Showing papers on "Fourier series published in 2013"

Classical and Multilinear Harmonic Analysis

Abstract: This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderon–Zygmund and Littlewood–Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman–Meyer theory; Carleson's resolution of the Lusin conjecture; Calderon's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form.

Bulk viscosity effects in event-by-event relativistic hydrodynamics

Abstract: Bulk viscosity effects on the collective flow harmonics in heavy-ion collisions are investigated, on an event-by-event basis, using a newly developed 2+1 Lagrangian hydrodynamic code named v-usphydro, which implements the smoothed particle hydrodynamics algorithm for viscous hydrodynamics. A new formula for the bulk viscous corrections present in the distribution function at freeze-out is derived, starting from the Boltzmann equation for multi-hadron species. Bulk viscosity is shown to enhance the collective flow Fourier coefficients from ${v}_{2}({p}_{T})$ to ${v}_{5}({p}_{T})$ when ${p}_{T}\ensuremath{\sim}1$--2.5 GeV, even when the bulk viscosity to entropy density ratio, $\ensuremath{\zeta}/s$, is significantly smaller than $1/(4\ensuremath{\pi})$.

A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions

Abstract: This paper describes a general formulation for free, steady-state and transient vibration analyses of functionally graded shells of revolution subjected to arbitrary boundary conditions. The formulation is derived by means of a modified variational principle in conjunction with a multi-segment partitioning procedure on the basis of the first-order shear deformation shell theory. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. Fourier series and polynomials are applied to expand the displacements and rotations of each shell segment. The versatility of the formulation is demonstrated through the application of the following polynomials: Chebyshev orthogonal polynomials, Legendre orthogonal polynomials, Hermite orthogonal polynomials and power polynomials. Numerical examples are given for the free vibrations of functionally graded cylindrical, conical and spherical shells with different combinations of free, shear-diaphragm, simply-supported, clamped and elastic-supported boundary conditions. Validity and accuracy of the present formulation are confirmed by comparing the present solutions with the existing results and those obtained from finite element analyses. As to the steady-state and transient vibration analyses, functionally graded conical shells subjected to axisymmetric line force and distributed surface pressure are investigated. The effects of the material power-law distribution, boundary condition and duration of blast loading on the transient responses of the conical shells are also examined.

Fourier series of atomic radial distribution functions: A molecular fingerprint for machine learning models of quantum chemical properties

Abstract: We introduce a fingerprint representation of molecules based on a Fourier series of atomic radial distribution functions. This fingerprint is unique (except for chirality), continuous, and differentiable with respect to atomic coordinates and nuclear charges. It is invariant with respect to translation, rotation, and nuclear permutation, and requires no pre-conceived knowledge about chemical bonding, topology, or electronic orbitals. As such it meets many important criteria for a good molecular representation, suggesting its usefulness for machine learning models of molecular properties trained across chemical compound space. To assess the performance of this new descriptor we have trained machine learning models of molecular enthalpies of atomization for training sets with up to 10k organic molecules, drawn at random from a published set of 134k organic molecules. We validate the descriptor on all remaining molecules of the 134k set. For a training set of 5k molecules the fingerprint descriptor achieves a mean absolute error of 8.0 kcal/mol, respectively. This is slightly worse than the performance attained using the Coulomb matrix, another popular alternative, reaching 6.2 kcal/mol for the same training and test sets.

