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We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigourous by using systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann (DNO) and Neumann-to-Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower-order Fourier coefficients of the shape perturbation of the obstacle., 23 pages

In 1929, Fisher proposed a test for periodicity based on the largest periodogram ordinate. If the true frequency lies between two consecutive Fourier frequencies and the signal to noise ratio is low, the test may conclude that there is no periodicity. This loss of power was noted by Whittle in 1952, as well as the necessary assumption that the noise be white. Whittle and subsequent authors suggested remedies for the white noise assumption. This paper proposes simple tests, based on the Fourier coefficients, that is, the Fourier transforms at the Fourier frequencies, that have good power properties at all frequencies.

Until now the problem of estimating circular densities when data are observed with errors has been mainly treated by Fourier series methods. We propose kernel-based estimators exhibiting simple construction and easy implementation. Specifically, we consider three different approaches: the first one is based on the equivalence between kernel estimators using data corrupted with different levels of error. This proposal appears to be totally unexplored, despite its potential for application also in the Euclidean setting. The second approach relies on estimators whose weight functions are circular deconvolution kernels. Due to the periodicity of the involved densities, it requires ad hoc mathematical tools. Finally, the third one is based on the idea of correcting extra bias of kernel estimators which use contaminated data and is essentially an adaptation of the standard theory to the circular case. For all the proposed estimators, we derive asymptotic properties, provide some simulation results, and also discuss some possible generalizations and extensions. Real data case studies are also included.

This chapter helps the reader to understand the difference between periodic and nonperiodic signals and to become familiar with the Fourier transform. It explains how to apply the Fourier transform to the spectral analysis of nonperiodic signals. The chapter examines the main Fourier transform pairs and the main properties of the Fourier transform. It provides study examples of applications of the Fourier transform to the spectral analysis of various nonperiodic signals and systems. The chapter also discusses four types of signals – continuous periodic, continuous nonperiodic, discrete periodic, and discrete nonperiodic. It also discusses four types of the Fourier tools: continuous‐time Fourier series, continuous‐time Fourier transform, discrete Fourier transform (DFT), and discrete‐time Fourier transform. The chapter also why only DFT can be employed for digital signal processing and how DFT can be applied for the spectral analysis of both discrete periodic and discrete nonperiodic signals.

This chapter explains how to analyze the frequency content of a periodic signal and shows how to reconstruct the signal if its frequency content is known, an operation called spectral synthesis. It introduces the basics of spectral analysis and spectral synthesis, distinguishes between time domain and frequency domain, and applies the Fourier series to find the spectrum of a periodic signal. The chapter describes how to find the Fourier series for various types of periodic signals and demonstrates the effect of filtering on signals from the standpoint of spectral analysis. It explains the concept of harmonic distortion and shows that the harmonic distortion is the phenomenon where an output signal contains more harmonics than an input signal. The chapter describes the mathematical foundation of the Fourier series and explains the power spectrum of periodic signals. It also introduces the concept of power‐bandwidth trade‐off in modern communications.

For conducting objects, this chapter first considers 2D‐approximate problems, which include large concave structures and open structures with incorporation of edge conditions, and illustrates how to solve them by using the Nystrom method. For regular 3D problems, it uses the electric field integral equation, the magnetic field integral equation, and the combined field integral equation, respectively, to describe the problems, and illustrates the solutions of those equations by the Nystrom method. The chapter also solves the body‐of‐revolution structures by using the Nystrom method. The key part in the solving process is the evaluation for singular Fourier expansion coefficients and the authors propose a different scheme for it by incorporating the Fourier coefficients with the integration over generating arc segments and performing the segment integration first with closed‐form expressions. Finally, the chapter presents appropriate numerical examples to demonstrate the method and provides necessary numerical analysis and comparison.

ith respect to the analytic‐algebraic dichotomy, the theory of Siegel modular forms of half‐integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p ‐adic L ‐function obtained by interpolating the complex L ‐function at special values. This is achieved through the Rankin–Selberg method and the explicit Fourier expansion of non‐holomorphic Siegel Eisenstein series. The construction of the p ‐stabilisation in this setting is also of independent interest.

In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are coordinates which allow to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the $L^2(\T)$ solutions of the Benjamin--Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin--Ono equation and on the use of a generating functional, which encodes the entire Benjamin--Ono hierarchy.

