Showing papers on "Fourier series published in 2010"
TL;DR: In this paper, the authors introduce the concepts of participant triangularity and triangular flow in heavy-ion collisions, analogous to the definitions of participant eccentricity and elliptic flow, and show that triangular flow is present in data.
Abstract: We introduce the concepts of participant triangularity and triangular flow in heavy-ion collisions, analogous to the definitions of participant eccentricity and elliptic flow. The participant triangularity characterizes the triangular anisotropy of the initial nuclear overlap geometry and arises from event-by-event fluctuations in the participant-nucleon collision points. In studies using a multiphase transport model (AMPT), a triangular flow signal is observed that is proportional to the participant triangularity and corresponds to a large third Fourier coefficient in two-particle azimuthal correlation functions. Using two-particle azimuthal correlations at large pseudorapidity separations measured by the PHOBOS and STAR experiments, we show that this Fourier component is also present in data. Ratios of the second and third Fourier coefficients in data exhibit similar trends as a function of centrality and transverse momentum as in AMPT calculations. These findings suggest a significant contribution of triangular flow to the ridge and broad away-side features observed in data. Triangular flow provides a new handle on the initial collision geometry and collective expansion dynamics in heavy-ion collisions.
587 citations
28 Feb 2010
TL;DR: In this article, the authors introduce the concepts of participant triangularity and triangular flow in heavy-ion collisions, analogous to the definitions of participant eccentricity and elliptic flow, and show that triangular flow is present in data.
Abstract: We introduce the concepts of participant triangularity and triangular flow in heavy-ion collisions, analogous to the definitions of participant eccentricity and elliptic flow. The participant triangularity characterizes the triangular anisotropy of the initial nuclear overlap geometry and arises from event-by-event fluctuations in the participant-nucleon collision points. In studies using a multiphase transport model (AMPT), a triangular flow signal is observed that is proportional to the participant triangularity and corresponds to a large third Fourier coefficient in two-particle azimuthal correlation functions. Using two-particle azimuthal correlations at large pseudorapidity separations measured by the PHOBOS and STAR experiments, we show that this Fourier component is also present in data. Ratios of the second and third Fourier coefficients in data exhibit similar trends as a function of centrality and transverse momentum as in AMPT calculations. These findings suggest a significant contribution of triangular flow to the ridge and broad away-side features observed in data. Triangular flow provides a new handle on the initial collision geometry and collective expansion dynamics in heavy-ion collisions.
406 citations
Book•
10 Mar 2010
TL;DR: The Analytical Theory of Heat as mentioned in this paper is an English translation of French mathematician Joseph Fourier's Theorie Analytique de la Chaleur, originally published in French in 1822.
Abstract: First published in 1878, The Analytical Theory of Heat is Alexander Freeman's English translation of French mathematician Joseph Fourier's Theorie Analytique de la Chaleur, originally published in French in 1822. In this groundbreaking study, arguing that previous theories of mechanics advanced by such scientific greats as Archimedes, Galileo, Newton and their successors did not explain the laws of heat, Fourier set out to study the mathematical laws governing heat diffusion and proposed that an infinite mathematical series may be used to analyse the conduction of heat in solids. Known in scientific circles as the 'Fourier Series', this work paved the way for modern mathematical physics. This translation, now reissued, contains footnotes that cross-reference other writings by Fourier and his contemporaries, along with 20 figures and an extensive bibliography. This book will be especially useful for mathematicians who are interested in trigonometric series and their applications.
372 citations
TL;DR: A new FFT-based scheme is proposed which is as simple as the basic scheme, while remaining valid for infinite contrasts, and provides an energetically consistent rule for the homogenization of boundary voxels.
223 citations
TL;DR: A novel-function approximator is constructed by combining a fuzzy-logic system with a Fourier series expansion in order to model unknown periodically disturbed system functions, and an adaptive backstepping tracking-control scheme is developed, where the dynamic-surface-control approach is used to solve the problem of “explosion of complexity” in the backstepped design procedure.
