Zernike Basis to Cartesian Transformations

NASA Astrophysics Data System (ADS)

Mathar, R. J.

2009-12-01

The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based on projections that take advantage of the orthogonality of the polynomials over the unit interval. They play a role in the expansion of products of the polynomials into sums, which is demonstrated by some examples. Multiplication of the polynomials by the angular bases (azimuth, polar angle) defines the Zernike functions, for which we derive transformations to and from the Cartesian coordinate system centered at the middle of the circle or sphere.

Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror

NASA Astrophysics Data System (ADS)

Dou, Yimeng; Yuan, Qun; Gao, Zhishan; Yin, Huimin; Chen, Lu; Yao, Yanxia; Cheng, Jinlong

2018-06-01

Partial null interferometry without using any null optics is proposed to measure a concave freeform Zernike mirror. Oblique incidence on the freeform mirror is used to compensate for astigmatism as the main component in its figure, and to constrain the divergence of the test beam as well. The phase demodulated from the partial nulled interferograms is divided into low-frequency phase and high-frequency phase by Zernike polynomial fitting. The low-frequency surface figure error of the freeform mirror represented by the coefficients of Zernike polynomials is reconstructed from the low-frequency phase, applying the reverse optimization reconstruction technology in the accurate model of the interferometric system. The high-frequency surface figure error of the freeform mirror is retrieved from the high-frequency phase adopting back propagating technology, according to the updated model in which the low-frequency surface figure error has been superimposed on the sag of the freeform mirror. Simulations verified that this method is capable of testing a wide variety of astigmatism-dominated freeform mirrors due to the high dynamic range. The experimental result using our proposed method for a concave freeform Zernike mirror is consistent with the null test result employing the computer-generated hologram.

Wavefront reconstruction for multi-lateral shearing interferometry using difference Zernike polynomials fitting

NASA Astrophysics Data System (ADS)

Liu, Ke; Wang, Jiannian; Wang, Hai; Li, Yanqiu

2018-07-01

For the multi-lateral shearing interferometers (multi-LSIs), the measurement accuracy can be enhanced by estimating the wavefront under test with the multidirectional phase information encoded in the shearing interferogram. Usually the multi-LSIs reconstruct the test wavefront from the phase derivatives in multiple directions using the discrete Fourier transforms (DFT) method, which is only suitable to small shear ratios and relatively sensitive to noise. To improve the accuracy of multi-LSIs, wavefront reconstruction from the multidirectional phase differences using the difference Zernike polynomials fitting (DZPF) method is proposed in this paper. For the DZPF method applied in the quadriwave LSI, difference Zernike polynomials in only two orthogonal shear directions are required to represent the phase differences in multiple shear directions. In this way, the test wavefront can be reconstructed from the phase differences in multiple shear directions using a noise-variance weighted least-squares method with almost no extra computational burden, compared with the usual recovery from the phase differences in two orthogonal directions. Numerical simulation results show that the DZPF method can maintain high reconstruction accuracy in a wider range of shear ratios and has much better anti-noise performance than the DFT method. A null test experiment of the quadriwave LSI has been conducted and the experimental results show that the measurement accuracy of the quadriwave LSI can be improved from 0.0054 λ rms to 0.0029 λ rms (λ = 632.8 nm) by substituting the DFT method with the proposed DZPF method in the wavefront reconstruction process.

Quantum superintegrable Zernike system

NASA Astrophysics Data System (ADS)

Pogosyan, George S.; Salto-Alegre, Cristina; Wolf, Kurt Bernardo; Yakhno, Alexander

2017-07-01

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account, the quantum Zernike system, where that differential equation is seen as a Schrödinger equation with a potential, is special in that it has a potential and a boundary condition that are not standard in quantum mechanics. We project the disk on a half-sphere and there we find that, in addition to polar coordinates, this system separates into two additional coordinate systems (non-orthogonal on the pupil disk), which lead to Schrödinger-type equations with Pöschl-Teller potentials, whose eigen-solutions involve Legendre, Gegenbauer, and Jacobi polynomials. This provides new expressions for separated polynomial solutions of the original Zernike system that are real. The operators which provide the separation constants are found to participate in a superintegrable cubic Higgs algebra.

Use Of Zernike Polynomials And Interferometry In The Optical Design And Assembly Of Large Carbon-Dioxide Laser Systems

NASA Astrophysics Data System (ADS)

Viswanathan, V. K.

1982-02-01

This paper describes the need for non-raytracing schemes in the optical design and analysis of large carbon-dioxide lasers like the Gigawatt,1 Gemini, 2 and Helios3 lasers currently operational at Los Alamos, and the Antares 4 laser fusion system under construction. The scheme currently used at Los Alamos involves characterizing the various optical components with a Zernike polynomial sets obtained by the digitization6 of experimentally produced interferograms of the components. A Fast Fourier Transform code then propagates the complex amplitude and phase of the beam through the whole system and computes the optical parameters of interest. The analysis scheme is illustrated through examples of the Gigawatt, Gemini, and Helios systems. A possible way of using the Zernike polynomials in optical design problems of this type is discussed. Comparisons between the computed values and experimentally obtained results are made and it is concluded that this appears to be a valid approach. As this is a review article, some previously published results are also used where relevant.

Solar occultation images analysis using Zernike polynomials ­— an ALTIUS imaging spectrometer application

NASA Astrophysics Data System (ADS)

Dekemper, Emmanuel; Fussen, Didier; Loodts, Nicolas; Neefs, Eddy

The ALTIUS (Atmospheric Limb Tracker for the Investigation of the Upcoming Stratosphere) instrument is a major project of the Belgian Institute for Space Aeronomy (BIRA-IASB) in Brussels, Belgium. It has been designed to profit from the benefits of the limb scattering ge-ometry (vertical resolution, global coverage,...), while providing better accuracy on the tangent height knowledge than classical "knee" methods used by scanning spectrometers. The optical concept is based on 3 AOTF's (UV-Vis-NIR) responsible for the instantaneous spectral filtering of the incoming image (complete FOV larger than 100km x 100km at tangent point), ranging from 250nm to 1800nm, with a moderate resolution of a few nm and a typical acquisition time of 1-10s per image. While the primary goal of the instrument is the measurement of ozone with a good vertical resolution, the ability to record full images of the limb can lead to other applications, like solar occultations. With a pixel FOV of 200rad, the full high-sun image is formed of 45x45 pixels, which is sufficient for pattern recognition using moments analysis for instance. The Zernike polynomials form a complete othogonal set of functions over the unit circle. It is well suited for images showing circular shape. Any such image can then be decomposed into a finite set of weighted polynomials, the weighting is called the moments. Due to atmospheric refraction, the sun shape is modified during apparent sunsets and sunrises. The sun appears more flattened which leads to a modification of its zernike moment description. A link between the pressure or the temperature profile (equivalent to air density through the perfect gas law and the hydrostatic equation) and the Zernike moments of a given image can then be made and used to retrieve these atmospheric parameters, with the advantage that the whole sun is used and not only central or edge pixels. Some retrievals will be performed for different conditions and the feasibility of the method

Ligand Electron Density Shape Recognition Using 3D Zernike Descriptors

NASA Astrophysics Data System (ADS)

Gunasekaran, Prasad; Grandison, Scott; Cowtan, Kevin; Mak, Lora; Lawson, David M.; Morris, Richard J.

We present a novel approach to crystallographic ligand density interpretation based on Zernike shape descriptors. Electron density for a bound ligand is expanded in an orthogonal polynomial series (3D Zernike polynomials) and the coefficients from this expansion are employed to construct rotation-invariant descriptors. These descriptors can be compared highly efficiently against large databases of descriptors computed from other molecules. In this manuscript we describe this process and show initial results from an electron density interpretation study on a dataset containing over a hundred OMIT maps. We could identify the correct ligand as the first hit in about 30 % of the cases, within the top five in a further 30 % of the cases, and giving rise to an 80 % probability of getting the correct ligand within the top ten matches. In all but a few examples, the top hit was highly similar to the correct ligand in both shape and chemistry. Further extensions and intrinsic limitations of the method are discussed.

Characteristic of entire corneal topography and tomography for the detection of sub-clinical keratoconus with Zernike polynomials using Pentacam.

PubMed

Xu, Zhe; Li, Weibo; Jiang, Jun; Zhuang, Xiran; Chen, Wei; Peng, Mei; Wang, Jianhua; Lu, Fan; Shen, Meixiao; Wang, Yuanyuan

2017-11-28

The study aimed to characterize the entire corneal topography and tomography for the detection of sub-clinical keratoconus (KC) with a Zernike application method. Normal subjects (n = 147; 147 eyes), sub-clinical KC patients (n = 77; 77 eyes), and KC patients (n = 139; 139 eyes) were imaged with the Pentacam HR system. The entire corneal data of pachymetry and elevation of both the anterior and posterior surfaces were exported from the Pentacam HR software. Zernike polynomials fitting was used to quantify the 3D distribution of the corneal thickness and surface elevation. The root mean square (RMS) values for each order and the total high-order irregularity were calculated. Multimeric discriminant functions combined with individual indices were built using linear step discriminant analysis. Receiver operating characteristic curves determined the diagnostic accuracy (area under the curve, AUC). The 3rd-order RMS of the posterior surface (AUC: 0.928) obtained the highest discriminating capability in sub-clinical KC eyes. The multimeric function, which consisted of the Zernike fitting indices of corneal posterior elevation, showed the highest discriminant ability (AUC: 0.951). Indices generated from the elevation of posterior surface and thickness measurements over the entire cornea using the Zernike method based on the Pentacam HR system were able to identify very early KC.

Cylinder surface test with Chebyshev polynomial fitting method

NASA Astrophysics Data System (ADS)

Yu, Kui-bang; Guo, Pei-ji; Chen, Xi

2017-10-01

Zernike polynomials fitting method is often applied in the test of optical components and systems, used to represent the wavefront and surface error in circular domain. Zernike polynomials are not orthogonal in rectangular region which results in its unsuitable for the test of optical element with rectangular aperture such as cylinder surface. Applying the Chebyshev polynomials which are orthogonal among the rectangular area as an substitution to the fitting method, can solve the problem. Corresponding to a cylinder surface with diameter of 50 mm and F number of 1/7, a measuring system has been designed in Zemax based on Fizeau Interferometry. The expressions of the two-dimensional Chebyshev polynomials has been given and its relationship with the aberration has been presented. Furthermore, Chebyshev polynomials are used as base items to analyze the rectangular aperture test data. The coefficient of different items are obtained from the test data through the method of least squares. Comparing the Chebyshev spectrum in different misalignment, it show that each misalignment is independence and has a certain relationship with the certain Chebyshev terms. The simulation results show that, through the Legendre polynomials fitting method, it will be a great improvement in the efficient of the detection and adjustment of the cylinder surface test.

Zernike analysis of all-sky night brightness maps.

PubMed

Bará, Salvador; Nievas, Miguel; Sánchez de Miguel, Alejandro; Zamorano, Jaime

2014-04-20

All-sky night brightness maps (calibrated images of the night sky with hemispherical field-of-view (FOV) taken at standard photometric bands) provide useful data to assess the light pollution levels at any ground site. We show that these maps can be efficiently described and analyzed using Zernike circle polynomials. The relevant image information can be compressed into a low-dimensional coefficients vector, giving an analytical expression for the sky brightness and alleviating the effects of noise. Moreover, the Zernike expansions allow us to quantify in a straightforward way the average and zenithal sky brightness and its variation across the FOV, providing a convenient framework to study the time course of these magnitudes. We apply this framework to analyze the results of a one-year campaign of night sky brightness measurements made at the UCM observatory in Madrid.

Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound.

PubMed

Kaye, Elena A; Hertzberg, Yoni; Marx, Michael; Werner, Beat; Navon, Gil; Levoy, Marc; Pauly, Kim Butts

2012-10-01

To study the phase aberrations produced by human skulls during transcranial magnetic resonance imaging guided focused ultrasound surgery (MRgFUS), to demonstrate the potential of Zernike polynomials (ZPs) to accelerate the adaptive focusing process, and to investigate the benefits of using phase corrections obtained in previous studies to provide the initial guess for correction of a new data set. The five phase aberration data sets, analyzed here, were calculated based on preoperative computerized tomography (CT) images of the head obtained during previous transcranial MRgFUS treatments performed using a clinical prototype hemispherical transducer. The noniterative adaptive focusing algorithm [Larrat et al., "MR-guided adaptive focusing of ultrasound," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57(8), 1734-1747 (2010)] was modified by replacing Hadamard encoding with Zernike encoding. The algorithm was tested in simulations to correct the patients' phase aberrations. MR acoustic radiation force imaging (MR-ARFI) was used to visualize the effect of the phase aberration correction on the focusing of a hemispherical transducer. In addition, two methods for constructing initial phase correction estimate based on previous patient's data were investigated. The benefits of the initial estimates in the Zernike-based algorithm were analyzed by measuring their effect on the ultrasound intensity at the focus and on the number of ZP modes necessary to achieve 90% of the intensity of the nonaberrated case. Covariance of the pairs of the phase aberrations data sets showed high correlation between aberration data of several patients and suggested that subgroups can be based on level of correlation. Simulation of the Zernike-based algorithm demonstrated the overall greater correction effectiveness of the low modes of ZPs. The focal intensity achieves 90% of nonaberrated intensity using fewer than 170 modes of ZPs. The initial estimates based on using the average of the

Zernike ultrasonic tomography for fluid velocity imaging based on pipeline intrusive time-of-flight measurements.

PubMed

Besic, Nikola; Vasile, Gabriel; Anghel, Andrei; Petrut, Teodor-Ion; Ioana, Cornel; Stankovic, Srdjan; Girard, Alexandre; d'Urso, Guy

2014-11-01

In this paper, we propose a novel ultrasonic tomography method for pipeline flow field imaging, based on the Zernike polynomial series. Having intrusive multipath time-offlight ultrasonic measurements (difference in flight time and speed of ultrasound) at the input, we provide at the output tomograms of the fluid velocity components (axial, radial, and orthoradial velocity). Principally, by representing these velocities as Zernike polynomial series, we reduce the tomography problem to an ill-posed problem of finding the coefficients of the series, relying on the acquired ultrasonic measurements. Thereupon, this problem is treated by applying and comparing Tikhonov regularization and quadratically constrained ℓ1 minimization. To enhance the comparative analysis, we additionally introduce sparsity, by employing SVD-based filtering in selecting Zernike polynomials which are to be included in the series. The first approach-Tikhonov regularization without filtering, is used because it is the most suitable method. The performances are quantitatively tested by considering a residual norm and by estimating the flow using the axial velocity tomogram. Finally, the obtained results show the relative residual norm and the error in flow estimation, respectively, ~0.3% and ~1.6% for the less turbulent flow and ~0.5% and ~1.8% for the turbulent flow. Additionally, a qualitative validation is performed by proximate matching of the derived tomograms with a flow physical model.

Characterization of bone collagen organization defects in murine hypophosphatasia using a Zernike model of optical aberrations

NASA Astrophysics Data System (ADS)

Tehrani, Kayvan Forouhesh; Pendleton, Emily G.; Leitmann, Bobby; Barrow, Ruth; Mortensen, Luke J.

2018-02-01

Bone growth and strength is severely impacted by Hypophosphatasia (HPP). It is a genetic disease that affects the mineralization of the bone. We hypothesize that it impacts overall organization, density, and porosity of collagen fibers. Lower density of fibers and higher porosity cause less absorption and scattering of light, and therefore a different regime of transport mean free path. To find a cure for this disease, a metric for the evaluation of bone is required. Here we present an evaluation method based on our Phase Accumulation Ray Tracing (PART) method. This method uses second harmonic generation (SHG) in bone collagen fiber to model bone indices of refraction, which is used to calculate phase retardation on the propagation path of light in bone. The calculated phase is then expanded using Zernike polynomials up to 15th order, to make a quantitative analysis of tissue anomalies. Because the Zernike modes are a complete set of orthogonal polynomials, we can compare low and high order modes in HPP, compare them with healthy wild type mice, to identify the differences between their geometry and structure. Larger coefficients of low order modes show more uniform fiber density and less porosity, whereas the opposite is shown with larger coefficients of higher order modes. Our analyses show significant difference between Zernike modes in different types of bone evidenced by Principal Components Analysis (PCA).

Application of Zernike polynomials towards accelerated adaptive focusing of transcranial high intensity focused ultrasound

PubMed Central

Kaye, Elena A.; Hertzberg, Yoni; Marx, Michael; Werner, Beat; Navon, Gil; Levoy, Marc; Pauly, Kim Butts

2012-01-01

Purpose: To study the phase aberrations produced by human skulls during transcranial magnetic resonance imaging guided focused ultrasound surgery (MRgFUS), to demonstrate the potential of Zernike polynomials (ZPs) to accelerate the adaptive focusing process, and to investigate the benefits of using phase corrections obtained in previous studies to provide the initial guess for correction of a new data set. Methods: The five phase aberration data sets, analyzed here, were calculated based on preoperative computerized tomography (CT) images of the head obtained during previous transcranial MRgFUS treatments performed using a clinical prototype hemispherical transducer. The noniterative adaptive focusing algorithm [Larrat , “MR-guided adaptive focusing of ultrasound,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57(8), 1734–1747 (2010)]10.1109/TUFFC.2010.1612 was modified by replacing Hadamard encoding with Zernike encoding. The algorithm was tested in simulations to correct the patients’ phase aberrations. MR acoustic radiation force imaging (MR-ARFI) was used to visualize the effect of the phase aberration correction on the focusing of a hemispherical transducer. In addition, two methods for constructing initial phase correction estimate based on previous patient's data were investigated. The benefits of the initial estimates in the Zernike-based algorithm were analyzed by measuring their effect on the ultrasound intensity at the focus and on the number of ZP modes necessary to achieve 90% of the intensity of the nonaberrated case. Results: Covariance of the pairs of the phase aberrations data sets showed high correlation between aberration data of several patients and suggested that subgroups can be based on level of correlation. Simulation of the Zernike-based algorithm demonstrated the overall greater correction effectiveness of the low modes of ZPs. The focal intensity achieves 90% of nonaberrated intensity using fewer than 170 modes of ZPs. The

On the coefficients of differentiated expansions of ultraspherical polynomials

NASA Technical Reports Server (NTRS)

Karageorghis, Andreas; Phillips, Timothy N.

1989-01-01

A formula expressing the coefficients of an expression of ultraspherical polynomials which has been differentiated an arbitrary number of times in terms of the coefficients of the original expansion is proved. The particular examples of Chebyshev and Legendre polynomials are considered.

Pattern placement errors: application of in-situ interferometer-determined Zernike coefficients in determining printed image deviations

NASA Astrophysics Data System (ADS)

Roberts, William R.; Gould, Christopher J.; Smith, Adlai H.; Rebitz, Ken

2000-08-01

Several ideas have recently been presented which attempt to measure and predict lens aberrations for new low k1 imaging systems. Abbreviated sets of Zernike coefficients have been produced and used to predict Across Chip Linewidth Variation. Empirical use of the wavefront aberrations can now be used in commercially available lithography simulators to predict pattern distortion and placement errors. Measurement and Determination of Zernike coefficients has been a significant effort of many. However the use of this data has generally been limited to matching lenses or picking best fit lense pairs. We will use wavefront aberration data collected using the Litel InspecStep in-situ Interferometer as input data for Prolith/3D to model and predict pattern placement errors and intrafield overlay variation. Experiment data will be collected and compared to the simulated predictions.

On Polynomial Solutions of Linear Differential Equations with Polynomial Coefficients

ERIC Educational Resources Information Center

Si, Do Tan

1977-01-01

Demonstrates a method for solving linear differential equations with polynomial coefficients based on the fact that the operators z and D + d/dz are known to be Hermitian conjugates with respect to the Bargman and Louck-Galbraith scalar products. (MLH)

On the coefficients of integrated expansions of Bessel polynomials

NASA Astrophysics Data System (ADS)

Doha, E. H.; Ahmed, H. M.

2006-03-01

A new formula expressing explicitly the integrals of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another new explicit formula relating the Bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.

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.

On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions

NASA Astrophysics Data System (ADS)

Becherer, Nico; Jödicke, Hanna; Schlosser, Gregor; Hesser, Jürgen; Zeilfelder, Frank; Männer, Reinhard

2006-02-01

Blur and noise originating from the physical imaging processes degrade the microscope data. Accurate deblurring techniques require, however, an accurate estimation of the underlying point-spread function (PSF). A good representation of PSFs can be achieved by Zernike Polynomials since they offer a compact representation where low-order coefficients represent typical aberrations of optical wavefronts while noise is represented in higher order coefficients. A quantitative description of the noise distribution (Gaussian) over the Zernike moments of various orders is given which is the basis for the new soft clipping approach for denoising of PSFs. Instead of discarding moments beyond a certain order, those Zernike moments that are more sensitive to noise are dampened according to the measured distribution and the present noise model. Further, a new scheme to combine experimental and theoretical PSFs in Zernike space is presented. According to our experimental reconstructions, using the new improved PSF the correlation between reconstructed and original volume is raised by 15% on average cases and up to 85% in the case of thin fibre structures, compared to reconstructions where a non improved PSF was used. Finally, we demonstrate the advantages of our approach on 3D images of confocal microscopes by generating visually improved volumes. Additionally, we are presenting a method to render the reconstructed results using a new volume rendering method that is almost artifact-free. The new approach is based on a Shear-Warp technique, wavelet data encoding techniques and a recent approach to approximate the gray value distribution by a Super spline model.

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.

Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials

NASA Astrophysics Data System (ADS)

Doha, E. H.; Ahmed, H. M.

2004-08-01

A formula expressing explicitly the derivatives of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another explicit formula, which expresses the Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Bessel coefficients, is also given. A formula for the Bessel coefficients of the moments of one single Bessel polynomial of certain degree is proved. A formula for the Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Bessel-Bessel polynomials is described. An explicit formula for these coefficients between Jacobi and Bessel polynomials is given, of which the ultraspherical polynomial and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre-Bessel and Hermite-Bessel are also developed.

A parallel implementation of 3D Zernike moment analysis

NASA Astrophysics Data System (ADS)

Berjón, Daniel; Arnaldo, Sergio; Morán, Francisco

2011-01-01

Zernike polynomials are a well known set of functions that find many applications in image or pattern characterization because they allow to construct shape descriptors that are invariant against translations, rotations or scale changes. The concepts behind them can be extended to higher dimension spaces, making them also fit to describe volumetric data. They have been less used than their properties might suggest due to their high computational cost. We present a parallel implementation of 3D Zernike moments analysis, written in C with CUDA extensions, which makes it practical to employ Zernike descriptors in interactive applications, yielding a performance of several frames per second in voxel datasets about 2003 in size. In our contribution, we describe the challenges of implementing 3D Zernike analysis in a general-purpose GPU. These include how to deal with numerical inaccuracies, due to the high precision demands of the algorithm, or how to deal with the high volume of input data so that it does not become a bottleneck for the system.

On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials

NASA Astrophysics Data System (ADS)

Doha, E. H.

2004-01-01

Formulae expressing explicitly the Jacobi coefficients of a general-order derivative (integral) of an infinitely differentiable function in terms of its original expansion coefficients, and formulae for the derivatives (integrals) of Jacobi polynomials in terms of Jacobi polynomials themselves are stated. A formula for the Jacobi coefficients of the moments of one single Jacobi polynomial of certain degree is proved. Another formula for the Jacobi coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given. A simple approach in order to construct and solve recursively for the connection coefficients between Jacobi-Jacobi polynomials is described. Explicit formulae for these coefficients between ultraspherical and Jacobi polynomials are deduced, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Two analytical formulae for the connection coefficients between Laguerre-Jacobi and Hermite-Jacobi are developed.

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

Application of field dependent polynomial model

NASA Astrophysics Data System (ADS)

Janout, Petr; Páta, Petr; Skala, Petr; Fliegel, Karel; Vítek, Stanislav; Bednář, Jan

2016-09-01

Extremely wide-field imaging systems have many advantages regarding large display scenes whether for use in microscopy, all sky cameras, or in security technologies. The Large viewing angle is paid by the amount of aberrations, which are included with these imaging systems. Modeling wavefront aberrations using the Zernike polynomials is known a longer time and is widely used. Our method does not model system aberrations in a way of modeling wavefront, but directly modeling of aberration Point Spread Function of used imaging system. This is a very complicated task, and with conventional methods, it was difficult to achieve the desired accuracy. Our optimization techniques of searching coefficients space-variant Zernike polynomials can be described as a comprehensive model for ultra-wide-field imaging systems. The advantage of this model is that the model describes the whole space-variant system, unlike the majority models which are partly invariant systems. The issue that this model is the attempt to equalize the size of the modeled Point Spread Function, which is comparable to the pixel size. Issues associated with sampling, pixel size, pixel sensitivity profile must be taken into account in the design. The model was verified in a series of laboratory test patterns, test images of laboratory light sources and consequently on real images obtained by an extremely wide-field imaging system WILLIAM. Results of modeling of this system are listed in this article.

Mathematical construction and perturbation analysis of Zernike discrete orthogonal points.

PubMed

Shi, Zhenguang; Sui, Yongxin; Liu, Zhenyu; Peng, Ji; Yang, Huaijiang

2012-06-20

Zernike functions are orthogonal within the unit circle, but they are not over the discrete points such as CCD arrays or finite element grids. This will result in reconstruction errors for loss of orthogonality. By using roots of Legendre polynomials, a set of points within the unit circle can be constructed so that Zernike functions over the set are discretely orthogonal. Besides that, the location tolerances of the points are studied by perturbation analysis, and the requirements of the positioning precision are not very strict. Computer simulations show that this approach provides a very accurate wavefront reconstruction with the proposed sampling set.

Wavefront aberrations: analytical method to convert Zernike coefficients from a pupil to a scaled arbitrarily decentered one

NASA Astrophysics Data System (ADS)

Comastri, S. A.; Perez, Liliana I.; Pérez, Gervasio D.; Bastida, K.; Martin, G.

2008-04-01

The wavefront aberration of any image forming system and, in particular, of a human eye, is often expanded in Zernike modes each mode being weighed by a coefficient that depends both on the image forming components of the system and on the contour, size and centering of the pupil. In the present article, expanding up to 7th order the wavefront aberration, an analytical method to compute a new set of Zernike coefficients corresponding to a pupil in terms of an original set evaluated via ray tracing for a dilated and transversally arbitrarily displaced pupil is developed. A transformation matrix of dimension 36×36 is attained multiplying the scaling-horizontal traslation matrix previously derived by appropriate rotation matrices. Multiplying the original coefficients by this transformation matrix, analytical formulas for each new coefficient are attained and supplied and, for the information concerning the wavefront aberration to be available, these formulas must be employed in cases in which the new pupil is contained in the original one. The use of these analytical formulas is exemplified applying them to study the effect of pupil contraction and/or decentering in 3 situations: calculation of corneal aberrations of a keratoconic subject for the natural photopic pupil size and various decenterings; coma compensation by means of pupil shift in a fictitious system solely having primary aberrations and evaluation of the amount of astigmatism and coma of a hypothetical system originally having spherical aberration alone.

Zernike-like systems in polygons and polygonal facets.

PubMed

Ferreira, Chelo; López, José L; Navarro, Rafael; Sinusía, Ester Pérez

2015-07-20

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Opt. Lett.32, 74 (2007)10.1364/OL.32.000074OPLEDP0146-9592] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piecewise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both the general form and the explicit expressions for a typical example of telescope optical aperture are provided.

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.

Alignment-independent comparison of binding sites based on DrugScore potential fields encoded by 3D Zernike descriptors.

PubMed

Nisius, Britta; Gohlke, Holger

2012-09-24

Analyzing protein binding sites provides detailed insights into the biological processes proteins are involved in, e.g., into drug-target interactions, and so is of crucial importance in drug discovery. Herein, we present novel alignment-independent binding site descriptors based on DrugScore potential fields. The potential fields are transformed to a set of information-rich descriptors using a series expansion in 3D Zernike polynomials. The resulting Zernike descriptors show a promising performance in detecting similarities among proteins with low pairwise sequence identities that bind identical ligands, as well as within subfamilies of one target class. Furthermore, the Zernike descriptors are robust against structural variations among protein binding sites. Finally, the Zernike descriptors show a high data compression power, and computing similarities between binding sites based on these descriptors is highly efficient. Consequently, the Zernike descriptors are a useful tool for computational binding site analysis, e.g., to predict the function of novel proteins, off-targets for drug candidates, or novel targets for known drugs.

A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or 6 - j Symbols.

DTIC Science & Technology

1978-03-01

Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966. [11] D. Stanton, Some basic hypergeometric polynomials arising from... Some bas ic hypergeometr ic an a logues of the classical orthogonal polynomials and applications , to appear. [3] C. de Boor and G. H. Golub , The...Report #1833 A SET OF ORTHOGONAL POLYNOMIALS THAT GENERALIZE THE RACAR COEFFICIENTS OR 6 — j SYMBOLS Richard Askey and James Wilson •

Eye micromotions influence on an error of Zernike coefficients reconstruction in the one-ray refractometry of an eye

NASA Astrophysics Data System (ADS)

Osipova, Irina Y.; Chyzh, Igor H.

2001-06-01

The influence of eye jumps on the accuracy of estimation of Zernike coefficients from eye transverse aberration measurements was investigated. By computer modeling the ametropy and astigmatism have been examined. The standard deviation of the wave aberration function was calculated. It was determined that the standard deviation of the wave aberration function achieves the minimum value if the number of scanning points is equal to the number of eye jumps in scanning period. The recommendations for duration of measurement were worked out.

Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials.

PubMed

Dai, Guang-ming; Campbell, Charles E; Chen, Li; Zhao, Huawei; Chernyak, Dimitri

2009-01-20

In wavefront-driven vision correction, ocular aberrations are often measured on the pupil plane and the correction is applied on a different plane. The problem with this practice is that any changes undergone by the wavefront as it propagates between planes are not currently included in devising customized vision correction. With some valid approximations, we have developed an analytical foundation based on geometric optics in which Zernike polynomials are used to characterize the propagation of the wavefront from one plane to another. Both the boundary and the magnitude of the wavefront change after the propagation. Taylor monomials were used to realize the propagation because of their simple form for this purpose. The method we developed to identify changes in low-order aberrations was verified with the classical vertex correction formula. The method we developed to identify changes in high-order aberrations was verified with ZEMAX ray-tracing software. Although the method may not be valid for highly irregular wavefronts and it was only proven for wavefronts with low-order or high-order aberrations, our analysis showed that changes in the propagating wavefront are significant and should, therefore, be included in calculating vision correction. This new approach could be of major significance in calculating wavefront-driven vision correction whether by refractive surgery, contact lenses, intraocular lenses, or spectacles.

On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations

NASA Astrophysics Data System (ADS)

Doha, E. H.

2002-02-01

An analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by Phillips and Karageorghis (27 (1990) 823). A new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given. The tensor product of ultraspherical polynomials is used to approximate a function of more than one variable. Formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved. Some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described.

Laguerre-Freud Equations for the Recurrence Coefficients of Some Discrete Semi-Classical Orthogonal Polynomials of Class Two

NASA Astrophysics Data System (ADS)

Hounga, C.; Hounkonnou, M. N.; Ronveaux, A.

2006-10-01

In this paper, we give Laguerre-Freud equations for the recurrence coefficients of discrete semi-classical orthogonal polynomials of class two, when the polynomials in the Pearson equation are of the same degree. The case of generalized Charlier polynomials is also presented.

Cubic Polynomials with Real or Complex Coefficients: The Full Picture

ERIC Educational Resources Information Center

Bardell, Nicholas S.

2016-01-01

The cubic polynomial with real coefficients has a rich and interesting history primarily associated with the endeavours of great mathematicians like del Ferro, Tartaglia, Cardano or Vieta who sought a solution for the roots (Katz, 1998; see Chapter 12.3: The Solution of the Cubic Equation). Suffice it to say that since the times of renaissance…

Zernike expansion coefficients: rescaling and decentring for different pupils and evaluation of corneal aberrations

NASA Astrophysics Data System (ADS)

Comastri, Silvia A.; Perez, Liliana I.; Pérez, Gervasio D.; Martin, Gabriel; Bastida, Karina

2007-03-01

An analytical method to convert the set of Zernike coefficients that fits the wavefront aberration for a pupil into another corresponding to a contracted and horizontally translated pupil is proposed. The underlying selection rules are provided and the resulting conversion formulae for a seventh-order expansion are given. These formulae are applied to calculate corneal aberrations referred to a given pupil centre in terms of those referred to the keratometric vertex supplied by the SN CT1000 topographer. Four typical cases are considered: a sphere and three eyes—normal, keratoconic and post-LASIK. When the pupil centre is fixed and the pupil diameter decreases from 6 mm to the photopic natural one, leaving aside piston, tilt and defocus, the difference between the root mean square wavefront error computed with the formulae and the topographer is less than 0.04 µm. When the pupil diameter is kept equal to the natural one and the pupil centre is displaced, coefficients vary according to the eye. For a 0.3 mm pupil shift, the variation of coma is at most 0.35 µm and that of spherical aberration 0.01 µm.

Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of Al-Salam Carlitz I polynomials

NASA Astrophysics Data System (ADS)

Doha, E. H.; Ahmed, H. M.

2005-12-01

Two formulae expressing explicitly the derivatives and moments of Al-Salam-Carlitz I polynomials of any degree and for any order in terms of Al-Salam-Carlitz I themselves are proved. Two other formulae for the expansion coefficients of general-order derivatives Dpqf(x), and for the moments xellDpqf(x), of an arbitrary function f(x) in terms of its original expansion coefficients are also obtained. Application of these formulae for solving q-difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Al-Salam-Carlitz I polynomials and any system of basic hypergeometric orthogonal polynomials, belonging to the q-Hahn class, is described.

The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation

NASA Astrophysics Data System (ADS)

Filipuk, Galina; Van Assche, Walter; Zhang, Lun

2012-05-01

We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painlevé equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre weight and show how it arises from the Bäcklund transformation of the fourth Painlevé equation.

Calculating Angle Lambda (λ) Using Zernike Tilt Measurements in Specular Reflection Corneal Topography

PubMed Central

Braaf, Boy; van de Watering, Thomas Christiaan; Spruijt, Kees; van der Heijde, Rob G.L.; Sicam, Victor Arni D.P.

2010-01-01

Purpose To develop a method to calculate the angle λ of the human eye using Zernike tilt measurements in specular reflection corneal topography. Methods The meaning of Zernike tilt in specular reflection corneal topography is demonstrated by measurements on translated artificial surfaces using the VU Topographer. The relationship derived from the translation experiments is used to determine the angle λ. Corneal surfaces are measured for a set of eight different fixation points, for which tilt angles ρ are obtained from the Zernike tilt coefficients. The angles ρ are used with respect to the fixation target angles to determine angle λ by fitting a geometrical model. This method is validated with Orbscan II's angle-κ measurements in 9 eyes. Results The translation experiments show that the Zernike tilt coefficient is directly related to an angle ρ, which describes a tilt orientation of the cornea and can therefore be used to derive a value for angle λ. A significant correlation exists between measured values for angle λ with the VU Topographer and the angle κ with the Orbscan II (r=0.95, P<0.001). A Bland-Altman plot indicates a mean difference of -0.52 degrees between the two instruments, but this is not statistically significant as indicated by a matched-pairs Wilcoxon signed-rank test (P≤0.1748). The mean precision for measuring angle λ using the VU topographer is 0.6±0.3 degrees. Conclusion The method described above to determine angle λ is sufficiently repeatable and performs similarly to the angle-κ measurements made with the Orbscan II.

Modeling corneal surfaces with rational functions for high-speed videokeratoscopy data compression.

PubMed

Schneider, Martin; Iskander, D Robert; Collins, Michael J

2009-02-01

High-speed videokeratoscopy is an emerging technique that enables study of the corneal surface and tear-film dynamics. Unlike its static predecessor, this new technique results in a very large amount of digital data for which storage needs become significant. We aimed to design a compression technique that would use mathematical functions to parsimoniously fit corneal surface data with a minimum number of coefficients. Since the Zernike polynomial functions that have been traditionally used for modeling corneal surfaces may not necessarily correctly represent given corneal surface data in terms of its optical performance, we introduced the concept of Zernike polynomial-based rational functions. Modeling optimality criteria were employed in terms of both the rms surface error as well as the point spread function cross-correlation. The parameters of approximations were estimated using a nonlinear least-squares procedure based on the Levenberg-Marquardt algorithm. A large number of retrospective videokeratoscopic measurements were used to evaluate the performance of the proposed rational-function-based modeling approach. The results indicate that the rational functions almost always outperform the traditional Zernike polynomial approximations with the same number of coefficients.

More on rotations as spin matrix polynomials

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

Curtright, Thomas L.

2015-09-15

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

Dual exponential polynomials and linear differential equations

NASA Astrophysics Data System (ADS)

Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne

2018-01-01

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.

Combining freeform optics and curved detectors for wide field imaging: a polynomial approach over squared aperture.

PubMed

Muslimov, Eduard; Hugot, Emmanuel; Jahn, Wilfried; Vives, Sebastien; Ferrari, Marc; Chambion, Bertrand; Henry, David; Gaschet, Christophe

2017-06-26

In the recent years a significant progress was achieved in the field of design and fabrication of optical systems based on freeform optical surfaces. They provide a possibility to build fast, wide-angle and high-resolution systems, which are very compact and free of obscuration. However, the field of freeform surfaces design techniques still remains underexplored. In the present paper we use the mathematical apparatus of orthogonal polynomials defined over a square aperture, which was developed before for the tasks of wavefront reconstruction, to describe shape of a mirror surface. Two cases, namely Legendre polynomials and generalization of the Zernike polynomials on a square, are considered. The potential advantages of these polynomials sets are demonstrated on example of a three-mirror unobscured telescope with F/# = 2.5 and FoV = 7.2x7.2°. In addition, we discuss possibility of use of curved detectors in such a design.

Analysis on the misalignment errors between Hartmann-Shack sensor and 45-element deformable mirror

NASA Astrophysics Data System (ADS)

Liu, Lihui; Zhang, Yi; Tao, Jianjun; Cao, Fen; Long, Yin; Tian, Pingchuan; Chen, Shangwu

2017-02-01

Aiming at 45-element adaptive optics system, the model of 45-element deformable mirror is truly built by COMSOL Multiphysics, and every actuator's influence function is acquired by finite element method. The process of this system correcting optical aberration is simulated by making use of procedure, and aiming for Strehl ratio of corrected diffraction facula, in the condition of existing different translation and rotation error between Hartmann-Shack sensor and deformable mirror, the system's correction ability for 3-20 Zernike polynomial wave aberration is analyzed. The computed result shows: the system's correction ability for 3-9 Zernike polynomial wave aberration is higher than that of 10-20 Zernike polynomial wave aberration. The correction ability for 3-20 Zernike polynomial wave aberration does not change with misalignment error changing. With rotation error between Hartmann-Shack sensor and deformable mirror increasing, the correction ability for 3-20 Zernike polynomial wave aberration gradually goes down, and with translation error increasing, the correction ability for 3-9 Zernike polynomial wave aberration gradually goes down, but the correction ability for 10-20 Zernike polynomial wave aberration behave up-and-down depression.

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

Solving the Rational Polynomial Coefficients Based on L Curve

NASA Astrophysics Data System (ADS)

Zhou, G.; Li, X.; Yue, T.; Huang, W.; He, C.; Huang, Y.

2018-05-01

The rational polynomial coefficients (RPC) model is a generalized sensor model, which can achieve high approximation accuracy. And it is widely used in the field of photogrammetry and remote sensing. Least square method is usually used to determine the optimal parameter solution of the rational function model. However the distribution of control points is not uniform or the model is over-parameterized, which leads to the singularity of the coefficient matrix of the normal equation. So the normal equation becomes ill conditioned equation. The obtained solutions are extremely unstable and even wrong. The Tikhonov regularization can effectively improve and solve the ill conditioned equation. In this paper, we calculate pathological equations by regularization method, and determine the regularization parameters by L curve. The results of the experiments on aerial format photos show that the accuracy of the first-order RPC with the equal denominators has the highest accuracy. The high order RPC model is not necessary in the processing of dealing with frame images, as the RPC model and the projective model are almost the same. The result shows that the first-order RPC model is basically consistent with the strict sensor model of photogrammetry. Orthorectification results both the firstorder RPC model and Camera Model (ERDAS9.2 platform) are similar to each other, and the maximum residuals of X and Y are 0.8174 feet and 0.9272 feet respectively. This result shows that RPC model can be used in the aerial photographic compensation replacement sensor model.

