, and three we require only 3, 2, and 1 Kharitonov's polynomials respectively. Only for sixth and higher order polynomials, a complete set of Kharitonov's polynomials are needed to determine the stability margin. Thus for lower order systems, the calculations are reduced to a large extent. This idea has been extended to determine the stability margin of fuzzy interval polynomials. It is also shown that the gain and phase margin of FPUS can be determined analytically without using graphical techniques. Copyright © 2011 ISA. Published by Elsevier Ltd. All rights reserved.

Polynomial Monogamy Relations for Entanglement Negativity.

PubMed

Allen, Grant W; Meyer, David A

2017-02-24

The notion of nonclassical correlations is a powerful contrivance for explaining phenomena exhibited in quantum systems. It is well known, however, that quantum systems are not free to explore arbitrary correlations-the church of the smaller Hilbert space only accepts monogamous congregants. We demonstrate how to characterize the limits of what is quantum mechanically possible with a computable measure, entanglement negativity. We show that negativity only saturates the standard linear monogamy inequality in trivial cases implied by its monotonicity under local operations and classical communication, and derive a necessary and sufficient inequality which, for the first time, is a nonlinear higher degree polynomial. For very large quantum systems, we prove that the negativity can be distributed at least linearly for the tightest constraint and conjecture that it is at most linear.

Polynomial Monogamy Relations for Entanglement Negativity

NASA Astrophysics Data System (ADS)

Allen, Grant W.; Meyer, David A.

2017-02-01

The notion of nonclassical correlations is a powerful contrivance for explaining phenomena exhibited in quantum systems. It is well known, however, that quantum systems are not free to explore arbitrary correlations—the church of the smaller Hilbert space only accepts monogamous congregants. We demonstrate how to characterize the limits of what is quantum mechanically possible with a computable measure, entanglement negativity. We show that negativity only saturates the standard linear monogamy inequality in trivial cases implied by its monotonicity under local operations and classical communication, and derive a necessary and sufficient inequality which, for the first time, is a nonlinear higher degree polynomial. For very large quantum systems, we prove that the negativity can be distributed at least linearly for the tightest constraint and conjecture that it is at most linear.

Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

NASA Astrophysics Data System (ADS)

Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

2017-04-01

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

A polynomial primal-dual Dikin-type algorithm for linear programming

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jansen, B.; Roos, R.; Terlaky, T.

1994-12-31

We present a new primal-dual affine scaling method for linear programming. The search direction is obtained by using Dikin`s original idea: minimize the objective function (which is the duality gap in a primal-dual algorithm) over a suitable ellipsoid. The search direction has no obvious relationship with the directions proposed in the literature so far. It guarantees a significant decrease in the duality gap in each iteration, and at the same time drives the iterates to the central path. The method admits a polynomial complexity bound that is better than the one for Monteiro et al.`s original primal-dual affine scaling method.

Efficient Craig Interpolation for Linear Diophantine (Dis)Equations and Linear Modular Equations

DTIC Science & Technology

2008-02-01

Craig interpolants has enabled the development of powerful hardware and software model checking techniques. Efficient algorithms are known for computing...interpolants in rational and real linear arithmetic. We focus on subsets of integer linear arithmetic. Our main results are polynomial time algorithms ...congruences), and linear diophantine disequations. We show the utility of the proposed interpolation algorithms for discovering modular/divisibility predicates

Detection and correction of laser induced breakdown spectroscopy spectral background based on spline interpolation method

NASA Astrophysics Data System (ADS)

Tan, Bing; Huang, Min; Zhu, Qibing; Guo, Ya; Qin, Jianwei

2017-12-01

Laser-induced breakdown spectroscopy (LIBS) is an analytical technique that has gained increasing attention because of many applications. The production of continuous background in LIBS is inevitable because of factors associated with laser energy, gate width, time delay, and experimental environment. The continuous background significantly influences the analysis of the spectrum. Researchers have proposed several background correction methods, such as polynomial fitting, Lorenz fitting and model-free methods. However, less of them apply these methods in the field of LIBS Technology, particularly in qualitative and quantitative analyses. This study proposes a method based on spline interpolation for detecting and estimating the continuous background spectrum according to its smooth property characteristic. Experiment on the background correction simulation indicated that, the spline interpolation method acquired the largest signal-to-background ratio (SBR) over polynomial fitting, Lorenz fitting and model-free method after background correction. These background correction methods all acquire larger SBR values than that acquired before background correction (The SBR value before background correction is 10.0992, whereas the SBR values after background correction by spline interpolation, polynomial fitting, Lorentz fitting, and model-free methods are 26.9576, 24.6828, 18.9770, and 25.6273 respectively). After adding random noise with different kinds of signal-to-noise ratio to the spectrum, spline interpolation method acquires large SBR value, whereas polynomial fitting and model-free method obtain low SBR values. All of the background correction methods exhibit improved quantitative results of Cu than those acquired before background correction (The linear correlation coefficient value before background correction is 0.9776. Moreover, the linear correlation coefficient values after background correction using spline interpolation, polynomial fitting, Lorentz fitting, and model-free methods are 0.9998, 0.9915, 0.9895, and 0.9940 respectively). The proposed spline interpolation method exhibits better linear correlation and smaller error in the results of the quantitative analysis of Cu compared with polynomial fitting, Lorentz fitting and model-free methods, The simulation and quantitative experimental results show that the spline interpolation method can effectively detect and correct the continuous background.

Mean-square state and parameter estimation for stochastic linear systems with Gaussian and Poisson noises

NASA Astrophysics Data System (ADS)

Basin, M.; Maldonado, J. J.; Zendejo, O.

2016-07-01

This paper proposes new mean-square filter and parameter estimator design for linear stochastic systems with unknown parameters over linear observations, where unknown parameters are considered as combinations of Gaussian and Poisson white noises. The problem is treated by reducing the original problem to a filtering problem for an extended state vector that includes parameters as additional states, modelled as combinations of independent Gaussian and Poisson processes. The solution to this filtering problem is based on the mean-square filtering equations for incompletely polynomial states confused with Gaussian and Poisson noises over linear observations. The resulting mean-square filter serves as an identifier for the unknown parameters. Finally, a simulation example shows effectiveness of the proposed mean-square filter and parameter estimator.

Exploiting structure: Introduction and motivation

NASA Technical Reports Server (NTRS)

Xu, Zhong Ling

1993-01-01

Research activities performed during the period of 29 June 1993 through 31 Aug. 1993 are summarized. The Robust Stability of Systems where transfer function or characteristic polynomial are multilinear affine functions of parameters of interest in two directions, Algorithmic and Theoretical, was developed. In the algorithmic direction, a new approach that reduces the computational burden of checking the robust stability of the system with multilinear uncertainty is found. This technique is called 'Stability by linear process.' In fact, the 'Stability by linear process' described gives an algorithm. In analysis, we obtained a robustness criterion for the family of polynomials with coefficients of multilinear affine function in the coefficient space and obtained the result for the robust stability of diamond families of polynomials with complex coefficients also. We obtained the limited results for SPR design and we provide a framework for solving ACS. Finally, copies of the outline of our results are provided in the appendix. Also, there is an administration issue in the appendix.

New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some connection formulae

NASA Astrophysics Data System (ADS)

Abd-Elhameed, W. M.

2017-07-01

In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type _4F3(1) . With the aid of some standard reduction formulae such as Pfaff-Saalschütz's and Watson's identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method.

Comparison of Linear and Non-linear Regression Analysis to Determine Pulmonary Pressure in Hyperthyroidism.

PubMed

Scarneciu, Camelia C; Sangeorzan, Livia; Rus, Horatiu; Scarneciu, Vlad D; Varciu, Mihai S; Andreescu, Oana; Scarneciu, Ioan

2017-01-01

This study aimed at assessing the incidence of pulmonary hypertension (PH) at newly diagnosed hyperthyroid patients and at finding a simple model showing the complex functional relation between pulmonary hypertension in hyperthyroidism and the factors causing it. The 53 hyperthyroid patients (H-group) were evaluated mainly by using an echocardiographical method and compared with 35 euthyroid (E-group) and 25 healthy people (C-group). In order to identify the factors causing pulmonary hypertension the statistical method of comparing the values of arithmetical means is used. The functional relation between the two random variables (PAPs and each of the factors determining it within our research study) can be expressed by linear or non-linear function. By applying the linear regression method described by a first-degree equation the line of regression (linear model) has been determined; by applying the non-linear regression method described by a second degree equation, a parabola-type curve of regression (non-linear or polynomial model) has been determined. We made the comparison and the validation of these two models by calculating the determination coefficient (criterion 1), the comparison of residuals (criterion 2), application of AIC criterion (criterion 3) and use of F-test (criterion 4). From the H-group, 47% have pulmonary hypertension completely reversible when obtaining euthyroidism. The factors causing pulmonary hypertension were identified: previously known- level of free thyroxin, pulmonary vascular resistance, cardiac output; new factors identified in this study- pretreatment period, age, systolic blood pressure. According to the four criteria and to the clinical judgment, we consider that the polynomial model (graphically parabola- type) is better than the linear one. The better model showing the functional relation between the pulmonary hypertension in hyperthyroidism and the factors identified in this study is given by a polynomial equation of second degree where the parabola is its graphical representation.

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

NASA Astrophysics Data System (ADS)

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

2018-01-01

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

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.

Global asymptotic stabilisation of rational dynamical systems based on solving BMI

NASA Astrophysics Data System (ADS)

Esmaili, Farhad; Kamyad, A. V.; Jahed-Motlagh, Mohammad Reza; Pariz, Naser

2017-08-01

In this paper, the global asymptotic stabiliser design of rational systems is studied in detail. To develop the idea, the state equations of the system are transformed to a new coordinate via polynomial transformation and the state feedback control law. This in turn is followed by the satisfaction of the linear growth condition (i.e. Lipschitz at zero). Based on a linear matrix inequality solution, the system in the new coordinate is globally asymptotically stabilised and then, leading to the global asymptotic stabilisation of the primary system. The polynomial transformation coefficients are derived by solving the bilinear matrix inequality problem. To confirm the capability of this method, three examples are highlighted.

Probabilistic risk assessment for CO2 storage in geological formations: robust design and support for decision making under uncertainty

NASA Astrophysics Data System (ADS)

Oladyshkin, Sergey; Class, Holger; Helmig, Rainer; Nowak, Wolfgang

2010-05-01

CO2 storage in geological formations is currently being discussed intensively as a technology for mitigating CO2 emissions. However, any large-scale application requires a thorough analysis of the potential risks. Current numerical simulation models are too expensive for probabilistic risk analysis and for stochastic approaches based on brute-force repeated simulation. Even single deterministic simulations may require parallel high-performance computing. The multiphase flow processes involved are too non-linear for quasi-linear error propagation and other simplified stochastic tools. As an alternative approach, we propose a massive stochastic model reduction based on the probabilistic collocation method. The model response is projected onto a orthogonal basis of higher-order polynomials to approximate dependence on uncertain parameters (porosity, permeability etc.) and design parameters (injection rate, depth etc.). This allows for a non-linear propagation of model uncertainty affecting the predicted risk, ensures fast computation and provides a powerful tool for combining design variables and uncertain variables into one approach based on an integrative response surface. Thus, the design task of finding optimal injection regimes explicitly includes uncertainty, which leads to robust designs of the non-linear system that minimize failure probability and provide valuable support for risk-informed management decisions. We validate our proposed stochastic approach by Monte Carlo simulation using a common 3D benchmark problem (Class et al. Computational Geosciences 13, 2009). A reasonable compromise between computational efforts and precision was reached already with second-order polynomials. In our case study, the proposed approach yields a significant computational speedup by a factor of 100 compared to Monte Carlo simulation. We demonstrate that, due to the non-linearity of the flow and transport processes during CO2 injection, including uncertainty in the analysis leads to a systematic and significant shift of predicted leakage rates towards higher values compared with deterministic simulations, affecting both risk estimates and the design of injection scenarios. This implies that, neglecting uncertainty can be a strong simplification for modeling CO2 injection, and the consequences can be stronger than when neglecting several physical phenomena (e.g. phase transition, convective mixing, capillary forces etc.). The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. Keywords: polynomial chaos; CO2 storage; multiphase flow; porous media; risk assessment; uncertainty; integrative response surfaces

Time-optimal Aircraft Pursuit-evasion with a Weapon Envelope Constraint

NASA Technical Reports Server (NTRS)

Menon, P. K. A.

1990-01-01

The optimal pursuit-evasion problem between two aircraft including a realistic weapon envelope is analyzed using differential game theory. Six order nonlinear point mass vehicle models are employed and the inclusion of an arbitrary weapon envelope geometry is allowed. The performance index is a linear combination of flight time and the square of the vehicle acceleration. Closed form solution to this high-order differential game is then obtained using feedback linearization. The solution is in the form of a feedback guidance law together with a quartic polynomial for time-to-go. Due to its modest computational requirements, this nonlinear guidance law is useful for on-board real-time implementation.

MagIC: Fluid dynamics in a spherical shell simulator

NASA Astrophysics Data System (ADS)

Wicht, J.; Gastine, T.; Barik, A.; Putigny, B.; Yadav, R.; Duarte, L.; Dintrans, B.

2017-09-01

MagIC simulates fluid dynamics in a spherical shell. It solves for the Navier-Stokes equation including Coriolis force, optionally coupled with an induction equation for Magneto-Hydro Dynamics (MHD), a temperature (or entropy) equation and an equation for chemical composition under both the anelastic and the Boussinesq approximations. MagIC uses either Chebyshev polynomials or finite differences in the radial direction and spherical harmonic decomposition in the azimuthal and latitudinal directions. The time-stepping scheme relies on a semi-implicit Crank-Nicolson for the linear terms of the MHD equations and a Adams-Bashforth scheme for the non-linear terms and the Coriolis force.

Shear-flexible finite-element models of laminated composite plates and shells

NASA Technical Reports Server (NTRS)

Noor, A. K.; Mathers, M. D.

1975-01-01

Several finite-element models are applied to the linear static, stability, and vibration analysis of laminated composite plates and shells. The study is based on linear shallow-shell theory, with the effects of shear deformation, anisotropic material behavior, and bending-extensional coupling included. Both stiffness (displacement) and mixed finite-element models are considered. Discussion is focused on the effects of shear deformation and anisotropic material behavior on the accuracy and convergence of different finite-element models. Numerical studies are presented which show the effects of increasing the order of the approximating polynomials, adding internal degrees of freedom, and using derivatives of generalized displacements as nodal parameters.

The generalized pole assignment problem. [dynamic output feedback problems

NASA Technical Reports Server (NTRS)

Djaferis, T. E.; Mitter, S. K.

1979-01-01

Two dynamic output feedback problems for a linear, strictly proper system are considered, along with their interrelationships. The problems are formulated in the frequency domain and investigated in terms of linear equations over rings of polynomials. Necessary and sufficient conditions are expressed using genericity.

Linear and nonlinear ARMA model parameter estimation using an artificial neural network

NASA Technical Reports Server (NTRS)

Chon, K. H.; Cohen, R. J.

1997-01-01

This paper addresses parametric system identification of linear and nonlinear dynamic systems by analysis of the input and output signals. Specifically, we investigate the relationship between estimation of the system using a feedforward neural network model and estimation of the system by use of linear and nonlinear autoregressive moving-average (ARMA) models. By utilizing a neural network model incorporating a polynomial activation function, we show the equivalence of the artificial neural network to the linear and nonlinear ARMA models. We compare the parameterization of the estimated system using the neural network and ARMA approaches by utilizing data generated by means of computer simulations. Specifically, we show that the parameters of a simulated ARMA system can be obtained from the neural network analysis of the simulated data or by conventional least squares ARMA analysis. The feasibility of applying neural networks with polynomial activation functions to the analysis of experimental data is explored by application to measurements of heart rate (HR) and instantaneous lung volume (ILV) fluctuations.

Model-based estimates of long-term persistence of induced HPV antibodies: a flexible subject-specific approach.

PubMed

Aregay, Mehreteab; Shkedy, Ziv; Molenberghs, Geert; David, Marie-Pierre; Tibaldi, Fabián

2013-01-01

In infectious diseases, it is important to predict the long-term persistence of vaccine-induced antibodies and to estimate the time points where the individual titers are below the threshold value for protection. This article focuses on HPV-16/18, and uses a so-called fractional-polynomial model to this effect, derived in a data-driven fashion. Initially, model selection was done from among the second- and first-order fractional polynomials on the one hand and from the linear mixed model on the other. According to a functional selection procedure, the first-order fractional polynomial was selected. Apart from the fractional polynomial model, we also fitted a power-law model, which is a special case of the fractional polynomial model. Both models were compared using Akaike's information criterion. Over the observation period, the fractional polynomials fitted the data better than the power-law model; this, of course, does not imply that it fits best over the long run, and hence, caution ought to be used when prediction is of interest. Therefore, we point out that the persistence of the anti-HPV responses induced by these vaccines can only be ascertained empirically by long-term follow-up analysis.

Linear precoding based on polynomial expansion: reducing complexity in massive MIMO.

PubMed

Mueller, Axel; Kammoun, Abla; Björnson, Emil; Debbah, Mérouane

Massive multiple-input multiple-output (MIMO) techniques have the potential to bring tremendous improvements in spectral efficiency to future communication systems. Counterintuitively, the practical issues of having uncertain channel knowledge, high propagation losses, and implementing optimal non-linear precoding are solved more or less automatically by enlarging system dimensions. However, the computational precoding complexity grows with the system dimensions. For example, the close-to-optimal and relatively "antenna-efficient" regularized zero-forcing (RZF) precoding is very complicated to implement in practice, since it requires fast inversions of large matrices in every coherence period. Motivated by the high performance of RZF, we propose to replace the matrix inversion and multiplication by a truncated polynomial expansion (TPE), thereby obtaining the new TPE precoding scheme which is more suitable for real-time hardware implementation and significantly reduces the delay to the first transmitted symbol. The degree of the matrix polynomial can be adapted to the available hardware resources and enables smooth transition between simple maximum ratio transmission and more advanced RZF. By deriving new random matrix results, we obtain a deterministic expression for the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding in massive MIMO systems. Furthermore, we provide a closed-form expression for the polynomial coefficients that maximizes this SINR. To maintain a fixed per-user rate loss as compared to RZF, the polynomial degree does not need to scale with the system, but it should be increased with the quality of the channel knowledge and the signal-to-noise ratio.

Global stability and quadratic Hamiltonian structure in Lotka-Volterra and quasi-polynomial systems

NASA Astrophysics Data System (ADS)

Szederkényi, Gábor; Hangos, Katalin M.

2004-04-01

We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities.

Couple stress theory of curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-01-01

New models for plane curved rods based on linear couple stress theory of elasticity have been developed.2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby, all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

Quadratically Convergent Method for Simultaneously Approaching the Roots of Polynomial Solutions of a Class of Differential Equations

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.

Robust consensus control with guaranteed rate of convergence using second-order Hurwitz polynomials

NASA Astrophysics Data System (ADS)

Fruhnert, Michael; Corless, Martin

2017-10-01

This paper considers homogeneous networks of general, linear time-invariant, second-order systems. We consider linear feedback controllers and require that the directed graph associated with the network contains a spanning tree and systems are stabilisable. We show that consensus with a guaranteed rate of convergence can always be achieved using linear state feedback. To achieve this, we provide a new and simple derivation of the conditions for a second-order polynomial with complex coefficients to be Hurwitz. We apply this result to obtain necessary and sufficient conditions to achieve consensus with networks whose graph Laplacian matrix may have complex eigenvalues. Based on the conditions found, methods to compute feedback gains are proposed. We show that gains can be chosen such that consensus is achieved robustly over a variety of communication structures and system dynamics. We also consider the use of static output feedback.

Lack of anomalous diffusion in linear translationally-invariant systems determined by only one initial condition

NASA Astrophysics Data System (ADS)

Khorrami, Mohammad; Shariati, Ahmad; Aghamohammadi, Amir; Fatollahi, Amir H.

2012-01-01

It is shown that as far as the linear diffusion equation meets both time- and space-translational invariance, the time dependence of a moment of degree α is a polynomial of degree at most equal to α, while all connected moments are at most linear functions of time. As a special case, the variance is an at most linear function of time.

Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion

NASA Astrophysics Data System (ADS)

Sánchez-Vizuet, Tonatiuh; Cerfon, Antoine J.

2018-02-01

We study the approximation and stability properties of a recently popularized discretization strategy for the speed variable in kinetic equations, based on pseudo-spectral collocation on a grid defined by the zeros of a non-standard family of orthogonal polynomials called Maxwell polynomials. Taking a one-dimensional equation describing energy diffusion due to Fokker-Planck collisions with a Maxwell-Boltzmann background distribution as the test bench for the performance of the scheme, we find that Maxwell based discretizations outperform other commonly used schemes in most situations, often by orders of magnitude. This provides a strong motivation for their use in high-dimensional gyrokinetic simulations. However, we also show that Maxwell based schemes are subject to a non-modal time stepping instability in their most straightforward implementation, so that special care must be given to the discrete representation of the linear operators in order to benefit from the advantages provided by Maxwell polynomials.

LMI-based stability analysis of fuzzy-model-based control systems using approximated polynomial membership functions.

PubMed

Narimani, Mohammand; Lam, H K; Dilmaghani, R; Wolfe, Charles

2011-06-01

Relaxed linear-matrix-inequality-based stability conditions for fuzzy-model-based control systems with imperfect premise matching are proposed. First, the derivative of the Lyapunov function, containing the product terms of the fuzzy model and fuzzy controller membership functions, is derived. Then, in the partitioned operating domain of the membership functions, the relations between the state variables and the mentioned product terms are represented by approximated polynomials in each subregion. Next, the stability conditions containing the information of all subsystems and the approximated polynomials are derived. In addition, the concept of the S-procedure is utilized to release the conservativeness caused by considering the whole operating region for approximated polynomials. It is shown that the well-known stability conditions can be special cases of the proposed stability conditions. Simulation examples are given to illustrate the validity of the proposed approach.

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.

Using polynomials to simplify fixed pattern noise and photometric correction of logarithmic CMOS image sensors.

PubMed

Li, Jing; Mahmoodi, Alireza; Joseph, Dileepan

2015-10-16

An important class of complementary metal-oxide-semiconductor (CMOS) image sensors are those where pixel responses are monotonic nonlinear functions of light stimuli. This class includes various logarithmic architectures, which are easily capable of wide dynamic range imaging, at video rates, but which are vulnerable to image quality issues. To minimize fixed pattern noise (FPN) and maximize photometric accuracy, pixel responses must be calibrated and corrected due to mismatch and process variation during fabrication. Unlike literature approaches, which employ circuit-based models of varying complexity, this paper introduces a novel approach based on low-degree polynomials. Although each pixel may have a highly nonlinear response, an approximately-linear FPN calibration is possible by exploiting the monotonic nature of imaging. Moreover, FPN correction requires only arithmetic, and an optimal fixed-point implementation is readily derived, subject to a user-specified number of bits per pixel. Using a monotonic spline, involving cubic polynomials, photometric calibration is also possible without a circuit-based model, and fixed-point photometric correction requires only a look-up table. The approach is experimentally validated with a logarithmic CMOS image sensor and is compared to a leading approach from the literature. The novel approach proves effective and efficient.

Assessing non-uniqueness: An algebraic approach

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vasco, Don W.

Geophysical inverse problems are endowed with a rich mathematical structure. When discretized, most differential and integral equations of interest are algebraic (polynomial) in form. Techniques from algebraic geometry and computational algebra provide a means to address questions of existence and uniqueness for both linear and non-linear inverse problem. In a sense, the methods extend ideas which have proven fruitful in treating linear inverse problems.

Application of overlay modeling and control with Zernike polynomials in an HVM environment

NASA Astrophysics Data System (ADS)

Ju, JaeWuk; Kim, MinGyu; Lee, JuHan; Nabeth, Jeremy; Jeon, Sanghuck; Heo, Hoyoung; Robinson, John C.; Pierson, Bill

2016-03-01

Shrinking technology nodes and smaller process margins require improved photolithography overlay control. Generally, overlay measurement results are modeled with Cartesian polynomial functions for both intra-field and inter-field models and the model coefficients are sent to an advanced process control (APC) system operating in an XY Cartesian basis. Dampened overlay corrections, typically via exponentially or linearly weighted moving average in time, are then retrieved from the APC system to apply on the scanner in XY Cartesian form for subsequent lot exposure. The goal of the above method is to process lots with corrections that target the least possible overlay misregistration in steady state as well as in change point situations. In this study, we model overlay errors on product using Zernike polynomials with same fitting capability as the process of reference (POR) to represent the wafer-level terms, and use the standard Cartesian polynomials to represent the field-level terms. APC calculations for wafer-level correction are performed in Zernike basis while field-level calculations use standard XY Cartesian basis. Finally, weighted wafer-level correction terms are converted to XY Cartesian space in order to be applied on the scanner, along with field-level corrections, for future wafer exposures. Since Zernike polynomials have the property of being orthogonal in the unit disk we are able to reduce the amount of collinearity between terms and improve overlay stability. Our real time Zernike modeling and feedback evaluation was performed on a 20-lot dataset in a high volume manufacturing (HVM) environment. The measured on-product results were compared to POR and showed a 7% reduction in overlay variation including a 22% terms variation. This led to an on-product raw overlay Mean + 3Sigma X&Y improvement of 5% and resulted in 0.1% yield improvement.

Periodic binary sequence generators: VLSI circuits considerations

NASA Technical Reports Server (NTRS)

Perlman, M.

1984-01-01

Feedback shift registers are efficient periodic binary sequence generators. Polynomials of degree r over a Galois field characteristic 2(GF(2)) characterize the behavior of shift registers with linear logic feedback. The algorithmic determination of the trinomial of lowest degree, when it exists, that contains a given irreducible polynomial over GF(2) as a factor is presented. This corresponds to embedding the behavior of an r-stage shift register with linear logic feedback into that of an n-stage shift register with a single two-input modulo 2 summer (i.e., Exclusive-OR gate) in its feedback. This leads to Very Large Scale Integrated (VLSI) circuit architecture of maximal regularity (i.e., identical cells) with intercell communications serialized to a maximal degree.

Spectral likelihood expansions for Bayesian inference

NASA Astrophysics Data System (ADS)

Nagel, Joseph B.; Sudret, Bruno

2016-03-01

A spectral approach to Bayesian inference is presented. It pursues the emulation of the posterior probability density. The starting point is a series expansion of the likelihood function in terms of orthogonal polynomials. From this spectral likelihood expansion all statistical quantities of interest can be calculated semi-analytically. The posterior is formally represented as the product of a reference density and a linear combination of polynomial basis functions. Both the model evidence and the posterior moments are related to the expansion coefficients. This formulation avoids Markov chain Monte Carlo simulation and allows one to make use of linear least squares instead. The pros and cons of spectral Bayesian inference are discussed and demonstrated on the basis of simple applications from classical statistics and inverse modeling.

Influence of surface error on electromagnetic performance of reflectors based on Zernike polynomials

NASA Astrophysics Data System (ADS)

Li, Tuanjie; Shi, Jiachen; Tang, Yaqiong

2018-04-01

This paper investigates the influence of surface error distribution on the electromagnetic performance of antennas. The normalized Zernike polynomials are used to describe a smooth and continuous deformation surface. Based on the geometrical optics and piecewise linear fitting method, the electrical performance of reflector described by the Zernike polynomials is derived to reveal the relationship between surface error distribution and electromagnetic performance. Then the relation database between surface figure and electric performance is built for ideal and deformed surfaces to realize rapidly calculation of far-field electric performances. The simulation analysis of the influence of Zernike polynomials on the electrical properties for the axis-symmetrical reflector with the axial mode helical antenna as feed is further conducted to verify the correctness of the proposed method. Finally, the influence rules of surface error distribution on electromagnetic performance are summarized. The simulation results show that some terms of Zernike polynomials may decrease the amplitude of main lobe of antenna pattern, and some may reduce the pointing accuracy. This work extracts a new concept for reflector's shape adjustment in manufacturing process.

Combinatorial Reliability and Repair

DTIC Science & Technology

1992-07-01

Press, Oxford, 1987. [2] G. Gordon and L. Traldi, Generalized activities and the Tutte polynomial, Discrete Math . 85 (1990), 167-176. [3] A. B. Huseby, A...Chromatic polynomials and network reliability, Discrete Math . 67 (1987), 57-79. [7] A. Satayanarayana and R. K. Wood, A linear-time algorithm for comput- ing...K-terminal reliability in series-parallel networks, SIAM J. Comput. 14 (1985), 818-832. [8] L. Traldi, Generalized activities and K-terminal reliability, Discrete Math . 96 (1991), 131-149. 4

New syndrome decoder for (n, 1) convolutional codes

NASA Technical Reports Server (NTRS)

Reed, I. S.; Truong, T. K.

1983-01-01

The letter presents a new syndrome decoding algorithm for the (n, 1) convolutional codes (CC) that is different and simpler than the previous syndrome decoding algorithm of Schalkwijk and Vinck. The new technique uses the general solution of the polynomial linear Diophantine equation for the error polynomial vector E(D). A recursive, Viterbi-like, algorithm is developed to find the minimum weight error vector E(D). An example is given for the binary nonsystematic (2, 1) CC.

Routh's algorithm - A centennial survey

NASA Technical Reports Server (NTRS)

Barnett, S.; Siljak, D. D.

1977-01-01

One hundred years have passed since the publication of Routh's fundamental work on determining the stability of constant linear systems. The paper presents an outline of the algorithm and considers such aspects of it as the distribution of zeros and applications of it that relate to the greatest common divisor, the abscissa of stability, continued fractions, canonical forms, the nonnegativity of polynomials and polynomial matrices, the absolute stability, optimality and passivity of dynamic systems, and the stability of two-dimensional circuits.

Eshelby's problem of polygonal inclusions with polynomial eigenstrains in an anisotropic magneto-electro-elastic full plane

PubMed Central

Lee, Y.-G.; Zou, W.-N.; Pan, E.

2015-01-01

This paper presents a closed-form solution for the arbitrary polygonal inclusion problem with polynomial eigenstrains of arbitrary order in an anisotropic magneto-electro-elastic full plane. The additional displacements or eigendisplacements, instead of the eigenstrains, are assumed to be a polynomial with general terms of order M+N. By virtue of the extended Stroh formulism, the induced fields are expressed in terms of a group of basic functions which involve boundary integrals of the inclusion domain. For the special case of polygonal inclusions, the boundary integrals are carried out explicitly, and their averages over the inclusion are also obtained. The induced fields under quadratic eigenstrains are mostly analysed in terms of figures and tables, as well as those under the linear and cubic eigenstrains. The connection between the present solution and the solution via the Green's function method is established and numerically verified. The singularity at the vertices of the arbitrary polygon is further analysed via the basic functions. The general solution and the numerical results for the constant, linear, quadratic and cubic eigenstrains presented in this paper enable us to investigate the features of the inclusion and inhomogeneity problem concerning polynomial eigenstrains in semiconductors and advanced composites, while the results can further serve as benchmarks for future analyses of Eshelby's inclusion problem. PMID:26345141

USING LINEAR AND POLYNOMIAL MODELS TO EXAMINE THE ENVIRONMENTAL STABILITY OF VIRUSES

EPA Science Inventory

The article presents the development of model equations for describing the fate of viral infectivity in environmental samples. Most of the models were based upon the use of a two-step linear regression approach. The first step employs regression of log base 10 transformed viral t...

Families of Linear Recurrences for Catalan Numbers

ERIC Educational Resources Information Center

Gauthier, N.

2011-01-01

Four different families of linear recurrences are derived for Catalan numbers. The derivations rest on John Riordan's 1973 generalization of Catalan numbers to a set of polynomials. Elementary differential and integral calculus techniques are used and the results should be of interest to teachers and students of introductory courses in calculus…

Linearized stability of extreme black holes

NASA Astrophysics Data System (ADS)

Burko, Lior M.; Khanna, Gaurav

2018-03-01

Extreme black holes have been argued to be unstable, in the sense that under linearized gravitational perturbations of the extreme Kerr spacetime the Weyl scalar ψ4 blows up along their event horizons at very late advanced times. We show numerically, by solving the Teukolsky equation in 2 +1 D , that all algebraically independent curvature scalar polynomials approach limits that exist when advanced time along the event horizon approaches infinity. Therefore, the horizons of extreme black holes are stable against linearized gravitational perturbations. We argue that the divergence of ψ4 is a consequence of the choice of a fixed tetrad, and that in a suitable dynamical tetrad all Weyl scalars, including ψ4, approach their background extreme Kerr values. We make similar conclusions also for the case of scalar field perturbations of extreme Kerr.

An Adaptive Prediction-Based Approach to Lossless Compression of Floating-Point Volume Data.

PubMed

Fout, N; Ma, Kwan-Liu

2012-12-01

In this work, we address the problem of lossless compression of scientific and medical floating-point volume data. We propose two prediction-based compression methods that share a common framework, which consists of a switched prediction scheme wherein the best predictor out of a preset group of linear predictors is selected. Such a scheme is able to adapt to different datasets as well as to varying statistics within the data. The first method, called APE (Adaptive Polynomial Encoder), uses a family of structured interpolating polynomials for prediction, while the second method, which we refer to as ACE (Adaptive Combined Encoder), combines predictors from previous work with the polynomial predictors to yield a more flexible, powerful encoder that is able to effectively decorrelate a wide range of data. In addition, in order to facilitate efficient visualization of compressed data, our scheme provides an option to partition floating-point values in such a way as to provide a progressive representation. We compare our two compressors to existing state-of-the-art lossless floating-point compressors for scientific data, with our data suite including both computer simulations and observational measurements. The results demonstrate that our polynomial predictor, APE, is comparable to previous approaches in terms of speed but achieves better compression rates on average. ACE, our combined predictor, while somewhat slower, is able to achieve the best compression rate on all datasets, with significantly better rates on most of the datasets.

Polynomial fuzzy observer designs: a sum-of-squares approach.

PubMed

Tanaka, Kazuo; Ohtake, Hiroshi; Seo, Toshiaki; Tanaka, Motoyasu; Wang, Hua O

2012-10-01

This paper presents a sum-of-squares (SOS) approach to polynomial fuzzy observer designs for three classes of polynomial fuzzy systems. The proposed SOS-based framework provides a number of innovations and improvements over the existing linear matrix inequality (LMI)-based approaches to Takagi-Sugeno (T-S) fuzzy controller and observer designs. First, we briefly summarize previous results with respect to a polynomial fuzzy system that is a more general representation of the well-known T-S fuzzy system. Next, we propose polynomial fuzzy observers to estimate states in three classes of polynomial fuzzy systems and derive SOS conditions to design polynomial fuzzy controllers and observers. A remarkable feature of the SOS design conditions for the first two classes (Classes I and II) is that they realize the so-called separation principle, i.e., the polynomial fuzzy controller and observer for each class can be separately designed without lack of guaranteeing the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. Although, for the last class (Class III), the separation principle does not hold, we propose an algorithm to design polynomial fuzzy controller and observer satisfying the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. All the design conditions in the proposed approach can be represented in terms of SOS and are symbolically and numerically solved via the recently developed SOSTOOLS and a semidefinite-program solver, respectively. To illustrate the validity and applicability of the proposed approach, three design examples are provided. The examples demonstrate the advantages of the SOS-based approaches for the existing LMI approaches to T-S fuzzy observer designs.

Polynomial dual energy inverse functions for bone Calcium/Phosphorus ratio determination and experimental evaluation.

PubMed

Sotiropoulou, P; Fountos, G; Martini, N; Koukou, V; Michail, C; Kandarakis, I; Nikiforidis, G

2016-12-01

An X-ray dual energy (XRDE) method was examined, using polynomial nonlinear approximation of inverse functions for the determination of the bone Calcium-to-Phosphorus (Ca/P) mass ratio. Inverse fitting functions with the least-squares estimation were used, to determine calcium and phosphate thicknesses. The method was verified by measuring test bone phantoms with a dedicated dual energy system and compared with previously published dual energy data. The accuracy in the determination of the calcium and phosphate thicknesses improved with the polynomial nonlinear inverse function method, introduced in this work, (ranged from 1.4% to 6.2%), compared to the corresponding linear inverse function method (ranged from 1.4% to 19.5%). Copyright © 2016 Elsevier Ltd. All rights reserved.

Minimizing Higgs potentials via numerical polynomial homotopy continuation

NASA Astrophysics Data System (ADS)

Maniatis, M.; Mehta, D.

2012-08-01

The study of models with extended Higgs sectors requires to minimize the corresponding Higgs potentials, which is in general very difficult. Here, we apply a recently developed method, called numerical polynomial homotopy continuation (NPHC), which guarantees to find all the stationary points of the Higgs potentials with polynomial-like non-linearity. The detection of all stationary points reveals the structure of the potential with maxima, metastable minima, saddle points besides the global minimum. We apply the NPHC method to the most general Higgs potential having two complex Higgs-boson doublets and up to five real Higgs-boson singlets. Moreover the method is applicable to even more involved potentials. Hence the NPHC method allows to go far beyond the limits of the Gröbner basis approach.

Homogenous polynomially parameter-dependent H∞ filter designs of discrete-time fuzzy systems.

PubMed

Zhang, Huaguang; Xie, Xiangpeng; Tong, Shaocheng

2011-10-01

This paper proposes a novel H(∞) filtering technique for a class of discrete-time fuzzy systems. First, a novel kind of fuzzy H(∞) filter, which is homogenous polynomially parameter dependent on membership functions with an arbitrary degree, is developed to guarantee the asymptotic stability and a prescribed H(∞) performance of the filtering error system. Second, relaxed conditions for H(∞) performance analysis are proposed by using a new fuzzy Lyapunov function and the Finsler lemma with homogenous polynomial matrix Lagrange multipliers. Then, based on a new kind of slack variable technique, relaxed linear matrix inequality-based H(∞) filtering conditions are proposed. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Deterministic absorbed dose estimation in computed tomography using a discrete ordinates method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Norris, Edward T.; Liu, Xin, E-mail: xinliu@mst.edu; Hsieh, Jiang

Purpose: Organ dose estimation for a patient undergoing computed tomography (CT) scanning is very important. Although Monte Carlo methods are considered gold-standard in patient dose estimation, the computation time required is formidable for routine clinical calculations. Here, the authors instigate a deterministic method for estimating an absorbed dose more efficiently. Methods: Compared with current Monte Carlo methods, a more efficient approach to estimating the absorbed dose is to solve the linear Boltzmann equation numerically. In this study, an axial CT scan was modeled with a software package, Denovo, which solved the linear Boltzmann equation using the discrete ordinates method. Themore » CT scanning configuration included 16 x-ray source positions, beam collimators, flat filters, and bowtie filters. The phantom was the standard 32 cm CT dose index (CTDI) phantom. Four different Denovo simulations were performed with different simulation parameters, including the number of quadrature sets and the order of Legendre polynomial expansions. A Monte Carlo simulation was also performed for benchmarking the Denovo simulations. A quantitative comparison was made of the simulation results obtained by the Denovo and the Monte Carlo methods. Results: The difference in the simulation results of the discrete ordinates method and those of the Monte Carlo methods was found to be small, with a root-mean-square difference of around 2.4%. It was found that the discrete ordinates method, with a higher order of Legendre polynomial expansions, underestimated the absorbed dose near the center of the phantom (i.e., low dose region). Simulations of the quadrature set 8 and the first order of the Legendre polynomial expansions proved to be the most efficient computation method in the authors’ study. The single-thread computation time of the deterministic simulation of the quadrature set 8 and the first order of the Legendre polynomial expansions was 21 min on a personal computer. Conclusions: The simulation results showed that the deterministic method can be effectively used to estimate the absorbed dose in a CTDI phantom. The accuracy of the discrete ordinates method was close to that of a Monte Carlo simulation, and the primary benefit of the discrete ordinates method lies in its rapid computation speed. It is expected that further optimization of this method in routine clinical CT dose estimation will improve its accuracy and speed.« less

Gravity Gradient Tensor of Arbitrary 3D Polyhedral Bodies with up to Third-Order Polynomial Horizontal and Vertical Mass Contrasts

NASA Astrophysics Data System (ADS)

Ren, Zhengyong; Zhong, Yiyuan; Chen, Chaojian; Tang, Jingtian; Kalscheuer, Thomas; Maurer, Hansruedi; Li, Yang

2018-03-01

During the last 20 years, geophysicists have developed great interest in using gravity gradient tensor signals to study bodies of anomalous density in the Earth. Deriving exact solutions of the gravity gradient tensor signals has become a dominating task in exploration geophysics or geodetic fields. In this study, we developed a compact and simple framework to derive exact solutions of gravity gradient tensor measurements for polyhedral bodies, in which the density contrast is represented by a general polynomial function. The polynomial mass contrast can continuously vary in both horizontal and vertical directions. In our framework, the original three-dimensional volume integral of gravity gradient tensor signals is transformed into a set of one-dimensional line integrals along edges of the polyhedral body by sequentially invoking the volume and surface gradient (divergence) theorems. In terms of an orthogonal local coordinate system defined on these edges, exact solutions are derived for these line integrals. We successfully derived a set of unified exact solutions of gravity gradient tensors for constant, linear, quadratic and cubic polynomial orders. The exact solutions for constant and linear cases cover all previously published vertex-type exact solutions of the gravity gradient tensor for a polygonal body, though the associated algorithms may differ in numerical stability. In addition, to our best knowledge, it is the first time that exact solutions of gravity gradient tensor signals are derived for a polyhedral body with a polynomial mass contrast of order higher than one (that is quadratic and cubic orders). Three synthetic models (a prismatic body with depth-dependent density contrasts, an irregular polyhedron with linear density contrast and a tetrahedral body with horizontally and vertically varying density contrasts) are used to verify the correctness and the efficiency of our newly developed closed-form solutions. Excellent agreements are obtained between our solutions and other published exact solutions. In addition, stability tests are performed to demonstrate that our exact solutions can safely be used to detect shallow subsurface targets.

Incomplete Gröbner basis as a preconditioner for polynomial systems

NASA Astrophysics Data System (ADS)

Sun, Yang; Tao, Yu-Hui; Bai, Feng-Shan

2009-04-01

Precondition plays a critical role in the numerical methods for large and sparse linear systems. It is also true for nonlinear algebraic systems. In this paper incomplete Gröbner basis (IGB) is proposed as a preconditioner of homotopy methods for polynomial systems of equations, which transforms a deficient system into a system with the same finite solutions, but smaller degree. The reduced system can thus be solved faster. Numerical results show the efficiency of the preconditioner.

Enumerative Algebraic Geometry of Conics

DTIC Science & Technology

2008-10-01

polynomial defining the conic factors into a product of linear polynomials, then the conic is just the union of two lines. Such a conic is said to be...corresponds to the union of two varieties, so [H ] + [H ] will be the class representing the union of two hyperplanes. But the union of two...sets form a topology, the union S′ = S ∪ [(P5)5 × E] is also closed. Now one great fact about projective varieties is that if we have a projection

Families of linear recurrences for Catalan numbers

NASA Astrophysics Data System (ADS)

Gauthier, N.

2011-01-01

Four different families of linear recurrences are derived for Catalan numbers. The derivations rest on John Riordan's 1973 generalization of Catalan numbers to a set of polynomials. Elementary differential and integral calculus techniques are used and the results should be of interest to teachers and students of introductory courses in calculus and number theory.

Breeding value accuracy estimates for growth traits using random regression and multi-trait models in Nelore cattle.

PubMed

Boligon, A A; Baldi, F; Mercadante, M E Z; Lobo, R B; Pereira, R J; Albuquerque, L G

2011-06-28

We quantified the potential increase in accuracy of expected breeding value for weights of Nelore cattle, from birth to mature age, using multi-trait and random regression models on Legendre polynomials and B-spline functions. A total of 87,712 weight records from 8144 females were used, recorded every three months from birth to mature age from the Nelore Brazil Program. For random regression analyses, all female weight records from birth to eight years of age (data set I) were considered. From this general data set, a subset was created (data set II), which included only nine weight records: at birth, weaning, 365 and 550 days of age, and 2, 3, 4, 5, and 6 years of age. Data set II was analyzed using random regression and multi-trait models. The model of analysis included the contemporary group as fixed effects and age of dam as a linear and quadratic covariable. In the random regression analyses, average growth trends were modeled using a cubic regression on orthogonal polynomials of age. Residual variances were modeled by a step function with five classes. Legendre polynomials of fourth and sixth order were utilized to model the direct genetic and animal permanent environmental effects, respectively, while third-order Legendre polynomials were considered for maternal genetic and maternal permanent environmental effects. Quadratic polynomials were applied to model all random effects in random regression models on B-spline functions. Direct genetic and animal permanent environmental effects were modeled using three segments or five coefficients, and genetic maternal and maternal permanent environmental effects were modeled with one segment or three coefficients in the random regression models on B-spline functions. For both data sets (I and II), animals ranked differently according to expected breeding value obtained by random regression or multi-trait models. With random regression models, the highest gains in accuracy were obtained at ages with a low number of weight records. The results indicate that random regression models provide more accurate expected breeding values than the traditionally finite multi-trait models. Thus, higher genetic responses are expected for beef cattle growth traits by replacing a multi-trait model with random regression models for genetic evaluation. B-spline functions could be applied as an alternative to Legendre polynomials to model covariance functions for weights from birth to mature age.

SOMBI: Bayesian identification of parameter relations in unstructured cosmological data

NASA Astrophysics Data System (ADS)

Frank, Philipp; Jasche, Jens; Enßlin, Torsten A.

2016-11-01

This work describes the implementation and application of a correlation determination method based on self organizing maps and Bayesian inference (SOMBI). SOMBI aims to automatically identify relations between different observed parameters in unstructured cosmological or astrophysical surveys by automatically identifying data clusters in high-dimensional datasets via the self organizing map neural network algorithm. Parameter relations are then revealed by means of a Bayesian inference within respective identified data clusters. Specifically such relations are assumed to be parametrized as a polynomial of unknown order. The Bayesian approach results in a posterior probability distribution function for respective polynomial coefficients. To decide which polynomial order suffices to describe correlation structures in data, we include a method for model selection, the Bayesian information criterion, to the analysis. The performance of the SOMBI algorithm is tested with mock data. As illustration we also provide applications of our method to cosmological data. In particular, we present results of a correlation analysis between galaxy and active galactic nucleus (AGN) properties provided by the SDSS catalog with the cosmic large-scale-structure (LSS). The results indicate that the combined galaxy and LSS dataset indeed is clustered into several sub-samples of data with different average properties (for example different stellar masses or web-type classifications). The majority of data clusters appear to have a similar correlation structure between galaxy properties and the LSS. In particular we revealed a positive and linear dependency between the stellar mass, the absolute magnitude and the color of a galaxy with the corresponding cosmic density field. A remaining subset of data shows inverted correlations, which might be an artifact of non-linear redshift distortions.

