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Legendre Polynomials (legendre + polynomial)
Selected AbstractsPhase Biaxility in Smectic-A Side-Chain Liquid Crystalline ElastomersMACROMOLECULAR RAPID COMMUNICATIONS, Issue 8 2009Rebekka Storz Abstract 2H NMR investigations on the biaxial phase behavior of smectic-A liquid crystalline side-chain elastomers are presented. Biaxiality parameters were determined by measuring the quadrupolar splitting of two spin probes, namely benzene-d6 and hexamethylbenzene-d18, at various angles between the principal director and the external magnetic field: while for a uniaxial sample the angular dependence can be described by the second Legendre polynomial, an additional asymmetric term needs to be included to fit the data of the two investigated biaxial systems. Two elastomers synthesized from mesogens that differ in the molecular geometry in order to study the molecular origin of biaxiality were compared. Biaxiality is observed for both elastomers when approaching the glass transition, suggesting that the network dynamics dominate the formation of the biaxial phase. [source] Fast Global Illumination on Dynamic Height FieldsCOMPUTER GRAPHICS FORUM, Issue 4 2009Derek Nowrouzezahrai Abstract We present a real-time method for rendering global illumination effects from large area and environmental lights on dynamic height fields. In contrast to previous work, our method handles inter-reflections (indirect lighting) and non-diffuse surfaces. To reduce sampling, we construct one multi-resolution pyramid for height variation to compute direct shadows, and another pyramid for each indirect bounce of incident radiance to compute inter-reflections. The basic principle is to sample the points blocking direct light, or shedding indirect light, from coarser levels of the pyramid the farther away they are from a given receiver point. We unify the representation of visibility and indirect radiance at discrete azimuthal directions (i.e., as a function of a single elevation angle) using the concept of a "casting set" of visible points along this direction whose contributions are collected in the basis of normalized Legendre polynomials. This analytic representation is compact, requires no precomputation, and allows efficient integration to produce the spherical visibility and indirect radiance signals. Sub-sampling visibility and indirect radiance, while shading with full-resolution surface normals, further increases performance without introducing noticeable artifacts. Our method renders 512×512 height fields (> 500K triangles) at 36Hz. [source] A simple method to calculate the signal-to-noise ratio of a circular-shaped coil for MRICONCEPTS IN MAGNETIC RESONANCE, Issue 6 2006K. Ocegueda Abstract The introduction of the ultrafast imaging sequences has renewed the interest in development of RF coils. The theoretical frame of the SNR of MRI coils is a challenge because it requires a deep mathematical background to master the associated concepts. Here, a simpler method is proposed based on Legendre polynomials. This approximation method, together with a quasi-static approach, was used to derive a signal-to-noise ratio expression for a circular-shaped coil. Legendre polynomials were used instead of a weighting function to simplify the vector potential of the power loss, and an SNR formula was then derived. The simplified version of the SNR formula of a circular coil was compared with the weighting function-derived SNR expression using the quasi-static approach. SNR-vs.-depth plots were computed to theoretically compare both SNR formulas. Results showed a strong agreement between SNR values for the circular-shaped coil. This approach can be used as a tool to derive SNR expressions for more complex geometries. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A 28A: 422,429, 2006 [source] Properties and performance of orthogonal neural network in function approximationINTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 12 2001Chieh F. Sher Backpropagation neural network has been applied successfully to solving uncertain problems in many fields. However, unsolved drawbacks still exist such as the problems of local minimum, slow convergence speed, and the determination of initial weights and the number of processing elements. In this paper, we introduce a single-layer orthogonal neural network (ONN) that is developed based on orthogonal functions. Since the processing elements are orthogonal to one another and there is no local minimum of the error function, the orthogonal neural network is able to avoid the above problems. Among the five existing orthogonal functions, Legendre polynomials and Chebyshev polynomials of the first kind have the properties of recursion and completeness. They are the most suitable to generate the neural network. Some typical examples are given to show their performance in function approximation. The results show that ONN has excellent convergence performance. Moreover, ONN is capable of approximating the mathematic model of backpropagation neural network. Therefore, it should be able to be applied to various applications that backpropagation neural network is suitable to solve. © 2001 John Wiley & Sons, Inc. [source] Design of nonlinear observers with approximately linear error dynamics using multivariable Legendre polynomialsINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 15 2006Joachim Deutscher Abstract This paper presents a numerical approach to the design of nonlinear observers by approximate error linearization. By using a Galerkin approach on the basis of multivariable Legendre polynomials an approximate solution to the singular PDE of the observer design technique proposed by Kazantzis and Krener (see (Syst. Control Lett. 1998; 34:241,247; SIAM J. Control Optim. 