Tolerance analysis of optical telescopes using coherent addition of wavefront errors

NASA Technical Reports Server (NTRS)

Davenport, J. W.

1982-01-01

A near diffraction-limited telescope requires that tolerance analysis be done on the basis of system wavefront error. One method of analyzing the wavefront error is to represent the wavefront error function in terms of its Zernike polynomial expansion. A Ramsey-Korsch ray trace package, a computer program that simulates the tracing of rays through an optical telescope system, was expanded to include the Zernike polynomial expansion up through the fifth-order spherical term. An option to determine a 3 dimensional plot of the wavefront error function was also included in the Ramsey-Korsch package. Several assimulation runs were analyzed to determine the particular set of coefficients in the Zernike expansion that are effected by various errors such as tilt, decenter and despace. A 3 dimensional plot of each error up through the fifth-order spherical term was also included in the study. Tolerance analysis data are presented.

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.

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.

Graphical Solution of Polynomial Equations

ERIC Educational Resources Information Center

Grishin, Anatole

2009-01-01

Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…

Statistical virtual eye model based on wavefront aberration

PubMed Central

Wang, Jie-Mei; Liu, Chun-Ling; Luo, Yi-Ning; Liu, Yi-Guang; Hu, Bing-Jie

2012-01-01

Wavefront aberration affects the quality of retinal image directly. This paper reviews the representation and reconstruction of wavefront aberration, as well as the construction of virtual eye model based on Zernike polynomial coefficients. In addition, the promising prospect of virtual eye model is emphasized. PMID:23173112

Frits Zernike--life and achievements

NASA Astrophysics Data System (ADS)

Ferwerda, Hedzer A.

1993-12-01

We present a review of the life and work of Frits Zernike (1888-1966), professor of mathematical and technical physics and theoretical mechanics at Groningen University, The Netherlands, inventor of phase contrast microscopy.

Polynomial compensation, inversion, and approximation of discrete time linear systems

NASA Technical Reports Server (NTRS)

Baram, Yoram

1987-01-01

The least-squares transformation of a discrete-time multivariable linear system into a desired one by convolving the first with a polynomial system yields optimal polynomial solutions to the problems of system compensation, inversion, and approximation. The polynomial coefficients are obtained from the solution to a so-called normal linear matrix equation, whose coefficients are shown to be the weighting patterns of certain linear systems. These, in turn, can be used in the recursive solution of the normal equation.

Simulation of aspheric tolerance with polynomial fitting

NASA Astrophysics Data System (ADS)

Li, Jing; Cen, Zhaofeng; Li, Xiaotong

2018-01-01

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

Correcting bias in the rational polynomial coefficients of satellite imagery using thin-plate smoothing splines

NASA Astrophysics Data System (ADS)

Shen, Xiang; Liu, Bin; Li, Qing-Quan

2017-03-01

The Rational Function Model (RFM) has proven to be a viable alternative to the rigorous sensor models used for geo-processing of high-resolution satellite imagery. Because of various errors in the satellite ephemeris and instrument calibration, the Rational Polynomial Coefficients (RPCs) supplied by image vendors are often not sufficiently accurate, and there is therefore a clear need to correct the systematic biases in order to meet the requirements of high-precision topographic mapping. In this paper, we propose a new RPC bias-correction method using the thin-plate spline modeling technique. Benefiting from its excellent performance and high flexibility in data fitting, the thin-plate spline model has the potential to remove complex distortions in vendor-provided RPCs, such as the errors caused by short-period orbital perturbations. The performance of the new method was evaluated by using Ziyuan-3 satellite images and was compared against the recently developed least-squares collocation approach, as well as the classical affine-transformation and quadratic-polynomial based methods. The results show that the accuracies of the thin-plate spline and the least-squares collocation approaches were better than the other two methods, which indicates that strong non-rigid deformations exist in the test data because they cannot be adequately modeled by simple polynomial-based methods. The performance of the thin-plate spline method was close to that of the least-squares collocation approach when only a few Ground Control Points (GCPs) were used, and it improved more rapidly with an increase in the number of redundant observations. In the test scenario using 21 GCPs (some of them located at the four corners of the scene), the correction residuals of the thin-plate spline method were about 36%, 37%, and 19% smaller than those of the affine transformation method, the quadratic polynomial method, and the least-squares collocation algorithm, respectively, which demonstrates

Graph characterization via Ihara coefficients.

PubMed

Ren, Peng; Wilson, Richard C; Hancock, Edwin R

2011-02-01

The novel contributions of this paper are twofold. First, we demonstrate how to characterize unweighted graphs in a permutation-invariant manner using the polynomial coefficients from the Ihara zeta function, i.e., the Ihara coefficients. Second, we generalize the definition of the Ihara coefficients to edge-weighted graphs. For an unweighted graph, the Ihara zeta function is the reciprocal of a quasi characteristic polynomial of the adjacency matrix of the associated oriented line graph. Since the Ihara zeta function has poles that give rise to infinities, the most convenient numerically stable representation is to work with the coefficients of the quasi characteristic polynomial. Moreover, the polynomial coefficients are invariant to vertex order permutations and also convey information concerning the cycle structure of the graph. To generalize the representation to edge-weighted graphs, we make use of the reduced Bartholdi zeta function. We prove that the computation of the Ihara coefficients for unweighted graphs is a special case of our proposed method for unit edge weights. We also present a spectral analysis of the Ihara coefficients and indicate their advantages over other graph spectral methods. We apply the proposed graph characterization method to capturing graph-class structure and clustering graphs. Experimental results reveal that the Ihara coefficients are more effective than methods based on Laplacian spectra.

Precipitate shape fitting and reconstruction by means of 3D Zernike functions

NASA Astrophysics Data System (ADS)

Callahan, P. G.; De Graef, M.

2012-01-01

3D Zernike functions are defined and used for the reconstruction of precipitate shapes. These functions are orthogonal over the unit ball and allow for an arbitrary shape, scaled to fit inside an embedding sphere, to be decomposed into 3D harmonics. Explicit expressions are given for the general Zernike moments, correcting typographical errors in the literature. Explicit expressions of the Zernike moments for the ellipsoid and the cube are given. The 3D Zernike functions and moments are applied to the reconstruction of γ' precipitate shapes in two Ni-based superalloys, one with nearly cuboidal precipitate shapes, and one with more complex dendritic shapes.

Schur Stability Regions for Complex Quadratic Polynomials

ERIC Educational Resources Information Center

Cheng, Sui Sun; Huang, Shao Yuan

2010-01-01

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

Conformal Galilei algebras, symmetric polynomials and singular vectors

NASA Astrophysics Data System (ADS)

Křižka, Libor; Somberg, Petr

2018-01-01

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras cga_ℓ (d,C) with d=1 for any integer value ℓ \\in N. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.

Single particle analysis based on Zernike phase contrast transmission electron microscopy.

PubMed

Danev, Radostin; Nagayama, Kuniaki

2008-02-01

We present the first application of Zernike phase-contrast transmission electron microscopy to single-particle 3D reconstruction of a protein, using GroEL chaperonin as the test specimen. We evaluated the performance of the technique by comparing 3D models derived from Zernike phase contrast imaging, with models from conventional underfocus phase contrast imaging. The same resolution, about 12A, was achieved by both imaging methods. The reconstruction based on Zernike phase contrast data required about 30% fewer particles. The advantages and prospects of each technique are discussed.

From sequences to polynomials and back, via operator orderings

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

Amdeberhan, Tewodros, E-mail: tamdeber@tulane.edu; Dixit, Atul, E-mail: adixit@tulane.edu; Moll, Victor H., E-mail: vhm@tulane.edu

2013-12-15

Bender and Dunne [“Polynomials and operator orderings,” J. Math. Phys. 29, 1727–1731 (1988)] showed that linear combinations of words q{sup k}p{sup n}q{sup n−k}, where p and q are subject to the relation qp − pq = ı, may be expressed as a polynomial in the symbol z=1/2 (qp+pq). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.

Gaussian quadrature for multiple orthogonal polynomials

NASA Astrophysics Data System (ADS)

Coussement, Jonathan; van Assche, Walter

2005-06-01

We study multiple orthogonal polynomials of type I and type II, which have orthogonality conditions with respect to r measures. These polynomials are connected by their recurrence relation of order r+1. First we show a relation with the eigenvalue problem of a banded lower Hessenberg matrix Ln, containing the recurrence coefficients. As a consequence, we easily find that the multiple orthogonal polynomials of type I and type II satisfy a generalized Christoffel-Darboux identity. Furthermore, we explain the notion of multiple Gaussian quadrature (for proper multi-indices), which is an extension of the theory of Gaussian quadrature for orthogonal polynomials and was introduced by Borges. In particular, we show that the quadrature points and quadrature weights can be expressed in terms of the eigenvalue problem of Ln.

A simple approach to quantitative analysis using three-dimensional spectra based on selected Zernike moments.

PubMed

Zhai, Hong Lin; Zhai, Yue Yuan; Li, Pei Zhen; Tian, Yue Li

2013-01-21

A very simple approach to quantitative analysis is proposed based on the technology of digital image processing using three-dimensional (3D) spectra obtained by high-performance liquid chromatography coupled with a diode array detector (HPLC-DAD). As the region-based shape features of a grayscale image, Zernike moments with inherently invariance property were employed to establish the linear quantitative models. This approach was applied to the quantitative analysis of three compounds in mixed samples using 3D HPLC-DAD spectra, and three linear models were obtained, respectively. The correlation coefficients (R(2)) for training and test sets were more than 0.999, and the statistical parameters and strict validation supported the reliability of established models. The analytical results suggest that the Zernike moment selected by stepwise regression can be used in the quantitative analysis of target compounds. Our study provides a new idea for quantitative analysis using 3D spectra, which can be extended to the analysis of other 3D spectra obtained by different methods or instruments.

A combinatorial model for the Macdonald polynomials.

PubMed

Haglund, J

2004-11-16

We introduce a polynomial C(mu)[Z; q, t], depending on a set of variables Z = z(1), z(2),..., a partition mu, and two extra parameters q, t. The definition of C(mu) involves a pair of statistics (maj(sigma, mu), inv(sigma, mu)) on words sigma of positive integers, and the coefficients of the z(i) are manifestly in N[q,t]. We conjecture that C(mu)[Z; q, t] is none other than the modified Macdonald polynomial H(mu)[Z; q, t]. We further introduce a general family of polynomials F(T)[Z; q, S], where T is an arbitrary set of squares in the first quadrant of the xy plane, and S is an arbitrary subset of T. The coefficients of the F(T)[Z; q, S] are in N[q], and C(mu)[Z; q, t] is a sum of certain F(T)[Z; q, S] times nonnegative powers of t. We prove F(T)[Z; q, S] is symmetric in the z(i) and satisfies other properties consistent with the conjecture. We also show how the coefficient of a monomial in F(T)[Z; q, S] can be expressed recursively. maple calculations indicate the F(T)[Z; q, S] are Schur-positive, and we present a combinatorial conjecture for their Schur coefficients when the set T is a partition with at most three columns.

Polynomial meta-models with canonical low-rank approximations: Numerical insights and comparison to sparse polynomial chaos expansions

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

Konakli, Katerina, E-mail: konakli@ibk.baug.ethz.ch; Sudret, Bruno

2016-09-15

The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely themore » exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the

Optical design of a novel instrument that uses the Hartmann-Shack sensor and Zernike polynomials to measure and simulate customized refraction correction surgery outcomes and patient satisfaction

NASA Astrophysics Data System (ADS)

Yasuoka, Fatima M. M.; Matos, Luciana; Cremasco, Antonio; Numajiri, Mirian; Marcato, Rafael; Oliveira, Otavio G.; Sabino, Luis G.; Castro N., Jarbas C.; Bagnato, Vanderlei S.; Carvalho, Luis A. V.

2016-03-01

An optical system that conjugates the patient's pupil to the plane of a Hartmann-Shack (HS) wavefront sensor has been simulated using optical design software. And an optical bench prototype is mounted using mechanical eye device, beam splitter, illumination system, lenses, mirrors, mirrored prism, movable mirror, wavefront sensor and camera CCD. The mechanical eye device is used to simulate aberrations of the eye. From this device the rays are emitted and travelled by the beam splitter to the optical system. Some rays fall on the camera CCD and others pass in the optical system and finally reach the sensor. The eye models based on typical in vivo eye aberrations is constructed using the optical design software Zemax. The computer-aided outcomes of each HS images for each case are acquired, and these images are processed using customized techniques. The simulated and real images for low order aberrations are compared using centroid coordinates to assure that the optical system is constructed precisely in order to match the simulated system. Afterwards a simulated version of retinal images is constructed to show how these typical eyes would perceive an optotype positioned 20 ft away. Certain personalized corrections are allowed by eye doctors based on different Zernike polynomial values and the optical images are rendered to the new parameters. Optical images of how that eye would see with or without corrections of certain aberrations are generated in order to allow which aberrations can be corrected and in which degree. The patient can then "personalize" the correction to their own satisfaction. This new approach to wavefront sensing is a promising change in paradigm towards the betterment of the patient-physician relationship.

The Zernike expansion--an example of a merit function for 2D/3D registration based on orthogonal functions.

PubMed

Dong, Shuo; Kettenbach, Joachim; Hinterleitner, Isabella; Bergmann, Helmar; Birkfellner, Wolfgang

2008-01-01

Current merit functions for 2D/3D registration usually rely on comparing pixels or small regions of images using some sort of statistical measure. Problems connected to this paradigm the sometimes problematic behaviour of the method if noise or artefacts (for instance a guide wire) are present on the projective image. We present a merit function for 2D/3D registration which utilizes the decomposition of the X-ray and the DRR under comparison into orthogonal Zernike moments; the quality of the match is assessed by an iterative comparison of expansion coefficients. Results in a imaging study on a physical phantom show that--compared to standard cross--correlation the Zernike moment based merit function shows better robustness if histogram content in images under comparison is different, and that time expenses are comparable if the merit function is constructed out of a few significant moments only.

Ricci polynomial gravity

NASA Astrophysics Data System (ADS)

Hao, Xin; Zhao, Liu

2017-12-01

We study a novel class of higher-curvature gravity models in n spacetime dimensions which we call Ricci polynomial gravity. The action consists purely of a polynomial in Ricci curvature of order N . In the absence of the second-order terms in the action, the models are ghost free around the Minkowski vacuum. By appropriately choosing the coupling coefficients in front of each term in the action, it is shown that the models can have multiple vacua with different effective cosmological constants, and can be made free of ghost and scalar degrees of freedom around at least one of the maximally symmetric vacua for any n >2 and any N ≥4 . We also discuss some of the physical implications of the existence of multiple vacua in the contexts of black hole physics and cosmology.

Comparison of the Performance of Modal Control Schemes for an Adaptive Optics System and Analysis of the Effect of Actuator Limitations

DTIC Science & Technology

2012-06-01

the open-loop path is established, the feedback system can be treated as a set of SISO feedback loops and a single SISO control law can be applied...Zernike polynomials are commonly referred to by the names, such as focus, coma, astigmatism , and etc. Zernike polynomials can be transformed into

Wavefront analysis from its slope data

NASA Astrophysics Data System (ADS)

Mahajan, Virendra N.; Acosta, Eva

2017-08-01

In the aberration analysis of a wavefront over a certain domain, the polynomials that are orthogonal over and represent balanced wave aberrations for this domain are used. For example, Zernike circle polynomials are used for the analysis of a circular wavefront. Similarly, the annular polynomials are used to analyze the annular wavefronts for systems with annular pupils, as in a rotationally symmetric two-mirror system, such as the Hubble space telescope. However, when the data available for analysis are the slopes of a wavefront, as, for example, in a Shack- Hartmann sensor, we can integrate the slope data to obtain the wavefront data, and then use the orthogonal polynomials to obtain the aberration coefficients. An alternative is to find vector functions that are orthogonal to the gradients of the wavefront polynomials, and obtain the aberration coefficients directly as the inner products of these functions with the slope data. In this paper, we show that an infinite number of vector functions can be obtained in this manner. We show further that the vector functions that are irrotational are unique and propagate minimum uncorrelated additive random noise from the slope data to the aberration coefficients.

Distortion theorems for polynomials on a circle

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

Dubinin, V N

2000-12-31

Inequalities for the derivatives with respect to {phi}=arg z the functions ReP(z), |P(z)|{sup 2} and arg P(z) are established for an algebraic polynomial P(z) at points on the circle |z|=1. These estimates depend, in particular, on the constant term and the leading coefficient of the polynomial P(z) and improve the classical Bernstein and Turan inequalities. The method of proof is based on the techniques of generalized reduced moduli.

Quantum Hurwitz numbers and Macdonald polynomials

NASA Astrophysics Data System (ADS)

Harnad, J.

2016-11-01

Parametric families in the center Z(C[Sn]) of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda τ-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of Sn generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.

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.

Combinatorial theory of Macdonald polynomials I: proof of Haglund's formula.

PubMed

Haglund, J; Haiman, M; Loehr, N

2005-02-22

Haglund recently proposed a combinatorial interpretation of the modified Macdonald polynomials H(mu). We give a combinatorial proof of this conjecture, which establishes the existence and integrality of H(mu). As corollaries, we obtain the cocharge formula of Lascoux and Schutzenberger for Hall-Littlewood polynomials, a formula of Sahi and Knop for Jack's symmetric functions, a generalization of this result to the integral Macdonald polynomials J(mu), a formula for H(mu) in terms of Lascoux-Leclerc-Thibon polynomials, and combinatorial expressions for the Kostka-Macdonald coefficients K(lambda,mu) when mu is a two-column shape.

Translation of Bernstein Coefficients Under an Affine Mapping of the Unit Interval

NASA Technical Reports Server (NTRS)

Alford, John A., II

2012-01-01

We derive an expression connecting the coefficients of a polynomial expanded in the Bernstein basis to the coefficients of an equivalent expansion of the polynomial under an affine mapping of the domain. The expression may be useful in the calculation of bounds for multi-variate polynomials.

Roots of polynomials by ratio of successive derivatives

NASA Technical Reports Server (NTRS)

Crouse, J. E.; Putt, C. W.

1972-01-01

An order of magnitude study of the ratios of successive polynomial derivatives yields information about the number of roots at an approached root point and the approximate location of a root point from a nearby point. The location approximation improves as a root is approached, so a powerful convergence procedure becomes available. These principles are developed into a computer program which finds the roots of polynomials with real number coefficients.

Differential modal Zernike wavefront sensor employing a computer-generated hologram: a proposal.

PubMed

Mishra, Sanjay K; Bhatt, Rahul; Mohan, Devendra; Gupta, Arun Kumar; Sharma, Anurag

2009-11-20

The process of Zernike mode detection with a Shack-Hartmann wavefront sensor is computationally extensive. A holographic modal wavefront sensor has therefore evolved to process the data optically by use of the concept of equal and opposite phase bias. Recently, a multiplexed computer-generated hologram (CGH) technique was developed in which the output is in the form of bright dots that specify the presence and strength of a specific Zernike mode. We propose a wavefront sensor using the concept of phase biasing in the latter technique such that the output is a pair of bright dots for each mode to be sensed. A normalized difference signal between the intensities of the two dots is proportional to the amplitude of the sensed Zernike mode. In our method the number of holograms to be multiplexed is decreased, thereby reducing the modal cross talk significantly. We validated the proposed method through simulation studies for several cases. The simulation results demonstrate simultaneous wavefront detection of lower-order Zernike modes with a resolution better than lambda/50 for the wide measurement range of +/-3.5lambda with much reduced cross talk at high speed.

Distortion theorems for polynomials on a circle

NASA Astrophysics Data System (ADS)

Dubinin, V. N.

2000-12-01

Inequalities for the derivatives with respect to \\varphi=\\arg z the functions \\operatorname{Re}P(z), \\vert P(z)\\vert^2 and \\arg P(z) are established for an algebraic polynomial P(z) at points on the circle \\vert z\\vert=1. These estimates depend, in particular, on the constant term and the leading coefficient of the polynomial P(z) and improve the classical Bernstein and Turan inequalities. The method of proof is based on the techniques of generalized reduced moduli.

Improving Zernike moments comparison for optimal similarity and rotation angle retrieval.

PubMed

Revaud, Jérôme; Lavoué, Guillaume; Baskurt, Atilla

2009-04-01

Zernike moments constitute a powerful shape descriptor in terms of robustness and description capability. However the classical way of comparing two Zernike descriptors only takes into account the magnitude of the moments and loses the phase information. The novelty of our approach is to take advantage of the phase information in the comparison process while still preserving the invariance to rotation. This new Zernike comparator provides a more accurate similarity measure together with the optimal rotation angle between the patterns, while keeping the same complexity as the classical approach. This angle information is particularly of interest for many applications, including 3D scene understanding through images. Experiments demonstrate that our comparator outperforms the classical one in terms of similarity measure. In particular the robustness of the retrieval against noise and geometric deformation is greatly improved. Moreover, the rotation angle estimation is also more accurate than state-of-the-art algorithms.

Inequalities for a polynomial and its derivative

NASA Astrophysics Data System (ADS)

Chanam, Barchand; Dewan, K. K.

2007-12-01

Let , 1[less-than-or-equals, slant][mu][less-than-or-equals, slant]n, be a polynomial of degree n such that p(z)[not equal to]0 in z0, then for 0polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved Equality holds for the polynomial where n is a multiple of [mu]E In this paper, we obtain an improvement of the above inequality by involving some of the coefficients. As an application of our result, we further improve upon a result recently proved by Aziz and Shah [A. Aziz, W.M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 7 (3) (2004) 379-391].

Orthogonal Polynomials Associated with Complementary Chain Sequences

NASA Astrophysics Data System (ADS)

Behera, Kiran Kumar; Sri Ranga, A.; Swaminathan, A.

2016-07-01

Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szegő polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carathéodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed.

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.

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

Estimation of Heat Transfer Coefficient in Squeeze Casting of Magnesium Alloy AM60 by Experimental Polynomial Extrapolation Method

NASA Astrophysics Data System (ADS)

Sun, Zhizhong; Niu, Xiaoping; Hu, Henry

In this work, a different wall-thickness 5-step (with thicknesses as 3, 5, 8, 12, 20 mm) casting mold was designed, and squeeze casting of magnesium alloy AM60 was performed in a hydraulic press. The casting-die interfacial heat transfer coefficients (IHTC) in 5-step casting were determined based on experimental thermal histories data throughout the die and inside the casting which were recorded by fine type-K thermocouples. With measured temperatures, heat flux and IHTC were evaluated using the polynomial curve fitting method. The results show that the wall thickness affects IHTC peak values significantly. The IHTC value for the thick step is higher than that for the thin steps.

Wave-aberration control with a liquid crystal on silicon (LCOS) spatial phase modulator.

PubMed

Fernández, Enrique J; Prieto, Pedro M; Artal, Pablo

2009-06-22

Liquid crystal on Silicon (LCOS) spatial phase modulators offer enhanced possibilities for adaptive optics applications in terms of response velocity and fidelity. Unlike deformable mirrors, they present a capability for reproducing discontinuous phase profiles. This ability also allows an increase in the effective stroke of the device by means of phase wrapping. The latter is only limited by the diffraction related effects that become noticeable as the number of phase cycles increase. In this work we estimated the ranges of generation of the Zernike polynomials as a means for characterizing the performance of the device. Sets of images systematically degraded with the different Zernike polynomials generated using a LCOS phase modulator have been recorded and compared with their theoretical digital counterparts. For each Zernike mode, we have found that image degradation reaches a limit for a certain coefficient value; further increase in the aberration amount has no additional effect in image quality. This behavior is attributed to the intensification of the 0-order diffraction. These results have allowed determining the usable limits of the phase modulator virtually free from diffraction artifacts. The results are particularly important for visual simulation and ophthalmic testing applications, although they are equally interesting for any adaptive optics application with liquid crystal based devices.

A Phase-Shifting Zernike Wavefront Sensor for the Palomar P3K Adaptive Optics System

NASA Technical Reports Server (NTRS)

Wallace, J. Kent; Crawford, Sam; Loya, Frank; Moore, James

2012-01-01

A phase-shifting Zernike wavefront sensor has distinct advantages over other types of wavefront sensors. Chief among them are: 1) improved sensitivity to low-order aberrations and 2) efficient use of photons (hence reduced sensitivity to photon noise). We are in the process of deploying a phase-shifting Zernike wavefront sensor to be used with the realtime adaptive optics system for Palomar. Here we present the current state of the Zernike wavefront sensor to be integrated into the high-order adaptive optics system at Mount Palomar's Hale Telescope.

Detection of Copy-Rotate-Move Forgery Using Zernike Moments

NASA Astrophysics Data System (ADS)

Ryu, Seung-Jin; Lee, Min-Jeong; Lee, Heung-Kyu

As forgeries have become popular, the importance of forgery detection is much increased. Copy-move forgery, one of the most commonly used methods, copies a part of the image and pastes it into another part of the the image. In this paper, we propose a detection method of copy-move forgery that localizes duplicated regions using Zernike moments. Since the magnitude of Zernike moments is algebraically invariant against rotation, the proposed method can detect a forged region even though it is rotated. Our scheme is also resilient to the intentional distortions such as additive white Gaussian noise, JPEG compression, and blurring. Experimental results demonstrate that the proposed scheme is appropriate to identify the forged region by copy-rotate-move forgery.

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.

Orthogonal sets of data windows constructed from trigonometric polynomials

NASA Technical Reports Server (NTRS)

Greenhall, C. A.

1989-01-01

Suboptimal, easily computable substitutes for the discrete prolate-spheroidal windows used by Thomson for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated.

On the derivatives of unimodular polynomials

NASA Astrophysics Data System (ADS)

Nevai, P.; Erdélyi, T.

2016-04-01

Let D be the open unit disk of the complex plane; its boundary, the unit circle of the complex plane, is denoted by \\partial D. Let \\mathscr P_n^c denote the set of all algebraic polynomials of degree at most n with complex coefficients. For λ ≥ 0, let {\\mathscr K}_n^λ \\stackrel{{def}}{=} \\biggl\\{P_n: P_n(z) = \\sumk=0^n{ak k^λ z^k}, ak \\in { C}, |a_k| = 1 \\biggr\\} \\subset {\\mathscr P}_n^c.The class \\mathscr K_n^0 is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (\\varepsilon_n) of positive numbers tending to 0, we say that a sequence (P_n) of polynomials P_n\\in\\mathscr K_n^λ is \\{λ, (\\varepsilon_n)\\}-ultraflat if \\displaystyle (1-\\varepsilon_n)\\frac{nλ+1/2}{\\sqrt{2λ+1}}≤\\ve......a +1/2}}{\\sqrt{2λ +1}},\\qquad z \\in \\partial D,\\quad n\\in N_0.Although we do not know, in general, whether or not \\{λ, (\\varepsilon_n)\\}-ultraflat sequences of polynomials P_n\\in\\mathscr K_n^λ exist for each fixed λ>0, we make an effort to prove various interesting properties of them. These allow us to conclude that there are no sequences (P_n) of either conjugate, or plain, or skew reciprocal unimodular polynomials P_n\\in\\mathscr K_n^0 such that (Q_n) with Q_n(z)\\stackrel{{def}}{=} zP_n'(z)+1 is a \\{1,(\\varepsilon_n)\\}-ultraflat sequence of polynomials.Bibliography: 18 titles.

A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials

NASA Astrophysics Data System (ADS)

Deift, P.; Kriecherbauer, T.; McLaughlin, K. T.-R.; Venakides, S.; Zhou, X.

2001-08-01

A few years ago the authors introduced a new approach to study asymptotic questions for orthogonal polynomials. In this paper we give an overview of our method and review the results which have been obtained in Deift et al. (Internat. Math. Res. Notices (1997) 759, Comm. Pure Appl. Math. 52 (1999) 1491, 1335), Deift (Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, Vol. 3, New York University, 1999), Kriecherbauer and McLaughlin (Internat. Math. Res. Notices (1999) 299) and Baik et al. (J. Amer. Math. Soc. 12 (1999) 1119). We mainly consider orthogonal polynomials with respect to weights on the real line which are either (1) Freud-type weights d[alpha](x)=e-Q(x) dx (Q polynomial or Q(x)=x[beta], [beta]>0), or (2) varying weights d[alpha]n(x)=e-nV(x) dx (V analytic, limx-->[infinity] V(x)/logx=[infinity]). We obtain Plancherel-Rotach-type asymptotics in the entire complex plane as well as asymptotic formulae with error estimates for the leading coefficients, for the recurrence coefficients, and for the zeros of the orthogonal polynomials. Our proof starts from an observation of Fokas et al. (Comm. Math. Phys. 142 (1991) 313) that the orthogonal polynomials can be determined as solutions of certain matrix valued Riemann-Hilbert problems. We analyze the Riemann-Hilbert problems by a steepest descent type method introduced by Deift and Zhou (Ann. Math. 137 (1993) 295) and further developed in Deift and Zhou (Comm. Pure Appl. Math. 48 (1995) 277) and Deift et al. (Proc. Nat. Acad. Sci. USA 95 (1998) 450). A crucial step in our analysis is the use of the well-known equilibrium measure which describes the asymptotic distribution of the zeros of the orthogonal polynomials.

Influence of each Zernike aberration on the propagation of laser beams through atmospheric turbulence

NASA Astrophysics Data System (ADS)

Azarian, Adrian; Gladysz, Szymon

2014-10-01

We study the influence of each Zernike mode on the propagation of a laser beam through the atmosphere by two different numerical methods. In the first method, an idealized adaptive optics system is modeled to subtract a certain number of Zernike modes from the beam. The effect of each aberration is quantified using the Strehl ratio of the longterm exposure in target/receiver plane. In the second method, the strength of each Zernike mode is varied using a numerical space-filling design during the generation of the phase screens. The resulting central intensity for each point of the design is then studied by a linear discriminant analysis, which yields to the importance of each Zernike mode. The results of the two methods are consistent. They indicate that, for a focused Gaussian beam and for certain geometries and turbulence strengths, the hypothesis of diminishing gains with correction of each new mode is not true. For such cases, we observe jumps in the calculated criteria, which indicate an increased importance of some particular modes, especially coma. The implications of these results for the design of adaptive optics systems are discussed.

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.

2D/3D registration using a rotation-invariant cost function based on Zernike moments

NASA Astrophysics Data System (ADS)

Birkfellner, Wolfgang; Yang, Xinhui; Burgstaller, Wolfgang; Baumann, Bernard; Jacob, Augustinus L.; Niederer, Peter F.; Regazzoni, Pietro; Messmer, Peter

2004-05-01

We present a novel in-plane rotation invariant cost function for 2D/3D registration utilizing projection-invariant transformation properties and the decomposition of the X-ray nad the DRR under comparision into orhogonal Zernike moments. As a result, only five dof have to be optimized, and the number of iteration necessary for registration can be significantly reduced. Results in a phantom study show that an accuracy of approximately 0.7° and 2 mm can be achieved using this method. We conclude that reduction of coupled dof and usage of linear independent coefficients for cost function evaluation provide intersting new perspectives for the field of 2D/3D registration.

Range image segmentation using Zernike moment-based generalized edge detector

NASA Technical Reports Server (NTRS)

Ghosal, S.; Mehrotra, R.

1992-01-01

The authors proposed a novel Zernike moment-based generalized step edge detection method which can be used for segmenting range and intensity images. A generalized step edge detector is developed to identify different kinds of edges in range images. These edge maps are thinned and linked to provide final segmentation. A generalized edge is modeled in terms of five parameters: orientation, two slopes, one step jump at the location of the edge, and the background gray level. Two complex and two real Zernike moment-based masks are required to determine all these parameters of the edge model. Theoretical noise analysis is performed to show that these operators are quite noise tolerant. Experimental results are included to demonstrate edge-based segmentation technique.

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

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.

Corneal aberrations in keratoconic eyes: influence of pupil size and centering

NASA Astrophysics Data System (ADS)

Comastri, S. A.; Perez, L. I.; Pérez, G. D.; Martin, G.; Bianchetti, A.

2011-01-01

Ocular aberrations vary among subjects and under different conditions and are commonly analyzed expanding the wavefront aberration function in Zernike polynomials. In previous articles, explicit analytical formulas to transform Zernike coefficients of up to 7th order corresponding to an original pupil into those related to a contracted displaced new pupil are obtained. In the present paper these formulas are applied to 20 keratoconic corneas of varying severity. Employing the SN CT1000 topographer, aberrations of the anterior corneal surface for a pupil of semi-diameter 3 mm centered on the keratometric axis are evaluated, the relation between the higher-order root mean square wavefront error and the index KISA% characterizing keratoconus is studied and the size and centering of the ocular photopic natural pupil are determined. Using these data and the transformation formulas, new coefficients associated to the photopic pupil size are computed and their variation when coordinates origin is shifted from the keratometric axis to the ocular pupil centre is analyzed.

Robust source and mask optimization compensating for mask topography effects in computational lithography.

PubMed

Li, Jia; Lam, Edmund Y

2014-04-21

Mask topography effects need to be taken into consideration for a more accurate solution of source mask optimization (SMO) in advanced optical lithography. However, rigorous 3D mask models generally involve intensive computation and conventional SMO fails to manipulate the mask-induced undesired phase errors that degrade the usable depth of focus (uDOF) and process yield. In this work, an optimization approach incorporating pupil wavefront aberrations into SMO procedure is developed as an alternative to maximize the uDOF. We first design the pupil wavefront function by adding primary and secondary spherical aberrations through the coefficients of the Zernike polynomials, and then apply the conjugate gradient method to achieve an optimal source-mask pair under the condition of aberrated pupil. We also use a statistical model to determine the Zernike coefficients for the phase control and adjustment. Rigorous simulations of thick masks show that this approach provides compensation for mask topography effects by improving the pattern fidelity and increasing uDOF.

DIFFERENTIAL CROSS SECTION ANALYSIS IN KAON PHOTOPRODUCTION USING ASSOCIATED LEGENDRE POLYNOMIALS

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

P. T. P. HUTAURUK, D. G. IRELAND, G. ROSNER

2009-04-01

Angular distributions of differential cross sections from the latest CLAS data sets,6 for the reaction γ + p→K+ + Λ have been analyzed using associated Legendre polynomials. This analysis is based upon theoretical calculations in Ref. 1 where all sixteen observables in kaon photoproduction can be classified into four Legendre classes. Each observable can be described by an expansion of associated Legendre polynomial functions. One of the questions to be addressed is how many associated Legendre polynomials are required to describe the data. In this preliminary analysis, we used data models with different numbers of associated Legendre polynomials. We thenmore » compared these models by calculating posterior probabilities of the models. We found that the CLAS data set needs no more than four associated Legendre polynomials to describe the differential cross section data. In addition, we also show the extracted coefficients of the best model.« less

A Polarimetric Extension of the van Cittert-Zernike Theorem for Use with Microwave Interferometers

NASA Technical Reports Server (NTRS)

Piepmeier, J. R.; Simon, N. K.

2004-01-01

The van Cittert-Zernike theorem describes the Fourier-transform relationship between an extended source and its visibility function. Developments in classical optics texts use scalar field formulations for the theorem. Here, we develop a polarimetric extension to the van Cittert-Zernike theorem with applications to passive microwave Earth remote sensing. The development provides insight into the mechanics of two-dimensional interferometric imaging, particularly the effects of polarization basis differences between the scene and the observer.

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.

Cubic Polynomials, Their Roots and the Perron-Frobenius Theorem

ERIC Educational Resources Information Center

Dealba, Luz Maria

2002-01-01

In this note several cubic polynomials and their roots are examined, in particular, how these roots move as some of the coefficients are modified. The results obtained are applied to eigenvalues of matrices. (Contains 8 figures and 1 footnote.)

Uniform versus Gaussian Beams: A Comparison of the Effects of Diffraction, Obscuration, and Aberations.

DTIC Science & Technology

1985-12-16

balancing is discussed for the two types of beams. Zernike polynomials representing balanced primary aberration for uniform and Gaussian annular beams...plotted on a logarithmic scale (Figs. 3c and 3d ). The positions of maxima and minima and the correspond- ing irradiance and encircled-power values are...aberration 2 4 (representing a term in the expansion of the aberration in terms of a set of " Zernike " polynomials which are orthonormal over the amplitude

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.

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…

Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes

DOE PAGES

Zlotnikov, Michael

2016-08-24

We develop a polynomial reduction procedure that transforms any gauge fixed CHY amplitude integrand for n scattering particles into a σ-moduli multivariate polynomial of what we call the standard form. We show that a standard form polynomial must have a specific ladder type monomial structure, which has finite size at any n, with highest multivariate degree given by (n – 3)(n – 4)/2. This set of monomials spans a complete basis for polynomials with rational coefficients in kinematic data on the support of scattering equations. Subsequently, at tree and one-loop level, we employ the global residue theorem to derive amore » prescription that evaluates any CHY amplitude by means of collecting simple residues at infinity only. Furthermore, the prescription is then applied explicitly to some tree and one-loop amplitude examples.« 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

Partial-fraction expansion and inverse Laplace transform of a rational function with real coefficients

NASA Technical Reports Server (NTRS)

Chang, F.-C.; Mott, H.

1974-01-01

This paper presents a technique for the partial-fraction expansion of functions which are ratios of polynomials with real coefficients. The expansion coefficients are determined by writing the polynomials as Taylor's series and obtaining the Laurent series expansion of the function. The general formula for the inverse Laplace transform is also derived.

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.

Measurement of EUV lithography pupil amplitude and phase variation via image-based methodology

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

Levinson, Zachary; Verduijn, Erik; Wood, Obert R.

2016-04-01

Here, an approach to image-based EUV aberration metrology using binary mask targets and iterative model-based solutions to extract both the amplitude and phase components of the aberrated pupil function is presented. The approach is enabled through previously developed modeling, fitting, and extraction algorithms. We seek to examine the behavior of pupil amplitude variation in real-optical systems. Optimized target images were captured under several conditions to fit the resulting pupil responses. Both the amplitude and phase components of the pupil function were extracted from a zone-plate-based EUV mask microscope. The pupil amplitude variation was expanded in three different bases: Zernike polynomials,more » Legendre polynomials, and Hermite polynomials. It was found that the Zernike polynomials describe pupil amplitude variation most effectively of the three.« less

Computer-aided diagnosis of malignant mammograms using Zernike moments and SVM.

PubMed

Sharma, Shubhi; Khanna, Pritee

2015-02-01

This work is directed toward the development of a computer-aided diagnosis (CAD) system to detect abnormalities or suspicious areas in digital mammograms and classify them as malignant or nonmalignant. Original mammogram is preprocessed to separate the breast region from its background. To work on the suspicious area of the breast, region of interest (ROI) patches of a fixed size of 128×128 are extracted from the original large-sized digital mammograms. For training, patches are extracted manually from a preprocessed mammogram. For testing, patches are extracted from a highly dense area identified by clustering technique. For all extracted patches corresponding to a mammogram, Zernike moments of different orders are computed and stored as a feature vector. A support vector machine (SVM) is used to classify extracted ROI patches. The experimental study shows that the use of Zernike moments with order 20 and SVM classifier gives better results among other studies. The proposed system is tested on Image Retrieval In Medical Application (IRMA) reference dataset and Digital Database for Screening Mammography (DDSM) mammogram database. On IRMA reference dataset, it attains 99% sensitivity and 99% specificity, and on DDSM mammogram database, it obtained 97% sensitivity and 96% specificity. To verify the applicability of Zernike moments as a fitting texture descriptor, the performance of the proposed CAD system is compared with the other well-known texture descriptors namely gray-level co-occurrence matrix (GLCM) and discrete cosine transform (DCT).

pyZELDA: Python code for Zernike wavefront sensors

NASA Astrophysics Data System (ADS)

Vigan, A.; N'Diaye, M.