Determination of many-electron basis functions for a quantum Hall ground state using Schur polynomials

NASA Astrophysics Data System (ADS)

Mandal, Sudhansu S.; Mukherjee, Sutirtha; Ray, Koushik

2018-03-01

A method for determining the ground state of a planar interacting many-electron system in a magnetic field perpendicular to the plane is described. The ground state wave-function is expressed as a linear combination of a set of basis functions. Given only the flux and the number of electrons describing an incompressible state, we use the combinatorics of partitioning the flux among the electrons to derive the basis wave-functions as linear combinations of Schur polynomials. The procedure ensures that the basis wave-functions form representations of the angular momentum algebra. We exemplify the method by deriving the basis functions for the 5/2 quantum Hall state with a few particles. We find that one of the basis functions is precisely the Moore-Read Pfaffian wave function.

Charge-based MOSFET model based on the Hermite interpolation polynomial

NASA Astrophysics Data System (ADS)

Colalongo, Luigi; Richelli, Anna; Kovacs, Zsolt

2017-04-01

An accurate charge-based compact MOSFET model is developed using the third order Hermite interpolation polynomial to approximate the relation between surface potential and inversion charge in the channel. This new formulation of the drain current retains the same simplicity of the most advanced charge-based compact MOSFET models such as BSIM, ACM and EKV, but it is developed without requiring the crude linearization of the inversion charge. Hence, the asymmetry and the non-linearity in the channel are accurately accounted for. Nevertheless, the expression of the drain current can be worked out to be analytically equivalent to BSIM, ACM and EKV. Furthermore, thanks to this new mathematical approach the slope factor is rigorously defined in all regions of operation and no empirical assumption is required.

Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Toeplitz systems

NASA Technical Reports Server (NTRS)

Freund, Roland W.; Zha, Hongyuan

1992-01-01

Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n(sub 2)) arithmetic operations, as compared to O(n(sub 3)) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices are nonsingular. An extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses look-ahead to skip over exactly singular, as well as ill-conditioned leading submatrices, and, at the same time, it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials.

Formal methods for modeling and analysis of hybrid systems

NASA Technical Reports Server (NTRS)

Tiwari, Ashish (Inventor); Lincoln, Patrick D. (Inventor)

2009-01-01

A technique based on the use of a quantifier elimination decision procedure for real closed fields and simple theorem proving to construct a series of successively finer qualitative abstractions of hybrid automata is taught. The resulting abstractions are always discrete transition systems which can then be used by any traditional analysis tool. The constructed abstractions are conservative and can be used to establish safety properties of the original system. The technique works on linear and non-linear polynomial hybrid systems: the guards on discrete transitions and the continuous flows in all modes can be specified using arbitrary polynomial expressions over the continuous variables. An exemplar tool in the SAL environment built over the theorem prover PVS is detailed. The technique scales well to large and complex hybrid systems.

A BiCGStab2 variant of the IDR(s) method for solving linear equations

NASA Astrophysics Data System (ADS)

Abe, Kuniyoshi; Sleijpen, Gerard L. G.

2012-09-01

The hybrid Bi-Conjugate Gradient (Bi-CG) methods, such as the BiCG STABilized (BiCGSTAB), BiCGstab(l), BiCGStab2 and BiCG×MR2 methods are well-known solvers for solving a linear equation with a nonsymmetric matrix. The Induced Dimension Reduction (IDR)(s) method has recently been proposed, and it has been reported that IDR(s) is often more effective than the hybrid BiCG methods. IDR(s) combining the stabilization polynomial of BiCGstab(l) has been designed to improve the convergence of the original IDR(s) method. We therefore propose IDR(s) combining the stabilization polynomial of BiCGStab2. Numerical experiments show that our proposed variant of IDR(s) is more effective than the original IDR(s) and BiCGStab2 methods.

Nonlocal theory of curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-09-01

New models for plane curved rods based on linear nonlocal theory of elasticity have been developed. The 2-D theory is developed from general 2-D equations of linear nonlocal elasticity using a special curvilinear system of coordinates related to the middle line of the rod along with special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including nonlocal constitutive relations have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of local elasticity, a system of differential equations in terms of displacements for Fourier coefficients has been obtained. First and second order approximations have been considered in detail. Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear nonlocal theory of elasticity which are considered in a special curvilinear system of coordinates related to the middle line of the rod. The obtained equations can be used to calculate stress-strain and to model thin walled structures in micro- and nanoscales when taking into account size dependent and nonlocal effects.

An Online Gravity Modeling Method Applied for High Precision Free-INS

PubMed Central

Wang, Jing; Yang, Gongliu; Li, Jing; Zhou, Xiao

2016-01-01

For real-time solution of inertial navigation system (INS), the high-degree spherical harmonic gravity model (SHM) is not applicable because of its time and space complexity, in which traditional normal gravity model (NGM) has been the dominant technique for gravity compensation. In this paper, a two-dimensional second-order polynomial model is derived from SHM according to the approximate linear characteristic of regional disturbing potential. Firstly, deflections of vertical (DOVs) on dense grids are calculated with SHM in an external computer. And then, the polynomial coefficients are obtained using these DOVs. To achieve global navigation, the coefficients and applicable region of polynomial model are both updated synchronously in above computer. Compared with high-degree SHM, the polynomial model takes less storage and computational time at the expense of minor precision. Meanwhile, the model is more accurate than NGM. Finally, numerical test and INS experiment show that the proposed method outperforms traditional gravity models applied for high precision free-INS. PMID:27669261

An Online Gravity Modeling Method Applied for High Precision Free-INS.

PubMed

Wang, Jing; Yang, Gongliu; Li, Jing; Zhou, Xiao

2016-09-23

For real-time solution of inertial navigation system (INS), the high-degree spherical harmonic gravity model (SHM) is not applicable because of its time and space complexity, in which traditional normal gravity model (NGM) has been the dominant technique for gravity compensation. In this paper, a two-dimensional second-order polynomial model is derived from SHM according to the approximate linear characteristic of regional disturbing potential. Firstly, deflections of vertical (DOVs) on dense grids are calculated with SHM in an external computer. And then, the polynomial coefficients are obtained using these DOVs. To achieve global navigation, the coefficients and applicable region of polynomial model are both updated synchronously in above computer. Compared with high-degree SHM, the polynomial model takes less storage and computational time at the expense of minor precision. Meanwhile, the model is more accurate than NGM. Finally, numerical test and INS experiment show that the proposed method outperforms traditional gravity models applied for high precision free-INS.

Kostant polynomials and the cohomology ring for G/B

PubMed Central

Billey, Sara C.

1997-01-01

The Schubert calculus for G/B can be completely determined by a certain matrix related to the Kostant polynomials introduced in section 5 of Bernstein, Gelfand, and Gelfand [Bernstein, I., Gelfand, I. & Gelfand, S. (1973) Russ. Math. Surv. 28, 1–26]. The polynomials are defined by vanishing properties on the orbit of a regular point under the action of the Weyl group. For each element w in the Weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the Bruhat order than w. The main theorem given here is an explicit formula for these values. The matrix of orbit values can be used to determine the cup product for the cohomology ring for G/B, using only linear algebra or as described by Lascoux and Schützenberger [Lascoux, A. & Schützenberger, M.-P. (1982) C. R. Seances Acad. Sci. Ser. A 294, 447–450]. Complete proofs of all the theorems will appear in a forthcoming paper. PMID:11038536

Peculiarities of stochastic regime of Arctic ice cover time evolution over 1987-2014 from microwave satellite sounding on the basis of NASA team 2 algorithm

NASA Astrophysics Data System (ADS)

Raev, M. D.; Sharkov, E. A.; Tikhonov, V. V.; Repina, I. A.; Komarova, N. Yu.

2015-12-01

The GLOBAL-RT database (DB) is composed of long-term radio heat multichannel observation data received from DMSP F08-F17 satellites; it is permanently supplemented with new data on the Earth's exploration from the space department of the Space Research Institute, Russian Academy of Sciences. Arctic ice-cover areas for regions higher than 60° N latitude were calculated using the DB polar version and NASA Team 2 algorithm, which is widely used in foreign scientific literature. According to the analysis of variability of Arctic ice cover during 1987-2014, 2 months were selected when the Arctic ice cover was maximal (February) and minimal (September), and the average ice cover area was calculated for these months. Confidence intervals of the average values are in the 95-98% limits. Several approximations are derived for the time dependences of the ice-cover maximum and minimum over the period under study. Regression dependences were calculated for polynomials from the first degree (linear) to sextic. It was ascertained that the minimal root-mean-square error of deviation from the approximated curve sharply decreased for the biquadratic polynomial and then varied insignificantly: from 0.5593 for the polynomial of third degree to 0.4560 for the biquadratic polynomial. Hence, the commonly used strictly linear regression with a negative time gradient for the September Arctic ice cover minimum over 30 years should be considered incorrect.

Study on the mapping of dark matter clustering from real space to redshift space

NASA Astrophysics Data System (ADS)

Zheng, Yi; Song, Yong-Seon

2016-08-01

The mapping of dark matter clustering from real space to redshift space introduces the anisotropic property to the measured density power spectrum in redshift space, known as the redshift space distortion effect. The mapping formula is intrinsically non-linear, which is complicated by the higher order polynomials due to indefinite cross correlations between the density and velocity fields, and the Finger-of-God effect due to the randomness of the peculiar velocity field. Whilst the full higher order polynomials remain unknown, the other systematics can be controlled consistently within the same order truncation in the expansion of the mapping formula, as shown in this paper. The systematic due to the unknown non-linear density and velocity fields is removed by separately measuring all terms in the expansion directly using simulations. The uncertainty caused by the velocity randomness is controlled by splitting the FoG term into two pieces, 1) the ``one-point" FoG term being independent of the separation vector between two different points, and 2) the ``correlated" FoG term appearing as an indefinite polynomials which is expanded in the same order as all other perturbative polynomials. Using 100 realizations of simulations, we find that the Gaussian FoG function with only one scale-independent free parameter works quite well, and that our new mapping formulation accurately reproduces the observed 2-dimensional density power spectrum in redshift space at the smallest scales by far, up to k~ 0.2 Mpc-1, considering the resolution of future experiments.

Investigating and Modelling Effects of Climatically and Hydrologically Indicators on the Urmia Lake Coastline Changes Using Time Series Analysis

NASA Astrophysics Data System (ADS)

Ahmadijamal, M.; Hasanlou, M.

2017-09-01

Study of hydrological parameters of lakes and examine the variation of water level to operate management on water resources are important. The purpose of this study is to investigate and model the Urmia Lake water level changes due to changes in climatically and hydrological indicators that affects in the process of level variation and area of this lake. For this purpose, Landsat satellite images, hydrological data, the daily precipitation, the daily surface evaporation and the daily discharge in total of the lake basin during the period of 2010-2016 have been used. Based on time-series analysis that is conducted on individual data independently with same procedure, to model variation of Urmia Lake level, we used polynomial regression technique and combined polynomial with periodic behavior. In the first scenario, we fit a multivariate linear polynomial to our datasets and determining RMSE, NRSME and R² value. We found that fourth degree polynomial can better fit to our datasets with lowest RMSE value about 9 cm. In the second scenario, we combine polynomial with periodic behavior for modeling. The second scenario has superiority comparing to the first one, by RMSE value about 3 cm.

Study of Randomness in AES Ciphertexts Produced by Randomly Generated S-Boxes and S-Boxes with Various Modulus and Additive Constant Polynomials

NASA Astrophysics Data System (ADS)

Das, Suman; Sadique Uz Zaman, J. K. M.; Ghosh, Ranjan

2016-06-01

In Advanced Encryption Standard (AES), the standard S-Box is conventionally generated by using a particular irreducible polynomial {11B} in GF(28) as the modulus and a particular additive constant polynomial {63} in GF(2), though it can be generated by many other polynomials. In this paper, it has been shown that it is possible to generate secured AES S-Boxes by using some other selected modulus and additive polynomials and also can be generated randomly, using a PRNG like BBS. A comparative study has been made on the randomness of corresponding AES ciphertexts generated, using these S-Boxes, by the NIST Test Suite coded for this paper. It has been found that besides using the standard one, other moduli and additive constants are also able to generate equally or better random ciphertexts; the same is true for random S-Boxes also. As these new types of S-Boxes are user-defined, hence unknown, they are able to prevent linear and differential cryptanalysis. Moreover, they act as additional key-inputs to AES, thus increasing the key-space.

Using Spherical-Harmonics Expansions for Optics Surface Reconstruction from Gradients.

PubMed

Solano-Altamirano, Juan Manuel; Vázquez-Otero, Alejandro; Khikhlukha, Danila; Dormido, Raquel; Duro, Natividad

2017-11-30

In this paper, we propose a new algorithm to reconstruct optics surfaces (aka wavefronts) from gradients, defined on a circular domain, by means of the Spherical Harmonics. The experimental results indicate that this algorithm renders the same accuracy, compared to the reconstruction based on classical Zernike polynomials, using a smaller number of polynomial terms, which potentially speeds up the wavefront reconstruction. Additionally, we provide an open-source C++ library, released under the terms of the GNU General Public License version 2 (GPLv2), wherein several polynomial sets are coded. Therefore, this library constitutes a robust software alternative for wavefront reconstruction in a high energy laser field, optical surface reconstruction, and, more generally, in surface reconstruction from gradients. The library is a candidate for being integrated in control systems for optical devices, or similarly to be used in ad hoc simulations. Moreover, it has been developed with flexibility in mind, and, as such, the implementation includes the following features: (i) a mock-up generator of various incident wavefronts, intended to simulate the wavefronts commonly encountered in the field of high-energy lasers production; (ii) runtime selection of the library in charge of performing the algebraic computations; (iii) a profiling mechanism to measure and compare the performance of different steps of the algorithms and/or third-party linear algebra libraries. Finally, the library can be easily extended to include additional dependencies, such as porting the algebraic operations to specific architectures, in order to exploit hardware acceleration features.

Using Spherical-Harmonics Expansions for Optics Surface Reconstruction from Gradients

PubMed Central

Solano-Altamirano, Juan Manuel; Khikhlukha, Danila

2017-01-01

In this paper, we propose a new algorithm to reconstruct optics surfaces (aka wavefronts) from gradients, defined on a circular domain, by means of the Spherical Harmonics. The experimental results indicate that this algorithm renders the same accuracy, compared to the reconstruction based on classical Zernike polynomials, using a smaller number of polynomial terms, which potentially speeds up the wavefront reconstruction. Additionally, we provide an open-source C++ library, released under the terms of the GNU General Public License version 2 (GPLv2), wherein several polynomial sets are coded. Therefore, this library constitutes a robust software alternative for wavefront reconstruction in a high energy laser field, optical surface reconstruction, and, more generally, in surface reconstruction from gradients. The library is a candidate for being integrated in control systems for optical devices, or similarly to be used in ad hoc simulations. Moreover, it has been developed with flexibility in mind, and, as such, the implementation includes the following features: (i) a mock-up generator of various incident wavefronts, intended to simulate the wavefronts commonly encountered in the field of high-energy lasers production; (ii) runtime selection of the library in charge of performing the algebraic computations; (iii) a profiling mechanism to measure and compare the performance of different steps of the algorithms and/or third-party linear algebra libraries. Finally, the library can be easily extended to include additional dependencies, such as porting the algebraic operations to specific architectures, in order to exploit hardware acceleration features. PMID:29189722

Using Polynomials to Simplify Fixed Pattern Noise and Photometric Correction of Logarithmic CMOS Image Sensors

PubMed Central

Li, Jing; Mahmoodi, Alireza; Joseph, Dileepan

2015-01-01

An important class of complementary metal-oxide-semiconductor (CMOS) image sensors are those where pixel responses are monotonic nonlinear functions of light stimuli. This class includes various logarithmic architectures, which are easily capable of wide dynamic range imaging, at video rates, but which are vulnerable to image quality issues. To minimize fixed pattern noise (FPN) and maximize photometric accuracy, pixel responses must be calibrated and corrected due to mismatch and process variation during fabrication. Unlike literature approaches, which employ circuit-based models of varying complexity, this paper introduces a novel approach based on low-degree polynomials. Although each pixel may have a highly nonlinear response, an approximately-linear FPN calibration is possible by exploiting the monotonic nature of imaging. Moreover, FPN correction requires only arithmetic, and an optimal fixed-point implementation is readily derived, subject to a user-specified number of bits per pixel. Using a monotonic spline, involving cubic polynomials, photometric calibration is also possible without a circuit-based model, and fixed-point photometric correction requires only a look-up table. The approach is experimentally validated with a logarithmic CMOS image sensor and is compared to a leading approach from the literature. The novel approach proves effective and efficient. PMID:26501287

Micropolar curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-01-01

New models for micropolar plane curved rods have been developed. 2-D theory is developed from general 2-D equations of linear micropolar elasticity using a special curvilinear system of coordinates related to the middle line of the rod and special hypothesis based on assumptions that take into account the fact that the rod is thin.High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First stress and strain tensors,vectors of displacements and rotation and body force shave been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then in the same way as in the theory of elasticity, system of differential equations in term of displacements and boundary conditions for Fourier coefficients have been obtained. The Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and 2-D equations of linear micropolar elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scale when taking in to account micropolar couple stress and rotation effects.

Higher-order Multivariable Polynomial Regression to Estimate Human Affective States

NASA Astrophysics Data System (ADS)

Wei, Jie; Chen, Tong; Liu, Guangyuan; Yang, Jiemin

2016-03-01

From direct observations, facial, vocal, gestural, physiological, and central nervous signals, estimating human affective states through computational models such as multivariate linear-regression analysis, support vector regression, and artificial neural network, have been proposed in the past decade. In these models, linear models are generally lack of precision because of ignoring intrinsic nonlinearities of complex psychophysiological processes; and nonlinear models commonly adopt complicated algorithms. To improve accuracy and simplify model, we introduce a new computational modeling method named as higher-order multivariable polynomial regression to estimate human affective states. The study employs standardized pictures in the International Affective Picture System to induce thirty subjects’ affective states, and obtains pure affective patterns of skin conductance as input variables to the higher-order multivariable polynomial model for predicting affective valence and arousal. Experimental results show that our method is able to obtain efficient correlation coefficients of 0.98 and 0.96 for estimation of affective valence and arousal, respectively. Moreover, the method may provide certain indirect evidences that valence and arousal have their brain’s motivational circuit origins. Thus, the proposed method can serve as a novel one for efficiently estimating human affective states.

Higher-order Multivariable Polynomial Regression to Estimate Human Affective States

PubMed Central

Wei, Jie; Chen, Tong; Liu, Guangyuan; Yang, Jiemin

2016-01-01

From direct observations, facial, vocal, gestural, physiological, and central nervous signals, estimating human affective states through computational models such as multivariate linear-regression analysis, support vector regression, and artificial neural network, have been proposed in the past decade. In these models, linear models are generally lack of precision because of ignoring intrinsic nonlinearities of complex psychophysiological processes; and nonlinear models commonly adopt complicated algorithms. To improve accuracy and simplify model, we introduce a new computational modeling method named as higher-order multivariable polynomial regression to estimate human affective states. The study employs standardized pictures in the International Affective Picture System to induce thirty subjects’ affective states, and obtains pure affective patterns of skin conductance as input variables to the higher-order multivariable polynomial model for predicting affective valence and arousal. Experimental results show that our method is able to obtain efficient correlation coefficients of 0.98 and 0.96 for estimation of affective valence and arousal, respectively. Moreover, the method may provide certain indirect evidences that valence and arousal have their brain’s motivational circuit origins. Thus, the proposed method can serve as a novel one for efficiently estimating human affective states. PMID:26996254

Matrix of moments of the Legendre polynomials and its application to problems of electrostatics

NASA Astrophysics Data System (ADS)

Savchenko, A. O.

2017-01-01

In this work, properties of the matrix of moments of the Legendre polynomials are presented and proven. In particular, the explicit form of the elements of the matrix inverse to the matrix of moments is found and theorems of the linear combination and orthogonality are proven. On the basis of these properties, the total charge and the dipole moment of a conducting ball in a nonuniform electric field, the charge distribution over the surface of the conducting ball, its multipole moments, and the force acting on a conducting ball situated on the axis of a nonuniform axisymmetric electric field are determined. All assertions are formulated in theorems, the proofs of which are based on the properties of the matrix of moments of the Legendre polynomials.

Polynomial mixture method of solving ordinary differential equations

NASA Astrophysics Data System (ADS)

Shahrir, Mohammad Shazri; Nallasamy, Kumaresan; Ratnavelu, Kuru; Kamali, M. Z. M.

2017-11-01

In this paper, a numerical solution of fuzzy quadratic Riccati differential equation is estimated using a proposed new approach that provides mixture of polynomials where iteratively the right mixture will be generated. This mixture provide a generalized formalism of traditional Neural Networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). This can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that Polynomial Mixture Method (PMM) shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over Mabood et al, RK-4, Multi-Agent NN and Neuro Method (NM).

An Exactly Solvable Spin Chain Related to Hahn Polynomials

NASA Astrophysics Data System (ADS)

Stoilova, Neli I.; van der Jeugt, Joris

2011-03-01

We study a linear spin chain which was originally introduced by Shi et al. [Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α and depends on the parity of the chain site. Extending the model by a second parameter β, it is shown that the single fermion eigenstates of the Hamiltonian can be computed in explicit form. The components of these eigenvectors turn out to be Hahn polynomials with parameters (α,β) and (α+1,β-1). The construction of the eigenvectors relies on two new difference equations for Hahn polynomials. The explicit knowledge of the eigenstates leads to a closed form expression for the correlation function of the spin chain. We also discuss some aspects of a q-extension of this model.

The Ponzano-Regge Model and Parametric Representation

NASA Astrophysics Data System (ADS)

Li, Dan

2014-04-01

We give a parametric representation of the effective noncommutative field theory derived from a -deformation of the Ponzano-Regge model and define a generalized Kirchhoff polynomial with -correction terms, obtained in a -linear approximation. We then consider the corresponding graph hypersurfaces and the question of how the presence of the correction term affects their motivic nature. We look in particular at the tetrahedron graph, which is the basic case of relevance to quantum gravity. With the help of computer calculations, we verify that the number of points over finite fields of the corresponding hypersurface does not fit polynomials with integer coefficients, hence the hypersurface of the tetrahedron is not polynomially countable. This shows that the correction term can change significantly the motivic properties of the hypersurfaces, with respect to the classical case.

A quadratic regression modelling on paddy production in the area of Perlis

NASA Astrophysics Data System (ADS)

Goh, Aizat Hanis Annas; Ali, Zalila; Nor, Norlida Mohd; Baharum, Adam; Ahmad, Wan Muhamad Amir W.

2017-08-01

Polynomial regression models are useful in situations in which the relationship between a response variable and predictor variables is curvilinear. Polynomial regression fits the nonlinear relationship into a least squares linear regression model by decomposing the predictor variables into a kth order polynomial. The polynomial order determines the number of inflexions on the curvilinear fitted line. A second order polynomial forms a quadratic expression (parabolic curve) with either a single maximum or minimum, a third order polynomial forms a cubic expression with both a relative maximum and a minimum. This study used paddy data in the area of Perlis to model paddy production based on paddy cultivation characteristics and environmental characteristics. The results indicated that a quadratic regression model best fits the data and paddy production is affected by urea fertilizer application and the interaction between amount of average rainfall and percentage of area defected by pest and disease. Urea fertilizer application has a quadratic effect in the model which indicated that if the number of days of urea fertilizer application increased, paddy production is expected to decrease until it achieved a minimum value and paddy production is expected to increase at higher number of days of urea application. The decrease in paddy production with an increased in rainfall is greater, the higher the percentage of area defected by pest and disease.

Parametric correlation functions to model the structure of permanent environmental (co)variances in milk yield random regression models.

PubMed

Bignardi, A B; El Faro, L; Cardoso, V L; Machado, P F; Albuquerque, L G

2009-09-01

The objective of the present study was to estimate milk yield genetic parameters applying random regression models and parametric correlation functions combined with a variance function to model animal permanent environmental effects. A total of 152,145 test-day milk yields from 7,317 first lactations of Holstein cows belonging to herds located in the southeastern region of Brazil were analyzed. Test-day milk yields were divided into 44 weekly classes of days in milk. Contemporary groups were defined by herd-test-day comprising a total of 2,539 classes. The model included direct additive genetic, permanent environmental, and residual random effects. The following fixed effects were considered: contemporary group, age of cow at calving (linear and quadratic regressions), and the population average lactation curve modeled by fourth-order orthogonal Legendre polynomial. Additive genetic effects were modeled by random regression on orthogonal Legendre polynomials of days in milk, whereas permanent environmental effects were estimated using a stationary or nonstationary parametric correlation function combined with a variance function of different orders. The structure of residual variances was modeled using a step function containing 6 variance classes. The genetic parameter estimates obtained with the model using a stationary correlation function associated with a variance function to model permanent environmental effects were similar to those obtained with models employing orthogonal Legendre polynomials for the same effect. A model using a sixth-order polynomial for additive effects and a stationary parametric correlation function associated with a seventh-order variance function to model permanent environmental effects would be sufficient for data fitting.

A comparison of different chemometrics approaches for the robust classification of electronic nose data.

PubMed

Gromski, Piotr S; Correa, Elon; Vaughan, Andrew A; Wedge, David C; Turner, Michael L; Goodacre, Royston

2014-11-01

Accurate detection of certain chemical vapours is important, as these may be diagnostic for the presence of weapons, drugs of misuse or disease. In order to achieve this, chemical sensors could be deployed remotely. However, the readout from such sensors is a multivariate pattern, and this needs to be interpreted robustly using powerful supervised learning methods. Therefore, in this study, we compared the classification accuracy of four pattern recognition algorithms which include linear discriminant analysis (LDA), partial least squares-discriminant analysis (PLS-DA), random forests (RF) and support vector machines (SVM) which employed four different kernels. For this purpose, we have used electronic nose (e-nose) sensor data (Wedge et al., Sensors Actuators B Chem 143:365-372, 2009). In order to allow direct comparison between our four different algorithms, we employed two model validation procedures based on either 10-fold cross-validation or bootstrapping. The results show that LDA (91.56% accuracy) and SVM with a polynomial kernel (91.66% accuracy) were very effective at analysing these e-nose data. These two models gave superior prediction accuracy, sensitivity and specificity in comparison to the other techniques employed. With respect to the e-nose sensor data studied here, our findings recommend that SVM with a polynomial kernel should be favoured as a classification method over the other statistical models that we assessed. SVM with non-linear kernels have the advantage that they can be used for classifying non-linear as well as linear mapping from analytical data space to multi-group classifications and would thus be a suitable algorithm for the analysis of most e-nose sensor data.

A 3D Ginibre Point Field

NASA Astrophysics Data System (ADS)

Kargin, Vladislav

2018-06-01

We introduce a family of three-dimensional random point fields using the concept of the quaternion determinant. The kernel of each field is an n-dimensional orthogonal projection on a linear space of quaternionic polynomials. We find explicit formulas for the basis of the orthogonal quaternion polynomials and for the kernel of the projection. For number of particles n → ∞, we calculate the scaling limits of the point field in the bulk and at the center of coordinates. We compare our construction with the previously introduced Fermi-sphere point field process.

Operational method of solution of linear non-integer ordinary and partial differential equations.

PubMed

Zhukovsky, K V

2016-01-01

We propose operational method with recourse to generalized forms of orthogonal polynomials for solution of a variety of differential equations of mathematical physics. Operational definitions of generalized families of orthogonal polynomials are used in this context. Integral transforms and the operational exponent together with some special functions are also employed in the solutions. The examples of solution of physical problems, related to such problems as the heat propagation in various models, evolutional processes, Black-Scholes-like equations etc. are demonstrated by the operational technique.

J3Gen: A PRNG for Low-Cost Passive RFID

PubMed Central

Melià-Seguí, Joan; Garcia-Alfaro, Joaquin; Herrera-Joancomartí, Jordi

2013-01-01

Pseudorandom number generation (PRNG) is the main security tool in low-cost passive radio-frequency identification (RFID) technologies, such as EPC Gen2. We present a lightweight PRNG design for low-cost passive RFID tags, named J3Gen. J3Gen is based on a linear feedback shift register (LFSR) configured with multiple feedback polynomials. The polynomials are alternated during the generation of sequences via a physical source of randomness. J3Gen successfully handles the inherent linearity of LFSR based PRNGs and satisfies the statistical requirements imposed by the EPC Gen2 standard. A hardware implementation of J3Gen is presented and evaluated with regard to different design parameters, defining the key-equivalence security and nonlinearity of the design. The results of a SPICE simulation confirm the power-consumption suitability of the proposal. PMID:23519344

Vehicle Sprung Mass Estimation for Rough Terrain

DTIC Science & Technology

2011-03-01

distributions are greater than zero. The multivariate polynomials are functions of the Legendre polynomials (Poularikas (1999...developed methods based on polynomial chaos theory and on the maximum likelihood approach to estimate the most likely value of the vehicle sprung...mass. The polynomial chaos estimator is compared to benchmark algorithms including recursive least squares, recursive total least squares, extended

Some comparisons of complexity in dictionary-based and linear computational models.

PubMed

Gnecco, Giorgio; Kůrková, Věra; Sanguineti, Marcello

2011-03-01

Neural networks provide a more flexible approximation of functions than traditional linear regression. In the latter, one can only adjust the coefficients in linear combinations of fixed sets of functions, such as orthogonal polynomials or Hermite functions, while for neural networks, one may also adjust the parameters of the functions which are being combined. However, some useful properties of linear approximators (such as uniqueness, homogeneity, and continuity of best approximation operators) are not satisfied by neural networks. Moreover, optimization of parameters in neural networks becomes more difficult than in linear regression. Experimental results suggest that these drawbacks of neural networks are offset by substantially lower model complexity, allowing accuracy of approximation even in high-dimensional cases. We give some theoretical results comparing requirements on model complexity for two types of approximators, the traditional linear ones and so called variable-basis types, which include neural networks, radial, and kernel models. We compare upper bounds on worst-case errors in variable-basis approximation with lower bounds on such errors for any linear approximator. Using methods from nonlinear approximation and integral representations tailored to computational units, we describe some cases where neural networks outperform any linear approximator. Copyright © 2010 Elsevier Ltd. All rights reserved.

Data driven discrete-time parsimonious identification of a nonlinear state-space model for a weakly nonlinear system with short data record

NASA Astrophysics Data System (ADS)

Relan, Rishi; Tiels, Koen; Marconato, Anna; Dreesen, Philippe; Schoukens, Johan

2018-05-01

Many real world systems exhibit a quasi linear or weakly nonlinear behavior during normal operation, and a hard saturation effect for high peaks of the input signal. In this paper, a methodology to identify a parsimonious discrete-time nonlinear state space model (NLSS) for the nonlinear dynamical system with relatively short data record is proposed. The capability of the NLSS model structure is demonstrated by introducing two different initialisation schemes, one of them using multivariate polynomials. In addition, a method using first-order information of the multivariate polynomials and tensor decomposition is employed to obtain the parsimonious decoupled representation of the set of multivariate real polynomials estimated during the identification of NLSS model. Finally, the experimental verification of the model structure is done on the cascaded water-benchmark identification problem.

A Test Strategy for High Resolution Image Scanners.

DTIC Science & Technology

1983-10-01

for multivariate analysis. Holt, Richart and Winston, Inc., New York. Graybill , F.A., 1961: An introduction to linear statistical models . SVolume I...i , j i -(7) 02 1 )2 y 4n .i ij 13 The linear estimation model for the polynomial coefficients can be set up as - =; =(8) with T = ( x’ . . X-nn "X...Resolution Image Scanner MTF Geometrical and radiometric performance Dynamic range, linearity , noise - Dynamic scanning errors Response uniformity Skewness of

Study on the mapping of dark matter clustering from real space to redshift space

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zheng, Yi; Song, Yong-Seon, E-mail: yizheng@kasi.re.kr, E-mail: ysong@kasi.re.kr

The mapping of dark matter clustering from real space to redshift space introduces the anisotropic property to the measured density power spectrum in redshift space, known as the redshift space distortion effect. The mapping formula is intrinsically non-linear, which is complicated by the higher order polynomials due to indefinite cross correlations between the density and velocity fields, and the Finger-of-God effect due to the randomness of the peculiar velocity field. Whilst the full higher order polynomials remain unknown, the other systematics can be controlled consistently within the same order truncation in the expansion of the mapping formula, as shown inmore » this paper. The systematic due to the unknown non-linear density and velocity fields is removed by separately measuring all terms in the expansion directly using simulations. The uncertainty caused by the velocity randomness is controlled by splitting the FoG term into two pieces, 1) the ''one-point' FoG term being independent of the separation vector between two different points, and 2) the ''correlated' FoG term appearing as an indefinite polynomials which is expanded in the same order as all other perturbative polynomials. Using 100 realizations of simulations, we find that the Gaussian FoG function with only one scale-independent free parameter works quite well, and that our new mapping formulation accurately reproduces the observed 2-dimensional density power spectrum in redshift space at the smallest scales by far, up to k ∼ 0.2 Mpc{sup -1}, considering the resolution of future experiments.« less

On a Family of Multivariate Modified Humbert Polynomials

PubMed Central

Aktaş, Rabia; Erkuş-Duman, Esra

2013-01-01

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

Measurement of the hyperelastic properties of 44 pathological ex vivo breast tissue samples

NASA Astrophysics Data System (ADS)

O'Hagan, Joseph J.; Samani, Abbas

2009-04-01

The elastic and hyperelastic properties of biological soft tissues have been of interest to the medical community. There are several biomedical applications where parameters characterizing such properties are critical for a reliable clinical outcome. These applications include surgery planning, needle biopsy and brachtherapy where tissue biomechanical modeling is involved. Another important application is interpreting nonlinear elastography images. While there has been considerable research on the measurement of the linear elastic modulus of small tissue samples, little research has been conducted for measuring parameters that characterize the nonlinear elasticity of tissues included in tissue slice specimens. This work presents hyperelastic measurement results of 44 pathological ex vivo breast tissue samples. For each sample, five hyperelastic models have been used, including the Yeoh, N = 2 polynomial, N = 1 Ogden, Arruda-Boyce, and Veronda-Westmann models. Results show that the Yeoh, polynomial and Ogden models are the most accurate in terms of fitting experimental data. The results indicate that almost all of the parameters corresponding to the pathological tissues are between two times to over two orders of magnitude larger than those of normal tissues, with C11 showing the most significant difference. Furthermore, statistical analysis indicates that C02 of the Yeoh model, and C11 and C20 of the polynomial model have very good potential for cancer classification as they show statistically significant differences for various cancer types, especially for invasive lobular carcinoma. In addition to the potential for use in cancer classification, the presented data are very important for applications such as surgery planning and virtual reality based clinician training systems where accurate nonlinear tissue response modeling is required.

Modelling the breeding of Aedes Albopictus species in an urban area in Pulau Pinang using polynomial regression

NASA Astrophysics Data System (ADS)

Salleh, Nur Hanim Mohd; Ali, Zalila; Noor, Norlida Mohd.; Baharum, Adam; Saad, Ahmad Ramli; Sulaiman, Husna Mahirah; Ahmad, Wan Muhamad Amir W.

2014-07-01

Polynomial regression is used to model a curvilinear relationship between a response variable and one or more predictor variables. It is a form of a least squares linear regression model that predicts a single response variable by decomposing the predictor variables into an nth order polynomial. In a curvilinear relationship, each curve has a number of extreme points equal to the highest order term in the polynomial. A quadratic model will have either a single maximum or minimum, whereas a cubic model has both a relative maximum and a minimum. This study used quadratic modeling techniques to analyze the effects of environmental factors: temperature, relative humidity, and rainfall distribution on the breeding of Aedes albopictus, a type of Aedes mosquito. Data were collected at an urban area in south-west Penang from September 2010 until January 2011. The results indicated that the breeding of Aedes albopictus in the urban area is influenced by all three environmental characteristics. The number of mosquito eggs is estimated to reach a maximum value at a medium temperature, a medium relative humidity and a high rainfall distribution.

Tensor calculus in polar coordinates using Jacobi polynomials

NASA Astrophysics Data System (ADS)

Vasil, Geoffrey M.; Burns, Keaton J.; Lecoanet, Daniel; Olver, Sheehan; Brown, Benjamin P.; Oishi, Jeffrey S.

2016-11-01

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.

Polynomial sequences for bond percolation critical thresholds

DOE PAGES

Scullard, Christian R.

2011-09-22

In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4, 6, 12) and (3 4, 6) using the linearity approximation described in (Scullard and Ziff, J. Stat. Mech. 03021), implemented as a branching process of lattices. I find the estimates for the bond percolation thresholds, pc(4, 6, 12) = 0.69377849... and p c(3 4, 6) = 0.43437077..., compared with Parviainen’s numerical results of p c = 0.69373383... and p c = 0.43430621... . These deviations are of the order 10 -5, as is standard for this method. Deriving thresholds in this way for a given latticemore » leads to a polynomial with integer coefficients, the root in [0, 1] of which gives the estimate for the bond threshold and I show how the method can be refined, leading to a series of higher order polynomials making predictions that likely converge to the exact answer. Finally, I discuss how this fact hints that for certain graphs, such as the kagome lattice, the exact bond threshold may not be the root of any polynomial with integer coefficients.« less

Identities associated with Milne-Thomson type polynomials and special numbers.

PubMed

Simsek, Yilmaz; Cakic, Nenad

2018-01-01

The purpose of this paper is to give identities and relations including the Milne-Thomson polynomials, the Hermite polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the central factorial numbers, and the Cauchy numbers. By using fermionic and bosonic p -adic integrals, we derive some new relations and formulas related to these numbers and polynomials, and also the combinatorial sums.

Decomposition Theory in the Teaching of Elementary Linear Algebra.

ERIC Educational Resources Information Center

London, R. R.; Rogosinski, H. P.

1990-01-01

Described is a decomposition theory from which the Cayley-Hamilton theorem, the diagonalizability of complex square matrices, and functional calculus can be developed. The theory and its applications are based on elementary polynomial algebra. (KR)

Orthogonal polynomials for refinable linear functionals

NASA Astrophysics Data System (ADS)

Laurie, Dirk; de Villiers, Johan

2006-12-01

A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments. Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. The algorithm requires O(n^2) rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic.

Universality of Generalized Bunching and Efficient Assessment of Boson Sampling.

PubMed

Shchesnovich, V S

2016-03-25

It is found that identical bosons (fermions) show a generalized bunching (antibunching) property in linear networks: the absolute maximum (minimum) of the probability that all N input particles are detected in a subset of K output modes of any nontrivial linear M-mode network is attained only by completely indistinguishable bosons (fermions). For fermions K is arbitrary; for bosons it is either (i) arbitrary for only classically correlated bosons or (ii) satisfies K≥N (or K=1) for arbitrary input states of N particles. The generalized bunching allows us to certify in a polynomial in N number of runs that a physical device realizing boson sampling with an arbitrary network operates in the regime of full quantum coherence compatible only with completely indistinguishable bosons. The protocol needs only polynomial classical computations for the standard boson sampling, whereas an analytic formula is available for the scattershot version.

Free and Forced Vibrations of Thick-Walled Anisotropic Cylindrical Shells

NASA Astrophysics Data System (ADS)

Marchuk, A. V.; Gnedash, S. V.; Levkovskii, S. A.

2017-03-01

Two approaches to studying the free and forced axisymmetric vibrations of cylindrical shell are proposed. They are based on the three-dimensional theory of elasticity and division of the original cylindrical shell with concentric cross-sectional circles into several coaxial cylindrical shells. One approach uses linear polynomials to approximate functions defined in plan and across the thickness. The other approach also uses linear polynomials to approximate functions defined in plan, but their variation with thickness is described by the analytical solution of a system of differential equations. Both approaches have approximation and arithmetic errors. When determining the natural frequencies by the semi-analytical finite-element method in combination with the divide and conqure method, it is convenient to find the initial frequencies by the finite-element method. The behavior of the shell during free and forced vibrations is analyzed in the case where the loading area is half the shell thickness

An Approach to Stable Gradient-Descent Adaptation of Higher Order Neural Units.

PubMed

Bukovsky, Ivo; Homma, Noriyasu

2017-09-01

Stability evaluation of a weight-update system of higher order neural units (HONUs) with polynomial aggregation of neural inputs (also known as classes of polynomial neural networks) for adaptation of both feedforward and recurrent HONUs by a gradient descent method is introduced. An essential core of the approach is based on the spectral radius of a weight-update system, and it allows stability monitoring and its maintenance at every adaptation step individually. Assuring the stability of the weight-update system (at every single adaptation step) naturally results in the adaptation stability of the whole neural architecture that adapts to the target data. As an aside, the used approach highlights the fact that the weight optimization of HONU is a linear problem, so the proposed approach can be generally extended to any neural architecture that is linear in its adaptable parameters.

A new third order finite volume weighted essentially non-oscillatory scheme on tetrahedral meshes

NASA Astrophysics Data System (ADS)

Zhu, Jun; Qiu, Jianxian

2017-11-01

In this paper a third order finite volume weighted essentially non-oscillatory scheme is designed for solving hyperbolic conservation laws on tetrahedral meshes. Comparing with other finite volume WENO schemes designed on tetrahedral meshes, the crucial advantages of such new WENO scheme are its simplicity and compactness with the application of only six unequal size spatial stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, and easy choice of the positive linear weights without considering the topology of the meshes. The original innovation of such scheme is to use a quadratic polynomial defined on a big central spatial stencil for obtaining third order numerical approximation at any points inside the target tetrahedral cell in smooth region and switch to at least one of five linear polynomials defined on small biased/central spatial stencils for sustaining sharp shock transitions and keeping essentially non-oscillatory property simultaneously. By performing such new procedures in spatial reconstructions and adopting a third order TVD Runge-Kutta time discretization method for solving the ordinary differential equation (ODE), the new scheme's memory occupancy is decreased and the computing efficiency is increased. So it is suitable for large scale engineering requirements on tetrahedral meshes. Some numerical results are provided to illustrate the good performance of such scheme.

Characterizing the Lyα forest flux probability distribution function using Legendre polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cieplak, Agnieszka M.; Slosar, Anze

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

Characterizing the Lyα forest flux probability distribution function using Legendre polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cieplak, Agnieszka M.; Slosar, Anže, E-mail: acieplak@bnl.gov, E-mail: anze@bnl.gov

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

Characterizing the Lyα forest flux probability distribution function using Legendre polynomials

DOE PAGES

Cieplak, Agnieszka M.; Slosar, Anze

2017-10-12

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

Direct localization of poles of a meromorphic function from measurements on an incomplete boundary

NASA Astrophysics Data System (ADS)

Nara, Takaaki; Ando, Shigeru

2010-01-01

This paper proposes an algebraic method to reconstruct the positions of multiple poles in a meromorphic function field from measurements on an arbitrary simple arc in it. A novel issue is the exactness of the algorithm depending on whether the arc is open or closed, and whether it encloses or does not enclose the poles. We first obtain a differential equation that can equivalently determine the meromorphic function field. From it, we derive linear equations that relate the elementary symmetric polynomials of the pole positions to weighted integrals of the field along the simple arc and end-point terms of the arc when it is an open one. Eliminating the end-point terms based on an appropriate choice of weighting functions and a combination of the linear equations, we obtain a simple system of linear equations for solving the elementary symmetric polynomials. We also show that our algorithm can be applied to a 2D electric impedance tomography problem. The effects of the proximity of the poles, the number of measurements and noise on the localization accuracy are numerically examined.

Characterizing the Lyα forest flux probability distribution function using Legendre polynomials

NASA Astrophysics Data System (ADS)

Cieplak, Agnieszka M.; Slosar, Anže

2017-10-01

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

Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations

NASA Astrophysics Data System (ADS)

Gerdt, Vladimir P.; Blinkov, Yuri A.; Mozzhilkin, Vladimir V.

2006-05-01

In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.

Kurtosis Approach for Nonlinear Blind Source Separation

NASA Technical Reports Server (NTRS)

Duong, Vu A.; Stubbemd, Allen R.

2005-01-01

In this paper, we introduce a new algorithm for blind source signal separation for post-nonlinear mixtures. The mixtures are assumed to be linearly mixed from unknown sources first and then distorted by memoryless nonlinear functions. The nonlinear functions are assumed to be smooth and can be approximated by polynomials. Both the coefficients of the unknown mixing matrix and the coefficients of the approximated polynomials are estimated by the gradient descent method conditional on the higher order statistical requirements. The results of simulation experiments presented in this paper demonstrate the validity and usefulness of our approach for nonlinear blind source signal separation.

Response Surface Modeling Tolerance and Inference Error Risk Specifications: Proposed Industry Standards

NASA Technical Reports Server (NTRS)

DeLoach, Richard

2012-01-01

This paper reviews the derivation of an equation for scaling response surface modeling experiments. The equation represents the smallest number of data points required to fit a linear regression polynomial so as to achieve certain specified model adequacy criteria. Specific criteria are proposed which simplify an otherwise rather complex equation, generating a practical rule of thumb for the minimum volume of data required to adequately fit a polynomial with a specified number of terms in the model. This equation and the simplified rule of thumb it produces can be applied to minimize the cost of wind tunnel testing.

Simplified Syndrome Decoding of (n, 1) Convolutional Codes

NASA Technical Reports Server (NTRS)

Reed, I. S.; Truong, T. K.

1983-01-01

A new syndrome decoding algorithm for the (n, 1) convolutional codes (CC) that is different and simpler than the previous syndrome decoding algorithm of Schalkwijk and Vinck is presented. The new algorithm uses the general solution of the polynomial linear Diophantine equation for the error polynomial vector E(D). This set of Diophantine solutions is a coset of the CC space. A recursive or Viterbi-like algorithm is developed to find the minimum weight error vector cirumflex E(D) in this error coset. An example illustrating the new decoding algorithm is given for the binary nonsymmetric (2,1)CC.

Smoothing optimization of supporting quadratic surfaces with Zernike polynomials

NASA Astrophysics Data System (ADS)

Zhang, Hang; Lu, Jiandong; Liu, Rui; Ma, Peifu

2018-03-01

A new optimization method to get a smooth freeform optical surface from an initial surface generated by the supporting quadratic method (SQM) is proposed. To smooth the initial surface, a 9-vertex system from the neighbor quadratic surface and the Zernike polynomials are employed to establish a linear equation system. A local optimized surface to the 9-vertex system can be build by solving the equations. Finally, a continuous smooth optimization surface is constructed by stitching the above algorithm on the whole initial surface. The spot corresponding to the optimized surface is no longer discrete pixels but a continuous distribution.

A linear and non-linear polynomial neural network modeling of dissolved oxygen content in surface water: Inter- and extrapolation performance with inputs' significance analysis.