Abstract: A novel-function approximator is constructed by combining a fuzzy-logic system with a Fourier series expansion in order to model unknown periodically disturbed system functions. Then, an adaptive backstepping tracking-control scheme is developed, where the dynamic-surface-control approach is used to solve the problem of “explosion of complexity” in the backstepping design procedure, and the time-varying parameter-dependent integral Lyapunov function is used to analyze the stability of the closed-loop system. The semiglobal uniform ultimate boundedness of all closed-loop signals is guaranteed, and the tracking error is proved to converge to a small residual set around the origin. Two simulation examples are provided to illustrate the effectiveness of the control scheme designed in this paper.
198 citations
TL;DR: This paper develops the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results and implies a simpler optimized version of the deterministic compressed sensing method previously developed in.
Abstract: We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N≫k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of $\hat{\mathbf{A}}$, and estimate their coefficients, in polynomial(k,log N) time. Randomized sublinear-time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem (Gilbert et al. in ACM STOC, pp. 152–161, 2002; Proceedings of SPIE Wavelets XI, 2005). In this paper we develop the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method (Gilbert et al. in Proceedings of SPIE Wavelets XI, 2005). Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in (Iwen in Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA’08), 2008).
170 citations
TL;DR: Pseudo-differential and Fourier series operators on the torus were analyzed in this paper by using global representations by Fourier-series instead of local representations in coordinate charts and the correspondence between toroidal and Euclidean symbols of pseudodifferential operators was established.
Abstract: Pseudo-differential and Fourier series operators on the torus ${{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n}$
are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on L
2 under certain conditions on their phases and amplitudes.
152 citations
TL;DR: It is obtained exponentially accurate Fourier series for nonperiodic functions on the interval $[-1,1]$ by extending these functions to periodic functions on a larger domain by means of the FFT.
Abstract: We obtain exponentially accurate Fourier series for nonperiodic functions on the interval $[-1,1]$ by extending these functions to periodic functions on a larger domain. The series may be evaluated, but not constructed, by means of the FFT. A complete convergence theory is given based on orthogonal polynomials that resemble Chebyshev polynomials of the first and second kinds. We analyze a previously proposed numerical method, which is unstable in theory but stable in practice. We propose a new numerical method that is stable both in theory and in practice.
138 citations
TL;DR: In this article, a harmonic-based numerical method was proposed to determine the local stability of periodic solutions of dynamical systems. But the harmonic-balance method and asymptotic numerical method continuation technique were combined with the developed Hill method to obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions.
137 citations
TL;DR: In this article, the Fourier coefficients of a modular form of half-integral weight were shown to be related to the special values of L-functions in an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over Q.
Abstract: We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over Q. This formula has applications to proving the nonvanishingof this lift and to an explicit version of the Rallis inner product formula.
115 citations
01 Jan 2010
TL;DR: It is shown that the moduli of the Fourier coefficients of the parameterizing function of the boundary of an object do not contain enough information to characterize the shape of a object.
Abstract: Astract-A setofFourier descriptors fortwo-dimensional shapes is defined whichiscomplete inthesensethattwoobjects havethesame shape ifandonly ifthey havethesamesetofFourier descriptors. Italso is shownthatthemoduli oftheFourier coefficients oftheparameterizing function oftheboundary ofanobject donotcontain enough information to characterize theshape ofanobject. Further arelationship isestablished between rotational symmetries ofanobject andthesetofintegers for whichthecorresponding Fourier coefficients oftheparameterizing function arenonzero.
TL;DR: An overall theoretical discussion of the FC-AD approach is provided and the methodology is extended to linear hyperbolic PDEs and an example concerning the Wave Equation shows a complete absence of the debilitating ''dispersion error'', that arises as finite-difference and finite-element solvers are applied to solution of wave propagation problems.
Book•
01 Jan 2010
TL;DR: In this paper, the authors present a solution to a set of problems in standard notation using the Fourier transform and Fourier series. But they do not specify a solution for the rest of the problems.