2018-06-01

pyZELDA analyzes data from Zernike wavefront sensors dedicated to high-contrast imaging applications. This modular software was originally designed to analyze data from the ZELDA wavefront sensor prototype installed in VLT/SPHERE; simple configuration files allow it to be extended to support several other instruments and testbeds. pyZELDA also includes simple simulation tools to measure the theoretical sensitivity of a sensor and to compare it to other sensors.

Combined invariants to similarity transformation and to blur using orthogonal Zernike moments

PubMed Central

Beijing, Chen; Shu, Huazhong; Zhang, Hui; Coatrieux, Gouenou; Luo, Limin; Coatrieux, Jean-Louis

2011-01-01

The derivation of moment invariants has been extensively investigated in the past decades. In this paper, we construct a set of invariants derived from Zernike moments which is simultaneously invariant to similarity transformation and to convolution with circularly symmetric point spread function (PSF). Two main contributions are provided: the theoretical framework for deriving the Zernike moments of a blurred image and the way to construct the combined geometric-blur invariants. The performance of the proposed descriptors is evaluated with various PSFs and similarity transformations. The comparison of the proposed method with the existing ones is also provided in terms of pattern recognition accuracy, template matching and robustness to noise. Experimental results show that the proposed descriptors perform on the overall better. PMID:20679028

Automated Decision Tree Classification of Corneal Shape

PubMed Central

Twa, Michael D.; Parthasarathy, Srinivasan; Roberts, Cynthia; Mahmoud, Ashraf M.; Raasch, Thomas W.; Bullimore, Mark A.

2011-01-01

Purpose The volume and complexity of data produced during videokeratography examinations present a challenge of interpretation. As a consequence, results are often analyzed qualitatively by subjective pattern recognition or reduced to comparisons of summary indices. We describe the application of decision tree induction, an automated machine learning classification method, to discriminate between normal and keratoconic corneal shapes in an objective and quantitative way. We then compared this method with other known classification methods. Methods The corneal surface was modeled with a seventh-order Zernike polynomial for 132 normal eyes of 92 subjects and 112 eyes of 71 subjects diagnosed with keratoconus. A decision tree classifier was induced using the C4.5 algorithm, and its classification performance was compared with the modified Rabinowitz–McDonnell index, Schwiegerling’s Z3 index (Z3), Keratoconus Prediction Index (KPI), KISA%, and Cone Location and Magnitude Index using recommended classification thresholds for each method. We also evaluated the area under the receiver operator characteristic (ROC) curve for each classification method. Results Our decision tree classifier performed equal to or better than the other classifiers tested: accuracy was 92% and the area under the ROC curve was 0.97. Our decision tree classifier reduced the information needed to distinguish between normal and keratoconus eyes using four of 36 Zernike polynomial coefficients. The four surface features selected as classification attributes by the decision tree method were inferior elevation, greater sagittal depth, oblique toricity, and trefoil. Conclusions Automated decision tree classification of corneal shape through Zernike polynomials is an accurate quantitative method of classification that is interpretable and can be generated from any instrument platform capable of raw elevation data output. This method of pattern classification is extendable to other classification

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.

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.

Molecular surface representation using 3D Zernike descriptors for protein shape comparison and docking.

PubMed

Kihara, Daisuke; Sael, Lee; Chikhi, Rayan; Esquivel-Rodriguez, Juan

2011-09-01

The tertiary structures of proteins have been solved in an increasing pace in recent years. To capitalize the enormous efforts paid for accumulating the structure data, efficient and effective computational methods need to be developed for comparing, searching, and investigating interactions of protein structures. We introduce the 3D Zernike descriptor (3DZD), an emerging technique to describe molecular surfaces. The 3DZD is a series expansion of mathematical three-dimensional function, and thus a tertiary structure is represented compactly by a vector of coefficients of terms in the series. A strong advantage of the 3DZD is that it is invariant to rotation of target object to be represented. These two characteristics of the 3DZD allow rapid comparison of surface shapes, which is sufficient for real-time structure database screening. In this article, we review various applications of the 3DZD, which have been recently proposed.

Information entropy of Gegenbauer polynomials and Gaussian quadrature

NASA Astrophysics Data System (ADS)

Sánchez-Ruiz, Jorge

2003-05-01

In a recent paper (Buyarov V S, López-Artés P, Martínez-Finkelshtein A and Van Assche W 2000 J. Phys. A: Math. Gen. 33 6549-60), an efficient method was provided for evaluating in closed form the information entropy of the Gegenbauer polynomials C(lambda)n(x) in the case when lambda = l in Bbb N. For given values of n and l, this method requires the computation by means of recurrence relations of two auxiliary polynomials, P(x) and H(x), of degrees 2l - 2 and 2l - 4, respectively. Here it is shown that P(x) is related to the coefficients of the Gaussian quadrature formula for the Gegenbauer weights wl(x) = (1 - x2)l-1/2, and this fact is used to obtain the explicit expression of P(x). From this result, an explicit formula is also given for the polynomial S(x) = limnrightarrowinfty P(1 - x/(2n2)), which is relevant to the study of the asymptotic (n rightarrow infty with l fixed) behaviour of the entropy.

Tsallis p, q-deformed Touchard polynomials and Stirling numbers

NASA Astrophysics Data System (ADS)

Herscovici, O.; Mansour, T.

2017-01-01

In this paper, we develop and investigate a new two-parametrized deformation of the Touchard polynomials, based on the definition of the NEXT q-exponential function of Tsallis. We obtain new generalizations of the Stirling numbers of the second kind and of the binomial coefficients and represent two new statistics for the set partitions.

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.

Explicit analytical expression for the condition number of polynomials in power form

NASA Astrophysics Data System (ADS)

Rack, Heinz-Joachim

2017-07-01

In his influential papers [1-3] W. Gautschi has defined and reshaped the condition number κ∞ of polynomials Pn of degree ≤ n which are represented in power form on a zero-symmetric interval [-ω, ω]. Basically, κ∞ is expressed as the product of two operator norms: an explicit factor times an implicit one (the l∞-norm of the coefficient vector of the n-th Chebyshev polynomial of the first kind relative to [-ω, ω]). We provide a new proof, economize the second factor and express it by an explicit analytical formula.

CLASSICAL AREAS OF PHENOMENOLOGY: Study on the design and Zernike aberrations of a segmented mirror telescope

NASA Astrophysics Data System (ADS)

Jiang, Zhen-Yu; Li, Lin; Huang, Yi-Fan

2009-07-01

The segmented mirror telescope is widely used. The aberrations of segmented mirror systems are different from single mirror systems. This paper uses the Fourier optics theory to analyse the Zernike aberrations of segmented mirror systems. It concludes that the Zernike aberrations of segmented mirror systems obey the linearity theorem. The design of a segmented space telescope and segmented schemes are discussed, and its optical model is constructed. The computer simulation experiment is performed with this optical model to verify the suppositions. The experimental results confirm the correctness of the model.

Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere

NASA Astrophysics Data System (ADS)

Miller, W., Jr.; Li, Q.

2015-04-01

The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L2 of H in terms of an eigenbasis of another symmetry operator L1, but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions.

Coupling coefficients for tensor product representations of quantum SU(2)

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

Groenevelt, Wolter, E-mail: w.g.m.groenevelt@tudelft.nl

2014-10-15

We study tensor products of infinite dimensional irreducible {sup *}-representations (not corepresentations) of the SU(2) quantum group. We obtain (generalized) eigenvectors of certain self-adjoint elements using spectral analysis of Jacobi operators associated to well-known q-hypergeometric orthogonal polynomials. We also compute coupling coefficients between different eigenvectors corresponding to the same eigenvalue. Since the continuous spectrum has multiplicity two, the corresponding coupling coefficients can be considered as 2 × 2-matrix-valued orthogonal functions. We compute explicitly the matrix elements of these functions. The coupling coefficients can be considered as q-analogs of Bessel functions. As a results we obtain several q-integral identities involving q-hypergeometricmore » orthogonal polynomials and q-Bessel-type functions.« less

On Generalized Continuous D Semi-Classical Hermite and Chebychev Orthogonal Polynomials of Class One

NASA Astrophysics Data System (ADS)

Azatassou, E.; Hounkonnou, M. N.

2002-10-01

In this contribution, starting from the system of equations for recurrence coefficients generated by continuous D semi-classical Laguerre-Freud equations of class 1, we deduce the β constant recurrence relation coefficient γn leading to the generalized D semi-classical Hermite and Chebychev orthogonal polynomials of class 1. Various interesting cases are pointed out.

Large-field high-contrast hard x-ray Zernike phase-contrast nano-imaging beamline at Pohang Light Source.

PubMed

Lim, Jun; Park, So Yeong; Huang, Jung Yun; Han, Sung Mi; Kim, Hong-Tae

2013-01-01

We developed an off-axis-illuminated zone-plate-based hard x-ray Zernike phase-contrast microscope beamline at Pohang Light Source. Owing to condenser optics-free and off-axis illumination, a large field of view was achieved. The pinhole-type Zernike phase plate affords high-contrast images of a cell with minimal artifacts such as the shade-off and halo effects. The setup, including the optics and the alignment, is simple and easy, and allows faster and easier imaging of large bio-samples.

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

NASA Technical Reports Server (NTRS)

Coutsias, Evangelos A.; Torres, David; Hagstrom, Thomas

1994-01-01

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

Spatial modeling and classification of corneal shape.

PubMed

Marsolo, Keith; Twa, Michael; Bullimore, Mark A; Parthasarathy, Srinivasan

2007-03-01

One of the most promising applications of data mining is in biomedical data used in patient diagnosis. Any method of data analysis intended to support the clinical decision-making process should meet several criteria: it should capture clinically relevant features, be computationally feasible, and provide easily interpretable results. In an initial study, we examined the feasibility of using Zernike polynomials to represent biomedical instrument data in conjunction with a decision tree classifier to distinguish between the diseased and non-diseased eyes. Here, we provide a comprehensive follow-up to that work, examining a second representation, pseudo-Zernike polynomials, to determine whether they provide any increase in classification accuracy. We compare the fidelity of both methods using residual root-mean-square (rms) error and evaluate accuracy using several classifiers: neural networks, C4.5 decision trees, Voting Feature Intervals, and Naïve Bayes. We also examine the effect of several meta-learning strategies: boosting, bagging, and Random Forests (RFs). We present results comparing accuracy as it relates to dataset and transformation resolution over a larger, more challenging, multi-class dataset. They show that classification accuracy is similar for both data transformations, but differs by classifier. We find that the Zernike polynomials provide better feature representation than the pseudo-Zernikes and that the decision trees yield the best balance of classification accuracy and interpretability.

New realisation of Preisach model using adaptive polynomial approximation

NASA Astrophysics Data System (ADS)

Liu, Van-Tsai; Lin, Chun-Liang; Wing, Home-Young

2012-09-01

Modelling system with hysteresis has received considerable attention recently due to the increasing accurate requirement in engineering applications. The classical Preisach model (CPM) is the most popular model to demonstrate hysteresis which can be represented by infinite but countable first-order reversal curves (FORCs). The usage of look-up tables is one way to approach the CPM in actual practice. The data in those tables correspond with the samples of a finite number of FORCs. This approach, however, faces two major problems: firstly, it requires a large amount of memory space to obtain an accurate prediction of hysteresis; secondly, it is difficult to derive efficient ways to modify the data table to reflect the timing effect of elements with hysteresis. To overcome, this article proposes the idea of using a set of polynomials to emulate the CPM instead of table look-up. The polynomial approximation requires less memory space for data storage. Furthermore, the polynomial coefficients can be obtained accurately by using the least-square approximation or adaptive identification algorithm, such as the possibility of accurate tracking of hysteresis model parameters.

Polynomial Chaos Based Acoustic Uncertainty Predictions from Ocean Forecast Ensembles

NASA Astrophysics Data System (ADS)

Dennis, S.

2016-02-01

Most significant ocean acoustic propagation occurs at tens of kilometers, at scales small compared basin and to most fine scale ocean modeling. To address the increased emphasis on uncertainty quantification, for example transmission loss (TL) probability density functions (PDF) within some radius, a polynomial chaos (PC) based method is utilized. In order to capture uncertainty in ocean modeling, Navy Coastal Ocean Model (NCOM) now includes ensembles distributed to reflect the ocean analysis statistics. Since the ensembles are included in the data assimilation for the new forecast ensembles, the acoustic modeling uses the ensemble predictions in a similar fashion for creating sound speed distribution over an acoustically relevant domain. Within an acoustic domain, singular value decomposition over the combined time-space structure of the sound speeds can be used to create Karhunen-Loève expansions of sound speed, subject to multivariate normality testing. These sound speed expansions serve as a basis for Hermite polynomial chaos expansions of derived quantities, in particular TL. The PC expansion coefficients result from so-called non-intrusive methods, involving evaluation of TL at multi-dimensional Gauss-Hermite quadrature collocation points. Traditional TL calculation from standard acoustic propagation modeling could be prohibitively time consuming at all multi-dimensional collocation points. This method employs Smolyak order and gridding methods to allow adaptive sub-sampling of the collocation points to determine only the most significant PC expansion coefficients to within a preset tolerance. Practically, the Smolyak order and grid sizes grow only polynomially in the number of Karhunen-Loève terms, alleviating the curse of dimensionality. The resulting TL PC coefficients allow the determination of TL PDF normality and its mean and standard deviation. In the non-normal case, PC Monte Carlo methods are used to rapidly establish the PDF. This work was

Genetic parameters of legendre polynomials for first parity lactation curves.

PubMed

Pool, M H; Janss, L L; Meuwissen, T H

2000-11-01

Variance components of the covariance function coefficients in a random regression test-day model were estimated by Legendre polynomials up to a fifth order for first-parity records of Dutch dairy cows using Gibbs sampling. Two Legendre polynomials of equal order were used to model the random part of the lactation curve, one for the genetic component and one for permanent environment. Test-day records from cows registered between 1990 to 1996 and collected by regular milk recording were available. For the data set, 23,700 complete lactations were selected from 475 herds sired by 262 sires. Because the application of a random regression model is limited by computing capacity, we investigated the minimum order needed to fit the variance structure in the data sufficiently. Predictions of genetic and permanent environmental variance structures were compared with bivariate estimates on 30-d intervals. A third-order or higher polynomial modeled the shape of variance curves over DIM with sufficient accuracy for the genetic and permanent environment part. Also, the genetic correlation structure was fitted with sufficient accuracy by a third-order polynomial, but, for the permanent environmental component, a fourth order was needed. Because equal orders are suggested in the literature, a fourth-order Legendre polynomial is recommended in this study. However, a rank of three for the genetic covariance matrix and of four for permanent environment allows a simpler covariance function with a reduced number of parameters based on the eigenvalues and eigenvectors.

Protein-protein docking using region-based 3D Zernike descriptors

PubMed Central

2009-01-01

Background Protein-protein interactions are a pivotal component of many biological processes and mediate a variety of functions. Knowing the tertiary structure of a protein complex is therefore essential for understanding the interaction mechanism. However, experimental techniques to solve the structure of the complex are often found to be difficult. To this end, computational protein-protein docking approaches can provide a useful alternative to address this issue. Prediction of docking conformations relies on methods that effectively capture shape features of the participating proteins while giving due consideration to conformational changes that may occur. Results We present a novel protein docking algorithm based on the use of 3D Zernike descriptors as regional features of molecular shape. The key motivation of using these descriptors is their invariance to transformation, in addition to a compact representation of local surface shape characteristics. Docking decoys are generated using geometric hashing, which are then ranked by a scoring function that incorporates a buried surface area and a novel geometric complementarity term based on normals associated with the 3D Zernike shape description. Our docking algorithm was tested on both bound and unbound cases in the ZDOCK benchmark 2.0 dataset. In 74% of the bound docking predictions, our method was able to find a near-native solution (interface C-αRMSD ≤ 2.5 Å) within the top 1000 ranks. For unbound docking, among the 60 complexes for which our algorithm returned at least one hit, 60% of the cases were ranked within the top 2000. Comparison with existing shape-based docking algorithms shows that our method has a better performance than the others in unbound docking while remaining competitive for bound docking cases. Conclusion We show for the first time that the 3D Zernike descriptors are adept in capturing shape complementarity at the protein-protein interface and useful for protein docking prediction

Protein-protein docking using region-based 3D Zernike descriptors.

PubMed

Venkatraman, Vishwesh; Yang, Yifeng D; Sael, Lee; Kihara, Daisuke

2009-12-09

Protein-protein interactions are a pivotal component of many biological processes and mediate a variety of functions. Knowing the tertiary structure of a protein complex is therefore essential for understanding the interaction mechanism. However, experimental techniques to solve the structure of the complex are often found to be difficult. To this end, computational protein-protein docking approaches can provide a useful alternative to address this issue. Prediction of docking conformations relies on methods that effectively capture shape features of the participating proteins while giving due consideration to conformational changes that may occur. We present a novel protein docking algorithm based on the use of 3D Zernike descriptors as regional features of molecular shape. The key motivation of using these descriptors is their invariance to transformation, in addition to a compact representation of local surface shape characteristics. Docking decoys are generated using geometric hashing, which are then ranked by a scoring function that incorporates a buried surface area and a novel geometric complementarity term based on normals associated with the 3D Zernike shape description. Our docking algorithm was tested on both bound and unbound cases in the ZDOCK benchmark 2.0 dataset. In 74% of the bound docking predictions, our method was able to find a near-native solution (interface C-alphaRMSD < or = 2.5 A) within the top 1000 ranks. For unbound docking, among the 60 complexes for which our algorithm returned at least one hit, 60% of the cases were ranked within the top 2000. Comparison with existing shape-based docking algorithms shows that our method has a better performance than the others in unbound docking while remaining competitive for bound docking cases. We show for the first time that the 3D Zernike descriptors are adept in capturing shape complementarity at the protein-protein interface and useful for protein docking prediction. Rigorous benchmark studies

Zernike phase-contrast electron cryotomography applied to marine cyanobacteria infected with cyanophages.

PubMed

Dai, Wei; Fu, Caroline; Khant, Htet A; Ludtke, Steven J; Schmid, Michael F; Chiu, Wah

2014-11-01

Advances in electron cryotomography have provided new opportunities to visualize the internal 3D structures of a bacterium. An electron microscope equipped with Zernike phase-contrast optics produces images with markedly increased contrast compared with images obtained by conventional electron microscopy. Here we describe a protocol to apply Zernike phase plate technology for acquiring electron tomographic tilt series of cyanophage-infected cyanobacterial cells embedded in ice, without staining or chemical fixation. We detail the procedures for aligning and assessing phase plates for data collection, and methods for obtaining 3D structures of cyanophage assembly intermediates in the host by subtomogram alignment, classification and averaging. Acquiring three or four tomographic tilt series takes ∼12 h on a JEM2200FS electron microscope. We expect this time requirement to decrease substantially as the technique matures. The time required for annotation and subtomogram averaging varies widely depending on the project goals and data volume.

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

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

Hampton, Jerrad; Doostan, Alireza, E-mail: alireza.doostan@colorado.edu

2015-01-01

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

Long-time uncertainty propagation using generalized polynomial chaos and flow map composition

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

Luchtenburg, Dirk M., E-mail: dluchten@cooper.edu; Brunton, Steven L.; Rowley, Clarence W.

2014-10-01

We present an efficient and accurate method for long-time uncertainty propagation in dynamical systems. Uncertain initial conditions and parameters are both addressed. The method approximates the intermediate short-time flow maps by spectral polynomial bases, as in the generalized polynomial chaos (gPC) method, and uses flow map composition to construct the long-time flow map. In contrast to the gPC method, this approach has spectral error convergence for both short and long integration times. The short-time flow map is characterized by small stretching and folding of the associated trajectories and hence can be well represented by a relatively low-degree basis. The compositionmore » of these low-degree polynomial bases then accurately describes the uncertainty behavior for long integration times. The key to the method is that the degree of the resulting polynomial approximation increases exponentially in the number of time intervals, while the number of polynomial coefficients either remains constant (for an autonomous system) or increases linearly in the number of time intervals (for a non-autonomous system). The findings are illustrated on several numerical examples including a nonlinear ordinary differential equation (ODE) with an uncertain initial condition, a linear ODE with an uncertain model parameter, and a two-dimensional, non-autonomous double gyre flow.« 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.

Umbral orthogonal polynomials

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

Lopez-Sendino, J. E.; del Olmo, M. A.

2010-12-23

We present an umbral operator version of the classical orthogonal polynomials. We obtain three families which are the umbral counterpart of the Jacobi, Laguerre and Hermite polynomials in the classical case.

Image based method for aberration measurement of lithographic tools

NASA Astrophysics Data System (ADS)

Xu, Shuang; Tao, Bo; Guo, Yongxing; Li, Gongfa

2018-01-01

Information of lens aberration of lithographic tools is important as it directly affects the intensity distribution in the image plane. Zernike polynomials are commonly used for a mathematical description of lens aberrations. Due to the advantage of lower cost and easier implementation of tools, image based measurement techniques have been widely used. Lithographic tools are typically partially coherent systems that can be described by a bilinear model, which entails time consuming calculations and does not lend a simple and intuitive relationship between lens aberrations and the resulted images. Previous methods for retrieving lens aberrations in such partially coherent systems involve through-focus image measurements and time-consuming iterative algorithms. In this work, we propose a method for aberration measurement in lithographic tools, which only requires measuring two images of intensity distribution. Two linear formulations are derived in matrix forms that directly relate the measured images to the unknown Zernike coefficients. Consequently, an efficient non-iterative solution is obtained.

Multiple zeros of polynomials

NASA Technical Reports Server (NTRS)

Wood, C. A.

1974-01-01

For polynomials of higher degree, iterative numerical methods must be used. Four iterative methods are presented for approximating the zeros of a polynomial using a digital computer. Newton's method and Muller's method are two well known iterative methods which are presented. They extract the zeros of a polynomial by generating a sequence of approximations converging to each zero. However, both of these methods are very unstable when used on a polynomial which has multiple zeros. That is, either they fail to converge to some or all of the zeros, or they converge to very bad approximations of the polynomial's zeros. This material introduces two new methods, the greatest common divisor (G.C.D.) method and the repeated greatest common divisor (repeated G.C.D.) method, which are superior methods for numerically approximating the zeros of a polynomial having multiple zeros. These methods were programmed in FORTRAN 4 and comparisons in time and accuracy are given.

Equations on knot polynomials and 3d/5d duality

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

Mironov, A.; Morozov, A.; ITEP, Moscow

2012-09-24

We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as 'differential' and (quantum) A-polynomial structures. We pay a special attention to identity between the A-polynomial equations for knots and Baxter equations for quantum relativistic integrable systems, related through Seiberg-Witten theory to 5d super-Yang-Mills models and through the AGT relation to the q-Virasoro algebra. This identity is an important ingredient of emerging a 3d- 5d generalization of the AGT relation. Themore » shape of the Baxter equation (including the values of coefficients) depend on the choice of the knot/braid. Thus, like the case of KP integrability, where (some, so far torus) knots parameterize particular points of the Universal Grassmannian, in this relation they parameterize particular points in the moduli space of many-body integrable systems of relativistic type.« less

Phase-Shifting Zernike Interferometer Wavefront Sensor

NASA Technical Reports Server (NTRS)

Wallace, J. Kent; Rao, Shanti; Jensen-Clemb, Rebecca M.; Serabyn, Gene

2011-01-01

The canonical Zernike phase-contrast technique1,2,3,4 transforms a phase object in one plane into an intensity object in the conjugate plane. This is done by applying a static pi/2 phase shift to the central core (approx. lambda/D) of the PSF which is intermediate between the input and output planes. Here we present a new architecture for this sensor. First, the optical system is simple and all reflective. Second, the phase shift in the central core of the PSF is dynamic and or arbitrary size. This common-path, all-reflective design makes it minimally sensitive to vibration, polarization and wavelength. We review the theory of operation, describe the optical system, summarize numerical simulations and sensitivities and review results from a laboratory demonstration of this novel instrument

Phase-Shifting Zernike Interferometer Wavefront Sensor

NASA Technical Reports Server (NTRS)

Wallace, J. Kent; Rao, Shanti; Jensen-Clem, Rebecca M.

2011-01-01

The canonical Zernike phase-contrast technique transforms a phase object in one plane into an intensity object in the conjugate plane. This is done by applying a static pi/2 phase shift to the central core (approx. lambda/diameter) of the PSF which is intermediate between the input and output plane. Here we present a new architecture for this sensor. First, the optical system is simple and all reflective, and second the phase shift in the central core of the PSF is dynamic and can be made arbitrarily large. This common-path, all-reflective design makes it minimally sensitive to vibration, polarization and wavelength. We review the theory of operation, describe the optical system, summarize numerical simulations and sensitivities and review results from a laboratory demonstration of this novel instrument.

A family of Nikishin systems with periodic recurrence coefficients

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

Delvaux, Steven; Lopez, Abey; Lopez, Guillermo L

2013-01-31

Suppose we have a Nikishin system of p measures with the kth generating measure of the Nikishin system supported on an interval {Delta}{sub k} subset of R with {Delta}{sub k} Intersection {Delta}{sub k+1} = Empty-Set for all k. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a (p+2)-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period p. (The limit values depend only on the positions of the intervals {Delta}{sub k}.) Taking these periodic limit values as the coefficients of a new (p+2)-term recurrence relation, wemore » construct a canonical sequence of monic polynomials {l_brace}P{sub n}{r_brace}{sub n=0}{sup {infinity}}, the so-called Chebyshev-Nikishin polynomials. We show that the polynomials P{sub n} themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the kth generating measure being absolutely continuous on {Delta}{sub k}. In this way we generalize a result of the third author and Rocha [22] for the case p=2. The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for functions of the second kind of the Nikishin system for {l_brace}P{sub n}{r_brace}{sub n=0}{sup {infinity}}. Bibliography: 27 titles.« less

A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials

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

Gashkov, Sergey B; Sergeev, Igor' S

2012-10-31

This work suggests a method for deriving lower bounds for the complexity of polynomials with positive real coefficients implemented by circuits of functional elements over the monotone arithmetic basis {l_brace}x+y, x {center_dot} y{r_brace} Union {l_brace}a {center_dot} x | a Element-Of R{sub +}{r_brace}. Using this method, several new results are obtained. In particular, we construct examples of polynomials of degree m-1 in each of the n variables with coefficients 0 and 1 having additive monotone complexity m{sup (1-o(1))n} and multiplicative monotone complexity m{sup (1/2-o(1))n} as m{sup n}{yields}{infinity}. In this form, the lower bounds derived here are sharp. Bibliography: 72 titles.

Influence of eye micromotions on spatially resolved refractometry

NASA Astrophysics Data System (ADS)

Chyzh, Igor H.; Sokurenko, Vyacheslav M.; Osipova, Irina Y.

2001-01-01

The influence eye micromotions on the accuracy of estimation of Zernike coefficients form eye transverse aberration measurements was investigated. By computer modeling, the following found eye aberrations have been examined: defocusing, primary astigmatism, spherical aberration of the 3rd and the 5th orders, as well as their combinations. It was determined that the standard deviation of estimated Zernike coefficients is proportional to the standard deviation of angular eye movements. Eye micromotions cause the estimation errors of Zernike coefficients of present aberrations and produce the appearance of Zernike coefficients of aberrations, absent in the eye. When solely defocusing is present, the biggest errors, cased by eye micromotions, are obtained for aberrations like coma and astigmatism. In comparison with other aberrations, spherical aberration of the 3rd and the 5th orders evokes the greatest increase of the standard deviation of other Zernike coefficients.

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.

Orthogonality of spherical harmonic coefficients

NASA Astrophysics Data System (ADS)

McLeod, M. G.

1980-08-01

Orthogonality relations are obtained for the spherical harmonic coefficients of functions defined on the surface of a sphere. Following a brief discussion of the orthogonality of Fourier series coefficients, consideration is given to the values averaged over all orientations of the coordinate system of the spherical harmonic coefficients of a function defined on the surface of a sphere that can be expressed in terms of Legendre polynomials for the special case where the function is the sum of two delta functions located at two different points on the sphere, and for the case of an essentially arbitrary function. It is noted that the orthogonality relations derived have found applications in statistical studies of the geomagnetic field.

More Zernike modes' open-loop measurement in the sub-aperture of the Shack-Hartmann wavefront sensor.

PubMed

Zhu, Zhaoyi; Mu, Quanquan; Li, Dayu; Yang, Chengliang; Cao, Zhaoliang; Hu, Lifa; Xuan, Li

2016-10-17

The centroid-based Shack-Hartmann wavefront sensor (SHWFS) treats the sampled wavefronts in the sub-apertures as planes, and the slopes of the sub-wavefronts are used to reconstruct the whole pupil wavefront. The problem is that the centroid method may fail to sense the high-order modes for strong turbulences, decreasing the precision of the whole pupil wavefront reconstruction. To solve this problem, we propose a sub-wavefront estimation method for SHWFS based on the focal plane sensing technique, by which more Zernike modes than the two slopes can be sensed in each sub-aperture. In this paper, the effects on the sub-wavefront estimation method of the related parameters, such as the spot size, the phase offset with its set amplitude and the pixels number in each sub-aperture, are analyzed and these parameters are optimized to achieve high efficiency. After the optimization, open-loop measurement is realized. For the sub-wavefront sensing, we achieve a large linearity range of 3.0 rad RMS for Zernike modes Z2 and Z3, and 2.0 rad RMS for Zernike modes Z4 to Z6 when the pixel number does not exceed 8 × 8 in each sub-aperture. The whole pupil wavefront reconstruction with the modified SHWFS is realized to analyze the improvements brought by the optimized sub-wavefront estimation method. Sixty-five Zernike modes can be reconstructed with a modified SHWFS containing only 7 × 7 sub-apertures, which could reconstruct only 35 modes by the centroid method, and the mean RMS errors of the residual phases are less than 0.2 rad², which is lower than the 0.35 rad² by the centroid method.

Algorithms for the computation of solutions of the Ornstein-Zernike equation.

PubMed

Peplow, A T; Beardmore, R E; Bresme, F

2006-10-01

We introduce a robust and efficient methodology to solve the Ornstein-Zernike integral equation using the pseudoarc length (PAL) continuation method that reformulates the integral equation in an equivalent but nonstandard form. This enables the computation of solutions in regions where the compressibility experiences large changes or where the existence of multiple solutions and so-called branch points prevents Newton's method from converging. We illustrate the use of the algorithm with a difficult problem that arises in the numerical solution of integral equations, namely the evaluation of the so-called no-solution line of the Ornstein-Zernike hypernetted chain (HNC) integral equation for the Lennard-Jones potential. We are able to use the PAL algorithm to solve the integral equation along this line and to connect physical and nonphysical solution branches (both isotherms and isochores) where appropriate. We also show that PAL continuation can compute solutions within the no-solution region that cannot be computed when Newton and Picard methods are applied directly to the integral equation. While many solutions that we find are new, some correspond to states with negative compressibility and consequently are not physical.

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

PubMed

Langley, Jason; Zhao, Qun

2009-09-07

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

Zernike Phase Contrast Electron Cryo-Tomography Applied to Marine Cyanobacteria Infected with Cyanophages

PubMed Central

Dai, Wei; Fu, Caroline; Khant, Htet A.; Ludtke, Steven J.; Schmid, Michael F.; Chiu, Wah

2015-01-01

Advances in electron cryo-tomography have provided a new opportunity to visualize the internal 3D structures of a bacterium. An electron microscope equipped with Zernike phase contrast optics produces images with dramatically increased contrast compared to images obtained by conventional electron microscopy. Here we describe a protocol to apply Zernike phase plate technology for acquiring electron tomographic tilt series of cyanophage-infected cyanobacterial cells embedded in ice, without staining or chemical fixation. We detail the procedures for aligning and assessing phase plates for data collection, and methods to obtain 3D structures of cyanophage assembly intermediates in the host, by subtomogram alignment, classification and averaging. Acquiring three to four tomographic tilt series takes approximately 12 h on a JEM2200FS electron microscope. We expect this time requirement to decrease substantially as the technique matures. Time required for annotation and subtomogram averaging varies widely depending on the project goals and data volume. PMID:25321408

Optical alignment procedure utilizing neural networks combined with Shack-Hartmann wavefront sensor

NASA Astrophysics Data System (ADS)

Adil, Fatime Zehra; Konukseven, Erhan İlhan; Balkan, Tuna; Adil, Ömer Faruk

2017-05-01

In the design of pilot helmets with night vision capability, to not limit or block the sight of the pilot, a transparent visor is used. The reflected image from the coated part of the visor must coincide with the physical human sight image seen through the nonreflecting regions of the visor. This makes the alignment of the visor halves critical. In essence, this is an alignment problem of two optical parts that are assembled together during the manufacturing process. Shack-Hartmann wavefront sensor is commonly used for the determination of the misalignments through wavefront measurements, which are quantified in terms of the Zernike polynomials. Although the Zernike polynomials provide very useful feedback about the misalignments, the corrective actions are basically ad hoc. This stems from the fact that there exists no easy inverse relation between the misalignment measurements and the physical causes of the misalignments. This study aims to construct this inverse relation by making use of the expressive power of the neural networks in such complex relations. For this purpose, a neural network is designed and trained in MATLAB® regarding which types of misalignments result in which wavefront measurements, quantitatively given by Zernike polynomials. This way, manual and iterative alignment processes relying on trial and error will be replaced by the trained guesses of a neural network, so the alignment process is reduced to applying the counter actions based on the misalignment causes. Such a training requires data containing misalignment and measurement sets in fine detail, which is hard to obtain manually on a physical setup. For that reason, the optical setup is completely modeled in Zemax® software, and Zernike polynomials are generated for misalignments applied in small steps. The performance of the neural network is experimented and found promising in the actual physical setup.

A new numerical treatment based on Lucas polynomials for 1D and 2D sinh-Gordon equation

NASA Astrophysics Data System (ADS)

Oruç, Ömer

2018-04-01

In this paper, a new mixed method based on Lucas and Fibonacci polynomials is developed for numerical solutions of 1D and 2D sinh-Gordon equations. Firstly time variable discretized by central finite difference and then unknown function and its derivatives are expanded to Lucas series. With the help of these series expansion and Fibonacci polynomials, matrices for differentiation are derived. With this approach, finding the solution of sinh-Gordon equation transformed to finding the solution of an algebraic system of equations. Lucas series coefficients are acquired by solving this system of algebraic equations. Then by plugginging these coefficients into Lucas series expansion numerical solutions can be obtained consecutively. The main objective of this paper is to demonstrate that Lucas polynomial based method is convenient for 1D and 2D nonlinear problems. By calculating L2 and L∞ error norms of some 1D and 2D test problems efficiency and performance of the proposed method is monitored. Acquired accurate results confirm the applicability of the method.

Mapping Landslides in Lunar Impact Craters Using Chebyshev Polynomials and Dem's

NASA Astrophysics Data System (ADS)

Yordanov, V.; Scaioni, M.; Brunetti, M. T.; Melis, M. T.; Zinzi, A.; Giommi, P.

2016-06-01

Geological slope failure processes have been observed on the Moon surface for decades, nevertheless a detailed and exhaustive lunar landslide inventory has not been produced yet. For a preliminary survey, WAC images and DEM maps from LROC at 100 m/pixels have been exploited in combination with the criteria applied by Brunetti et al. (2015) to detect the landslides. These criteria are based on the visual analysis of optical images to recognize mass wasting features. In the literature, Chebyshev polynomials have been applied to interpolate crater cross-sections in order to obtain a parametric characterization useful for classification into different morphological shapes. Here a new implementation of Chebyshev polynomial approximation is proposed, taking into account some statistical testing of the results obtained during Least-squares estimation. The presence of landslides in lunar craters is then investigated by analyzing the absolute values off odd coefficients of estimated Chebyshev polynomials. A case study on the Cassini A crater has demonstrated the key-points of the proposed methodology and outlined the required future development to carry out.

Hadamard Factorization of Stable Polynomials

NASA Astrophysics Data System (ADS)

Loredo-Villalobos, Carlos Arturo; Aguirre-Hernández, Baltazar

2011-11-01

The stable (Hurwitz) polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p,q ∈ R[x]:p(x) = anxn+an-1xn-1+...+a1x+a0q(x) = bmx m+bm-1xm-1+...+b1x+b0the Hadamard product (p × q) is defined as (p×q)(x) = akbkxk+ak-1bk-1xk-1+...+a1b1x+a0b0where k = min(m,n). Some results (see [16]) shows that if p,q ∈R[x] are stable polynomials then (p×q) is stable, also, i.e. the Hadamard product is closed; however, the reciprocal is not always true, that is, not all stable polynomial has a factorization into two stable polynomials the same degree n, if n> 4 (see [15]).In this work we will give some conditions to Hadamard factorization existence for stable polynomials.

Extending a Property of Cubic Polynomials to Higher-Degree Polynomials

ERIC Educational Resources Information Center

Miller, David A.; Moseley, James

2012-01-01

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

Simple analysis of scattering data with the Ornstein-Zernike equation

NASA Astrophysics Data System (ADS)

Kats, E. I.; Muratov, A. R.

2018-01-01

In this paper we propose and explore a method of analysis of the scattering experimental data for uniform liquidlike systems. In our pragmatic approach we are not trying to introduce by hands an artificial small parameter to work out a perturbation theory with respect to the known results, e.g., for hard spheres or sticky hard spheres (all the more that in the agreement with the notorious Landau statement, there is no physical small parameter for liquids). Instead of it being guided by the experimental data we are solving the Ornstein-Zernike equation with a trial (variational) form of the interparticle interaction potential. To find all needed correlation functions this variational input is iterated numerically to satisfy the Ornstein-Zernike equation supplemented by a closure relation. Our method is developed for spherically symmetric scattering objects, and our numeric code is written for such a case. However, it can be extended (at the expense of more involved computations and a larger amount of required experimental input information) for nonspherical particles. What is important for our approach is that it is sufficient to know experimental data in a relatively narrow range of the scattering wave vectors (q ) to compute the static structure factor in a much broader range of q . We illustrate by a few model and real experimental examples of the x-ray and neutron scattering data how the approach works.

On universal knot polynomials

NASA Astrophysics Data System (ADS)

Mironov, A.; Mkrtchyan, R.; Morozov, A.

2016-02-01

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, respectively and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.

A Unified Methodology for Computing Accurate Quaternion Color Moments and Moment Invariants.

PubMed

Karakasis, Evangelos G; Papakostas, George A; Koulouriotis, Dimitrios E; Tourassis, Vassilios D

2014-02-01

In this paper, a general framework for computing accurate quaternion color moments and their corresponding invariants is proposed. The proposed unified scheme arose by studying the characteristics of different orthogonal polynomials. These polynomials are used as kernels in order to form moments, the invariants of which can easily be derived. The resulted scheme permits the usage of any polynomial-like kernel in a unified and consistent way. The resulted moments and moment invariants demonstrate robustness to noisy conditions and high discriminative power. Additionally, in the case of continuous moments, accurate computations take place to avoid approximation errors. Based on this general methodology, the quaternion Tchebichef, Krawtchouk, Dual Hahn, Legendre, orthogonal Fourier-Mellin, pseudo Zernike and Zernike color moments, and their corresponding invariants are introduced. A selected paradigm presents the reconstruction capability of each moment family, whereas proper classification scenarios evaluate the performance of color moment invariants.

Comparison of organs' shapes with geometric and Zernike 3D moments.