PubMed

Šiljić Tomić, Aleksandra; Antanasijević, Davor; Ristić, Mirjana; Perić-Grujić, Aleksandra; Pocajt, Viktor

2018-01-01

Accurate prediction of water quality parameters (WQPs) is an important task in the management of water resources. Artificial neural networks (ANNs) are frequently applied for dissolved oxygen (DO) prediction, but often only their interpolation performance is checked. The aims of this research, beside interpolation, were the determination of extrapolation performance of ANN model, which was developed for the prediction of DO content in the Danube River, and the assessment of relationship between the significance of inputs and prediction error in the presence of values which were of out of the range of training. The applied ANN is a polynomial neural network (PNN) which performs embedded selection of most important inputs during learning, and provides a model in the form of linear and non-linear polynomial functions, which can then be used for a detailed analysis of the significance of inputs. Available dataset that contained 1912 monitoring records for 17 water quality parameters was split into a "regular" subset that contains normally distributed and low variability data, and an "extreme" subset that contains monitoring records with outlier values. The results revealed that the non-linear PNN model has good interpolation performance (R 2 =0.82), but it was not robust in extrapolation (R 2 =0.63). The analysis of extrapolation results has shown that the prediction errors are correlated with the significance of inputs. Namely, the out-of-training range values of the inputs with low importance do not affect significantly the PNN model performance, but their influence can be biased by the presence of multi-outlier monitoring records. Subsequently, linear PNN models were successfully applied to study the effect of water quality parameters on DO content. It was observed that DO level is mostly affected by temperature, pH, biological oxygen demand (BOD) and phosphorus concentration, while in extreme conditions the importance of alkalinity and bicarbonates rises over pH and BOD. Copyright © 2017 Elsevier B.V. All rights reserved.

Random regression analyses using B-spline functions to model growth of Nellore cattle.

PubMed

Boligon, A A; Mercadante, M E Z; Lôbo, R B; Baldi, F; Albuquerque, L G

2012-02-01

The objective of this study was to estimate (co)variance components using random regression on B-spline functions to weight records obtained from birth to adulthood. A total of 82 064 weight records of 8145 females obtained from the data bank of the Nellore Breeding Program (PMGRN/Nellore Brazil) which started in 1987, were used. The models included direct additive and maternal genetic effects and animal and maternal permanent environmental effects as random. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of age (cubic regression) were considered as random covariate. The random effects were modeled using B-spline functions considering linear, quadratic and cubic polynomials for each individual segment. Residual variances were grouped in five age classes. Direct additive genetic and animal permanent environmental effects were modeled using up to seven knots (six segments). A single segment with two knots at the end points of the curve was used for the estimation of maternal genetic and maternal permanent environmental effects. A total of 15 models were studied, with the number of parameters ranging from 17 to 81. The models that used B-splines were compared with multi-trait analyses with nine weight traits and to a random regression model that used orthogonal Legendre polynomials. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most appropriate and parsimonious model to describe the covariance structure of the data. Selection for higher weight, such as at young ages, should be performed taking into account an increase in mature cow weight. Particularly, this is important in most of Nellore beef cattle production systems, where the cow herd is maintained on range conditions. There is limited modification of the growth curve of Nellore cattle with respect to the aim of selecting them for rapid growth at young ages while maintaining constant adult weight.

Adaptive nonlinear polynomial neural networks for control of boundary layer/structural interaction

NASA Technical Reports Server (NTRS)

Parker, B. Eugene, Jr.; Cellucci, Richard L.; Abbott, Dean W.; Barron, Roger L.; Jordan, Paul R., III; Poor, H. Vincent

1993-01-01

The acoustic pressures developed in a boundary layer can interact with an aircraft panel to induce significant vibration in the panel. Such vibration is undesirable due to the aerodynamic drag and structure-borne cabin noises that result. The overall objective of this work is to develop effective and practical feedback control strategies for actively reducing this flow-induced structural vibration. This report describes the results of initial evaluations using polynomial, neural network-based, feedback control to reduce flow induced vibration in aircraft panels due to turbulent boundary layer/structural interaction. Computer simulations are used to develop and analyze feedback control strategies to reduce vibration in a beam as a first step. The key differences between this work and that going on elsewhere are as follows: that turbulent and transitional boundary layers represent broadband excitation and thus present a more complex stochastic control scenario than that of narrow band (e.g., laminar boundary layer) excitation; and secondly, that the proposed controller structures are adaptive nonlinear infinite impulse response (IIR) polynomial neural network, as opposed to the traditional adaptive linear finite impulse response (FIR) filters used in most studies to date. The controllers implemented in this study achieved vibration attenuation of 27 to 60 dB depending on the type of boundary layer established by laminar, turbulent, and intermittent laminar-to-turbulent transitional flows. Application of multi-input, multi-output, adaptive, nonlinear feedback control of vibration in aircraft panels based on polynomial neural networks appears to be feasible today. Plans are outlined for Phase 2 of this study, which will include extending the theoretical investigation conducted in Phase 2 and verifying the results in a series of laboratory experiments involving both bum and plate models.

Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ho, Choon-Lin, E-mail: hcl@mail.tku.edu.tw

2011-04-15

Research Highlights: > Physical examples involving exceptional orthogonal polynomials. > Exceptional polynomials as deformations of classical orthogonal polynomials. > Exceptional polynomials from Darboux-Crum transformation. - Abstract: An interesting discovery in the last two years in the field of mathematical physics has been the exceptional X{sub l} Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree l = 1, 2, and ..., and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new X{sub l} polynomials deserve further analysis, it ismore » also of interest to see if they play any role in physical systems. In this paper we indicate some physical models in which these new polynomials appear as the main part of the eigenfunctions. The systems we consider include the Dirac equations coupled minimally and non-minimally with some external fields, and the Fokker-Planck equations. The systems presented here have enlarged the number of exactly solvable physical systems known so far.« less

Use of polynomial expressions to describe the bioconcentration of hydrophobic chemicals by fish

DOE Office of Scientific and Technical Information (OSTI.GOV)

Connell, D.W.; Hawker, D.W.

1988-12-01

For the bioconcentration of hydrophobic chemicals by fish, relationships have been previously established between uptake rate constants (k1) and the octanol/water partition coefficient (Kow), and also between the clearance rate constant (k2) and Kow. These have been refined and extended on the basis of data for chlorinated hydrocarbons, and closely related compounds including polychlorinated dibenzodioxins, that covered a wider range of hydrophobicity (2.5 less than log Kow less than 9.5). This has allowed the development of new relationships between log Kow and various factors, including the bioconcentration factor (as log KB), equilibrium time (as log teq), and maximum biotic concentrationmore » (as log CB), which include extremely hydrophobic compounds previously not taken into account. The shape of the curves generated by these equations are in qualitative agreement with theoretical prediction and are described by polynomial expressions which are generally approximately linear over the more limited range of log Kow values used to develop previous relationships. The influences of factors such as hydrophobicity, aqueous solubility, molecular weight, lipid solubility, and also exposure time were considered. Decreasing lipid solubilities of extremely hydrophobic chemicals were found to result in increasing clearance rate constants, as well decreasing equilibrium times and bioconcentration factors.« less

A 2 epoch proper motion catalogue from the UKIDSS Large Area Survey

NASA Astrophysics Data System (ADS)

Smith, Leigh; Lucas, Phil; Burningham, Ben; Jones, Hugh; Pinfield, David; Smart, Ricky; Andrei, Alexandre

2013-04-01

The UKIDSS Large Area Survey (LAS) began in 2005, with the start of the UKIDSS program as a 7 year effort to survey roughly 4000 square degrees at high galactic latitudes in Y, J, H and K bands. The survey also included a significant quantity of 2-epoch J band observations, with epoch baselines ranging from 2 to 7 years. We present a proper motion catalogue for the 1500 square degrees of the 2 epoch LAS data, which includes some 800,000 sources with motions detected above the 5σ level. We developed a bespoke proper motion pipeline which applies a source-unique second order polynomial transformation to UKIDSS array coordinates of each source to counter potential local non-uniformity in the focal plane. Our catalogue agrees well with the proper motion data supplied in the current WFCAM Science Archive (WSA) DR9 catalogue where there is overlap, and in various optical catalogues, but it benefits from some improvements. One improvement is that we provide absolute proper motions, using LAS galaxies for the relative to absolute correction. Also, by using unique, local, 2nd order polynomial tranformations, as opposed to the linear transformations in the WSA, we correct better for any local distortions in the focal plane, not including the radial distortion that is removed by their pipeline.

Classification of Phylogenetic Profiles for Protein Function Prediction: An SVM Approach

NASA Astrophysics Data System (ADS)

Kotaru, Appala Raju; Joshi, Ramesh C.

Predicting the function of an uncharacterized protein is a major challenge in post-genomic era due to problems complexity and scale. Having knowledge of protein function is a crucial link in the development of new drugs, better crops, and even the development of biochemicals such as biofuels. Recently numerous high-throughput experimental procedures have been invented to investigate the mechanisms leading to the accomplishment of a protein’s function and Phylogenetic profile is one of them. Phylogenetic profile is a way of representing a protein which encodes evolutionary history of proteins. In this paper we proposed a method for classification of phylogenetic profiles using supervised machine learning method, support vector machine classification along with radial basis function as kernel for identifying functionally linked proteins. We experimentally evaluated the performance of the classifier with the linear kernel, polynomial kernel and compared the results with the existing tree kernel. In our study we have used proteins of the budding yeast saccharomyces cerevisiae genome. We generated the phylogenetic profiles of 2465 yeast genes and for our study we used the functional annotations that are available in the MIPS database. Our experiments show that the performance of the radial basis kernel is similar to polynomial kernel is some functional classes together are better than linear, tree kernel and over all radial basis kernel outperformed the polynomial kernel, linear kernel and tree kernel. In analyzing these results we show that it will be feasible to make use of SVM classifier with radial basis function as kernel to predict the gene functionality using phylogenetic profiles.

Necessary and sufficient conditions for the complete controllability and observability of systems in series using the coprime factorization of a rational matrix

NASA Technical Reports Server (NTRS)

Callier, F. M.; Nahum, C. D.

1975-01-01

The series connection of two linear time-invariant systems that have minimal state space system descriptions is considered. From these descriptions, strict-system-equivalent polynomial matrix system descriptions in the manner of Rosenbrock are derived. They are based on the factorization of the transfer matrix of the subsystems as a ratio of two right or left coprime polynomial matrices. They give rise to a simple polynomial matrix system description of the tandem connection. Theorem 1 states that for the complete controllability and observability of the state space system description of the series connection, it is necessary and sufficient that certain 'denominator' and 'numerator' groups are coprime. Consequences for feedback systems are drawn in Corollary 1. The role of pole-zero cancellations is explained by Lemma 3 and Corollaires 2 and 3.

On conjugate gradient type methods and polynomial preconditioners for a class of complex non-Hermitian matrices

NASA Technical Reports Server (NTRS)

Freund, Roland

1988-01-01

Conjugate gradient type methods are considered for the solution of large linear systems Ax = b with complex coefficient matrices of the type A = T + i(sigma)I where T is Hermitian and sigma, a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidian error minimization, respectively, are investigated. In particular, numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices are proposed. Error bounds for all three methods are derived. It is shown how the special shift structure of A can be preserved by using polynomial preconditioning. Results on the optimal choice of the polynomial preconditioner are given. Also, some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation are reported.

Convex optimisation approach to constrained fuel optimal control of spacecraft in close relative motion

NASA Astrophysics Data System (ADS)

Massioni, Paolo; Massari, Mauro

2018-05-01

This paper describes an interesting and powerful approach to the constrained fuel-optimal control of spacecraft in close relative motion. The proposed approach is well suited for problems under linear dynamic equations, therefore perfectly fitting to the case of spacecraft flying in close relative motion. If the solution of the optimisation is approximated as a polynomial with respect to the time variable, then the problem can be approached with a technique developed in the control engineering community, known as "Sum Of Squares" (SOS), and the constraints can be reduced to bounds on the polynomials. Such a technique allows rewriting polynomial bounding problems in the form of convex optimisation problems, at the cost of a certain amount of conservatism. The principles of the techniques are explained and some application related to spacecraft flying in close relative motion are shown.

Comparison of random regression models with Legendre polynomials and linear splines for production traits and somatic cell score of Canadian Holstein cows.

PubMed

Bohmanova, J; Miglior, F; Jamrozik, J; Misztal, I; Sullivan, P G

2008-09-01

A random regression model with both random and fixed regressions fitted by Legendre polynomials of order 4 was compared with 3 alternative models fitting linear splines with 4, 5, or 6 knots. The effects common for all models were a herd-test-date effect, fixed regressions on days in milk (DIM) nested within region-age-season of calving class, and random regressions for additive genetic and permanent environmental effects. Data were test-day milk, fat and protein yields, and SCS recorded from 5 to 365 DIM during the first 3 lactations of Canadian Holstein cows. A random sample of 50 herds consisting of 96,756 test-day records was generated to estimate variance components within a Bayesian framework via Gibbs sampling. Two sets of genetic evaluations were subsequently carried out to investigate performance of the 4 models. Models were compared by graphical inspection of variance functions, goodness of fit, error of prediction of breeding values, and stability of estimated breeding values. Models with splines gave lower estimates of variances at extremes of lactations than the model with Legendre polynomials. Differences among models in goodness of fit measured by percentages of squared bias, correlations between predicted and observed records, and residual variances were small. The deviance information criterion favored the spline model with 6 knots. Smaller error of prediction and higher stability of estimated breeding values were achieved by using spline models with 5 and 6 knots compared with the model with Legendre polynomials. In general, the spline model with 6 knots had the best overall performance based upon the considered model comparison criteria.

Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights

NASA Astrophysics Data System (ADS)

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

2009-12-01

We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form , with [gamma]>0, which include as particular cases the counterparts of the so-called Freud (i.e., when [phi] has a polynomial growth at infinity) and Erdös (when [phi] grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.

Optimizing Support Vector Machine Parameters with Genetic Algorithm for Credit Risk Assessment

NASA Astrophysics Data System (ADS)

Manurung, Jonson; Mawengkang, Herman; Zamzami, Elviawaty

2017-12-01

Support vector machine (SVM) is a popular classification method known to have strong generalization capabilities. SVM can solve the problem of classification and linear regression or nonlinear kernel which can be a learning algorithm for the ability of classification and regression. However, SVM also has a weakness that is difficult to determine the optimal parameter value. SVM calculates the best linear separator on the input feature space according to the training data. To classify data which are non-linearly separable, SVM uses kernel tricks to transform the data into a linearly separable data on a higher dimension feature space. The kernel trick using various kinds of kernel functions, such as : linear kernel, polynomial, radial base function (RBF) and sigmoid. Each function has parameters which affect the accuracy of SVM classification. To solve the problem genetic algorithms are proposed to be applied as the optimal parameter value search algorithm thus increasing the best classification accuracy on SVM. Data taken from UCI repository of machine learning database: Australian Credit Approval. The results show that the combination of SVM and genetic algorithms is effective in improving classification accuracy. Genetic algorithms has been shown to be effective in systematically finding optimal kernel parameters for SVM, instead of randomly selected kernel parameters. The best accuracy for data has been upgraded from kernel Linear: 85.12%, polynomial: 81.76%, RBF: 77.22% Sigmoid: 78.70%. However, for bigger data sizes, this method is not practical because it takes a lot of time.

Analytic Development of a Reference Profile for the First Entry in a Skip Atmospheric Entry

NASA Technical Reports Server (NTRS)

Garcia-Llama, Eduardo

2010-01-01

This note shows that a feasible reference drag profile for the first entry portion of a skip entry can be generated as a polynomial expression of the velocity. The coefficients of that polynomial are found through the resolution of a system composed of m + 1 equations, where m is the degree of the drag polynomial. It has been shown that a minimum of five equations (m = 4) are required to establish the range and the initial and final conditions on velocity and flight path angle. It has been shown that at least one constraint on the trajectory can be imposed through the addition of one extra equation in the system, which must be accompanied by the increase in the degree of the drag polynomial. In order to simplify the resolution of the system of equations, the drag was considered as being a probability density function of the velocity, with the velocity as a distribution function of the drag. Combining this notion with the introduction of empirically derived constants, it has been shown that the system of equations required to generate the drag profile can be successfully reduced to a system of linear algebraic equations. For completeness, the resulting drag profiles have been flown using the feedback linearization method of differential geometric control as a guidance law with the error dynamics of a second order homogeneous equation in the form of a damped oscillator. Satisfactory results were achieved when the gains in the error dynamics were changed at a certain point along the trajectory that is dependent on the velocity and the curvature of the drag as a function of the velocity. Future work should study the capacity to update the drag profile in flight when dispersions are introduced. Also, future studies should attempt to link the first entry, as presented and controlled in this note, with a more standard control concept for the second entry, such as the Apollo entry guidance, to try to assess the overall skip entry performance. A guidance law that includes an integral feedback term, as is the case in the actual Space Shuttle entry guidance and as is proposed in Ref 29, could be tried in future studies to assess whether its use results in an improvement of the tracking performance, and to evaluate the design needs when determining the control gains.

Genome-wide association study on legendre random regression coefficients for the growth and feed intake trajectory on Duroc Boars.

PubMed

Howard, Jeremy T; Jiao, Shihui; Tiezzi, Francesco; Huang, Yijian; Gray, Kent A; Maltecca, Christian

2015-05-30

Feed intake and growth are economically important traits in swine production. Previous genome wide association studies (GWAS) have utilized average daily gain or daily feed intake to identify regions that impact growth and feed intake across time. The use of longitudinal models in GWAS studies, such as random regression, allows for SNPs having a heterogeneous effect across the trajectory to be characterized. The objective of this study is therefore to conduct a single step GWAS (ssGWAS) on the animal polynomial coefficients for feed intake and growth. Corrected daily feed intake (DFI Adj) and average daily weight measurements (DBW Avg) on 8981 (n=525,240 observations) and 5643 (n=283,607 observations) animals were utilized in a random regression model using Legendre polynomials (order=2) and a relationship matrix that included genotyped and un-genotyped animals. A ssGWAS was conducted on the animal polynomials coefficients (intercept, linear and quadratic) for animals with genotypes (DFIAdj: n=855; DBWAvg: n=590). Regions were characterized based on the variance of 10-SNP sliding windows GEBV (WGEBV). A bootstrap analysis (n=1000) was conducted to declare significance. Heritability estimates for the traits trajectory ranged from 0.34-0.52 to 0.07-0.23 for DBWAvg and DFIAdj, respectively. Genetic correlations across age classes were large and positive for both DBWAvg and DFIAdj, albeit age classes at the beginning had a small to moderate genetic correlation with age classes towards the end of the trajectory for both traits. The WGEBV variance explained by significant regions (P<0.001) for each polynomial coefficient ranged from 0.2-0.9 to 0.3-1.01% for DBWAvg and DFIAdj, respectively. The WGEBV variance explained by significant regions for the trajectory was 1.54 and 1.95% for DBWAvg and DFIAdj. Both traits identified candidate genes with functions related to metabolite and energy homeostasis, glucose and insulin signaling and behavior. We have identified regions of the genome that have an impact on the intercept, linear and quadratic terms for DBWAvg and DFIAdj. These results provide preliminary evidence that individual growth and feed intake trajectories are impacted by different regions of the genome at different times.

Quantum calculus of classical vortex images, integrable models and quantum states

NASA Astrophysics Data System (ADS)

Pashaev, Oktay K.

2016-10-01

From two circle theorem described in terms of q-periodic functions, in the limit q→1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrodinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.

Sum-of-squares-based fuzzy controller design using quantum-inspired evolutionary algorithm

NASA Astrophysics Data System (ADS)

Yu, Gwo-Ruey; Huang, Yu-Chia; Cheng, Chih-Yung

2016-07-01

In the field of fuzzy control, control gains are obtained by solving stabilisation conditions in linear-matrix-inequality-based Takagi-Sugeno fuzzy control method and sum-of-squares-based polynomial fuzzy control method. However, the optimal performance requirements are not considered under those stabilisation conditions. In order to handle specific performance problems, this paper proposes a novel design procedure with regard to polynomial fuzzy controllers using quantum-inspired evolutionary algorithms. The first contribution of this paper is a combination of polynomial fuzzy control and quantum-inspired evolutionary algorithms to undertake an optimal performance controller design. The second contribution is the proposed stability condition derived from the polynomial Lyapunov function. The proposed design approach is dissimilar to the traditional approach, in which control gains are obtained by solving the stabilisation conditions. The first step of the controller design uses the quantum-inspired evolutionary algorithms to determine the control gains with the best performance. Then, the stability of the closed-loop system is analysed under the proposed stability conditions. To illustrate effectiveness and validity, the problem of balancing and the up-swing of an inverted pendulum on a cart is used.

Kurtosis Approach Nonlinear Blind Source Separation

NASA Technical Reports Server (NTRS)

Duong, Vu A.; Stubbemd, Allen R.

2005-01-01

In this paper, we introduce a new algorithm for blind source signal separation for post-nonlinear mixtures. The mixtures are assumed to be linearly mixed from unknown sources first and then distorted by memoryless nonlinear functions. The nonlinear functions are assumed to be smooth and can be approximated by polynomials. Both the coefficients of the unknown mixing matrix and the coefficients of the approximated polynomials are estimated by the gradient descent method conditional on the higher order statistical requirements. The results of simulation experiments presented in this paper demonstrate the validity and usefulness of our approach for nonlinear blind source signal separation Keywords: Independent Component Analysis, Kurtosis, Higher order statistics.

Model-based multi-fringe interferometry using Zernike polynomials

NASA Astrophysics Data System (ADS)

Gu, Wei; Song, Weihong; Wu, Gaofeng; Quan, Haiyang; Wu, Yongqian; Zhao, Wenchuan

2018-06-01

In this paper, a general phase retrieval method is proposed, which is based on one single interferogram with a small amount of fringes (either tilt or power). Zernike polynomials are used to characterize the phase to be measured; the phase distribution is reconstructed by a non-linear least squares method. Experiments show that the proposed method can obtain satisfactory results compared to the standard phase-shifting interferometry technique. Additionally, the retrace errors of proposed method can be neglected because of the few fringes; it does not need any auxiliary phase shifting facilities (low cost) and it is easy to implement without the process of phase unwrapping.

Active exterior cloaking for the 2D Laplace and Helmholtz equations.

PubMed

Vasquez, Fernando Guevara; Milton, Graeme W; Onofrei, Daniel

2009-08-14

A new cloaking method is presented for 2D quasistatics and the 2D Helmholtz equation that we speculate extends to other linear wave equations. For 2D quasistatics it is proven how a single active exterior cloaking device can be used to shield an object from surrounding fields, yet produce very small scattered fields. The problem is reduced to finding a polynomial which is close to 1 in a disk and close to 0 in another disk, and such a polynomial is constructed. For the 2D Helmholtz equation it is numerically shown that three exterior cloaking devices placed around the object suffice to hide it.

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

PubMed

Mahajan, Virendra N

2012-06-20

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

Exact zeros of entanglement for arbitrary rank-two mixtures derived from a geometric view of the zero polytope

NASA Astrophysics Data System (ADS)

Osterloh, Andreas

2016-12-01

Here I present a method for how intersections of a certain density matrix of rank 2 with the zero polytope can be calculated exactly. This is a purely geometrical procedure which thereby is applicable to obtaining the zeros of SL- and SU-invariant entanglement measures of arbitrary polynomial degree. I explain this method in detail for a recently unsolved problem. In particular, I show how a three-dimensional view, namely, in terms of the Bloch-sphere analogy, solves this problem immediately. To this end, I determine the zero polytope of the three-tangle, which is an exact result up to computer accuracy, and calculate upper bounds to its convex roof which are below the linearized upper bound. The zeros of the three-tangle (in this case) induced by the zero polytope (zero simplex) are exact values. I apply this procedure to a superposition of the four-qubit Greenberger-Horne-Zeilinger and W state. It can, however, be applied to every case one has under consideration, including an arbitrary polynomial convex-roof measure of entanglement and for arbitrary local dimension.

High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pieper, Andreas; Kreutzer, Moritz; Alvermann, Andreas, E-mail: alvermann@physik.uni-greifswald.de

2016-11-15

We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need formore » matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the 10{sup 2} innermost eigenpairs of a topological insulator matrix with dimension 10{sup 9} derived from quantum physics applications.« less

Dual-energy X-ray analysis using synchrotron computed tomography at 35 and 60 keV for the estimation of photon interaction coefficients describing attenuation and energy absorption.

PubMed

Midgley, Stewart; Schleich, Nanette

2015-05-01

A novel method for dual-energy X-ray analysis (DEXA) is tested using measurements of the X-ray linear attenuation coefficient μ. The key is a mathematical model that describes elemental cross sections using a polynomial in atomic number. The model is combined with the mixture rule to describe μ for materials, using the same polynomial coefficients. Materials are characterized by their electron density Ne and statistical moments Rk describing their distribution of elements, analogous to the concept of effective atomic number. In an experiment with materials of known density and composition, measurements of μ are written as a system of linear simultaneous equations, which is solved for the polynomial coefficients. DEXA itself involves computed tomography (CT) scans at two energies to provide a system of non-linear simultaneous equations that are solved for Ne and the fourth statistical moment R4. Results are presented for phantoms containing dilute salt solutions and for a biological specimen. The experiment identifies 1% systematic errors in the CT measurements, arising from third-harmonic radiation, and 20-30% noise, which is reduced to 3-5% by pre-processing with the median filter and careful choice of reconstruction parameters. DEXA accuracy is quantified for the phantom as the mean absolute differences for Ne and R4: 0.8% and 1.0% for soft tissue and 1.2% and 0.8% for bone-like samples, respectively. The DEXA results for the biological specimen are combined with model coefficients obtained from the tabulations to predict μ and the mass energy absorption coefficient at energies of 10 keV to 20 MeV.

AZTEC: A parallel iterative package for the solving linear systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hutchinson, S.A.; Shadid, J.N.; Tuminaro, R.S.

1996-12-31

We describe a parallel linear system package, AZTEC. The package incorporates a number of parallel iterative methods (e.g. GMRES, biCGSTAB, CGS, TFQMR) and preconditioners (e.g. Jacobi, Gauss-Seidel, polynomial, domain decomposition with LU or ILU within subdomains). Additionally, AZTEC allows for the reuse of previous preconditioning factorizations within Newton schemes for nonlinear methods. Currently, a number of different users are using this package to solve a variety of PDE applications.

The Shock and Vibration Digest, Volume 18, Number 3

DTIC Science & Technology

1986-03-01

Linear Distributed Parameter Des., Proc. Intl. Symp., 11th ONR Naval Struc. Systems by Shifted Legendre Polynomial Func- Mech. Symp., Tucson, AZ, pp...University, Atlanta, Georgia nonlinear problems with elementary algebra . It J. Sound Vib., 102 (2), pp 247-257 (Sept 22, uses i = -1, the Pascal’s...eigenvalues specified. The optimal avoid failure due to resonance under the action control problem of a linear distributed parameter 0School of Mechanical

Betti numbers of graded modules and cohomology of vector bundles

NASA Astrophysics Data System (ADS)

Eisenbud, David; Schreyer, Frank-Olaf

2009-07-01

In the remarkable paper Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture, Mats Boij and Jonas Soederberg conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring is a positive linear combination of Betti tables of modules with pure resolutions. We prove a strengthened form of their conjectures. Applications include a proof of the Multiplicity Conjecture of Huneke and Srinivasan and a proof of the convexity of a fan naturally associated to the Young lattice. With the same tools we show that the cohomology table of any vector bundle on projective space is a positive rational linear combination of the cohomology tables of what we call supernatural vector bundles. Using this result we give new bounds on the slope of a vector bundle in terms of its cohomology.

Anisotropic k-essence cosmologies

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chimento, Luis P.; Forte, Monica

We investigate a Bianchi type-I cosmology with k-essence and find the set of models which dissipate the initial anisotropy. There are cosmological models with extended tachyon fields and k-essence having a constant barotropic index. We obtain the conditions leading to a regular bounce of the average geometry and the residual anisotropy on the bounce. For constant potential, we develop purely kinetic k-essence models which are dust dominated in their early stages, dissipate the initial anisotropy, and end in a stable de Sitter accelerated expansion scenario. We show that linear k-field and polynomial kinetic function models evolve asymptotically to Friedmann-Robertson-Walker cosmologies.more » The linear case is compatible with an asymptotic potential interpolating between V{sub l}{proportional_to}{phi}{sup -{gamma}{sub l}}, in the shear dominated regime, and V{sub l}{proportional_to}{phi}{sup -2} at late time. In the polynomial case, the general solution contains cosmological models with an oscillatory average geometry. For linear k-essence, we find the general solution in the Bianchi type-I cosmology when the k field is driven by an inverse square potential. This model shares the same geometry as a quintessence field driven by an exponential potential.« less

Invariant algebraic surfaces for a virus dynamics

NASA Astrophysics Data System (ADS)

Valls, Claudia

2015-08-01

In this paper, we provide a complete classification of the invariant algebraic surfaces and of the rational first integrals for a well-known virus system. In the proofs, we use the weight-homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.

A point-value enhanced finite volume method based on approximate delta functions

NASA Astrophysics Data System (ADS)

Xuan, Li-Jun; Majdalani, Joseph

2018-02-01

We revisit the concept of an approximate delta function (ADF), introduced by Huynh (2011) [1], in the form of a finite-order polynomial that holds identical integral properties to the Dirac delta function when used in conjunction with a finite-order polynomial integrand over a finite domain. We show that the use of generic ADF polynomials can be effective at recovering and generalizing several high-order methods, including Taylor-based and nodal-based Discontinuous Galerkin methods, as well as the Correction Procedure via Reconstruction. Based on the ADF concept, we then proceed to formulate a Point-value enhanced Finite Volume (PFV) method, which stores and updates the cell-averaged values inside each element as well as the unknown quantities and, if needed, their derivatives on nodal points. The sharing of nodal information with surrounding elements saves the number of degrees of freedom compared to other compact methods at the same order. To ensure conservation, cell-averaged values are updated using an identical approach to that adopted in the finite volume method. Here, the updating of nodal values and their derivatives is achieved through an ADF concept that leverages all of the elements within the domain of integration that share the same nodal point. The resulting scheme is shown to be very stable at successively increasing orders. Both accuracy and stability of the PFV method are verified using a Fourier analysis and through applications to the linear wave and nonlinear Burgers' equations in one-dimensional space.

Application of Statistic Experimental Design to Assess the Effect of Gammairradiation Pre-Treatment on the Drying Characteristics and Qualities of Wheat

NASA Astrophysics Data System (ADS)

Yu, Yong; Wang, Jun

Wheat, pretreated by 60Co gamma irradiation, was dried by hot-air with irradiation dosage 0-3 kGy, drying temperature 40-60 °C, and initial moisture contents 19-25% (drying basis). The drying characteristics and dried qualities of wheat were evaluated based on drying time, average dehydration rate, wet gluten content (WGC), moisture content of wet gluten (MCWG)and titratable acidity (TA). A quadratic rotation-orthogonal composite experimental design, with three variables (at five levels) and five response functions, and analysis method were employed to study the effect of three variables on the individual response functions. The five response functions (drying time, average dehydration rate, WGC, MCWG, TA) correlated with these variables by second order polynomials consisting of linear, quadratic and interaction terms. A high correlation coefficient indicated the suitability of the second order polynomial to predict these response functions. The linear, interaction and quadratic effects of three variables on the five response functions were all studied.

A new basis set for molecular bending degrees of freedom.

PubMed

Jutier, Laurent

2010-07-21

We present a new basis set as an alternative to Legendre polynomials for the variational treatment of bending vibrational degrees of freedom in order to highly reduce the number of basis functions. This basis set is inspired from the harmonic oscillator eigenfunctions but is defined for a bending angle in the range theta in [0:pi]. The aim is to bring the basis functions closer to the final (ro)vibronic wave functions nature. Our methodology is extended to complicated potential energy surfaces, such as quasilinearity or multiequilibrium geometries, by using several free parameters in the basis functions. These parameters allow several density maxima, linear or not, around which the basis functions will be mainly located. Divergences at linearity in integral computations are resolved as generalized Legendre polynomials. All integral computations required for the evaluation of molecular Hamiltonian matrix elements are given for both discrete variable representation and finite basis representation. Convergence tests for the low energy vibronic states of HCCH(++), HCCH(+), and HCCS are presented.

Nonlinear channel equalization for QAM signal constellation using artificial neural networks.

PubMed

Patra, J C; Pal, R N; Baliarsingh, R; Panda, G

1999-01-01

Application of artificial neural networks (ANN's) to adaptive channel equalization in a digital communication system with 4-QAM signal constellation is reported in this paper. A novel computationally efficient single layer functional link ANN (FLANN) is proposed for this purpose. This network has a simple structure in which the nonlinearity is introduced by functional expansion of the input pattern by trigonometric polynomials. Because of input pattern enhancement, the FLANN is capable of forming arbitrarily nonlinear decision boundaries and can perform complex pattern classification tasks. Considering channel equalization as a nonlinear classification problem, the FLANN has been utilized for nonlinear channel equalization. The performance of the FLANN is compared with two other ANN structures [a multilayer perceptron (MLP) and a polynomial perceptron network (PPN)] along with a conventional linear LMS-based equalizer for different linear and nonlinear channel models. The effect of eigenvalue ratio (EVR) of input correlation matrix on the equalizer performance has been studied. The comparison of computational complexity involved for the three ANN structures is also provided.

Interpolation problem for the solutions of linear elasticity equations based on monogenic functions

NASA Astrophysics Data System (ADS)

Grigor'ev, Yuri; Gürlebeck, Klaus; Legatiuk, Dmitrii

2017-11-01

Interpolation is an important tool for many practical applications, and very often it is beneficial to interpolate not only with a simple basis system, but rather with solutions of a certain differential equation, e.g. elasticity equation. A typical example for such type of interpolation are collocation methods widely used in practice. It is known, that interpolation theory is fully developed in the framework of the classical complex analysis. However, in quaternionic analysis, which shows a lot of analogies to complex analysis, the situation is more complicated due to the non-commutative multiplication. Thus, a fundamental theorem of algebra is not available, and standard tools from linear algebra cannot be applied in the usual way. To overcome these problems, a special system of monogenic polynomials the so-called Pseudo Complex Polynomials, sharing some properties of complex powers, is used. In this paper, we present an approach to deal with the interpolation problem, where solutions of elasticity equations in three dimensions are used as an interpolation basis.

Flat bases of invariant polynomials and P-matrices of E{sub 7} and E{sub 8}

DOE Office of Scientific and Technical Information (OSTI.GOV)

Talamini, Vittorino

2010-02-15

Let G be a compact group of linear transformations of a Euclidean space V. The G-invariant C{sup {infinity}} functions can be expressed as C{sup {infinity}} functions of a finite basic set of G-invariant homogeneous polynomials, sometimes called an integrity basis. The mathematical description of the orbit space V/G depends on the integrity basis too: it is realized through polynomial equations and inequalities expressing rank and positive semidefiniteness conditions of the P-matrix, a real symmetric matrix determined by the integrity basis. The choice of the basic set of G-invariant homogeneous polynomials forming an integrity basis is not unique, so it ismore » not unique the mathematical description of the orbit space too. If G is an irreducible finite reflection group, Saito et al. [Commun. Algebra 8, 373 (1980)] characterized some special basic sets of G-invariant homogeneous polynomials that they called flat. They also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection groups except of the two largest groups E{sub 7} and E{sub 8}. In this paper the flat basic sets of invariant homogeneous polynomials of E{sub 7} and E{sub 8} and the corresponding P-matrices are determined explicitly. Using the results here reported one is able to determine easily the P-matrices corresponding to any other integrity basis of E{sub 7} or E{sub 8}. From the P-matrices one may then write down the equations and inequalities defining the orbit spaces of E{sub 7} and E{sub 8} relatively to a flat basis or to any other integrity basis. The results here obtained may be employed concretely to study analytically the symmetry breaking in all theories where the symmetry group is one of the finite reflection groups E{sub 7} and E{sub 8} or one of the Lie groups E{sub 7} and E{sub 8} in their adjoint representations.« less

Quasi-topological Ricci polynomial gravities

NASA Astrophysics Data System (ADS)

Li, Yue-Zhou; Liu, Hai-Shan; Lü, H.

2018-02-01

Quasi-topological terms in gravity can be viewed as those that give no contribution to the equations of motion for a special subclass of metric ansätze. They therefore play no rôle in constructing these solutions, but can affect the general perturbations. We consider Einstein gravity extended with Ricci tensor polynomial invariants, which admits Einstein metrics with appropriate effective cosmological constants as its vacuum solutions. We construct three types of quasi-topological gravities. The first type is for the most general static metrics with spherical, toroidal or hyperbolic isometries. The second type is for the special static metrics where g tt g rr is constant. The third type is the linearized quasitopological gravities on the Einstein metrics. We construct and classify results that are either dependent on or independent of dimensions, up to the tenth order. We then consider a subset of these three types and obtain Lovelock-like quasi-topological gravities, that are independent of the dimensions. The linearized gravities on Einstein metrics on all dimensions are simply Einstein and hence ghost free. The theories become quasi-topological on static metrics in one specific dimension, but non-trivial in others. We also focus on the quasi-topological Ricci cubic invariant in four dimensions as a specific example to study its effect on holography, including shear viscosity, thermoelectric DC conductivities and butterfly velocity. In particular, we find that the holographic diffusivity bounds can be violated by the quasi-topological terms, which can induce an extra massive mode that yields a butterfly velocity unbound above.

Characterization of high order spatial discretizations and lumping techniques for discontinuous finite element SN transport

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maginot, P. G.; Ragusa, J. C.; Morel, J. E.

2013-07-01

We examine several possible methods of mass matrix lumping for discontinuous finite element discrete ordinates transport using a Lagrange interpolatory polynomial trial space. Though positive outflow angular flux is guaranteed with traditional mass matrix lumping in a purely absorbing 1-D slab cell for the linear discontinuous approximation, we show that when used with higher degree interpolatory polynomial trial spaces, traditional lumping does yield strictly positive outflows and does not increase in accuracy with an increase in trial space polynomial degree. As an alternative, we examine methods which are 'self-lumping'. Self-lumping methods yield diagonal mass matrices by using numerical quadrature restrictedmore » to the Lagrange interpolatory points. Using equally-spaced interpolatory points, self-lumping is achieved through the use of closed Newton-Cotes formulas, resulting in strictly positive outflows in pure absorbers for odd power polynomials in 1-D slab geometry. By changing interpolatory points from the traditional equally-spaced points to the quadrature points of the Gauss-Legendre or Lobatto-Gauss-Legendre quadratures, it is possible to generate solution representations with a diagonal mass matrix and a strictly positive outflow for any degree polynomial solution representation in a pure absorber medium in 1-D slab geometry. Further, there is no inherent limit to local truncation error order of accuracy when using interpolatory points that correspond to the quadrature points of high order accuracy numerical quadrature schemes. (authors)« less

Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chin, Alex W.; Rivas, Angel; Huelga, Susana F.

2010-09-15

By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbor interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain systemmore » for a wide range of spectral functions and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short-range interactions of the effective chain system permit these open-quantum systems to be efficiently simulated by the density matrix renormalization group methods.« less

A two-step, fourth-order method with energy preserving properties

NASA Astrophysics Data System (ADS)

Brugnano, Luigi; Iavernaro, Felice; Trigiante, Donato

2012-09-01

We introduce a family of fourth-order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. As is the case with linear mutistep and one-leg methods, a prerogative of the new formulae is that the associated nonlinear systems to be solved at each step of the integration procedure have the very same dimension of the underlying continuous problem. The key tools in the new methods are the line integral associated with a conservative vector field (such as the one defined by a Hamiltonian dynamical system) and its discretization obtained by the aid of a quadrature formula. Energy conservation is equivalent to the requirement that the quadrature is exact, which turns out to be always the case in the event that the Hamiltonian function is a polynomial and the degree of precision of the quadrature formula is high enough. The non-polynomial case is also discussed and a number of test problems are finally presented in order to compare the behavior of the new methods to the theoretical results.

Evaluation of more general integrals involving universal associated Legendre polynomials

NASA Astrophysics Data System (ADS)

You, Yuan; Chen, Chang-Yuan; Tahir, Farida; Dong, Shi-Hai

2017-05-01

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. We present a popular integral formula which includes universal associated Legendre polynomials and we also evaluate some important integrals involving the product of two universal associated Legendre polynomials Pl' m'(x ) , Pk' n'(x ) and x2 a(1-x2 ) -p -1, xb(1±x2 ) -p, and xc(1-x2 ) -p(1±x ) -1, where l'≠k' and m'≠n'. Their selection rules are also mentioned.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei

The algebra H of the dual -1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl{sub -1}(2). The dual -1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q{yields}-1 limit of the dual q-Hahn polynomials. The Hopf algebra sl{sub -1}(2) has four generators including an involution, it is also a q{yields}-1 limit of the quantum algebra sl{sub q}(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of u(2) with an involution as additional generator, is first derived from the recurrence relation of themore » -1 Hahn polynomials. It is then shown that H can be realized in terms of the generators of two added sl{sub -1}(2) algebras, so that the Clebsch-Gordan coefficients of sl{sub -1}(2) are dual -1 Hahn polynomials. An irreducible representation of H involving five-diagonal matrices and connected to the difference equation of the dual -1 Hahn polynomials is constructed.« less

Coherent orthogonal polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Celeghini, E., E-mail: celeghini@fi.infn.it; Olmo, M.A. del, E-mail: olmo@fta.uva.es

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 relatemore » 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 the corresponding OP family. •Generalized coherent polynomials are obtained from OP.« less

General-Purpose Software For Computer Graphics

NASA Technical Reports Server (NTRS)

Rogers, Joseph E.

1992-01-01

NASA Device Independent Graphics Library (NASADIG) is general-purpose computer-graphics package for computer-based engineering and management applications which gives opportunity to translate data into effective graphical displays for presentation. Features include two- and three-dimensional plotting, spline and polynomial interpolation, control of blanking of areas, multiple log and/or linear axes, control of legends and text, control of thicknesses of curves, and multiple text fonts. Included are subroutines for definition of areas and axes of plots; setup and display of text; blanking of areas; setup of style, interpolation, and plotting of lines; control of patterns and of shading of colors; control of legends, blocks of text, and characters; initialization of devices; and setting of mixed alphabets. Written in FORTRAN 77.

Pulse transmission transmitter including a higher order time derivate filter

DOEpatents

Dress, Jr., William B.; Smith, Stephen F.

2003-09-23

Systems and methods for pulse-transmission low-power communication modes are disclosed. A pulse transmission transmitter includes: a clock; a pseudorandom polynomial generator coupled to the clock, the pseudorandom polynomial generator having a polynomial load input; an exclusive-OR gate coupled to the pseudorandom polynomial generator, the exclusive-OR gate having a serial data input; a programmable delay circuit coupled to both the clock and the exclusive-OR gate; a pulse generator coupled to the programmable delay circuit; and a higher order time derivative filter coupled to the pulse generator. The systems and methods significantly reduce lower-frequency emissions from pulse transmission spread-spectrum communication modes, which reduces potentially harmful interference to existing radio frequency services and users and also simultaneously permit transmission of multiple data bits by utilizing specific pulse shapes.

Image distortion analysis using polynomial series expansion.

PubMed

Baggenstoss, Paul M

2004-11-01

In this paper, we derive a technique for analysis of local distortions which affect data in real-world applications. In the paper, we focus on image data, specifically handwritten characters. Given a reference image and a distorted copy of it, the method is able to efficiently determine the rotations, translations, scaling, and any other distortions that have been applied. Because the method is robust, it is also able to estimate distortions for two unrelated images, thus determining the distortions that would be required to cause the two images to resemble each other. The approach is based on a polynomial series expansion using matrix powers of linear transformation matrices. The technique has applications in pattern recognition in the presence of distortions.

A frequency domain global parameter estimation method for multiple reference frequency response measurements

NASA Astrophysics Data System (ADS)

Shih, C. Y.; Tsuei, Y. G.; Allemang, R. J.; Brown, D. L.

1988-10-01

A method of using the matrix Auto-Regressive Moving Average (ARMA) model in the Laplace domain for multiple-reference global parameter identification is presented. This method is particularly applicable to the area of modal analysis where high modal density exists. The method is also applicable when multiple reference frequency response functions are used to characterise linear systems. In order to facilitate the mathematical solution, the Forsythe orthogonal polynomial is used to reduce the ill-conditioning of the formulated equations and to decouple the normal matrix into two reduced matrix blocks. A Complex Mode Indicator Function (CMIF) is introduced, which can be used to determine the proper order of the rational polynomials.

Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

NASA Astrophysics Data System (ADS)

Pazner, Will; Persson, Per-Olof

2018-02-01

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O (p2d) storage and O (p3d) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O (p d + 1) storage, O (p d + 1) work in two spatial dimensions, and O (p d + 2) work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O (p9) to O (p5) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier-Stokes equations, using polynomials of degree up to p = 30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p.

Genetic analyses of stillbirth in relation to litter size using random regression models.

PubMed

Chen, C Y; Misztal, I; Tsuruta, S; Herring, W O; Holl, J; Culbertson, M

2010-12-01

Estimates of genetic parameters for number of stillborns (NSB) in relation to litter size (LS) were obtained with random regression models (RRM). Data were collected from 4 purebred Duroc nucleus farms between 2004 and 2008. Two data sets with 6,575 litters for the first parity (P1) and 6,259 litters for the second to fifth parity (P2-5) with a total of 8,217 and 5,066 animals in the pedigree were analyzed separately. Number of stillborns was studied as a trait on sow level. Fixed effects were contemporary groups (farm-year-season) and fixed cubic regression coefficients on LS with Legendre polynomials. Models for P2-5 included the fixed effect of parity. Random effects were additive genetic effects for both data sets with permanent environmental effects included for P2-5. Random effects modeled with Legendre polynomials (RRM-L), linear splines (RRM-S), and degree 0 B-splines (RRM-BS) with regressions on LS were used. For P1, the order of polynomial, the number of knots, and the number of intervals used for respective models were quadratic, 3, and 3, respectively. For P2-5, the same parameters were linear, 2, and 2, respectively. Heterogeneous residual variances were considered in the models. For P1, estimates of heritability were 12 to 15%, 5 to 6%, and 6 to 7% in LS 5, 9, and 13, respectively. For P2-5, estimates were 15 to 17%, 4 to 5%, and 4 to 6% in LS 6, 9, and 12, respectively. For P1, average estimates of genetic correlations between LS 5 to 9, 5 to 13, and 9 to 13 were 0.53, -0.29, and 0.65, respectively. For P2-5, same estimates averaged for RRM-L and RRM-S were 0.75, -0.21, and 0.50, respectively. For RRM-BS with 2 intervals, the correlation was 0.66 between LS 5 to 7 and 8 to 13. Parameters obtained by 3 RRM revealed the nonlinear relationship between additive genetic effect of NSB and the environmental deviation of LS. The negative correlations between the 2 extreme LS might possibly indicate different genetic bases on incidence of stillbirth.

Pointwise convergence of derivatives of Lagrange interpolation polynomials for exponential weights

NASA Astrophysics Data System (ADS)

Damelin, S. B.; Jung, H. S.