Abstract: Preface.- Standard Notation.- 1 Motivation .- Problems.- 2 Test Functions.- Problems.- 3 Distributions.- Problems.- 4 Differentiation of Distributions.- Problems.- 5 Convergence of Distributions.- Problems.- 6 Taylor Expansion in Several Variables.- Problems.- 7 Localization.- Problems.- 8 Distributions with Compact Support.- Problems.- 9 Multiplication by Functions.- Problems.- 10 Transposition: Pullback and Pushforward.- Problems.- 11 Convolution of Distributions.- Problems.- 12 Fundamental Solutions.- Problems.- 13 Fractional Integration and Differentiation .- 13.1 The Case of Dimension One.- 13.2 Wave Family.- 13.3 Appendix: Euler's Gamma Function.- Problems.- 14 Fourier Transform.- Problems.- 15 Distribution Kernels.- Problems.- 16 Fourier Series.- Problems.- 17 Fundamental Solutions and Fourier Transform.- 17.1 Appendix: Fundamental Solution of .I?/k.- Problems.- 18 Supports and Fourier Transform.- Problems.- 19 Sobolev Spaces.- Problems.- 20 Appendix: Integration.- 21 Solutions to Selected Problems.- References.- Index of Notation.- Index.
Patent•
06 Apr 2010TL;DR: In this article, a generic, re-useable, parameterized Fourier series is derived to generate realistic waveforms with controllable voltage noise and timing jitter in a computer-based simulation environment.
Abstract: Methods for generating realistic waveforms with controllable voltage noise and timing jitter in a computer-based simulation environment and the simulation of a subset of those waveforms with system elements along the signal path is disclosed. By deriving a generic, re-useable, parameterized Fourier series, time-domain clock and pseudo-random data signals are generated from a subset of their true harmonic components. Time-domain signal parameters including high, low, and common-mode voltage levels, transition slew-rates, transition timing, period and/or frequency, may be designated by the user, and the computer calculates the harmonic components that will combine through the inverse Fourier transform to provide the required time-domain characteristics. By computing the frequency content of the signal directly it is possible to simulate the interaction of the signal with various system blocks while remaining in the frequency domain, thereby reducing simulation time and memory requirements. By allowing the parameters of the signal model to vary on a cycle-to-cycle basis, signal characteristics such as voltage noise and timing jitter may be modeled with flexibility and precision down to the numerical limitations of the simulator.
22 Sep 2010
TL;DR: The concept of translation-invariance and the random Fourier feature methodology is extended to arbitrary, locally compact Abelian groups and a new group called the skewed-multiplicative group is proposed and designed to design translation- Invariant kernels on it.
Abstract: Approximations based on random Fourier features have recently emerged as an efficient and elegant methodology for designing large-scale kernel machines [4]. By expressing the kernel as a Fourier expansion, features are generated based on a finite set of random basis projections with inner products that are Monte Carlo approximations to the original kernel. However, the original Fourier features are only applicable to translation-invariant kernels and are not suitable for histograms that are always non-negative. This paper extends the concept of translation-invariance and the random Fourier feature methodology to arbitrary, locally compact Abelian groups. Based on empirical observations drawn from the exponentiated χ2 kernel, the state-of-the-art for histogram descriptors, we propose a new group called the skewed-multiplicative group and design translation-invariant kernels on it. Experiments show that the proposed kernels outperform other kernels that can be similarly approximated. In a semantic segmentation experiment on the PASCAL VOC 2009 dataset, the approximation allows us to train large-scale learning machines more than two orders of magnitude faster than previous nonlinear SVMs.
TL;DR: In this article, an adaptive harmonic balance method is proposed based on the relative variation of an approximate strain energy for two consecutive numbers of harmonics, which is also expressed in a condensation case.
TL;DR: In this paper, a kinetic equation for the collisional evolution of stable, bound, self-gravitating and slowly relaxing systems is established, which is valid when the number of constituents is very large.
Abstract: A kinetic equation for the collisional evolution of stable, bound, self-gravitating and slowly relaxing systems is established, which is valid when the number of constituents is very large. It accounts for the detailed dynamics and self-consistent dressing by collective gravitational interaction of the colliding particles, for the system's inhomogeneity and for different constituents' masses. It describes the coupled evolution of collisionally interacting populations, such as stars in a thick disc and the molecular clouds off which they scatter.