PubMed

Broggio, D; Moignier, A; Ben Brahim, K; Gardumi, A; Grandgirard, N; Pierrat, N; Chea, M; Derreumaux, S; Desbrée, A; Boisserie, G; Aubert, B; Mazeron, J-J; Franck, D

2013-09-01

The morphological similarity of organs is studied with feature vectors based on geometric and Zernike 3D moments. It is particularly investigated if outliers and average models can be identified. For this purpose, the relative proximity to the mean feature vector is defined, principal coordinate and clustering analyses are also performed. To study the consistency and usefulness of this approach, 17 livers and 76 hearts voxel models from several sources are considered. In the liver case, models with similar morphological feature are identified. For the limited amount of studied cases, the liver of the ICRP male voxel model is identified as a better surrogate than the female one. For hearts, the clustering analysis shows that three heart shapes represent about 80% of the morphological variations. The relative proximity and clustering analysis rather consistently identify outliers and average models. For the two cases, identification of outliers and surrogate of average models is rather robust. However, deeper classification of morphological feature is subject to caution and can only be performed after cross analysis of at least two kinds of feature vectors. Finally, the Zernike moments contain all the information needed to re-construct the studied objects and thus appear as a promising tool to derive statistical organ shapes. Copyright © 2013 Elsevier Ireland Ltd. All rights reserved.

Accurate polynomial expressions for the density and specific volume of seawater using the TEOS-10 standard

NASA Astrophysics Data System (ADS)

Roquet, F.; Madec, G.; McDougall, Trevor J.; Barker, Paul M.

2015-06-01

A new set of approximations to the standard TEOS-10 equation of state are presented. These follow a polynomial form, making it computationally efficient for use in numerical ocean models. Two versions are provided, the first being a fit of density for Boussinesq ocean models, and the second fitting specific volume which is more suitable for compressible models. Both versions are given as the sum of a vertical reference profile (6th-order polynomial) and an anomaly (52-term polynomial, cubic in pressure), with relative errors of ∼0.1% on the thermal expansion coefficients. A 75-term polynomial expression is also presented for computing specific volume, with a better accuracy than the existing TEOS-10 48-term rational approximation, especially regarding the sound speed, and it is suggested that this expression represents a valuable approximation of the TEOS-10 equation of state for hydrographic data analysis. In the last section, practical aspects about the implementation of TEOS-10 in ocean models are discussed.

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

PubMed

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

2015-08-10

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

Advantages of intermediate X-ray energies in Zernike phase contrast X-ray microscopy.

PubMed

Wang, Zhili; Gao, Kun; Chen, Jian; Hong, Youli; Ge, Xin; Wang, Dajiang; Pan, Zhiyun; Zhu, Peiping; Yun, Wenbing; Jacobsen, Chris; Wu, Ziyu

2013-01-01

Understanding the hierarchical organizations of molecules and organelles within the interior of large eukaryotic cells is a challenge of fundamental interest in cell biology. Light microscopy is a powerful tool for observations of the dynamics of live cells, its resolution attainable is limited and insufficient. While electron microscopy can produce images with astonishing resolution and clarity of ultra-thin (<1 μm thick) sections of biological specimens, many questions involve the three-dimensional organization of a cell or the interconnectivity of cells. X-ray microscopy offers superior imaging resolution compared to light microscopy, and unique capability of nondestructive three-dimensional imaging of hydrated unstained biological cells, complementary to existing light and electron microscopy. Until now, X-ray microscopes operating in the "water window" energy range between carbon and oxygen k-shell absorption edges have produced outstanding 3D images of cryo-preserved cells. The relatively low X-ray energy (<540 eV) of the water window imposes two important limitations: limited penetration (<10 μm) not suitable for imaging larger cells or tissues, and small depth of focus (DoF) for high resolution 3D imaging (e.g., ~1 μm DoF for 20 nm resolution). An X-ray microscope operating at intermediate energy around 2.5 keV using Zernike phase contrast can overcome the above limitations and reduces radiation dose to the specimen. Using a hydrated model cell with an average chemical composition reported in literature, we calculated the image contrast and the radiation dose for absorption and Zernike phase contrast, respectively. The results show that an X-ray microscope operating at ~2.5 keV using Zernike phase contrast offers substantial advantages in terms of specimen size, radiation dose and depth-of-focus. Copyright © 2012 Elsevier Inc. All rights reserved.

Determinants with orthogonal polynomial entries

NASA Astrophysics Data System (ADS)

Ismail, Mourad E. H.

2005-06-01

We use moment representations of orthogonal polynomials to evaluate the corresponding Hankel determinants formed by the orthogonal polynomials. We also study the Hankel determinants which start with pn on the top left-hand corner. As examples we evaluate the Hankel determinants whose entries are q-ultraspherical or Al-Salam-Chihara polynomials.

Policy Iteration for $H_\\infty $ Optimal Control of Polynomial Nonlinear Systems via Sum of Squares Programming.

PubMed

Zhu, Yuanheng; Zhao, Dongbin; Yang, Xiong; Zhang, Qichao

2018-02-01

Sum of squares (SOS) polynomials have provided a computationally tractable way to deal with inequality constraints appearing in many control problems. It can also act as an approximator in the framework of adaptive dynamic programming. In this paper, an approximate solution to the optimal control of polynomial nonlinear systems is proposed. Under a given attenuation coefficient, the Hamilton-Jacobi-Isaacs equation is relaxed to an optimization problem with a set of inequalities. After applying the policy iteration technique and constraining inequalities to SOS, the optimization problem is divided into a sequence of feasible semidefinite programming problems. With the converged solution, the attenuation coefficient is further minimized to a lower value. After iterations, approximate solutions to the smallest -gain and the associated optimal controller are obtained. Four examples are employed to verify the effectiveness of the proposed algorithm.

Uncertainty Quantification in Simulations of Epidemics Using Polynomial Chaos

PubMed Central

Santonja, F.; Chen-Charpentier, B.

2012-01-01

Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the first-and the second-order moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model. PMID:22927889

Artificial neural network for the determination of Hubble Space Telescope aberration from stellar images

NASA Technical Reports Server (NTRS)

Barrett, Todd K.; Sandler, David G.

1993-01-01

An artificial-neural-network method, first developed for the measurement and control of atmospheric phase distortion, using stellar images, was used to estimate the optical aberration of the Hubble Space Telescope. A total of 26 estimates of distortion was obtained from 23 stellar images acquired at several secondary-mirror axial positions. The results were expressed as coefficients of eight orthogonal Zernike polynomials: focus through third-order spherical. For all modes other than spherical the measured aberration was small. The average spherical aberration of the estimates was -0.299 micron rms, which is in good agreement with predictions obtained when iterative phase-retrieval algorithms were used.

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.

Interpolation and Polynomial Curve Fitting

ERIC Educational Resources Information Center

Yang, Yajun; Gordon, Sheldon P.

2014-01-01

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

Stochastic Estimation via Polynomial Chaos

DTIC Science & Technology

2015-10-01

AFRL-RW-EG-TR-2015-108 Stochastic Estimation via Polynomial Chaos Douglas V. Nance Air Force Research...COVERED (From - To) 20-04-2015 – 07-08-2015 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Stochastic Estimation via Polynomial Chaos ...This expository report discusses fundamental aspects of the polynomial chaos method for representing the properties of second order stochastic

PIZZA: a phase-induced zonal Zernike apodization designed for stellar coronagraphy

NASA Astrophysics Data System (ADS)

Martinache, Frantz

2004-08-01

I explore here the possibilities offered by the general formalism of coronagraphy for the very special case of phase contrast. This technique, invented by Zernike, is commonly used in microscopy, to see phase objects such as micro-organisms, and in strioscopy, to control the quality of optics polishing. It may find application in telescope pupil apodization with significant advantages over classical pupil apodization techniques, including high throughput and no off-axis resolution loss, which is essential for exoplanet imaging.

Chaos, Fractals, and Polynomials.

ERIC Educational Resources Information Center

Tylee, J. Louis; Tylee, Thomas B.

1996-01-01

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

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

Aberration influence and active compensation on laser mode properties for asymmetric folded resonators

NASA Astrophysics Data System (ADS)

Zhang, Xiang; Hu, Zhiqiu; Yang, Wentao; Su, Likun

2017-09-01

We demonstrate the influence on mode features with introducing typical intracavity perturbation and results of aberrated wavefront compensation in a folded-type unstable resonator used in high energy lasers. The mode properties and aberration coefficient with intracavity misalignment are achieved by iterative calculation and Zernike polynomial fitting. Experimental results for the relation of intracavity maladjustment and mode characteristics are further obtained in terms of S-H detection and model wavefront reconstruction. It indicates that intracavity phase perturbation has significant influence on out coupling beam properties, and the uniform and symmetry of the mode is rapidly disrupted even by a slight misalignment of the resonator mirrors. Meanwhile, the far-field beam patterns will obviously degrade with increasing the distance between the convex mirror and the phase perturbation position even if the equivalent disturbation is inputted into such the resonator. The closed-loop device for compensating intracavity low order aberration is successfully fabricated. Moreover, Zernike defocus aberration is also effectively controlled by precisely adjusting resonator length, and the beam quality is noticeably improved.

Characterising a holographic modal phase mask for the detection of ocular aberrations

NASA Astrophysics Data System (ADS)

Corbett, A. D.; Leyva, D. Gil; Diaz-Santana, L.; Wilkinson, T. D.; Zhong, J. J.

2005-12-01

The accurate measurement of the double-pass ocular wave front has been shown to have a broad range of applications from LASIK surgery to adaptively corrected retinal imaging. The ocular wave front can be accurately described by a small number of Zernike circle polynomials. The modal wave front sensor was first proposed by Neil et al. and allows the coefficients of the individual Zernike modes to be measured directly. Typically the aberrations measured with the modal sensor are smaller than those seen in the ocular wave front. In this work, we investigated a technique for adapting a modal phase mask for the sensing of the ocular wave front. This involved extending the dynamic range of the sensor by increasing the pinhole size to 2.4mm and optimising the mask bias to 0.75λ. This was found to decrease the RMS error by up to a factor of three for eye-like aberrations with amplitudes up to 0.2μm. For aberrations taken from a sample of real-eye measurements a 20% decrease in the RMS error was observed.

Multi-indexed (q-)Racah polynomials

NASA Astrophysics Data System (ADS)

Odake, Satoru; Sasaki, Ryu

2012-09-01

As the second stage of the project multi-indexed orthogonal polynomials, we present, in the framework of ‘discrete quantum mechanics’ with real shifts in one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from the (q-)Racah polynomials by the multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of ‘virtual state’ vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the ‘solutions’ of the matrix Schrödinger equation with negative ‘eigenvalues’, except for one of the two boundary points.

Uncertainty propagation through an aeroelastic wind turbine model using polynomial surrogates

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

Murcia, Juan Pablo; Réthoré, Pierre-Elouan; Dimitrov, Nikolay

Polynomial surrogates are used to characterize the energy production and lifetime equivalent fatigue loads for different components of the DTU 10 MW reference wind turbine under realistic atmospheric conditions. The variability caused by different turbulent inflow fields are captured by creating independent surrogates for the mean and standard deviation of each output with respect to the inflow realizations. A global sensitivity analysis shows that the turbulent inflow realization has a bigger impact on the total distribution of equivalent fatigue loads than the shear coefficient or yaw miss-alignment. The methodology presented extends the deterministic power and thrust coefficient curves to uncertaintymore » models and adds new variables like damage equivalent fatigue loads in different components of the turbine. These surrogate models can then be implemented inside other work-flows such as: estimation of the uncertainty in annual energy production due to wind resource variability and/or robust wind power plant layout optimization. It can be concluded that it is possible to capture the global behavior of a modern wind turbine and its uncertainty under realistic inflow conditions using polynomial response surfaces. In conclusion, the surrogates are a way to obtain power and load estimation under site specific characteristics without sharing the proprietary aeroelastic design.« less

Uncertainty propagation through an aeroelastic wind turbine model using polynomial surrogates

DOE PAGES

Murcia, Juan Pablo; Réthoré, Pierre-Elouan; Dimitrov, Nikolay; ...

2017-07-17

Polynomial surrogates are used to characterize the energy production and lifetime equivalent fatigue loads for different components of the DTU 10 MW reference wind turbine under realistic atmospheric conditions. The variability caused by different turbulent inflow fields are captured by creating independent surrogates for the mean and standard deviation of each output with respect to the inflow realizations. A global sensitivity analysis shows that the turbulent inflow realization has a bigger impact on the total distribution of equivalent fatigue loads than the shear coefficient or yaw miss-alignment. The methodology presented extends the deterministic power and thrust coefficient curves to uncertaintymore » models and adds new variables like damage equivalent fatigue loads in different components of the turbine. These surrogate models can then be implemented inside other work-flows such as: estimation of the uncertainty in annual energy production due to wind resource variability and/or robust wind power plant layout optimization. It can be concluded that it is possible to capture the global behavior of a modern wind turbine and its uncertainty under realistic inflow conditions using polynomial response surfaces. In conclusion, the surrogates are a way to obtain power and load estimation under site specific characteristics without sharing the proprietary aeroelastic design.« less

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.

Bond lifetime and diffusion coefficient in colloids with short-range interactions.

PubMed

Ndong Mintsa, E; Germain, Ph; Amokrane, S

2015-03-01

We use molecular dynamics simulations to study the influence of short-range structures in the interaction potential between hard-sphere-like colloidal particles. Starting from model potentials and effective potentials in binary mixtures computed from the Ornstein-Zernike equations, we investigate the influence of the range and strength of a possible tail beyond the usual core repulsion or the presence of repulsive barriers. The diffusion coefficient and mean "bond" lifetimes are used as indicators of the effect of this structure on the dynamics. The existence of correlations between the variations of these quantities with the physical parameters is discussed to assess the interpretation of dynamics slowing down in terms of long-lived bonds. We also discuss the question of a universal behaviour determined by the second virial coefficient B ((2)) and the interplay of attraction and repulsion. While the diffusion coefficient follows the B ((2)) law for purely attractive tails, this is no longer true in the presence of repulsive barriers. Furthermore, the bond lifetime shows a dependence on the physical parameters that differs from that of the diffusion coefficient. This raises the question of the precise role of bonds on the dynamics slowing down in colloidal gels.

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.

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

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.

A computer program to find the kernel of a polynomial operator

NASA Technical Reports Server (NTRS)

Gejji, R. R.

1976-01-01

This paper presents a FORTRAN program written to solve for the kernel of a matrix of polynomials with real coefficients. It is an implementation of Sain's free modular algorithm for solving the minimal design problem of linear multivariable systems. The structure of the program is discussed, together with some features as they relate to questions of implementing the above method. An example of the use of the program to solve a design problem is included.

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.

Developing the Polynomial Expressions for Fields in the ITER Tokamak

NASA Astrophysics Data System (ADS)

Sharma, Stephen

2017-10-01

The two most important problems to be solved in the development of working nuclear fusion power plants are: sustained partial ignition and turbulence. These two phenomena are the subject of research and investigation through the development of analytic functions and computational models. Ansatz development through Gaussian wave-function approximations, dielectric quark models, field solutions using new elliptic functions, and better descriptions of the polynomials of the superconducting current loops are the critical theoretical developments that need to be improved. Euler-Lagrange equations of motion in addition to geodesic formulations generate the particle model which should correspond to the Dirac dispersive scattering coefficient calculations and the fluid plasma model. Feynman-Hellman formalism and Heaviside step functional forms are introduced to the fusion equations to produce simple expressions for the kinetic energy and loop currents. Conclusively, a polynomial description of the current loops, the Biot-Savart field, and the Lagrangian must be uncovered before there can be an adequate computational and iterative model of the thermonuclear plasma.

Astigmatism and defocus wavefront correction via Zernike modes produced with fluidic lenses

PubMed Central

Marks, Randall; Mathine, David L.; Schwiegerling, Jim; Peyman, Gholam; Peyghambarian, Nasser

2010-01-01

Fluidic lenses have been developed for ophthalmic applications with continuously varying optical powers for second order Zernike modes. Continuously varying corrections for both myopic and hyperopic defocus have been demonstrated over a range of three diopters using a fluidic lens with a circular retaining aperture. Likewise, a six diopter range of astigmatism has been continuously corrected using fluidic lenses with rectangular apertures. Imaging results have been characterized using a model eye. PMID:19571912

Extending Romanovski polynomials in quantum mechanics

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

Quesne, C.

2013-12-15

Some extensions of the (third-class) Romanovski polynomials (also called Romanovski/pseudo-Jacobi polynomials), which appear in bound-state wavefunctions of rationally extended Scarf II and Rosen-Morse I potentials, are considered. For the former potentials, the generalized polynomials satisfy a finite orthogonality relation, while for the latter an infinite set of relations among polynomials with degree-dependent parameters is obtained. Both types of relations are counterparts of those known for conventional polynomials. In the absence of any direct information on the zeros of the Romanovski polynomials present in denominators, the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties ofmore » second-order differential equations of Schrödinger type. It is also shown that on going from Scarf I to Scarf II or from Rosen-Morse II to Rosen-Morse I potentials, the variety of rational extensions is narrowed down from types I, II, and III to type III only.« less

Astigmatism error modification for absolute shape reconstruction using Fourier transform method

NASA Astrophysics Data System (ADS)

He, Yuhang; Li, Qiang; Gao, Bo; Liu, Ang; Xu, Kaiyuan; Wei, Xiaohong; Chai, Liqun

2014-12-01

A method is proposed to modify astigmatism errors in absolute shape reconstruction of optical plane using Fourier transform method. If a transmission and reflection flat are used in an absolute test, two translation measurements lead to obtain the absolute shapes by making use of the characteristic relationship between the differential and original shapes in spatial frequency domain. However, because the translation device cannot guarantee the test and reference flats rigidly parallel to each other after the translations, a tilt error exists in the obtained differential data, which caused power and astigmatism errors in the reconstructed shapes. In order to modify the astigmatism errors, a rotation measurement is added. Based on the rotation invariability of the form of Zernike polynomial in circular domain, the astigmatism terms are calculated by solving polynomial coefficient equations related to the rotation differential data, and subsequently the astigmatism terms including error are modified. Computer simulation proves the validity of the proposed method.

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.

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 --> ∞.

On Certain Wronskians of Multiple Orthogonal Polynomials

NASA Astrophysics Data System (ADS)

Zhang, Lun; Filipuk, Galina

2014-11-01

We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szegő. There are two applications of our results. The first application arises from the observation that the m-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even m in a special multiple orthogonal ensemble - the AT ensemble. As the second application, we derive some Turán type inequalities for m! ultiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest.

Polynomial Graphs and Symmetry

ERIC Educational Resources Information Center

Goehle, Geoff; Kobayashi, Mitsuo

2013-01-01

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

Chain mapping approach of Hamiltonian for FMO complex using associated, generalized and exceptional Jacobi polynomials

NASA Astrophysics Data System (ADS)

Mahdian, M.; Arjmandi, M. B.; Marahem, F.

2016-06-01

The excitation energy transfer (EET) in photosynthesis complex has been widely investigated in recent years. However, one of the main problems is simulation of this complex under realistic condition. In this paper by using the associated, generalized and exceptional Jacobi polynomials, firstly, we introduce the spectral density of Fenna-Matthews-Olson (FMO) complex. Afterward, we obtain a map that transforms the Hamiltonian of FMO complex as an open quantum system to a one-dimensional chain of oscillatory modes with only nearest neighbor interaction in which the system is coupled only to first mode of chain. The frequency and coupling strength of each mode can be analytically obtained from recurrence coefficient of mentioned orthogonal polynomials.

Lattice Boltzmann method for bosons and fermions and the fourth-order Hermite polynomial expansion.

PubMed

Coelho, Rodrigo C V; Ilha, Anderson; Doria, Mauro M; Pereira, R M; Aibe, Valter Yoshihiko

2014-04-01

The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried to the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by Yang et al. [Shi and Yang, J. Comput. Phys. 227, 9389 (2008); Yang and Hung, Phys. Rev. E 79, 056708 (2009)] through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded to fourth order in the Hermite polynomials.

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.

An Artificial Intelligence Approach to the Symbolic Factorization of Multivariable Polynomials. Technical Report No. CS74019-R.

ERIC Educational Resources Information Center

Claybrook, Billy G.

A new heuristic factorization scheme uses learning to improve the efficiency of determining the symbolic factorization of multivariable polynomials with interger coefficients and an arbitrary number of variables and terms. The factorization scheme makes extensive use of artificial intelligence techniques (e.g., model-building, learning, and…

Percolation critical polynomial as a graph invariant

DOE PAGES

Scullard, Christian R.

2012-10-18

Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0; 1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer withmore » increasing subgraph size. In this paper, I show how the critical polynomial can be viewed as a graph invariant like the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction p c = 0:52440572:::, which differs from the numerical value, p c = 0:52440503(5), by only 6:9 X 10 -7.« less

Prediction of Aerodynamic Coefficients using Neural Networks for Sparse Data

NASA Technical Reports Server (NTRS)

Rajkumar, T.; Bardina, Jorge; Clancy, Daniel (Technical Monitor)

2002-01-01

Basic aerodynamic coefficients are modeled as functions of angles of attack and sideslip with vehicle lateral symmetry and compressibility effects. Most of the aerodynamic parameters can be well-fitted using polynomial functions. In this paper a fast, reliable way of predicting aerodynamic coefficients is produced using a neural network. The training data for the neural network is derived from wind tunnel test and numerical simulations. The coefficients of lift, drag, pitching moment are expressed as a function of alpha (angle of attack) and Mach number. The results produced from preliminary neural network analysis are very good.

Assessing an ensemble Kalman filter inference of Manning's n coefficient of an idealized tidal inlet against a polynomial chaos-based MCMC

NASA Astrophysics Data System (ADS)

Siripatana, Adil; Mayo, Talea; Sraj, Ihab; Knio, Omar; Dawson, Clint; Le Maitre, Olivier; Hoteit, Ibrahim

2017-08-01

Bayesian estimation/inversion is commonly used to quantify and reduce modeling uncertainties in coastal ocean model, especially in the framework of parameter estimation. Based on Bayes rule, the posterior probability distribution function (pdf) of the estimated quantities is obtained conditioned on available data. It can be computed either directly, using a Markov chain Monte Carlo (MCMC) approach, or by sequentially processing the data following a data assimilation approach, which is heavily exploited in large dimensional state estimation problems. The advantage of data assimilation schemes over MCMC-type methods arises from the ability to algorithmically accommodate a large number of uncertain quantities without significant increase in the computational requirements. However, only approximate estimates are generally obtained by this approach due to the restricted Gaussian prior and noise assumptions that are generally imposed in these methods. This contribution aims at evaluating the effectiveness of utilizing an ensemble Kalman-based data assimilation method for parameter estimation of a coastal ocean model against an MCMC polynomial chaos (PC)-based scheme. We focus on quantifying the uncertainties of a coastal ocean ADvanced CIRCulation (ADCIRC) model with respect to the Manning's n coefficients. Based on a realistic framework of observation system simulation experiments (OSSEs), we apply an ensemble Kalman filter and the MCMC method employing a surrogate of ADCIRC constructed by a non-intrusive PC expansion for evaluating the likelihood, and test both approaches under identical scenarios. We study the sensitivity of the estimated posteriors with respect to the parameters of the inference methods, including ensemble size, inflation factor, and PC order. A full analysis of both methods, in the context of coastal ocean model, suggests that an ensemble Kalman filter with appropriate ensemble size and well-tuned inflation provides reliable mean estimates and

Parallel multigrid smoothing: polynomial versus Gauss-Seidel

NASA Astrophysics Data System (ADS)

Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray

2003-07-01

Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.

Orthogonal Polynomials on the Unit Circle with Fibonacci Verblunsky Coefficients, II. Applications

NASA Astrophysics Data System (ADS)

Damanik, David; Munger, Paul; Yessen, William N.

2013-10-01

We consider CMV matrices with Verblunsky coefficients determined in an appropriate way by the Fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics. In our first application we estimate the spreading rates of quantum walks on the line with time-independent coins following the Fibonacci sequence. The estimates we obtain are explicit in terms of the parameters of the system. In our second application, we establish a connection between the classical nearest neighbor Ising model on the one-dimensional lattice in the complex magnetic field regime, and CMV operators. In particular, given a sequence of nearest-neighbor interaction couplings, we construct a sequence of Verblunsky coefficients, such that the support of the Lee-Yang zeros of the partition function for the Ising model in the thermodynamic limit coincides with the essential spectrum of the CMV matrix with the constructed Verblunsky coefficients. Under certain technical conditions, we also show that the zeros distribution measure coincides with the density of states measure for the CMV matrix.

Statistics of Data Fitting: Flaws and Fixes of Polynomial Analysis of Channeled Spectra

NASA Astrophysics Data System (ADS)

Karstens, William; Smith, David

2013-03-01

Starting from general statistical principles, we have critically examined Baumeister's procedure* for determining the refractive index of thin films from channeled spectra. Briefly, the method assumes that the index and interference fringe order may be approximated by polynomials quadratic and cubic in photon energy, respectively. The coefficients of the polynomials are related by differentiation, which is equivalent to comparing energy differences between fringes. However, we find that when the fringe order is calculated from the published IR index for silicon* and then analyzed with Baumeister's procedure, the results do not reproduce the original index. This problem has been traced to 1. Use of unphysical powers in the polynomials (e.g., time-reversal invariance requires that the index is an even function of photon energy), and 2. Use of insufficient terms of the correct parity. Exclusion of unphysical terms and addition of quartic and quintic terms to the index and order polynomials yields significantly better fits with fewer parameters. This represents a specific example of using statistics to determine if the assumed fitting model adequately captures the physics contained in experimental data. The use of analysis of variance (ANOVA) and the Durbin-Watson statistic to test criteria for the validity of least-squares fitting will be discussed. *D.F. Edwards and E. Ochoa, Appl. Opt. 19, 4130 (1980). Supported in part by the US Department of Energy, Office of Nuclear Physics under contract DE-AC02-06CH11357.

Image object recognition based on the Zernike moment and neural networks

NASA Astrophysics Data System (ADS)

Wan, Jianwei; Wang, Ling; Huang, Fukan; Zhou, Liangzhu

1998-03-01

This paper first give a comprehensive discussion about the concept of artificial neural network its research methods and the relations with information processing. On the basis of such a discussion, we expound the mathematical similarity of artificial neural network and information processing. Then, the paper presents a new method of image recognition based on invariant features and neural network by using image Zernike transform. The method not only has the invariant properties for rotation, shift and scale of image object, but also has good fault tolerance and robustness. Meanwhile, it is also compared with statistical classifier and invariant moments recognition method.

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.

An extended UTD analysis for the scattering and diffraction from cubic polynomial strips

NASA Technical Reports Server (NTRS)

Constantinides, E. D.; Marhefka, R. J.

1993-01-01

Spline and polynomial type surfaces are commonly used in high frequency modeling of complex structures such as aircraft, ships, reflectors, etc. It is therefore of interest to develop an efficient and accurate solution to describe the scattered fields from such surfaces. An extended Uniform Geometrical Theory of Diffraction (UTD) solution for the scattering and diffraction from perfectly conducting cubic polynomial strips is derived and involves the incomplete Airy integrals as canonical functions. This new solution is universal in nature and can be used to effectively describe the scattered fields from flat, strictly concave or convex, and concave convex boundaries containing edges. The classic UTD solution fails to describe the more complicated field behavior associated with higher order phase catastrophes and therefore a new set of uniform reflection and first-order edge diffraction coefficients is derived. Also, an additional diffraction coefficient associated with a zero-curvature (inflection) point is presented. Higher order effects such as double edge diffraction, creeping waves, and whispering gallery modes are not examined. The extended UTD solution is independent of the scatterer size and also provides useful physical insight into the various scattering and diffraction processes. Its accuracy is confirmed via comparison with some reference moment method results.

Novel Image Encryption Scheme Based on Chebyshev Polynomial and Duffing Map

PubMed Central

2014-01-01

We present a novel image encryption algorithm using Chebyshev polynomial based on permutation and substitution and Duffing map based on substitution. Comprehensive security analysis has been performed on the designed scheme using key space analysis, visual testing, histogram analysis, information entropy calculation, correlation coefficient analysis, differential analysis, key sensitivity test, and speed test. The study demonstrates that the proposed image encryption algorithm shows advantages of more than 10113 key space and desirable level of security based on the good statistical results and theoretical arguments. PMID:25143970

Discrete Tchebycheff orthonormal polynomials and applications

NASA Technical Reports Server (NTRS)

Lear, W. M.

1980-01-01

Discrete Tchebycheff orthonormal polynomials offer a convenient way to make least squares polynomial fits of uniformly spaced discrete data. Computer programs to do so are simple and fast, and appear to be less affected by computer roundoff error, for the higher order fits, than conventional least squares programs. They are useful for any application of polynomial least squares fits: approximation of mathematical functions, noise analysis of radar data, and real time smoothing of noisy data, to name a few.

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.

Subjective and Quantitative Measurement of Wavefront Aberrations in Nuclear Cataracts – A Retrospective Case Controlled Study

PubMed Central

Wali, Upender K.; Bialasiewicz, Alexander A.; Al-Kharousi, Nadia; Rizvi, Syed G.; Baloushi, Habiba

2009-01-01

Purpose: To measure, quantify and compare Ocular Aberrations due to nuclear cataracts. Setting: Department of ophthalmology and school for ophthalmic technicians, college of medicine and health sciences, Sultan Qaboos University, Muscat, Oman. Design: Retrospective case controlled study. Methods: 113 eyes of 77 patients with nuclear cataract (NC) were recruited from outpatient clinic of a major tertiary referral center for Ophthalmology. Patients having NC with no co-existing ocular pathologies were selected. All patients were subjected to wavefront aberrometry (make) using Hartmann-Shack (HS) aberrometer. Consents were taken from all patients. Higher order Aberrations (HOA) were calculated with Zernike polynomials up to the fourth order. For comparison 28 eyes of 15 subjects with no lenticular opacities (control group) were recruited and evaluated in an identical manner. No pupillary mydriasis was done in both groups. Results: Total aberrations were almost six times higher in NC group compared to control (normal) subjects. The HOA were 21 times higher in NC group, and coma was significantly higher in NC eyes compared to normal (control) group. The pupillary diameter was significantly larger in control group (5.48mm ± 1.0024, p<.001) compared to NC (3.05mm ± 1.9145) subjects (probably due to younger control age group). Amongst Zernike coefficients up to fourth order, two polynomials, defocus (Z20) and spherical aberration (Z42) were found to be significantly greater amongst NC group, compared to normal control group. Conclusion: Nuclear cataracts predominantly produce increased defocus and spherical aberrations. This could explain visual symptoms like image deterioration in spite of normal Visual acuity. PMID:20142953

3D scene reconstruction based on multi-view distributed video coding in the Zernike domain for mobile applications

NASA Astrophysics Data System (ADS)

Palma, V.; Carli, M.; Neri, A.

2011-02-01

In this paper a Multi-view Distributed Video Coding scheme for mobile applications is presented. Specifically a new fusion technique between temporal and spatial side information in Zernike Moments domain is proposed. Distributed video coding introduces a flexible architecture that enables the design of very low complex video encoders compared to its traditional counterparts. The main goal of our work is to generate at the decoder the side information that optimally blends temporal and interview data. Multi-view distributed coding performance strongly depends on the side information quality built at the decoder. At this aim for improving its quality a spatial view compensation/prediction in Zernike moments domain is applied. Spatial and temporal motion activity have been fused together to obtain the overall side-information. The proposed method has been evaluated by rate-distortion performances for different inter-view and temporal estimation quality conditions.

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,

Effect of fringe-artifact correction on sub-tomogram averaging from Zernike phase-plate cryo-TEM

PubMed Central

Kishchenko, Gregory P.; Danev, Radostin; Fisher, Rebecca; He, Jie; Hsieh, Chyongere; Marko, Michael; Sui, Haixin

2015-01-01

Zernike phase-plate (ZPP) imaging greatly increases contrast in cryo-electron microscopy, however fringe artifacts appear in the images. A computational de-fringing method has been proposed, but it has not been widely employed, perhaps because the importance of de-fringing has not been clearly demonstrated. For testing purposes, we employed Zernike phase-plate imaging in a cryo-electron tomographic study of radial-spoke complexes attached to microtubule doublets. We found that the contrast enhancement by ZPP imaging made nonlinear denoising insensitive to the filtering parameters, such that simple low-frequency band-pass filtering made the same improvement in map quality. We employed sub-tomogram averaging, which compensates for the effect of the “missing wedge” and considerably improves map quality. We found that fringes (caused by the abrupt cut-on of the central hole in the phase plate) can lead to incorrect representation of a structure that is well-known from the literature. The expected structure was restored by amplitude scaling, as proposed in the literature. Our results show that de-fringing is an important part of image-processing for cryo-electron tomography of macromolecular complexes with ZPP imaging. PMID:26210582

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

Approximating smooth functions using algebraic-trigonometric polynomials

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

Sharapudinov, Idris I

2011-01-14

The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p{sub n}(t)+{tau}{sub m}(t), where p{sub n}(t) is an algebraic polynomial of degree n and {tau}{sub m}(t)=a{sub 0}+{Sigma}{sub k=1}{sup m}a{sub k} cos k{pi}t + b{sub k} sin k{pi}t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W{sup r}{sub {infinity}(}M) and an upper bound for similar approximations in the class W{sup r}{sub p}(M) with 4/3polynomials which the author has introduced andmore » investigated previously. Bibliography: 13 titles.« 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, 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

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.

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

Low-Complexity Polynomial Channel Estimation in Large-Scale MIMO With Arbitrary Statistics

NASA Astrophysics Data System (ADS)

Shariati, Nafiseh; Bjornson, Emil; Bengtsson, Mats; Debbah, Merouane

2014-10-01

This paper considers pilot-based channel estimation in large-scale multiple-input multiple-output (MIMO) communication systems, also known as massive MIMO, where there are hundreds of antennas at one side of the link. Motivated by the fact that computational complexity is one of the main challenges in such systems, a set of low-complexity Bayesian channel estimators, coined Polynomial ExpAnsion CHannel (PEACH) estimators, are introduced for arbitrary channel and interference statistics. While the conventional minimum mean square error (MMSE) estimator has cubic complexity in the dimension of the covariance matrices, due to an inversion operation, our proposed estimators significantly reduce this to square complexity by approximating the inverse by a L-degree matrix polynomial. The coefficients of the polynomial are optimized to minimize the mean square error (MSE) of the estimate. We show numerically that near-optimal MSEs are achieved with low polynomial degrees. We also derive the exact computational complexity of the proposed estimators, in terms of the floating-point operations (FLOPs), by which we prove that the proposed estimators outperform the conventional estimators in large-scale MIMO systems of practical dimensions while providing a reasonable MSEs. Moreover, we show that L needs not scale with the system dimensions to maintain a certain normalized MSE. By analyzing different interference scenarios, we observe that the relative MSE loss of using the low-complexity PEACH estimators is smaller in realistic scenarios with pilot contamination. On the other hand, PEACH estimators are not well suited for noise-limited scenarios with high pilot power; therefore, we also introduce the low-complexity diagonalized estimator that performs well in this regime. Finally, we ...

Cosmographic analysis with Chebyshev polynomials

NASA Astrophysics Data System (ADS)

Capozziello, Salvatore; D'Agostino, Rocco; Luongo, Orlando

2018-05-01

The limits of standard cosmography are here revised addressing the problem of error propagation during statistical analyses. To do so, we propose the use of Chebyshev polynomials to parametrize cosmic distances. In particular, we demonstrate that building up rational Chebyshev polynomials significantly reduces error propagations with respect to standard Taylor series. This technique provides unbiased estimations of the cosmographic parameters and performs significatively better than previous numerical approximations. To figure this out, we compare rational Chebyshev polynomials with Padé series. In addition, we theoretically evaluate the convergence radius of (1,1) Chebyshev rational polynomial and we compare it with the convergence radii of Taylor and Padé approximations. We thus focus on regions in which convergence of Chebyshev rational functions is better than standard approaches. With this recipe, as high-redshift data are employed, rational Chebyshev polynomials remain highly stable and enable one to derive highly accurate analytical approximations of Hubble's rate in terms of the cosmographic series. Finally, we check our theoretical predictions by setting bounds on cosmographic parameters through Monte Carlo integration techniques, based on the Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic data, using the Joint Light-curve Analysis supernovae sample and the most recent versions of Hubble parameter and baryon acoustic oscillation measurements. We find that cosmography with Taylor series fails to be predictive with the aforementioned data sets, while turns out to be much more stable using the Chebyshev approach.

Joint optimization of source, mask, and pupil in optical lithography

NASA Astrophysics Data System (ADS)

Li, Jia; Lam, Edmund Y.

2014-03-01

Mask topography effects need to be taken into consideration for more advanced resolution enhancement techniques in optical lithography. However, rigorous 3D mask model achieves high accuracy at a large computational cost. This work develops a combined source, mask and pupil optimization (SMPO) approach by taking advantage of the fact that pupil phase manipulation is capable of partially compensating for mask topography effects. We first design the pupil wavefront function by incorporating primary and secondary spherical aberration through the coefficients of the Zernike polynomials, and achieve optimal source-mask pair under the condition of aberrated pupil. Evaluations against conventional source mask optimization (SMO) without incorporating pupil aberrations show that SMPO provides improved performance in terms of pattern fidelity and process window sizes.

Alzheimer's Disease Detection by Pseudo Zernike Moment and Linear Regression Classification.

PubMed

Wang, Shui-Hua; Du, Sidan; Zhang, Yin; Phillips, Preetha; Wu, Le-Nan; Chen, Xian-Qing; Zhang, Yu-Dong

2017-01-01

This study presents an improved method based on "Gorji et al. Neuroscience. 2015" by introducing a relatively new classifier-linear regression classification. Our method selects one axial slice from 3D brain image, and employed pseudo Zernike moment with maximum order of 15 to extract 256 features from each image. Finally, linear regression classification was harnessed as the classifier. The proposed approach obtains an accuracy of 97.51%, a sensitivity of 96.71%, and a specificity of 97.73%. Our method performs better than Gorji's approach and five other state-of-the-art approaches. Copyright© Bentham Science Publishers; For any queries, please email at epub@benthamscience.org.

Tutte polynomial in functional magnetic resonance imaging

NASA Astrophysics Data System (ADS)

García-Castillón, Marlly V.

2015-09-01

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

Polynomial coefficients for calculating O2 Schumann-Runge cross sections at 0.5/cm resolution

NASA Technical Reports Server (NTRS)

Minschwaner, K.; Anderson, G. P.; Hall, L. A.; Yoshino, K.

1992-01-01

O2 cross sections from 49,000 to 57,000/cm have been fitted with temperature dependent polynomial expressions, providing an accurate and efficient means of determining Schumann-Runge band cross sections for temperatures between 130 and 500 K. The least squares fits were carried out on a 0.5/cm spectral grid, using cross sections obtained from a Schumann-Runge line-by-line model that incorporates the most recent spectroscopic data. The O2 cross sections do not include the underlying Herzberg continuum, but they do contain contributions from the temperature dependent Schumann-Runge continuum. The cross sections are suitable for use in UV transmission calculations at high spectral resolution. They should also prove useful for updating existing parameterizations of ultraviolet transmission and O2 photolysis.

From Jack to Double Jack Polynomials via the Supersymmetric Bridge

NASA Astrophysics Data System (ADS)

Lapointe, Luc; Mathieu, Pierre

2015-07-01

The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine {widehat{sl}_2} algebra.

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

Zernike phase contrast cryo-electron tomography of whole bacterial cells.

PubMed

Guerrero-Ferreira, Ricardo C; Wright, Elizabeth R

2014-01-01

Cryo-electron tomography (cryo-ET) provides three-dimensional (3D) structural information of bacteria preserved in a native, frozen-hydrated state. The typical low contrast of tilt-series images, a result of both the need for a low electron dose and the use of conventional defocus phase-contrast imaging, is a challenge for high-quality tomograms. We show that Zernike phase-contrast imaging allows the electron dose to be reduced. This limits movement of gold fiducials during the tilt series, which leads to better alignment and a higher-resolution reconstruction. Contrast is also enhanced, improving visibility of weak features. The reduced electron dose also means that more images at more tilt angles could be recorded, further increasing resolution. Copyright © 2013 Elsevier Inc. All rights reserved.

Zernike phase contrast cryo-electron tomography of whole bacterial cells

PubMed Central

Guerrero-Ferreira, Ricardo C.; Wright, Elizabeth R.