2005-01-01

For a general class of exponential weights on the line and on (-1,1), we study pointwise convergence of the derivatives of Lagrange interpolation. Our weights include even weights of smooth polynomial decay near +/-[infinity] (Freud weights), even weights of faster than smooth polynomial decay near +/-[infinity] (Erdos weights) and even weights which vanish strongly near +/-1, for example Pollaczek type weights.

Pre-Service Teachers' Perceptions and Beliefs of Technological Pedagogical Content Knowledge on Algebra

ERIC Educational Resources Information Center

Lin, Cheng-Yao; Kuo, Yu-Chun; Ko, Yi-Yin

2015-01-01

The purpose of this study was to investigate elementary pre-service teachers' content knowledge in algebra (Linear Equation, Quadratic Equation, Functions, System Equations and Polynomials) as well as their technological pedagogical content knowledge (TPACK) in teaching algebra. Participants were 79 undergraduate pre-service teachers who were…

Meta-Regression Approximations to Reduce Publication Selection Bias

ERIC Educational Resources Information Center

Stanley, T. D.; Doucouliagos, Hristos

2014-01-01

Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression approximations to reduce this bias. Our approach employs Taylor polynomial approximations to the conditional mean of a truncated distribution. A quadratic approximation without a linear term, precision-effect estimate with…

Prospective Mathematics Teachers' Sense Making of Polynomial Multiplication and Factorization Modeled with Algebra Tiles

ERIC Educational Resources Information Center

Caglayan, Günhan

2013-01-01

This study is about prospective secondary mathematics teachers' understanding and sense making of representational quantities generated by algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations--referent preserving versus referent…

Bi-cubic interpolation for shift-free pan-sharpening

NASA Astrophysics Data System (ADS)

Aiazzi, Bruno; Baronti, Stefano; Selva, Massimo; Alparone, Luciano

2013-12-01

Most of pan-sharpening techniques require the re-sampling of the multi-spectral (MS) image for matching the size of the panchromatic (Pan) image, before the geometric details of Pan are injected into the MS image. This operation is usually performed in a separable fashion by means of symmetric digital low-pass filtering kernels with odd lengths that utilize piecewise local polynomials, typically implementing linear or cubic interpolation functions. Conversely, constant, i.e. nearest-neighbour, and quadratic kernels, implementing zero and two degree polynomials, respectively, introduce shifts in the magnified images, that are sub-pixel in the case of interpolation by an even factor, as it is the most usual case. However, in standard satellite systems, the point spread functions (PSF) of the MS and Pan instruments are centered in the middle of each pixel. Hence, commercial MS and Pan data products, whose scale ratio is an even number, are relatively shifted by an odd number of half pixels. Filters of even lengths may be exploited to compensate the half-pixel shifts between the MS and Pan sampling grids. In this paper, it is shown that separable polynomial interpolations of odd degrees are feasible with linear-phase kernels of even lengths. The major benefit is that bi-cubic interpolation, which is known to represent the best trade-off between performances and computational complexity, can be applied to commercial MS + Pan datasets, without the need of performing a further half-pixel registration after interpolation, to align the expanded MS with the Pan image.

Algebraic special functions and SO(3,2)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Celeghini, E., E-mail: celeghini@fi.infn.it; Olmo, M.A. del, E-mail: olmo@fta.uva.es

2013-06-15

A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L{sup 2} functions defined on (−1,1)×Z and on the sphere S{sup 2}, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining inmore » this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L{sup 2} functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to get a Lie group representation defining “algebraic special functions”. •The “algebraic special functions” connect Lie algebras and L{sup 2} functions.« less

Beampattern control of a microphone array to minimize secondary source contamination.

PubMed

Jordan, Peter; Fitzpatrick, John A; Meskell, Craig

2003-10-01

A null-steering technique is adapted and applied to a linear delay-and-sum beamformer in order to measure the noise generated by one of the propellers of a 1/8 scale twin propeller aircraft model. The technique involves shading the linear array using a set of weights, which are calculated according to the locations onto which the nulls need to be steered (in this case onto the second propeller). The technique is based on an established microwave antenna theory, and uses a plane-wave, or far field formulation in order to represent the response of the array by an nth-order polynomial, where n is the number of array elements. The roots of this polynomial correspond to the minima of the array response, and so by an appropriate choice of roots, a polynomial can be generated, the coefficients of which are the weights needed to achieve the prespecified set of null positions. It is shown that, for the technique to work with actual data, the cross-spectral matrix must be conditioned before array shading is implemented. This ensures that the shading function is not distorted by the intrinsic element weighting which can occur as a result of the directional nature of aeroacoustic systems. A difference of 6 dB between measurements before and after null steering shows the technique to have been effective in eliminating the contribution from one of the propellers, thus providing a quantitative measure of the acoustic energy from the other.

On-line estimation of nonlinear physical systems

USGS Publications Warehouse

Christakos, G.

1988-01-01

Recursive algorithms for estimating states of nonlinear physical systems are presented. Orthogonality properties are rediscovered and the associated polynomials are used to linearize state and observation models of the underlying random processes. This requires some key hypotheses regarding the structure of these processes, which may then take account of a wide range of applications. The latter include streamflow forecasting, flood estimation, environmental protection, earthquake engineering, and mine planning. The proposed estimation algorithm may be compared favorably to Taylor series-type filters, nonlinear filters which approximate the probability density by Edgeworth or Gram-Charlier series, as well as to conventional statistical linearization-type estimators. Moreover, the method has several advantages over nonrecursive estimators like disjunctive kriging. To link theory with practice, some numerical results for a simulated system are presented, in which responses from the proposed and extended Kalman algorithms are compared. ?? 1988 International Association for Mathematical Geology.

Exponential Sensitivity and its Cost in Quantum Physics

PubMed Central

Gilyén, András; Kiss, Tamás; Jex, Igor

2016-01-01

State selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system has to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schrödinger microscope) is possible; however, there is a strict bound on the number of copies needed. PMID:26861076

Exponential Sensitivity and its Cost in Quantum Physics.

PubMed

Gilyén, András; Kiss, Tamás; Jex, Igor

2016-02-10

State selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system has to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schrödinger microscope) is possible; however, there is a strict bound on the number of copies needed.

FIT: Computer Program that Interactively Determines Polynomial Equations for Data which are a Function of Two Independent Variables

NASA Technical Reports Server (NTRS)

Arbuckle, P. D.; Sliwa, S. M.; Roy, M. L.; Tiffany, S. H.

1985-01-01

A computer program for interactively developing least-squares polynomial equations to fit user-supplied data is described. The program is characterized by the ability to compute the polynomial equations of a surface fit through data that are a function of two independent variables. The program utilizes the Langley Research Center graphics packages to display polynomial equation curves and data points, facilitating a qualitative evaluation of the effectiveness of the fit. An explanation of the fundamental principles and features of the program, as well as sample input and corresponding output, are included.

Analytical potential-density pairs for bars

NASA Astrophysics Data System (ADS)

Vogt, D.; Letelier, P. S.

2010-11-01

An identity that relates multipolar solutions of the Einstein equations to Newtonian potentials of bars with linear densities proportional to Legendre polynomials is used to construct analytical potential-density pairs of infinitesimally thin bars with a given linear density profile. By means of a suitable transformation, softened bars that are free of singularities are also obtained. As an application we study the equilibrium points and stability for the motion of test particles in the gravitational field for three models of rotating bars.

Grid Effect on Spherical Shallow Water Jets Using Continuous and Discontinuous Galerkin Methods

DTIC Science & Technology

2013-01-01

The high-order Legendre -Gauss-Lobatto (LGL) points are added to the linear grid by projecting the linear elements onto the auxiliary gnomonic space...mapping, the triangles are subdivided into smaller ones by a Lagrange polynomial of order nI . The number of quadrilateral elements and grid points of...of the acceleration of gravity and the vertical height of the fluid), ν∇2 is the artificial viscosity term of viscous coefficient ν = 1× 105 m2 s−1

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

DTIC Science & Technology

2014-11-01

linear hybrid systems by linear algebraic methods. In SAS, volume 6337 of LNCS, pages 373–389. Springer, 2010. [19] E. W. Mayr. Membership in polynomial...383–394, 2009. [31] A. Tarski. A decision method for elementary algebra and geometry. Bull. Amer. Math. Soc., 59, 1951. [32] A. Tiwari. Abstractions...A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 November 2014 CMU

Genetic parameters for test-day yield of milk, fat and protein in buffaloes estimated by random regression models.

PubMed

Aspilcueta-Borquis, Rúsbel R; Araujo Neto, Francisco R; Baldi, Fernando; Santos, Daniel J A; Albuquerque, Lucia G; Tonhati, Humberto

2012-08-01

The test-day yields of milk, fat and protein were analysed from 1433 first lactations of buffaloes of the Murrah breed, daughters of 113 sires from 12 herds in the state of São Paulo, Brazil, born between 1985 and 2007. For the test-day yields, 10 monthly classes of lactation days were considered. The contemporary groups were defined as the herd-year-month of the test day. Random additive genetic, permanent environmental and residual effects were included in the model. The fixed effects considered were the contemporary group, number of milkings (1 or 2 milkings), linear and quadratic effects of the covariable cow age at calving and the mean lactation curve of the population (modelled by third-order Legendre orthogonal polynomials). The random additive genetic and permanent environmental effects were estimated by means of regression on third- to sixth-order Legendre orthogonal polynomials. The residual variances were modelled with a homogenous structure and various heterogeneous classes. According to the likelihood-ratio test, the best model for milk and fat production was that with four residual variance classes, while a third-order Legendre polynomial was best for the additive genetic effect for milk and fat yield, a fourth-order polynomial was best for the permanent environmental effect for milk production and a fifth-order polynomial was best for fat production. For protein yield, the best model was that with three residual variance classes and third- and fourth-order Legendre polynomials were best for the additive genetic and permanent environmental effects, respectively. The heritability estimates for the characteristics analysed were moderate, varying from 0·16±0·05 to 0·29±0·05 for milk yield, 0·20±0·05 to 0·30±0·08 for fat yield and 0·18±0·06 to 0·27±0·08 for protein yield. The estimates of the genetic correlations between the tests varied from 0·18±0·120 to 0·99±0·002; from 0·44±0·080 to 0·99±0·004; and from 0·41±0·080 to 0·99±0·004, for milk, fat and protein production, respectively, indicating that whatever the selection criterion used, indirect genetic gains can be expected throughout the lactation curve.

Universal Racah matrices and adjoint knot polynomials: Arborescent knots

NASA Astrophysics Data System (ADS)

Mironov, A.; Morozov, A.

2016-04-01

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SUN) and Kauffman (SON) polynomials. For E8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the "eigenvalue conjecture", which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6 × 6 case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.

Random regression models on Legendre polynomials to estimate genetic parameters for weights from birth to adult age in Canchim cattle.

PubMed

Baldi, F; Albuquerque, L G; Alencar, M M

2010-08-01

The objective of this work was to estimate covariance functions for direct and maternal genetic effects, animal and maternal permanent environmental effects, and subsequently, to derive relevant genetic parameters for growth traits in Canchim cattle. Data comprised 49,011 weight records on 2435 females from birth to adult age. The model of analysis included fixed effects of contemporary groups (year and month of birth and at weighing) and age of dam as quadratic covariable. Mean trends were taken into account by a cubic regression on orthogonal polynomials of animal age. Residual variances were allowed to vary and were modelled by a step function with 1, 4 or 11 classes based on animal's age. The model fitting four classes of residual variances was the best. A total of 12 random regression models from second to seventh order were used to model direct and maternal genetic effects, animal and maternal permanent environmental effects. The model with direct and maternal genetic effects, animal and maternal permanent environmental effects fitted by quadric, cubic, quintic and linear Legendre polynomials, respectively, was the most adequate to describe the covariance structure of the data. Estimates of direct and maternal heritability obtained by multi-trait (seven traits) and random regression models were very similar. Selection for higher weight at any age, especially after weaning, will produce an increase in mature cow weight. The possibility to modify the growth curve in Canchim cattle to obtain animals with rapid growth at early ages and moderate to low mature cow weight is limited.

SU-E-J-85: Leave-One-Out Perturbation (LOOP) Fitting Algorithm for Absolute Dose Film Calibration

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chu, A; Ahmad, M; Chen, Z

2014-06-01

Purpose: To introduce an outliers-recognition fitting routine for film dosimetry. It cannot only be flexible with any linear and non-linear regression but also can provide information for the minimal number of sampling points, critical sampling distributions and evaluating analytical functions for absolute film-dose calibration. Methods: The technique, leave-one-out (LOO) cross validation, is often used for statistical analyses on model performance. We used LOO analyses with perturbed bootstrap fitting called leave-one-out perturbation (LOOP) for film-dose calibration . Given a threshold, the LOO process detects unfit points (“outliers”) compared to other cohorts, and a bootstrap fitting process follows to seek any possibilitiesmore » of using perturbations for further improvement. After that outliers were reconfirmed by a traditional t-test statistics and eliminated, then another LOOP feedback resulted in the final. An over-sampled film-dose- calibration dataset was collected as a reference (dose range: 0-800cGy), and various simulated conditions for outliers and sampling distributions were derived from the reference. Comparisons over the various conditions were made, and the performance of fitting functions, polynomial and rational functions, were evaluated. Results: (1) LOOP can prove its sensitive outlier-recognition by its statistical correlation to an exceptional better goodness-of-fit as outliers being left-out. (2) With sufficient statistical information, the LOOP can correct outliers under some low-sampling conditions that other “robust fits”, e.g. Least Absolute Residuals, cannot. (3) Complete cross-validated analyses of LOOP indicate that the function of rational type demonstrates a much superior performance compared to the polynomial. Even with 5 data points including one outlier, using LOOP with rational function can restore more than a 95% value back to its reference values, while the polynomial fitting completely failed under the same conditions. Conclusion: LOOP can cooperate with any fitting routine functioning as a “robust fit”. In addition, it can be set as a benchmark for film-dose calibration fitting performance.« less

? and ? nonquadratic stabilisation of discrete-time Takagi-Sugeno systems based on multi-instant fuzzy Lyapunov functions

NASA Astrophysics Data System (ADS)

Tognetti, Eduardo S.; Oliveira, Ricardo C. L. F.; Peres, Pedro L. D.

2015-01-01

The problem of state feedback control design for discrete-time Takagi-Sugeno (TS) (T-S) fuzzy systems is investigated in this paper. A Lyapunov function, which is quadratic in the state and presents a multi-polynomial dependence on the fuzzy weighting functions at the current and past instants of time, is proposed.This function contains, as particular cases, other previous Lyapunov functions already used in the literature, being able to provide less conservative conditions of control design for TS fuzzy systems. The structure of the proposed Lyapunov function also motivates the design of a new stabilising compensator for Takagi-Sugeno fuzzy systems. The main novelty of the proposed state feedback control law is that the gain is composed of matrices with multi-polynomial dependence on the fuzzy weighting functions at a set of past instants of time, including the current one. The conditions for the existence of a stabilising state feedback control law that minimises an upper bound to the ? or ? norms are given in terms of linear matrix inequalities. Numerical examples show that the approach can be less conservative and more efficient than other methods available in the literature.

A rational fraction polynomials model to study vertical dynamic wheel-rail interaction

NASA Astrophysics Data System (ADS)

Correa, N.; Vadillo, E. G.; Santamaria, J.; Gómez, J.

2012-04-01

This paper presents a model designed to study vertical interactions between wheel and rail when the wheel moves over a rail welding. The model focuses on the spatial domain, and is drawn up in a simple fashion from track receptances. The paper obtains the receptances from a full track model in the frequency domain already developed by the authors, which includes deformation of the rail section and propagation of bending, elongation and torsional waves along an infinite track. Transformation between domains was secured by applying a modified rational fraction polynomials method. This obtains a track model with very few degrees of freedom, and thus with minimum time consumption for integration, with a good match to the original model over a sufficiently broad range of frequencies. Wheel-rail interaction is modelled on a non-linear Hertzian spring, and consideration is given to parametric excitation caused by the wheel moving over a sleeper, since this is a moving wheel model and not a moving irregularity model. The model is used to study the dynamic loads and displacements emerging at the wheel-rail contact passing over a welding defect at different speeds.

Classes of exact Einstein Maxwell solutions

NASA Astrophysics Data System (ADS)

Komathiraj, K.; Maharaj, S. D.

2007-12-01

We find new classes of exact solutions to the Einstein Maxwell system of equations for a charged sphere with a particular choice of the electric field intensity and one of the gravitational potentials. The condition of pressure isotropy is reduced to a linear, second order differential equation which can be solved in general. Consequently we can find exact solutions to the Einstein Maxwell field equations corresponding to a static spherically symmetric gravitational potential in terms of hypergeometric functions. It is possible to find exact solutions which can be written explicitly in terms of elementary functions, namely polynomials and product of polynomials and algebraic functions. Uncharged solutions are regainable with our choice of electric field intensity; in particular we generate the Einstein universe for particular parameter values.

Nonlinear Structured Growth Mixture Models in M"plus" and OpenMx

ERIC Educational Resources Information Center

Grimm, Kevin J.; Ram, Nilam; Estabrook, Ryne

2010-01-01

Growth mixture models (GMMs; B. O. Muthen & Muthen, 2000; B. O. Muthen & Shedden, 1999) are a combination of latent curve models (LCMs) and finite mixture models to examine the existence of latent classes that follow distinct developmental patterns. GMMs are often fit with linear, latent basis, multiphase, or polynomial change models…

Ghosts of Mathematicians Past: Paolo Ruffini

ERIC Educational Resources Information Center

Fitzherbert, John

2016-01-01

Paolo Ruffini (1765-1822) may be something of an unknown in high school mathematics; however his contributions to the world of mathematics are a rich source of inspiration. Ruffini's rule (often known as "synthetic division") is an efficient method of dividing a polynomial by a linear factor, with or without a remainder. The process can…

Final Report - Subcontract B623760

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bank, R.

2017-11-17

During my visit to LLNL during July 17{27, 2017, I worked on linear system solvers. The two level hierarchical solver that initiated our study was developed to solve linear systems arising from hp adaptive finite element calculations, and is implemented in the PLTMG software package, version 12. This preconditioner typically requires 3-20% of the space used by the stiffness matrix for higher order elements. It has multigrid like convergence rates for a wide variety of PDEs (self-adjoint positive de nite elliptic equations, convection dominated convection-diffusion equations, and highly indefinite Helmholtz equations, among others). The convergence rate is not independent ofmore » the polynomial degree p as p ! 1, but but remains strong for p 9, which is the highest polynomial degree allowed in PLTMG, due to limitations of the numerical quadrature rules implemented in the software package. A more complete description of the method and some numerical experiments illustrating its effectiveness appear in. Like traditional geometric multilevel methods, this scheme relies on knowledge of the underlying finite element space in order to construct the smoother and the coarse grid correction.« less

Learning epistatic interactions from sequence-activity data to predict enantioselectivity

NASA Astrophysics Data System (ADS)

Zaugg, Julian; Gumulya, Yosephine; Malde, Alpeshkumar K.; Bodén, Mikael

2017-12-01

Enzymes with a high selectivity are desirable for improving economics of chemical synthesis of enantiopure compounds. To improve enzyme selectivity mutations are often introduced near the catalytic active site. In this compact environment epistatic interactions between residues, where contributions to selectivity are non-additive, play a significant role in determining the degree of selectivity. Using support vector machine regression models we map mutations to the experimentally characterised enantioselectivities for a set of 136 variants of the epoxide hydrolase from the fungus Aspergillus niger (AnEH). We investigate whether the influence a mutation has on enzyme selectivity can be accurately predicted through linear models, and whether prediction accuracy can be improved using higher-order counterparts. Comparing linear and polynomial degree = 2 models, mean Pearson coefficients (r) from 50 {× } 5 -fold cross-validation increase from 0.84 to 0.91 respectively. Equivalent models tested on interaction-minimised sequences achieve values of r=0.90 and r=0.93 . As expected, testing on a simulated control data set with no interactions results in no significant improvements from higher-order models. Additional experimentally derived AnEH mutants are tested with linear and polynomial degree = 2 models, with values increasing from r=0.51 to r=0.87 respectively. The study demonstrates that linear models perform well, however the representation of epistatic interactions in predictive models improves identification of selectivity-enhancing mutations. The improvement is attributed to higher-order kernel functions that represent epistatic interactions between residues.

Learning epistatic interactions from sequence-activity data to predict enantioselectivity

NASA Astrophysics Data System (ADS)

Zaugg, Julian; Gumulya, Yosephine; Malde, Alpeshkumar K.; Bodén, Mikael

2017-12-01

Enzymes with a high selectivity are desirable for improving economics of chemical synthesis of enantiopure compounds. To improve enzyme selectivity mutations are often introduced near the catalytic active site. In this compact environment epistatic interactions between residues, where contributions to selectivity are non-additive, play a significant role in determining the degree of selectivity. Using support vector machine regression models we map mutations to the experimentally characterised enantioselectivities for a set of 136 variants of the epoxide hydrolase from the fungus Aspergillus niger ( AnEH). We investigate whether the influence a mutation has on enzyme selectivity can be accurately predicted through linear models, and whether prediction accuracy can be improved using higher-order counterparts. Comparing linear and polynomial degree = 2 models, mean Pearson coefficients ( r) from 50 {× } 5-fold cross-validation increase from 0.84 to 0.91 respectively. Equivalent models tested on interaction-minimised sequences achieve values of r=0.90 and r=0.93. As expected, testing on a simulated control data set with no interactions results in no significant improvements from higher-order models. Additional experimentally derived AnEH mutants are tested with linear and polynomial degree = 2 models, with values increasing from r=0.51 to r=0.87 respectively. The study demonstrates that linear models perform well, however the representation of epistatic interactions in predictive models improves identification of selectivity-enhancing mutations. The improvement is attributed to higher-order kernel functions that represent epistatic interactions between residues.

Learning epistatic interactions from sequence-activity data to predict enantioselectivity.

PubMed

Zaugg, Julian; Gumulya, Yosephine; Malde, Alpeshkumar K; Bodén, Mikael

2017-12-01

Enzymes with a high selectivity are desirable for improving economics of chemical synthesis of enantiopure compounds. To improve enzyme selectivity mutations are often introduced near the catalytic active site. In this compact environment epistatic interactions between residues, where contributions to selectivity are non-additive, play a significant role in determining the degree of selectivity. Using support vector machine regression models we map mutations to the experimentally characterised enantioselectivities for a set of 136 variants of the epoxide hydrolase from the fungus Aspergillus niger (AnEH). We investigate whether the influence a mutation has on enzyme selectivity can be accurately predicted through linear models, and whether prediction accuracy can be improved using higher-order counterparts. Comparing linear and polynomial degree = 2 models, mean Pearson coefficients (r) from [Formula: see text]-fold cross-validation increase from 0.84 to 0.91 respectively. Equivalent models tested on interaction-minimised sequences achieve values of [Formula: see text] and [Formula: see text]. As expected, testing on a simulated control data set with no interactions results in no significant improvements from higher-order models. Additional experimentally derived AnEH mutants are tested with linear and polynomial degree = 2 models, with values increasing from [Formula: see text] to [Formula: see text] respectively. The study demonstrates that linear models perform well, however the representation of epistatic interactions in predictive models improves identification of selectivity-enhancing mutations. The improvement is attributed to higher-order kernel functions that represent epistatic interactions between residues.

Anomalous dielectric relaxation with linear reaction dynamics in space-dependent force fields.

PubMed

Hong, Tao; Tang, Zhengming; Zhu, Huacheng

2016-12-28

The anomalous dielectric relaxation of disordered reaction with linear reaction dynamics is studied via the continuous time random walk model in the presence of space-dependent electric field. Two kinds of modified reaction-subdiffusion equations are derived for different linear reaction processes by the master equation, including the instantaneous annihilation reaction and the noninstantaneous annihilation reaction. If a constant proportion of walkers is added or removed instantaneously at the end of each step, there will be a modified reaction-subdiffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term. If the walkers are added or removed at a constant per capita rate during the waiting time between steps, there will be a standard linear reaction kinetics term but a fractional order temporal derivative operating on an anomalous diffusion term. The dielectric polarization is analyzed based on the Legendre polynomials and the dielectric properties of both reactions can be expressed by the effective rotational diffusion function and component concentration function, which is similar to the standard reaction-diffusion process. The results show that the effective permittivity can be used to describe the dielectric properties in these reactions if the chemical reaction time is much longer than the relaxation time.

Generalised Transfer Functions of Neural Networks

NASA Astrophysics Data System (ADS)

Fung, C. F.; Billings, S. A.; Zhang, H.

1997-11-01

When artificial neural networks are used to model non-linear dynamical systems, the system structure which can be extremely useful for analysis and design, is buried within the network architecture. In this paper, explicit expressions for the frequency response or generalised transfer functions of both feedforward and recurrent neural networks are derived in terms of the network weights. The derivation of the algorithm is established on the basis of the Taylor series expansion of the activation functions used in a particular neural network. This leads to a representation which is equivalent to the non-linear recursive polynomial model and enables the derivation of the transfer functions to be based on the harmonic expansion method. By mapping the neural network into the frequency domain information about the structure of the underlying non-linear system can be recovered. Numerical examples are included to demonstrate the application of the new algorithm. These examples show that the frequency response functions appear to be highly sensitive to the network topology and training, and that the time domain properties fail to reveal deficiencies in the trained network structure.

Direct discriminant locality preserving projection with Hammerstein polynomial expansion.

PubMed

Chen, Xi; Zhang, Jiashu; Li, Defang

2012-12-01

Discriminant locality preserving projection (DLPP) is a linear approach that encodes discriminant information into the objective of locality preserving projection and improves its classification ability. To enhance the nonlinear description ability of DLPP, we can optimize the objective function of DLPP in reproducing kernel Hilbert space to form a kernel-based discriminant locality preserving projection (KDLPP). However, KDLPP suffers the following problems: 1) larger computational burden; 2) no explicit mapping functions in KDLPP, which results in more computational burden when projecting a new sample into the low-dimensional subspace; and 3) KDLPP cannot obtain optimal discriminant vectors, which exceedingly optimize the objective of DLPP. To overcome the weaknesses of KDLPP, in this paper, a direct discriminant locality preserving projection with Hammerstein polynomial expansion (HPDDLPP) is proposed. The proposed HPDDLPP directly implements the objective of DLPP in high-dimensional second-order Hammerstein polynomial space without matrix inverse, which extracts the optimal discriminant vectors for DLPP without larger computational burden. Compared with some other related classical methods, experimental results for face and palmprint recognition problems indicate the effectiveness of the proposed HPDDLPP.

Pedestrian detection in crowded scenes with the histogram of gradients principle

NASA Astrophysics Data System (ADS)

Sidla, O.; Rosner, M.; Lypetskyy, Y.

2006-10-01

This paper describes a close to real-time scale invariant implementation of a pedestrian detector system which is based on the Histogram of Oriented Gradients (HOG) principle. Salient HOG features are first selected from a manually created very large database of samples with an evolutionary optimization procedure that directly trains a polynomial Support Vector Machine (SVM). Real-time operation is achieved by a cascaded 2-step classifier which uses first a very fast linear SVM (with the same features as the polynomial SVM) to reject most of the irrelevant detections and then computes the decision function with a polynomial SVM on the remaining set of candidate detections. Scale invariance is achieved by running the detector of constant size on scaled versions of the original input images and by clustering the results over all resolutions. The pedestrian detection system has been implemented in two versions: i) fully body detection, and ii) upper body only detection. The latter is especially suited for very busy and crowded scenarios. On a state-of-the-art PC it is able to run at a frequency of 8 - 20 frames/sec.

Sensor selection cost optimisation for tracking structurally cyclic systems: a P-order solution

NASA Astrophysics Data System (ADS)

Doostmohammadian, M.; Zarrabi, H.; Rabiee, H. R.

2017-08-01

Measurements and sensing implementations impose certain cost in sensor networks. The sensor selection cost optimisation is the problem of minimising the sensing cost of monitoring a physical (or cyber-physical) system. Consider a given set of sensors tracking states of a dynamical system for estimation purposes. For each sensor assume different costs to measure different (realisable) states. The idea is to assign sensors to measure states such that the global cost is minimised. The number and selection of sensor measurements need to ensure the observability to track the dynamic state of the system with bounded estimation error. The main question we address is how to select the state measurements to minimise the cost while satisfying the observability conditions. Relaxing the observability condition for structurally cyclic systems, the main contribution is to propose a graph theoretic approach to solve the problem in polynomial time. Note that polynomial time algorithms are suitable for large-scale systems as their running time is upper-bounded by a polynomial expression in the size of input for the algorithm. We frame the problem as a linear sum assignment with solution complexity of ?.

Mixed kernel function support vector regression for global sensitivity analysis

NASA Astrophysics Data System (ADS)

Cheng, Kai; Lu, Zhenzhou; Wei, Yuhao; Shi, Yan; Zhou, Yicheng

2017-11-01

Global sensitivity analysis (GSA) plays an important role in exploring the respective effects of input variables on an assigned output response. Amongst the wide sensitivity analyses in literature, the Sobol indices have attracted much attention since they can provide accurate information for most models. In this paper, a mixed kernel function (MKF) based support vector regression (SVR) model is employed to evaluate the Sobol indices at low computational cost. By the proposed derivation, the estimation of the Sobol indices can be obtained by post-processing the coefficients of the SVR meta-model. The MKF is constituted by the orthogonal polynomials kernel function and Gaussian radial basis kernel function, thus the MKF possesses both the global characteristic advantage of the polynomials kernel function and the local characteristic advantage of the Gaussian radial basis kernel function. The proposed approach is suitable for high-dimensional and non-linear problems. Performance of the proposed approach is validated by various analytical functions and compared with the popular polynomial chaos expansion (PCE). Results demonstrate that the proposed approach is an efficient method for global sensitivity analysis.

Polynomial Expressions for Estimating Elastic Constants From the Resonance of Circular Plates

NASA Technical Reports Server (NTRS)

Salem, Jonathan A.; Singh, Abhishek

2005-01-01

Two approaches were taken to make convenient spread sheet calculations of elastic constants from resonance data and the tables in ASTM C1259 and E1876: polynomials were fit to the tables; and an automated spread sheet interpolation routine was generated. To compare the approaches, the resonant frequencies of circular plates made of glass, hardened maraging steel, alpha silicon carbide, silicon nitride, tungsten carbide, tape cast NiO-YSZ, and zinc selenide were measured. The elastic constants, as calculated via the polynomials and linear interpolation of the tabular data in ASTM C1259 and E1876, were found comparable for engineering purposes, with the differences typically being less than 0.5 percent. Calculation of additional v values at t/R between 0 and 0.2 would allow better curve fits. This is not necessary for common engineering purposes, however, it might benefit the testing of emerging thin structures such as fuel cell electrolytes, gas conversion membranes, and coatings when Poisson s ratio is less than 0.15 and high precision is needed.

Evaluating convex roof entanglement measures.

PubMed

Tóth, Géza; Moroder, Tobias; Gühne, Otfried

2015-04-24

We show a powerful method to compute entanglement measures based on convex roof constructions. In particular, our method is applicable to measures that, for pure states, can be written as low order polynomials of operator expectation values. We show how to compute the linear entropy of entanglement, the linear entanglement of assistance, and a bound on the dimension of the entanglement for bipartite systems. We discuss how to obtain the convex roof of the three-tangle for three-qubit states. We also show how to calculate the linear entropy of entanglement and the quantum Fisher information based on partial information or device independent information. We demonstrate the usefulness of our method by concrete examples.

Stability margin of linear systems with parameters described by fuzzy numbers.

PubMed

Husek, Petr

2011-10-01

This paper deals with the linear systems with uncertain parameters described by fuzzy numbers. The problem of determining the stability margin of those systems with linear affine dependence of the coefficients of a characteristic polynomial on system parameters is studied. Fuzzy numbers describing the system parameters are allowed to be characterized by arbitrary nonsymmetric membership functions. An elegant solution, graphical in nature, based on generalization of the Tsypkin-Polyak plot is presented. The advantage of the presented approach over the classical robust concept is demonstrated on a control of the Fiat Dedra engine model and a control of the quarter car suspension model.

Separation of the long-term thermal effects from the strain measurements in the Geodynamics Laboratory of Lanzarote

NASA Astrophysics Data System (ADS)

Venedikov, A. P.; Arnoso, J.; Cai, W.; Vieira, R.; Tan, S.; Velez, E. J.

2006-01-01

A 12-year series (1992-2004) of strain measurements recorded in the Geodynamics Laboratory of Lanzarote is investigated. Through a tidal analysis the non-tidal component of the data is separated in order to use it for studying signals, useful for monitoring of the volcanic activity on the island. This component contains various perturbations of meteorological and oceanic origin, which should be eliminated in order to make the useful signals discernible. The paper is devoted to the estimation and elimination of the effect of the air temperature inside the station, which strongly dominates the strainmeter data. For solving this task, a regression model is applied, which includes a linear relation with the temperature and time-dependant polynomials. The regression includes nonlinearly a set of parameters, which are estimated by a properly applied Bayesian approach. The results obtained are: the regression coefficient of the strain data on temperature is equal to (-367.4 ± 0.8) × 10 -9 °C -1, the curve of the non-tidal component reduced by the effect of the temperature and a polynomial approximation of the reduced curve. The technique used here can be helpful to investigators in the domain of the earthquake and volcano monitoring. However, the fundamental and extremely difficult problem of what kind of signals in the reduced curves might be useful in this field is not considered here.

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

DOE Office of Scientific and Technical Information (OSTI.GOV)

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

2014-01-15

Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge–Kutta-like time-steps to advance the parabolic terms by a time-step that is s{sup 2} times larger than amore » single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge–Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems – a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very useful in parabolic problems with variable diffusion coefficients. This includes variable coefficient parabolic equations that might give rise to skew symmetric terms. The RKC1 and RKC2 schemes do not share this convex monotonicity preserving property. One-dimensional and two-dimensional von Neumann stability analyses of RKC1, RKC2, RKL1 and RKL2 are also presented, showing that the latter two have some advantages. The paper includes several details to facilitate implementation. A detailed accuracy analysis is presented to show that the methods reach their design accuracies. A stringent set of test problems is also presented. To demonstrate the robustness and versatility of our methods, we show their successful operation on problems involving linear and non-linear heat conduction and viscosity, resistive magnetohydrodynamics, ambipolar diffusion dominated magnetohydrodynamics, level set methods and flux limited radiation diffusion. In a prior paper (Meyer, Balsara and Aslam 2012 [36]) we have also presented an extensive test-suite showing that the RKL2 method works robustly in the presence of shocks in an anisotropically conducting, magnetized plasma.« less

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

NASA Astrophysics Data System (ADS)

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

2014-01-01

Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge-Kutta-like time-steps to advance the parabolic terms by a time-step that is s2 times larger than a single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge-Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems - a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very useful in parabolic problems with variable diffusion coefficients. This includes variable coefficient parabolic equations that might give rise to skew symmetric terms. The RKC1 and RKC2 schemes do not share this convex monotonicity preserving property. One-dimensional and two-dimensional von Neumann stability analyses of RKC1, RKC2, RKL1 and RKL2 are also presented, showing that the latter two have some advantages. The paper includes several details to facilitate implementation. A detailed accuracy analysis is presented to show that the methods reach their design accuracies. A stringent set of test problems is also presented. To demonstrate the robustness and versatility of our methods, we show their successful operation on problems involving linear and non-linear heat conduction and viscosity, resistive magnetohydrodynamics, ambipolar diffusion dominated magnetohydrodynamics, level set methods and flux limited radiation diffusion. In a prior paper (Meyer, Balsara and Aslam 2012 [36]) we have also presented an extensive test-suite showing that the RKL2 method works robustly in the presence of shocks in an anisotropically conducting, magnetized plasma.

A Parallel Implicit Reconstructed Discontinuous Galerkin Method for Compressible Flows on Hybrid Grids

NASA Astrophysics Data System (ADS)

Xia, Yidong

The objective this work is to develop a parallel, implicit reconstructed discontinuous Galerkin (RDG) method using Taylor basis for the solution of the compressible Navier-Stokes equations on 3D hybrid grids. This third-order accurate RDG method is based on a hierarchical weighed essentially non- oscillatory reconstruction scheme, termed as HWENO(P1P 2) to indicate that a quadratic polynomial solution is obtained from the underlying linear polynomial DG solution via a hierarchical WENO reconstruction. The HWENO(P1P2) is designed not only to enhance the accuracy of the underlying DG(P1) method but also to ensure non-linear stability of the RDG method. In this reconstruction scheme, a quadratic polynomial (P2) solution is first reconstructed using a least-squares approach from the underlying linear (P1) discontinuous Galerkin solution. The final quadratic solution is then obtained using a Hermite WENO reconstruction, which is necessary to ensure the linear stability of the RDG method on 3D unstructured grids. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the non-linear stability of the RDG method. The parallelization in the RDG method is based on a message passing interface (MPI) programming paradigm, where the METIS library is used for the partitioning of a mesh into subdomain meshes of approximately the same size. Both multi-stage explicit Runge-Kutta and simple implicit backward Euler methods are implemented for time advancement in the RDG method. In the implicit method, three approaches: analytical differentiation, divided differencing (DD), and automatic differentiation (AD) are developed and implemented to obtain the resulting flux Jacobian matrices. The automatic differentiation is a set of techniques based on the mechanical application of the chain rule to obtain derivatives of a function given as a computer program. By using an AD tool, the manpower can be significantly reduced for deriving the flux Jacobians, which can be quite complicated, tedious, and error-prone if done by hand or symbolic arithmetic software, depending on the complexity of the numerical flux scheme. In addition, the workload for code maintenance can also be largely reduced in case the underlying flux scheme is updated. The approximate system of linear equations arising from the Newton linearization is solved by the general minimum residual (GMRES) algorithm with lower-upper symmetric gauss-seidel (LUSGS) preconditioning. This GMRES+LU-SGS linear solver is the most robust and efficient for implicit time integration of the discretized Navier-Stokes equations when the AD-based flux Jacobians are provided other than the other two approaches. The developed HWENO(P1P2) method is used to compute a variety of well-documented compressible inviscid and viscous flow test cases on 3D hybrid grids, including some standard benchmark test cases such as the Sod shock tube, flow past a circular cylinder, and laminar flow past a at plate. The computed solutions are compared with either analytical solutions or experimental data, if available to assess the accuracy of the HWENO(P 1P2) method. Numerical results demonstrate that the HWENO(P 1P2) method is able to not only enhance the accuracy of the underlying HWENO(P1) method, but also ensure the linear and non-linear stability at the presence of strong discontinuities. An extensive study of grid convergence analysis on various types of elements: tetrahedron, prism, hexahedron, and hybrid prism/hexahedron, for a number of test cases indicates that the developed HWENO(P1P2) method is able to achieve the designed third-order accuracy of spatial convergence for smooth inviscid flows: one order higher than the underlying second-order DG(P1) method without significant increase in computing costs and storage requirements. The performance of the the developed GMRES+LU-SGS implicit method is compared with the multi-stage Runge-Kutta time stepping scheme for a number of test cases in terms of the timestep and CPU time. Numerical results indicate that the overall performance of the implicit method with AD-based Jacobians is order of magnitude better than the its explicit counterpart. Finally, a set of parallel scaling tests for both explicit and implicit methods is conducted on North Carolina State University's ARC cluster, demonstrating almost an ideal scalability of the RDG method. (Abstract shortened by UMI.)

Dielectric and conformational studies of hydrogen bonded 2-ethoxyethanol and ethyl methyl ketone system

NASA Astrophysics Data System (ADS)

Pattebahadur, Kanchan. L.; Deshmukh, S. D.; Mohod, A. G.; Undre, P. B.; Patil, S. S.; Khirade, P. W.

2018-05-01

The Dielectric constant, density and refractive index of binary mixture of 2-ethoxy ethanol (2-EE) with ethyl methyl ketone (EMK) including those of the pure liquids were measured for 11 concentrations at 25°C temperature. The experimental data is used to calculate the Excess molar volume, Excess dielectric constant, Kirkwood correlation factor and Bruggemann factor. The excess parameters results were fitted to the Redlich-Kister type polynomial equation to derive its fitting coefficient. The Kirkwood correlation factor of the mixture has been discussed to yield information about solute solvent interaction. The Bruggeman plot shows a deviation from linearity. The FT-IR spectra of pure and their binary mixtures are also studied.

On the concept of a filtered bundle

NASA Astrophysics Data System (ADS)

Bruce, Andrew James; Grabowska, Katarzyna; Grabowski, Janusz

We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalizations of double vector bundles and double affine bundles. Furthermore, we show that the linearization of a filtered bundle — which can be seen as a partial polarization of the admissible changes of local coordinates — is well defined.

New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.; Bassuony, M. A.

2013-03-01

This paper is concerned with spectral Galerkin algorithms for solving high even-order two point boundary value problems in one dimension subject to homogeneous and nonhomogeneous boundary conditions. The proposed algorithms are extended to solve two-dimensional high even-order differential equations. The key to the efficiency of these algorithms is to construct compact combinations of Chebyshev polynomials of the third and fourth kinds as basis functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithms, and some comparisons with some other methods are made.

The challenges of transitioning from linear to high-order overlay control in advanced lithography

NASA Astrophysics Data System (ADS)

Adel, M.; Izikson, P.; Tien, D.; Huang, C. K.; Robinson, J. C.; Eichelberger, B.

2008-03-01

In the lithography section of the ITRS 2006 update, at the top of the list of difficult challenges appears the text "overlay of multiple exposures including mask image placement". This is a reflection of the fact that today overlay is becoming a major yield risk factor in semiconductor manufacturing. Historically, lithographers have achieved sufficient alignment accuracy and hence layer to layer overlay control by relying on models which define overlay as a linear function of the field and wafer coordinates. These linear terms were easily translated to correctibles in the available exposure tool degrees of freedom on the wafer and reticle stages. However, as the 45 nm half pitch node reaches production, exposure tool vendors have begun to make available, and lithographers have begun to utilize so called high order wafer and field control, in which either look up table or high order polynomial models are modified on a product by product basis. In this paper, the major challenges of this transition will be described. It will include characterization of the sources of variation which need to be controlled by these new models and the overlay and alignment sampling optimization problem which needs to be addressed, while maintaining the ever tightening demands on productivity and cost of ownership.

Random regression analyses using B-splines functions to model growth from birth to adult age in Canchim cattle.

PubMed

Baldi, F; Alencar, M M; Albuquerque, L G

2010-12-01

The objective of this work was to estimate covariance functions using random regression models on B-splines functions of animal age, for weights from birth to adult age in Canchim cattle. Data comprised 49,011 records on 2435 females. The model of analysis included fixed effects of contemporary groups, age of dam as quadratic covariable and the population mean trend taken into account by a cubic regression on orthogonal polynomials of animal age. Residual variances were modelled through a step function with four classes. The direct and maternal additive genetic effects, and animal and maternal permanent environmental effects were included as random effects in the model. A total of seventeen analyses, considering linear, quadratic and cubic B-splines functions and up to seven knots, were carried out. B-spline functions of the same order were considered for all random effects. Random regression models on B-splines functions were compared to a random regression model on Legendre polynomials and with a multitrait model. Results from different models of analyses were compared using the REML form of the Akaike Information criterion and Schwarz' Bayesian Information criterion. In addition, the variance components and genetic parameters estimated for each random regression model were also used as criteria to choose the most adequate model to describe the covariance structure of the data. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most adequate to describe the covariance structure of the data. Random regression models using B-spline functions as base functions fitted the data better than Legendre polynomials, especially at mature ages, but higher number of parameters need to be estimated with B-splines functions. © 2010 Blackwell Verlag GmbH.

Random regression models using different functions to model test-day milk yield of Brazilian Holstein cows.

PubMed

Bignardi, A B; El Faro, L; Torres Júnior, R A A; Cardoso, V L; Machado, P F; Albuquerque, L G

2011-10-31

We analyzed 152,145 test-day records from 7317 first lactations of Holstein cows recorded from 1995 to 2003. Our objective was to model variations in test-day milk yield during the first lactation of Holstein cows by random regression model (RRM), using various functions in order to obtain adequate and parsimonious models for the estimation of genetic parameters. Test-day milk yields were grouped into weekly classes of days in milk, ranging from 1 to 44 weeks. The contemporary groups were defined as herd-test-day. The analyses were performed using a single-trait RRM, including the direct additive, permanent environmental and residual random effects. In addition, contemporary group and linear and quadratic effects of the age of cow at calving were included as fixed effects. The mean trend of milk yield was modeled with a fourth-order orthogonal Legendre polynomial. The additive genetic and permanent environmental covariance functions were estimated by random regression on two parametric functions, Ali and Schaeffer and Wilmink, and on B-spline functions of days in milk. The covariance components and the genetic parameters were estimated by the restricted maximum likelihood method. Results from RRM parametric and B-spline functions were compared to RRM on Legendre polynomials and with a multi-trait analysis, using the same data set. Heritability estimates presented similar trends during mid-lactation (13 to 31 weeks) and between week 37 and the end of lactation, for all RRM. Heritabilities obtained by multi-trait analysis were of a lower magnitude than those estimated by RRM. The RRMs with a higher number of parameters were more useful to describe the genetic variation of test-day milk yield throughout the lactation. RRM using B-spline and Legendre polynomials as base functions appears to be the most adequate to describe the covariance structure of the data.

On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on ℝ

NASA Astrophysics Data System (ADS)

Alvarez-Nodarse, R.; Atakishiyeva, M. K.; Atakishiyev, N. M.

2005-11-01

We discuss a q-analogue of the linear harmonic oscillator in quantum mechanics based on a q-extension of the classical Hermite polynomials H n ( x) recently introduced by us in R. Alvarez-Nodarse et al.: Boletin de la Sociedad Matematica Mexicana (3) 8 (2002) 127. The wave functions in this q-model of the quantum harmonic oscillator possess the continuous orthogonality property on the whole real line ℝ with respect to a positive weight function. A detailed description of the corresponding q-system is carried out.

A linear model of population dynamics

NASA Astrophysics Data System (ADS)

Lushnikov, A. A.; Kagan, A. I.

2016-08-01

The Malthus process of population growth is reformulated in terms of the probability w(n,t) to find exactly n individuals at time t assuming that both the birth and the death rates are linear functions of the population size. The master equation for w(n,t) is solved exactly. It is shown that w(n,t) strongly deviates from the Poisson distribution and is expressed in terms either of Laguerre’s polynomials or a modified Bessel function. The latter expression allows for considerable simplifications of the asymptotic analysis of w(n,t).

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.

Assessment of Hybrid High-Order methods on curved meshes and comparison with discontinuous Galerkin methods

NASA Astrophysics Data System (ADS)

Botti, Lorenzo; Di Pietro, Daniele A.