Defining nonlinear rheological material functions for oscillatory shear

Abstract: Material functions underlie our understanding of rheology. They form the descriptive language of rheologists and require clear definitions. Here, it is shown that the definitions of oscillatory material functions depend on how the oscillating input is mathematically referenced, as a sine or cosine. Depending on this seemingly arbitrary trigonometric reference choice, the (3rd, 7th, 11th, etc.) Fourier coefficients of a nonlinear shear response change sign. Additionally, the even harmonic coefficients of a shear normal stress response are transposed. This impacts large-amplitude oscillatory shear (LAOS) characterization in both shear strain-control (LAOStrain) and shear stress-control (LAOStress) modes. It is important to resolve this issue, because it involves the leading-order nonlinearities and the signs of these higher harmonics convey important information. This paper provides a resolution, in two parts. First, it is shown that the deformation-domain Chebyshev coefficients are immune to the arbitrary trigonometric reference in the time domain, and therefore the Chebyshev-coefficient material functions can be used and interpreted without risk of inconsistency. Second, this paper proposes the convention of referencing to a sine input for strain-control tests (currently the typical convention) and using a cosine input for stress-control (where there is not currently a convention). Finally, clarity is brought to the practical issue of data processing a digital signal, which is required for numerical simulations and every instrument that performs oscillatory characterization.

A modified variational approach for vibration analysis of ring-stiffened conical–cylindrical shell combinations

Abstract: This work presents a modified variational method for dynamic analysis of ring-stiffened conical–cylindrical shells subjected to different boundary conditions. The method involves partitioning of the stiffened shell into appropriate shell segments in order to accommodate the computing requirement of high-order vibration modes and responses. All essential continuity constraints on segment interfaces are imposed by means of a modified variational principle and least-squares weighted residual method. Reissner-Naghdi's thin shell theory combined with the discrete element stiffener theory to consider the ring-stiffening effect is employed to formulate the theoretical model. Double mixed series, i.e., the Fourier series and Chebyshev orthogonal polynomials, are adopted as admissible displacement functions for each shell segment. To test the convergence, efficiency and accuracy of the present method, both free and forced vibrations of non-stiffened and stiffened shells are examined under different combinations of edge support conditions. Two types of external excitation forces are considered for the forced vibration analysis, i.e., the axisymmetric line force and concentrated point force. The numerical results obtained from the present method show good agreement with previously published results and those from the finite element program ANSYS. Effects of structural damping on the harmonic vibration responses of the stiffened conical–cylindrical–conical shell are also presented.

Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Abstract: Chapter 1 Flat Space Fourier Analysis on R^m- 11 Distributions or Generalized Functions- 12 Fourier Integrals- 13 Fourier Series and the Poisson Summation Formula- 14 Mellin Transforms, Epstein and Dedekind Zeta Functions- 15 Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyl's Criterion for Uniform Distribution- Chapter 2 A Compact Symmetric Space--The Sphere- 21 Fourier Analysis on the Sphere- 22 O(3) and R^3 The Radon Transform- Chapter 3 The Poincare Upper Half-Plane- 31 Hyperbolic Geometry- 32 Harmonic Analysis on H- 33 Fundamental Domains for Discrete Subgroups ? of G = SL(2, R)- 34 Modular of Automorphic Forms--Classical- 35 Automorphic Forms--Not So Classical--Maass Waveforms- 36 Modular Forms and Dirichlet Series Hecke Theory and Generalizations- References- Index

Fluctuating-surface-current formulation of radiative heat transfer: Theory and applications

Abstract: We describe a fluctuating-surface current formulation of radiative heat transfer between bodies of arbitrary shape that exploits efficient and sophisticated techniques from the surface-integral-equation formulation of classical electromagnetic scattering. Unlike previous approaches to nonequilibrium fluctuations that involve scattering matrices---relating ``incoming'' and ``outgoing'' waves from each body---our approach is formulated in terms of ``unknown'' surface currents, laying at the surfaces of the bodies, that need not satisfy any wave equation. We show that our formulation can be applied as a spectral method to obtain fast-converging semianalytical formulas in high-symmetry geometries using specialized spectral bases that conform to the surfaces of the bodies (e.g., Fourier series for planar bodies or spherical harmonics for spherical bodies), and can also be employed as a numerical method by exploiting the generality of surface meshes/grids to obtain results in more complicated geometries (e.g., interleaved bodies as well as bodies with sharp corners). In particular, our formalism allows direct application of the boundary-element method, a robust and powerful numerical implementation of the surface-integral formulation of classical electromagnetism, which we use to obtain results in new geometries, such as the heat transfer between finite slabs, cylinders, and cones.