2014-01-01

Cryo-electron tomography (cryo-ET) provides three-dimensional (3D) structural information of bacteria preserved in a native, frozen-hydrated state. The typical low contrast of tilt-series images, a result of both the need for a low electron dose and the use of conventional defocus phase-contrast imaging, is a challenge for high-quality tomograms. We show that Zernike phase-contrast imaging allows the electron dose to be reduced. This limits movement of gold fiducials during the tilt series, which leads to better alignment and a higher-resolution reconstruction. Contrast is also enhanced, improving visibility of weak features. The reduced electron dose also means that more images at more tilt angles could be recorded, further increasing resolution. PMID:24075950

The Gibbs Phenomenon for Series of Orthogonal Polynomials

ERIC Educational Resources Information Center

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

2006-01-01

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

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/.

Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation

ERIC Educational Resources Information Center

Gordon, Sheldon P.; Yang, Yajun

2017-01-01

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

Scattering amplitudes from multivariate polynomial division

NASA Astrophysics Data System (ADS)

Mastrolia, Pierpaolo; Mirabella, Edoardo; Ossola, Giovanni; Peraro, Tiziano

2012-11-01

We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the weak Nullstellensatz theorem and on the division modulo the Gröbner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrized by a number of coefficients equal to the number of solutions of the cut itself.

The Translated Dowling Polynomials and Numbers.

PubMed

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

2014-01-01

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

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.

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.

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 solutions of the Monge-Ampère equation

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

Aminov, Yu A

2014-11-30

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

Calculation of Thermal Conductivity Coefficients of Electrons in Magnetized Dense Matter

NASA Astrophysics Data System (ADS)

Bisnovatyi-Kogan, G. S.; Glushikhina, M. V.

2018-04-01

The solution of Boltzmann equation for plasma in magnetic field with arbitrarily degenerate electrons and nondegenerate nuclei is obtained by Chapman-Enskog method. Functions generalizing Sonine polynomials are used for obtaining an approximate solution. Fully ionized plasma is considered. The tensor of the heat conductivity coefficients in nonquantized magnetic field is calculated. For nondegenerate and strongly degenerate plasma the asymptotic analytic formulas are obtained and compared with results of previous authors. The Lorentz approximation with neglecting of electron-electron encounters is asymptotically exact for strongly degenerate plasma. For the first time, analytical expressions for the heat conductivity tensor for nondegenerate electrons in the presence of a magnetic field are obtained in the three-polynomial approximation with account of electron-electron collisions. Account of the third polynomial improved substantially the precision of results. In the two-polynomial approximation, the obtained solution coincides with the published results. For strongly degenerate electrons, an asymptotically exact analytical solution for the heat conductivity tensor in the presence of a magnetic field is obtained for the first time. This solution has a considerably more complicated dependence on the magnetic field than those in previous publications and gives a several times smaller relative value of the thermal conductivity across the magnetic field at ωτ * 0.8.

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

PubMed

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

2013-11-01

Head-related transfer functions (HRTFs) are generally large datasets, which can be an important constraint for embedded real-time applications. A method is proposed here to reduce redundancy and compress the datasets. In this method, HRTFs are first compressed by conversion into autoregressive-moving-average (ARMA) filters whose coefficients are calculated using Prony's method. Such filters are specified by a few coefficients which can generate the full head-related impulse responses (HRIRs). Next, Legendre polynomials (LPs) are used to compress the ARMA filter coefficients. LPs are derived on the sphere and form an orthonormal basis set for spherical functions. Higher-order LPs capture increasingly fine spatial details. The number of LPs needed to represent an HRTF, therefore, is indicative of its spatial complexity. The results indicate that compression ratios can exceed 98% while maintaining a spectral error of less than 4 dB in the recovered HRTFs.

Transport coefficients in ultrarelativistic kinetic theory

NASA Astrophysics Data System (ADS)

Ambruş, Victor E.

2018-02-01

A spatially periodic longitudinal wave is considered in relativistic dissipative hydrodynamics. At sufficiently small wave amplitudes, an analytic solution is obtained in the linearized limit of the macroscopic conservation equations within the first- and second-order relativistic hydrodynamics formulations. A kinetic solver is used to obtain the numerical solution of the relativistic Boltzmann equation for massless particles in the Anderson-Witting approximation for the collision term. It is found that, at small values of the Anderson-Witting relaxation time τ , the transport coefficients emerging from the relativistic Boltzmann equation agree with those predicted through the Chapman-Enskog procedure, while the relaxation times of the heat flux and shear pressure are equal to τ . These claims are further strengthened by considering a moment-type approximation based on orthogonal polynomials under which the Chapman-Enskog results for the transport coefficients are exactly recovered.

Corridor of existence of thermodynamically consistent solution of the Ornstein-Zernike equation.

PubMed

Vorob'ev, V S; Martynov, G A

2007-07-14

We obtain the exact equation for a correction to the Ornstein-Zernike (OZ) equation based on the assumption of the uniqueness of thermodynamical functions. We show that this equation is reduced to a differential equation with one arbitrary parameter for the hard sphere model. The compressibility factor within narrow limits of this parameter variation can either coincide with one of the formulas obtained on the basis of analytical solutions of the OZ equation or assume all intermediate values lying in a corridor between these solutions. In particular, we find the value of this parameter when the thermodynamically consistent compressibility factor corresponds to the Carnahan-Stirling formula.

Orbifold E-functions of dual invertible polynomials

NASA Astrophysics Data System (ADS)

Ebeling, Wolfgang; Gusein-Zade, Sabir M.; Takahashi, Atsushi

2016-08-01

An invertible polynomial is a weighted homogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau-Ginzburg models, P. Berglund and M. Henningson considered a pair (f , G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair (f ˜ , G ˜) . We consider the so-called orbifold E-function of such a pair (f , G) which is a generating function for the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of f. We prove that the orbifold E-functions of Berglund-Henningson dual pairs coincide up to a sign depending on the number of variables and a simple change of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial.

Neck curve polynomials in neck rupture model

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

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

2012-06-06

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

A FAST POLYNOMIAL TRANSFORM PROGRAM WITH A MODULARIZED STRUCTURE

NASA Technical Reports Server (NTRS)

Truong, T. K.

1994-01-01

This program utilizes a fast polynomial transformation (FPT) algorithm applicable to two-dimensional mathematical convolutions. Two-dimensional convolution has many applications, particularly in image processing. Two-dimensional cyclic convolutions can be converted to a one-dimensional convolution in a polynomial ring. Traditional FPT methods decompose the one-dimensional cyclic polynomial into polynomial convolutions of different lengths. This program will decompose a cyclic polynomial into polynomial convolutions of the same length. Thus, only FPTs and Fast Fourier Transforms of the same length are required. This modular approach can save computational resources. To further enhance its appeal, the program is written in the transportable 'C' language. The steps in the algorithm are: 1) formulate the modulus reduction equations, 2) calculate the polynomial transforms, 3) multiply the transforms using a generalized fast Fourier transformation, 4) compute the inverse polynomial transforms, and 5) reconstruct the final matrices using the Chinese remainder theorem. Input to this program is comprised of the row and column dimensions and the initial two matrices. The matrices are printed out at all steps, ending with the final reconstruction. This program is written in 'C' for batch execution and has been implemented on the IBM PC series of computers under DOS with a central memory requirement of approximately 18K of 8 bit bytes. This program was developed in 1986.

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

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.

Target recognition of ladar range images using even-order Zernike moments.

PubMed

Liu, Zheng-Jun; Li, Qi; Xia, Zhi-Wei; Wang, Qi

2012-11-01

Ladar range images have attracted considerable attention in automatic target recognition fields. In this paper, Zernike moments (ZMs) are applied to classify the target of the range image from an arbitrary azimuth angle. However, ZMs suffer from high computational costs. To improve the performance of target recognition based on small samples, even-order ZMs with serial-parallel backpropagation neural networks (BPNNs) are applied to recognize the target of the range image. It is found that the rotation invariance and classified performance of the even-order ZMs are both better than for odd-order moments and for moments compressed by principal component analysis. The experimental results demonstrate that combining the even-order ZMs with serial-parallel BPNNs can significantly improve the recognition rate for small samples.

Towards a unification of the hierarchical reference theory and the self-consistent Ornstein-Zernike approximation.

PubMed

Reiner, A; Høye, J S

2005-12-01

The hierarchical reference theory and the self-consistent Ornstein-Zernike approximation are two liquid state theories that both furnish a largely satisfactory description of the critical region as well as phase coexistence and the equation of state in general. Furthermore, there are a number of similarities that suggest the possibility of a unification of both theories. As a first step towards this goal, we consider the problem of combining the lowest order gamma expansion result for the incorporation of a Fourier component of the interaction with the requirement of consistency between internal and free energies, leaving aside the compressibility relation. For simplicity, we restrict ourselves to a simplified lattice gas that is expected to display the same qualitative behavior as more elaborate models. It turns out that the analytically tractable mean spherical approximation is a solution to this problem, as are several of its generalizations. Analysis of the characteristic equations shows the potential for a practical scheme and yields necessary conditions that any closure to the Ornstein-Zernike relation must fulfill for the consistency problem to be well posed and to have a unique differentiable solution. These criteria are expected to remain valid for more general discrete and continuous systems, even if consistency with the compressibility route is also enforced where possible explicit solutions will require numerical evaluations.

A gradient-based model parametrization using Bernstein polynomials in Bayesian inversion of surface wave dispersion

NASA Astrophysics Data System (ADS)

Gosselin, Jeremy M.; Dosso, Stan E.; Cassidy, John F.; Quijano, Jorge E.; Molnar, Sheri; Dettmer, Jan

2017-10-01

This paper develops and applies a Bernstein-polynomial parametrization to efficiently represent general, gradient-based profiles in nonlinear geophysical inversion, with application to ambient-noise Rayleigh-wave dispersion data. Bernstein polynomials provide a stable parametrization in that small perturbations to the model parameters (basis-function coefficients) result in only small perturbations to the geophysical parameter profile. A fully nonlinear Bayesian inversion methodology is applied to estimate shear wave velocity (VS) profiles and uncertainties from surface wave dispersion data extracted from ambient seismic noise. The Bayesian information criterion is used to determine the appropriate polynomial order consistent with the resolving power of the data. Data error correlations are accounted for in the inversion using a parametric autoregressive model. The inversion solution is defined in terms of marginal posterior probability profiles for VS as a function of depth, estimated using Metropolis-Hastings sampling with parallel tempering. This methodology is applied to synthetic dispersion data as well as data processed from passive array recordings collected on the Fraser River Delta in British Columbia, Canada. Results from this work are in good agreement with previous studies, as well as with co-located invasive measurements. The approach considered here is better suited than `layered' modelling approaches in applications where smooth gradients in geophysical parameters are expected, such as soil/sediment profiles. Further, the Bernstein polynomial representation is more general than smooth models based on a fixed choice of gradient type (e.g. power-law gradient) because the form of the gradient is determined objectively by the data, rather than by a subjective parametrization choice.

Orthogonal basis with a conicoid first mode for shape specification of optical surfaces.

PubMed

Ferreira, Chelo; López, José L; Navarro, Rafael; Sinusía, Ester Pérez

2016-03-07

A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.

Applications of polynomial optimization in financial risk investment

NASA Astrophysics Data System (ADS)

Zeng, Meilan; Fu, Hongwei

2017-09-01

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

Optimal Chebyshev polynomials on ellipses in the complex plane

NASA Technical Reports Server (NTRS)

Fischer, Bernd; Freund, Roland

1989-01-01

The design of iterative schemes for sparse matrix computations often leads to constrained polynomial approximation problems on sets in the complex plane. For the case of ellipses, we introduce a new class of complex polynomials which are in general very good approximations to the best polynomials and even optimal in most cases.

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.

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.

The development of a revised version of multi-center molecular Ornstein-Zernike equation

NASA Astrophysics Data System (ADS)

Kido, Kentaro; Yokogawa, Daisuke; Sato, Hirofumi

2012-04-01

Ornstein-Zernike (OZ)-type theory is a powerful tool to obtain 3-dimensional solvent distribution around solute molecule. Recently, we proposed multi-center molecular OZ method, which is suitable for parallel computing of 3D solvation structure. The distribution function in this method consists of two components, namely reference and residue parts. Several types of the function were examined as the reference part to investigate the numerical robustness of the method. As the benchmark, the method is applied to water, benzene in aqueous solution and single-walled carbon nanotube in chloroform solution. The results indicate that fully-parallelization is achieved by utilizing the newly proposed reference functions.

A comparison of companion matrix methods to find roots of a trigonometric polynomial

NASA Astrophysics Data System (ADS)

Boyd, John P.

2013-08-01

A trigonometric polynomial is a truncated Fourier series of the form fN(t)≡∑j=0Naj cos(jt)+∑j=1N bj sin(jt). It has been previously shown by the author that zeros of such a polynomial can be computed as the eigenvalues of a companion matrix with elements which are complex valued combinations of the Fourier coefficients, the "CCM" method. However, previous work provided no examples, so one goal of this new work is to experimentally test the CCM method. A second goal is introduce a new alternative, the elimination/Chebyshev algorithm, and experimentally compare it with the CCM scheme. The elimination/Chebyshev matrix (ECM) algorithm yields a companion matrix with real-valued elements, albeit at the price of usefulness only for real roots. The new elimination scheme first converts the trigonometric rootfinding problem to a pair of polynomial equations in the variables (c,s) where c≡cos(t) and s≡sin(t). The elimination method next reduces the system to a single univariate polynomial P(c). We show that this same polynomial is the resultant of the system and is also a generator of the Groebner basis with lexicographic ordering for the system. Both methods give very high numerical accuracy for real-valued roots, typically at least 11 decimal places in Matlab/IEEE 754 16 digit floating point arithmetic. The CCM algorithm is typically one or two decimal places more accurate, though these differences disappear if the roots are "Newton-polished" by a single Newton's iteration. The complex-valued matrix is accurate for complex-valued roots, too, though accuracy decreases with the magnitude of the imaginary part of the root. The cost of both methods scales as O(N3) floating point operations. In spite of intimate connections of the elimination/Chebyshev scheme to two well-established technologies for solving systems of equations, resultants and Groebner bases, and the advantages of using only real-valued arithmetic to obtain a companion matrix with real-valued elements

Polynomial solution of quantum Grassmann matrices

NASA Astrophysics Data System (ADS)

Tierz, Miguel

2017-05-01

We study a model of quantum mechanical fermions with matrix-like index structure (with indices N and L) and quartic interactions, recently introduced by Anninos and Silva. We compute the partition function exactly with q-deformed orthogonal polynomials (Stieltjes-Wigert polynomials), for different values of L and arbitrary N. From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given L, the number of states of different energy is quadratic in N, which implies an exponential degeneracy of the energy levels. We also show that at high-temperature we have a Gaussian matrix model, which implies a symmetry that swaps N and L, together with a Wick rotation of the spectral parameter. In this limit, we also write the partition function, for generic L and N, in terms of a single generalized Hermite polynomial.

Social Security Polynomials.

ERIC Educational Resources Information Center

Gordon, Sheldon P.

1992-01-01

Demonstrates how the uniqueness and anonymity of a student's Social Security number can be utilized to create individualized polynomial equations that students can investigate using computers or graphing calculators. Students write reports of their efforts to find and classify all real roots of their equation. (MDH)

Accurate Iris Recognition at a Distance Using Stabilized Iris Encoding and Zernike Moments Phase Features.

PubMed

Tan, Chun-Wei; Kumar, Ajay

2014-07-10

Accurate iris recognition from the distantly acquired face or eye images requires development of effective strategies which can account for significant variations in the segmented iris image quality. Such variations can be highly correlated with the consistency of encoded iris features and the knowledge that such fragile bits can be exploited to improve matching accuracy. A non-linear approach to simultaneously account for both local consistency of iris bit and also the overall quality of the weight map is proposed. Our approach therefore more effectively penalizes the fragile bits while simultaneously rewarding more consistent bits. In order to achieve more stable characterization of local iris features, a Zernike moment-based phase encoding of iris features is proposed. Such Zernike moments-based phase features are computed from the partially overlapping regions to more effectively accommodate local pixel region variations in the normalized iris images. A joint strategy is adopted to simultaneously extract and combine both the global and localized iris features. The superiority of the proposed iris matching strategy is ascertained by providing comparison with several state-of-the-art iris matching algorithms on three publicly available databases: UBIRIS.v2, FRGC, CASIA.v4-distance. Our experimental results suggest that proposed strategy can achieve significant improvement in iris matching accuracy over those competing approaches in the literature, i.e., average improvement of 54.3%, 32.7% and 42.6% in equal error rates, respectively for UBIRIS.v2, FRGC, CASIA.v4-distance.

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

Symmetric polynomials in information theory: Entropy and subentropy

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

Jozsa, Richard; Mitchison, Graeme

2015-06-15

Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore, we see that H and the intrinsically quantum informational quantitymore » Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials, we also derive a series of further properties of H and Q.« less

Temperature dependence of electron impact ionization coefficient in bulk silicon

NASA Astrophysics Data System (ADS)

Ahmed, Mowfaq Jalil

2017-09-01

This work exhibits a modified procedure to compute the electron impact ionization coefficient of silicon for temperatures between 77 and 800K and electric fields ranging from 70 to 400 kV/cm. The ionization coefficients are computed from the electron momentum distribution function through solving the Boltzmann transport equation (BTE). The arrangement is acquired by joining Legendre polynomial extension with BTE. The resulting BTE is solved by differences-differential method using MATLAB®. Six (X) equivalent ellipsoidal and non-parabolic valleys of the conduction band of silicon are taken into account. Concerning the scattering mechanisms, the interval acoustic scattering, non-polar optical scattering and II scattering are taken into consideration. This investigation showed that the ionization coefficients decrease with increasing temperature. The overall results are in good agreement with previous experimental and theoretical reported data predominantly at high electric fields.

On the Waring problem for polynomial rings

PubMed Central

Fröberg, Ralf; Ottaviani, Giorgio; Shapiro, Boris

2012-01-01

In this note we discuss an analog of the classical Waring problem for . Namely, we show that a general homogeneous polynomial of degree divisible by k≥2 can be represented as a sum of at most kn k-th powers of homogeneous polynomials in . Noticeably, kn coincides with the number obtained by naive dimension count. PMID:22460787

Planar harmonic polynomials of type B

NASA Astrophysics Data System (ADS)

Dunkl, Charles F.

1999-11-01

The hyperoctahedral group acting on icons/Journals/Common/BbbR" ALT="BbbR" ALIGN="TOP"/>N is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators {Ti:1icons/Journals/Common/leq" ALT="leq" ALIGN="TOP"/> iicons/Journals/Common/leq" ALT="leq" ALIGN="TOP"/> N}. These operators are analogous to partial derivative operators. This paper finds all the polynomials h on icons/Journals/Common/BbbR" ALT="BbbR" ALIGN="TOP"/>N which are harmonic, icons/Journals/Common/Delta" ALT="Delta" ALIGN="TOP"/>Bh = 0 and annihilated by Ti for i>2, where the Laplacian 0305-4470/32/46/308/img1" ALT="(sum). They are given explicitly in terms of a novel basis of polynomials, defined by generating functions. The harmonic polynomials can be used to find wavefunctions for the quantum many-body spin Calogero model.

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.

Applicability of the polynomial chaos expansion method for personalization of a cardiovascular pulse wave propagation model.

PubMed

Huberts, W; Donders, W P; Delhaas, T; van de Vosse, F N

2014-12-01

Patient-specific modeling requires model personalization, which can be achieved in an efficient manner by parameter fixing and parameter prioritization. An efficient variance-based method is using generalized polynomial chaos expansion (gPCE), but it has not been applied in the context of model personalization, nor has it ever been compared with standard variance-based methods for models with many parameters. In this work, we apply the gPCE method to a previously reported pulse wave propagation model and compare the conclusions for model personalization with that of a reference analysis performed with Saltelli's efficient Monte Carlo method. We furthermore differentiate two approaches for obtaining the expansion coefficients: one based on spectral projection (gPCE-P) and one based on least squares regression (gPCE-R). It was found that in general the gPCE yields similar conclusions as the reference analysis but at much lower cost, as long as the polynomial metamodel does not contain unnecessary high order terms. Furthermore, the gPCE-R approach generally yielded better results than gPCE-P. The weak performance of the gPCE-P can be attributed to the assessment of the expansion coefficients using the Smolyak algorithm, which might be hampered by the high number of model parameters and/or by possible non-smoothness in the output space. Copyright © 2014 John Wiley & Sons, Ltd.

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.

Low-order auditory Zernike moment: a novel approach for robust music identification in the compressed domain

NASA Astrophysics Data System (ADS)

Li, Wei; Xiao, Chuan; Liu, Yaduo

2013-12-01

Audio identification via fingerprint has been an active research field for years. However, most previously reported methods work on the raw audio format in spite of the fact that nowadays compressed format audio, especially MP3 music, has grown into the dominant way to store music on personal computers and/or transmit it over the Internet. It will be interesting if a compressed unknown audio fragment could be directly recognized from the database without decompressing it into the wave format at first. So far, very few algorithms run directly on the compressed domain for music information retrieval, and most of them take advantage of the modified discrete cosine transform coefficients or derived cepstrum and energy type of features. As a first attempt, we propose in this paper utilizing compressed domain auditory Zernike moment adapted from image processing techniques as the key feature to devise a novel robust audio identification algorithm. Such fingerprint exhibits strong robustness, due to its statistically stable nature, against various audio signal distortions such as recompression, noise contamination, echo adding, equalization, band-pass filtering, pitch shifting, and slight time scale modification. Experimental results show that in a music database which is composed of 21,185 MP3 songs, a 10-s long music segment is able to identify its original near-duplicate recording, with average top-5 hit rate up to 90% or above even under severe audio signal distortions.

Dynamic Bidirectional Reflectance Distribution Functions: Measurement and Representation

DTIC Science & Technology

2008-02-01

be included in the harmonic fits. Other sets of orthogonal functions such as Zernike polynomials have also been used to characterize BRDF and could...reflectance spectra of 3D objects,” Proc. SPIE 4663, 370–378 2001. 13J. R. Shell II, C. Salvagio, and J. R. Schott, “A novel BRDF measurement technique

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.

A note on the zeros of Freud-Sobolev orthogonal polynomials

NASA Astrophysics Data System (ADS)

Moreno-Balcazar, Juan J.

2007-10-01

We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e-x4 on are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e-x4. Some numerical examples are shown.

A positivity result in the theory of Macdonald polynomials

PubMed Central

Garsia, A. M.; Haglund, J.

2001-01-01

We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection 𝒟n of Dyck paths Π of length 2n yielding Cn(q, t) = ∑π ta(Π)qb(Π). Our proof is based on a recursion for Cn(q, t) suggested by a pair of statistics recently proposed by J. Haglund. One of the byproducts of our results is a proof of the validity of Haglund's conjecture. PMID:11274351

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.

Fast beampattern evaluation by polynomial rooting

NASA Astrophysics Data System (ADS)

Häcker, P.; Uhlich, S.; Yang, B.

2011-07-01

Current automotive radar systems measure the distance, the relative velocity and the direction of objects in their environment. This information enables the car to support the driver. The direction estimation capabilities of a sensor array depend on its beampattern. To find the array configuration leading to the best angle estimation by a global optimization algorithm, a huge amount of beampatterns have to be calculated to detect their maxima. In this paper, a novel algorithm is proposed to find all maxima of an array's beampattern fast and reliably, leading to accelerated array optimizations. The algorithm works for arrays having the sensors on a uniformly spaced grid. We use a general version of the gcd (greatest common divisor) function in order to write the problem as a polynomial. We differentiate and root the polynomial to get the extrema of the beampattern. In addition, we show a method to reduce the computational burden even more by decreasing the order of the polynomial.

Reduced-order modeling with sparse polynomial chaos expansion and dimension reduction for evaluating the impact of CO2 and brine leakage on groundwater

NASA Astrophysics Data System (ADS)

Liu, Y.; Zheng, L.; Pau, G. S. H.

2016-12-01

A careful assessment of the risk associated with geologic CO2 storage is critical to the deployment of large-scale storage projects. While numerical modeling is an indispensable tool for risk assessment, there has been increasing need in considering and addressing uncertainties in the numerical models. However, uncertainty analyses have been significantly hindered by the computational complexity of the model. As a remedy, reduced-order models (ROM), which serve as computationally efficient surrogates for high-fidelity models (HFM), have been employed. The ROM is constructed at the expense of an initial set of HFM simulations, and afterwards can be relied upon to predict the model output values at minimal cost. The ROM presented here is part of National Risk Assessment Program (NRAP) and intends to predict the water quality change in groundwater in response to hypothetical CO2 and brine leakage. The HFM based on which the ROM is derived is a multiphase flow and reactive transport model, with 3-D heterogeneous flow field and complex chemical reactions including aqueous complexation, mineral dissolution/precipitation, adsorption/desorption via surface complexation and cation exchange. Reduced-order modeling techniques based on polynomial basis expansion, such as polynomial chaos expansion (PCE), are widely used in the literature. However, the accuracy of such ROMs can be affected by the sparse structure of the coefficients of the expansion. Failing to identify vanishing polynomial coefficients introduces unnecessary sampling errors, the accumulation of which deteriorates the accuracy of the ROMs. To address this issue, we treat the PCE as a sparse Bayesian learning (SBL) problem, and the sparsity is obtained by detecting and including only the non-zero PCE coefficients one at a time by iteratively selecting the most contributing coefficients. The computational complexity due to predicting the entire 3-D concentration fields is further mitigated by a dimension

Numerical solutions for Helmholtz equations using Bernoulli polynomials

NASA Astrophysics Data System (ADS)

Bicer, Kubra Erdem; Yalcinbas, Salih

2017-07-01

This paper reports a new numerical method based on Bernoulli polynomials for the solution of Helmholtz equations. The method uses matrix forms of Bernoulli polynomials and their derivatives by means of collocation points. Aim of this paper is to solve Helmholtz equations using this matrix relations.

Computing Tutte polynomials of contact networks in classrooms

NASA Astrophysics Data System (ADS)

Hincapié, Doracelly; Ospina, Juan

2013-05-01

Objective: The topological complexity of contact networks in classrooms and the potential transmission of an infectious disease were analyzed by sex and age. Methods: The Tutte polynomials, some topological properties and the number of spanning trees were used to algebraically compute the topological complexity. Computations were made with the Maple package GraphTheory. Published data of mutually reported social contacts within a classroom taken from primary school, consisting of children in the age ranges of 4-5, 7-8 and 10-11, were used. Results: The algebraic complexity of the Tutte polynomial and the probability of disease transmission increases with age. The contact networks are not bipartite graphs, gender segregation was observed especially in younger children. Conclusion: Tutte polynomials are tools to understand the topology of the contact networks and to derive numerical indexes of such topologies. It is possible to establish relationships between the Tutte polynomial of a given contact network and the potential transmission of an infectious disease within such network

Polynomial interpolation and sums of powers of integers

NASA Astrophysics Data System (ADS)

Cereceda, José Luis

2017-02-01

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, Pk(n) and Qk(n), such that Pk(n) = Qk(n) = fk(n) for n = 1, 2,… , k, where fk(1), fk(2),… , fk(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that fk(n) is given by the sum of powers of the first n positive integers Sk(n) = 1k + 2k + ṡṡṡ + nk, and show that Sk(n) admits the polynomial representations Sk(n) = Pk(n) and Sk(n) = Qk(n) for all n = 1, 2,… , and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for Sk(n) alternative to the well-known formula of Bernoulli.

Learning polynomial feedforward neural networks by genetic programming and backpropagation.

PubMed

Nikolaev, N Y; Iba, H

2003-01-01

This paper presents an approach to learning polynomial feedforward neural networks (PFNNs). The approach suggests, first, finding the polynomial network structure by means of a population-based search technique relying on the genetic programming paradigm, and second, further adjustment of the best discovered network weights by an especially derived backpropagation algorithm for higher order networks with polynomial activation functions. These two stages of the PFNN learning process enable us to identify networks with good training as well as generalization performance. Empirical results show that this approach finds PFNN which outperform considerably some previous constructive polynomial network algorithms on processing benchmark time series.

Higher order derivatives of R-Jacobi polynomials

NASA Astrophysics Data System (ADS)

Das, Sourav; Swaminathan, A.

2016-06-01

In this work, the R-Jacobi polynomials defined on the nonnegative real axis related to F-distribution are considered. Using their Sturm-Liouville system higher order derivatives are constructed. Orthogonality property of these higher ordered R-Jacobi polynomials are obtained besides their normal form, self-adjoint form and hypergeometric representation. Interesting results on the Interpolation formula and Gaussian quadrature formulae are obtained with numerical examples.

Torus Knot Polynomials and Susy Wilson Loops

NASA Astrophysics Data System (ADS)

Giasemidis, Georgios; Tierz, Miguel

2014-12-01

We give, using an explicit expression obtained in (Jones V, Ann Math 126:335, 1987), a basic hypergeometric representation of the HOMFLY polynomial of ( n, m) torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result, the symmetry and the leading polynomial at large N are explicit. We show the latter to be the Wilson loop of 2d Yang-Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang-Mills theory, which is known to give averages of Wilson loops in = 4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones-Rosso representation in terms of q-harmonic oscillators.

Fuzzy logic controller for the LOLA AO tip-tilt corrector system

NASA Astrophysics Data System (ADS)

Sotelo, Pablo D.; Flores, Ruben; Garfias, Fernando; Cuevas, Salvador

1998-09-01

At the INSTITUTO DE ASTRONOMIA we developed an adaptive optics system for the correction of the two first orders of the Zernike polynomials measuring the image controid. Here we discus the two system modalities based in two different control strategies and we present simulations comparing the systems. For the classic control system we present telescope results.

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.

Grid and basis adaptive polynomial chaos techniques for sensitivity and uncertainty analysis

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

Perkó, Zoltán, E-mail: Z.Perko@tudelft.nl; Gilli, Luca, E-mail: Gilli@nrg.eu; Lathouwers, Danny, E-mail: D.Lathouwers@tudelft.nl

2014-03-01

The demand for accurate and computationally affordable sensitivity and uncertainty techniques is constantly on the rise and has become especially pressing in the nuclear field with the shift to Best Estimate Plus Uncertainty methodologies in the licensing of nuclear installations. Besides traditional, already well developed methods – such as first order perturbation theory or Monte Carlo sampling – Polynomial Chaos Expansion (PCE) has been given a growing emphasis in recent years due to its simple application and good performance. This paper presents new developments of the research done at TU Delft on such Polynomial Chaos (PC) techniques. Our work ismore » focused on the Non-Intrusive Spectral Projection (NISP) approach and adaptive methods for building the PCE of responses of interest. Recent efforts resulted in a new adaptive sparse grid algorithm designed for estimating the PC coefficients. The algorithm is based on Gerstner's procedure for calculating multi-dimensional integrals but proves to be computationally significantly cheaper, while at the same it retains a similar accuracy as the original method. More importantly the issue of basis adaptivity has been investigated and two techniques have been implemented for constructing the sparse PCE of quantities of interest. Not using the traditional full PC basis set leads to further reduction in computational time since the high order grids necessary for accurately estimating the near zero expansion coefficients of polynomial basis vectors not needed in the PCE can be excluded from the calculation. Moreover the sparse PC representation of the response is easier to handle when used for sensitivity analysis or uncertainty propagation due to the smaller number of basis vectors. The developed grid and basis adaptive methods have been implemented in Matlab as the Fully Adaptive Non-Intrusive Spectral Projection (FANISP) algorithm and were tested on four analytical problems. These show consistent good

On direct theorems for best polynomial approximation

NASA Astrophysics Data System (ADS)

Auad, A. A.; AbdulJabbar, R. S.

2018-05-01

This paper is to obtain similarity for the best approximation degree of functions, which are unbounded in L p,α (A = [0,1]), which called weighted space by algebraic polynomials. {E}nH{(f)}p,α and the best approximation degree in the same space on the interval [0,2π] by trigonometric polynomials {E}nT{(f)}p,α of direct wellknown theorems in forms the average modules.

Spline-based high-accuracy piecewise-polynomial phase-to-sinusoid amplitude converters.

PubMed

Petrinović, Davor; Brezović, Marko

2011-04-01

We propose a method for direct digital frequency synthesis (DDS) using a cubic spline piecewise-polynomial model for a phase-to-sinusoid amplitude converter (PSAC). This method offers maximum smoothness of the output signal. Closed-form expressions for the cubic polynomial coefficients are derived in the spectral domain and the performance analysis of the model is given in the time and frequency domains. We derive the closed-form performance bounds of such DDS using conventional metrics: rms and maximum absolute errors (MAE) and maximum spurious free dynamic range (SFDR) measured in the discrete time domain. The main advantages of the proposed PSAC are its simplicity, analytical tractability, and inherent numerical stability for high table resolutions. Detailed guidelines for a fixed-point implementation are given, based on the algebraic analysis of all quantization effects. The results are verified on 81 PSAC configurations with the output resolutions from 5 to 41 bits by using a bit-exact simulation. The VHDL implementation of a high-accuracy DDS based on the proposed PSAC with 28-bit input phase word and 32-bit output value achieves SFDR of its digital output signal between 180 and 207 dB, with a signal-to-noise ratio of 192 dB. Its implementation requires only one 18 kB block RAM and three 18-bit embedded multipliers in a typical field-programmable gate array (FPGA) device. © 2011 IEEE

Maximal aggregation of polynomial dynamical systems

PubMed Central

Cardelli, Luca; Tschaikowski, Max

2017-01-01

Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory. PMID:28878023

Animating Nested Taylor Polynomials to Approximate a Function

ERIC Educational Resources Information Center

Mazzone, Eric F.; Piper, Bruce R.

2010-01-01

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

Optical analysis of thermal induced structural distortions

NASA Technical Reports Server (NTRS)

Weinswig, Shepard; Hookman, Robert A.

1991-01-01

The techniques used for the analysis of thermally induced structural distortions of optical components such as scanning mirrors and telescope optics are outlined. Particular attention is given to the methodology used in the thermal and structural analysis of the GOES scan mirror, the optical analysis using Zernike coefficients, and the optical system performance evaluation. It is pointed out that the use of Zernike coefficients allows an accurate, effective, and simple linkage between thermal/mechanical effects and the optical design.

Quantization of gauge fields, graph polynomials and graph homology

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

Kreimer, Dirk, E-mail: kreimer@physik.hu-berlin.de; Sars, Matthias; Suijlekom, Walter D. van

2013-09-15

We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial–we call it cycle homology–and by graph homology.more » -- Highlights: •We derive gauge theory Feynman from scalar field theory with 3-valent vertices. •We clarify the role of graph homology and cycle homology. •We use parametric renormalization and the new corolla polynomial.« less

From Chebyshev to Bernstein: A Tour of Polynomials Small and Large

ERIC Educational Resources Information Center

Boelkins, Matthew; Miller, Jennifer; Vugteveen, Benjamin

2006-01-01

Consider the family of monic polynomials of degree n having zeros at -1 and +1 and all their other real zeros in between these two values. This article explores the size of these polynomials using the supremum of the absolute value on [-1, 1], showing that scaled Chebyshev and Bernstein polynomials give the extremes.

Polynomial equations for science orbits around Europa

NASA Astrophysics Data System (ADS)

Cinelli, Marco; Circi, Christian; Ortore, Emiliano

2015-07-01

In this paper, the design of science orbits for the observation of a celestial body has been carried out using polynomial equations. The effects related to the main zonal harmonics of the celestial body and the perturbation deriving from the presence of a third celestial body have been taken into account. The third body describes a circular and equatorial orbit with respect to the primary body and, for its disturbing potential, an expansion in Legendre polynomials up to the second order has been considered. These polynomial equations allow the determination of science orbits around Jupiter's satellite Europa, where the third body gravitational attraction represents one of the main forces influencing the motion of an orbiting probe. Thus, the retrieved relationships have been applied to this moon and periodic sun-synchronous and multi-sun-synchronous orbits have been determined. Finally, numerical simulations have been carried out to validate the analytical results.

Solving the interval type-2 fuzzy polynomial equation using the ranking method

NASA Astrophysics Data System (ADS)

Rahman, Nurhakimah Ab.; Abdullah, Lazim

2014-07-01

Polynomial equations with trapezoidal and triangular fuzzy numbers have attracted some interest among researchers in mathematics, engineering and social sciences. There are some methods that have been developed in order to solve these equations. In this study we are interested in introducing the interval type-2 fuzzy polynomial equation and solving it using the ranking method of fuzzy numbers. The ranking method concept was firstly proposed to find real roots of fuzzy polynomial equation. Therefore, the ranking method is applied to find real roots of the interval type-2 fuzzy polynomial equation. We transform the interval type-2 fuzzy polynomial equation to a system of crisp interval type-2 fuzzy polynomial equation. This transformation is performed using the ranking method of fuzzy numbers based on three parameters, namely value, ambiguity and fuzziness. Finally, we illustrate our approach by numerical example.

Path-integral and Ornstein-Zernike study of quantum fluid structures on the crystallization line

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

Sesé, Luis M., E-mail: msese@ccia.uned.es

2016-03-07

Liquid neon, liquid para-hydrogen, and the quantum hard-sphere fluid are studied with path integral Monte Carlo simulations and the Ornstein-Zernike pair equation on their respective crystallization lines. The results cover the whole sets of structures in the r-space and the k-space and, for completeness, the internal energies, pressures and isothermal compressibilities. Comparison with experiment is made wherever possible, and the possibilities of establishing k-space criteria for quantum crystallization based on the path-integral centroids are discussed. In this regard, the results show that the centroid structure factor contains two significant parameters related to its main peak features (amplitude and shape) thatmore » can be useful to characterize freezing.« less

Riemann-Liouville Fractional Calculus of Certain Finite Class of Classical Orthogonal Polynomials

NASA Astrophysics Data System (ADS)

Malik, Pradeep; Swaminathan, A.

2010-11-01

In this work we consider certain class of classical orthogonal polynomials defined on the positive real line. These polynomials have their weight function related to the probability density function of F distribution and are finite in number up to orthogonality. We generalize these polynomials for fractional order by considering the Riemann-Liouville type operator on these polynomials. Various properties like explicit representation in terms of hypergeometric functions, differential equations, recurrence relations are derived.

Precise starshade stationkeeping and pointing with a Zernike wavefront sensor

NASA Astrophysics Data System (ADS)

Bottom, Michael; Martin, Stefan; Seubert, Carl; Cady, Eric; Zareh, Shannon Kian; Shaklan, Stuart

2017-09-01

Starshades, large occulters positioned tens of thousands of kilometers in front of space telescopes, offer one of the few paths to imaging and characterizing Earth-like extrasolar planets. However, for a starshade to generate a sufficiently dark shadow on the telescope, the two must be coaligned to just 1 meter laterally, even at these large separations. The principal challenge to achieving this level of control is in determining the position of the starshade with respect to the space telescope. In this paper, we present numerical simulations and laboratory results demonstrating that a Zernike wavefront sensor coupled to a WFIRST-type telescope is able to deliver the stationkeeping precision required, by measuring light outside of the science wavelengths. The sensor can determine the starshade lateral position to centimeter level in seconds of open shutter time for stars brighter than eighth magnitude, with a capture range of 10 meters. We discuss the potential for fast (ms) tip/tilt pointing control at the milli-arcsecond level by illuminating the sensor with a laser mounted on the starshade. Finally, we present early laboratory results.

Generalized Freud's equation and level densities with polynomial potential

NASA Astrophysics Data System (ADS)

Boobna, Akshat; Ghosh, Saugata

2013-08-01

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

Asymptotic formulae for the zeros of orthogonal polynomials

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

Badkov, V M

2012-09-30

Let p{sub n}(t) be an algebraic polynomial that is orthonormal with weight p(t) on the interval [-1, 1]. When p(t) is a perturbation (in certain limits) of the Chebyshev weight of the first kind, the zeros of the polynomial p{sub n}( cos {tau}) and the differences between pairs of (not necessarily consecutive) zeros are shown to satisfy asymptotic formulae as n{yields}{infinity}, which hold uniformly with respect to the indices of the zeros. Similar results are also obtained for perturbations of the Chebyshev weight of the second kind. First, some preliminary results on the asymptotic behaviour of the difference between twomore » zeros of an orthogonal trigonometric polynomial, which are needed, are established. Bibliography: 15 titles.« less

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.

Multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials

NASA Astrophysics Data System (ADS)

Odake, Satoru; Sasaki, Ryu

2017-04-01

As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials in the framework of ‘discrete quantum mechanics’ with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual state vectors. The virtual state vectors are the solutions of the matrix Schrödinger equation on all the lattice points having negative energies and infinite norm. This is in good contrast to the (q-)Racah systems defined on a finite lattice, in which the ‘virtual state’ vectors satisfy the matrix Schrödinger equation except for one of the two boundary points.

Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials

PubMed Central

Corteel, Sylvie; Williams, Lauren K.

2010-01-01

We introduce some combinatorial objects called staircase tableaux, which have cardinality 4nn !, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics introduced in the late 1960s, which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and translation in protein synthesis. In its most general form, particles may enter and exit at the left with probabilities α and γ, and they may exit and enter at the right with probabilities β and δ. In the bulk, the probability of hopping left is q times the probability of hopping right. Our first result is a formula for the stationary distribution of the ASEP with all parameters general, in terms of staircase tableaux. Our second result is a formula for the moments of (the weight function of) Askey-Wilson polynomials, also in terms of staircase tableaux. Since the 1980s there has been a great deal of work giving combinatorial formulas for moments of classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre); among these polynomials, the Askey-Wilson polynomials are the most important, because they are at the top of the hierarchy of classical orthogonal polynomials. PMID:20348417

Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials.

PubMed

Corteel, Sylvie; Williams, Lauren K

2010-04-13

We introduce some combinatorial objects called staircase tableaux, which have cardinality 4(n)n!, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics introduced in the late 1960s, which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and translation in protein synthesis. In its most general form, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. Our first result is a formula for the stationary distribution of the ASEP with all parameters general, in terms of staircase tableaux. Our second result is a formula for the moments of (the weight function of) Askey-Wilson polynomials, also in terms of staircase tableaux. Since the 1980s there has been a great deal of work giving combinatorial formulas for moments of classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre); among these polynomials, the Askey-Wilson polynomials are the most important, because they are at the top of the hierarchy of classical orthogonal polynomials.

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.

Euler polynomials and identities for non-commutative operators

NASA Astrophysics Data System (ADS)

De Angelis, Valerio; Vignat, Christophe

2015-12-01

Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt [Phys. Rev. D 54(12), 7710-7723 (1996)], expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, by Pain [J. Phys. A: Math. Theor. 46, 035304 (2013)], links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring [J. Phys. A: Math. Gen. 39, 9269 (2006)] in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.

On polynomial selection for the general number field sieve

NASA Astrophysics Data System (ADS)

Kleinjung, Thorsten

2006-12-01

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

Calculators and Polynomial Evaluation.

ERIC Educational Resources Information Center

Weaver, J. F.

The intent of this paper is to suggest and illustrate how electronic hand-held calculators, especially non-programmable ones with limited data-storage capacity, can be used to advantage by students in one particular aspect of work with polynomial functions. The basic mathematical background upon which calculator application is built is summarized.…

Polynomial Conjoint Analysis of Similarities: A Model for Constructing Polynomial Conjoint Measurement Algorithms.

ERIC Educational Resources Information Center

Young, Forrest W.

A model permitting construction of algorithms for the polynomial conjoint analysis of similarities is presented. This model, which is based on concepts used in nonmetric scaling, permits one to obtain the best approximate solution. The concepts used to construct nonmetric scaling algorithms are reviewed. Finally, examples of algorithmic models for…

Thermodynamic characterization of networks using graph polynomials

NASA Astrophysics Data System (ADS)

Ye, Cheng; Comin, César H.; Peron, Thomas K. DM.; Silva, Filipi N.; Rodrigues, Francisco A.; Costa, Luciano da F.; Torsello, Andrea; Hancock, Edwin R.

2015-09-01

In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.

Polynomial Interpolation and Sums of Powers of Integers

ERIC Educational Resources Information Center

Cereceda, José Luis

2017-01-01

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P[subscript k](n) and Q[subscript k](n), such that P[subscript k](n) = Q[subscript k](n) = f[subscript k](n) for n = 1, 2,… , k, where f[subscript k](1), f[subscript k](2),… , f[subscript k](k) are k arbitrarily chosen…

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.

Accurate Estimation of Solvation Free Energy Using Polynomial Fitting Techniques

PubMed Central

Shyu, Conrad; Ytreberg, F. Marty

2010-01-01

This report details an approach to improve the accuracy of free energy difference estimates using thermodynamic integration data (slope of the free energy with respect to the switching variable λ) and its application to calculating solvation free energy. The central idea is to utilize polynomial fitting schemes to approximate the thermodynamic integration data to improve the accuracy of the free energy difference estimates. Previously, we introduced the use of polynomial regression technique to fit thermodynamic integration data (Shyu and Ytreberg, J Comput Chem 30: 2297–2304, 2009). In this report we introduce polynomial and spline interpolation techniques. Two systems with analytically solvable relative free energies are used to test the accuracy of the interpolation approach. We also use both interpolation and regression methods to determine a small molecule solvation free energy. Our simulations show that, using such polynomial techniques and non-equidistant λ values, the solvation free energy can be estimated with high accuracy without using soft-core scaling and separate simulations for Lennard-Jones and partial charges. The results from our study suggest these polynomial techniques, especially with use of non-equidistant λ values, improve the accuracy for ΔF estimates without demanding additional simulations. We also provide general guidelines for use of polynomial fitting to estimate free energy. To allow researchers to immediately utilize these methods, free software and documentation is provided via http://www.phys.uidaho.edu/ytreberg/software. PMID:20623657

Using Tutte polynomials to analyze the structure of the benzodiazepines

NASA Astrophysics Data System (ADS)

Cadavid Muñoz, Juan José

2014-05-01

Graph theory in general and Tutte polynomials in particular, are implemented for analyzing the chemical structure of the benzodiazepines. Similarity analysis are used with the Tutte polynomials for finding other molecules that are similar to the benzodiazepines and therefore that might show similar psycho-active actions for medical purpose, in order to evade the drawbacks associated to the benzodiazepines based medicine. For each type of benzodiazepines, Tutte polynomials are computed and some numeric characteristics are obtained, such as the number of spanning trees and the number of spanning forests. Computations are done using the computer algebra Maple's GraphTheory package. The obtained analytical results are of great importance in pharmaceutical engineering. As a future research line, the usage of the chemistry computational program named Spartan, will be used to extent and compare it with the obtained results from the Tutte polynomials of benzodiazepines.

A polynomial based model for cell fate prediction in human diseases.

PubMed

Ma, Lichun; Zheng, Jie

2017-12-21

Cell fate regulation directly affects tissue homeostasis and human health. Research on cell fate decision sheds light on key regulators, facilitates understanding the mechanisms, and suggests novel strategies to treat human diseases that are related to abnormal cell development. In this study, we proposed a polynomial based model to predict cell fate. This model was derived from Taylor series. As a case study, gene expression data of pancreatic cells were adopted to test and verify the model. As numerous features (genes) are available, we employed two kinds of feature selection methods, i.e. correlation based and apoptosis pathway based. Then polynomials of different degrees were used to refine the cell fate prediction function. 10-fold cross-validation was carried out to evaluate the performance of our model. In addition, we analyzed the stability of the resultant cell fate prediction model by evaluating the ranges of the parameters, as well as assessing the variances of the predicted values at randomly selected points. Results show that, within both the two considered gene selection methods, the prediction accuracies of polynomials of different degrees show little differences. Interestingly, the linear polynomial (degree 1 polynomial) is more stable than others. When comparing the linear polynomials based on the two gene selection methods, it shows that although the accuracy of the linear polynomial that uses correlation analysis outcomes is a little higher (achieves 86.62%), the one within genes of the apoptosis pathway is much more stable. Considering both the prediction accuracy and the stability of polynomial models of different degrees, the linear model is a preferred choice for cell fate prediction with gene expression data of pancreatic cells. The presented cell fate prediction model can be extended to other cells, which may be important for basic research as well as clinical study of cell development related diseases.

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.

Robust stability of fractional order polynomials with complicated uncertainty structure

PubMed Central

Şenol, Bilal; Pekař, Libor

2017-01-01

The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition. PMID:28662173

Polynomial probability distribution estimation using the method of moments

PubMed Central

Mattsson, Lars; Rydén, Jesper

2017-01-01

We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram–Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation. PMID:28394949

Polynomial probability distribution estimation using the method of moments.

PubMed

Munkhammar, Joakim; Mattsson, Lars; Rydén, Jesper

2017-01-01

We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram-Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation.

Fast human pose estimation using 3D Zernike descriptors

NASA Astrophysics Data System (ADS)

Berjón, Daniel; Morán, Francisco

2012-03-01

Markerless video-based human pose estimation algorithms face a high-dimensional problem that is frequently broken down into several lower-dimensional ones by estimating the pose of each limb separately. However, in order to do so they need to reliably locate the torso, for which they typically rely on time coherence and tracking algorithms. Their losing track usually results in catastrophic failure of the process, requiring human intervention and thus precluding their usage in real-time applications. We propose a very fast rough pose estimation scheme based on global shape descriptors built on 3D Zernike moments. Using an articulated model that we configure in many poses, a large database of descriptor/pose pairs can be computed off-line. Thus, the only steps that must be done on-line are the extraction of the descriptors for each input volume and a search against the database to get the most likely poses. While the result of such process is not a fine pose estimation, it can be useful to help more sophisticated algorithms to regain track or make more educated guesses when creating new particles in particle-filter-based tracking schemes. We have achieved a performance of about ten fps on a single computer using a database of about one million entries.

Processing short-term and long-term information with a combination of polynomial approximation techniques and time-delay neural networks.

PubMed

Fuchs, Erich; Gruber, Christian; Reitmaier, Tobias; Sick, Bernhard

2009-09-01

Neural networks are often used to process temporal information, i.e., any kind of information related to time series. In many cases, time series contain short-term and long-term trends or behavior. This paper presents a new approach to capture temporal information with various reference periods simultaneously. A least squares approximation of the time series with orthogonal polynomials will be used to describe short-term trends contained in a signal (average, increase, curvature, etc.). Long-term behavior will be modeled with the tapped delay lines of a time-delay neural network (TDNN). This network takes the coefficients of the orthogonal expansion of the approximating polynomial as inputs such considering short-term and long-term information efficiently. The advantages of the method will be demonstrated by means of artificial data and two real-world application examples, the prediction of the user number in a computer network and online tool wear classification in turning.

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.

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.

Design and Use of a Learning Object for Finding Complex Polynomial Roots

ERIC Educational Resources Information Center

Benitez, Julio; Gimenez, Marcos H.; Hueso, Jose L.; Martinez, Eulalia; Riera, Jaime

2013-01-01

Complex numbers are essential in many fields of engineering, but students often fail to have a natural insight of them. We present a learning object for the study of complex polynomials that graphically shows that any complex polynomials has a root and, furthermore, is useful to find the approximate roots of a complex polynomial. Moreover, we…

Segmented Mirror Telescope Model and Simulation

DTIC Science & Technology

2011-06-01

mirror surface is treated as a grid of masses and springs. The actuators have surface normal forces applied to individual masses. The equation to...are not widely treated in the literature. The required modifications for the wavefront reconstruction algorithm of a circular aperture to correctly...Zernike polynomials, which are particularly suitable to describe the common optical character- izations of astigmatism , coma, defocus and others [9

Zernike Wavefront Sensor Modeling Development for LOWFS on WFIRST-AFTA

NASA Technical Reports Server (NTRS)

Wang, Xu; Wallace, J. Kent; Shi, Fang

2015-01-01

WFIRST-AFTA design makes use of an existing 2.4m telescope for direct imaging of exoplanets. To maintain the high contrast needed for the coronagraph, wavefront error (WFE) of the optical system needs to be continuously sensed and controlled. Low Order Wavefront Sensing (LOWFS) uses the rejected starlight from an immediate focal plane to sense wavefront changes (mostly thermally induced low order WFE) by combining the LOWFS mask (a phase plate located at the small center region with reflective layer) with the starlight rejection masks, i.e. Hybrid Lyot Coronagraph (HLC)'s occulter or Shaped Pupil Coronagraph (SPC)'s field stop. Zernike wavefront sensor (ZWFS) measures phase via the phase-contrast method and is known to be photon noise optimal for measuring low order aberrations. Recently, ZWFS was selected as the baseline LOWFS technology on WFIST/AFTA for its good sensitivity, accuracy, and its easy integration with the starlight rejection mask. In this paper, we review the theory of ZWFS operation, describe the ZWFS algorithm development, and summarize various numerical sensitivity studies on the sensor performance. In the end, the predicted sensor performance on SPC and HLC configurations are presented.

A weighted ℓ{sub 1}-minimization approach for sparse polynomial chaos expansions

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

Peng, Ji; Hampton, Jerrad; Doostan, Alireza, E-mail: alireza.doostan@colorado.edu

2014-06-15

This work proposes a method for sparse polynomial chaos (PC) approximation of high-dimensional stochastic functions based on non-adapted random sampling. We modify the standard ℓ{sub 1}-minimization algorithm, originally proposed in the context of compressive sampling, using a priori information about the decay of the PC coefficients, when available, and refer to the resulting algorithm as weightedℓ{sub 1}-minimization. We provide conditions under which we may guarantee recovery using this weighted scheme. Numerical tests are used to compare the weighted and non-weighted methods for the recovery of solutions to two differential equations with high-dimensional random inputs: a boundary value problem with amore » random elliptic operator and a 2-D thermally driven cavity flow with random boundary condition.« less

Polynomial chaos expansion with random and fuzzy variables

NASA Astrophysics Data System (ADS)

Jacquelin, E.; Friswell, M. I.; Adhikari, S.; Dessombz, O.; Sinou, J.-J.

2016-06-01

A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.

Polynomial algebra of discrete models in systems biology.

PubMed

Veliz-Cuba, Alan; Jarrah, Abdul Salam; Laubenbacher, Reinhard

2010-07-01

An increasing number of discrete mathematical models are being published in Systems Biology, ranging from Boolean network models to logical models and Petri nets. They are used to model a variety of biochemical networks, such as metabolic networks, gene regulatory networks and signal transduction networks. There is increasing evidence that such models can capture key dynamic features of biological networks and can be used successfully for hypothesis generation. This article provides a unified framework that can aid the mathematical analysis of Boolean network models, logical models and Petri nets. They can be represented as polynomial dynamical systems, which allows the use of a variety of mathematical tools from computer algebra for their analysis. Algorithms are presented for the translation into polynomial dynamical systems. Examples are given of how polynomial algebra can be used for the model analysis. alanavc@vt.edu Supplementary data are available at Bioinformatics online.

High degree interpolation polynomial in Newton form

NASA Technical Reports Server (NTRS)

Tal-Ezer, Hillel

1988-01-01

Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4.

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.

Design of hybrid radial basis function neural networks (HRBFNNs) realized with the aid of hybridization of fuzzy clustering method (FCM) and polynomial neural networks (PNNs).

PubMed

Huang, Wei; Oh, Sung-Kwun; Pedrycz, Witold

2014-12-01

In this study, we propose Hybrid Radial Basis Function Neural Networks (HRBFNNs) realized with the aid of fuzzy clustering method (Fuzzy C-Means, FCM) and polynomial neural networks. Fuzzy clustering used to form information granulation is employed to overcome a possible curse of dimensionality, while the polynomial neural network is utilized to build local models. Furthermore, genetic algorithm (GA) is exploited here to optimize the essential design parameters of the model (including fuzzification coefficient, the number of input polynomial fuzzy neurons (PFNs), and a collection of the specific subset of input PFNs) of the network. To reduce dimensionality of the input space, principal component analysis (PCA) is considered as a sound preprocessing vehicle. The performance of the HRBFNNs is quantified through a series of experiments, in which we use several modeling benchmarks of different levels of complexity (different number of input variables and the number of available data). A comparative analysis reveals that the proposed HRBFNNs exhibit higher accuracy in comparison to the accuracy produced by some models reported previously in the literature. Copyright © 2014 Elsevier Ltd. All rights reserved.

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.

Transfer matrix computation of generalized critical polynomials in percolation

DOE PAGES

Scullard, Christian R.; Jacobsen, Jesper Lykke

2012-09-27

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

a Unified Matrix Polynomial Approach to Modal Identification

NASA Astrophysics Data System (ADS)

Allemang, R. J.; Brown, D. L.

1998-04-01

One important current focus of modal identification is a reformulation of modal parameter estimation algorithms into a single, consistent mathematical formulation with a corresponding set of definitions and unifying concepts. Particularly, a matrix polynomial approach is used to unify the presentation with respect to current algorithms such as the least-squares complex exponential (LSCE), the polyreference time domain (PTD), Ibrahim time domain (ITD), eigensystem realization algorithm (ERA), rational fraction polynomial (RFP), polyreference frequency domain (PFD) and the complex mode indication function (CMIF) methods. Using this unified matrix polynomial approach (UMPA) allows a discussion of the similarities and differences of the commonly used methods. the use of least squares (LS), total least squares (TLS), double least squares (DLS) and singular value decomposition (SVD) methods is discussed in order to take advantage of redundant measurement data. Eigenvalue and SVD transformation methods are utilized to reduce the effective size of the resulting eigenvalue-eigenvector problem as well.

Hermite polynomials and quasi-classical asymptotics

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

Ali, S. Twareque, E-mail: twareque.ali@concordia.ca; Engliš, Miroslav, E-mail: englis@math.cas.cz

2014-04-15

We study an unorthodox variant of the Berezin-Toeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated quasi-classical asymptotics.

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

A simple method for finding the scattering coefficients of quantum graphs

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

Cottrell, Seth S.

2015-09-15

Quantum walks are roughly analogous to classical random walks, and similar to classical walks they have been used to find new (quantum) algorithms. When studying the behavior of large graphs or combinations of graphs, it is useful to find the response of a subgraph to signals of different frequencies. In doing so, we can replace an entire subgraph with a single vertex with variable scattering coefficients. In this paper, a simple technique for quickly finding the scattering coefficients of any discrete-time quantum graph will be presented. These scattering coefficients can be expressed entirely in terms of the characteristic polynomial ofmore » the graph’s time step operator. This is a marked improvement over previous techniques which have traditionally required finding eigenstates for a given eigenvalue, which is far more computationally costly. With the scattering coefficients we can easily derive the “impulse response” which is the key to predicting the response of a graph to any signal. This gives us a powerful set of tools for rapidly understanding the behavior of graphs or for reducing a large graph into its constituent subgraphs regardless of how they are connected.« less

Weierstrass method for quaternionic polynomial root-finding

NASA Astrophysics Data System (ADS)

Falcão, M. Irene; Miranda, Fernando; Severino, Ricardo; Soares, M. Joana

2018-01-01

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas which motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper we propose a Weierstrass-like method for finding simultaneously {\\sl all} the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.

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.

Discovery Channel Telescope active optics system early integration and test

NASA Astrophysics Data System (ADS)

Venetiou, Alexander J.; Bida, Thomas A.

2012-09-01

The Discovery Channel Telescope (DCT) is a 4.3-meter telescope with a thin meniscus primary mirror (M1) and a honeycomb secondary mirror (M2). The optical design is an f/6.1 Ritchey-Chrétien (RC) with an unvignetted 0.5° Field of View (FoV) at the Cassegrain focus. We describe the design, implementation and performance of the DCT active optics system (AOS). The DCT AOS maintains collimation and controls the figure of the mirror to provide seeing-limited images across the focal plane. To minimize observing overhead, rapid settling times are achieved using a combination of feed-forward and low-bandwidth feedback control using a wavefront sensing system. In 2011, we mounted a Shack-Hartmann wavefront sensor at the prime focus of M1, the Prime Focus Test Assembly (PFTA), to test the AOS with the wavefront sensor, and the feedback loop. The incoming wavefront is decomposed using Zernike polynomials, and the mirror figure is corrected with a set of bending modes. Components of the system that we tested and tuned included the Zernike to Bending Mode transformations. We also started open-loop feed-forward coefficients determination. In early 2012, the PFTA was replaced by M2, and the wavefront sensor moved to its normal location on the Cassegrain instrument assembly. We present early open loop wavefront test results with the full optical system and instrument cube, along with refinements to the overall control loop operating at RC Cassegrain focus.

Uncertainty Quantification of Turbulence Model Closure Coefficients for Transonic Wall-Bounded Flows

NASA Technical Reports Server (NTRS)

Schaefer, John; West, Thomas; Hosder, Serhat; Rumsey, Christopher; Carlson, Jan-Renee; Kleb, William

2015-01-01

The goal of this work was to quantify the uncertainty and sensitivity of commonly used turbulence models in Reynolds-Averaged Navier-Stokes codes due to uncertainty in the values of closure coefficients for transonic, wall-bounded flows and to rank the contribution of each coefficient to uncertainty in various output flow quantities of interest. Specifically, uncertainty quantification of turbulence model closure coefficients was performed for transonic flow over an axisymmetric bump at zero degrees angle of attack and the RAE 2822 transonic airfoil at a lift coefficient of 0.744. Three turbulence models were considered: the Spalart-Allmaras Model, Wilcox (2006) k-w Model, and the Menter Shear-Stress Trans- port Model. The FUN3D code developed by NASA Langley Research Center was used as the flow solver. The uncertainty quantification analysis employed stochastic expansions based on non-intrusive polynomial chaos as an efficient means of uncertainty propagation. Several integrated and point-quantities are considered as uncertain outputs for both CFD problems. All closure coefficients were treated as epistemic uncertain variables represented with intervals. Sobol indices were used to rank the relative contributions of each closure coefficient to the total uncertainty in the output quantities of interest. This study identified a number of closure coefficients for each turbulence model for which more information will reduce the amount of uncertainty in the output significantly for transonic, wall-bounded flows.

Algorithms for Solvents and Spectral Factors of Matrix Polynomials

DTIC Science & Technology

1981-01-01

spectral factors of matrix polynomials LEANG S. SHIEHt, YIH T. TSAYt and NORMAN P. COLEMANt A generalized Newton method , based on the contracted gradient...of a matrix poly- nomial, is derived for solving the right (left) solvents and spectral factors of matrix polynomials. Two methods of selecting initial...estimates for rapid convergence of the newly developed numerical method are proposed. Also, new algorithms for solving complete sets of the right

Enhanced Interferometry with Programmable Spatial Light Modulator

DTIC Science & Technology

2010-06-07

metrolaserinc.com6-7-2010-Monday 6 Simulated by Zemax  Lenslet diameters, d, define spatial resolution over the wavefront being measured.  (sensitivity...MetroLaser Irvine, California Fitted Zernike Polynomials upto 36 terms, found and put into Zemax Simulated Cats’ eye wavefronts by ZEMAX Experimental...measurement Simulated Fringes Leftover < 0.1λ 23 Cat’s eye wavefronts by ZEMAX based on Experimental results Jtrolinger@metrolaserinc.com6-7-2010

The neighbourhood polynomial of some families of dendrimers

NASA Astrophysics Data System (ADS)

Nazri Husin, Mohamad; Hasni, Roslan

2018-04-01

The neighbourhood polynomial N(G,x) is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph and it is defined as (G,x)={\\sum }U\\in N(G){x}|U|, where N(G) is neighbourhood complex of a graph, whose vertices of the graph and faces are subsets of vertices that have a common neighbour. A dendrimers is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, we compute this polynomial for some families of dendrimer.

Thermal/structural/optical integrated design for optical sensor mounted on unmanned aerial vehicle

NASA Astrophysics Data System (ADS)

Zhang, Gaopeng; Yang, Hongtao; Mei, Chao; Wu, Dengshan; Shi, Kui

2016-01-01

With the rapid development of science and technology and the promotion of many local wars in the world, altitude optical sensor mounted on unmanned aerial vehicle is more widely applied in the airborne remote sensing, measurement and detection. In order to obtain high quality image of the aero optical remote sensor, it is important to analysis its thermal-optical performance on the condition of high speed and high altitude. Especially for the key imaging assembly, such as optical window, the temperature variation and temperature gradient can result in defocus and aberrations in optical system, which will lead to the poor quality image. In order to improve the optical performance of a high speed aerial camera optical window, the thermal/structural/optical integrated design method is developed. Firstly, the flight environment of optical window is analyzed. Based on the theory of aerodynamics and heat transfer, the convection heat transfer coefficient is calculated. The temperature distributing of optical window is simulated by the finite element analysis software. The maximum difference in temperature of the inside and outside of optical window is obtained. Then the deformation of optical window under the boundary condition of the maximum difference in temperature is calculated. The optical window surface deformation is fitted in Zernike polynomial as the interface, the calculated Zernike fitting coefficients is brought in and analyzed by CodeV Optical Software. At last, the transfer function diagrams of the optical system on temperature field are comparatively analyzed. By comparing and analyzing the result, it can be obtained that the optical path difference caused by thermal deformation of the optical window is 138.2 nm, which is under PV ≤1 4λ . The above study can be used as an important reference for other optical window designs.

Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion

NASA Astrophysics Data System (ADS)

Sikora, Roman; Markiewicz, Przemysław; Pabjańczyk, Wiesława

2018-04-01

The power systems usually include a number of nonlinear receivers. Nonlinear receivers are the source of disturbances generated to the power system in the form of higher harmonics. The level of these disturbances describes the total harmonic distortion coefficient THD. Its value depends on many factors. One of them are the deformation and change in RMS value of supply voltage. A modern LED luminaire is a nonlinear receiver as well. The paper presents the results of the analysis of the influence of change in RMS value of supply voltage and the level of dimming of the tested luminaire on the value of the current THD. The analysis was made using a mathematical model based on multivariable polynomial fitting.

Recurrence approach and higher order polynomial algebras for superintegrable monopole systems

NASA Astrophysics Data System (ADS)

Hoque, Md Fazlul; Marquette, Ian; Zhang, Yao-Zhong

2018-05-01

We revisit the MIC-harmonic oscillator in flat space with monopole interaction and derive the polynomial algebra satisfied by the integrals of motion and its energy spectrum using the ad hoc recurrence approach. We introduce a superintegrable monopole system in a generalized Taub-Newman-Unti-Tamburino (NUT) space. The Schrödinger equation of this model is solved in spherical coordinates in the framework of Stäckel transformation. It is shown that wave functions of the quantum system can be expressed in terms of the product of Laguerre and Jacobi polynomials. We construct ladder and shift operators based on the corresponding wave functions and obtain the recurrence formulas. By applying these recurrence relations, we construct higher order algebraically independent integrals of motion. We show that the integrals form a polynomial algebra. We construct the structure functions of the polynomial algebra and obtain the degenerate energy spectra of the model.

Application of polynomial su(1, 1) algebra to Pöschl-Teller potentials

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

Zhang, Hong-Biao, E-mail: zhanghb017@nenu.edu.cn; Lu, Lu

2013-12-15

Two novel polynomial su(1, 1) algebras for the physical systems with the first and second Pöschl-Teller (PT) potentials are constructed, and their specific representations are presented. Meanwhile, these polynomial su(1, 1) algebras are used as an algebraic technique to solve eigenvalues and eigenfunctions of the Hamiltonians associated with the first and second PT potentials. The algebraic approach explores an appropriate new pair of raising and lowing operators K-circumflex{sub ±} of polynomial su(1, 1) algebra as a pair of shift operators of our Hamiltonians. In addition, two usual su(1, 1) algebras associated with the first and second PT potentials are derivedmore » naturally from the polynomial su(1, 1) algebras built by us.« less

Efficient computer algebra algorithms for polynomial matrices in control design

NASA Technical Reports Server (NTRS)

Baras, J. S.; Macenany, D. C.; Munach, R.

1989-01-01

The theory of polynomial matrices plays a key role in the design and analysis of multi-input multi-output control and communications systems using frequency domain methods. Examples include coprime factorizations of transfer functions, cannonical realizations from matrix fraction descriptions, and the transfer function design of feedback compensators. Typically, such problems abstract in a natural way to the need to solve systems of Diophantine equations or systems of linear equations over polynomials. These and other problems involving polynomial matrices can in turn be reduced to polynomial matrix triangularization procedures, a result which is not surprising given the importance of matrix triangularization techniques in numerical linear algebra. Matrices with entries from a field and Gaussian elimination play a fundamental role in understanding the triangularization process. In the case of polynomial matrices, matrices with entries from a ring for which Gaussian elimination is not defined and triangularization is accomplished by what is quite properly called Euclidean elimination. Unfortunately, the numerical stability and sensitivity issues which accompany floating point approaches to Euclidean elimination are not very well understood. New algorithms are presented which circumvent entirely such numerical issues through the use of exact, symbolic methods in computer algebra. The use of such error-free algorithms guarantees that the results are accurate to within the precision of the model data--the best that can be hoped for. Care must be taken in the design of such algorithms due to the phenomenon of intermediate expressions swell.

Generalized clustering conditions of Jack polynomials at negative Jack parameter {alpha}

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

Bernevig, B. Andrei; Department of Physics, Princeton University, Princeton, New Jersey 08544; Haldane, F. D. M.

We present several conjectures on the behavior and clustering properties of Jack polynomials at a negative parameter {alpha}=-(k+1/r-1), with partitions that violate the (k,r,N)- admissibility rule of [Feigin et al. [Int. Math. Res. Notices 23, 1223 (2002)]. We find that the ''highest weight'' Jack polynomials of specific partitions represent the minimum degree polynomials in N variables that vanish when s distinct clusters of k+1 particles are formed, where s and k are positive integers. Explicit counting formulas are conjectured. The generalized clustering conditions are useful in a forthcoming description of fractional quantum Hall quasiparticles.

Perceptually informed synthesis of bandlimited classical waveforms using integrated polynomial interpolation.

PubMed

Välimäki, Vesa; Pekonen, Jussi; Nam, Juhan

2012-01-01

Digital subtractive synthesis is a popular music synthesis method, which requires oscillators that are aliasing-free in a perceptual sense. It is a research challenge to find computationally efficient waveform generation algorithms that produce similar-sounding signals to analog music synthesizers but which are free from audible aliasing. A technique for approximately bandlimited waveform generation is considered that is based on a polynomial correction function, which is defined as the difference of a non-bandlimited step function and a polynomial approximation of the ideal bandlimited step function. It is shown that the ideal bandlimited step function is equivalent to the sine integral, and that integrated polynomial interpolation methods can successfully approximate it. Integrated Lagrange interpolation and B-spline basis functions are considered for polynomial approximation. The polynomial correction function can be added onto samples around each discontinuity in a non-bandlimited waveform to suppress aliasing. Comparison against previously known methods shows that the proposed technique yields the best tradeoff between computational cost and sound quality. The superior method amongst those considered in this study is the integrated third-order B-spline correction function, which offers perceptually aliasing-free sawtooth emulation up to the fundamental frequency of 7.8 kHz at the sample rate of 44.1 kHz. © 2012 Acoustical Society of America.

Potential for protein surface shape analysis using spherical harmonics and 3D Zernike descriptors.

PubMed

Venkatraman, Vishwesh; Sael, Lee; Kihara, Daisuke

2009-01-01

With structure databases expanding at a rapid rate, the task at hand is to provide reliable clues to their molecular function and to be able to do so on a large scale. This, however, requires suitable encodings of the molecular structure which are amenable to fast screening. To this end, moment-based representations provide a compact and nonredundant description of molecular shape and other associated properties. In this article, we present an overview of some commonly used representations with specific focus on two schemes namely spherical harmonics and their extension, the 3D Zernike descriptors. Key features and differences of the two are reviewed and selected applications are highlighted. We further discuss recent advances covering aspects of shape and property-based comparison at both global and local levels and demonstrate their applicability through some of our studies.

Soliton interactions and Bäcklund transformation for a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili equation in fluid dynamics

NASA Astrophysics Data System (ADS)

Xiao, Zi-Jian; Tian, Bo; Sun, Yan

2018-01-01

In this paper, we investigate a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili (mKP) equation in fluid dynamics. With the binary Bell-polynomial and an auxiliary function, bilinear forms for the equation are constructed. Based on the bilinear forms, multi-soliton solutions and Bell-polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton interactions are presented. Based on the graphic analysis, Parametric conditions for the existence of the shock waves, elevation solitons and depression solitons are given, and it is shown that under the condition of keeping the wave vectors invariable, the change of α(t) and β(t) can lead to the change of the solitonic velocities, but the shape of each soliton remains unchanged, where α(t) and β(t) are the variable coefficients in the equation. Oblique elastic interactions can exist between the (i) two shock waves, (ii) two elevation solitons, and (iii) elevation and depression solitons. However, oblique interactions between (i) shock waves and elevation solitons, (ii) shock waves and depression solitons are inelastic.

Prediction of random-regression coefficient for daily milk yield after 305 days in milk by using the regression-coefficient estimates from the first 305 days.

PubMed

Yamazaki, Takeshi; Takeda, Hisato; Hagiya, Koichi; Yamaguchi, Satoshi; Sasaki, Osamu

2018-03-13

Because lactation periods in dairy cows lengthen with increasing total milk production, it is important to predict individual productivities after 305 days in milk (DIM) to determine the optimal lactation period. We therefore examined whether the random regression (RR) coefficient from 306 to 450 DIM (M2) can be predicted from those during the first 305 DIM (M1) by using a random regression model. We analyzed test-day milk records from 85690 Holstein cows in their first lactations and 131727 cows in their later (second to fifth) lactations. Data in M1 and M2 were analyzed separately by using different single-trait RR animal models. We then performed a multiple regression analysis of the RR coefficients of M2 on those of M1 during the first and later lactations. The first-order Legendre polynomials were practical covariates of random regression for the milk yields of M2. All RR coefficients for the additive genetic (AG) effect and the intercept for the permanent environmental (PE) effect of M2 had moderate to strong correlations with the intercept for the AG effect of M1. The coefficients of determination for multiple regression of the combined intercepts for the AG and PE effects of M2 on the coefficients for the AG effect of M1 were moderate to high. The daily milk yields of M2 predicted by using the RR coefficients for the AG effect of M1 were highly correlated with those obtained by using the coefficients of M2. Milk production after 305 DIM can be predicted by using the RR coefficient estimates of the AG effect during the first 305 DIM.

Comparison of parametric methods for modeling corneal surfaces

NASA Astrophysics Data System (ADS)

Bouazizi, Hala; Brunette, Isabelle; Meunier, Jean

2017-02-01

Corneal topography is a medical imaging technique to get the 3D shape of the cornea as a set of 3D points of its anterior and posterior surfaces. From these data, topographic maps can be derived to assist the ophthalmologist in the diagnosis of disorders. In this paper, we compare three different mathematical parametric representations of the corneal surfaces leastsquares fitted to the data provided by corneal topography. The parameters obtained from these models reduce the dimensionality of the data from several thousand 3D points to only a few parameters and could eventually be useful for diagnosis, biometry, implant design etc. The first representation is based on Zernike polynomials that are commonly used in optics. A variant of these polynomials, named Bhatia-Wolf will also be investigated. These two sets of polynomials are defined over a circular domain which is convenient to model the elevation (height) of the corneal surface. The third representation uses Spherical Harmonics that are particularly well suited for nearly-spherical object modeling, which is the case for cornea. We compared the three methods using the following three criteria: the root-mean-square error (RMSE), the number of parameters and the visual accuracy of the reconstructed topographic maps. A large dataset of more than 2000 corneal topographies was used. Our results showed that Spherical Harmonics were superior with a RMSE mean lower than 2.5 microns with 36 coefficients (order 5) for normal corneas and lower than 5 microns for two diseases affecting the corneal shapes: keratoconus and Fuchs' dystrophy.

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.

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

On Arithmetic-Geometric-Mean Polynomials

ERIC Educational Resources Information Center

Griffiths, Martin; MacHale, Des

2017-01-01

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

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.

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.

Inversion of Robin coefficient by a spectral stochastic finite element approach

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

Jin Bangti; Zou Jun

2008-03-01

This paper investigates a variational approach to the nonlinear stochastic inverse problem of probabilistically calibrating the Robin coefficient from boundary measurements for the steady-state heat conduction. The problem is formulated into an optimization problem, and mathematical properties relevant to its numerical computations are investigated. The spectral stochastic finite element method using polynomial chaos is utilized for the discretization of the optimization problem, and its convergence is analyzed. The nonlinear conjugate gradient method is derived for the optimization system. Numerical results for several two-dimensional problems are presented to illustrate the accuracy and efficiency of the stochastic finite element method.

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

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

Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk; Belkouchi, B.; Montomoli, F.

2016-09-01

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrixmore » is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

NASA Astrophysics Data System (ADS)

Ahlfeld, R.; Belkouchi, B.; Montomoli, F.

2016-09-01

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10

Fast template matching with polynomials.

PubMed

Omachi, Shinichiro; Omachi, Masako

2007-08-01

Template matching is widely used for many applications in image and signal processing. This paper proposes a novel template matching algorithm, called algebraic template matching. Given a template and an input image, algebraic template matching efficiently calculates similarities between the template and the partial images of the input image, for various widths and heights. The partial image most similar to the template image is detected from the input image for any location, width, and height. In the proposed algorithm, a polynomial that approximates the template image is used to match the input image instead of the template image. The proposed algorithm is effective especially when the width and height of the template image differ from the partial image to be matched. An algorithm using the Legendre polynomial is proposed for efficient approximation of the template image. This algorithm not only reduces computational costs, but also improves the quality of the approximated image. It is shown theoretically and experimentally that the computational cost of the proposed algorithm is much smaller than the existing methods.

Bayer Demosaicking with Polynomial Interpolation.

PubMed

Wu, Jiaji; Anisetti, Marco; Wu, Wei; Damiani, Ernesto; Jeon, Gwanggil

2016-08-30

Demosaicking is a digital image process to reconstruct full color digital images from incomplete color samples from an image sensor. It is an unavoidable process for many devices incorporating camera sensor (e.g. mobile phones, tablet, etc.). In this paper, we introduce a new demosaicking algorithm based on polynomial interpolation-based demosaicking (PID). Our method makes three contributions: calculation of error predictors, edge classification based on color differences, and a refinement stage using a weighted sum strategy. Our new predictors are generated on the basis of on the polynomial interpolation, and can be used as a sound alternative to other predictors obtained by bilinear or Laplacian interpolation. In this paper we show how our predictors can be combined according to the proposed edge classifier. After populating three color channels, a refinement stage is applied to enhance the image quality and reduce demosaicking artifacts. Our experimental results show that the proposed method substantially improves over existing demosaicking methods in terms of objective performance (CPSNR, S-CIELAB E, and FSIM), and visual performance.

Fitting Multimeric Protein Complexes into Electron Microscopy Maps Using 3D Zernike Descriptors

PubMed Central

Esquivel-Rodríguez, Juan; Kihara, Daisuke

2012-01-01

A novel computational method for fitting high-resolution structures of multiple proteins into a cryoelectron microscopy map is presented. The method named EMLZerD generates a pool of candidate multiple protein docking conformations of component proteins, which are later compared with a provided electron microscopy (EM) density map to select the ones that fit well into the EM map. The comparison of docking conformations and the EM map is performed using the 3D Zernike descriptor (3DZD), a mathematical series expansion of three-dimensional functions. The 3DZD provides a unified representation of the surface shape of multimeric protein complex models and EM maps, which allows a convenient, fast quantitative comparison of the three dimensional structural data. Out of 19 multimeric complexes tested, near native complex structures with a root mean square deviation of less than 2.5 Å were obtained for 14 cases while medium range resolution structures with correct topology were computed for the additional 5 cases. PMID:22417139

Fitting multimeric protein complexes into electron microscopy maps using 3D Zernike descriptors.

PubMed

Esquivel-Rodríguez, Juan; Kihara, Daisuke

2012-06-14

A novel computational method for fitting high-resolution structures of multiple proteins into a cryoelectron microscopy map is presented. The method named EMLZerD generates a pool of candidate multiple protein docking conformations of component proteins, which are later compared with a provided electron microscopy (EM) density map to select the ones that fit well into the EM map. The comparison of docking conformations and the EM map is performed using the 3D Zernike descriptor (3DZD), a mathematical series expansion of three-dimensional functions. The 3DZD provides a unified representation of the surface shape of multimeric protein complex models and EM maps, which allows a convenient, fast quantitative comparison of the three-dimensional structural data. Out of 19 multimeric complexes tested, near native complex structures with a root-mean-square deviation of less than 2.5 Å were obtained for 14 cases while medium range resolution structures with correct topology were computed for the additional 5 cases.