2018-10-01

We propose and validate a novel extension of Hybrid High-Order (HHO) methods to meshes featuring curved elements. HHO methods are based on discrete unknowns that are broken polynomials on the mesh and its skeleton. We propose here the use of physical frame polynomials over mesh elements and reference frame polynomials over mesh faces. With this choice, the degree of face unknowns must be suitably selected in order to recover on curved meshes the same convergence rates as on straight meshes. We provide an estimate of the optimal face polynomial degree depending on the element polynomial degree and on the so-called effective mapping order. The estimate is numerically validated through specifically crafted numerical tests. All test cases are conducted considering two- and three-dimensional pure diffusion problems, and include comparisons with discontinuous Galerkin discretizations. The extension to agglomerated meshes with curved boundaries is also considered.

Heuristics for Understanding the Concepts of Interaction, Polynomial Trend, and the General Linear Model.

ERIC Educational Resources Information Center

Thompson, Bruce

The relationship between analysis of variance (ANOVA) methods and their analogs (analysis of covariance and multiple analyses of variance and covariance--collectively referred to as OVA methods) and the more general analytic case is explored. A small heuristic data set is used, with a hypothetical sample of 20 subjects, randomly assigned to five…

Prediction of leaf area in individual leaves of cherrybark oak seedlings (Quercus pagoda Raf.)

Treesearch

Yanfei Guo; Brian Lockhart; John Hodges

1995-01-01

The prediction of leaf area for cherrybark oak (Quercus pagoda Raf.) seedlings is important for studying the physiology of the species. Linear and polynomial models involving leaf length, width, fresh weight, dry weight, and internodal length were tested independently and collectively to predict leaf area. Twenty-nine cherrybark oak seedlings were...

Relationship between age and elite marathon race time in world single age records from 5 to 93 years

PubMed Central

2014-01-01

Background The aims of the study were (i) to investigate the relationship between elite marathon race times and age in 1-year intervals by using the world single age records in marathon running from 5 to 93 years and (ii) to evaluate the sex difference in elite marathon running performance with advancing age. Methods World single age records in marathon running in 1-year intervals for women and men were analysed regarding changes across age for both men and women using linear and non-linear regression analyses for each age for women and men. Results The relationship between elite marathon race time and age was non-linear (i.e. polynomial regression 4th degree) for women and men. The curve was U-shaped where performance improved from 5 to ~20 years. From 5 years to ~15 years, boys and girls performed very similar. Between ~20 and ~35 years, performance was quite linear, but started to decrease at the age of ~35 years in a curvilinear manner with increasing age in both women and men. The sex difference increased non-linearly (i.e. polynomial regression 7th degree) from 5 to ~20 years, remained unchanged at ~20 min from ~20 to ~50 years and increased thereafter. The sex difference was lowest (7.5%, 10.5 min) at the age of 49 years. Conclusion Elite marathon race times improved from 5 to ~20 years, remained linear between ~20 and ~35 years, and started to increase at the age of ~35 years in a curvilinear manner with increasing age in both women and men. The sex difference in elite marathon race time increased non-linearly and was lowest at the age of ~49 years. PMID:25120915

Colorimetric characterization models based on colorimetric characteristics evaluation for active matrix organic light emitting diode panels.

PubMed

Gong, Rui; Xu, Haisong; Tong, Qingfen

2012-10-20

The colorimetric characterization of active matrix organic light emitting diode (AMOLED) panels suffers from their poor channel independence. Based on the colorimetric characteristics evaluation of channel independence and chromaticity constancy, an accurate colorimetric characterization method, namely, the polynomial compensation model (PC model) considering channel interactions was proposed for AMOLED panels. In this model, polynomial expressions are employed to calculate the relationship between the prediction errors of XYZ tristimulus values and the digital inputs to compensate the XYZ prediction errors of the conventional piecewise linear interpolation assuming the variable chromaticity coordinates (PLVC) model. The experimental results indicated that the proposed PC model outperformed other typical characterization models for the two tested AMOLED smart-phone displays and for the professional liquid crystal display monitor as well.

The value of continuity: Refined isogeometric analysis and fast direct solvers

DOE PAGES

Garcia, Daniel; Pardo, David; Dalcin, Lisandro; ...

2016-08-24

Here, we propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce C0-separators to reduce the interconnection between degrees of freedom in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method “refined Isogeometric Analysis (rIGA)”. To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing themore » Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3, with pp being the polynomial order of the discretization. Numerical results indicate that our proposed refined isogeometric analysis delivers a speed-up factor proportional to p 2. In a 2D mesh with four million elements and p=5, the linear system resulting from rIGA is solved 22 times faster than the one from highly continuous IGA. In a 3D mesh with one million elements and p=3, the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis.« less

Estimation of genetic parameters for milk yield in Murrah buffaloes by Bayesian inference.

PubMed

Breda, F C; Albuquerque, L G; Euclydes, R F; Bignardi, A B; Baldi, F; Torres, R A; Barbosa, L; Tonhati, H

2010-02-01

Random regression models were used to estimate genetic parameters for test-day milk yield in Murrah buffaloes using Bayesian inference. Data comprised 17,935 test-day milk records from 1,433 buffaloes. Twelve models were tested using different combinations of third-, fourth-, fifth-, sixth-, and seventh-order orthogonal polynomials of weeks of lactation for additive genetic and permanent environmental effects. All models included the fixed effects of contemporary group, number of daily milkings and age of cow at calving as covariate (linear and quadratic effect). In addition, residual variances were considered to be heterogeneous with 6 classes of variance. Models were selected based on the residual mean square error, weighted average of residual variance estimates, and estimates of variance components, heritabilities, correlations, eigenvalues, and eigenfunctions. Results indicated that changes in the order of fit for additive genetic and permanent environmental random effects influenced the estimation of genetic parameters. Heritability estimates ranged from 0.19 to 0.31. Genetic correlation estimates were close to unity between adjacent test-day records, but decreased gradually as the interval between test-days increased. Results from mean squared error and weighted averages of residual variance estimates suggested that a model considering sixth- and seventh-order Legendre polynomials for additive and permanent environmental effects, respectively, and 6 classes for residual variances, provided the best fit. Nevertheless, this model presented the largest degree of complexity. A more parsimonious model, with fourth- and sixth-order polynomials, respectively, for these same effects, yielded very similar genetic parameter estimates. Therefore, this last model is recommended for routine applications. Copyright 2010 American Dairy Science Association. Published by Elsevier Inc. All rights reserved.

Linearity versus Nonlinearity of Offspring-Parent Regression: An Experimental Study of Drosophila Melanogaster

PubMed Central

Gimelfarb, A.; Willis, J. H.

1994-01-01

An experiment was conducted to investigate the offspring-parent regression for three quantitative traits (weight, abdominal bristles and wing length) in Drosophila melanogaster. Linear and polynomial models were fitted for the regressions of a character in offspring on both parents. It is demonstrated that responses by the characters to selection predicted by the nonlinear regressions may differ substantially from those predicted by the linear regressions. This is true even, and especially, if selection is weak. The realized heritability for a character under selection is shown to be determined not only by the offspring-parent regression but also by the distribution of the character and by the form and strength of selection. PMID:7828818

Elastic and viscoelastic mechanical properties of brain tissues on the implanting trajectory of sub-thalamic nucleus stimulation.

PubMed

Li, Yan; Deng, Jianxin; Zhou, Jun; Li, Xueen

2016-11-01

Corresponding to pre-puncture and post-puncture insertion, elastic and viscoelastic mechanical properties of brain tissues on the implanting trajectory of sub-thalamic nucleus stimulation are investigated, respectively. Elastic mechanical properties in pre-puncture are investigated through pre-puncture needle insertion experiments using whole porcine brains. A linear polynomial and a second order polynomial are fitted to the average insertion force in pre-puncture. The Young's modulus in pre-puncture is calculated from the slope of the two fittings. Viscoelastic mechanical properties of brain tissues in post-puncture insertion are investigated through indentation stress relaxation tests for six interested regions along a planned trajectory. A linear viscoelastic model with a Prony series approximation is fitted to the average load trace of each region using Boltzmann hereditary integral. Shear relaxation moduli of each region are calculated using the parameters of the Prony series approximation. The results show that, in pre-puncture insertion, needle force almost increases linearly with needle displacement. Both fitting lines can perfectly fit the average insertion force. The Young's moduli calculated from the slope of the two fittings are worthy of trust to model linearly or nonlinearly instantaneous elastic responses of brain tissues, respectively. In post-puncture insertion, both region and time significantly affect the viscoelastic behaviors. Six tested regions can be classified into three categories in stiffness. Shear relaxation moduli decay dramatically in short time scales but equilibrium is never truly achieved. The regional and temporal viscoelastic mechanical properties in post-puncture insertion are valuable for guiding probe insertion into each region on the implanting trajectory.

Random regression models for the prediction of days to weight, ultrasound rib eye area, and ultrasound back fat depth in beef cattle.

PubMed

Speidel, S E; Peel, R K; Crews, D H; Enns, R M

2016-02-01

Genetic evaluation research designed to reduce the required days to a specified end point has received very little attention in pertinent scientific literature, given that its economic importance was first discussed in 1957. There are many production scenarios in today's beef industry, making a prediction for the required number of days to a single end point a suboptimal option. Random regression is an attractive alternative to calculate days to weight (DTW), days to ultrasound back fat (DTUBF), and days to ultrasound rib eye area (DTUREA) genetic predictions that could overcome weaknesses of a single end point prediction. The objective of this study was to develop random regression approaches for the prediction of the DTW, DTUREA, and DTUBF. Data were obtained from the Agriculture and Agri-Food Canada Research Centre, Lethbridge, AB, Canada. Data consisted of records on 1,324 feedlot cattle spanning 1999 to 2007. Individual animals averaged 5.77 observations with weights, ultrasound rib eye area (UREA), ultrasound back fat depth (UBF), and ages ranging from 293 to 863 kg, 73.39 to 129.54 cm, 1.53 to 30.47 mm, and 276 to 519 d, respectively. Random regression models using Legendre polynomials were used to regress age of the individual on weight, UREA, and UBF. Fixed effects in the model included an overall fixed regression of age on end point (weight, UREA, and UBF) nested within breed to account for the mean relationship between age and weight as well as a contemporary group effect consisting of breed of the animal (Angus, Charolais, and Charolais sired), feedlot pen, and year of measure. Likelihood ratio tests were used to determine the appropriate random polynomial order. Use of the quadratic polynomial did not account for any additional genetic variation in days for DTW ( > 0.11), for DTUREA ( > 0.18), and for DTUBF ( > 0.20) when compared with the linear random polynomial. Heritability estimates from the linear random regression for DTW ranged from 0.54 to 0.74, corresponding to end points of 293 and 863 kg, respectively. Heritability for DTUREA ranged from 0.51 to 0.34 and for DTUBF ranged from 0.55 to 0.37. These estimates correspond to UREA end points of 35 and 125 cm and UBF end points of 1.53 and 30 mm, respectively. This range of heritability shows DTW, DTUREA, and DTUBF to be highly heritable and indicates that selection pressure aimed at reducing the number of days to reach a finish weight end point can result in genetic change given sufficient data.

Modeling State-Space Aeroelastic Systems Using a Simple Matrix Polynomial Approach for the Unsteady Aerodynamics

NASA Technical Reports Server (NTRS)

Pototzky, Anthony S.

2008-01-01

A simple matrix polynomial approach is introduced for approximating unsteady aerodynamics in the s-plane and ultimately, after combining matrix polynomial coefficients with matrices defining the structure, a matrix polynomial of the flutter equations of motion (EOM) is formed. A technique of recasting the matrix-polynomial form of the flutter EOM into a first order form is also presented that can be used to determine the eigenvalues near the origin and everywhere on the complex plane. An aeroservoelastic (ASE) EOM have been generalized to include the gust terms on the right-hand side. The reasons for developing the new matrix polynomial approach are also presented, which are the following: first, the "workhorse" methods such as the NASTRAN flutter analysis lack the capability to consistently find roots near the origin, along the real axis or accurately find roots farther away from the imaginary axis of the complex plane; and, second, the existing s-plane methods, such as the Roger s s-plane approximation method as implemented in ISAC, do not always give suitable fits of some tabular data of the unsteady aerodynamics. A method available in MATLAB is introduced that will accurately fit generalized aerodynamic force (GAF) coefficients in a tabular data form into the coefficients of a matrix polynomial form. The root-locus results from the NASTRAN pknl flutter analysis, the ISAC-Roger's s-plane method and the present matrix polynomial method are presented and compared for accuracy and for the number and locations of roots.

A Galerkin method for linear PDE systems in circular geometries with structural acoustic applications

NASA Technical Reports Server (NTRS)

Smith, Ralph C.

1994-01-01

A Galerkin method for systems of PDE's in circular geometries is presented with motivating problems being drawn from structural, acoustic, and structural acoustic applications. Depending upon the application under consideration, piecewise splines or Legendre polynomials are used when approximating the system dynamics with modifications included to incorporate the analytic solution decay near the coordinate singularity. This provides an efficient method which retains its accuracy throughout the circular domain without degradation at singularity. Because the problems under consideration are linear or weakly nonlinear with constant or piecewise constant coefficients, transform methods for the problems are not investigated. While the specific method is developed for the two dimensional wave equations on a circular domain and the equation of transverse motion for a thin circular plate, examples demonstrating the extension of the techniques to a fully coupled structural acoustic system are used to illustrate the flexibility of the method when approximating the dynamics of more complex systems.

Groebner Basis Solutions to Satellite Trajectory Control by Pole Placement

NASA Astrophysics Data System (ADS)

Kukelova, Z.; Krsek, P.; Smutny, V.; Pajdla, T.

2013-09-01

Satellites play an important role, e.g., in telecommunication, navigation and weather monitoring. Controlling their trajectories is an important problem. In [1], an approach to the pole placement for the synthesis of a linear controller has been presented. It leads to solving five polynomial equations in nine unknown elements of the state space matrices of a compensator. This is an underconstrained system and therefore four of the unknown elements need to be considered as free parameters and set to some prior values to obtain a system of five equations in five unknowns. In [1], this system was solved for one chosen set of free parameters with the help of Dixon resultants. In this work, we study and present Groebner basis solutions to this problem of computation of a dynamic compensator for the satellite for different combinations of input free parameters. We show that the Groebner basis method for solving systems of polynomial equations leads to very simple solutions for all combinations of free parameters. These solutions require to perform only the Gauss-Jordan elimination of a small matrix and computation of roots of a single variable polynomial. The maximum degree of this polynomial is not greater than six in general but for most combinations of the input free parameters its degree is even lower. [1] B. Palancz. Application of Dixon resultant to satellite trajectory control by pole placement. Journal of Symbolic Computation, Volume 50, March 2013, Pages 79-99, Elsevier.

A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shao, Yan-Lin, E-mail: yanlin.shao@dnvgl.com; Faltinsen, Odd M.

2014-10-01

We propose a new efficient and accurate numerical method based on harmonic polynomials to solve boundary value problems governed by 3D Laplace equation. The computational domain is discretized by overlapping cells. Within each cell, the velocity potential is represented by the linear superposition of a complete set of harmonic polynomials, which are the elementary solutions of Laplace equation. By its definition, the method is named as Harmonic Polynomial Cell (HPC) method. The characteristics of the accuracy and efficiency of the HPC method are demonstrated by studying analytical cases. Comparisons will be made with some other existing boundary element based methods,more » e.g. Quadratic Boundary Element Method (QBEM) and the Fast Multipole Accelerated QBEM (FMA-QBEM) and a fourth order Finite Difference Method (FDM). To demonstrate the applications of the method, it is applied to some studies relevant for marine hydrodynamics. Sloshing in 3D rectangular tanks, a fully-nonlinear numerical wave tank, fully-nonlinear wave focusing on a semi-circular shoal, and the nonlinear wave diffraction of a bottom-mounted cylinder in regular waves are studied. The comparisons with the experimental results and other numerical results are all in satisfactory agreement, indicating that the present HPC method is a promising method in solving potential-flow problems. The underlying procedure of the HPC method could also be useful in other fields than marine hydrodynamics involved with solving Laplace equation.« less

Bohnenblust-Hille inequalities: analytical and computational aspects.

PubMed

Cavalcante, Wasthenny V; Pellegrino, Daniel M

2018-02-01

The Bohnenblust-Hille polynomial and multilinear inequalities were proved in 1931 and the determination of exact values of their constants is still an open and challenging problem, pursued by various authors. The present paper briefly surveys recent attempts to attack/solve this problem; it also presents new results, like connections with classical results of the linear theory of absolutely summing operators, and new perspectives.

Kleinberg Complex Networks

DTIC Science & Technology

2014-10-21

linear combinations of paths. This project featured research on two classes of routing problems , namely traveling salesman problems and multicommodity...flows. One highlight of this research was our discovery of a polynomial-time algorithm for the metric traveling salesman s-t path problem whose...metric TSP would resolve one of the most venerable open problems in the theory of approximation algorithms. Our research on traveling salesman

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

NASA Astrophysics Data System (ADS)

Cieplak, Agnieszka; Slosar, Anze

2017-01-01

The Lyman-alpha forest has become a powerful cosmological probe of the underlying matter distribution at high redshift. It is a highly non-linear field with much information present beyond the two-point statistics of the power spectrum. The flux probability distribution function (PDF) in particular has been used as a successful probe of small-scale physics. In addition to the cosmological evolution however, it is also sensitive to pixel noise, spectrum resolution, and continuum fitting, all of which lead to possible biased estimators. Here we argue that measuring coefficients of the Legendre polynomial expansion of the PDF offers several advantages over the binned PDF as is commonly done. Since the n-th coefficient can be expressed as a linear combination of the first n moments of the field, this allows for the coefficients to be measured in the presence of noise and allows for a clear route towards marginalization over the mean flux. In addition, we use hydrodynamic cosmological simulations to demonstrate that in the presence of noise, a finite number of these coefficients are well measured with a very sharp transition into noise dominance. This compresses the information into a finite small number of well-measured quantities.

Anisotropy of susceptibility in rocks which are magnetically nonlinear even in low fields

NASA Astrophysics Data System (ADS)

Hrouda, František; Chadima, Martin; Ježek, Josef

2018-06-01

Theory of the low-field anisotropy of magnetic susceptibility (AMS) assumes a linear relationship between magnetization and magnetizing field, resulting in field-independent susceptibility. This is valid for diamagnetic and paramagnetic minerals by definition and also for pure magnetite, while in titanomagnetite, pyrrhotite and hematite the susceptibility may be clearly field-dependent even in low fields used in common AMS meter. Consequently, the use of the linear AMS theory is fully legitimate in the former minerals, but in principle incorrect in the latter ones. Automated measurement of susceptibility in 320 directions in variable low-fields ranging from 5 to 700 A m-1 was applied to more than 100 specimens of various pyrrhotite-bearing and titanomagnetite-bearing rocks. Data analysis showed that the anisotropic susceptibility remains well represented by an ellipsoid in the entire low-field span even though the ellipsoid increases its volume and eccentricity. The principal directions do not change their orientations with low-field in most specimens. Expressions for susceptibility as a function of field were found in the form of diagonal tensor whose elements are polynomials of low order. In a large proportion of samples, the susceptibility expressions can be further simplified to have one common skeleton polynomial.

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

NASA Astrophysics Data System (ADS)

Cieplak, Agnieszka; Slosar, Anze

2018-01-01

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

Polynomial asymptotes of the second kind

NASA Astrophysics Data System (ADS)

Dobbs, David E.

2011-03-01

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 conics. Prerequisites include the division algorithm for polynomials with coefficients in the field of real numbers and elementary facts about limits from calculus. This note could be used as enrichment material in courses ranging from Calculus to Real Analysis to Abstract Algebra.

Characterization of a dielectric phantom for high-field magnetic resonance imaging applications

DOE Office of Scientific and Technical Information (OSTI.GOV)

Duan, Qi, E-mail: Qi.Duan@nih.gov; Duyn, Jeff H.; Gudino, Natalia

2014-10-15

Purpose: In this work, a generic recipe for an inexpensive and nontoxic phantom was developed within a range of biologically relevant dielectric properties from 150 MHz to 4.5 GHz. Methods: The recipe includes deionized water as the solvent, NaCl to primarily control conductivity, sucrose to primarily control permittivity, agar–agar to gel the solution and reduce heat diffusivity, and benzoic acid to preserve the gel. Two hundred and seventeen samples were prepared to cover the feasible range of NaCl and sucrose concentrations. Their dielectric properties were measured using a commercial dielectric probe and were fitted to a 3D polynomial to generatemore » a recipe describing the properties as a function of NaCl concentration, sucrose concentration, and frequency. Results: Results indicated that the intuitive linear and independent relationships between NaCl and conductivity and between sucrose and permittivity are not valid. A generic polynomial recipe was developed to characterize the complex relationship between the solutes and the resulting dielectric values and has been made publicly available as a web application. In representative mixtures developed to mimic brain and muscle tissue, less than 2% difference was observed between the predicted and measured conductivity and permittivity values. Conclusions: It is expected that the recipe will be useful for generating dielectric phantoms for general magnetic resonance imaging (MRI) coil development at high magnetic field strength, including coil safety evaluation as well as pulse sequence evaluation (including B{sub 1}{sup +} mapping, B{sub 1}{sup +} shimming, and selective excitation pulse design), and other non-MRI applications which require biologically equivalent dielectric properties.« less

Adaptive optics with a magnetic deformable mirror: applications in the human eye

NASA Astrophysics Data System (ADS)

Fernandez, Enrique J.; Vabre, Laurent; Hermann, Boris; Unterhuber, Angelika; Povazay, Boris; Drexler, Wolfgang

2006-10-01

A novel deformable mirror using 52 independent magnetic actuators (MIRAO 52, Imagine Eyes) is presented and characterized for ophthalmic applications. The capabilities of the device to reproduce different surfaces, in particular Zernike polynomials up to the fifth order, are investigated in detail. The study of the influence functions of the deformable mirror reveals a significant linear response with the applied voltage. The correcting device also presents a high fidelity in the generation of surfaces. The ranges of production of Zernike polynomials fully cover those typically found in the human eye, even for the cases of highly aberrated eyes. Data from keratoconic eyes are confronted with the obtained ranges, showing that the deformable mirror is able to compensate for these strong aberrations. Ocular aberration correction with polychromatic light, using a near Gaussian spectrum of 130 nm full width at half maximum centered at 800 nm, in five subjects is accomplished by simultaneously using the deformable mirror and an achromatizing lens, in order to compensate for the monochromatic and chromatic aberrations, respectively. Results from living eyes, including one exhibiting 4.66 D of myopia and a near pathologic cornea with notable high order aberrations, show a practically perfect aberration correction. Benefits and applications of simultaneous monochromatic and chromatic aberration correction are finally discussed in the context of retinal imaging and vision.

All-in-one visual and computer decoding of multiple secrets: translated-flip VC with polynomial-style sharing

NASA Astrophysics Data System (ADS)

Wu, Chia-Hua; Lee, Suiang-Shyan; Lin, Ja-Chen

2017-06-01

This all-in-one hiding method creates two transparencies that have several decoding options: visual decoding with or without translation flipping and computer decoding. In visual decoding, two less-important (or fake) binary secret images S1 and S2 can be revealed. S1 is viewed by the direct stacking of two transparencies. S2 is viewed by flipping one transparency and translating the other to a specified coordinate before stacking. Finally, important/true secret files can be decrypted by a computer using the information extracted from transparencies. The encoding process to hide this information includes the translated-flip visual cryptography, block types, the ways to use polynomial-style sharing, and linear congruential generator. If a thief obtained both transparencies, which are stored in distinct places, he still needs to find the values of keys used in computer decoding to break through after viewing S1 and/or S2 by stacking. However, the thief might just try every other kind of stacking and finally quit finding more secrets; for computer decoding is totally different from stacking decoding. Unlike traditional image hiding that uses images as host media, our method hides fine gray-level images in binary transparencies. Thus, our host media are transparencies. Comparisons and analysis are provided.

The background model in the energy range from 0.1 MeV up to several MeV for low altitude and high inclination satellites.

NASA Astrophysics Data System (ADS)

Arkhangelskaja, I. V.; Arkhangelskiy, A. I.

2016-02-01

The gamma-ray background physical origin for low altitude orbits defined by: diffuse cosmic gamma-emission, atmospheric gamma-rays, gamma-emission formed in interactions of charged particles (both prompt and activation) and transient events such as electrons precipitations and solar flares. The background conditions in the energy range from 0.1 MeV up to several MeV for low altitude orbits differ due to frequency of Earth Radiation Belts - ERBs (included South Atlantic Anomaly - SAA) passes and cosmic rays rigidity. The detectors and satellite constructive elements are activated by trapped in ERBs and moving along magnetic lines charged particles. In this case we propose simplified polynomial model separately for polar and equatorial orbits parts: background count rate temporal profile approximation by 4-5 order polynomials in equatorial regions, and linear approximations, parabolas or constants in polar caps. The polynomials’ coefficients supposed to be similar for identical spectral channels for each analyzed equatorial part taken into account normalization coefficients defined due to Kp-indexes study within period corresponding to calibration coefficients being approximately constants. The described model was successfully applied for the solar flares hard X-ray and gamma-ray emission characteristic studies by AVS-F apparatus data onboard CORONAS-F satellite.

Modeling of driver's collision avoidance maneuver based on controller switching model.

PubMed

Kim, Jong-Hae; Hayakawa, Soichiro; Suzuki, Tatsuya; Hayashi, Koji; Okuma, Shigeru; Tsuchida, Nuio; Shimizu, Masayuki; Kido, Shigeyuki

2005-12-01

This paper presents a modeling strategy of human driving behavior based on the controller switching model focusing on the driver's collision avoidance maneuver. The driving data are collected by using the three-dimensional (3-D) driving simulator based on the CAVE Automatic Virtual Environment (CAVE), which provides stereoscopic immersive virtual environment. In our modeling, the control scenario of the human driver, that is, the mapping from the driver's sensory information to the operation of the driver such as acceleration, braking, and steering, is expressed by Piecewise Polynomial (PWP) model. Since the PWP model includes both continuous behaviors given by polynomials and discrete logical conditions, it can be regarded as a class of Hybrid Dynamical System (HDS). The identification problem for the PWP model is formulated as the Mixed Integer Linear Programming (MILP) by transforming the switching conditions into binary variables. From the obtained results, it is found that the driver appropriately switches the "control law" according to the sensory information. In addition, the driving characteristics of the beginner driver and the expert driver are compared and discussed. These results enable us to capture not only the physical meaning of the driving skill but the decision-making aspect (switching conditions) in the driver's collision avoidance maneuver as well.

Light field creating and imaging with different order intensity derivatives

NASA Astrophysics Data System (ADS)

Wang, Yu; Jiang, Huan

2014-10-01

Microscopic image restoration and reconstruction is a challenging topic in the image processing and computer vision, which can be widely applied to life science, biology and medicine etc. A microscopic light field creating and three dimensional (3D) reconstruction method is proposed for transparent or partially transparent microscopic samples, which is based on the Taylor expansion theorem and polynomial fitting. Firstly the image stack of the specimen is divided into several groups in an overlapping or non-overlapping way along the optical axis, and the first image of every group is regarded as reference image. Then different order intensity derivatives are calculated using all the images of every group and polynomial fitting method based on the assumption that the structure of the specimen contained by the image stack in a small range along the optical axis are possessed of smooth and linear property. Subsequently, new images located any position from which to reference image the distance is Δz along the optical axis can be generated by means of Taylor expansion theorem and the calculated different order intensity derivatives. Finally, the microscopic specimen can be reconstructed in 3D form using deconvolution technology and all the images including both the observed images and the generated images. The experimental results show the effectiveness and feasibility of our method.

A new root-based direction-finding algorithm

NASA Astrophysics Data System (ADS)

Wasylkiwskyj, Wasyl; Kopriva, Ivica; DoroslovačKi, Miloš; Zaghloul, Amir I.

2007-04-01

Polynomial rooting direction-finding (DF) algorithms are a computationally efficient alternative to search-based DF algorithms and are particularly suitable for uniform linear arrays of physically identical elements provided that mutual interaction among the array elements can be either neglected or compensated for. A popular algorithm in such situations is Root Multiple Signal Classification (Root MUSIC (RM)), wherein the estimation of the directions of arrivals (DOA) requires the computation of the roots of a (2N - 2) -order polynomial, where N represents number of array elements. The DOA are estimated from the L pairs of roots closest to the unit circle, where L represents number of sources. In this paper we derive a modified root polynomial (MRP) algorithm requiring the calculation of only L roots in order to estimate the L DOA. We evaluate the performance of the MRP algorithm numerically and show that it is as accurate as the RM algorithm but with a significantly simpler algebraic structure. In order to demonstrate that the theoretically predicted performance can be achieved in an experimental setting, a decoupled array is emulated in hardware using phase shifters. The results are in excellent agreement with theory.

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.

Using Chebyshev polynomials and approximate inverse triangular factorizations for preconditioning the conjugate gradient method

NASA Astrophysics Data System (ADS)

Kaporin, I. E.

2012-02-01

In order to precondition a sparse symmetric positive definite matrix, its approximate inverse is examined, which is represented as the product of two sparse mutually adjoint triangular matrices. In this way, the solution of the corresponding system of linear algebraic equations (SLAE) by applying the preconditioned conjugate gradient method (CGM) is reduced to performing only elementary vector operations and calculating sparse matrix-vector products. A method for constructing the above preconditioner is described and analyzed. The triangular factor has a fixed sparsity pattern and is optimal in the sense that the preconditioned matrix has a minimum K-condition number. The use of polynomial preconditioning based on Chebyshev polynomials makes it possible to considerably reduce the amount of scalar product operations (at the cost of an insignificant increase in the total number of arithmetic operations). The possibility of an efficient massively parallel implementation of the resulting method for solving SLAEs is discussed. For a sequential version of this method, the results obtained by solving 56 test problems from the Florida sparse matrix collection (which are large-scale and ill-conditioned) are presented. These results show that the method is highly reliable and has low computational costs.

Integration of system identification and finite element modelling of nonlinear vibrating structures

NASA Astrophysics Data System (ADS)

Cooper, Samson B.; DiMaio, Dario; Ewins, David J.

2018-03-01

The Finite Element Method (FEM), Experimental modal analysis (EMA) and other linear analysis techniques have been established as reliable tools for the dynamic analysis of engineering structures. They are often used to provide solutions to small and large structures and other variety of cases in structural dynamics, even those exhibiting a certain degree of nonlinearity. Unfortunately, when the nonlinear effects are substantial or the accuracy of the predicted response is of vital importance, a linear finite element model will generally prove to be unsatisfactory. As a result, the validated linear FE model requires further enhancement so that it can represent and predict the nonlinear behaviour exhibited by the structure. In this paper, a pragmatic approach to integrating test-based system identification and FE modelling of a nonlinear structure is presented. This integration is based on three different phases: the first phase involves the derivation of an Underlying Linear Model (ULM) of the structure, the second phase includes experiment-based nonlinear identification using measured time series and the third phase covers augmenting the linear FE model and experimental validation of the nonlinear FE model. The proposed case study is demonstrated on a twin cantilever beam assembly coupled with a flexible arch shaped beam. In this case, polynomial-type nonlinearities are identified and validated with force-controlled stepped-sine test data at several excitation levels.

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jakeman, John D.; Narayan, Akil; Zhou, Tao

We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditionedmore » $$\\ell^1$$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. In conclusion, numerical examples are also provided to demonstrate that our proposed algorithm leads to comparable or improved accuracy even when compared with Legendre- and Hermite-specific algorithms.« less

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

DOE PAGES

Jakeman, John D.; Narayan, Akil; Zhou, Tao

2017-06-22

We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditionedmore » $$\\ell^1$$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. In conclusion, numerical examples are also provided to demonstrate that our proposed algorithm leads to comparable or improved accuracy even when compared with Legendre- and Hermite-specific algorithms.« less

Quantum solvability of a general ordered position dependent mass system: Mathews-Lakshmanan oscillator

NASA Astrophysics Data System (ADS)

Karthiga, S.; Chithiika Ruby, V.; Senthilvelan, M.; Lakshmanan, M.

2017-10-01

In position dependent mass (PDM) problems, the quantum dynamics of the associated systems have been understood well in the literature for particular orderings. However, no efforts seem to have been made to solve such PDM problems for general orderings to obtain a global picture. In this connection, we here consider the general ordered quantum Hamiltonian of an interesting position dependent mass problem, namely, the Mathews-Lakshmanan oscillator, and try to solve the quantum problem for all possible orderings including Hermitian and non-Hermitian ones. The other interesting point in our study is that for all possible orderings, although the Schrödinger equation of this Mathews-Lakshmanan oscillator is uniquely reduced to the associated Legendre differential equation, their eigenfunctions cannot be represented in terms of the associated Legendre polynomials with integral degree and order. Rather the eigenfunctions are represented in terms of associated Legendre polynomials with non-integral degree and order. We here explore such polynomials and represent the discrete and continuum states of the system. We also exploit the connection between associated Legendre polynomials with non-integral degree with other orthogonal polynomials such as Jacobi and Gegenbauer polynomials.

Creating a non-linear total sediment load formula using polynomial best subset regression model

NASA Astrophysics Data System (ADS)

Okcu, Davut; Pektas, Ali Osman; Uyumaz, Ali

2016-08-01

The aim of this study is to derive a new total sediment load formula which is more accurate and which has less application constraints than the well-known formulae of the literature. 5 most known stream power concept sediment formulae which are approved by ASCE are used for benchmarking on a wide range of datasets that includes both field and flume (lab) observations. The dimensionless parameters of these widely used formulae are used as inputs in a new regression approach. The new approach is called Polynomial Best subset regression (PBSR) analysis. The aim of the PBRS analysis is fitting and testing all possible combinations of the input variables and selecting the best subset. Whole the input variables with their second and third powers are included in the regression to test the possible relation between the explanatory variables and the dependent variable. While selecting the best subset a multistep approach is used that depends on significance values and also the multicollinearity degrees of inputs. The new formula is compared to others in a holdout dataset and detailed performance investigations are conducted for field and lab datasets within this holdout data. Different goodness of fit statistics are used as they represent different perspectives of the model accuracy. After the detailed comparisons are carried out we figured out the most accurate equation that is also applicable on both flume and river data. Especially, on field dataset the prediction performance of the proposed formula outperformed the benchmark formulations.

Qrtzgeotherm: An ActiveX component for the quartz solubility geothermometer

NASA Astrophysics Data System (ADS)

Verma, Mahendra P.

2008-12-01

An ActiveX component, QrtzGeotherm, to calculate temperature and vapor fraction in a geothermal reservoir using quartz solubility geothermometry was written in Visual Basic 6.0. Four quartz solubility equations along the liquid-vapor saturation curve: (i) a quadratic equation of 1/ T and pressure, (ii) a linear equation relating log SiO 2 to the inverse of absolute temperature ( T), (iii) a polynomial of T including logarithmic terms and (iv) temperature as a polynomial of SiO 2 including logarithmic terms are programmed. The QrtzGeotherm has input parameters: (i) HRes—the reservoir enthalpy (kJ/kg), (ii) SiO2TD—silica concentration in total discharge (ppm), (iii) GeoEq—number of quartz solubility equation and (iv) TempGuess—a guess value of the reservoir temperature (°C). The reservoir enthalpy Hres is assumed to be the same as the total discharge enthalpy HR. The output parameters are (i) TempRes—reservoir temperature (°C) and (ii) VapRes—reservoir vapor fraction. The first step is to calculate the total discharge concentration of silica SiO2TD from the concentration of silica SiO2Col of separated water, sampled after N-separations of vapor and water. To use QrtzGeotherm in MS-Excel, three functions SiO2TD, GeoResTemp and GeoResVap for an N-stage separation of geothermal reservoir fluid are written in Visual Basic for Application (VBA). Similarly, a demonstration program, QrtzGeothrm, is written in Visual Basic 6.0.

On the parallel solution of parabolic equations

NASA Technical Reports Server (NTRS)

Gallopoulos, E.; Saad, Youcef

1989-01-01

Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented.

Linear FBG Temperature Sensor Interrogation with Fabry-Perot ITU Multi-wavelength Reference.

PubMed

Park, Hyoung-Jun; Song, Minho

2008-10-29

The equidistantly spaced multi-passbands of a Fabry-Perot ITU filter are used as an efficient multi-wavelength reference for fiber Bragg grating sensor demodulation. To compensate for the nonlinear wavelength tuning effect in the FBG sensor demodulator, a polynomial fitting algorithm was applied to the temporal peaks of the wavelength-scanned ITU filter. The fitted wavelength values are assigned to the peak locations of the FBG sensor reflections, obtaining constant accuracy, regardless of the wavelength scan range and frequency. A linearity error of about 0.18% against a reference thermocouple thermometer was obtained with the suggested method.

Algebraic methods for the solution of some linear matrix equations

NASA Technical Reports Server (NTRS)

Djaferis, T. E.; Mitter, S. K.

1979-01-01

The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.

An Exponential Stability Result of a Timoshenko System with Thermoelasticity with Second Sound and in the Presence of Delay

DOE Office of Scientific and Technical Information (OSTI.GOV)

Apalara, Tijani A., E-mail: tijani@kfupm.edu.sa; Messaoudi, Salim A., E-mail: messaoud@kfupm.edu.sa

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay, where the heat flux is given by Cattaneo’s law. We prove an exponential decay result under a smallness condition on the delay and a stability number introduced first in Santos et al. (J Diff Eqs 253:2715–2733, 2012), using a method different from that of Santos et al. (J Diff Eqs 253:2715–2733, 2012). We also reproduce the polynomial decay of Santos et al. (J Diff Eqs 253:2715–2733, 2012) using the multiplier method in the case of absence of delay. The polynomial decay issue in themore » presence of a small delay is an open question.« less

A ROM-Less Direct Digital Frequency Synthesizer Based on Hybrid Polynomial Approximation

PubMed Central

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

Causal properties of nonlinear gravitational waves in modified gravity

NASA Astrophysics Data System (ADS)

Suvorov, Arthur George; Melatos, Andrew

2017-09-01

Some exact, nonlinear, vacuum gravitational wave solutions are derived for certain polynomial f (R ) gravities. We show that the boundaries of the gravitational domain of dependence, associated with events in polynomial f (R ) gravity, are not null as they are in general relativity. The implication is that electromagnetic and gravitational causality separate into distinct notions in modified gravity, which may have observable astrophysical consequences. The linear theory predicts that tachyonic instabilities occur, when the quadratic coefficient a2 of the Taylor expansion of f (R ) is negative, while the exact, nonlinear, cylindrical wave solutions presented here can be superluminal for all values of a2. Anisotropic solutions are found, whose wave fronts trace out time- or spacelike hypersurfaces with complicated geometric properties. We show that the solutions exist in f (R ) theories that are consistent with Solar System and pulsar timing experiments.

New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov–Galerkin method

PubMed Central

Doha, E.H.; Abd-Elhameed, W.M.; Youssri, Y.H.

2014-01-01

Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method. The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted. The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms. The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient. PMID:26425358

Design of a wearable hand exoskeleton for exercising flexion/extension of the fingers.

PubMed

Jo, Inseong; Lee, Jeongsoo; Park, Yeongyu; Bae, Joonbum

2017-07-01

In this paper, design of a wearable hand exoskeleton system for exercising flexion/extension of the fingers, is proposed. The exoskeleton was designed with a simple and wearable structure to aid finger motions in 1 degree of freedom (DOF). A hand grasping experiment by fully-abled people was performed to investigate general hand flexion/extension motions and the polynomial curve of general hand motions was obtained. To customize the hand exoskeleton for the user, the polynomial curve was adjusted to the joint range of motion (ROM) of the user and the optimal design of the exoskeleton structure was obtained using the optimization algorithm. A prototype divided into two parts (one part for the thumb, the other for rest fingers) was actuated by only two linear motors for compact size and light weight.

Digital Materials - Evaluation of the Possibilities of using Selected Hyperelastic Models to Describe Constitutive Relations

NASA Astrophysics Data System (ADS)

Mańkowski, J.; Lipnicki, J.

2017-08-01

The authors tried to identify the parameters of numerical models of digital materials, which are a kind of composite resulting from the manufacture of the product in 3D printers. With the arrangement of several heads of the printer, the new material can result from mixing of materials with radically different properties, during the process of producing single layer of the product. The new material has properties dependent on the base materials properties and their proportions. Digital materials tensile characteristics are often non-linear and qualify to be described by hyperelastic materials models. The identification was conducted based on the results of tensile tests models, its various degrees coefficients of the polynomials to various degrees coefficients of the polynomials. The Drucker's stability criterion was also examined. Fourteen different materials were analyzed.

Multiple regression technique for Pth degree polynominals with and without linear cross products

NASA Technical Reports Server (NTRS)

Davis, J. W.

1973-01-01

A multiple regression technique was developed by which the nonlinear behavior of specified independent variables can be related to a given dependent variable. The polynomial expression can be of Pth degree and can incorporate N independent variables. Two cases are treated such that mathematical models can be studied both with and without linear cross products. The resulting surface fits can be used to summarize trends for a given phenomenon and provide a mathematical relationship for subsequent analysis. To implement this technique, separate computer programs were developed for the case without linear cross products and for the case incorporating such cross products which evaluate the various constants in the model regression equation. In addition, the significance of the estimated regression equation is considered and the standard deviation, the F statistic, the maximum absolute percent error, and the average of the absolute values of the percent of error evaluated. The computer programs and their manner of utilization are described. Sample problems are included to illustrate the use and capability of the technique which show the output formats and typical plots comparing computer results to each set of input data.

Two-dimensional orthonormal trend surfaces for prospecting

NASA Astrophysics Data System (ADS)

Sarma, D. D.; Selvaraj, J. B.

Orthonormal polynomials have distinct advantages over conventional polynomials: the equations for evaluating trend coefficients are not ill-conditioned and the convergence power of this method is greater compared to the least-squares approximation and therefore the approach by orthonormal functions provides a powerful alternative to the least-squares method. In this paper, orthonormal polynomials in two dimensions are obtained using the Gram-Schmidt method for a polynomial series of the type: Z = 1 + x + y + x2 + xy + y2 + … + yn, where x and y are the locational coordinates and Z is the value of the variable under consideration. Trend-surface analysis, which has wide applications in prospecting, has been carried out using the orthonormal polynomial approach for two sample sets of data from India concerned with gold accumulation from the Kolar Gold Field, and gravity data. A comparison of the orthonormal polynomial trend surfaces with those obtained by the classical least-squares method has been made for the two data sets. In both the situations, the orthonormal polynomial surfaces gave an improved fit to the data. A flowchart and a FORTRAN-IV computer program for deriving orthonormal polynomials of any order and for using them to fit trend surfaces is included. The program has provision for logarithmic transformation of the Z variable. If log-transformation is performed the predicted Z values are reconverted to the original units and the trend-surface map generated for use. The illustration of gold assay data related to the Champion lode system of Kolar Gold Fields, for which a 9th-degree orthonormal trend surface was fit, could be used for further prospecting the area.

Blending Velocities In Task Space In Computing Robot Motions

NASA Technical Reports Server (NTRS)

Volpe, Richard A.

1995-01-01

Blending of linear and angular velocities between sequential specified points in task space constitutes theoretical basis of improved method of computing trajectories followed by robotic manipulators. In method, generalized velocity-vector-blending technique provides relatively simple, common conceptual framework for blending linear, angular, and other parametric velocities. Velocity vectors originate from straight-line segments connecting specified task-space points, called "via frames" and represent specified robot poses. Linear-velocity-blending functions chosen from among first-order, third-order-polynomial, and cycloidal options. Angular velocities blended by use of first-order approximation of previous orientation-matrix-blending formulation. Angular-velocity approximation yields small residual error, quantified and corrected. Method offers both relative simplicity and speed needed for generation of robot-manipulator trajectories in real time.

Principal Curves and Surfaces

DTIC Science & Technology

1984-11-01

welL The subipace is found by using the usual linear eigenv’ctor solution in th3 new enlarged space. This technique was first suggested by Gnanadesikan ...Wilk (1966, 1968), and a good description can be found in Gnanadesikan (1977). They suggested using polynomial functions’ of the original p co...Heidelberg, Springer Ver- lag. Gnanadesikan , R. (1977), Methods for Statistical Data Analysis of Multivariate Observa- tions, Wiley, New York

Linear Modulation Techniques for Digital Microwave

DTIC Science & Technology

1979-08-01

impulse response. Following Forney, a polynomial R(D) is defined such that +0o R(D) - Rh (iT)0i (2-2) i00 The coefficients of R(D) are symnetrical...EQUALIZATION: 8/ I - NYQUIST EQUALIZED / 5- -- DUOINARY EQUALIZED NOTE: 6 MODIFIED 6-QAM I- 4 / 4 -2 2 0 5 10 15 20 25 30 35 40 PEAK AMPLIFIER Eb/N0 Ift 103M

APPLICATION OF NEURAL NETWORK ALGORITHMS FOR BPM LINEARIZATION

DOE Office of Scientific and Technical Information (OSTI.GOV)

Musson, John C.; Seaton, Chad; Spata, Mike F.

2012-11-01

Stripline BPM sensors contain inherent non-linearities, as a result of field distortions from the pickup elements. Many methods have been devised to facilitate corrections, often employing polynomial fitting. The cost of computation makes real-time correction difficult, particulalry when integer math is utilized. The application of neural-network technology, particularly the multi-layer perceptron algorithm, is proposed as an efficient alternative for electrode linearization. A process of supervised learning is initially used to determine the weighting coefficients, which are subsequently applied to the incoming electrode data. A non-linear layer, known as an activation layer, is responsible for the removal of saturation effects. Implementationmore » of a perceptron in an FPGA-based software-defined radio (SDR) is presented, along with performance comparisons. In addition, efficient calculation of the sigmoidal activation function via the CORDIC algorithm is presented.« less

Polynomial approximations of thermodynamic properties of arbitrary gas mixtures over wide pressure and density ranges

NASA Technical Reports Server (NTRS)

Allison, D. O.

1972-01-01

Computer programs for flow fields around planetary entry vehicles require real-gas equilibrium thermodynamic properties in a simple form which can be evaluated quickly. To fill this need, polynomial approximations were found for thermodynamic properties of air and model planetary atmospheres. A coefficient-averaging technique was used for curve fitting in lieu of the usual least-squares method. The polynomials consist of terms up to the ninth degree in each of two variables (essentially pressure and density) including all cross terms. Four of these polynomials can be joined to cover, for example, a range of about 1000 to 11000 K and 0.00001 to 1 atmosphere (1 atm = 1.0133 x 100,000 N/m sq) for a given thermodynamic property. Relative errors of less than 1 percent are found over most of the applicable range.

Concerning the Development of the Wide-Field Optics for WFXT Including Methods of Optimizing X-Ray Optical Prescriptions for Wide-Field Applications

NASA Technical Reports Server (NTRS)

Weisskopf, M. C.; Elsner, R. F.; O'Dell, S. L.; Ramsey, B. D.