Numerical solution of an inverse diffraction grating problem from phaseless data

Abstract: This paper is concerned with the numerical solution of an inverse diffraction grating problem, which is to reconstruct a periodic grating profile from measurements of the phaseless diffracted field at a constant height above the grating structure. An efficient continuation method is developed to recover the Fourier coefficients of the periodic grating profile. The continuation proceeds along the wavenumber and updates are obtained from the Landweber iteration at each step. Numerical results are presented to show that the method can effectively reconstruct the shape of the grating profile.

A variational method for free vibration analysis of joined cylindrical-conical shells

Abstract: A variational method is proposed to study the free vibration of joined cylindrical-conical shells (JCCSs) subjected to classical and non-classical boundary conditions. A JCCS is divided into its components (i.e., conical and cylindrical shells) at the cone-cylinder junction. The interface continuity and geometric boundary conditions are approximately enforced by means of a modified variational principle and least-squares weighted residual method. No constraints need to be imposed a priori in the admissible displacement functions for each shell component. Reissner-Naghdi's thin shell theory is used to formulate the theoretical model. Double mixed series, i.e. the Fourier series and Chebyshev orthogonal polynomials, are adopted as admissible displacement functions for each shell component. To test the convergence, efficiency and accuracy of the present method, free vibrations of JCCSs are examined under various combinations of edge support conditions. The results obtained in this study are found to be in a ...

Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity

Abstract: This paper studies vibration behavior of single-walled carbon nanotubes based on three-dimensional theory of elasticity To accounting for the size effect of carbon nanotubes, nonlocal theory is adopted to the shell model The nonlocal parameter is incorporated into all constitutive equations in three dimensions Governing differential equations of motion are reduced to the ordinary differential equations in thickness direction by using Fourier series expansion in axial and circumferential direction The state equations obtained from constitutive relations and governing equations are solved analytically by making use of the state space method A detailed parametric study is carried out to show the influences of the nonlocal parameter, thickness-to-radius ratio and length-to-radius ratio Results reveal that excluding small-scale effects caused decreasing accuracy of natural frequencies Furthermore, the obtained closed form solution can be used to assess the accuracy of conventional two-dimensional theories

Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

Abstract: The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations.

Fluctuating surface-current formulation of radiative heat transfer: theory and applications

Abstract: We describe a fluctuating-surface current formulation of radiative heat transfer between bodies of arbitrary shape that exploits efficient and sophisticated techniques from the surface-integral-equation formulation of classical electromagnetic scattering. Unlike previous approaches to nonequilibrium fluctuations that involve scattering matrices—relating “incoming” and “outgoing” waves from each body—our approach is formulated in terms of “unknown” surface currents, laying at the surfaces of the bodies, that need not satisfy any wave equation. We show that our formulation can be applied as a spectral method to obtain fast-converging semianalytical formulas in high-symmetry geometries using specialized spectral bases that conform to the surfaces of the bodies (e.g., Fourier series for planar bodies or spherical harmonics for spherical bodies), and can also be employed as a numerical method by exploiting the generality of surface meshes/grids to obtain results in more complicated geometries (e.g., interleaved bodies as well as bodies with sharp corners). In particular, our formalism allows direct application of the boundary-element method, a robust and powerful numerical implementation of the surface-integral formulation of classical electromagnetism, which we use to obtain results in new geometries, such as the heat transfer between finite slabs, cylinders, and cones.

Divergence of the mock and scrambled Fourier series on fractal measures

Abstract: We study divergence properties of the Fourier series on Cantortype fractal measures, also called the mock Fourier series. We show that in some cases the L1-norm of the corresponding Dirichlet kernel grows exponentially fast, and therefore the Fourier series are not even pointwise convergent. We apply these results to the Lebesgue measure to show that a certain rearrangement of the exponential functions, with affine structure, which we call a scrambled Fourier series, have a corresponding Dirichlet kernel whose L1norm grows exponentially fast, which is much worse than the known logarithmic bound. The divergence properties are related to the Mahler measure of certain polynomials and to spectral properties of Ruelle operators.