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.

On-sky performance of the Zernike phase contrast sensor for the phasing of segmented telescopes.

PubMed

Surdej, Isabelle; Yaitskova, Natalia; Gonte, Frederic

2010-07-20

The Zernike phase contrast method is a novel technique to phase the primary mirrors of segmented telescopes. It has been tested on-sky on a unit telescope of the Very Large Telescope with a segmented mirror conjugated to the primary mirror to emulate a segmented telescope. The theoretical background of this sensor and the algorithm used to retrieve the piston, tip, and tilt information are described. The performance of the sensor as a function of parameters such as star magnitude, seeing, and integration time is discussed. The phasing accuracy has always been below 15 nm root mean square wavefront error under normal conditions of operation and the limiting star magnitude achieved on-sky with this sensor is 15.7 in the red, which would be sufficient to phase segmented telescopes in closed-loop during observations.

Non-stationary component extraction in noisy multicomponent signal using polynomial chirping Fourier transform.

PubMed

Lu, Wenlong; Xie, Junwei; Wang, Heming; Sheng, Chuan

2016-01-01

Inspired by track-before-detection technology in radar, a novel time-frequency transform, namely polynomial chirping Fourier transform (PCFT), is exploited to extract components from noisy multicomponent signal. The PCFT combines advantages of Fourier transform and polynomial chirplet transform to accumulate component energy along a polynomial chirping curve in the time-frequency plane. The particle swarm optimization algorithm is employed to search optimal polynomial parameters with which the PCFT will achieve a most concentrated energy ridge in the time-frequency plane for the target component. The component can be well separated in the polynomial chirping Fourier domain with a narrow-band filter and then reconstructed by inverse PCFT. Furthermore, an iterative procedure, involving parameter estimation, PCFT, filtering and recovery, is introduced to extract components from a noisy multicomponent signal successively. The Simulations and experiments show that the proposed method has better performance in component extraction from noisy multicomponent signal as well as provides more time-frequency details about the analyzed signal than conventional methods.

Principal polynomial analysis.

PubMed

Laparra, Valero; Jiménez, Sandra; Tuia, Devis; Camps-Valls, Gustau; Malo, Jesus

2014-11-01

This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet-Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository.

Polynomial time blackbox identity testers for depth-3 circuits : the field doesn't matter.

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

Seshadhri, Comandur; Saxena, Nitin

Let C be a depth-3 circuit with n variables, degree d and top fanin k (called {Sigma}{Pi}{Sigma}(k, d, n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005). We give the first polynomial time blackbox algorithm for this problem. Our algorithm runsmore » in time poly(n)d{sup k}, regardless of the base field. The only field for which polynomial time algorithms were previously known is F = Q (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth-3 circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008). We prove an important tool for the study of depth-3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a {Sigma}{Pi}{Sigma}(k, d, n) circuit to k variables, but preserves the identity structure. Polynomial identity testing (PIT) is a major open problem in theoretical computer science. The input is an arithmetic circuit that computes a polynomial p(x{sub 1}, x{sub 2},..., x{sub n}) over a base field F. We wish to check if p is the zero polynomial, or in other words, is identically zero. We may be provided with an explicit circuit, or may only have blackbox access. In the latter case, we can only evaluate the polynomial p at various domain points. The main goal is to devise a deterministic blackbox polynomial time algorithm for PIT.« less

Multivariable Hermite polynomials and phase-space dynamics

NASA Technical Reports Server (NTRS)

Dattoli, G.; Torre, Amalia; Lorenzutta, S.; Maino, G.; Chiccoli, C.

1994-01-01

The phase-space approach to classical and quantum systems demands for advanced analytical tools. Such an approach characterizes the evolution of a physical system through a set of variables, reducing to the canonically conjugate variables in the classical limit. It often happens that phase-space distributions can be written in terms of quadratic forms involving the above quoted variables. A significant analytical tool to treat these problems may come from the generalized many-variables Hermite polynomials, defined on quadratic forms in R(exp n). They form an orthonormal system in many dimensions and seem the natural tool to treat the harmonic oscillator dynamics in phase-space. In this contribution we discuss the properties of these polynomials and present some applications to physical problems.

The explicit computation of integration algorithms and first integrals for ordinary differential equations with polynomials coefficients using trees

NASA Technical Reports Server (NTRS)

Crouch, P. E.; Grossman, Robert

1992-01-01

This note is concerned with the explicit symbolic computation of expressions involving differential operators and their actions on functions. The derivation of specialized numerical algorithms, the explicit symbolic computation of integrals of motion, and the explicit computation of normal forms for nonlinear systems all require such computations. More precisely, if R = k(x(sub 1),...,x(sub N)), where k = R or C, F denotes a differential operator with coefficients from R, and g member of R, we describe data structures and algorithms for efficiently computing g. The basic idea is to impose a multiplicative structure on the vector space with basis the set of finite rooted trees and whose nodes are labeled with the coefficients of the differential operators. Cancellations of two trees with r + 1 nodes translates into cancellation of O(N(exp r)) expressions involving the coefficient functions and their derivatives.

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

NASA Astrophysics Data System (ADS)

Attorps, Iiris; Björk, Kjell; Radic, Mirko

2016-01-01

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

Non-Abelian integrable hierarchies: matrix biorthogonal polynomials and perturbations

NASA Astrophysics Data System (ADS)

Ariznabarreta, Gerardo; García-Ardila, Juan C.; Mañas, Manuel; Marcellán, Francisco

2018-05-01

In this paper, Geronimus–Uvarov perturbations for matrix orthogonal polynomials on the real line are studied and then applied to the analysis of non-Abelian integrable hierarchies. The orthogonality is understood in full generality, i.e. in terms of a nondegenerate continuous sesquilinear form, determined by a quasidefinite matrix of bivariate generalized functions with a well-defined support. We derive Christoffel-type formulas that give the perturbed matrix biorthogonal polynomials and their norms in terms of the original ones. The keystone for this finding is the Gauss–Borel factorization of the Gram matrix. Geronimus–Uvarov transformations are considered in the context of the 2D non-Abelian Toda lattice and noncommutative KP hierarchies. The interplay between transformations and integrable flows is discussed. Miwa shifts, τ-ratio matrix functions and Sato formulas are given. Bilinear identities, involving Geronimus–Uvarov transformations, first for the Baker functions, then secondly for the biorthogonal polynomials and its second kind functions, and finally for the τ-ratio matrix functions, are found.

Recognition of Arabic Sign Language Alphabet Using Polynomial Classifiers

NASA Astrophysics Data System (ADS)

Assaleh, Khaled; Al-Rousan, M.

2005-12-01

Building an accurate automatic sign language recognition system is of great importance in facilitating efficient communication with deaf people. In this paper, we propose the use of polynomial classifiers as a classification engine for the recognition of Arabic sign language (ArSL) alphabet. Polynomial classifiers have several advantages over other classifiers in that they do not require iterative training, and that they are highly computationally scalable with the number of classes. Based on polynomial classifiers, we have built an ArSL system and measured its performance using real ArSL data collected from deaf people. We show that the proposed system provides superior recognition results when compared with previously published results using ANFIS-based classification on the same dataset and feature extraction methodology. The comparison is shown in terms of the number of misclassified test patterns. The reduction in the rate of misclassified patterns was very significant. In particular, we have achieved a 36% reduction of misclassifications on the training data and 57% on the test data.

Correlations between corneal and total wavefront aberrations

NASA Astrophysics Data System (ADS)

Mrochen, Michael; Jankov, Mirko; Bueeler, Michael; Seiler, Theo

2002-06-01

Purpose: Corneal topography data expressed as corneal aberrations are frequently used to report corneal laser surgery results. However, the optical image quality at the retina depends on all optical elements of the eye such as the human lens. Thus, the aim of this study was to investigate the correlations between the corneal and total wavefront aberrations and to discuss the importance of corneal aberrations for representing corneal laser surgery results. Methods: Thirty three eyes of 22 myopic subjects were measured with a corneal topography system and a Tschernig-type wavefront analyzer after the pupils were dilated to at least 6 mm in diameter. All measurements were centered with respect to the line of sight. Corneal and total wavefront aberrations were calculated up to the 6th Zernike order in the same reference plane. Results: Statistically significant correlations (p < 0.05) between the corneal and total wavefront aberrations were found for the astigmatism (C3,C5) and all 3rd Zernike order coefficients such as coma (C7,C8). No statistically significant correlations were found for all 4th to 6th order Zernike coefficients except for the 5th order horizontal coma C18 (p equals 0.003). On average, all Zernike coefficients for the corneal aberrations were found to be larger compared to Zernike coefficients for the total wavefront aberrations. Conclusions: Corneal aberrations are only of limited use for representing the optical quality of the human eye after corneal laser surgery. This is due to the lack of correlation between corneal and total wavefront aberrations in most of the higher order aberrations. Besides this, the data present in this study yield towards an aberration balancing between corneal aberrations and the optical elements within the eye that reduces the aberration from the cornea by a certain degree. Consequently, ideal customized ablations have to take both, corneal and total wavefront aberrations, into consideration.

A general U-block model-based design procedure for nonlinear polynomial control systems

NASA Astrophysics Data System (ADS)

Zhu, Q. M.; Zhao, D. Y.; Zhang, Jianhua

2016-10-01

The proposition of U-model concept (in terms of 'providing concise and applicable solutions for complex problems') and a corresponding basic U-control design algorithm was originated in the first author's PhD thesis. The term of U-model appeared (not rigorously defined) for the first time in the first author's other journal paper, which established a framework for using linear polynomial control system design approaches to design nonlinear polynomial control systems (in brief, linear polynomial approaches → nonlinear polynomial plants). This paper represents the next milestone work - using linear state-space approaches to design nonlinear polynomial control systems (in brief, linear state-space approaches → nonlinear polynomial plants). The overall aim of the study is to establish a framework, defined as the U-block model, which provides a generic prototype for using linear state-space-based approaches to design the control systems with smooth nonlinear plants/processes described by polynomial models. For analysing the feasibility and effectiveness, sliding mode control design approach is selected as an exemplary case study. Numerical simulation studies provide a user-friendly step-by-step procedure for the readers/users with interest in their ad hoc applications. In formality, this is the first paper to present the U-model-oriented control system design in a formal way and to study the associated properties and theorems. The previous publications, in the main, have been algorithm-based studies and simulation demonstrations. In some sense, this paper can be treated as a landmark for the U-model-based research from intuitive/heuristic stage to rigour/formal/comprehensive studies.

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

NUMERICAL INTEGRAL OF RESISTANCE COEFFICIENTS IN DIFFUSION

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

Zhang, Q. S., E-mail: zqs@ynao.ac.cn

2017-01-10

The resistance coefficients in the screened Coulomb potential of stellar plasma are evaluated to high accuracy. I have analyzed the possible singularities in the integral of scattering angle. There are possible singularities in the case of an attractive potential. This may result in a problem for the numerical integral. In order to avoid the problem, I have used a proper scheme, e.g., splitting into many subintervals where the width of each subinterval is determined by the variation of the integrand, to calculate the scattering angle. The collision integrals are calculated by using Romberg’s method, therefore the accuracy is high (i.e.,more » ∼10{sup −12}). The results of collision integrals and their derivatives for −7 ≤ ψ ≤ 5 are listed. By using Hermite polynomial interpolation from those data, the collision integrals can be obtained with an accuracy of 10{sup −10}. For very weakly coupled plasma ( ψ ≥ 4.5), analytical fittings for collision integrals are available with an accuracy of 10{sup −11}. I have compared the final results of resistance coefficients with other works and found that, for a repulsive potential, the results are basically the same as others’; for an attractive potential, the results in cases of intermediate and strong coupling show significant differences. The resulting resistance coefficients are tested in the solar model. Comparing with the widely used models of Cox et al. and Thoul et al., the resistance coefficients in the screened Coulomb potential lead to a slightly weaker effect in the solar model, which is contrary to the expectation of attempts to solve the solar abundance problem.« less

Polynomial approximation of the Lense-Thirring rigid precession frequency

NASA Astrophysics Data System (ADS)

De Falco, Vittorio; Motta, Sara

2018-05-01

We propose a polynomial approximation of the global Lense-Thirring rigid precession frequency to study low-frequency quasi-periodic oscillations around spinning black holes. This high-performing approximation allows to determine the expected frequencies of a precessing thick accretion disc with fixed inner radius and variable outer radius around a black hole with given mass and spin. We discuss the accuracy and the applicability regions of our polynomial approximation, showing that the computational times are reduced by a factor of ≈70 in the range of minutes.

Generalized recursive solutions to Ornstein-Zernike integral equations

NASA Astrophysics Data System (ADS)

Rossky, Peter J.; Dale, William D. T.

1980-09-01

Recursive procedures for the solution of a class of integral equations based on the Ornstein-Zernike equation are developed; the hypernetted chain and Percus-Yevick equations are two special cases of the class considered. It is shown that certain variants of the new procedures developed here are formally equivalent to those recently developed by Dale and Friedman, if the new recursive expressions are initialized in the same way as theirs. However, the computational solution of the new equations is significantly more efficient. Further, the present analysis leads to the identification of various graphical quantities arising in the earlier study with more familiar quantities related to pair correlation functions. The analysis is greatly facilitated by the use of several identities relating simple chain sums whose graphical elements can be written as a sum of two or more parts. In particular, the use of these identities permits renormalization of the equivalent series solution to the integral equation to be directly incorporated into the recursive solution in a straightforward manner. Formulas appropriate to renormalization with respect to long and short range parts of the pair potential, as well as more general components of the direct correlation function, are obtained. To further illustrate the utility of this approach, we show that a simple generalization of the hypernetted chain closure relation for the direct correlation function leads directly to the reference hypernetted chain (RHNC) equation due to Lado. The form of the correlation function used in the exponential approximation of Andersen and Chandler is then seen to be equivalent to the first estimate obtained from a renormalized RHNC equation.

Significantly Reduced Blood Pressure Measurement Variability for Both Normotensive and Hypertensive Subjects: Effect of Polynomial Curve Fitting of Oscillometric Pulses

PubMed Central

Zhu, Mingping; Chen, Aiqing

2017-01-01

This study aimed to compare within-subject blood pressure (BP) variabilities from different measurement techniques. Cuff pressures from three repeated BP measurements were obtained from 30 normotensive and 30 hypertensive subjects. Automatic BPs were determined from the pulses with normalised peak amplitude larger than a threshold (0.5 for SBP, 0.7 for DBP, and 1.0 for MAP). They were also determined from cuff pressures associated with the above thresholds on a fitted curve polynomial curve of the oscillometric pulse peaks. Finally, the standard deviation (SD) of three repeats and its coefficient of variability (CV) were compared between the two automatic techniques. For the normotensive group, polynomial curve fitting significantly reduced SD of repeats from 3.6 to 2.5 mmHg for SBP and from 3.7 to 2.1 mmHg for MAP and reduced CV from 3.0% to 2.2% for SBP and from 4.3% to 2.4% for MAP (all P < 0.01). For the hypertensive group, SD of repeats decreased from 6.5 to 5.5 mmHg for SBP and from 6.7 to 4.2 mmHg for MAP, and CV decreased from 4.2% to 3.6% for SBP and from 5.8% to 3.8% for MAP (all P < 0.05). In conclusion, polynomial curve fitting of oscillometric pulses had the ability to reduce automatic BP measurement variability. PMID:28785580

Polynomial elimination theory and non-linear stability analysis for the Euler equations

NASA Technical Reports Server (NTRS)

Kennon, S. R.; Dulikravich, G. S.; Jespersen, D. C.

1986-01-01

Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.

Georeferencing CAMS data: Polynomial rectification and beyond

NASA Astrophysics Data System (ADS)

Yang, Xinghe

The Calibrated Airborne Multispectral Scanner (CAMS) is a sensor used in the commercial remote sensing program at NASA Stennis Space Center. In geographic applications of the CAMS data, accurate geometric rectification is essential for the analysis of the remotely sensed data and for the integration of the data into Geographic Information Systems (GIS). The commonly used rectification techniques such as the polynomial transformation and ortho rectification have been very successful in the field of remote sensing and GIS for most remote sensing data such as Landsat imagery, SPOT imagery and aerial photos. However, due to the geometric nature of the airborne line scanner which has high spatial frequency distortions, the polynomial model and the ortho rectification technique in current commercial software packages such as Erdas Imagine are not adequate for obtaining sufficient geometric accuracy. In this research, the geometric nature, especially the major distortions, of the CAMS data has been described. An analytical step-by-step geometric preprocessing has been utilized to deal with the potential high frequency distortions of the CAMS data. A generic sensor-independent photogrammetric model has been developed for the ortho-rectification of the CAMS data. Three generalized kernel classes and directional elliptical basis have been formulated into a rectification model of summation of multisurface functions, which is a significant extension to the traditional radial basis functions. The preprocessing mechanism has been fully incorporated into the polynomial, the triangle-based finite element analysis as well as the summation of multisurface functions. While the multisurface functions and the finite element analysis have the characteristics of localization, piecewise logic has been applied to the polynomial and photogrammetric methods, which can produce significant accuracy improvement over the global approach. A software module has been implemented with full

Orthogonal polynomials, Laguerre Fock space, and quasi-classical asymptotics

NASA Astrophysics Data System (ADS)

Engliš, Miroslav; Ali, S. Twareque

2015-07-01

Continuing our earlier investigation of the Hermite case [S. T. Ali and M. Engliš, J. Math. Phys. 55, 042102 (2014)], we study an unorthodox variant of the Berezin-Toeplitz quantization scheme associated with Laguerre polynomials. In particular, we describe a "Laguerre analogue" of the classical Fock (Segal-Bargmann) space and the relevant semi-classical asymptotics of its Toeplitz operators; the former actually turns out to coincide with the Hilbert space appearing in the construction of the well-known Barut-Girardello coherent states. Further extension to the case of Legendre polynomials is likewise discussed.

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

NASA Astrophysics Data System (ADS)

Othmani, Cherif; Takali, Farid; Njeh, Anouar

2017-06-01

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

A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

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

Chen, Luoping; Zheng, Bin; Lin, Guang

In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse meshmore » $$\\mathcal{T}_H$$ with a low level stochastic collocation (corresponding to the polynomial space $$\\mathcal{P}_{P}$$) and solve linearized equations on a fine mesh $$\\mathcal{T}_h$$ using high level stochastic collocation (corresponding to the polynomial space $$\\mathcal{P}_p$$). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with $$\\mathcal{T}_h$$ and $$\\mathcal{P}_p$$. The two-level method is computationally more efficient, especially for nonlinear problems with high random dimensions. Numerical experiments are also provided to verify the theoretical results.« less

Automatic bone outer contour extraction from B-modes ultrasound images based on local phase symmetry and quadratic polynomial fitting

NASA Astrophysics Data System (ADS)

Karlita, Tita; Yuniarno, Eko Mulyanto; Purnama, I. Ketut Eddy; Purnomo, Mauridhi Hery

2017-06-01

Analyzing ultrasound (US) images to get the shapes and structures of particular anatomical regions is an interesting field of study since US imaging is a non-invasive method to capture internal structures of a human body. However, bone segmentation of US images is still challenging because it is strongly influenced by speckle noises and it has poor image quality. This paper proposes a combination of local phase symmetry and quadratic polynomial fitting methods to extract bone outer contour (BOC) from two dimensional (2D) B-modes US image as initial steps of three-dimensional (3D) bone surface reconstruction. By using local phase symmetry, the bone is initially extracted from US images. BOC is then extracted by scanning one pixel on the bone boundary in each column of the US images using first phase features searching method. Quadratic polynomial fitting is utilized to refine and estimate the pixel location that fails to be detected during the extraction process. Hole filling method is then applied by utilize the polynomial coefficients to fill the gaps with new pixel. The proposed method is able to estimate the new pixel position and ensures smoothness and continuity of the contour path. Evaluations are done using cow and goat bones by comparing the resulted BOCs with the contours produced by manual segmentation and contours produced by canny edge detection. The evaluation shows that our proposed methods produces an excellent result with average MSE before and after hole filling at the value of 0.65.

Training Data Requirement for a Neural Network to Predict Aerodynamic Coefficients

NASA Technical Reports Server (NTRS)

Korsmeyer, David (Technical Monitor); Rajkumar, T.; Bardina, Jorge

2003-01-01

Basic aerodynamic coefficients are modeled as functions of angle of attack, speed brake deflection angle, Mach number, and side slip angle. Most of the aerodynamic parameters can be well-fitted using polynomial functions. We previously demonstrated that a neural network is a fast, reliable way of predicting aerodynamic coefficients. We encountered few under fitted and/or over fitted results during prediction. The training data for the neural network are derived from wind tunnel test measurements and numerical simulations. The basic questions that arise are: how many training data points are required to produce an efficient neural network prediction, and which type of transfer functions should be used between the input-hidden layer and hidden-output layer. In this paper, a comparative study of the efficiency of neural network prediction based on different transfer functions and training dataset sizes is presented. The results of the neural network prediction reflect the sensitivity of the architecture, transfer functions, and training dataset size.

Lifting q-difference operators for Askey-Wilson polynomials and their weight function

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

Atakishiyeva, M. K.; Atakishiyev, N. M., E-mail: natig_atakishiyev@hotmail.com

2011-06-15

We determine an explicit form of a q-difference operator that transforms the continuous q-Hermite polynomials H{sub n}(x | q) of Rogers into the Askey-Wilson polynomials p{sub n}(x; a, b, c, d | q) on the top level in the Askey q-scheme. This operator represents a special convolution-type product of four one-parameter q-difference operators of the form {epsilon}{sub q}(c{sub q}D{sub q}) (where c{sub q} are some constants), defined as Exton's q-exponential function {epsilon}{sub q}(z) in terms of the Askey-Wilson divided q-difference operator D{sub q}. We also determine another q-difference operator that lifts the orthogonality weight function for the continuous q-Hermite polynomialsH{submore » n}(x | q) up to the weight function, associated with the Askey-Wilson polynomials p{sub n}(x; a, b, c, d | q).« less

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.

Representing Lumped Markov Chains by Minimal Polynomials over Field GF(q)

NASA Astrophysics Data System (ADS)

Zakharov, V. M.; Shalagin, S. V.; Eminov, B. F.

2018-05-01

A method has been proposed to represent lumped Markov chains by minimal polynomials over a finite field. The accuracy of representing lumped stochastic matrices, the law of lumped Markov chains depends linearly on the minimum degree of polynomials over field GF(q). The method allows constructing the realizations of lumped Markov chains on linear shift registers with a pre-defined “linear complexity”.

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

Effect of Wigner energy on the symmetry energy coefficient in nuclei

NASA Astrophysics Data System (ADS)

Tian, Jun-Long; Cui, Hai-Tao; Gao, Teng; Wang, Ning

2016-09-01

The nuclear symmetry energy coefficient (including the coefficient of the I4 term) of finite nuclei is extracted by using the differences of available experimental binding energies of isobaric nuclei. It is found that the extracted symmetry energy coefficient decreases with increasing isospin asymmetry I, which is mainly caused by Wigner correction, since is the summation of the traditional symmetry energy esym and the Wigner energy eW. We obtain the optimal values J = 30.25 ± 0.10 MeV, ass = 56.18 ± 1.25 MeV, and the Wigner parameter x = 2.38 ± 0.12 through a polynomial fit to 2240 measured binding energies for nuclei with 20 ⩽ A ⩽ 261 with an rms deviation of 23.42 keV. We also find that the volume symmetry coefficient J ≃ 30 MeV is insensitive to the value x, whereas the surface symmetry coefficient ass and the coefficient are very sensitive to the value of x in the range 1 ⩽ x ⩽ 4. The contribution of the term increases rapidly with increasing isospin asymmetry I. For very neutron-rich nuclei, the contribution of the term will play an important role. Supported by National Natural Science Foundation of China (11475004, 11275052, 11305003, 11375094 and 11465005), Natural Science Foundation of He'nan Educational Committee (2011A140001 and 2011GGJS-147), Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences (Y4KF041CJ1)

Automatic differentiation for Fourier series and the radii polynomial approach

NASA Astrophysics Data System (ADS)

Lessard, Jean-Philippe; Mireles James, J. D.; Ransford, Julian

2016-11-01

In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem (PCRTBP).

Interpolation Hermite Polynomials For Finite Element Method

NASA Astrophysics Data System (ADS)

Gusev, Alexander; Vinitsky, Sergue; Chuluunbaatar, Ochbadrakh; Chuluunbaatar, Galmandakh; Gerdt, Vladimir; Derbov, Vladimir; Góźdź, Andrzej; Krassovitskiy, Pavel

2018-02-01

We describe a new algorithm for analytic calculation of high-order Hermite interpolation polynomials of the simplex and give their classification. A typical example of triangle element, to be built in high accuracy finite element schemes, is given.

Soliton interactions, Bäcklund transformations, Lax pair for a variable-coefficient generalized dispersive water-wave system

NASA Astrophysics Data System (ADS)

Liu, Lei; Tian, Bo; Zhen, Hui-Ling; Liu, De-Yin; Xie, Xi-Yang

2018-04-01

Under investigation in this paper is a variable-coefficient generalized dispersive water-wave system, which can simulate the propagation of the long weakly non-linear and weakly dispersive surface waves of variable depth in the shallow water. Under certain variable-coefficient constraints, by virtue of the Bell polynomials, Hirota method and symbolic computation, the bilinear forms, one- and two-soliton solutions are obtained. Bäcklund transformations and new Lax pair are also obtained. Our Lax pair is different from that previously reported. Based on the asymptotic and graphic analysis, with different forms of the variable coefficients, we find that there exist the elastic interactions for u, while either the elastic or inelastic interactions for v, with u and v as the horizontal velocity field and deviation height from the equilibrium position of the water, respectively. When the interactions are inelastic, we see the fission and fusion phenomena.

Limitations of the paraxial Debye approximation.

PubMed

Sheppard, Colin J R

2013-04-01

In the paraxial form of the Debye integral for focusing, higher order defocus terms are ignored, which can result in errors in dealing with aberrations, even for low numerical aperture. These errors can be avoided by using a different integration variable. The aberrations of a glass slab, such as a coverslip, are expanded in terms of the new variable, and expressed in terms of Zernike polynomials to assist with aberration balancing. Tube length error is also discussed.

High quality adaptive optics zoom with adaptive lenses

NASA Astrophysics Data System (ADS)

Quintavalla, M.; Santiago, F.; Bonora, S.; Restaino, S.

2018-02-01

We present the combined use of large aperture adaptive lens with large optical power modulation with a multi actuator adaptive lens. The Multi-actuator Adaptive Lens (M-AL) can correct up to the 4th radial order of Zernike polynomials, without any obstructions (electrodes and actuators) placed inside its clear aperture. We demonstrated that the use of both lenses together can lead to better image quality and to the correction of aberrations of adaptive optics optical systems.

New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials

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

Marquette, Ian; Quesne, Christiane

2013-04-15

In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequencesmore » of EOP.« less

Dynamic changes in ocular Zernike aberrations and tear menisci measured with a wavefront sensor and an anterior segment OCT.

PubMed

Xu, Jingjing; Bao, Jinhua; Deng, Jun; Lu, Fan; He, Ji C

2011-07-29

To measure dynamic change characteristics of spatial and temporal variations in the post-blink tear film of normal eyes. A wavefront sensor was used to measure dynamic changes in wavefront aberrations, up to the seventh order, for 10 seconds in a group of 33 normal young adults. Tear menisci were imaged with an anterior segment optical coherence tomography (AS-OCT) system and tear film break-up times (TFBUTs) were determined. Systematic changes in main axis astigmatism (R(2) = 0.933, P < 0.0001), vertical coma (R(2) = 0.935, P < 0.0001) and spherical aberrations (R(2) = 0.879, P = 0.0002) occurred during the 10-second post-blink period. Both lower tear meniscus height and area increased by 10 seconds compared with the initial levels (P < 0.0001 for each). The change of vertical coma had significant correlation with the increase of lower tear meniscus areas during the 10-second post-blink period (R(2) = 0.181, P = 0.014). Subjects with TFBUTs < 15 seconds had significantly increased main axis astigmatism, vertical coma, and spherical aberrations by 10 seconds. Subjects with longer TFBUTs did not have any significant wavefront aberrations during that period. Systematic changes in some Zernike aberrations after blinking are associated with the changes in tear menisci and TFBUT. There was a substantial individual variation in dynamic changes of Zernike aberrations, suggesting the necessity to explore individual differences in tear quality and tear performance. Dynamic wavefront measurement combined with anterior segment optical coherence tomography could provide a useful tool to understand spatial and temporal processes of the tear film in clinical practice.

A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials

NASA Astrophysics Data System (ADS)

Wu, Jinglai; Luo, Zhen; Zhang, Nong; Zhang, Yunqing

2015-09-01

The accuracy of metamodelling is determined by both the sampling and approximation. This article proposes a new sampling method based on the zeros of Chebyshev polynomials to capture the sampling information effectively. First, the zeros of one-dimensional Chebyshev polynomials are applied to construct Chebyshev tensor product (CTP) sampling, and the CTP is then used to construct high-order multi-dimensional metamodels using the 'hypercube' polynomials. Secondly, the CTP sampling is further enhanced to develop Chebyshev collocation method (CCM) sampling, to construct the 'simplex' polynomials. The samples of CCM are randomly and directly chosen from the CTP samples. Two widely studied sampling methods, namely the Smolyak sparse grid and Hammersley, are used to demonstrate the effectiveness of the proposed sampling method. Several numerical examples are utilized to validate the approximation accuracy of the proposed metamodel under different dimensions.

Carrier and aberrations removal in interferometric fringe projection profilometry

NASA Astrophysics Data System (ADS)

Blain, P.; Michel, F.; Renotte, Y.; Habraken, S.

2012-04-01

A profilometer which takes advantage of polarization states splitting technique and monochromatic light projection method as a way to overcome ambient lighting for in-situ measurement is under development [1, 2]. Because of the Savart plate which refracts two out of axis beams, the device suffers from aberrations (mostly coma and astigmatism). These aberrations affect the quality of the sinusoidal fringe pattern. In fringe projection profilometry, the unwrapped phase distribution map contains the sum of the object's shape-related phase and carrier-fringe-related phase. In order to extract the 3D shape of the object, the carrier phase has to be removed [3, 4]. An easy way to remove both the fringe carrier and the aberrations of the optical system is to measure the phases of the test object and to measure the phase of a reference plane with the same set up and to subtract both phase maps. This time consuming technique is suitable for laboratory but not for industry. We propose a method to numerically remove both the fringe carrier and the aberrations. A first reference phase of a calibration plane is evaluated knowing the position of the different elements in the set up and the orientation of the fringes. Then a fitting of the phase map by Zernike polynomials is computed [5]. As the triangulation parameters are known during the calibration, the computation of Zernike coefficients has only to be made once. The wavefront error can be adjusted by a scale factor which depends on the position of the test object.

Colored knot polynomials for arbitrary pretzel knots and links

DOE PAGES

Galakhov, D.; Melnikov, D.; Mironov, A.; ...

2015-04-01

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g+1)-parametric family of pretzel knots and links. The answer for the Jones and HOMFLY is fully and explicitly expressed through the Racah matrix of Uq(SU N), and looks related to a modular transformation of toric conformal block. Knot polynomials are among the hottest topics in modern theory. They are supposed to summarize nicely representation theory of quantum algebras and modular properties of conformal blocks. The result reported in the present letter, provides a spectacular illustration and support to this general expectation.

A comparison between space-time video descriptors

NASA Astrophysics Data System (ADS)

Costantini, Luca; Capodiferro, Licia; Neri, Alessandro

2013-02-01

The description of space-time patches is a fundamental task in many applications such as video retrieval or classification. Each space-time patch can be described by using a set of orthogonal functions that represent a subspace, for example a sphere or a cylinder, within the patch. In this work, our aim is to investigate the differences between the spherical descriptors and the cylindrical descriptors. In order to compute the descriptors, the 3D spherical and cylindrical Zernike polynomials are employed. This is important because both the functions are based on the same family of polynomials, and only the symmetry is different. Our experimental results show that the cylindrical descriptor outperforms the spherical descriptor. However, the performances of the two descriptors are similar.

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.

Exploring the potential of 3D Zernike descriptors and SVM for protein-protein interface prediction.

PubMed

Daberdaku, Sebastian; Ferrari, Carlo

2018-02-06

The correct determination of protein-protein interaction interfaces is important for understanding disease mechanisms and for rational drug design. To date, several computational methods for the prediction of protein interfaces have been developed, but the interface prediction problem is still not fully understood. Experimental evidence suggests that the location of binding sites is imprinted in the protein structure, but there are major differences among the interfaces of the various protein types: the characterising properties can vary a lot depending on the interaction type and function. The selection of an optimal set of features characterising the protein interface and the development of an effective method to represent and capture the complex protein recognition patterns are of paramount importance for this task. In this work we investigate the potential of a novel local surface descriptor based on 3D Zernike moments for the interface prediction task. Descriptors invariant to roto-translations are extracted from circular patches of the protein surface enriched with physico-chemical properties from the HQI8 amino acid index set, and are used as samples for a binary classification problem. Support Vector Machines are used as a classifier to distinguish interface local surface patches from non-interface ones. The proposed method was validated on 16 classes of proteins extracted from the Protein-Protein Docking Benchmark 5.0 and compared to other state-of-the-art protein interface predictors (SPPIDER, PrISE and NPS-HomPPI). The 3D Zernike descriptors are able to capture the similarity among patterns of physico-chemical and biochemical properties mapped on the protein surface arising from the various spatial arrangements of the underlying residues, and their usage can be easily extended to other sets of amino acid properties. The results suggest that the choice of a proper set of features characterising the protein interface is crucial for the interface prediction

Polynomial Asymptotes of the Second Kind

ERIC Educational Resources Information Center

Dobbs, David E.

2011-01-01

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

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

Generalized linear mixed models with varying coefficients for longitudinal data.

PubMed

Zhang, Daowen

2004-03-01

The routinely assumed parametric functional form in the linear predictor of a generalized linear mixed model for longitudinal data may be too restrictive to represent true underlying covariate effects. We relax this assumption by representing these covariate effects by smooth but otherwise arbitrary functions of time, with random effects used to model the correlation induced by among-subject and within-subject variation. Due to the usually intractable integration involved in evaluating the quasi-likelihood function, the double penalized quasi-likelihood (DPQL) approach of Lin and Zhang (1999, Journal of the Royal Statistical Society, Series B61, 381-400) is used to estimate the varying coefficients and the variance components simultaneously by representing a nonparametric function by a linear combination of fixed effects and random effects. A scaled chi-squared test based on the mixed model representation of the proposed model is developed to test whether an underlying varying coefficient is a polynomial of certain degree. We evaluate the performance of the procedures through simulation studies and illustrate their application with Indonesian children infectious disease data.

Quadratic polynomial interpolation on triangular domain

NASA Astrophysics Data System (ADS)

Li, Ying; Zhang, Congcong; Yu, Qian

2018-04-01

In the simulation of natural terrain, the continuity of sample points are not in consonance with each other always, traditional interpolation methods often can't faithfully reflect the shape information which lie in data points. So, a new method for constructing the polynomial interpolation surface on triangular domain is proposed. Firstly, projected the spatial scattered data points onto a plane and then triangulated them; Secondly, A C1 continuous piecewise quadric polynomial patch was constructed on each vertex, all patches were required to be closed to the line-interpolation one as far as possible. Lastly, the unknown quantities were gotten by minimizing the object functions, and the boundary points were treated specially. The result surfaces preserve as many properties of data points as possible under conditions of satisfying certain accuracy and continuity requirements, not too convex meantime. New method is simple to compute and has a good local property, applicable to shape fitting of mines and exploratory wells and so on. The result of new surface is given in experiments.

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.

Polynomial chaos representation of databases on manifolds

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

Soize, C., E-mail: christian.soize@univ-paris-est.fr; Ghanem, R., E-mail: ghanem@usc.edu

2017-04-15

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

Some Curious Properties and Loci Problems Associated with Cubics and Other Polynomials

ERIC Educational Resources Information Center

de Alwis, Amal

2012-01-01

The article begins with a well-known property regarding tangent lines to a cubic polynomial that has distinct, real zeros. We were then able to generalize this property to any polynomial with distinct, real zeros. We also considered a certain family of cubics with two fixed zeros and one variable zero, and explored the loci of centroids of…

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.

RBCs as microlenses: wavefront analysis and applications

NASA Astrophysics Data System (ADS)

Merola, Francesco; Barroso, Álvaro; Miccio, Lisa; Memmolo, Pasquale; Mugnano, Martina; Ferraro, Pietro; Denz, Cornelia

2017-06-01

Developing the recently discovered concept of RBCs as microlenses, we demonstrate further applications in wavefront analysis and diagnostics. Correlation between RBC's morphology and its behavior as a refractive optical element has been established. In fact, any deviation from the healthy RBC morphology can be seen as additional aberration in the optical wavefront passing through the cell. By this concept, accurate localization of focal spots of RBCs can become very useful in blood disorders identification. Moreover, By modelling RBC as bio-lenses through Zernike polynomials it is possible to identify a series of orthogonal parameters able to recognise RBC shapes. The main improvement concerns the possibility to combine such parameters because of their independence conversely to standard image-based analysis where morphological factors are dependent each-others. We investigate the three-dimensional positioning of such focal spots over time for samples with two different osmolarity conditions, i.e. discocytes and spherocytes. Finally, Zernike polynomials wavefront analysis allows us to study the optical behavior of RBCs under an optically-induced mechanical stress. Detailed wavefront analysis provides comprehensive information about the aberrations induced by the deformation obtained using optical tweezers. This could open new routes for analyzing cell elasticity by examining optical parameters instead of direct but with low resolution strain analysis, thanks to the high sensitivity of the interferometric tool.

Temperature dependence of the radiative recombination coefficient in crystalline silicon from spectral photoluminescence

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

Nguyen, Hieu T., E-mail: hieu.nguyen@anu.edu.au; Macdonald, Daniel; Baker-Finch, Simeon C.

2014-03-17

The radiative recombination coefficient B(T) in crystalline silicon is determined for the temperature range 90–363 K, and in particular from 270 to 350 K with an interval of 10 K, where only sparse data are available at present. The band-band absorption coefficient established recently by Nguyen et al. [J. Appl. Phys. 115, 043710 (2014)] via photoluminescence spectrum measurements is employed to compute the values of B(T) at various temperatures. The results agree very well with literature data from Trupke et al. [J. Appl. Phys. 94, 4930 (2003).] We present a polynomial parameterization describing the temperature dependence of the product of B(T) and themore » square of the intrinsic carrier density. We also find that B(T) saturates at a near constant value at room temperature and above for silicon samples with relatively low free carrier densities.« less

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

PubMed

Mahajan, Virendra N

2010-12-20

The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.

Derivatives of random matrix characteristic polynomials with applications to elliptic curves

NASA Astrophysics Data System (ADS)

Snaith, N. C.

2005-12-01

The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have at least n eigenvalues equal to 1 and investigate the first non-zero derivative of the characteristic polynomial at that point. The connection between the values of random matrix characteristic polynomials and values of L-functions in families has been well established. The motivation for this work is the expectation that through this connection with L-functions derived from families of elliptic curves, and using the Birch and Swinnerton-Dyer conjecture to relate values of the L-functions to the rank of elliptic curves, random matrix theory will be useful in probing important questions concerning these ranks.

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.

The polynomial form of the scattering equations is an H -basis

NASA Astrophysics Data System (ADS)

Bosma, Jorrit; Søgaard, Mads; Zhang, Yang

2016-08-01

We prove that the polynomial form of the scattering equations is a Macaulay H -basis. We demonstrate that this H -basis facilitates integrand reduction and global residue computations in a way very similar to using a Gröbner basis, but circumvents the heavy computation of the latter. As an example, we apply the H -basis to prove the conjecture that the dual basis of the polynomial scattering equations must contain one constant term.