2010-01-01

We present a progress report on the various endeavors we are undertaking at MSFC in support of the Wide Field X-Ray Telescope development. In particular we discuss assembly and alignment techniques, in-situ polishing corrections, and the results of our efforts to optimize mirror prescriptions including polynomial coefficients, relative shell displacements, detector placements and tilts. This optimization does not require a blind search through the multi-dimensional parameter space. Under the assumption that the parameters are small enough so that second order expansions are valid, we show that the performance at the detector can be expressed as a quadratic function with numerical coefficients derived from a ray trace through the underlying Wolter I optic. The optimal values for the parameters are found by solving the linear system of equations creating by setting derivatives of this function with respect to each parameter to zero.

ADS: A FORTRAN program for automated design synthesis: Version 1.10

NASA Technical Reports Server (NTRS)

Vanderplaats, G. N.

1985-01-01

A new general-purpose optimization program for engineering design is described. ADS (Automated Design Synthesis - Version 1.10) is a FORTRAN program for solution of nonlinear constrained optimization problems. The program is segmented into three levels: strategy, optimizer, and one-dimensional search. At each level, several options are available so that a total of over 100 possible combinations can be created. Examples of available strategies are sequential unconstrained minimization, the Augmented Lagrange Multiplier method, and Sequential Linear Programming. Available optimizers include variable metric methods and the Method of Feasible Directions as examples, and one-dimensional search options include polynomial interpolation and the Golden Section method as examples. Emphasis is placed on ease of use of the program. All information is transferred via a single parameter list. Default values are provided for all internal program parameters such as convergence criteria, and the user is given a simple means to over-ride these, if desired.

Experimental Modal Analysis and Dynamic Component Synthesis. Volume 6. Software User’s Guide.

DTIC Science & Technology

1987-12-01

generate a Complex Mode Indication Function ( CMIF ) from the measurement directory, including modifications from the measurement selection option. This...reference measurements are - included in the data set to be analyzed. The peaks in the CMIF chart indicate existing modes. Thus, the order of the the...polynomials is determined by the number of peaks found in the CMIF chart. Then, the order of the polynomials can be determined before the estimation process

Defining the "older" crash victim: the relationship between age and serious injury in motor vehicle crashes.

PubMed

Newgard, Craig D

2008-07-01

Age is often used as a predictor of injury and mortality in motor vehicle crashes (MVCs), however, the age that defines an "older" occupant in terms of injury-risk remains unclear, as do specific injury patterns associated with increasing age. The objective of this study was to evaluate the relationship between age and serious injury (including injury patterns) for occupants involved in MVCs. This was a retrospective cohort study using a national population-based cohort of adult front-seat occupants involved in MVCs and included in the National Automotive Sampling System Crashworthiness Data System database from 1995 to 2006. The primary outcome was serious injury, defined as an abbreviated injury scale (AIS) score >/=3 in any body region. Anatomic injury patterns were also assessed by age. One hundred thousand one hundred and fifty-six adult front-seat occupants were included in the analysis, of which 14,128 (2%) were seriously injured. Age was a strong predictor of serious injury using a variety of different age covariates (categorical, continuous, and polynomial) in multivariable regression models (p<0.0001 for all). There was evidence of a strong non-linear relationship between age and serious injury (p<0.001 for comparison of non-linear to linear representation of age). There was no age that clearly defined an "older" occupant by injury risk, as the odds of injury increased with increasing age across all age groups. The proportion of serious head and extremity injuries gradually increased with increasing age, while serious chest injuries markedly increased after 60 years. Age is a strong predictor of serious injury from motor vehicle trauma, the risk of which increases in non-linear fashion as age increases. There is no specific age that clearly defines an "older" occupant by injury risk.

Jack Polynomials as Fractional Quantum Hall States and the Betti Numbers of the ( k + 1)-Equals Ideal

NASA Astrophysics Data System (ADS)

Zamaere, Christine Berkesch; Griffeth, Stephen; Sam, Steven V.

2014-08-01

We show that for Jack parameter α = -( k + 1)/( r - 1), certain Jack polynomials studied by Feigin-Jimbo-Miwa-Mukhin vanish to order r when k + 1 of the coordinates coincide. This result was conjectured by Bernevig and Haldane, who proposed that these Jack polynomials are model wavefunctions for fractional quantum Hall states. Special cases of these Jack polynomials include the wavefunctions of Laughlin and Read-Rezayi. In fact, along these lines we prove several vanishing theorems known as clustering properties for Jack polynomials in the mathematical physics literature, special cases of which had previously been conjectured by Bernevig and Haldane. Motivated by the method of proof, which in the case r = 2 identifies the span of the relevant Jack polynomials with the S n -invariant part of a unitary representation of the rational Cherednik algebra, we conjecture that unitary representations of the type A Cherednik algebra have graded minimal free resolutions of Bernstein-Gelfand-Gelfand type; we prove this for the ideal of the ( k + 1)-equals arrangement in the case when the number of coordinates n is at most 2 k + 1. In general, our conjecture predicts the graded S n -equivariant Betti numbers of the ideal of the ( k + 1)-equals arrangement with no restriction on the number of ambient dimensions.

Local Composite Quantile Regression Smoothing for Harris Recurrent Markov Processes

PubMed Central

Li, Degui; Li, Runze

2016-01-01

In this paper, we study the local polynomial composite quantile regression (CQR) smoothing method for the nonlinear and nonparametric models under the Harris recurrent Markov chain framework. The local polynomial CQR regression method is a robust alternative to the widely-used local polynomial method, and has been well studied in stationary time series. In this paper, we relax the stationarity restriction on the model, and allow that the regressors are generated by a general Harris recurrent Markov process which includes both the stationary (positive recurrent) and nonstationary (null recurrent) cases. Under some mild conditions, we establish the asymptotic theory for the proposed local polynomial CQR estimator of the mean regression function, and show that the convergence rate for the estimator in nonstationary case is slower than that in stationary case. Furthermore, a weighted type local polynomial CQR estimator is provided to improve the estimation efficiency, and a data-driven bandwidth selection is introduced to choose the optimal bandwidth involved in the nonparametric estimators. Finally, we give some numerical studies to examine the finite sample performance of the developed methodology and theory. PMID:27667894

Simulated quantum computation of molecular energies.

PubMed

Aspuru-Guzik, Alán; Dutoi, Anthony D; Love, Peter J; Head-Gordon, Martin

2005-09-09

The calculation time for the energy of atoms and molecules scales exponentially with system size on a classical computer but polynomially using quantum algorithms. We demonstrate that such algorithms can be applied to problems of chemical interest using modest numbers of quantum bits. Calculations of the water and lithium hydride molecular ground-state energies have been carried out on a quantum computer simulator using a recursive phase-estimation algorithm. The recursive algorithm reduces the number of quantum bits required for the readout register from about 20 to 4. Mappings of the molecular wave function to the quantum bits are described. An adiabatic method for the preparation of a good approximate ground-state wave function is described and demonstrated for a stretched hydrogen molecule. The number of quantum bits required scales linearly with the number of basis functions, and the number of gates required grows polynomially with the number of quantum bits.

Spline based least squares integration for two-dimensional shape or wavefront reconstruction

DOE PAGES

Huang, Lei; Xue, Junpeng; Gao, Bo; ...

2016-12-21

In this paper, we present a novel method to handle two-dimensional shape or wavefront reconstruction from its slopes. The proposed integration method employs splines to fit the measured slope data with piecewise polynomials and uses the analytical polynomial functions to represent the height changes in a lateral spacing with the pre-determined spline coefficients. The linear least squares method is applied to estimate the height or wavefront as a final result. Numerical simulations verify that the proposed method has less algorithm errors than two other existing methods used for comparison. Especially at the boundaries, the proposed method has better performance. Themore » noise influence is studied by adding white Gaussian noise to the slope data. Finally, experimental data from phase measuring deflectometry are tested to demonstrate the feasibility of the new method in a practical measurement.« less

Spline based least squares integration for two-dimensional shape or wavefront reconstruction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Huang, Lei; Xue, Junpeng; Gao, Bo

In this paper, we present a novel method to handle two-dimensional shape or wavefront reconstruction from its slopes. The proposed integration method employs splines to fit the measured slope data with piecewise polynomials and uses the analytical polynomial functions to represent the height changes in a lateral spacing with the pre-determined spline coefficients. The linear least squares method is applied to estimate the height or wavefront as a final result. Numerical simulations verify that the proposed method has less algorithm errors than two other existing methods used for comparison. Especially at the boundaries, the proposed method has better performance. Themore » noise influence is studied by adding white Gaussian noise to the slope data. Finally, experimental data from phase measuring deflectometry are tested to demonstrate the feasibility of the new method in a practical measurement.« less

Temperature Effects and Compensation-Control Methods

PubMed Central

Xia, Dunzhu; Chen, Shuling; Wang, Shourong; Li, Hongsheng

2009-01-01

In the analysis of the effects of temperature on the performance of microgyroscopes, it is found that the resonant frequency of the microgyroscope decreases linearly as the temperature increases, and the quality factor changes drastically at low temperatures. Moreover, the zero bias changes greatly with temperature variations. To reduce the temperature effects on the microgyroscope, temperature compensation-control methods are proposed. In the first place, a BP (Back Propagation) neural network and polynomial fitting are utilized for building the temperature model of the microgyroscope. Considering the simplicity and real-time requirements, piecewise polynomial fitting is applied in the temperature compensation system. Then, an integral-separated PID (Proportion Integration Differentiation) control algorithm is adopted in the temperature control system, which can stabilize the temperature inside the microgyrocope in pursuing its optimal performance. Experimental results reveal that the combination of microgyroscope temperature compensation and control methods is both realizable and effective in a miniaturized microgyroscope prototype. PMID:22408509

An hp-adaptivity and error estimation for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Bey, Kim S.

1995-01-01

This paper presents an hp-adaptive discontinuous Galerkin method for linear hyperbolic conservation laws. A priori and a posteriori error estimates are derived in mesh-dependent norms which reflect the dependence of the approximate solution on the element size (h) and the degree (p) of the local polynomial approximation. The a posteriori error estimate, based on the element residual method, provides bounds on the actual global error in the approximate solution. The adaptive strategy is designed to deliver an approximate solution with the specified level of error in three steps. The a posteriori estimate is used to assess the accuracy of a given approximate solution and the a priori estimate is used to predict the mesh refinements and polynomial enrichment needed to deliver the desired solution. Numerical examples demonstrate the reliability of the a posteriori error estimates and the effectiveness of the hp-adaptive strategy.

Differential Galois theory and non-integrability of planar polynomial vector fields

NASA Astrophysics Data System (ADS)

Acosta-Humánez, Primitivo B.; Lázaro, J. Tomás; Morales-Ruiz, Juan J.; Pantazi, Chara

2018-06-01

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.

Gabor-based kernel PCA with fractional power polynomial models for face recognition.

PubMed

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 polynomial models, the Gabor wavelet-based PCA method, and the Gabor wavelet-based kernel PCA method with polynomial kernels.

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

PubMed

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

2014-10-01

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

Key-Generation Algorithms for Linear Piece In Hand Matrix Method

NASA Astrophysics Data System (ADS)

Tadaki, Kohtaro; Tsujii, Shigeo

The linear Piece In Hand (PH, for short) matrix method with random variables was proposed in our former work. It is a general prescription which can be applicable to any type of multivariate public-key cryptosystems for the purpose of enhancing their security. Actually, we showed, in an experimental manner, that the linear PH matrix method with random variables can certainly enhance the security of HFE against the Gröbner basis attack, where HFE is one of the major variants of multivariate public-key cryptosystems. In 1998 Patarin, Goubin, and Courtois introduced the plus method as a general prescription which aims to enhance the security of any given MPKC, just like the linear PH matrix method with random variables. In this paper we prove the equivalence between the plus method and the primitive linear PH matrix method, which is introduced by our previous work to explain the notion of the PH matrix method in general in an illustrative manner and not for a practical use to enhance the security of any given MPKC. Based on this equivalence, we show that the linear PH matrix method with random variables has the substantial advantage over the plus method with respect to the security enhancement. In the linear PH matrix method with random variables, the three matrices, including the PH matrix, play a central role in the secret-key and public-key. In this paper, we clarify how to generate these matrices and thus present two probabilistic polynomial-time algorithms to generate these matrices. In particular, the second one has a concise form, and is obtained as a byproduct of the proof of the equivalence between the plus method and the primitive linear PH matrix method.

Intra-Operative Dosimetry in Prostate Brachytherapy

DTIC Science & Technology

2007-11-01

of the focal spot. 2.1. Model for Reconstruction Space Transformation As illustrated in Figure 8, let A & B ( with reference frames FA & FB) be the two...simplex optimization method in MATLAB 7.0 with the search space being defined by the distortion modes from PCA. A linear combination of the modes would...arm is tracked with an X-ray fiducial system called FTRAC that is composed of optimally selected polynomial

Conflicting Frames: A Case of Misalignment between Professional Development Efforts and a Teacher's Practice in a High School Mathematics Classroom

ERIC Educational Resources Information Center

Heyd-Metzuyanim, Einat; Munter, Charles; Greeno, James

2018-01-01

We examine the case of a lesson planning session within the context of professional development for dialogic instruction, and the lesson enacted following this session, which was intended to provide opportunities to 11th and 12th grade algebra students to explore polynomial functions in terms of their roots and linear factors. Our goal was,…

Stochastic Galerkin methods for the steady-state Navier–Stokes equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sousedík, Bedřich, E-mail: sousedik@umbc.edu; Elman, Howard C., E-mail: elman@cs.umd.edu

2016-07-01

We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmarkmore » problems.« less

Algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations with the use of parallel computations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Moryakov, A. V., E-mail: sailor@orc.ru

2016-12-15

An algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations is presented. The algorithm for systems of first-order differential equations is implemented in the EDELWEISS code with the possibility of parallel computations on supercomputers employing the MPI (Message Passing Interface) standard for the data exchange between parallel processes. The solution is represented by a series of orthogonal polynomials on the interval [0, 1]. The algorithm is characterized by simplicity and the possibility to solve nonlinear problems with a correction of the operator in accordance with the solution obtained in the previous iterative process.

Exponential stabilization of magnetoelastic waves in a Mindlin-Timoshenko plate by localized internal damping

NASA Astrophysics Data System (ADS)

Grobbelaar-Van Dalsen, Marié

2015-08-01

This article is a continuation of our earlier work in Grobbelaar-Van Dalsen (Z Angew Math Phys 63:1047-1065, 2012) on the polynomial stabilization of a linear model for the magnetoelastic interactions in a two-dimensional electrically conducting Mindlin-Timoshenko plate. We introduce nonlinear damping that is effective only in a small portion of the interior of the plate. It turns out that the model is uniformly exponentially stable when the function , that represents the locally distributed damping, behaves linearly near the origin. However, the use of Mindlin-Timoshenko plate theory in the model enforces a restriction on the region occupied by the plate.

Experimentally validated quantitative linear model for the device physics of elastomeric microfluidic valves

NASA Astrophysics Data System (ADS)

Kartalov, Emil P.; Scherer, Axel; Quake, Stephen R.; Taylor, Clive R.; Anderson, W. French

2007-03-01

A systematic experimental study and theoretical modeling of the device physics of polydimethylsiloxane "pushdown" microfluidic valves are presented. The phase space is charted by 1587 dimension combinations and encompasses 45-295μm lateral dimensions, 16-39μm membrane thickness, and 1-28psi closing pressure. Three linear models are developed and tested against the empirical data, and then combined into a fourth-power-polynomial superposition. The experimentally validated final model offers a useful quantitative prediction for a valve's properties as a function of its dimensions. Typical valves (80-150μm width) are shown to behave like thin springs.

Linear FBG Temperature Sensor Interrogation with Fabry-Perot ITU Multi-wavelength Reference

PubMed Central

Park, Hyoung-Jun; Song, Minho

2008-01-01

The equidistantly spaced multi-passbands of a Fabry-Perot ITU filter are used as an efficient multi-wavelength reference for fiber Bragg grating sensor demodulation. To compensate for the nonlinear wavelength tuning effect in the FBG sensor demodulator, a polynomial fitting algorithm was applied to the temporal peaks of the wavelength-scanned ITU filter. The fitted wavelength values are assigned to the peak locations of the FBG sensor reflections, obtaining constant accuracy, regardless of the wavelength scan range and frequency. A linearity error of about 0.18% against a reference thermocouple thermometer was obtained with the suggested method. PMID:27873898

Stochastic Galerkin methods for the steady-state Navier–Stokes equations

DOE PAGES

Sousedík, Bedřich; Elman, Howard C.

2016-04-12

We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmarkmore » problems.« less

Effects of longitudinal asymmetry in heavy-ion collisions

NASA Astrophysics Data System (ADS)

Raniwala, Rashmi; Raniwala, Sudhir; Loizides, Constantin

2018-02-01

In collisions of identical nuclei at a given impact parameter, the number of nucleons participating in the overlap region of each nucleus can be unequal due to nuclear density fluctuations. The asymmetry due to the unequal number of participating nucleons, referred to as longitudinal asymmetry, causes a shift in the center-of-mass rapidity of the participant zone. The information of the event asymmetry allows us to isolate and study the effect of longitudinal asymmetry on rapidity distribution of final state particles. In a Monte Carlo Glauber model the average rapidity shift is found to be almost linearly related to the asymmetry. Using toy models, as well as Monte Carlo data for Pb-Pb collisions at 2.76 TeV generated with hijing, two different versions of ampt and dpmjet models, we demonstrate that the effect of asymmetry on final state rapidity distribution can be quantitatively related to the average rapidity shift via a third-order polynomial with a dominantly linear term. The coefficients of the polynomial are proportional to the rapidity shift with the dependence being sensitive to the details of the rapidity distribution. Experimental estimates of the spectator asymmetry through the measurement of spectator nucleons in a zero-degree calorimeter may hence be used to further constrain the initial conditions in ultra-relativistic heavy-ion collisions.

Modeling the effect of temperature on survival rate of Salmonella Enteritidis in yogurt.

PubMed

Szczawiński, J; Szczawińska, M E; Łobacz, A; Jackowska-Tracz, A

2014-01-01

The aim of the study was to determine the inactivation rates of Salmonella Enteritidis in commercially produced yogurt and to generate primary and secondary mathematical models to predict the behaviour of these bacteria during storage at different temperatures. The samples were inoculated with the mixture of three S. Enteritidis strains and stored at 5 degrees C, 10 degrees C, 15 degrees C, 20 degrees C and 25 degrees C for 24 h. The number of salmonellae was determined every two hours. It was found that the number of bacteria decreased linearly with storage time in all samples. Storage temperature and pH of yogurt significantly influenced survival rate of S. Enteritidis (p < 0.05). In samples kept at 5 degrees C the number of salmonellae decreased at the lowest rate, whereas at 25 degrees C the reduction in number of bacteria was the most dynamic. The natural logarithm of mean inactivation rates of Salmonella calculated from primary model was fitted to two secondary models: linear and polynomial. Equations obtained from both secondary models can be applied as a tool for prediction of inactivation rate of Salmonella in yogurt stored under temperature range from 5 to 25 degrees C; however, polynomial model gave the better fit to the experimental data.

Simple, fast, and low-cost camera-based water content measurement with colorimetric fluorescent indicator

NASA Astrophysics Data System (ADS)

Song, Seok-Jeong; Kim, Tae-Il; Kim, Youngmi; Nam, Hyoungsik

2018-05-01

Recently, a simple, sensitive, and low-cost fluorescent indicator has been proposed to determine water contents in organic solvents, drugs, and foodstuffs. The change of water content leads to the change of the indicator's fluorescence color under the ultra-violet (UV) light. Whereas the water content values could be estimated from the spectrum obtained by a bulky and expensive spectrometer in the previous research, this paper demonstrates a simple and low-cost camera-based water content measurement scheme with the same fluorescent water indicator. Water content is calculated over the range of 0-30% by quadratic polynomial regression models with color information extracted from the captured images of samples. Especially, several color spaces such as RGB, xyY, L∗a∗b∗, u‧v‧, HSV, and YCBCR have been investigated to establish the optimal color information features over both linear and nonlinear RGB data given by a camera before and after gamma correction. In the end, a 2nd order polynomial regression model along with HSV in a linear domain achieves the minimum mean square error of 1.06% for a 3-fold cross validation method. Additionally, the resultant water content estimation model is implemented and evaluated in an off-the-shelf Android-based smartphone.

Progress Report on Optimizing X-ray Optical Prescriptions for Wide-Field Applications

NASA Technical Reports Server (NTRS)

Elsner, R. F.; O'Dell, S. L.; Ramsey, B. D.; Weisskopf, M. C.

2011-01-01

We report on the present status of our continuing efforts to develop a method for optimizing wide-field nested x-ray telescope mirror prescriptions. Utilizing extensive Monte-Carlo ray trace simulations, we find an analytic form for the root-mean-square dispersion of rays from a Wolter I optic on the surface of a flat focal plane detector as a function of detector tilt away from the nominal focal plane and detector displacement along the optical axis. The configuration minimizing the ray dispersion from a nested array of Wolter I telescopes is found by solving a linear system of equations for tilt and individual mirror pair displacement. Finally we outline our initial efforts at expanding this method to include higher order polynomial terms in the mirror prescriptions.

Exact models for isotropic matter

NASA Astrophysics Data System (ADS)

Thirukkanesh, S.; Maharaj, S. D.

2006-04-01

We study the Einstein-Maxwell system of equations in spherically symmetric gravitational fields for static interior spacetimes. The condition for pressure isotropy is reduced to a recurrence equation with variable, rational coefficients. We demonstrate that this difference equation can be solved in general using mathematical induction. Consequently, we can find an explicit exact solution to the Einstein-Maxwell field equations. The metric functions, energy density, pressure and the electric field intensity can be found explicitly. Our result contains models found previously, including the neutron star model of Durgapal and Bannerji. By placing restrictions on parameters arising in the general series, we show that the series terminate and there exist two linearly independent solutions. Consequently, it is possible to find exact solutions in terms of elementary functions, namely polynomials and algebraic functions.

A method for the selection of a functional form for a thermodynamic equation of state using weighted linear least squares stepwise regression

NASA Technical Reports Server (NTRS)

Jacobsen, R. T.; Stewart, R. B.; Crain, R. W., Jr.; Rose, G. L.; Myers, A. F.

1976-01-01

A method was developed for establishing a rational choice of the terms to be included in an equation of state with a large number of adjustable coefficients. The methods presented were developed for use in the determination of an equation of state for oxygen and nitrogen. However, a general application of the methods is possible in studies involving the determination of an optimum polynomial equation for fitting a large number of data points. The data considered in the least squares problem are experimental thermodynamic pressure-density-temperature data. Attention is given to a description of stepwise multiple regression and the use of stepwise regression in the determination of an equation of state for oxygen and nitrogen.

Fitting by Orthonormal Polynomials of Silver Nanoparticles Spectroscopic Data

NASA Astrophysics Data System (ADS)

Bogdanova, Nina; Koleva, Mihaela

2018-02-01

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

Linear and curvilinear correlations of brain gray matter volume and density with age using voxel-based morphometry with the Akaike information criterion in 291 healthy children.

PubMed

Taki, Yasuyuki; Hashizume, Hiroshi; Thyreau, Benjamin; Sassa, Yuko; Takeuchi, Hikaru; Wu, Kai; Kotozaki, Yuka; Nouchi, Rui; Asano, Michiko; Asano, Kohei; Fukuda, Hiroshi; Kawashima, Ryuta

2013-08-01

We examined linear and curvilinear correlations of gray matter volume and density in cortical and subcortical gray matter with age using magnetic resonance images (MRI) in a large number of healthy children. We applied voxel-based morphometry (VBM) and region-of-interest (ROI) analyses with the Akaike information criterion (AIC), which was used to determine the best-fit model by selecting which predictor terms should be included. We collected data on brain structural MRI in 291 healthy children aged 5-18 years. Structural MRI data were segmented and normalized using a custom template by applying the diffeomorphic anatomical registration using exponentiated lie algebra (DARTEL) procedure. Next, we analyzed the correlations of gray matter volume and density with age in VBM with AIC by estimating linear, quadratic, and cubic polynomial functions. Several regions such as the prefrontal cortex, the precentral gyrus, and cerebellum showed significant linear or curvilinear correlations between gray matter volume and age on an increasing trajectory, and between gray matter density and age on a decreasing trajectory in VBM and ROI analyses with AIC. Because the trajectory of gray matter volume and density with age suggests the progress of brain maturation, our results may contribute to clarifying brain maturation in healthy children from the viewpoint of brain structure. Copyright © 2012 Wiley Periodicals, Inc.

Explaining variation in tropical plant community composition: influence of environmental and spatial data quality.

PubMed

Jones, Mirkka M; Tuomisto, Hanna; Borcard, Daniel; Legendre, Pierre; Clark, David B; Olivas, Paulo C

2008-03-01

The degree to which variation in plant community composition (beta-diversity) is predictable from environmental variation, relative to other spatial processes, is of considerable current interest. We addressed this question in Costa Rican rain forest pteridophytes (1,045 plots, 127 species). We also tested the effect of data quality on the results, which has largely been overlooked in earlier studies. To do so, we compared two alternative spatial models [polynomial vs. principal coordinates of neighbour matrices (PCNM)] and ten alternative environmental models (all available environmental variables vs. four subsets, and including their polynomials vs. not). Of the environmental data types, soil chemistry contributed most to explaining pteridophyte community variation, followed in decreasing order of contribution by topography, soil type and forest structure. Environmentally explained variation increased moderately when polynomials of the environmental variables were included. Spatially explained variation increased substantially when the multi-scale PCNM spatial model was used instead of the traditional, broad-scale polynomial spatial model. The best model combination (PCNM spatial model and full environmental model including polynomials) explained 32% of pteridophyte community variation, after correcting for the number of sampling sites and explanatory variables. Overall evidence for environmental control of beta-diversity was strong, and the main floristic gradients detected were correlated with environmental variation at all scales encompassed by the study (c. 100-2,000 m). Depending on model choice, however, total explained variation differed more than fourfold, and the apparent relative importance of space and environment could be reversed. Therefore, we advocate a broader recognition of the impacts that data quality has on analysis results. A general understanding of the relative contributions of spatial and environmental processes to species distributions and beta-diversity requires that methodological artefacts are separated from real ecological differences.

Identifying the nonlinear mechanical behaviour of micro-speakers from their quasi-linear electrical response

NASA Astrophysics Data System (ADS)

Zilletti, Michele; Marker, Arthur; Elliott, Stephen John; Holland, Keith

2017-05-01

In this study model identification of the nonlinear dynamics of a micro-speaker is carried out by purely electrical measurements, avoiding any explicit vibration measurements. It is shown that a dynamic model of the micro-speaker, which takes into account the nonlinear damping characteristic of the device, can be identified by measuring the response between the voltage input and the current flowing into the coil. An analytical formulation of the quasi-linear model of the micro-speaker is first derived and an optimisation method is then used to identify a polynomial function which describes the mechanical damping behaviour of the micro-speaker. The analytical results of the quasi-linear model are compared with numerical results. This study potentially opens up the possibility of efficiently implementing nonlinear echo cancellers.

The Application of Various Nonlinear Models to Describe Academic Growth Trajectories: An Empirical Analysis Using Four-Wave Longitudinal Achievement Data from a Large Urban School District

ERIC Educational Resources Information Center

Shin, Tacksoo

2012-01-01

This study introduced various nonlinear growth models, including the quadratic conventional polynomial model, the fractional polynomial model, the Sigmoid model, the growth model with negative exponential functions, the multidimensional scaling technique, and the unstructured growth curve model. It investigated which growth models effectively…

Poly-Frobenius-Euler polynomials

NASA Astrophysics Data System (ADS)

Kurt, Burak

2017-07-01

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

Analytic Theory for the Yarkovsky-O Effect on Obliquity

NASA Astrophysics Data System (ADS)

Nesvorný, David; Vokrouhlický, David

2008-07-01

The Yarkovsky-O'Keefe-Radzievski-Paddack (YORP) effect is a thermal radiation torque that causes small objects to speed up or slow down their rotation and modify their spin vector orientation. This effect has important implications for spin dynamics of diameter D lsim 50 km asteroids. In our previous work we developed an analytic theory for the component of the YORP torque that affects the spin rate. Here we extend these calculations to determine the effect of the YORP torque on obliquity. Our theory is limited to objects with near-spherical shapes. Two limiting cases are studied: (1) immediate emission of the thermal energy that occurs for surface thermal conductivity K = 0; (2) the effects of K ≠ 0 in the limit of small temporal variations of the surface temperature. We use the linearized heat transport equation to model (2). The results include explicit scaling of the YORP torque on obliquity with physical and dynamical parameters such as the thermal conductivity and spin rate. The dependence of torques on the obliquity is given as series of the Legendre polynomials. Comparisons show excellent agreement of the analytic results with the numerically calculated YORP torques for objects such as asteroids 1998 KY26 and (66391) 1999 KW4. We suggest that an important fraction of main belt asteroids may have specific obliquity values (generalized Slivan states) arising from the roots of the Legendre polynomials.

Application of Vector Spherical Harmonics and Kernel Regression to the Computations of OMM Parameters

NASA Astrophysics Data System (ADS)

Marco, F. J.; Martínez, M. J.; López, J. A.

2015-04-01

The high quality of Hipparcos data in position, proper motion, and parallax has allowed for studies about stellar kinematics with the aim of achieving a better physical understanding of our galaxy, based on accurate calculus of the Ogorodnikov-Milne model (OMM) parameters. The use of discrete least squares is the most common adjustment method, but it may lead to errors mainly because of the inhomogeneous spatial distribution of the data. We present an example of the instability of this method using the case of a function given by a linear combination of Legendre polynomials. These polynomials are basic in the use of vector spherical harmonics, which have been used to compute the OMM parameters by several authors, such as Makarov & Murphy, Mignard & Klioner, and Vityazev & Tsvetkov. To overcome the former problem, we propose the use of a mixed method (see Marco et al.) that includes the extension of the functions of residuals to any point on the celestial sphere. The goal is to be able to work with continuous variables in the calculation of the coefficients of the vector spherical harmonic developments with stability and efficiency. We apply this mixed procedure to the study of the kinematics of the stars in our Galaxy, employing the Hipparcos velocity field data to obtain the OMM parameters. Previously, we tested the method by perturbing the Vectorial Spherical Harmonics model as well as the velocity vector field.

Sum-of-Squares-Based Region of Attraction Analysis for Gain-Scheduled Three-Loop Autopilot

NASA Astrophysics Data System (ADS)

Seo, Min-Won; Kwon, Hyuck-Hoon; Choi, Han-Lim

2018-04-01

A conventional method of designing a missile autopilot is to linearize the original nonlinear dynamics at several trim points, then to determine linear controllers for each linearized model, and finally implement gain-scheduling technique. The validation of such a controller is often based on linear system analysis for the linear closed-loop system at the trim conditions. Although this type of gain-scheduled linear autopilot works well in practice, validation based solely on linear analysis may not be sufficient to fully characterize the closed-loop system especially when the aerodynamic coefficients exhibit substantial nonlinearity with respect to the flight condition. The purpose of this paper is to present a methodology for analyzing the stability of a gain-scheduled controller in a setting close to the original nonlinear setting. The method is based on sum-of-squares (SOS) optimization that can be used to characterize the region of attraction of a polynomial system by solving convex optimization problems. The applicability of the proposed SOS-based methodology is verified on a short-period autopilot of a skid-to-turn missile.

On the Development of an Efficient Parallel Hybrid Solver with Application to Acoustically Treated Aero-Engine Nacelles

NASA Technical Reports Server (NTRS)

Watson, Willie R.; Nark, Douglas M.; Nguyen, Duc T.; Tungkahotara, Siroj

2006-01-01

A finite element solution to the convected Helmholtz equation in a nonuniform flow is used to model the noise field within 3-D acoustically treated aero-engine nacelles. Options to select linear or cubic Hermite polynomial basis functions and isoparametric elements are included. However, the key feature of the method is a domain decomposition procedure that is based upon the inter-mixing of an iterative and a direct solve strategy for solving the discrete finite element equations. This procedure is optimized to take full advantage of sparsity and exploit the increased memory and parallel processing capability of modern computer architectures. Example computations are presented for the Langley Flow Impedance Test facility and a rectangular mapping of a full scale, generic aero-engine nacelle. The accuracy and parallel performance of this new solver are tested on both model problems using a supercomputer that contains hundreds of central processing units. Results show that the method gives extremely accurate attenuation predictions, achieves super-linear speedup over hundreds of CPUs, and solves upward of 25 million complex equations in a quarter of an hour.

Consequences of Laughter Upon Trunk Compression and Cortical Activation: Linear and Polynomial Relations

PubMed Central

Svebak, Sven

2016-01-01

Results from two studies of biological consequences of laughter are reported. A proposed inhibitory brain mechanism was tested in Study 1. It aims to protect against trunk compression that can cause health hazards during vigorous laughter. Compression may be maximal during moderate durations and, for protective reasons, moderate in enduring vigorous laughs. Twenty-five university students volunteered to see a candid camera film. Laughter responses (LR) and the superimposed ha-responses were operationally assessed by mercury-filled strain gauges strapped around the trunk. On average, the thorax compression amplitudes exceeded those of the abdomen, and greater amplitudes were seen in the males than in the females after correction for resting trunk circumference. Regression analyses supported polynomial relations because medium LR durations were associated with particularly high thorax amplitudes. In Study 2, power changes were computed in the beta and alpha EEG frequency bands of the parietal cortex from before to after exposure to the comedy “Dinner for one” in 56 university students. Highly significant linear relations were calculated between the number of laughs and post-exposure cortical activation (increase of beta, decrease of alpha) due to high activation after frequent laughter. The results from Study 1 supported the hypothesis of a protective brain mechanism that is activated during long LRs to reduce the risk of harm to vital organs in the trunk cavity. The results in Study 2 supported a linear cortical activation and, thus, provided evidence for a biological correlate to the subjective experience of mental refreshment after laughter. PMID:27547260

A refinement of the combination equations for evaporation

USGS Publications Warehouse

Milly, P.C.D.

1991-01-01

Most combination equations for evaporation rely on a linear expansion of the saturation vapor-pressure curve around the air temperature. Because the temperature at the surface may differ from this temperature by several degrees, and because the saturation vapor-pressure curve is nonlinear, this approximation leads to a certain degree of error in those evaporation equations. It is possible, however, to introduce higher-order polynomial approximations for the saturation vapor-pressure curve and to derive a family of explicit equations for evaporation, having any desired degree of accuracy. Under the linear approximation, the new family of equations for evaporation reduces, in particular cases, to the combination equations of H. L. Penman (Natural evaporation from open water, bare soil and grass, Proc. R. Soc. London, Ser. A193, 120-145, 1948) and of subsequent workers. Comparison of the linear and quadratic approximations leads to a simple approximate expression for the error associated with the linear case. Equations based on the conventional linear approximation consistently underestimate evaporation, sometimes by a substantial amount. ?? 1991 Kluwer Academic Publishers.

Stochastic Modeling of Flow-Structure Interactions using Generalized Polynomial Chaos

DTIC Science & Technology

2001-09-11

Some basic hypergeometric polynomials that generalize Jacobi polynomials . Memoirs Amer. Math. Soc...scheme, which is represented as a tree structure in figure 1 (following [24]), classifies the hypergeometric orthogonal polynomials and indicates the...2F0(1) 2F0(0) Figure 1: The Askey scheme of orthogonal polynomials The orthogonal polynomials associated with the generalized polynomial chaos,

Comparative Study of SVM Methods Combined with Voxel Selection for Object Category Classification on fMRI Data

PubMed Central

Song, Sutao; Zhan, Zhichao; Long, Zhiying; Zhang, Jiacai; Yao, Li

2011-01-01

Background Support vector machine (SVM) has been widely used as accurate and reliable method to decipher brain patterns from functional MRI (fMRI) data. Previous studies have not found a clear benefit for non-linear (polynomial kernel) SVM versus linear one. Here, a more effective non-linear SVM using radial basis function (RBF) kernel is compared with linear SVM. Different from traditional studies which focused either merely on the evaluation of different types of SVM or the voxel selection methods, we aimed to investigate the overall performance of linear and RBF SVM for fMRI classification together with voxel selection schemes on classification accuracy and time-consuming. Methodology/Principal Findings Six different voxel selection methods were employed to decide which voxels of fMRI data would be included in SVM classifiers with linear and RBF kernels in classifying 4-category objects. Then the overall performances of voxel selection and classification methods were compared. Results showed that: (1) Voxel selection had an important impact on the classification accuracy of the classifiers: in a relative low dimensional feature space, RBF SVM outperformed linear SVM significantly; in a relative high dimensional space, linear SVM performed better than its counterpart; (2) Considering the classification accuracy and time-consuming holistically, linear SVM with relative more voxels as features and RBF SVM with small set of voxels (after PCA) could achieve the better accuracy and cost shorter time. Conclusions/Significance The present work provides the first empirical result of linear and RBF SVM in classification of fMRI data, combined with voxel selection methods. Based on the findings, if only classification accuracy was concerned, RBF SVM with appropriate small voxels and linear SVM with relative more voxels were two suggested solutions; if users concerned more about the computational time, RBF SVM with relative small set of voxels when part of the principal components were kept as features was a better choice. PMID:21359184

Comparative study of SVM methods combined with voxel selection for object category classification on fMRI data.

PubMed

Song, Sutao; Zhan, Zhichao; Long, Zhiying; Zhang, Jiacai; Yao, Li

2011-02-16

Support vector machine (SVM) has been widely used as accurate and reliable method to decipher brain patterns from functional MRI (fMRI) data. Previous studies have not found a clear benefit for non-linear (polynomial kernel) SVM versus linear one. Here, a more effective non-linear SVM using radial basis function (RBF) kernel is compared with linear SVM. Different from traditional studies which focused either merely on the evaluation of different types of SVM or the voxel selection methods, we aimed to investigate the overall performance of linear and RBF SVM for fMRI classification together with voxel selection schemes on classification accuracy and time-consuming. Six different voxel selection methods were employed to decide which voxels of fMRI data would be included in SVM classifiers with linear and RBF kernels in classifying 4-category objects. Then the overall performances of voxel selection and classification methods were compared. Results showed that: (1) Voxel selection had an important impact on the classification accuracy of the classifiers: in a relative low dimensional feature space, RBF SVM outperformed linear SVM significantly; in a relative high dimensional space, linear SVM performed better than its counterpart; (2) Considering the classification accuracy and time-consuming holistically, linear SVM with relative more voxels as features and RBF SVM with small set of voxels (after PCA) could achieve the better accuracy and cost shorter time. The present work provides the first empirical result of linear and RBF SVM in classification of fMRI data, combined with voxel selection methods. Based on the findings, if only classification accuracy was concerned, RBF SVM with appropriate small voxels and linear SVM with relative more voxels were two suggested solutions; if users concerned more about the computational time, RBF SVM with relative small set of voxels when part of the principal components were kept as features was a better choice.

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

PubMed

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

2013-12-01

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

Adaptive surrogate modeling by ANOVA and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation

NASA Astrophysics Data System (ADS)

Tang, Kunkun; Congedo, Pietro M.; Abgrall, Rémi

2016-06-01

The Polynomial Dimensional Decomposition (PDD) is employed in this work for the global sensitivity analysis and uncertainty quantification (UQ) of stochastic systems subject to a moderate to large number of input random variables. Due to the intimate connection between the PDD and the Analysis of Variance (ANOVA) approaches, PDD is able to provide a simpler and more direct evaluation of the Sobol' sensitivity indices, when compared to the Polynomial Chaos expansion (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 standard methods unaffordable for real engineering applications. In order to address the problem of the curse of dimensionality, this work proposes essentially variance-based adaptive strategies aiming to build a cheap meta-model (i.e. surrogate model) by employing the sparse PDD approach with its coefficients computed by regression. Three levels of adaptivity are carried out in this paper: 1) the truncated dimensionality for ANOVA component functions, 2) the active dimension technique especially for second- and higher-order parameter interactions, and 3) the stepwise regression approach designed to retain only the most influential polynomials in the PDD expansion. During this adaptive procedure featuring stepwise regressions, the surrogate model representation keeps containing few terms, so that the cost to resolve repeatedly the linear systems of the least-squares regression problem is negligible. The size of the finally obtained sparse PDD representation is much smaller than the one of the full expansion, since only significant terms are eventually retained. Consequently, a much smaller number of calls to the deterministic model is required to compute the final PDD coefficients.

Detection and Classification of Whale Acoustic Signals

NASA Astrophysics Data System (ADS)

Xian, Yin

This dissertation focuses on two vital challenges in relation to whale acoustic signals: detection and classification. In detection, we evaluated the influence of the uncertain ocean environment on the spectrogram-based detector, and derived the likelihood ratio of the proposed Short Time Fourier Transform detector. Experimental results showed that the proposed detector outperforms detectors based on the spectrogram. The proposed detector is more sensitive to environmental changes because it includes phase information. In classification, our focus is on finding a robust and sparse representation of whale vocalizations. Because whale vocalizations can be modeled as polynomial phase signals, we can represent the whale calls by their polynomial phase coefficients. In this dissertation, we used the Weyl transform to capture chirp rate information, and used a two dimensional feature set to represent whale vocalizations globally. Experimental results showed that our Weyl feature set outperforms chirplet coefficients and MFCC (Mel Frequency Cepstral Coefficients) when applied to our collected data. Since whale vocalizations can be represented by polynomial phase coefficients, it is plausible that the signals lie on a manifold parameterized by these coefficients. We also studied the intrinsic structure of high dimensional whale data by exploiting its geometry. Experimental results showed that nonlinear mappings such as Laplacian Eigenmap and ISOMAP outperform linear mappings such as PCA and MDS, suggesting that the whale acoustic data is nonlinear. We also explored deep learning algorithms on whale acoustic data. We built each layer as convolutions with either a PCA filter bank (PCANet) or a DCT filter bank (DCTNet). With the DCT filter bank, each layer has different a time-frequency scale representation, and from this, one can extract different physical information. Experimental results showed that our PCANet and DCTNet achieve high classification rate on the whale vocalization data set. The word error rate of the DCTNet feature is similar to the MFSC in speech recognition tasks, suggesting that the convolutional network is able to reveal acoustic content of speech signals.

Strong stabilization servo controller with optimization of performance criteria.

PubMed

Sarjaš, Andrej; Svečko, Rajko; Chowdhury, Amor

2011-07-01

Synthesis of a simple robust controller with a pole placement technique and a H(∞) metrics is the method used for control of a servo mechanism with BLDC and BDC electric motors. The method includes solving a polynomial equation on the basis of the chosen characteristic polynomial using the Manabe standard polynomial form and parametric solutions. Parametric solutions are introduced directly into the structure of the servo controller. On the basis of the chosen parametric solutions the robustness of a closed-loop system is assessed through uncertainty models and assessment of the norm ‖•‖(∞). The design procedure and the optimization are performed with a genetic algorithm differential evolution - DE. The DE optimization method determines a suboptimal solution throughout the optimization on the basis of a spectrally square polynomial and Šiljak's absolute stability test. The stability of the designed controller during the optimization is being checked with Lipatov's stability condition. Both utilized approaches: Šiljak's test and Lipatov's condition, check the robustness and stability characteristics on the basis of the polynomial's coefficients, and are very convenient for automated design of closed-loop control and for application in optimization algorithms such as DE. Copyright © 2011 ISA. Published by Elsevier Ltd. All rights reserved.

Polynomial expressions of electron depth dose as a function of energy in various materials: application to thermoluminescence (TL) dosimetry

NASA Astrophysics Data System (ADS)

Deogracias, E. C.; Wood, J. L.; Wagner, E. C.; Kearfott, K. J.

1999-02-01

The CEPXS/ONEDANT code package was used to produce a library of depth-dose profiles for monoenergetic electrons in various materials for energies ranging from 500 keV to 5 MeV in 10 keV increments. The various materials for which depth-dose functions were derived include: lithium fluoride (LiF), aluminum oxide (Al 2O 3), beryllium oxide (BeO), calcium sulfate (CaSO 4), calcium fluoride (CaF 2), lithium boron oxide (LiBO), soft tissue, lens of the eye, adiopose, muscle, skin, glass and water. All materials data sets were fit to five polynomials, each covering a different range of electron energies, using a least squares method. The resultant three dimensional, fifth-order polynomials give the dose as a function of depth and energy for the monoenergetic electrons in each material. The polynomials can be used to describe an energy spectrum by summing the doses at a given depth for each energy, weighted by the spectral intensity for that energy. An application of the polynomial is demonstrated by explaining the energy dependence of thermoluminescent detectors (TLDs) and illustrating the relationship between TLD signal and actual shallow dose due to beta particles.

Functional Form of the Radiometric Equation for the SNPP VIIRS Reflective Solar Bands: An Initial Study

NASA Technical Reports Server (NTRS)

Lei, Ning; Xiong, Xiaoxiong

2016-01-01

The Visible Infrared Imaging Radiometer Suite (VIIRS) aboard the Suomi National Polar-orbiting Partnership (SNPP) satellite is a passive scanning radiometer and an imager, observing radiative energy from the Earth in 22 spectral bands from 0.41 to 12 microns which include 14 reflective solar bands (RSBs). Extending the formula used by the Moderate Resolution Imaging Spectroradiometer instruments, currently the VIIRS determines the sensor aperture spectral radiance through a quadratic polynomial of its detector digital count. It has been known that for the RSBs the quadratic polynomial is not adequate in the design specified spectral radiance region and using a quadratic polynomial could drastically increase the errors in the polynomial coefficients, leading to possible large errors in the determined aperture spectral radiance. In addition, it is very desirable to be able to extend the radiance calculation formula to correctly retrieve the aperture spectral radiance with the level beyond the design specified range. In order to more accurately determine the aperture spectral radiance from the observed digital count, we examine a few polynomials of the detector digital count to calculate the sensor aperture spectral radiance.

Generalized neurofuzzy network modeling algorithms using Bézier-Bernstein polynomial functions and additive decomposition.

PubMed

Hong, X; Harris, C J

2000-01-01

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bézier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bézier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bézier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bézier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.

Existence and energy decay of a nonuniform Timoshenko system with second sound

NASA Astrophysics Data System (ADS)

Hamadouche, Taklit; Messaoudi, Salim A.

2018-02-01

In this paper, we consider a linear thermoelastic Timoshenko system with variable physical parameters, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We discuss the well-posedness and the regularity of solution using the semigroup theory. Moreover, we establish the exponential decay result provided that the stability function χ r(x)=0. Otherwise, we show that the solution decays polynomially.

Term Cancellations in Computing Floating-Point Gröbner Bases

NASA Astrophysics Data System (ADS)

Sasaki, Tateaki; Kako, Fujio

We discuss the term cancellation which makes the floating-point Gröbner basis computation unstable, and show that error accumulation is never negligible in our previous method. Then, we present a new method, which removes accumulated errors as far as possible by reducing matrices constructed from coefficient vectors by the Gaussian elimination. The method manifests amounts of term cancellations caused by the existence of approximate linearly dependent relations among input polynomials.