The kinetic equation derives from the BBGKY hierarchy in the limit of weak, but non-vanishing, binary correlations, an approximation which is well justified for large stellar systems. The evolution of the 1-body distribution function is described in action–angle space. The collective response is calculated using a biorthogonal basis of pairs of density–potential functions.
The collision operators are expressed in terms of the collective response function allowed by the existing distribution functions at any given time and involve particles in resonant motion. These equations are shown to satisfy an H theorem. Because of the inhomogeneous character of the system, the relaxation causes the potential as well as the orbits of the particles to secularly evolve. The changing orbits also cause the angle Fourier coefficients of the basis potentials to change with time. We derive the set of equations which describes this coupled evolution of distribution functions, potential and basis Fourier coefficients for spherically symmetric systems. In the homogeneous limit, which sacrifices the description of the evolution of the spatial structure of the system but retains the effect of collective gravitational dressing, the kinetic equation reduces to a form similar to the Balescu–Lenard equation of plasma physics.
TL;DR: In this article, the authors present an analytical method for the vibration analysis of plates reinforced by any number of beams of arbitrary lengths and placement angles, and demonstrate the effect on modal parameters for a stiffened plate with various boundary conditions, coupling conditions, and reinforcement configurations.
TL;DR: The Inverse Polynomial Reconstruction Method is generalized by reconstructing a function from its m lowest Fourier coefficients as an algebraic polynomial of degree at most n-1(m>=n) and approximate Legendre coefficients of the function are computed by solving a linear least squares problem.
TL;DR: In this article, a mathematical model of an elastic material with constant parameters fills the half-space and the governing equations will be taken into the context of the fractional order generalized thermoelasticity theory.
Posted Content•
TL;DR: In this paper, modified variants of the sparse Fourier transform algorithms from [14] are presented which improve on the approximation error bounds of the original algorithms. And simple methods for extending the improved sparse transform algorithms to higher dimensional settings are developed.
Abstract: In this paper modified variants of the sparse Fourier transform algorithms from [14] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, $f : [0, 2 pi] -> C, in O(k^2 \cdot D^4)$ time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexities that scale linearly in the sparsity parameter k are also presented.
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01 Nov 2010
TL;DR: In this paper, the authors considered the problem of determining the first sign change of Hecke eigenvalues of a given set of primes in a modular form and gave both individual and statistical results, improving the estimate of Iwaniec, Kohnen and Sengupta.
Abstract: We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first sign-change of Hecke eigenvalues. Here we improve the recent estimate of Iwaniec, Kohnen and Sengupta.
TL;DR: In this article, the approximation properties of Norlund (Nn) and Riesz (Rn) means of trigono- metric Fourier series were investigated in generalized Lebesgue spaces.
Abstract: The approximation properties of Norlund (Nn) and Riesz (Rn) means of trigono- metric Fourier series are investigated in generalized Lebesgue spaces L p(x) . The deviations � f −Nn(f)� p(x) andf −Rn(f)� p(x) are estimated by n −α for f ∈ Lip(α,p(x)) (0 < α 1).
TL;DR: This work proposes a nonequispaced hyperbolic cross FFT based on onehyperbolicCross FFT and a dedicated interpolation by splines on sparse grids and allows for the efficient evaluation of trigonometric polynomials with Fourier coefficients supported on the hyperbolics cross at arbitrary spatial sampling nodes.
Abstract: A straightforward discretization of problems in $d$ spatial dimensions often leads to an exponential growth in the number of degrees of freedom. Thus, even efficient algorithms like the fast Fourier transform (FFT) have high computational costs. Hyperbolic cross approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives. We propose a nonequispaced hyperbolic cross FFT based on one hyperbolic cross FFT and a dedicated interpolation by splines on sparse grids. Analogously to the nonequispaced FFT for trigonometric polynomials with Fourier coefficients supported on the full grid, this allows for the efficient evaluation of trigonometric polynomials with Fourier coefficients supported on the hyperbolic cross at arbitrary spatial sampling nodes.