Exact traveling-wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional Schrödinger equation with polynomial nonlinearity of arbitrary order.

PubMed

Petrović, Nikola Z; Belić, Milivoj; Zhong, Wei-Ping

2011-02-01

We obtain exact traveling wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with variable coefficients and polynomial Kerr nonlinearity of an arbitrarily high order. Exact solutions, given in terms of Jacobi elliptic functions, are presented for the special cases of cubic-quintic and septic models. We demonstrate that the widely used method for finding exact solutions in terms of Jacobi elliptic functions is not applicable to the nonlinear Schrödinger equation with saturable nonlinearity. ©2011 American Physical Society

Genetic modelling of test day records in dairy sheep using orthogonal Legendre polynomials.

PubMed

Kominakis, A; Volanis, M; Rogdakis, E

2001-03-01

Test day milk yields of three lactations in Sfakia sheep were analyzed fitting a random regression (RR) model, regressing on orthogonal polynomials of the stage of the lactation period, i.e. days in milk. Univariate (UV) and multivariate (MV) analyses were also performed for four stages of the lactation period, represented by average days in milk, i.e. 15, 45, 70 and 105 days, to compare estimates obtained from RR models with estimates from UV and MV analyses. The total number of test day records were 790, 1314 and 1041 obtained from 214, 342 and 303 ewes in the first, second and third lactation, respectively. Error variances and covariances between regression coefficients were estimated by restricted maximum likelihood. Models were compared using likelihood ratio tests (LRTs). Log likelihoods were not significantly reduced when the rank of the orthogonal Legendre polynomials (LPs) of lactation stage was reduced from 4 to 2 and homogenous variances for lactation stages within lactations were considered. Mean weighted heritability estimates with RR models were 0.19, 0.09 and 0.08 for first, second and third lactation, respectively. The respective estimates obtained from UV analyses were 0.14, 0.12 and 0.08, respectively. Mean permanent environmental variance, as a proportion of the total, was high at all stages and lactations ranging from 0.54 to 0.71. Within lactations, genetic and permanent environmental correlations between lactation stages were in the range from 0.36 to 0.99 and 0.76 to 0.99, respectively. Genetic parameters for additive genetic and permanent environmental effects obtained from RR models were different from those obtained from UV and MV analyses.

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).

Quality factor analysis for aberrated laser beam

NASA Astrophysics Data System (ADS)

Ghafary, B.; Alavynejad, M.; Kashani, F. D.

2006-12-01

The quality factor of laser beams has attracted considerable attention and some different approaches have been reported to treat the problem. In this paper we analyze quality factor of laser beam and compare the effect of different aberrations on beam quality by expanding pure phase term of wavefront in terms of Zernike polynomials. Also we analyze experimentally the change of beam quality for different Astigmatism aberrations, and compare theoretical results with experimentally results. The experimental and theoretical results are in good agreement.

Triangulation-based 3D surveying borescope

NASA Astrophysics Data System (ADS)

Pulwer, S.; Steglich, P.; Villringer, C.; Bauer, J.; Burger, M.; Franz, M.; Grieshober, K.; Wirth, F.; Blondeau, J.; Rautenberg, J.; Mouti, S.; Schrader, S.

2016-04-01

In this work, a measurement concept based on triangulation was developed for borescopic 3D-surveying of surface defects. The integration of such measurement system into a borescope environment requires excellent space utilization. The triangulation angle, the projected pattern, the numerical apertures of the optical system, and the viewing angle were calculated using partial coherence imaging and geometric optical raytracing methods. Additionally, optical aberrations and defocus were considered by the integration of Zernike polynomial coefficients. The measurement system is able to measure objects with a size of 50 μm in all dimensions with an accuracy of +/- 5 μm. To manage the issue of a low depth of field while using an optical high resolution system, a wavelength dependent aperture was integrated. Thereby, we are able to control depth of field and resolution of the optical system and can use the borescope in measurement mode with high resolution and low depth of field or in inspection mode with low resolution and higher depth of field. First measurements of a demonstrator system are in good agreement with our simulations.

Application of the aberration ring test (ARTEMIS) to determine lens quality and predict its lithographic performance

NASA Astrophysics Data System (ADS)

Moers, Marco H. P.; van der Laan, Hans; Zellenrath, Mark; de Boeij, Wim; Beaudry, Neil A.; Cummings, Kevin D.; van Zwol, Adriaan; Brecht, Arthur; Willekers, Rob

2001-09-01

ARTEMISTM (Aberration Ring Test Exposed at Multiple Illumination Settings) is a technique to determine in-situ, full-field, low and high order lens aberrations. In this paper we are analyzing the ARTEMISTM data of PAS5500/750TM DUV Step & Scan systems and its use as a lithographic prediction tool. ARTEMISTM is capable of determining Zernike coefficients up to Z25 with a 3(sigma) reproducibility range from 1.5 to 4.5 nm depending on the aberration type. 3D electric field simulations, that take the extended geometry of the phase shift feature into account, have been used for an improved treatment of the extraction of the spherical Zernike coefficients. Knowledge of the extracted Zernike coefficients allows an accurate prediction of the lithographic performance of the scanner system. This ability is demonstrated for a two bar pattern and an isolation pattern. The RMS difference between the ARTEMISTM-based lithographic prediction and the lithographic measurement is 2.5 nm for the two bar pattern and 3 nm for the isolation pattern. The 3(sigma) reproducibility of the prediction for the two bar pattern is 2.5 nm and 1 nm for the isolation pattern. This is better than the reproducibility of the lithographic measurements themselves.

A general method for computing Tutte polynomials of self-similar graphs

NASA Astrophysics Data System (ADS)

Gong, Helin; Jin, Xian'an

2017-10-01

Self-similar graphs were widely studied in both combinatorics and statistical physics. Motivated by the construction of the well-known 3-dimensional Sierpiński gasket graphs, in this paper we introduce a family of recursively constructed self-similar graphs whose inner duals are of the self-similar property. By combining the dual property of the Tutte polynomial and the subgraph-decomposition trick, we show that the Tutte polynomial of this family of graphs can be computed in an iterative way and in particular the exact expression of the formula of the number of their spanning trees is derived. Furthermore, we show our method is a general one that is easily extended to compute Tutte polynomials for other families of self-similar graphs such as Farey graphs, 2-dimensional Sierpiński gasket graphs, Hanoi graphs, modified Koch graphs, Apollonian graphs, pseudofractal scale-free web, fractal scale-free network, etc.

Explicit bounds for the positive root of classes of polynomials with applications

NASA Astrophysics Data System (ADS)

Herzberger, Jürgen

2003-03-01

We consider a certain type of polynomial equations for which there exists--according to Descartes' rule of signs--only one simple positive root. These equations are occurring in Numerical Analysis when calculating or estimating the R-order or Q-order of convergence of certain iterative processes with an error-recursion of special form. On the other hand, these polynomial equations are very common as defining equations for the effective rate of return for certain cashflows like bonds or annuities in finance. The effective rate of interest i* for those cashflows is i*=q*-1, where q* is the unique positive root of such polynomial. We construct bounds for i* for a special problem concerning an ordinary simple annuity which is obtained by changing the conditions of such an annuity with given data applying the German rule (Preisangabeverordnung or short PAngV). Moreover, we consider a number of results for such polynomial roots in Numerical Analysis showing that by a simple variable transformation we can derive several formulas out of earlier results by applying this transformation. The same is possible in finance in order to generalize results to more complicated cashflows.

A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models

NASA Technical Reports Server (NTRS)

Giunta, Anthony A.; Watson, Layne T.

1998-01-01

Two methods of creating approximation models are compared through the calculation of the modeling accuracy on test problems involving one, five, and ten independent variables. Here, the test problems are representative of the modeling challenges typically encountered in realistic engineering optimization problems. The first approximation model is a quadratic polynomial created using the method of least squares. This type of polynomial model has seen considerable use in recent engineering optimization studies due to its computational simplicity and ease of use. However, quadratic polynomial models may be of limited accuracy when the response data to be modeled have multiple local extrema. The second approximation model employs an interpolation scheme known as kriging developed in the fields of spatial statistics and geostatistics. This class of interpolating model has the flexibility to model response data with multiple local extrema. However, this flexibility is obtained at an increase in computational expense and a decrease in ease of use. The intent of this study is to provide an initial exploration of the accuracy and modeling capabilities of these two approximation methods.

Structure of highly asymmetric hard-sphere mixtures: an efficient closure of the Ornstein-Zernike equations.

PubMed

Amokrane, S; Ayadim, A; Malherbe, J G

2005-11-01

A simple modification of the reference hypernetted chain (RHNC) closure of the multicomponent Ornstein-Zernike equations with bridge functions taken from Rosenfeld's hard-sphere bridge functional is proposed. Its main effect is to remedy the major limitation of the RHNC closure in the case of highly asymmetric mixtures--the wide domain of packing fractions in which it has no solution. The modified closure is also much faster, while being of similar complexity. This is achieved with a limited loss of accuracy, mainly for the contact value of the big sphere correlation functions. Comparison with simulation shows that inside the RHNC no-solution domain, it provides a good description of the structure, while being clearly superior to all the other closures used so far to study highly asymmetric mixtures. The generic nature of this closure and its good accuracy combined with a reduced no-solution domain open up the possibility to study the phase diagram of complex fluids beyond the hard-sphere model.

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)

Hypernetted-chain-like closure of Ornstein-Zernike equation in multibody dissipative particle dynamics

NASA Astrophysics Data System (ADS)

Mo, Chao-jie; Qin, Li-zi; Yang, Li-jun

2017-10-01

We have derived a hypernetted-chain-like (HNC-like) approximate closure of the Ornstein-Zernike equation for multibody dissipative particle dynamics (MDPD) system in which the classic closures are not directly practicable. We first point out that the Percus's method is applicable to MDPD system in which particles interact with a density-dependent potential. And then an HNC-like closure is derived using Percus's idea and the saddle-point approximation of particle free energy. This HNC-like closure is compared with results of previous researchers, and in many cases, it demonstrates better agreement with computer simulation results. The HNC-like closure is used to predict the cluster crystallization in MDPD. We determine whether the cluster crystallization will happen in a system utilizing the widely applicable Hansen-Verlet freezing criterion and by observing the radial distribution function. The conclusions drawn from the results of the HNC-like closure are in agreement with computer simulation results. We evaluate different weight functions to determine whether they are prone to cluster crystallization. A new effective density-dependent pairwise potential is also proposed to help to explain the tendency to cluster crystallization of MDPD systems.

New graph polynomials in parametric QED Feynman integrals

NASA Astrophysics Data System (ADS)

Golz, Marcel

2017-10-01

In recent years enormous progress has been made in perturbative quantum field theory by applying methods of algebraic geometry to parametric Feynman integrals for scalar theories. The transition to gauge theories is complicated not only by the fact that their parametric integrand is much larger and more involved. It is, moreover, only implicitly given as the result of certain differential operators applied to the scalar integrand exp(-ΦΓ /ΨΓ) , where ΨΓ and ΦΓ are the Kirchhoff and Symanzik polynomials of the Feynman graph Γ. In the case of quantum electrodynamics we find that the full parametric integrand inherits a rich combinatorial structure from ΨΓ and ΦΓ. In the end, it can be expressed explicitly as a sum over products of new types of graph polynomials which have a combinatoric interpretation via simple cycle subgraphs of Γ.

A Formally-Verified Decision Procedure for Univariate Polynomial Computation Based on Sturm's Theorem

NASA Technical Reports Server (NTRS)

Narkawicz, Anthony J.; Munoz, Cesar A.

2014-01-01

Sturm's Theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semiopen interval. This paper presents a formalization of this theorem in the PVS theorem prover, as well as a decision procedure that checks whether a polynomial is always positive, nonnegative, nonzero, negative, or nonpositive on any input interval. The soundness and completeness of the decision procedure is proven in PVS. The procedure and its correctness properties enable the implementation of a PVS strategy for automatically proving existential and universal univariate polynomial inequalities. Since the decision procedure is formally verified in PVS, the soundness of the strategy depends solely on the internal logic of PVS rather than on an external oracle. The procedure itself uses a combination of Sturm's Theorem, an interval bisection procedure, and the fact that a polynomial with exactly one root in a bounded interval is always nonnegative on that interval if and only if it is nonnegative at both endpoints.

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

A new method to calculate unsteady particle kinematics and drag coefficient in a subsonic post-shock flow

NASA Astrophysics Data System (ADS)

Bordoloi, Ankur D.; Ding, Liuyang; Martinez, Adam A.; Prestridge, Katherine; Adrian, Ronald J.

2018-07-01

We introduce a new method (piecewise integrated dynamics equation fit, PIDEF) that uses the particle dynamics equation to determine unsteady kinematics and drag coefficient (C D) for a particle in subsonic post-shock flow. The uncertainty of this method is assessed based on simulated trajectories for both quasi-steady and unsteady flow conditions. Traditional piecewise polynomial fitting (PPF) shows high sensitivity to measurement error and the function used to describe C D, creating high levels of relative error (1) when applied to unsteady shock-accelerated flows. The PIDEF method provides reduced uncertainty in calculations of unsteady acceleration and drag coefficient for both quasi-steady and unsteady flows. This makes PIDEF a preferable method over PPF for complex flows where the temporal response of C D is unknown. We apply PIDEF to experimental measurements of particle trajectories from 8-pulse particle tracking and determine the effect of incident Mach number on relaxation kinematics and drag coefficient of micron-sized particles.

Method for Expressing Clinical and Statistical Significance of Ocular and Corneal Wavefront Error Aberrations

PubMed Central

Smolek, Michael K.

2011-01-01

Purpose The significance of ocular or corneal aberrations may be subject to misinterpretation whenever eyes with different pupil sizes or the application of different Zernike expansion orders are compared. A method is shown that uses simple mathematical interpolation techniques based on normal data to rapidly determine the clinical significance of aberrations, without concern for pupil and expansion order. Methods Corneal topography (Tomey, Inc.; Nagoya, Japan) from 30 normal corneas was collected and the corneal wavefront error analyzed by Zernike polynomial decomposition into specific aberration types for pupil diameters of 3, 5, 7, and 10 mm and Zernike expansion orders of 6, 8, 10 and 12. Using this 4×4 matrix of pupil sizes and fitting orders, best-fitting 3-dimensional functions were determined for the mean and standard deviation of the RMS error for specific aberrations. The functions were encoded into software to determine the significance of data acquired from non-normal cases. Results The best-fitting functions for 6 types of aberrations were determined: defocus, astigmatism, prism, coma, spherical aberration, and all higher-order aberrations. A clinical screening method of color-coding the significance of aberrations in normal, postoperative LASIK, and keratoconus cases having different pupil sizes and different expansion orders is demonstrated. Conclusions A method to calibrate wavefront aberrometry devices by using a standard sample of normal cases was devised. This method could be potentially useful in clinical studies involving patients with uncontrolled pupil sizes or in studies that compare data from aberrometers that use different Zernike fitting-order algorithms. PMID:22157570

Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials

PubMed Central

Khan, Rahmat Ali; Tajadodi, Haleh; Johnston, Sarah Jane

2014-01-01

In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques. PMID:25485293

Estimates for the Aerodynamic Coefficients of Ringsail and Disk-Gap-Band Parachutes Operating on Mars

NASA Technical Reports Server (NTRS)

Cruz, Juan R.; Snyder, Miranda L.

2017-01-01

Models are presented for the aerodynamic coefficients of Supersonic Ringsail and Disk-Gap-Band parachutes as functions of total porosity, Lambda(sub t), Mach number, M, and total angle of attack, Alpha(sub t) (when necessary). The source aerodynamic coefficients data used for creating these models were obtained from a wind tunnel test of subscale parachutes. In this wind tunnel test, subscale parachutes of both parachute types were fabricated from two different fabrics with very different permeabilities. By varying the fabric permeability, while maintaining the parachute geometry constant, it was possible to vary Alpha(sub t). The fabric permeability test data necessary for the calculation of Alpha(sub t) were obtained from samples of the same fabrics used to fabricate the subscale parachutes. Although the models for the aerodynamic coefficients are simple polynomial functions of Alpha(sub t) and M, they are capable of producing good reproductions of the source data. The (Alpha(sub t), M) domains over which these models are applicable are clearly defined. The models are applicable to flight operations on Mars.

On Using Homogeneous Polynomials To Design Anisotropic Yield Functions With Tension/Compression Symmetry/Assymetry

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

Soare, S.; Cazacu, O.; Yoon, J. W.

With few exceptions, non-quadratic homogeneous polynomials have received little attention as possible candidates for yield functions. One reason might be that not every such polynomial is a convex function. In this paper we show that homogeneous polynomials can be used to develop powerful anisotropic yield criteria, and that imposing simple constraints on the identification process leads, aposteriori, to the desired convexity property. It is shown that combinations of such polynomials allow for modeling yielding properties of metallic materials with any crystal structure, i.e. both cubic and hexagonal which display strength differential effects. Extensions of the proposed criteria to 3D stressmore » states are also presented. We apply these criteria to the description of the aluminum alloy AA2090T3. We prove that a sixth order orthotropic homogeneous polynomial is capable of a satisfactory description of this alloy. Next, applications to the deep drawing of a cylindrical cup are presented. The newly proposed criteria were implemented as UMAT subroutines into the commercial FE code ABAQUS. We were able to predict six ears on the AA2090T3 cup's profile. Finally, we show that a tension/compression asymmetry in yielding can have an important effect on the earing profile.« less

On Using Homogeneous Polynomials To Design Anisotropic Yield Functions With Tension/Compression Symmetry/Assymetry

NASA Astrophysics Data System (ADS)

Soare, S.; Yoon, J. W.; Cazacu, O.

2007-05-01

With few exceptions, non-quadratic homogeneous polynomials have received little attention as possible candidates for yield functions. One reason might be that not every such polynomial is a convex function. In this paper we show that homogeneous polynomials can be used to develop powerful anisotropic yield criteria, and that imposing simple constraints on the identification process leads, aposteriori, to the desired convexity property. It is shown that combinations of such polynomials allow for modeling yielding properties of metallic materials with any crystal structure, i.e. both cubic and hexagonal which display strength differential effects. Extensions of the proposed criteria to 3D stress states are also presented. We apply these criteria to the description of the aluminum alloy AA2090T3. We prove that a sixth order orthotropic homogeneous polynomial is capable of a satisfactory description of this alloy. Next, applications to the deep drawing of a cylindrical cup are presented. The newly proposed criteria were implemented as UMAT subroutines into the commercial FE code ABAQUS. We were able to predict six ears on the AA2090T3 cup's profile. Finally, we show that a tension/compression asymmetry in yielding can have an important effect on the earing profile.

Evaluation of a global algorithm for wavefront reconstruction for Shack-Hartmann wave-front sensors and thick fundus reflectors.

PubMed

Liu, Tao; Thibos, Larry; Marin, Gildas; Hernandez, Martha

2014-01-01

Conventional aberration analysis by a Shack-Hartmann aberrometer is based on the implicit assumption that an injected probe beam reflects from a single fundus layer. In fact, the biological fundus is a thick reflector and therefore conventional analysis may produce errors of unknown magnitude. We developed a novel computational method to investigate this potential failure of conventional analysis. The Shack-Hartmann wavefront sensor was simulated by computer software and used to recover by two methods the known wavefront aberrations expected from a population of normally-aberrated human eyes and bi-layer fundus reflection. The conventional method determines the centroid of each spot in the SH data image, from which wavefront slopes are computed for least-squares fitting with derivatives of Zernike polynomials. The novel 'global' method iteratively adjusted the aberration coefficients derived from conventional centroid analysis until the SH image, when treated as a unitary picture, optimally matched the original data image. Both methods recovered higher order aberrations accurately and precisely, but only the global algorithm correctly recovered the defocus coefficients associated with each layer of fundus reflection. The global algorithm accurately recovered Zernike coefficients for mean defocus and bi-layer separation with maximum error <0.1%. The global algorithm was robust for bi-layer separation up to 2 dioptres for a typical SH wavefront sensor design. For 100 randomly generated test wavefronts with 0.7 D axial separation, the retrieved mean axial separation was 0.70 D with standard deviations (S.D.) of 0.002 D. Sufficient information is contained in SH data images to measure the dioptric thickness of dual-layer fundus reflection. The global algorithm is superior since it successfully recovered the focus value associated with both fundus layers even when their separation was too small to produce clearly separated spots, while the conventional analysis

Wavefront aberrations of x-ray dynamical diffraction beams.

PubMed

Liao, Keliang; Hong, Youli; Sheng, Weifan

2014-10-01

The effects of dynamical diffraction in x-ray diffractive optics with large numerical aperture render the wavefront aberrations difficult to describe using the aberration polynomials, yet knowledge of them plays an important role in a vast variety of scientific problems ranging from optical testing to adaptive optics. Although the diffraction theory of optical aberrations was established decades ago, its application in the area of x-ray dynamical diffraction theory (DDT) is still lacking. Here, we conduct a theoretical study on the aberration properties of x-ray dynamical diffraction beams. By treating the modulus of the complex envelope as the amplitude weight function in the orthogonalization procedure, we generalize the nonrecursive matrix method for the determination of orthonormal aberration polynomials, wherein Zernike DDT and Legendre DDT polynomials are proposed. As an example, we investigate the aberration evolution inside a tilted multilayer Laue lens. The corresponding Legendre DDT polynomials are obtained numerically, which represent balanced aberrations yielding minimum variance of the classical aberrations of an anamorphic optical system. The balancing of classical aberrations and their standard deviations are discussed. We also present the Strehl ratio of the primary and secondary balanced aberrations.

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.

Correlations of RMT characteristic polynomials and integrability: Hermitean matrices

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

Osipov, Vladimir Al., E-mail: Vladimir.Osipov@uni-due.d; Kanzieper, Eugene, E-mail: Eugene.Kanzieper@hit.ac.i; Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100

Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of {tau} functions, we (i) identify a zoo of hierarchical relations satisfied by {tau} functions in an abstract infinite-dimensional space and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasismore » is placed on the phenomenon of fermionic-bosonic factorisation of random-matrix-theory correlation functions.« less

Polynomial complexity despite the fermionic sign

NASA Astrophysics Data System (ADS)

Rossi, R.; Prokof'ev, N.; Svistunov, B.; Van Houcke, K.; Werner, F.

2017-04-01

It is commonly believed that in unbiased quantum Monte Carlo approaches to fermionic many-body problems, the infamous sign problem generically implies prohibitively large computational times for obtaining thermodynamic-limit quantities. We point out that for convergent Feynman diagrammatic series evaluated with a recently introduced Monte Carlo algorithm (see Rossi R., arXiv:1612.05184), the computational time increases only polynomially with the inverse error on thermodynamic-limit quantities.

Optimization of Cubic Polynomial Functions without Calculus

ERIC Educational Resources Information Center

Taylor, Ronald D., Jr.; Hansen, Ryan

2008-01-01

In algebra and precalculus courses, students are often asked to find extreme values of polynomial functions in the context of solving an applied problem; but without the notion of derivative, something is lost. Either the functions are reduced to quadratics, since students know the formula for the vertex of a parabola, or solutions are…

The hit problem for symmetric polynomials over the Steenrod algebra

NASA Astrophysics Data System (ADS)

Janfada, A. S.; Wood, R. M. W.

2002-09-01

We cite [18] for references to work on the hit problem for the polynomial algebra P(n) = [open face F]2[x1, ;…, xn] = [oplus B: plus sign in circle]d[gt-or-equal, slanted]0 Pd(n), viewed as a graded left module over the Steenrod algebra [script A] at the prime 2. The grading is by the homogeneous polynomials Pd(n) of degree d in the n variables x1, …, xn of grading 1. The present article investigates the hit problem for the [script A]-submodule of symmetric polynomials B(n) = P(n)[sum L: summation operator]n , where [sum L: summation operator]n denotes the symmetric group on n letters acting on the right of P(n). Among the main results is the symmetric version of the well-known Peterson conjecture. For a positive integer d, let [mu](d) denote the smallest value of k for which d = [sum L: summation operator]ki=1(2[lambda]i[minus sign]1), where [lambda]i [gt-or-equal, slanted] 0.

Charactering baseline shift with 4th polynomial function for portable biomedical near-infrared spectroscopy device

NASA Astrophysics Data System (ADS)

Zhao, Ke; Ji, Yaoyao; Pan, Boan; Li, Ting

2018-02-01

The continuous-wave Near-infrared spectroscopy (NIRS) devices have been highlighted for its clinical and health care applications in noninvasive hemodynamic measurements. The baseline shift of the deviation measurement attracts lots of attentions for its clinical importance. Nonetheless current published methods have low reliability or high variability. In this study, we found a perfect polynomial fitting function for baseline removal, using NIRS. Unlike previous studies on baseline correction for near-infrared spectroscopy evaluation of non-hemodynamic particles, we focused on baseline fitting and corresponding correction method for NIRS and found that the polynomial fitting function at 4th order is greater than the function at 2nd order reported in previous research. Through experimental tests of hemodynamic parameters of the solid phantom, we compared the fitting effect between the 4th order polynomial and the 2nd order polynomial, by recording and analyzing the R values and the SSE (the sum of squares due to error) values. The R values of the 4th order polynomial function fitting are all higher than 0.99, which are significantly higher than the corresponding ones of 2nd order, while the SSE values of the 4th order are significantly smaller than the corresponding ones of the 2nd order. By using the high-reliable and low-variable 4th order polynomial fitting function, we are able to remove the baseline online to obtain more accurate NIRS measurements.

General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory

NASA Astrophysics Data System (ADS)

van Haver, Sven; Janssen, Olaf T. A.; Braat, Joseph J. M.; Janssen, Augustus J. E. M.; Urbach, H. Paul; Pereira, Silvania F.

2008-03-01

In this paper we introduce a new mask imaging algorithm that is based on the source point integration method (or Abbe method). The method presented here distinguishes itself from existing methods by exploiting the through-focus imaging feature of the Extended Nijboer-Zernike (ENZ) theory of diffraction. An introduction to ENZ-theory and its application in general imaging is provided after which we describe the mask imaging scheme that can be derived from it. The remainder of the paper is devoted to illustrating the advantages of the new method over existing methods (Hopkins-based). To this extent several simulation results are included that illustrate advantages arising from: the accurate incorporation of isolated structures, the rigorous treatment of the object (mask topography) and the fully vectorial through-focus image formation of the ENZ-based algorithm.

Polynomial stability of a magneto-thermoelastic Mindlin-Timoshenko plate model

NASA Astrophysics Data System (ADS)

Ferreira, Marcio V.; Muñoz Rivera, Jaime E.

2018-02-01

In this paper, we consider the magneto-thermoelastic interactions in a two-dimensional Mindlin-Timoshenko plate. Our main result is concerned with the strong asymptotic stabilization of the model. In particular, we determine the rate of polynomial decay of the associated energy. In contrast with what was observed in other related articles, geometrical hypotheses on the plate configuration (such as radial symmetry) are not imposed in this study nor any kind of frictional damping mechanism. A suitable multiplier is instrumental in establishing the polynomial stability with the aid of a recent result due to Borichev and Tomilov (Math Ann 347(2):455-478, 2010).

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

PubMed

Mahmood, Zahid; Ning, Huansheng; Ghafoor, AtaUllah

2017-03-24

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

Optical Performance Modeling of FUSE Telescope Mirror

NASA Technical Reports Server (NTRS)

Saha, Timo T.; Ohl, Raymond G.; Friedman, Scott D.; Moos, H. Warren

2000-01-01

We describe the Metrology Data Processor (METDAT), the Optical Surface Analysis Code (OSAC), and their application to the image evaluation of the Far Ultraviolet Spectroscopic Explorer (FUSE) mirrors. The FUSE instrument - designed and developed by the Johns Hopkins University and launched in June 1999 is an astrophysics satellite which provides high resolution spectra (lambda/Delta(lambda) = 20,000 - 25,000) in the wavelength region from 90.5 to 118.7 nm The FUSE instrument is comprised of four co-aligned, normal incidence, off-axis parabolic mirrors, four Rowland circle spectrograph channels with holographic gratings, and delay line microchannel plate detectors. The OSAC code provides a comprehensive analysis of optical system performance, including the effects of optical surface misalignments, low spatial frequency deformations described by discrete polynomial terms, mid- and high-spatial frequency deformations (surface roughness), and diffraction due to the finite size of the aperture. Both normal incidence (traditionally infrared, visible, and near ultraviolet mirror systems) and grazing incidence (x-ray mirror systems) systems can be analyzed. The code also properly accounts for reflectance losses on the mirror surfaces. Low frequency surface errors are described in OSAC by using Zernike polynomials for normal incidence mirrors and Legendre-Fourier polynomials for grazing incidence mirrors. The scatter analysis of the mirror is based on scalar scatter theory. The program accepts simple autocovariance (ACV) function models or power spectral density (PSD) models derived from mirror surface metrology data as input to the scatter calculation. The end product of the program is a user-defined pixel array containing the system Point Spread Function (PSF). The METDAT routine is used in conjunction with the OSAC program. This code reads in laboratory metrology data in a normalized format. The code then fits the data using Zernike polynomials for normal incidence

Dynamics of a new family of iterative processes for quadratic polynomials

NASA Astrophysics Data System (ADS)

Gutiérrez, J. M.; Hernández, M. A.; Romero, N.

2010-03-01

In this work we show the presence of the well-known Catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations. In fact, we introduce a family of methods, depending on a parameter . These methods reach the order of convergence m+2 when they are applied to quadratic polynomials with different roots. Newton's and Chebyshev's methods appear as particular choices of the family appear for m=0 and m=1, respectively. We make both analytical and graphical studies of these methods, which give rise to rational functions defined in the extended complex plane. Firstly, we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the Catalan numbers. Secondly, we make an incursion into its dynamical behavior. In fact, we show that the rational maps related to these methods can be written in terms of the entries of the Catalan triangle. Next we analyze its general convergence, by including some computer plots showing the intricate structure of the Universal Julia sets associated with the methods.

Transfer matrix computation of critical polynomials for two-dimensional Potts models

DOE PAGES

Jacobsen, Jesper Lykke; Scullard, Christian R.

2013-02-04

We showed, In our previous work, that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial P B(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = e K — 1 of P B(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasingmore » the size of B in an appropriate way. In earlier work, P B(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of P B(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible.We present results for the critical polynomial on the (4, 8 2), kagome, and (3, 12 2) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures v c obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain v c(4, 8 2) = 3.742 489 (4), v c(kagome) = 1.876 459 7 (2), and v c(3, 12 2) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.« less

Introduction to Real Orthogonal Polynomials

DTIC Science & Technology

1992-06-01

uses Green’s functions. As motivation , consider the Dirichlet problem for the unit circle in the plane, which involves finding a harmonic function u(r...xv ; a, b ; q) - TO [q-N ab+’q ; q, xq b. Orthogoy RMotion O0 (bq :q)x p.(q* ; a, b ; q) pg(q’ ; a, b ; q) (q "q), (aq)x (q ; q), (I -abq) (bq ; q... motivation and justi- fication for continued study of the intrinsic structure of orthogonal polynomials. 99 LIST OF REFERENCES 1. Deyer, W. M., ed., CRC

Polynomial decay rate of a thermoelastic Mindlin-Timoshenko plate model with Dirichlet boundary conditions

NASA Astrophysics Data System (ADS)

Grobbelaar-Van Dalsen, Marié

2015-02-01

In this article, we are concerned with the polynomial stabilization of a two-dimensional thermoelastic Mindlin-Timoshenko plate model with no mechanical damping. The model is subject to Dirichlet boundary conditions on the elastic as well as the thermal variables. The work complements our earlier work in Grobbelaar-Van Dalsen (Z Angew Math Phys 64:1305-1325, 2013) on the polynomial stabilization of a Mindlin-Timoshenko model in a radially symmetric domain under Dirichlet boundary conditions on the displacement and thermal variables and free boundary conditions on the shear angle variables. In particular, our aim is to investigate the effect of the Dirichlet boundary conditions on all the variables on the polynomial decay rate of the model. By once more applying a frequency domain method in which we make critical use of an inequality for the trace of Sobolev functions on the boundary of a bounded, open connected set we show that the decay is slower than in the model considered in the cited work. A comparison of our result with our polynomial decay result for a magnetoelastic Mindlin-Timoshenko model subject to Dirichlet boundary conditions on the elastic variables in Grobbelaar-Van Dalsen (Z Angew Math Phys 63:1047-1065, 2012) also indicates a correlation between the robustness of the coupling between parabolic and hyperbolic dynamics and the polynomial decay rate in the two models.

Advanced Stochastic Collocation Methods for Polynomial Chaos in RAVEN

NASA Astrophysics Data System (ADS)

Talbot, Paul W.

As experiment complexity in fields such as nuclear engineering continually increases, so does the demand for robust computational methods to simulate them. In many simulations, input design parameters and intrinsic experiment properties are sources of uncertainty. Often small perturbations in uncertain parameters have significant impact on the experiment outcome. For instance, in nuclear fuel performance, small changes in fuel thermal conductivity can greatly affect maximum stress on the surrounding cladding. The difficulty quantifying input uncertainty impact in such systems has grown with the complexity of numerical models. Traditionally, uncertainty quantification has been approached using random sampling methods like Monte Carlo. For some models, the input parametric space and corresponding response output space is sufficiently explored with few low-cost calculations. For other models, it is computationally costly to obtain good understanding of the output space. To combat the expense of random sampling, this research explores the possibilities of using advanced methods in Stochastic Collocation for generalized Polynomial Chaos (SCgPC) as an alternative to traditional uncertainty quantification techniques such as Monte Carlo (MC) and Latin Hypercube Sampling (LHS) methods for applications in nuclear engineering. We consider traditional SCgPC construction strategies as well as truncated polynomial spaces using Total Degree and Hyperbolic Cross constructions. We also consider applying anisotropy (unequal treatment of different dimensions) to the polynomial space, and offer methods whereby optimal levels of anisotropy can be approximated. We contribute development to existing adaptive polynomial construction strategies. Finally, we consider High-Dimensional Model Reduction (HDMR) expansions, using SCgPC representations for the subspace terms, and contribute new adaptive methods to construct them. We apply these methods on a series of models of increasing

Prediction of Aerodynamic Coefficients for Wind Tunnel Data using a Genetic Algorithm Optimized Neural Network

NASA Technical Reports Server (NTRS)

Rajkumar, T.; Aragon, Cecilia; Bardina, Jorge; Britten, Roy

2002-01-01

A fast, reliable way of predicting aerodynamic coefficients is produced using a neural network optimized by a genetic algorithm. Basic aerodynamic coefficients (e.g. lift, drag, pitching moment) are modelled as functions of angle of attack and Mach number. The neural network is first trained on a relatively rich set of data from wind tunnel tests of numerical simulations to learn an overall model. Most of the aerodynamic parameters can be well-fitted using polynomial functions. A new set of data, which can be relatively sparse, is then supplied to the network to produce a new model consistent with the previous model and the new data. Because the new model interpolates realistically between the sparse test data points, it is suitable for use in piloted simulations. The genetic algorithm is used to choose a neural network architecture to give best results, avoiding over-and under-fitting of the test data.

Polynomial Phase Estimation Based on Adaptive Short-Time Fourier Transform

PubMed Central

Jing, Fulong; Zhang, Chunjie; Si, Weijian; Wang, Yu; Jiao, Shuhong

2018-01-01

Polynomial phase signals (PPSs) have numerous applications in many fields including radar, sonar, geophysics, and radio communication systems. Therefore, estimation of PPS coefficients is very important. In this paper, a novel approach for PPS parameters estimation based on adaptive short-time Fourier transform (ASTFT), called the PPS-ASTFT estimator, is proposed. Using the PPS-ASTFT estimator, both one-dimensional and multi-dimensional searches and error propagation problems, which widely exist in PPSs field, are avoided. In the proposed algorithm, the instantaneous frequency (IF) is estimated by S-transform (ST), which can preserve information on signal phase and provide a variable resolution similar to the wavelet transform (WT). The width of the ASTFT analysis window is equal to the local stationary length, which is measured by the instantaneous frequency gradient (IFG). The IFG is calculated by the principal component analysis (PCA), which is robust to the noise. Moreover, to improve estimation accuracy, a refinement strategy is presented to estimate signal parameters. Since the PPS-ASTFT avoids parameter search, the proposed algorithm can be computed in a reasonable amount of time. The estimation performance, computational cost, and implementation of the PPS-ASTFT are also analyzed. The conducted numerical simulations support our theoretical results and demonstrate an excellent statistical performance of the proposed algorithm. PMID:29438317

Polynomial Phase Estimation Based on Adaptive Short-Time Fourier Transform.

PubMed

Jing, Fulong; Zhang, Chunjie; Si, Weijian; Wang, Yu; Jiao, Shuhong

2018-02-13

Polynomial phase signals (PPSs) have numerous applications in many fields including radar, sonar, geophysics, and radio communication systems. Therefore, estimation of PPS coefficients is very important. In this paper, a novel approach for PPS parameters estimation based on adaptive short-time Fourier transform (ASTFT), called the PPS-ASTFT estimator, is proposed. Using the PPS-ASTFT estimator, both one-dimensional and multi-dimensional searches and error propagation problems, which widely exist in PPSs field, are avoided. In the proposed algorithm, the instantaneous frequency (IF) is estimated by S-transform (ST), which can preserve information on signal phase and provide a variable resolution similar to the wavelet transform (WT). The width of the ASTFT analysis window is equal to the local stationary length, which is measured by the instantaneous frequency gradient (IFG). The IFG is calculated by the principal component analysis (PCA), which is robust to the noise. Moreover, to improve estimation accuracy, a refinement strategy is presented to estimate signal parameters. Since the PPS-ASTFT avoids parameter search, the proposed algorithm can be computed in a reasonable amount of time. The estimation performance, computational cost, and implementation of the PPS-ASTFT are also analyzed. The conducted numerical simulations support our theoretical results and demonstrate an excellent statistical performance of the proposed algorithm.

Polynomial expansions of single-mode motions around equilibrium points in the circular restricted three-body problem

NASA Astrophysics Data System (ADS)

Lei, Hanlun; Xu, Bo; Circi, Christian

2018-05-01

In this work, the single-mode motions around the collinear and triangular libration points in the circular restricted three-body problem are studied. To describe these motions, we adopt an invariant manifold approach, which states that a suitable pair of independent variables are taken as modal coordinates and the remaining state variables are expressed as polynomial series of them. Based on the invariant manifold approach, the general procedure on constructing polynomial expansions up to a certain order is outlined. Taking the Earth-Moon system as the example dynamical model, we construct the polynomial expansions up to the tenth order for the single-mode motions around collinear libration points, and up to order eight and six for the planar and vertical-periodic motions around triangular libration point, respectively. The application of the polynomial expansions constructed lies in that they can be used to determine the initial states for the single-mode motions around equilibrium points. To check the validity, the accuracy of initial states determined by the polynomial expansions is evaluated.

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

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

Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu; Department of Physics, Indiana University Northwest, 3400 Broadway, Gary IN 46408; Roy, Pinaki, E-mail: pinaki@isical.ac.in

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

The Julia sets of basic uniCremer polynomials of arbitrary degree

NASA Astrophysics Data System (ADS)

Blokh, Alexander; Oversteegen, Lex

Let P be a polynomial of degree d with a Cremer point p and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets J_P . The red dwarf J_P are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing p and the orbits of all critical images. The solar J_P are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and J_P is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of J_P and show that such sets J_P appear through polynomial-like maps for generic polynomials with Cremer points. Since known tools break down for d>2 (if d>2 , it is not known if there are small cycles near p , while if d=2 , this result is due to Yoccoz), we introduce wandering ray continua in J_P and provide a new application of Thurston laminations.

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.

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

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

Balondo Iyela, Daddy; Centre for Cosmology, Particle Physics and Phenomenology; Département de Physique, Université de Kinshasa

2013-09-15

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

Hydrodynamics-based functional forms of activity metabolism: a case for the power-law polynomial function in animal swimming energetics.

PubMed

Papadopoulos, Anthony

2009-01-01

The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.