Structural deformation measurement via efficient tensor polynomial calibrated electro-active glass targets

NASA Astrophysics Data System (ADS)

Gugg, Christoph; Harker, Matthew; O'Leary, Paul

2013-03-01

This paper describes the physical setup and mathematical modelling of a device for the measurement of structural deformations over large scales, e.g., a mining shaft. Image processing techniques are used to determine the deformation by measuring the position of a target relative to a reference laser beam. A particular novelty is the incorporation of electro-active glass; the polymer dispersion liquid crystal shutters enable the simultaneous calibration of any number of consecutive measurement units without manual intervention, i.e., the process is fully automatic. It is necessary to compensate for optical distortion if high accuracy is to be achieved in a compact hardware design where lenses with short focal lengths are used. Wide-angle lenses exhibit significant distortion, which are typically characterized using Zernike polynomials. Radial distortion models assume that the lens is rotationally symmetric; such models are insufficient in the application at hand. This paper presents a new coordinate mapping procedure based on a tensor product of discrete orthogonal polynomials. Both lens distortion and the projection are compensated by a single linear transformation. Once calibrated, to acquire the measurement data, it is necessary to localize a single laser spot in the image. For this purpose, complete interpolation and rectification of the image is not required; hence, we have developed a new hierarchical approach based on a quad-tree subdivision. Cross-validation tests verify the validity, demonstrating that the proposed method accurately models both the optical distortion as well as the projection. The achievable accuracy is e <= +/-0.01 [mm] in a field of view of 150 [mm] x 150 [mm] at a distance of the laser source of 120 [m]. Finally, a Kolmogorov Smirnov test shows that the error distribution in localizing a laser spot is Gaussian. Consequently, due to the linearity of the proposed method, this also applies for the algorithm's output. Therefore, first-order covariance propagation provides an accurate estimate of the measurement uncertainty, which is essential for any measurement device.

CFORM- LINEAR CONTROL SYSTEM DESIGN AND ANALYSIS: CLOSED FORM SOLUTION AND TRANSIENT RESPONSE OF THE LINEAR DIFFERENTIAL EQUATION

NASA Technical Reports Server (NTRS)

Jamison, J. W.

1994-01-01

CFORM was developed by the Kennedy Space Center Robotics Lab to assist in linear control system design and analysis using closed form and transient response mechanisms. The program computes the closed form solution and transient response of a linear (constant coefficient) differential equation. CFORM allows a choice of three input functions: the Unit Step (a unit change in displacement); the Ramp function (step velocity); and the Parabolic function (step acceleration). It is only accurate in cases where the differential equation has distinct roots, and does not handle the case for roots at the origin (s=0). Initial conditions must be zero. Differential equations may be input to CFORM in two forms - polynomial and product of factors. In some linear control analyses, it may be more appropriate to use a related program, Linear Control System Design and Analysis (KSC-11376), which uses root locus and frequency response methods. CFORM was written in VAX FORTRAN for a VAX 11/780 under VAX VMS 4.7. It has a central memory requirement of 30K. CFORM was developed in 1987.

A Novel Approach to Solve Linearized Stellar Pulsation Equations

NASA Astrophysics Data System (ADS)

Bard, Christopher; Teitler, S.

2011-01-01

We present a new approach to modeling linearized, non-radial pulsations in differentially rotating, massive stars. As a first step in this direction, we consider adiabatic pulsations and adopt the Cowling approximation that perturbations of the gravitational potential and its radial derivative are negligible. The angular dependence of the pulsation modes is expressed as a series expansion of associated Legendre polynomials; the resulting coupled system of differential equations is then solved by finding the eigenfrequencies at which the determinant of a characteristic matrix vanishes. Our method improves on previous treatments by removing the requirement that an arbitrary normalization be applied to the eigenfunctions; this brings the benefit of improved numerical robustness.

An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy

NASA Astrophysics Data System (ADS)

Matsushima, Masatomo; Ohmiya, Mayumi

2009-09-01

The stationary KdV hierarchy is constructed using a kind of recursion operator called Λ-operator. The notion of the maximal solution of the n-th stationary KdV equation is introduced. Using this maximal solution, a specific differential polynomial with the auxiliary spectral parameter called the spectral M-function is constructed as the quadratic form of the fundamental system of the eigenvalue problem for the 2-nd order linear ordinary differential equation which is related to the linearizing operator of the hierarchy. By calculating a perfect square condition of the quadratic form by an elementary algebraic method, the complete set of first integrals of this hierarchy is constructed.

Instability of the cored barotropic disc: the linear eigenvalue formulation

NASA Astrophysics Data System (ADS)

Polyachenko, E. V.

2018-05-01

Gaseous rotating razor-thin discs are a testing ground for theories of spiral structure that try to explain appearance and diversity of disc galaxy patterns. These patterns are believed to arise spontaneously under the action of gravitational instability, but calculations of its characteristics in the gas are mostly obscured. The paper suggests a new method for finding the spiral patterns based on an expansion of small amplitude perturbations over Lagrange polynomials in small radial elements. The final matrix equation is extracted from the original hydrodynamical equations without the use of an approximate theory and has a form of the linear algebraic eigenvalue problem. The method is applied to a galactic model with the cored exponential density profile.

Exact Solution of Klein-Gordon and Dirac Equations with Snyder-de Sitter Algebra

NASA Astrophysics Data System (ADS)

Merad, M.; Hadj Moussa, M.

2018-01-01

In this paper, we present the exact solution of the (1+1)-dimensional relativistic Klein-Gordon and Dirac equations with linear vector and scalar potentials in the framework of deformed Snyder-de Sitter model. We introduce some changes of variables, we show that a one-dimensional linear potential for the relativistic system in a space deformed can be equivalent to the trigonometric Rosen-Morse potential in a regular space. In both cases, we determine explicitly the energy eigenvalues and their corresponding eigenfunctions expressed in terms of Romonovski polynomials. The limiting cases are analyzed for α 1 and α 2 → 0 and are compared with those of literature.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jakeman, John D.; Narayan, Akil; Zhou, Tao

We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditionedmore » $$\\ell^1$$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. In conclusion, numerical examples are also provided to demonstrate that our proposed algorithm leads to comparable or improved accuracy even when compared with Legendre- and Hermite-specific algorithms.« less

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.

Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials

NASA Astrophysics Data System (ADS)

Volkmer, Hans

2008-04-01

Sequences of polynomials, orthogonal with respect to signed measures, are associated with a class of differential equations including the Mathieu, Lame and Whittaker-Hill equation. It is shown that the zeros of pn form sequences which converge to the eigenvalues of the corresponding differential equations. Moreover, interlacing properties of the zeros of pn are found. Applications to the numerical treatment of eigenvalue problems are given.

Feasibility study on the least square method for fitting non-Gaussian noise data

NASA Astrophysics Data System (ADS)

Xu, Wei; Chen, Wen; Liang, Yingjie

2018-02-01

This study is to investigate the feasibility of least square method in fitting non-Gaussian noise data. We add different levels of the two typical non-Gaussian noises, Lévy and stretched Gaussian noises, to exact value of the selected functions including linear equations, polynomial and exponential equations, and the maximum absolute and the mean square errors are calculated for the different cases. Lévy and stretched Gaussian distributions have many applications in fractional and fractal calculus. It is observed that the non-Gaussian noises are less accurately fitted than the Gaussian noise, but the stretched Gaussian cases appear to perform better than the Lévy noise cases. It is stressed that the least-squares method is inapplicable to the non-Gaussian noise cases when the noise level is larger than 5%.

Methods of Optimizing X-Ray Optical Prescriptions for Wide-Field Applications

NASA Technical Reports Server (NTRS)

Elsner, R. F.; O'Dell, S. L.; Ramsey, B. D.; Weisskopf, M. C.

2010-01-01

We are working on the development of a method for optimizing wide-field x-ray telescope mirror prescriptions, including polynomial coefficients, mirror shell relative displacements, and (assuming 4 focal plane detectors) detector placement and tilt that does not require a search through the multi-dimensional parameter space. Under the assumption that the parameters are small enough that second order expansions are valid, we show that the performance at the detector surface can be expressed as a quadratic function of the parameters with numerical coefficients derived from a ray trace through the underlying Wolter I optic. The best values for the parameters are found by solving the linear system of equations creating by setting derivatives of this function with respect to each parameter to zero. We describe the present status of this development effort.

Dynamic graphs, community detection, and Riemannian geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bakker, Craig; Halappanavar, Mahantesh; Visweswara Sathanur, Arun

A community is a subset of a wider network where the members of that subset are more strongly connected to each other than they are to the rest of the network. In this paper, we consider the problem of identifying and tracking communities in graphs that change over time {dynamic community detection} and present a framework based on Riemannian geometry to aid in this task. Our framework currently supports several important operations such as interpolating between and averaging over graph snapshots. We compare these Riemannian methods with entry-wise linear interpolation and that the Riemannian methods are generally better suited tomore » dynamic community detection. Next steps with the Riemannian framework include developing higher-order interpolation methods (e.g. the analogues of polynomial and spline interpolation) and a Riemannian least-squares regression method for working with noisy data.« less

Interpolation by new B-splines on a four directional mesh of the plane

NASA Astrophysics Data System (ADS)

Nouisser, O.; Sbibih, D.

2004-01-01

In this paper we construct new simple and composed B-splines on the uniform four directional mesh of the plane, in order to improve the approximation order of B-splines studied in Sablonniere (in: Program on Spline Functions and the Theory of Wavelets, Proceedings and Lecture Notes, Vol. 17, University of Montreal, 1998, pp. 67-78). If φ is such a simple B-spline, we first determine the space of polynomials with maximal total degree included in , and we prove some results concerning the linear independence of the family . Next, we show that the cardinal interpolation with φ is correct and we study in S(φ) a Lagrange interpolation problem. Finally, we define composed B-splines by repeated convolution of φ with the characteristic functions of a square or a lozenge, and we give some of their properties.

Paradoxical Trend for Improvement in Mental Health with Aging: A Community-Based Study of 1,546 Adults Aged 21–100 Years

PubMed Central

Thomas, Michael L.; Kaufmann, Christopher N.; Palmer, Barton W.; Depp, Colin A.; Martin, Averria Sirkin; Glorioso, Danielle K.; Thompson, Wesley K.; Jeste, Dilip V.

2017-01-01

Objective Studies of aging usually focus on trajectories of physical and cognitive function, with far less emphasis on overall mental health, despite its impact on general health and mortality. This study examined linear and non-linear trends of physical, cognitive, and mental health over the entire adult lifespan. Method Cross-sectional data were obtained from 1,546 individuals aged 21 to 100 years, selected using random digit dialing for the Successful AGing Evaluation (SAGE) study, a structured multi-cohort investigation, that included telephone interviews and in-home surveys of community-based adults without dementia. Data were collected from 1/26/2010 to 10/07/2011 targeting participants aged 50 to 100 years, and 6/25/2012 to 7/15/2013 targeting participants aged 21 to 50 years. Data included self-report measures of physical health, measures of both positive and negative attributes of mental health, and a phone interview-based measure of cognition. Results Comparison of age cohorts using polynomial regression suggested a possible accelerated deterioration in physical and cognitive functioning, averaging one-and-a-half to two standard deviations over the adult lifespan. In contrast, there appeared to be a linear improvement of about one standard deviation in various attributes of mental health over the same life period. Conclusion These cross-sectional findings suggest the possibility of a linear improvement in mental health beginning in young adulthood rather than a U-shaped curve reported in some prior studies. Lifespan research combining psychosocial and biological markers may improve our understanding of resilience to mental disability in older age, and lead to broad-based interventions promoting mental health in all age groups. PMID:27561149

Quadratic Polynomial Regression using Serial Observation Processing:Implementation within DART

NASA Astrophysics Data System (ADS)

Hodyss, D.; Anderson, J. L.; Collins, N.; Campbell, W. F.; Reinecke, P. A.

2017-12-01

Many Ensemble-Based Kalman ltering (EBKF) algorithms process the observations serially. Serial observation processing views the data assimilation process as an iterative sequence of scalar update equations. What is useful about this data assimilation algorithm is that it has very low memory requirements and does not need complex methods to perform the typical high-dimensional inverse calculation of many other algorithms. Recently, the push has been towards the prediction, and therefore the assimilation of observations, for regions and phenomena for which high-resolution is required and/or highly nonlinear physical processes are operating. For these situations, a basic hypothesis is that the use of the EBKF is sub-optimal and performance gains could be achieved by accounting for aspects of the non-Gaussianty. To this end, we develop here a new component of the Data Assimilation Research Testbed [DART] to allow for a wide-variety of users to test this hypothesis. This new version of DART allows one to run several variants of the EBKF as well as several variants of the quadratic polynomial lter using the same forecast model and observations. Dierences between the results of the two systems will then highlight the degree of non-Gaussianity in the system being examined. We will illustrate in this work the differences between the performance of linear versus quadratic polynomial regression in a hierarchy of models from Lorenz-63 to a simple general circulation model.

An Extension of the Time-Spectral Method to Overset Solvers

NASA Technical Reports Server (NTRS)

Leffell, Joshua Isaac; Murman, Scott M.; Pulliam, Thomas

2013-01-01

Relative motion in the Cartesian or overset framework causes certain spatial nodes to move in and out of the physical domain as they are dynamically blanked by moving solid bodies. This poses a problem for the conventional Time-Spectral approach, which expands the solution at every spatial node into a Fourier series spanning the period of motion. The proposed extension to the Time-Spectral method treats unblanked nodes in the conventional manner but expands the solution at dynamically blanked nodes in a basis of barycentric rational polynomials spanning partitions of contiguously defined temporal intervals. Rational polynomials avoid Runge's phenomenon on the equidistant time samples of these sub-periodic intervals. Fourier- and rational polynomial-based differentiation operators are used in tandem to provide a consistent hybrid Time-Spectral overset scheme capable of handling relative motion. The hybrid scheme is tested with a linear model problem and implemented within NASA's OVERFLOW Reynolds-averaged Navier- Stokes (RANS) solver. The hybrid Time-Spectral solver is then applied to inviscid and turbulent RANS cases of plunging and pitching airfoils and compared to time-accurate and experimental data. A limiter was applied in the turbulent case to avoid undershoots in the undamped turbulent eddy viscosity while maintaining accuracy. The hybrid scheme matches the performance of the conventional Time-Spectral method and converges to the time-accurate results with increased temporal resolution.

Well-conditioning global-local analysis using stable generalized/extended finite element method for linear elastic fracture mechanics

NASA Astrophysics Data System (ADS)

Malekan, Mohammad; Barros, Felicio Bruzzi

2016-11-01

Using the locally-enriched strategy to enrich a small/local part of the problem by generalized/extended finite element method (G/XFEM) leads to non-optimal convergence rate and ill-conditioning system of equations due to presence of blending elements. The local enrichment can be chosen from polynomial, singular, branch or numerical types. The so-called stable version of G/XFEM method provides a well-conditioning approach when only singular functions are used in the blending elements. This paper combines numeric enrichment functions obtained from global-local G/XFEM method with the polynomial enrichment along with a well-conditioning approach, stable G/XFEM, in order to show the robustness and effectiveness of the approach. In global-local G/XFEM, the enrichment functions are constructed numerically from the solution of a local problem. Furthermore, several enrichment strategies are adopted along with the global-local enrichment. The results obtained with these enrichments strategies are discussed in detail, considering convergence rate in strain energy, growth rate of condition number, and computational processing. Numerical experiments show that using geometrical enrichment along with stable G/XFEM for global-local strategy improves the convergence rate and the conditioning of the problem. In addition, results shows that using polynomial enrichment for global problem simultaneously with global-local enrichments lead to ill-conditioned system matrices and bad convergence rate.

Near constant-time optimal piecewise LDR to HDR inverse tone mapping

NASA Astrophysics Data System (ADS)

Chen, Qian; Su, Guan-Ming; Yin, Peng

2015-02-01

In a backward compatible HDR image/video compression, it is a general approach to reconstruct HDR from compressed LDR as a prediction to original HDR, which is referred to as inverse tone mapping. Experimental results show that 2- piecewise 2nd order polynomial has the best mapping accuracy than 1 piece high order or 2-piecewise linear, but it is also the most time-consuming method because to find the optimal pivot point to split LDR range to 2 pieces requires exhaustive search. In this paper, we propose a fast algorithm that completes optimal 2-piecewise 2nd order polynomial inverse tone mapping in near constant time without quality degradation. We observe that in least square solution, each entry in the intermediate matrix can be written as the sum of some basic terms, which can be pre-calculated into look-up tables. Since solving the matrix becomes looking up values in tables, computation time barely differs regardless of the number of points searched. Hence, we can carry out the most thorough pivot point search to find the optimal pivot that minimizes MSE in near constant time. Experiment shows that our proposed method achieves the same PSNR performance while saving 60 times computation time compared to the traditional exhaustive search in 2-piecewise 2nd order polynomial inverse tone mapping with continuous constraint.

Determination of Nerve Fiber Diameter Distribution From Compound Action Potential: A Continuous Approach.

PubMed

Un, M Kerem; Kaghazchi, Hamed

2018-01-01

When a signal is initiated in the nerve, it is transmitted along each nerve fiber via an action potential (called single fiber action potential (SFAP)) which travels with a velocity that is related with the diameter of the fiber. The additive superposition of SFAPs constitutes the compound action potential (CAP) of the nerve. The fiber diameter distribution (FDD) in the nerve can be computed from the CAP data by solving an inverse problem. This is usually achieved by dividing the fibers into a finite number of diameter groups and solve a corresponding linear system to optimize FDD. However, number of fibers in a nerve can be measured sometimes in thousands and it is possible to assume a continuous distribution for the fiber diameters which leads to a gradient optimization problem. In this paper, we have evaluated this continuous approach to the solution of the inverse problem. We have utilized an analytical function for SFAP and an assumed a polynomial form for FDD. The inverse problem involves the optimization of polynomial coefficients to obtain the best estimate for the FDD. We have observed that an eighth order polynomial for FDD can capture both unimodal and bimodal fiber distributions present in vivo, even in case of noisy CAP data. The assumed FDD distribution regularizes the ill-conditioned inverse problem and produces good results.

Six-Inch Shock Tube Characterization

DTIC Science & Technology

2016-12-09

Change of Address Organizations receiving reports from the U.S. Army Aeromedical Research Laboratory on automatic mailing lists should confirm...92A Figure 2 summarizes the peak levels for shots using 92A Mylar® as a membrane with a linear trend line overlaid on the data, which produced the...peak levels for shots using 500A Mylar® as a membrane with a 6th-order polynomial trend line overlaid on the data, which produced the highest R2 value

Use of Log-Linear Models in Classification Problems.

DTIC Science & Technology

1981-12-01

polynomials. The second example involves infant hypoxic trauma, and many cells are empty. The existence conditions are used to find a model for which esti...mates of cell frequencies exist and are in good agreement with the ob- served data. Key Words: Classification problem, log-difference models, minimum 8...variates define k states, which are labeled consecutively. Thus, while MB define cells in their tables by an I-vector Z, we simply take Z to be a

Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos

DTIC Science & Technology

2002-07-25

Some basic hypergeometric polynomials that generalize Jacobi polynomials . Memoirs Amer. Math. Soc., AMS... orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener (1938). A Galerkin projection...1) by generalized polynomial chaos expansion, where the uncertainties can be introduced through κ, f , or g, or some combinations. It is worth

An analysis of value function learning with piecewise linear control

NASA Astrophysics Data System (ADS)

Tutsoy, Onder; Brown, Martin

2016-05-01

Reinforcement learning (RL) algorithms attempt to learn optimal control actions by iteratively estimating a long-term measure of system performance, the so-called value function. For example, RL algorithms have been applied to walking robots to examine the connection between robot motion and the brain, which is known as embodied cognition. In this paper, RL algorithms are analysed using an exemplar test problem. A closed form solution for the value function is calculated and this is represented in terms of a set of basis functions and parameters, which is used to investigate parameter convergence. The value function expression is shown to have a polynomial form where the polynomial terms depend on the plant's parameters and the value function's discount factor. It is shown that the temporal difference error introduces a null space for the differenced higher order basis associated with the effects of controller switching (saturated to linear control or terminating an experiment) apart from the time of the switch. This leads to slow convergence in the relevant subspace. It is also shown that badly conditioned learning problems can occur, and this is a function of the value function discount factor and the controller switching points. Finally, a comparison is performed between the residual gradient and TD(0) learning algorithms, and it is shown that the former has a faster rate of convergence for this test problem.

CS-AMPPred: An Updated SVM Model for Antimicrobial Activity Prediction in Cysteine-Stabilized Peptides

PubMed Central

Porto, William F.; Pires, Állan S.; Franco, Octavio L.

2012-01-01

The antimicrobial peptides (AMP) have been proposed as an alternative to control resistant pathogens. However, due to multifunctional properties of several AMP classes, until now there has been no way to perform efficient AMP identification, except through in vitro and in vivo tests. Nevertheless, an indication of activity can be provided by prediction methods. In order to contribute to the AMP prediction field, the CS-AMPPred (Cysteine-Stabilized Antimicrobial Peptides Predictor) is presented here, consisting of an updated version of the Support Vector Machine (SVM) model for antimicrobial activity prediction in cysteine-stabilized peptides. The CS-AMPPred is based on five sequence descriptors: indexes of (i) α-helix and (ii) loop formation; and averages of (iii) net charge, (iv) hydrophobicity and (v) flexibility. CS-AMPPred was based on 310 cysteine-stabilized AMPs and 310 sequences extracted from PDB. The polynomial kernel achieves the best accuracy on 5-fold cross validation (85.81%), while the radial and linear kernels achieve 84.19%. Testing in a blind data set, the polynomial and radial kernels achieve an accuracy of 90.00%, while the linear model achieves 89.33%. The three models reach higher accuracies than previously described methods. A standalone version of CS-AMPPred is available for download at and runs on any Linux machine. PMID:23240023

Multi-Party Privacy-Preserving Set Intersection with Quasi-Linear Complexity

NASA Astrophysics Data System (ADS)

Cheon, Jung Hee; Jarecki, Stanislaw; Seo, Jae Hong

Secure computation of the set intersection functionality allows n parties to find the intersection between their datasets without revealing anything else about them. An efficient protocol for such a task could have multiple potential applications in commerce, health care, and security. However, all currently known secure set intersection protocols for n>2 parties have computational costs that are quadratic in the (maximum) number of entries in the dataset contributed by each party, making secure computation of the set intersection only practical for small datasets. In this paper, we describe the first multi-party protocol for securely computing the set intersection functionality with both the communication and the computation costs that are quasi-linear in the size of the datasets. For a fixed security parameter, our protocols require O(n2k) bits of communication and Õ(n2k) group multiplications per player in the malicious adversary setting, where k is the size of each dataset. Our protocol follows the basic idea of the protocol proposed by Kissner and Song, but we gain efficiency by using different representations of the polynomials associated with users' datasets and careful employment of algorithms that interpolate or evaluate polynomials on multiple points more efficiently. Moreover, the proposed protocol is robust. This means that the protocol outputs the desired result even if some corrupted players leave during the execution of the protocol.

Mathematics of Zernike polynomials: a review.

PubMed

McAlinden, Colm; McCartney, Mark; Moore, Jonathan

2011-11-01

Monochromatic aberrations of the eye principally originate from the cornea and the crystalline lens. Aberrometers operate via differing principles but function by either analysing the reflected wavefront from the retina or by analysing an image on the retina. Aberrations may be described as lower order or higher order aberrations with Zernike polynomials being the most commonly employed fitting method. The complex mathematical aspects with regards the Zernike polynomial expansion series are detailed in this review. Refractive surgery has been a key clinical application of aberrometers; however, more recently aberrometers have been used in a range of other areas ophthalmology including corneal diseases, cataract and retinal imaging. © 2011 The Authors. Clinical and Experimental Ophthalmology © 2011 Royal Australian and New Zealand College of Ophthalmologists.

Prospective Measurement of Daily Health Behaviors: Modeling Temporal Patterns in Missing Data, Sexual Behavior, and Substance Use in an Online Daily Diary Study of Gay and Bisexual Men.

PubMed

Rendina, H Jonathon; Ventuneac, Ana; Mustanski, Brian; Grov, Christian; Parsons, Jeffrey T

2016-08-01

Daily diary and other intensive longitudinal methods are increasingly being used to investigate fluctuations in psychological and behavioral processes. To inform the development of this methodology, we sought to explore predictors of and patterns in diary compliance and behavioral reports. We used multilevel modeling to analyze data from an online daily diary study of 371 gay and bisexual men focused on sexual behavior and substance use. We found that greater education and older age as well as lower frequency of substance use were associated with higher compliance. Using polynomial and trigonometric functions, we found evidence for circaseptan patterns in compliance, sexual behavior, and substance use, as well as linear declines in compliance and behavior over time. The results suggest potential sources of non-random patterns of missing data and suggest that trigonometric terms provide a similar but more parsimonious investigation of circaseptan rhythms than do third-order polynomial terms.

On the modular structure of the genus-one Type II superstring low energy expansion

NASA Astrophysics Data System (ADS)

D'Hoker, Eric; Green, Michael B.; Vanhove, Pierre

2015-08-01

The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.

The Boundary Function Method. Fundamentals

NASA Astrophysics Data System (ADS)

Kot, V. A.

2017-03-01

The boundary function method is proposed for solving applied problems of mathematical physics in the region defined by a partial differential equation of the general form involving constant or variable coefficients with a Dirichlet, Neumann, or Robin boundary condition. In this method, the desired function is defined by a power polynomial, and a boundary function represented in the form of the desired function or its derivative at one of the boundary points is introduced. Different sequences of boundary equations have been set up with the use of differential operators. Systems of linear algebraic equations constructed on the basis of these sequences allow one to determine the coefficients of a power polynomial. Constitutive equations have been derived for initial boundary-value problems of all the main types. With these equations, an initial boundary-value problem is transformed into the Cauchy problem for the boundary function. The determination of the boundary function by its derivative with respect to the time coordinate completes the solution of the problem.

Polynomial Size Formulations for the Distance and Capacity Constrained Vehicle Routing Problem

NASA Astrophysics Data System (ADS)

Kara, Imdat; Derya, Tusan

2011-09-01

The Distance and Capacity Constrained Vehicle Routing Problem (DCVRP) is an extension of the well known Traveling Salesman Problem (TSP). DCVRP arises in distribution and logistics problems. It would be beneficial to construct new formulations, which is the main motivation and contribution of this paper. We focused on two indexed integer programming formulations for DCVRP. One node based and one arc (flow) based formulation for DCVRP are presented. Both formulations have O(n2) binary variables and O(n2) constraints, i.e., the number of the decision variables and constraints grows with a polynomial function of the nodes of the underlying graph. It is shown that proposed arc based formulation produces better lower bound than the existing one (this refers to the Water's formulation in the paper). Finally, various problems from literature are solved with the node based and arc based formulations by using CPLEX 8.0. Preliminary computational analysis shows that, arc based formulation outperforms the node based formulation in terms of linear programming relaxation.

Prospective Measurement of Daily Health Behaviors: Modeling Temporal Patterns in Missing Data, Sexual Behavior, and Substance Use in an Online Daily Diary Study of Gay and Bisexual Men

PubMed Central

Rendina, H. Jonathon; Ventuneac, Ana; Mustanski, Brian; Grov, Christian; Parsons, Jeffrey T.

2016-01-01

Daily diary and other intensive longitudinal methods are increasingly being used to investigate fluctuations in psychological and behavioral processes. To inform the development of this methodology, we sought to explore predictors of and patterns in diary compliance and behavioral reports. We used multilevel modeling to analyze data from an online daily diary study of 371 gay and bisexual men focused on sexual behavior and substance use. We found that greater education and older age as well as lower frequency of substance use were associated with higher compliance. Using polynomial and trigonometric functions, we found evidence for circaseptan patterns in compliance, sexual behavior, and substance use, as well as linear declines in compliance and behavior over time. The results suggest potential sources of non-random patterns of missing data and suggest that trigonometric terms provide a similar but more parsimonious investigation of circaseptan rhythms than do third-order polynomial terms. PMID:26992392

More accurate depiction of adsorption energy on transition metals using work function as one additional descriptor.

PubMed

Shen, Xiaochen; Pan, Yanbo; Liu, Bin; Yang, Jinlong; Zeng, Jie; Peng, Zhenmeng

2017-05-24

The reaction mechanism and properties of a catalytic process are primarily determined by the interactions between reacting species and catalysts. However, the interactions are often challenging to be experimentally measured, especially for unstable intermediates. Therefore, it is of significant importance to establish an exact relationship between chemical-catalyst interactions and catalyst parameters, which will allow calculation of these interactions and thus advance their mechanistic understanding. Herein we report the description of adsorption energy on transition metals by considering both ionic bonding and covalent bonding contributions and introduce the work function as one additional responsible parameter. We find that the adsorption energy can be more accurately described using a two-dimensional (2D) polynomial model, which shows a significant improvement compared with the current adsorption energy-d-band center linear correlation. We also demonstrate the utilization of this new 2D polynomial model to calculate oxygen binding energy of different transition metals to help understand their catalytic properties in oxygen reduction reactions.

A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hong Luo; Luqing Luo; Robert Nourgaliev

2012-11-01

A reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, a variant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems onmore » arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements.« less

Additive-Multiplicative Approximation of Genotype-Environment Interaction

PubMed Central

Gimelfarb, A.

1994-01-01

A model of genotype-environment interaction in quantitative traits is considered. The model represents an expansion of the traditional additive (first degree polynomial) approximation of genotypic and environmental effects to a second degree polynomial incorporating a multiplicative term besides the additive terms. An experimental evaluation of the model is suggested and applied to a trait in Drosophila melanogaster. The environmental variance of a genotype in the model is shown to be a function of the genotypic value: it is a convex parabola. The broad sense heritability in a population depends not only on the genotypic and environmental variances, but also on the position of the genotypic mean in the population relative to the minimum of the parabola. It is demonstrated, using the model, that GXE interaction rectional may cause a substantial non-linearity in offspring-parent regression and a reversed response to directional selection. It is also shown that directional selection may be accompanied by an increase in the heritability. PMID:7896113

Stability analysis of nonlinear Roesser-type two-dimensional systems via a homogenous polynomial technique

NASA Astrophysics Data System (ADS)

Zhang, Tie-Yan; Zhao, Yan; Xie, Xiang-Peng

2012-12-01

This paper is concerned with the problem of stability analysis of nonlinear Roesser-type two-dimensional (2D) systems. Firstly, the fuzzy modeling method for the usual one-dimensional (1D) systems is extended to the 2D case so that the underlying nonlinear 2D system can be represented by the 2D Takagi—Sugeno (TS) fuzzy model, which is convenient for implementing the stability analysis. Secondly, a new kind of fuzzy Lyapunov function, which is a homogeneous polynomially parameter dependent on fuzzy membership functions, is developed to conceive less conservative stability conditions for the TS Roesser-type 2D system. In the process of stability analysis, the obtained stability conditions approach exactness in the sense of convergence by applying some novel relaxed techniques. Moreover, the obtained result is formulated in the form of linear matrix inequalities, which can be easily solved via standard numerical software. Finally, a numerical example is also given to demonstrate the effectiveness of the proposed approach.

Conversion from Engineering Units to Telemetry Counts on Dryden Flight Simulators

NASA Technical Reports Server (NTRS)

Fantini, Jay A.

1998-01-01

Dryden real-time flight simulators encompass the simulation of pulse code modulation (PCM) telemetry signals. This paper presents a new method whereby the calibration polynomial (from first to sixth order), representing the conversion from counts to engineering units (EU), is numerically inverted in real time. The result is less than one-count error for valid EU inputs. The Newton-Raphson method is used to numerically invert the polynomial. A reverse linear interpolation between the EU limits is used to obtain an initial value for the desired telemetry count. The method presented here is not new. What is new is how classical numerical techniques are optimized to take advantage of modem computer power to perform the desired calculations in real time. This technique makes the method simple to understand and implement. There are no interpolation tables to store in memory as in traditional methods. The NASA F-15 simulation converts and transmits over 1000 parameters at 80 times/sec. This paper presents algorithm development, FORTRAN code, and performance results.

Approximating exponential and logarithmic functions using polynomial interpolation

NASA Astrophysics Data System (ADS)

Gordon, Sheldon P.; Yang, Yajun

2017-04-01

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

Exact and heuristic algorithms for Space Information Flow.

PubMed

Uwitonze, Alfred; Huang, Jiaqing; Ye, Yuanqing; Cheng, Wenqing; Li, Zongpeng

2018-01-01

Space Information Flow (SIF) is a new promising research area that studies network coding in geometric space, such as Euclidean space. The design of algorithms that compute the optimal SIF solutions remains one of the key open problems in SIF. This work proposes the first exact SIF algorithm and a heuristic SIF algorithm that compute min-cost multicast network coding for N (N ≥ 3) given terminal nodes in 2-D Euclidean space. Furthermore, we find that the Butterfly network in Euclidean space is the second example besides the Pentagram network where SIF is strictly better than Euclidean Steiner minimal tree. The exact algorithm design is based on two key techniques: Delaunay triangulation and linear programming. Delaunay triangulation technique helps to find practically good candidate relay nodes, after which a min-cost multicast linear programming model is solved over the terminal nodes and the candidate relay nodes, to compute the optimal multicast network topology, including the optimal relay nodes selected by linear programming from all the candidate relay nodes and the flow rates on the connection links. The heuristic algorithm design is also based on Delaunay triangulation and linear programming techniques. The exact algorithm can achieve the optimal SIF solution with an exponential computational complexity, while the heuristic algorithm can achieve the sub-optimal SIF solution with a polynomial computational complexity. We prove the correctness of the exact SIF algorithm. The simulation results show the effectiveness of the heuristic SIF algorithm.

Normality of raw data in general linear models: The most widespread myth in statistics

USGS Publications Warehouse

Kery, Marc; Hatfield, Jeff S.

2003-01-01

In years of statistical consulting for ecologists and wildlife biologists, by far the most common misconception we have come across has been the one about normality in general linear models. These comprise a very large part of the statistical models used in ecology and include t tests, simple and multiple linear regression, polynomial regression, and analysis of variance (ANOVA) and covariance (ANCOVA). There is a widely held belief that the normality assumption pertains to the raw data rather than to the model residuals. We suspect that this error may also occur in countless published studies, whenever the normality assumption is tested prior to analysis. This may lead to the use of nonparametric alternatives (if there are any), when parametric tests would indeed be appropriate, or to use of transformations of raw data, which may introduce hidden assumptions such as multiplicative effects on the natural scale in the case of log-transformed data. Our aim here is to dispel this myth. We very briefly describe relevant theory for two cases of general linear models to show that the residuals need to be normally distributed if tests requiring normality are to be used, such as t and F tests. We then give two examples demonstrating that the distribution of the response variable may be nonnormal, and yet the residuals are well behaved. We do not go into the issue of how to test normality; instead we display the distributions of response variables and residuals graphically.

Uncertainty Modeling for Robustness Analysis of Control Upset Prevention and Recovery Systems

NASA Technical Reports Server (NTRS)

Belcastro, Christine M.; Khong, Thuan H.; Shin, Jong-Yeob; Kwatny, Harry; Chang, Bor-Chin; Balas, Gary J.

2005-01-01

Formal robustness analysis of aircraft control upset prevention and recovery systems could play an important role in their validation and ultimate certification. Such systems (developed for failure detection, identification, and reconfiguration, as well as upset recovery) need to be evaluated over broad regions of the flight envelope and under extreme flight conditions, and should include various sources of uncertainty. However, formulation of linear fractional transformation (LFT) models for representing system uncertainty can be very difficult for complex parameter-dependent systems. This paper describes a preliminary LFT modeling software tool which uses a matrix-based computational approach that can be directly applied to parametric uncertainty problems involving multivariate matrix polynomial dependencies. Several examples are presented (including an F-16 at an extreme flight condition, a missile model, and a generic example with numerous crossproduct terms), and comparisons are given with other LFT modeling tools that are currently available. The LFT modeling method and preliminary software tool presented in this paper are shown to compare favorably with these methods.

Analytical approximate solutions for a general class of nonlinear delay differential equations.

PubMed

Căruntu, Bogdan; Bota, Constantin

2014-01-01

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

Polynomial modal analysis of slanted lamellar gratings.

PubMed

Granet, Gérard; Randriamihaja, Manjakavola Honore; Raniriharinosy, Karyl

2017-06-01

The problem of diffraction by slanted lamellar dielectric and metallic gratings in classical mounting is formulated as an eigenvalue eigenvector problem. The numerical solution is obtained by using the moment method with Legendre polynomials as expansion and test functions, which allows us to enforce in an exact manner the boundary conditions which determine the eigensolutions. Our method is successfully validated by comparison with other methods including in the case of highly slanted gratings.

On the Rate of Relaxation for the Landau Kinetic Equation and Related Models

NASA Astrophysics Data System (ADS)

Bobylev, Alexander; Gamba, Irene M.; Zhang, Chenglong

2017-08-01

We study the rate of relaxation to equilibrium for Landau kinetic equation and some related models by considering the relatively simple case of radial solutions of the linear Landau-type equations. The well-known difficulty is that the evolution operator has no spectral gap, i.e. its spectrum is not separated from zero. Hence we do not expect purely exponential relaxation for large values of time t>0. One of the main goals of our work is to numerically identify the large time asymptotics for the relaxation to equilibrium. We recall the work of Strain and Guo (Arch Rat Mech Anal 187:287-339 2008, Commun Partial Differ Equ 31:17-429 2006), who rigorously show that the expected law of relaxation is \\exp (-ct^{2/3}) with some c > 0. In this manuscript, we find an heuristic way, performed by asymptotic methods, that finds this "law of two thirds", and then study this question numerically. More specifically, the linear Landau equation is approximated by a set of ODEs based on expansions in generalized Laguerre polynomials. We analyze the corresponding quadratic form and the solution of these ODEs in detail. It is shown that the solution has two different asymptotic stages for large values of time t and maximal order of polynomials N: the first one focus on intermediate asymptotics which agrees with the "law of two thirds" for moderately large values of time t and then the second one on absolute, purely exponential asymptotics for very large t, as expected for linear ODEs. We believe that appearance of intermediate asymptotics in finite dimensional approximations must be a generic behavior for different classes of equations in functional spaces (some PDEs, Boltzmann equations for soft potentials, etc.) and that our methods can be applied to related problems.

A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

NASA Astrophysics Data System (ADS)

Wang, Yan; Zhang, Yufeng; Zhang, Xiangzhi

2016-09-01

We first introduced a linear stationary equation with a quadratic operator in ∂x and ∂y, then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

Variability simulations with a steady, linearized primitive equations model

NASA Technical Reports Server (NTRS)

Kinter, J. L., III; Nigam, S.

1985-01-01

Solutions of the steady, primitive equations on a sphere, linearized about a zonally symmetric basic state are computed for the purpose of simulating monthly mean variability in the troposphere. The basic states are observed, winter monthly mean, zonal means of zontal and meridional velocities, temperatures and surface pressures computed from the 15 year NMC time series. A least squares fit to a series of Legendre polynomials is used to compute the basic states between 20 H and the equator, and the hemispheres are assumed symmetric. The model is spectral in the zonal direction, and centered differences are employed in the meridional and vertical directions. Since the model is steady and linear, the solution is obtained by inversion of a block, pente-diagonal matrix. The model simulates the climatology of the GFDL nine level, spectral general circulation model quite closely, particularly in middle latitudes above the boundary layer. This experiment is an extension of that simulation to examine variability of the steady, linear solution.

Quantum Linear System Algorithm for Dense Matrices.

PubMed

Wossnig, Leonard; Zhao, Zhikuan; Prakash, Anupam

2018-02-02

Solving linear systems of equations is a frequently encountered problem in machine learning and optimization. Given a matrix A and a vector b the task is to find the vector x such that Ax=b. We describe a quantum algorithm that achieves a sparsity-independent runtime scaling of O(κ^{2}sqrt[n]polylog(n)/ε) for an n×n dimensional A with bounded spectral norm, where κ denotes the condition number of A, and ε is the desired precision parameter. This amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices, and poses a new state of the art for solving dense linear systems on a quantum computer. Furthermore, an exponential improvement is achievable if the rank of A is polylogarithmic in the matrix dimension. Our algorithm is built upon a singular value estimation subroutine, which makes use of a memory architecture that allows for efficient preparation of quantum states that correspond to the rows of A and the vector of Euclidean norms of the rows of A.

Detecting and isolating abrupt changes in linear switching systems

NASA Astrophysics Data System (ADS)

Nazari, Sohail; Zhao, Qing; Huang, Biao

2015-04-01

In this paper, a novel fault detection and isolation (FDI) method for switching linear systems is developed. All input and output signals are assumed to be corrupted with measurement noises. In the proposed method, a 'lifted' linear model named as stochastic hybrid decoupling polynomial (SHDP) is introduced. The SHDP model governs the dynamics of the switching linear system with all different modes, and is independent of the switching sequence. The error-in-variable (EIV) representation of SHDP is derived, and is used for the fault residual generation and isolation following the well-adopted local approach. The proposed FDI method can detect and isolate the fault-induced abrupt changes in switching models' parameters without estimating the switching modes. Furthermore, in this paper, the analytical expressions of the gradient vector and Hessian matrix are obtained based on the EIV SHDP formulation, so that they can be used to implement the online fault detection scheme. The performance of the proposed method is then illustrated by simulation examples.

Integration of epidemiology into the genetic analysis of mastitis in Swedish Holstein.

PubMed

Windig, Jack J; Urioste, Jorge I; Strandberg, Erling

2013-04-01

Heritability of mastitis (and diseases in general) tends to be low. One possible cause is that no clear distinction can be made between resistant and nonresistant animals, because healthy animals include animals that have not been exposed to pathogens and resistant animals. To account for this, we quantified the prevalence of clinical mastitis (CM) and subclinical mastitis (SCM) in 2,069 Swedish Holstein herds as a measure of exposure. Herd prevalence averaged 26.5% for SCM and 6.4% for CM; 61% of the first lactations of 177,309 cows were classified as having at least one case of SCM and 10% as having CM. In a reaction norm approach, heritability of (S)CM was quantified as a function of herd prevalence of (S)CM. The best-fitting model was a second-order polynomial of first-lactation cow SCM as a function of herd prevalence SCM, and a first-order (linear) polynomial of first-lactation cow CM as a function of CM herd prevalence. Heritability for SCM ranged from 0.069 to 0.105 and for CM from 0.016 to 0.032. For both, we found no clear effect of herd prevalence on their heritability. Genetic correlations within traits across herd prevalences were all greater than 0.92. Whether relationships among prevalence, exposure, disease, and genetics were as expected is a matter of discussion, but reaction norm analyses may be a valuable tool for epidemiological genetics. Copyright © 2013 American Dairy Science Association. Published by Elsevier Inc. All rights reserved.

Equivalences of the multi-indexed orthogonal polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Odake, Satoru

2014-01-15

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

Consensus Algorithms for Networks of Systems with Second- and Higher-Order Dynamics

NASA Astrophysics Data System (ADS)

Fruhnert, Michael

This thesis considers homogeneous networks of linear systems. We consider linear feedback controllers and require that the directed graph associated with the network contains a spanning tree and systems are stabilizable. We show that, in continuous-time, consensus with a guaranteed rate of convergence can always be achieved using linear state feedback. For networks of continuous-time second-order systems, we provide a new and simple derivation of the conditions for a second-order polynomials with complex coefficients to be Hurwitz. We apply this result to obtain necessary and sufficient conditions to achieve consensus with networks whose graph Laplacian matrix may have complex eigenvalues. Based on the conditions found, methods to compute feedback gains are proposed. We show that gains can be chosen such that consensus is achieved robustly over a variety of communication structures and system dynamics. We also consider the use of static output feedback. For networks of discrete-time second-order systems, we provide a new and simple derivation of the conditions for a second-order polynomials with complex coefficients to be Schur. We apply this result to obtain necessary and sufficient conditions to achieve consensus with networks whose graph Laplacian matrix may have complex eigenvalues. We show that consensus can always be achieved for marginally stable systems and discretized systems. Simple conditions for consensus achieving controllers are obtained when the Laplacian eigenvalues are all real. For networks of continuous-time time-variant higher-order systems, we show that uniform consensus can always be achieved if systems are quadratically stabilizable. In this case, we provide a simple condition to obtain a linear feedback control. For networks of discrete-time higher-order systems, we show that constant gains can be chosen such that consensus is achieved for a variety of network topologies. First, we develop simple results for networks of time-invariant systems and networks of time-variant systems that are given in controllable canonical form. Second, we formulate the problem in terms of Linear Matrix Inequalities (LMIs). The condition found simplifies the design process and avoids the parallel solution of multiple LMIs. The result yields a modified Algebraic Riccati Equation (ARE) for which we present an equivalent LMI condition.

Cytogenetic effect of low dose gamma-radiation in Hordeum vulgare seedlings: non-linear dose-effect relationship.

PubMed

Geras'kin, Stanislav A; Oudalova, Alla A; Kim, Jin Kyu; Dikarev, Vladimir G; Dikareva, Nina S

2007-03-01

The induction of chromosome aberrations in Hordeum vulgare germinated seeds was studied after ionizing irradiation with doses in the range of 10-1,000 mGy. The relationship between the frequency of aberrant cells and the absorbed dose was found to be nonlinear. A dose-independent plateau in the dose range from about 50 to 500 mGy was observed, where the level of cytogenetic damage was significantly different from the spontaneous level. The comparison of the goodness of the experimental data fitting with mathematical models of different complexity, using the most common quantitative criteria, demonstrated the advantage of a piecewise linear model over linear and polynomial models in approximating the frequency of cytogenetical disturbances. The results of the study support the hypothesis of indirect mechanisms of mutagenesis induced by low doses. Fundamental and applied implications of these findings are discussed.

The use of Galerkin finite-element methods to solve mass-transport equations

USGS Publications Warehouse

Grove, David B.

1977-01-01

The partial differential equation that describes the transport and reaction of chemical solutes in porous media was solved using the Galerkin finite-element technique. These finite elements were superimposed over finite-difference cells used to solve the flow equation. Both convection and flow due to hydraulic dispersion were considered. Linear and Hermite cubic approximations (basis functions) provided satisfactory results: however, the linear functions were computationally more efficient for two-dimensional problems. Successive over relaxation (SOR) and iteration techniques using Tchebyschef polynomials were used to solve the sparce matrices generated using the linear and Hermite cubic functions, respectively. Comparisons of the finite-element methods to the finite-difference methods, and to analytical results, indicated that a high degree of accuracy may be obtained using the method outlined. The technique was applied to a field problem involving an aquifer contaminated with chloride, tritium, and strontium-90. (Woodard-USGS)

Research on numerical algorithms for large space structures

NASA Technical Reports Server (NTRS)

Denman, E. D.

1981-01-01

Numerical algorithms for analysis and design of large space structures are investigated. The sign algorithm and its application to decoupling of differential equations are presented. The generalized sign algorithm is given and its application to several problems discussed. The Laplace transforms of matrix functions and the diagonalization procedure for a finite element equation are discussed. The diagonalization of matrix polynomials is considered. The quadrature method and Laplace transforms is discussed and the identification of linear systems by the quadrature method investigated.