Buckling analysis of rectangular functionally graded plates under various edge conditions using Fourier series expansion

Abstract: In this paper, the buckling problem of thin rectangular functionally graded plates subjected to proportional biaxial compressive loadings with arbitrary edge supports is investigated Classical plate theory (CPT) based on the physical neutral plane is applied to derive the stability equations Mechanical properties of the FGM plate are assumed to vary continuously along its thickness according to a power law function The displacement function is considered to be in the form of a double Fourier series whose derivatives are determined using Stokes' transformation The advantage of this method is capability of considering any possible combination of boundary conditions with no necessity to be satisfied in the Fourier series To give generality to the problem, the plate is assumed to be elastically restrained by means of rotational and translational springs at the four edges Numerical examples are presented, and the effects of the plate aspect ratio, the FGM power index, and the loading proportionality factor on the buckling load of an FGM plate with different usual boundary conditions are studied The present results are compared with those have been previously reported by other analytical and numerical methods, and very good agreement is seen between the findings indicating validity and accuracy of the proposed approach in the buckling analysis of FGM plates

Optimal observation of the one-dimensional wave equation

Abstract: In this paper, we consider the homogeneous one-dimensional wave equation on [0,π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0,π]. Let L∈(0,1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets ω of [0,π] of Lebesgue measure Lπ. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for L=1/2. When L≠1/2 we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following Hebrard and Henrot (Syst. Control Lett., 48:199–209, 2003; SIAM J. Control Optim., 44:349–366, 2005), we then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.

Simulating Spatial Averages of Stationary Random Field Using the Fourier Series Method

Abstract: A Fourier series method (FSM) of simulating spatial averages of stationary Gaussian random fields is presented The FSM is able to simulate spatial averages over nonequally spaced rectangular cells, and by adopting Gauss quadrature, it can be further applied to nonrectangular cells It is also capable of simulating line averages over any prescribed line segment in two dimensions The former (spatial averaging over cells) is essential for finite-element analysis, while the latter (line averaging over line segments) is essential for slope-stability analysis using limit equilibrium To resolve the issue of unrealistic periodical correlation pertaining to the FSM, a rule of thumb is provided to extend the simulation space For cases with nonrectangular cells, the required number of Gauss points to achieve a prescribed accuracy is calibrated for both one-dimensional (1D) and two-dimensional (2D) cases and for both the single-exponential (SExp) and squared-exponential (QExp) autocorrelation models The

The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian H\"ormander theory

Abstract: We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46-57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron-Martin paths and complementary Young regularity (CYR) of the Cameron-Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Ito-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hormander theory.

Multivariate integration of infinitely many times differentiable functions in weighted Korobov spaces

Abstract: We study multivariate integration for a weighted Korobov space of periodic infinitely many times differentiable functions for which the Fourier coefficients decay exponentially fast. The weights are defined in terms of two non-decreasing sequences a = {ai} and b = {bi} of numbers no less than one and a parameter ω ∈ (0, 1). Let e(n, s) be the minimal worst-case error of all algorithms that use n function values in the s-variate case. We would like to check conditions on a, b and ω such that e(n, s) decays exponentially fast, i.e., for some q ∈ (0, 1) and p > 0 we have e(n, s) = O(q n p) as n goes to infinity. The factor in the O notation may depend on s in an arbitrary way. We prove that exponential convergence holds iff B := ∑∞ i=1 1/bi < ∞ independently of a and ω. Furthermore, the largest p of exponential convergence is 1/B. We also study exponential convergence with weak, polynomial and strong polynomial tractability. This means that e(n, s) ≤ C(s) q n p for all n and s and with log C(s) = exp(o(s)) for weak tractability, with a polynomial bound on log C(s) for polynomial tractability, and with uniformly bounded C(s) for strong polynomial tractability. We prove that the notions of weak, polynomial and strong polynomial tractability are equivalent, and hold iff B < ∞ and ai are exponentially growing with i. We also prove that the largest (or the supremum of) p for exponential convergence with strong polynomial tractability belongs to [1/(2B), 1/B].