TL;DR: In this paper, a probabilistic analysis of buckling loads is performed regarding Fourier coefficients as random variables, and Monte Carlo simulations are executed for a set of six similarly manufactured composite shells.
Abstract: The discrepancy between the analytically determined buckling load of perfect cylindrical shells and experimental test results is traced back to imperfections. The most frequently used guideline for design of cylindrical shells, NASA SP-8007, proposes a deterministic calculation of a knockdown factor with respect to the ratio of radius and wall thickness, which turned out to be very conservative in numerous cases and which is not intended for composite shells. In order to determine a lower bound for the buckling load of an arbitrary type of shell, probabilistic design methods have been developed. Measured imperfection patterns are described using double Fourier series, whereas the Fourier coefficients characterize the scattering of geometry. In this paper, probabilistic analyses of buckling loads are performed regarding Fourier coefficients as random variables. A nonlinear finite element model is used to determine buckling loads, and Monte Carlo simulations are executed. The probabilistic approach is used for a set of six similarly manufactured composite shells. The results indicate that not only geometric but also nontraditional imperfections like loading imperfections have to be considered for obtaining a reliable lower limit of the buckling load. Finally, further Monte Carlo simulations are executed including traditional as well as loading imperfections, and lower bounds of buckling loads are obtained, which are less conservative than NASA SP-8007.
TL;DR: Cubic Schrödinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed and it is shown that along both the solution of the nonlinear Schr Ödinger equation as well as the solutions of the semi-Discretized equation the actions of the linear SchröDinger equation are approximately conserved over long times.
Abstract: Cubic Schrodinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed. It is shown that along both the solution of the nonlinear Schrodinger equation as well as the solution of the semi-discretized equation the actions of the linear Schrodinger equation are approximately conserved over long times. This also allows us to show approximate conservation of energy and momentum along the solution of the semi-discretized equation over long times. These results are obtained by analyzing a modulated Fourier expansion in time. They are valid in arbitrary spatial dimension.
Posted Content•
TL;DR: In this article, a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis is introduced.
Abstract: We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be numerically implemented in a stable manner. Moreover, the accuracy of the resulting approximation is completely determined by the choice of reconstruction basis, meaning that the reconstruction vectors can be tailored to the particular problem at hand.
An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis, we obtain an approximation with root exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction function. Numerical examples illustrate the advantage of this approach over other existing methods.
TL;DR: In this article, the Fourier expansion of general Eisenstein series and local Whittaker functions over SL2 is given, which is used to compute both the value and derivatives of these functions at critical points.
Abstract: This paper gives explicit formulas for the Fourier expansion of general Eisenstein series and local Whittaker functions over SL2. They are used to compute both the value and derivatives of these functions at critical points.
TL;DR: A frequency-domain finite element technique is presented that enables the complete characterization of a finite-sized scatterer using a minimum number of separate model executions and a relatively small spatial modeling domain.
Abstract: A frequency-domain finite element technique is presented that enables the complete characterization of a finite-sized scatterer using a minimum number of separate model executions and a relatively small spatial modeling domain. The technique is implemented using a commercial finite element package. A certain forcing profile is applied at a set of points surrounding the scatterer to cause a uni-modal plane wave to be incident on the scatterer from a specified direction. The scattered field is recorded and decomposed first into modes and then into far-field scattering coefficients in different directions. The data obtained from the model are represented in a scattering matrix that describes the far-field scattering response for all combinations of incident and scattering angles. The information in the scattering matrix can be efficiently represented in the Fourier domain by another matrix containing a finite number of Fourier coefficients. It is shown how the complete scattering behavior in both the near- and far-field can be extracted from the matrix of Fourier coefficients. Modeling accuracy is examined in various ways, including a comparison with the analytical solution for a circular cavity, and guidelines for the selection of modeling parameters are given.
TL;DR: The paper is dedicated to the component method for estimating the periodically correlated random processes mean and covariance functions, when number of harmonics is finite, based on the decomposition of these time periodic functions into trigonometric polynomials and the estimation of their Fourier coefficients.