Hybrid Solution of Stochastic Optimal Control Problems Using Gauss Pseudospectral Method and Generalized Polynomial Chaos Algorithms

DTIC Science & Technology

2012-03-01

0-486-41183-4. 15. Brown , Robert G. and Patrick Y. C. Hwang . Introduction to Random Signals and Applied Kalman Filtering. Wiley, New York, 1996. ISBN...stability and perfor- mance criteria. In the 1960’s, Kalman introduced the Linear Quadratic Regulator (LQR) method using an integral performance index...feedback of the state variables and was able to apply this method to time-varying and Multi-Input Multi-Output (MIMO) systems. Kalman further showed

The use of Lyapunov differential inequalities for estimating the transients of mechanical systems

NASA Astrophysics Data System (ADS)

Alyshev, A. S.; Dudarenko, N. A.; Melnikov, V. G.; Melnikov, G. I.

2018-05-01

In this paper we consider an autonomous mechanical system in a finite neighborhood of the zero of the phase space of states. The system is given as a matrix differential equation in the Cauchy form with the right-hand side of the polynomial structure. We propose a method for constructing a sequence of linear inhomogeneous differential inequalities for Lyapunov functions. As a result, we obtain estimates of transient processes in the form of functional inequalities.

Koopman Mode Decomposition Methods in Dynamic Stall: Reduced Order Modeling and Control

DTIC Science & Technology

2015-11-10

the flow phenomena by separating them into individual modes. The technique of Proper Orthogonal Decomposition (POD), see [ Holmes : 1998] is a popular...sampled values h(k), k = 0,…,2M-1, of the exponential sum 1. Solve the following linear system where 2. Compute all zeros zj  D, j = 1,…,M...of the Prony polynomial i.e., calculate all eigenvalues of the associated companion matrix and form fj = log zj for j = 1,…,M, where log is the

Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces

NASA Technical Reports Server (NTRS)

Carpenter, Mark H.; Fisher, Travis C.; Nielsen, Eric J.; Frankel, Steven H.

2013-01-01

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the new schemes are applicable to a broad class of linear and nonlinear conservation laws, emphasis herein is placed on the entropy stability of the compressible Navier-Stokes equations.

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.

Method of Characteristics Calculations and Computer Code for Materials with Arbitrary Equations of State and Using Orthogonal Polynomial Least Square Surface Fits

NASA Technical Reports Server (NTRS)

Chang, T. S.

1974-01-01

A numerical scheme using the method of characteristics to calculate the flow properties and pressures behind decaying shock waves for materials under hypervelocity impact is developed. Time-consuming double interpolation subroutines are replaced by a technique based on orthogonal polynomial least square surface fits. Typical calculated results are given and compared with the double interpolation results. The complete computer program is included.

Horizontal vestibuloocular reflex evoked by high-acceleration rotations in the squirrel monkey. I. Normal responses

NASA Technical Reports Server (NTRS)

Minor, L. B.; Lasker, D. M.; Backous, D. D.; Hullar, T. E.; Shelhamer, M. J. (Principal Investigator)

1999-01-01

The horizontal angular vestibuloocular reflex (VOR) evoked by high-frequency, high-acceleration rotations was studied in five squirrel monkeys with intact vestibular function. The VOR evoked by steps of acceleration in darkness (3,000 degrees /s(2) reaching a velocity of 150 degrees /s) began after a latency of 7.3 +/- 1.5 ms (mean +/- SD). Gain of the reflex during the acceleration was 14.2 +/- 5.2% greater than that measured once the plateau head velocity had been reached. A polynomial regression was used to analyze the trajectory of the responses to steps of acceleration. A better representation of the data was obtained from a polynomial that included a cubic term in contrast to an exclusively linear fit. For sinusoidal rotations of 0.5-15 Hz with a peak velocity of 20 degrees /s, the VOR gain measured 0.83 +/- 0.06 and did not vary across frequencies or animals. The phase of these responses was close to compensatory except at 15 Hz where a lag of 5.0 +/- 0.9 degrees was noted. The VOR gain did not vary with head velocity at 0.5 Hz but increased with velocity for rotations at frequencies of >/=4 Hz (0. 85 +/- 0.04 at 4 Hz, 20 degrees /s; 1.01 +/- 0.05 at 100 degrees /s, P < 0.0001). No responses to these rotations were noted in two animals that had undergone bilateral labyrinthectomy indicating that inertia of the eye had a negligible effect for these stimuli. We developed a mathematical model of VOR dynamics to account for these findings. The inputs to the reflex come from linear and nonlinear pathways. The linear pathway is responsible for the constant gain across frequencies at peak head velocity of 20 degrees /s and also for the phase lag at higher frequencies being less than that expected based on the reflex delay. The frequency- and velocity-dependent nonlinearity in VOR gain is accounted for by the dynamics of the nonlinear pathway. A transfer function that increases the gain of this pathway with frequency and a term related to the third power of head velocity are used to represent the dynamics of this pathway. This model accounts for the experimental findings and provides a method for interpreting responses to these stimuli after vestibular lesions.

An exponential decay model for mediation.

PubMed

Fritz, Matthew S

2014-10-01

Mediation analysis is often used to investigate mechanisms of change in prevention research. Results finding mediation are strengthened when longitudinal data are used because of the need for temporal precedence. Current longitudinal mediation models have focused mainly on linear change, but many variables in prevention change nonlinearly across time. The most common solution to nonlinearity is to add a quadratic term to the linear model, but this can lead to the use of the quadratic function to explain all nonlinearity, regardless of theory and the characteristics of the variables in the model. The current study describes the problems that arise when quadratic functions are used to describe all nonlinearity and how the use of nonlinear functions, such as exponential decay, address many of these problems. In addition, nonlinear models provide several advantages over polynomial models including usefulness of parameters, parsimony, and generalizability. The effects of using nonlinear functions for mediation analysis are then discussed and a nonlinear growth curve model for mediation is presented. An empirical example using data from a randomized intervention study is then provided to illustrate the estimation and interpretation of the model. Implications, limitations, and future directions are also discussed.

Uncertainty Quantification of Non-linear Oscillation Triggering in a Multi-injector Liquid-propellant Rocket Combustion Chamber

NASA Astrophysics Data System (ADS)

Popov, Pavel; Sideris, Athanasios; Sirignano, William

2014-11-01

We examine the non-linear dynamics of the transverse modes of combustion-driven acoustic instability in a liquid-propellant rocket engine. Triggering can occur, whereby small perturbations from mean conditions decay, while larger disturbances grow to a limit-cycle of amplitude that may compare to the mean pressure. For a deterministic perturbation, the system is also deterministic, computed by coupled finite-volume solvers at low computational cost for a single realization. The randomness of the triggering disturbance is captured by treating the injector flow rates, local pressure disturbances, and sudden acceleration of the entire combustion chamber as random variables. The combustor chamber with its many sub-fields resulting from many injector ports may be viewed as a multi-scale complex system wherein the developing acoustic oscillation is the emergent structure. Numerical simulation of the resulting stochastic PDE system is performed using the polynomial chaos expansion method. The overall probability of unstable growth is assessed in different regions of the parameter space. We address, in particular, the seven-injector, rectangular Purdue University experimental combustion chamber. In addition to the novel geometry, new features include disturbances caused by engine acceleration and unsteady thruster nozzle flow.

An Exponential Decay Model for Mediation

PubMed Central

Fritz, Matthew S.

2013-01-01

Mediation analysis is often used to investigate mechanisms of change in prevention research. Results finding mediation are strengthened when longitudinal data are used because of the need for temporal precedence. Current longitudinal mediation models have focused mainly on linear change, but many variables in prevention change nonlinearly across time. The most common solution to nonlinearity is to add a quadratic term to the linear model, but this can lead to the use of the quadratic function to explain all nonlinearity, regardless of theory and the characteristics of the variables in the model. The current study describes the problems that arise when quadratic functions are used to describe all nonlinearity and how the use of nonlinear functions, such as exponential decay, addresses many of these problems. In addition, nonlinear models provide several advantages over polynomial models including usefulness of parameters, parsimony, and generalizability. The effects of using nonlinear functions for mediation analysis are then discussed and a nonlinear growth curve model for mediation is presented. An empirical example using data from a randomized intervention study is then provided to illustrate the estimation and interpretation of the model. Implications, limitations, and future directions are also discussed. PMID:23625557

Application of Semi-Definite Programming for Many-Fermion Systems

NASA Astrophysics Data System (ADS)

Zhao, Zhengji; Braams, Bastiaan; Fukuda, Mituhiro; Overton, Michael

2003-03-01

The ground state energy and other important observables of a many-fermion system with one- and two-body interactions only can all be obtained from the first order and second order Reduced Density Matrices (RDM's) of the system. Using these density matrices and a family of associated representability conditions one may obtain an approximation method for electronic structure theory that is in the mathematical form of Semi-Definite Programming (SDP): minimize a linear matrix functional over a space of positive semidefinite matrices subject to linear constraints. The representability conditions are some known necessary conditions, starting with the well-known P, Q, and G conditions [Claude Garrod and Jerome K. Percus, Reducation of the N-Particle Variational Problem, J. Math. Phys. 5 (1964) 1756-1776]. The RDM method with SDP has great potential advantages over the wave function method when the particle number N is large. The dimension of the full configuration space increases exponentially with N, but in RDM method with SDP the dimension of the objective matrix (which includes RDM's) increases only polynomially with N. We will report on the effect of adding the generalized three-index conditions proposed in [R. M. Erdahl, Representability, Int. J. Quantum Chem. 13 (1978) 697-718].

Simple Proof of Jury Test for Complex Polynomials

NASA Astrophysics Data System (ADS)

Choo, Younseok; Kim, Dongmin

Recently some attempts have been made in the literature to give simple proofs of Jury test for real polynomials. This letter presents a similar result for complex polynomials. A simple proof of Jury test for complex polynomials is provided based on the Rouché's Theorem and a single-parameter characterization of Schur stability property for complex polynomials.

The expression and comparison of healthy and ptotic upper eyelid contours using a polynomial mathematical function.

PubMed

Mocan, Mehmet C; Ilhan, Hacer; Gurcay, Hasmet; Dikmetas, Ozlem; Karabulut, Erdem; Erdener, Ugur; Irkec, Murat

2014-06-01

To derive a mathematical expression for the healthy upper eyelid (UE) contour and to use this expression to differentiate the normal UE curve from its abnormal configuration in the setting of blepharoptosis. The study was designed as a cross-sectional study. Fifty healthy subjects (26M/24F) and 50 patients with blepharoptosis (28M/22F) with a margin-reflex distance (MRD1) of ≤2.5 mm were recruited. A polynomial interpolation was used to approximate UE curve. The polynomial coefficients were calculated from digital eyelid images of all participants using a set of operator defined points along the UE curve. Coefficients up to the fourth-order polynomial, iris area covered by the UE, iris area covered by the lower eyelid and total iris area covered by both the upper and the lower eyelids were defined using the polynomial function and used in statistical comparisons. The t-test, Mann-Whitney U test and the Spearman's correlation test were used for statistical comparisons. The mathematical expression derived from the data of 50 healthy subjects aged 24.1 ± 2.6 years was defined as y = 22.0915 + (-1.3213)x + 0.0318x(2 )+ (-0.0005x)(3). The fifth and the consecutive coefficients were <0.00001 in all cases and were not included in the polynomial function. None of the first fourth-order coefficients of the equation were found to be significantly different in male versus female subjects. In normal subjects, the percentage of the iris area covered by upper and lower lids was 6.46 ± 5.17% and 0.66% ± 1.62%, respectively. All coefficients and mean iris area covered by the UE were significantly different between healthy and ptotic eyelids. The healthy and abnormal eyelid contour can be defined and differentiated using a polynomial mathematical function.

PRECONDITIONED CONJUGATE-GRADIENT 2 (PCG2), a computer program for solving ground-water flow equations

USGS Publications Warehouse

Hill, Mary C.

1990-01-01

This report documents PCG2 : a numerical code to be used with the U.S. Geological Survey modular three-dimensional, finite-difference, ground-water flow model . PCG2 uses the preconditioned conjugate-gradient method to solve the equations produced by the model for hydraulic head. Linear or nonlinear flow conditions may be simulated. PCG2 includes two reconditioning options : modified incomplete Cholesky preconditioning, which is efficient on scalar computers; and polynomial preconditioning, which requires less computer storage and, with modifications that depend on the computer used, is most efficient on vector computers . Convergence of the solver is determined using both head-change and residual criteria. Nonlinear problems are solved using Picard iterations. This documentation provides a description of the preconditioned conjugate gradient method and the two preconditioners, detailed instructions for linking PCG2 to the modular model, sample data inputs, a brief description of PCG2, and a FORTRAN listing.

Massively parallel sparse matrix function calculations with NTPoly

NASA Astrophysics Data System (ADS)

Dawson, William; Nakajima, Takahito

2018-04-01

We present NTPoly, a massively parallel library for computing the functions of sparse, symmetric matrices. The theory of matrix functions is a well developed framework with a wide range of applications including differential equations, graph theory, and electronic structure calculations. One particularly important application area is diagonalization free methods in quantum chemistry. When the input and output of the matrix function are sparse, methods based on polynomial expansions can be used to compute matrix functions in linear time. We present a library based on these methods that can compute a variety of matrix functions. Distributed memory parallelization is based on a communication avoiding sparse matrix multiplication algorithm. OpenMP task parallellization is utilized to implement hybrid parallelization. We describe NTPoly's interface and show how it can be integrated with programs written in many different programming languages. We demonstrate the merits of NTPoly by performing large scale calculations on the K computer.

Multigrid methods for isogeometric discretization

PubMed Central

Gahalaut, K.P.S.; Kraus, J.K.; Tomar, S.K.

2013-01-01

We present (geometric) multigrid methods for isogeometric discretization of scalar second order elliptic problems. The smoothing property of the relaxation method, and the approximation property of the intergrid transfer operators are analyzed. These properties, when used in the framework of classical multigrid theory, imply uniform convergence of two-grid and multigrid methods. Supporting numerical results are provided for the smoothing property, the approximation property, convergence factor and iterations count for V-, W- and F-cycles, and the linear dependence of V-cycle convergence on the smoothing steps. For two dimensions, numerical results include the problems with variable coefficients, simple multi-patch geometry, a quarter annulus, and the dependence of convergence behavior on refinement levels ℓ, whereas for three dimensions, only the constant coefficient problem in a unit cube is considered. The numerical results are complete up to polynomial order p=4, and for C0 and Cp-1 smoothness. PMID:24511168

Tikekar superdense stars in electric fields

NASA Astrophysics Data System (ADS)

Komathiraj, K.; Maharaj, S. D.

2007-04-01

We present exact solutions to the Einstein-Maxwell system of equations with a specified form of the electric field intensity by assuming that the hypersurface {t=constant} are spheroidal. The solution of the Einstein-Maxwell system is reduced to a recurrence relation with variable rational coefficients which can be solved in general using mathematical induction. New classes of solutions of linearly independent functions are obtained by restricting the spheroidal parameter K and the electric field intensity parameter α. Consequently, it is possible to find exact solutions in terms of elementary functions, namely, polynomials and algebraic functions. Our result contains models found previously including the superdense Tikekar neutron star model [J. Math. Phys. 31, 2454 (1990)] when K=-7 and α=0. Our class of charged spheroidal models generalize the uncharged isotropic Maharaj and Leach solutions [J. Math. Phys. 37, 430 (1996)]. In particular, we find an explicit relationship directly relating the spheroidal parameter K to the electromagnetic field.

Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials

NASA Astrophysics Data System (ADS)

Chen, Zhixiang; Fu, Bin

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a ΠΣΠ polynomial. We first prove that the first problem is #P-hard and then devise a O *(3 n s(n)) upper bound for this problem for any polynomial represented by an arithmetic circuit of size s(n). Later, this upper bound is improved to O *(2 n ) for ΠΣΠ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for ΠΣ polynomials. On the negative side, we prove that, even for ΠΣΠ polynomials with terms of degree ≤ 2, the first problem cannot be approximated at all for any approximation factor ≥ 1, nor "weakly approximated" in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time λ-approximation algorithm for ΠΣΠ polynomials with terms of degrees no more a constant λ ≥ 2. On the inapproximability side, we give a n (1 - ɛ)/2 lower bound, for any ɛ> 0, on the approximation factor for ΠΣΠ polynomials. When the degrees of the terms in these polynomials are constrained as ≤ 2, we prove a 1.0476 lower bound, assuming Pnot=NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.

The Extrapolar SWIFT model (version 1.0): fast stratospheric ozone chemistry for global climate models

NASA Astrophysics Data System (ADS)

Kreyling, Daniel; Wohltmann, Ingo; Lehmann, Ralph; Rex, Markus

2018-03-01

The Extrapolar SWIFT model is a fast ozone chemistry scheme for interactive calculation of the extrapolar stratospheric ozone layer in coupled general circulation models (GCMs). In contrast to the widely used prescribed ozone, the SWIFT ozone layer interacts with the model dynamics and can respond to atmospheric variability or climatological trends.The Extrapolar SWIFT model employs a repro-modelling approach, in which algebraic functions are used to approximate the numerical output of a full stratospheric chemistry and transport model (ATLAS). The full model solves a coupled chemical differential equation system with 55 initial and boundary conditions (mixing ratio of various chemical species and atmospheric parameters). Hence the rate of change of ozone over 24 h is a function of 55 variables. Using covariances between these variables, we can find linear combinations in order to reduce the parameter space to the following nine basic variables: latitude, pressure altitude, temperature, overhead ozone column and the mixing ratio of ozone and of the ozone-depleting families (Cly, Bry, NOy and HOy). We will show that these nine variables are sufficient to characterize the rate of change of ozone. An automated procedure fits a polynomial function of fourth degree to the rate of change of ozone obtained from several simulations with the ATLAS model. One polynomial function is determined per month, which yields the rate of change of ozone over 24 h. A key aspect for the robustness of the Extrapolar SWIFT model is to include a wide range of stratospheric variability in the numerical output of the ATLAS model, also covering atmospheric states that will occur in a future climate (e.g. temperature and meridional circulation changes or reduction of stratospheric chlorine loading).For validation purposes, the Extrapolar SWIFT model has been integrated into the ATLAS model, replacing the full stratospheric chemistry scheme. Simulations with SWIFT in ATLAS have proven that the systematic error is small and does not accumulate during the course of a simulation. In the context of a 10-year simulation, the ozone layer simulated by SWIFT shows a stable annual cycle, with inter-annual variations comparable to the ATLAS model. The application of Extrapolar SWIFT requires the evaluation of polynomial functions with 30-100 terms. Computers can currently calculate such polynomial functions at thousands of model grid points in seconds. SWIFT provides the desired numerical efficiency and computes the ozone layer 104 times faster than the chemistry scheme in the ATLAS CTM.

Discrete-time state estimation for stochastic polynomial systems over polynomial observations

NASA Astrophysics Data System (ADS)

Hernandez-Gonzalez, M.; Basin, M.; Stepanov, O.

2018-07-01

This paper presents a solution to the mean-square state estimation problem for stochastic nonlinear polynomial systems over polynomial observations confused with additive white Gaussian noises. The solution is given in two steps: (a) computing the time-update equations and (b) computing the measurement-update equations for the state estimate and error covariance matrix. A closed form of this filter is obtained by expressing conditional expectations of polynomial terms as functions of the state estimate and error covariance. As a particular case, the mean-square filtering equations are derived for a third-degree polynomial system with second-degree polynomial measurements. Numerical simulations show effectiveness of the proposed filter compared to the extended Kalman filter.

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 as the polynomials of degree for which Eq. (1) has a polynomial solution of some degree k. In this paper, we compute the limiting distribution, as well as the limiting mean level spacings distribution of the zeros of any Van Vleck polynomial as N --> ∞.

Theoretical study of the dynamic magnetic response of ferrofluid to static and alternating magnetic fields

NASA Astrophysics Data System (ADS)

Batrudinov, Timur M.; Ambarov, Alexander V.; Elfimova, Ekaterina A.; Zverev, Vladimir S.; Ivanov, Alexey O.

2017-06-01

The dynamic magnetic response of ferrofluid in a static uniform external magnetic field to a weak, linear polarized, alternating magnetic field is investigated theoretically. The ferrofluid is modeled as a system of dipolar hard spheres, suspended in a long cylindrical tube whose long axis is parallel to the direction of the static and alternating magnetic fields. The theory is based on the Fokker-Planck-Brown equation formulated for the case when the both static and alternating magnetic fields are applied. The solution of the Fokker-Planck-Brown equation describing the orientational probability density of a randomly chosen dipolar particle is expressed as a series in terms of the spherical Legendre polynomials. The obtained analytical expression connecting three neighboring coefficients of the series makes possible to determine the probability density with any order of accuracy in terms of Legendre polynomials. The analytical formula for the probability density truncated at the first Legendre polynomial is evaluated and used for the calculation of the magnetization and dynamic susceptibility spectra. In the absence of the static magnetic field the presented theory gives the correct single-particle Debye-theory result, which is the exact solution of the Fokker-Planck-Brown equation for the case of applied weak alternating magnetic field. The influence of the static magnetic field on the dynamic susceptibility is analyzed in terms of the low-frequency behavior of the real part and the position of the peak in the imaginary part.

Histogram-driven cupping correction (HDCC) in CT

NASA Astrophysics Data System (ADS)

Kyriakou, Y.; Meyer, M.; Lapp, R.; Kalender, W. A.

2010-04-01

Typical cupping correction methods are pre-processing methods which require either pre-calibration measurements or simulations of standard objects to approximate and correct for beam hardening and scatter. Some of them require the knowledge of spectra, detector characteristics, etc. The aim of this work was to develop a practical histogram-driven cupping correction (HDCC) method to post-process the reconstructed images. We use a polynomial representation of the raw-data generated by forward projection of the reconstructed images; forward and backprojection are performed on graphics processing units (GPU). The coefficients of the polynomial are optimized using a simplex minimization of the joint entropy of the CT image and its gradient. The algorithm was evaluated using simulations and measurements of homogeneous and inhomogeneous phantoms. For the measurements a C-arm flat-detector CT (FD-CT) system with a 30×40 cm2 detector, a kilovoltage on board imager (radiation therapy simulator) and a micro-CT system were used. The algorithm reduced cupping artifacts both in simulations and measurements using a fourth-order polynomial and was in good agreement to the reference. The minimization algorithm required less than 70 iterations to adjust the coefficients only performing a linear combination of basis images, thus executing without time consuming operations. HDCC reduced cupping artifacts without the necessity of pre-calibration or other scan information enabling a retrospective improvement of CT image homogeneity. However, the method can work with other cupping correction algorithms or in a calibration manner, as well.

A Maximum Likelihood Approach to Determine Sensor Radiometric Response Coefficients for NPP VIIRS Reflective Solar Bands

NASA Technical Reports Server (NTRS)

Lei, Ning; Chiang, Kwo-Fu; Oudrari, Hassan; Xiong, Xiaoxiong

2011-01-01

Optical sensors aboard Earth orbiting satellites such as the next generation Visible/Infrared Imager/Radiometer Suite (VIIRS) assume that the sensors radiometric response in the Reflective Solar Bands (RSB) is described by a quadratic polynomial, in relating the aperture spectral radiance to the sensor Digital Number (DN) readout. For VIIRS Flight Unit 1, the coefficients are to be determined before launch by an attenuation method, although the linear coefficient will be further determined on-orbit through observing the Solar Diffuser. In determining the quadratic polynomial coefficients by the attenuation method, a Maximum Likelihood approach is applied in carrying out the least-squares procedure. Crucial to the Maximum Likelihood least-squares procedure is the computation of the weight. The weight not only has a contribution from the noise of the sensor s digital count, with an important contribution from digitization error, but also is affected heavily by the mathematical expression used to predict the value of the dependent variable, because both the independent and the dependent variables contain random noise. In addition, model errors have a major impact on the uncertainties of the coefficients. The Maximum Likelihood approach demonstrates the inadequacy of the attenuation method model with a quadratic polynomial for the retrieved spectral radiance. We show that using the inadequate model dramatically increases the uncertainties of the coefficients. We compute the coefficient values and their uncertainties, considering both measurement and model errors.

Legendre modified moments for Euler's constant

NASA Astrophysics Data System (ADS)

Prévost, Marc

2008-10-01

Polynomial moments are often used for the computation of Gauss quadrature to stabilize the numerical calculation of the orthogonal polynomials, see [W. Gautschi, Computational aspects of orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials-Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 294. Kluwer, Dordrecht, 1990, pp. 181-216 [6]; W. Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48(4) (1986) 369-382 [5]; W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3(3) (1982) 289-317 [4

Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation

NASA Astrophysics Data System (ADS)

Milovanovic, Gradimir V.

2001-01-01

Quadrature formulas with multiple nodes, power orthogonality, and some applications of such quadratures to moment-preserving approximation by defective splines are considered. An account on power orthogonality (s- and [sigma]-orthogonal polynomials) and generalized Gaussian quadratures with multiple nodes, including stable algorithms for numerical construction of the corresponding polynomials and Cotes numbers, are given. In particular, the important case of Chebyshev weight is analyzed. Finally, some applications in moment-preserving approximation of functions by defective splines are discussed.

Monograph on the use of the multivariate Gram Charlier series Type A

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hatayodom, T.; Heydt, G.

1978-01-01

The Gram-Charlier series in an infinite series expansion for a probability density function (pdf) in which terms of the series are Hermite polynomials. There are several Gram-Charlier series - the best known is Type A. The Gram-Charlier series, Type A (GCA) exists for both univariate and multivariate random variables. This monograph introduces the multivariate GCA and illustrates its use through several examples. A brief bibliography and discussion of Hermite polynomials is also included. 9 figures, 2 tables.

Multi Objective Controller Design for Linear System via Optimal Interpolation

NASA Technical Reports Server (NTRS)

Ozbay, Hitay

1996-01-01

We propose a methodology for the design of a controller which satisfies a set of closed-loop objectives simultaneously. The set of objectives consists of: (1) pole placement, (2) decoupled command tracking of step inputs at steady-state, and (3) minimization of step response transients with respect to envelope specifications. We first obtain a characterization of all controllers placing the closed-loop poles in a prescribed region of the complex plane. In this characterization, the free parameter matrix Q(s) is to be determined to attain objectives (2) and (3). Objective (2) is expressed as determining a Pareto optimal solution to a vector valued optimization problem. The solution of this problem is obtained by transforming it to a scalar convex optimization problem. This solution determines Q(O) and the remaining freedom in choosing Q(s) is used to satisfy objective (3). We write Q(s) = (l/v(s))bar-Q(s) for a prescribed polynomial v(s). Bar-Q(s) is a polynomial matrix which is arbitrary except that Q(O) and the order of bar-Q(s) are fixed. Obeying these constraints bar-Q(s) is now to be 'shaped' to minimize the step response characteristics of specific input/output pairs according to the maximum envelope violations. This problem is expressed as a vector valued optimization problem using the concept of Pareto optimality. We then investigate a scalar optimization problem associated with this vector valued problem and show that it is convex. The organization of the report is as follows. The next section includes some definitions and preliminary lemmas. We then give the problem statement which is followed by a section including a detailed development of the design procedure. We then consider an aircraft control example. The last section gives some concluding remarks. The Appendix includes the proofs of technical lemmas, printouts of computer programs, and figures.

Genetic analysis of resistance to ticks, gastrointestinal nematodes and Eimeria spp. in Nellore cattle.

PubMed

Passafaro, Tiago Luciano; Carrera, Juan Pablo Botero; dos Santos, Livia Loiola; Raidan, Fernanda Santos Silva; dos Santos, Dalinne Chrystian Carvalho; Cardoso, Eduardo Penteado; Leite, Romário Cerqueira; Toral, Fabio Luiz Buranelo

2015-06-15

The aim of the present study was to obtain genetic parameters for resistance to ticks, gastrointestinal nematodes (worms) and Eimeria spp. in Nellore cattle, analyze the inclusion of resistance traits in Nellore breeding programs and evaluate genetic selection as a complementary tool in parasite control programs. Counting of ticks, gastrointestinal nematode eggs and Eimeria spp. oocysts per gram of feces totaling 4270; 3872 and 3872 records from 1188; 1142 and 1142 animals, respectively, aged 146 to 597 days were used. The animals were classified as resistant (counts equal to zero) or susceptible (counts above zero) to each parasite. The statistical models included systematics effects of contemporary groups and the mean trajectory. The random effects included additive genetic effects, direct permanent environmental effects and residual. The mean trajectory and random effects were modeled with linear Legendre polynomials for all traits except for the mean trajectory of resistance to Eimeria spp., which employed the cubic polynomial. Heritability estimates were of low to moderate magnitude and ranged from 0.06 to 0.30, 0.06 to 0.33 and 0.04 to 0.33 for resistance to ticks, gastrointestinal nematodes and Eimeria spp., respectively. The posterior mean of genetic and environmental correlations for the same trait at different ages (205, 365, 450 and 550 days) were favorable at adjacent ages and unfavorable at distant ages. In general, the posterior mean of the genetic and environmental correlations between traits of resistance were low and high-density intervals were large and included zero in many cases. The heritability estimates support the inclusion of resistance to ticks, gastrointestinal nematodes and Eimeria spp. in Nellore breeding programs. Genetic selection can increase the frequency of resistant animals and be used as a complementary tool in parasite control programs. Copyright © 2015 Elsevier B.V. All rights reserved.

A {1,2}-Order Plate Theory Accounting for Three-Dimensional Thermoelastic Deformations in Thick Composite and Sandwich Laminates

NASA Technical Reports Server (NTRS)

Tessler, A.; Annett, M. S.; Gendron, G.

2001-01-01

A {1,2}-order theory for laminated composite and sandwich plates is extended to include thermoelastic effects. The theory incorporates all three-dimensional strains and stresses. Mixed-field assumptions are introduced which include linear in-plane displacements, parabolic transverse displacement and shear strains, and a cubic distribution of the transverse normal stress. Least squares strain compatibility conditions and exact traction boundary conditions are enforced to yield higher polynomial degree distributions for the transverse shear strains and transverse normal stress through the plate thickness. The principle of virtual work is used to derive a 10th-order system of equilibrium equations and associated Poisson boundary conditions. The predictive capability of the theory is demonstrated using a closed-form analytic solution for a simply-supported rectangular plate subjected to a linearly varying temperature field across the thickness. Several thin and moderately thick laminated composite and sandwich plates are analyzed. Numerical comparisons are made with corresponding solutions of the first-order shear deformation theory and three-dimensional elasticity theory. These results, which closely approximate the three-dimensional elasticity solutions, demonstrate that through - the - thickness deformations even in relatively thin and, especially in thick. composite and sandwich laminates can be significant under severe thermal gradients. The {1,2}-order kinematic assumptions insure an overall accurate theory that is in general superior and, in some cases, equivalent to the first-order theory.

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

PubMed

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

2018-04-01

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

Non-linear transfer characteristics of stimulation and recording hardware account for spurious low-frequency artifacts during amplitude modulated transcranial alternating current stimulation (AM-tACS).

PubMed

Kasten, Florian H; Negahbani, Ehsan; Fröhlich, Flavio; Herrmann, Christoph S

2018-05-31

Amplitude modulated transcranial alternating current stimulation (AM-tACS) has been recently proposed as a possible solution to overcome the pronounced stimulation artifact encountered when recording brain activity during tACS. In theory, AM-tACS does not entail power at its modulating frequency, thus avoiding the problem of spectral overlap between brain signal of interest and stimulation artifact. However, the current study demonstrates how weak non-linear transfer characteristics inherent to stimulation and recording hardware can reintroduce spurious artifacts at the modulation frequency. The input-output transfer functions (TFs) of different stimulation setups were measured. Setups included recordings of signal-generator and stimulator outputs and M/EEG phantom measurements. 6 th -degree polynomial regression models were fitted to model the input-output TFs of each setup. The resulting TF models were applied to digitally generated AM-tACS signals to predict the frequency of spurious artifacts in the spectrum. All four setups measured for the study exhibited low-frequency artifacts at the modulation frequency and its harmonics when recording AM-tACS. Fitted TF models showed non-linear contributions significantly different from zero (all p < .05) and successfully predicted the frequency of artifacts observed in AM-signal recordings. Results suggest that even weak non-linearities of stimulation and recording hardware can lead to spurious artifacts at the modulation frequency and its harmonics. These artifacts were substantially larger than alpha-oscillations of a human subject in the MEG. Findings emphasize the need for more linear stimulation devices for AM-tACS and careful analysis procedures, taking into account low-frequency artifacts to avoid confusion with effects of AM-tACS on the brain. Copyright © 2018 Elsevier Inc. All rights reserved.

On multiple orthogonal polynomials for discrete Meixner measures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sorokin, Vladimir N

2010-12-07

The paper examines two examples of multiple orthogonal polynomials generalizing orthogonal polynomials of a discrete variable, meaning thereby the Meixner polynomials. One example is bound up with a discrete Nikishin system, and the other leads to essentially new effects. The limit distribution of the zeros of polynomials is obtained in terms of logarithmic equilibrium potentials and in terms of algebraic curves. Bibliography: 9 titles.

Permutation invariant polynomial neural network approach to fitting potential energy surfaces. II. Four-atom systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Li, Jun; Jiang, Bin; Guo, Hua, E-mail: hguo@unm.edu

2013-11-28

A rigorous, general, and simple method to fit global and permutation invariant potential energy surfaces (PESs) using neural networks (NNs) is discussed. This so-called permutation invariant polynomial neural network (PIP-NN) method imposes permutation symmetry by using in its input a set of symmetry functions based on PIPs. For systems with more than three atoms, it is shown that the number of symmetry functions in the input vector needs to be larger than the number of internal coordinates in order to include both the primary and secondary invariant polynomials. This PIP-NN method is successfully demonstrated in three atom-triatomic reactive systems, resultingmore » in full-dimensional global PESs with average errors on the order of meV. These PESs are used in full-dimensional quantum dynamical calculations.« less

Correction factors for on-line microprobe analysis of multielement alloy systems

NASA Technical Reports Server (NTRS)

Unnam, J.; Tenney, D. R.; Brewer, W. D.

1977-01-01

An on-line correction technique was developed for the conversion of electron probe X-ray intensities into concentrations of emitting elements. This technique consisted of off-line calculation and representation of binary interaction data which were read into an on-line minicomputer to calculate variable correction coefficients. These coefficients were used to correct the X-ray data without significantly increasing computer core requirements. The binary interaction data were obtained by running Colby's MAGIC 4 program in the reverse mode. The data for each binary interaction were represented by polynomial coefficients obtained by least-squares fitting a third-order polynomial. Polynomial coefficients were generated for most of the common binary interactions at different accelerating potentials and are included. Results are presented for the analyses of several alloy standards to demonstrate the applicability of this correction procedure.

Direct calculation of modal parameters from matrix orthogonal polynomials

NASA Astrophysics Data System (ADS)

El-Kafafy, Mahmoud; Guillaume, Patrick

2011-10-01

The object of this paper is to introduce a new technique to derive the global modal parameter (i.e. system poles) directly from estimated matrix orthogonal polynomials. This contribution generalized the results given in Rolain et al. (1994) [5] and Rolain et al. (1995) [6] for scalar orthogonal polynomials to multivariable (matrix) orthogonal polynomials for multiple input multiple output (MIMO) system. Using orthogonal polynomials improves the numerical properties of the estimation process. However, the derivation of the modal parameters from the orthogonal polynomials is in general ill-conditioned if not handled properly. The transformation of the coefficients from orthogonal polynomials basis to power polynomials basis is known to be an ill-conditioned transformation. In this paper a new approach is proposed to compute the system poles directly from the multivariable orthogonal polynomials. High order models can be used without any numerical problems. The proposed method will be compared with existing methods (Van Der Auweraer and Leuridan (1987) [4] Chen and Xu (2003) [7]). For this comparative study, simulated as well as experimental data will be used.

Hierarchical cluster-based partial least squares regression (HC-PLSR) is an efficient tool for metamodelling of nonlinear dynamic models.

PubMed

Tøndel, Kristin; Indahl, Ulf G; Gjuvsland, Arne B; Vik, Jon Olav; Hunter, Peter; Omholt, Stig W; Martens, Harald

2011-06-01

Deterministic dynamic models of complex biological systems contain a large number of parameters and state variables, related through nonlinear differential equations with various types of feedback. A metamodel of such a dynamic model is a statistical approximation model that maps variation in parameters and initial conditions (inputs) to variation in features of the trajectories of the state variables (outputs) throughout the entire biologically relevant input space. A sufficiently accurate mapping can be exploited both instrumentally and epistemically. Multivariate regression methodology is a commonly used approach for emulating dynamic models. However, when the input-output relations are highly nonlinear or non-monotone, a standard linear regression approach is prone to give suboptimal results. We therefore hypothesised that a more accurate mapping can be obtained by locally linear or locally polynomial regression. We present here a new method for local regression modelling, Hierarchical Cluster-based PLS regression (HC-PLSR), where fuzzy C-means clustering is used to separate the data set into parts according to the structure of the response surface. We compare the metamodelling performance of HC-PLSR with polynomial partial least squares regression (PLSR) and ordinary least squares (OLS) regression on various systems: six different gene regulatory network models with various types of feedback, a deterministic mathematical model of the mammalian circadian clock and a model of the mouse ventricular myocyte function. Our results indicate that multivariate regression is well suited for emulating dynamic models in systems biology. The hierarchical approach turned out to be superior to both polynomial PLSR and OLS regression in all three test cases. The advantage, in terms of explained variance and prediction accuracy, was largest in systems with highly nonlinear functional relationships and in systems with positive feedback loops. HC-PLSR is a promising approach for metamodelling in systems biology, especially for highly nonlinear or non-monotone parameter to phenotype maps. The algorithm can be flexibly adjusted to suit the complexity of the dynamic model behaviour, inviting automation in the metamodelling of complex systems.

Hierarchical Cluster-based Partial Least Squares Regression (HC-PLSR) is an efficient tool for metamodelling of nonlinear dynamic models

PubMed Central

2011-01-01

Background Deterministic dynamic models of complex biological systems contain a large number of parameters and state variables, related through nonlinear differential equations with various types of feedback. A metamodel of such a dynamic model is a statistical approximation model that maps variation in parameters and initial conditions (inputs) to variation in features of the trajectories of the state variables (outputs) throughout the entire biologically relevant input space. A sufficiently accurate mapping can be exploited both instrumentally and epistemically. Multivariate regression methodology is a commonly used approach for emulating dynamic models. However, when the input-output relations are highly nonlinear or non-monotone, a standard linear regression approach is prone to give suboptimal results. We therefore hypothesised that a more accurate mapping can be obtained by locally linear or locally polynomial regression. We present here a new method for local regression modelling, Hierarchical Cluster-based PLS regression (HC-PLSR), where fuzzy C-means clustering is used to separate the data set into parts according to the structure of the response surface. We compare the metamodelling performance of HC-PLSR with polynomial partial least squares regression (PLSR) and ordinary least squares (OLS) regression on various systems: six different gene regulatory network models with various types of feedback, a deterministic mathematical model of the mammalian circadian clock and a model of the mouse ventricular myocyte function. Results Our results indicate that multivariate regression is well suited for emulating dynamic models in systems biology. The hierarchical approach turned out to be superior to both polynomial PLSR and OLS regression in all three test cases. The advantage, in terms of explained variance and prediction accuracy, was largest in systems with highly nonlinear functional relationships and in systems with positive feedback loops. Conclusions HC-PLSR is a promising approach for metamodelling in systems biology, especially for highly nonlinear or non-monotone parameter to phenotype maps. The algorithm can be flexibly adjusted to suit the complexity of the dynamic model behaviour, inviting automation in the metamodelling of complex systems. PMID:21627852

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.

Linear spline multilevel models for summarising childhood growth trajectories: A guide to their application using examples from five birth cohorts.

PubMed

Howe, Laura D; Tilling, Kate; Matijasevich, Alicia; Petherick, Emily S; Santos, Ana Cristina; Fairley, Lesley; Wright, John; Santos, Iná S; Barros, Aluísio Jd; Martin, Richard M; Kramer, Michael S; Bogdanovich, Natalia; Matush, Lidia; Barros, Henrique; Lawlor, Debbie A

2016-10-01

Childhood growth is of interest in medical research concerned with determinants and consequences of variation from healthy growth and development. Linear spline multilevel modelling is a useful approach for deriving individual summary measures of growth, which overcomes several data issues (co-linearity of repeat measures, the requirement for all individuals to be measured at the same ages and bias due to missing data). Here, we outline the application of this methodology to model individual trajectories of length/height and weight, drawing on examples from five cohorts from different generations and different geographical regions with varying levels of economic development. We describe the unique features of the data within each cohort that have implications for the application of linear spline multilevel models, for example, differences in the density and inter-individual variation in measurement occasions, and multiple sources of measurement with varying measurement error. After providing example Stata syntax and a suggested workflow for the implementation of linear spline multilevel models, we conclude with a discussion of the advantages and disadvantages of the linear spline approach compared with other growth modelling methods such as fractional polynomials, more complex spline functions and other non-linear models. © The Author(s) 2013.

Linear spline multilevel models for summarising childhood growth trajectories: A guide to their application using examples from five birth cohorts

PubMed Central

Tilling, Kate; Matijasevich, Alicia; Petherick, Emily S; Santos, Ana Cristina; Fairley, Lesley; Wright, John; Santos, Iná S.; Barros, Aluísio JD; Martin, Richard M; Kramer, Michael S; Bogdanovich, Natalia; Matush, Lidia; Barros, Henrique; Lawlor, Debbie A

2013-01-01

Childhood growth is of interest in medical research concerned with determinants and consequences of variation from healthy growth and development. Linear spline multilevel modelling is a useful approach for deriving individual summary measures of growth, which overcomes several data issues (co-linearity of repeat measures, the requirement for all individuals to be measured at the same ages and bias due to missing data). Here, we outline the application of this methodology to model individual trajectories of length/height and weight, drawing on examples from five cohorts from different generations and different geographical regions with varying levels of economic development. We describe the unique features of the data within each cohort that have implications for the application of linear spline multilevel models, for example, differences in the density and inter-individual variation in measurement occasions, and multiple sources of measurement with varying measurement error. After providing example Stata syntax and a suggested workflow for the implementation of linear spline multilevel models, we conclude with a discussion of the advantages and disadvantages of the linear spline approach compared with other growth modelling methods such as fractional polynomials, more complex spline functions and other non-linear models. PMID:24108269

Orbital component extraction by time-variant sinusoidal modeling.

NASA Astrophysics Data System (ADS)

Sinnesael, Matthias; Zivanovic, Miroslav; De Vleeschouwer, David; Claeys, Philippe; Schoukens, Johan

2016-04-01

Accurately deciphering periodic variations in paleoclimate proxy signals is essential for cyclostratigraphy. Classical spectral analysis often relies on methods based on the (Fast) Fourier Transformation. This technique has no unique solution separating variations in amplitude and frequency. This characteristic makes it difficult to correctly interpret a proxy's power spectrum or to accurately evaluate simultaneous changes in amplitude and frequency in evolutionary analyses. Here, we circumvent this drawback by using a polynomial approach to estimate instantaneous amplitude and frequency in orbital components. This approach has been proven useful to characterize audio signals (music and speech), which are non-stationary in nature (Zivanovic and Schoukens, 2010, 2012). Paleoclimate proxy signals and audio signals have in nature similar dynamics; the only difference is the frequency relationship between the different components. A harmonic frequency relationship exists in audio signals, whereas this relation is non-harmonic in paleoclimate signals. However, the latter difference is irrelevant for the problem at hand. Using a sliding window approach, the model captures time variations of an orbital component by modulating a stationary sinusoid centered at its mean frequency, with a single polynomial. Hence, the parameters that determine the model are the mean frequency of the orbital component and the polynomial coefficients. The first parameter depends on geologic interpretation, whereas the latter are estimated by means of linear least-squares. As an output, the model provides the orbital component waveform, either in the depth or time domain. Furthermore, it allows for a unique decomposition of the signal into its instantaneous amplitude and frequency. Frequency modulation patterns can be used to reconstruct changes in accumulation rate, whereas amplitude modulation can be used to reconstruct e.g. eccentricity-modulated precession. The time-variant sinusoidal model is applied to well-established Pleistocene benthic isotope records to evaluate its performance. Zivanovic M. and Schoukens J. (2010) On The Polynomial Approximation for Time-Variant Harmonic Signal Modeling. IEEE Transactions On Audio, Speech, and Language Processing vol. 19, no. 3, pp. 458-467. Doi: 10.1109/TASL.2010.2049673. Zivanovic M. and Schoukens J. (2012) Single and Piecewise Polynomials for Modeling of Pitched Sounds. IEEE Transactions On Audio, Speech, and Language Processing vol. 20, no. 4, pp. 1270-1281. Doi: 10.1109/TASL.2011.2174228.

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

NASA Astrophysics Data System (ADS)

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

2008-10-01

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

A study of the orthogonal polynomials associated with the quantum harmonic oscillator on constant curvature spaces

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vignat, C.; Lamberti, P. W.

2009-10-15

Recently, Carinena, et al. [Ann. Phys. 322, 434 (2007)] introduced a new family of orthogonal polynomials that appear in the wave functions of the quantum harmonic oscillator in two-dimensional constant curvature spaces. They are a generalization of the Hermite polynomials and will be called curved Hermite polynomials in the following. We show that these polynomials are naturally related to the relativistic Hermite polynomials introduced by Aldaya et al. [Phys. Lett. A 156, 381 (1991)], and thus are Jacobi polynomials. Moreover, we exhibit a natural bijection between the solutions of the quantum harmonic oscillator on negative curvature spaces and on positivemore » curvature spaces. At last, we show a maximum entropy property for the ground states of these oscillators.« less

Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach

NASA Astrophysics Data System (ADS)

Nasiri, Alireza; Nguang, Sing Kiong; Swain, Akshya; Almakhles, Dhafer

2018-02-01

This paper deals with the problem of designing a controller for a class of discrete-time nonlinear systems which is represented by discrete-time polynomial fuzzy model. Most of the existing control design methods for discrete-time fuzzy polynomial systems cannot guarantee their Lyapunov function to be a radially unbounded polynomial function, hence the global stability cannot be assured. The proposed control design in this paper guarantees a radially unbounded polynomial Lyapunov functions which ensures global stability. In the proposed design, state feedback structure is considered and non-convexity problem is solved by incorporating an integrator into the controller. Sufficient conditions of stability are derived in terms of polynomial matrix inequalities which are solved via SOSTOOLS in MATLAB. A numerical example is presented to illustrate the effectiveness of the proposed controller.

Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D

NASA Astrophysics Data System (ADS)

Arad, Itai; Landau, Zeph; Vazirani, Umesh; Vidick, Thomas

2017-11-01

One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly( n) degenerate ground spaces and an n O(log n) algorithm for the poly( n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is {\\tilde{O}(nM(n))} , where M( n) is the time required to multiply two n × n matrices.