2002; 41:932,953)) is determined. It is shown that the L2 -norm of the remaining nonlinearity in the resulting error dynamics can be made small on a specified multivariable interval in the state space. Furthermore, a linear matrix equation is derived for determining the corresponding change of co-ordinates and output injection such that the proposed design procedure can easily be implemented in a numerical software package. A simple example demonstrates the properties of the new numerical observer design. Copyright © 2006 John Wiley & Sons, Ltd. [source] Selection of locations of knots for linear splines in random regression test-day modelsJOURNAL OF ANIMAL BREEDING AND GENETICS, Issue 2 2010J. Jamrozik Summary Using spline functions (segmented polynomials) in regression models requires the knowledge of the location of the knots. Knots are the points at which independent linear segments are connected. Optimal positions of knots for linear splines of different orders were determined in this study for different scenarios, using existing estimates of covariance functions and an optimization algorithm. The traits considered were test-day milk, fat and protein yields, and somatic cell score (SCS) in the first three lactations of Canadian Holsteins. Two ranges of days in milk (from 5 to 305 and from 5 to 365) were taken into account. In addition, four different populations of Holstein cows, from Australia, Canada, Italy and New Zealand, were examined with respect to first lactation (305 days) milk only. The estimates of genetic and permanent environmental covariance functions were based on single- and multiple-trait test-day models, with Legendre polynomials of order 4 as random regressions. A differential evolution algorithm was applied to find the best location of knots for splines of orders 4 to 7 and the criterion for optimization was the goodness-of-fit of the spline covariance function. Results indicated that the optimal position of knots for linear splines differed between genetic and permanent environmental effects, as well as between traits and lactations. Different populations also exhibited different patterns of optimal knot locations. With linear splines, different positions of knots should therefore be used for different effects and traits in random regression test-day models when analysing milk production traits. [source] Bayesian comparison of test-day models under different assumptions of heterogeneity for the residual variance: the change point technique versus arbitrary intervalsJOURNAL OF ANIMAL BREEDING AND GENETICS, Issue 1 2004P. López-Romero Summary Test-day milk yields from Spanish Holstein cows were analysed with two random regression models based on Legendre polynomials under two different assumptions of heterogeneity of residual variance which aim to describe the variability of temporary measurement errors along days in milk with a reduced number of parameters, such as (i) the change point identification technique with two unknown change points and (ii) using 10 arbitrary intervals of residual variance. Both implementations were based on a previous study where the trajectory of the residual variance was estimated using 30 intervals. The change point technique has been previously implemented in the analysis of the heterogeneity of the residual variance in the Spanish population, yet no comparisons with other methods have been reported so far. This study aims to compare the change point technique identification versus the use of arbitrary intervals as two possible techniques to deal with the characterization of the residual variance in random regression test-day models. The Bayes factor and the cross-validation predictive densities were employed for the model assessment. The two model-selecting tools revealed a strong consistency between them. Both specifications for the residual variance were close to each other. The 10 intervals modelling showed a slightly better performance probably because the change point function overestimates the residual variance values at the very early lactation. Zusammenfassung Testtagsgemelke von Spanischen Holstein-Kühen wurden mittels zweier zufälliger Regressionsmodelle, basierend auf Legendre Polynomen, unter zwei unterschiedlichen Voraussetzungen von Heterogenität der Residualvarianz, untersucht, um die Variabilität der Restvarianz der Milchleistung der Testtage durch so wenig Parameter wie möglich beschreiben zu können: 1) dem Verfahren des Wechsel-Identifikationspunktes mit zwei unbekannten Änderungspunkten und 2) der Verwendung von 10 frei gewählten Intervallen der Residualvarianz. Beide Anwendungen beruhen auf einer vorherigen Untersuchung, in der der Verlauf der Residualvarianz durch die Verwendung von 30 Intervallen geschätzt wurde. Das Wechsel-Identifikationspunkt Verfahren wurde bereits bei der Untersuchung der Residualvarianz in der spanischen Population verwendet, aber das Verfahren wurde noch nicht mit anderen Methoden verglichen. Das Ziel dieser Studie war der Vergleich des Wechsel-Identifikationspunkt Verfahrens mit dem Gebrauch von frei wählbaren Intervallen als zwei Möglichkeiten zur Charakterisierung der Residualvarianz in zufälligen Testtags-Regressionsmodellen. Der Bayes'sche Faktor und die Vorhersage der Vergleichsprüfungsdichten wurden zur Bewertung der Modelle verwandt. Beide Verfahren zeigten eine überzeugende Konsistenz der Modelle und die Beschreibung der Residualvarianzen stimmte in beiden Fällen überein. Die Modellierung mit 10 Intervallen zeigte eine etwas bessere Leistung, möglicherweise weil die Wechsel-Identifikationspunkt Funktion die Residualvarianz in der sehr frühen Laktation überbewertet. [source] | |||||||||||||