Purpose Grating-based x-ray phase-contrast imaging (GPCI) is a promising technique for clinical applications as it can provide two newly emerging imaging modalities (differential phase-contrast and dark-field contrast) in addition to the conventional absorption contrast. As far, phase-stepping strategy is the most commonly used approach in GPCI to indirectly acquire differential phase-contrast and dark-field contrast. It is known that the obtained phase-stepping curves (PSCs) have the cosine property and the convolution property, leading to two types of information retrieval approaches in literature: the Fourier component analysis and the multi-order moment analysis. The purpose of this paper is to derive a new property of PSCs and apply the property to noise optimization for information retrieval. Methods Based on the cosine expression of the flat PSC without the sample and the well-established convolution relationship between the flat PSC and the sample PSC, we reveal an important integral property of PSCs: the inner product of PSCs and an arbitrary function contains only zero-order and first-order components in the Fourier series. Furthermore, we apply the property to the direct multi-order moment analysis and propose a set of generalized forms including an optimal one in the presence of noise. Results To validate the effectiveness of our analysis, we compared the simulated and real experiment results retrieved by the original direct multi-order moment analysis with the ones retrieved by our proposed noise-optimal form. A significant improvement of noise performance by our method is observed and the improvement ratio in differential phase-contrast is consistent with our theoretical calculation (39.2%). Conclusions In this paper, we reveal a new integral property of the acquired PSCs with and without samples in GPCI, which can be applied to information retrieval approaches like the direct multi-order moment analysis. Then we optimize these approaches to improve the noise performance, offering great potentials of dose reduction in practical applications.

This chapter presents signal processing in the frequency domain, which has the ability to divulge information based on frequency characteristics that are not easy to observe in the time domain. It describes Fourier analysis, including Fourier series, discrete Fourier transform, and fast Fourier transform (FFT), which are the most commonly used signal transformation techniques and allow one to transform time domain signals to the frequency domain. With the invention of FFT and digital computers, the efficient computation of the signal's power spectrum became feasible. The spectrum of the frequency components generated from the time domain waveforms makes it easier to see each source of vibration. The chapter provides an explanation of different techniques that can be used to extract various frequency spectrum features that can more efficiently represent a machine's health. These include: envelope analysis, also called high‐frequency resonance analysis or resonance demodulation; and frequency domain features.

The conventional photoacoustic microscopy usually uses mechanical raster scanning to obtain three-dimensional information, and the imaging speed is limited. Meanwhile, the conventional photoacoustic microscopy can only be performed at one single scale due to fixed resolution, it cannot make full use of multiscale information for integrated imaging. Here, we proposed a multiscale photoacoustic microscopy based on single-pixel imaging. A sequence of sinusoidal fringes with varying spatial frequencies is used to obtain the Fourier coefficients in the case of a single ultrasonic transducer. By controlling the spatial frequency of fringe, the acquisition of Fourier coefficients can be controlled and multiscale imaging can be achieved. The feasibility of this method is verified by theory and simulation. The results show that the lateral resolution can be tuned from several microns to tens of microns without mechanical scanning. This method will expand the application of photoacoustic imaging in biomedical research.

The aim of this article is to study the derivative of "incoherent" Siegel-Eisenstein series on symplectic groups over function fields. By the Siegel-Weil formula for "coherent" Siegel-Eisenstein series, we can relate the non-singular Fourier coefficients of the derivative in question to the arithmetic of quadratic forms. Restricting to the special case when the incoherent quadratic space has dimension 2, we explicitly compute all the Fourier coefficients, and connect the derivative with the special cycles on the coarse moduli schemes of rank 2 Drinfeld modules with "complex multiplication.", Comment: 25 pages

We study the fine-scale -mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

Real interest rate differentials usually exhibit two properties; structural breaks and asymmetric dynamics. In this paper, we use various types of Quantile Unit Root Test (QURT) which accounts for both properties. Unlike previous research, we reject the unit root in the real interest rate differentials in 18 out of 21 OECD countries as well as in 4 out of 5 BRICS countries using QURT with sharp and smooth breaks.

PURPOSE To develop a rapid Fourier decomposition (FD) free-breathing pulmonary 1 H MRI (FDMRI) image processing and biomarker pipeline for research use. METHODS We acquired MRI in 20 asthmatic subjects using a balanced steady-state free precession (bSSFP) sequence optimized for ventilation imaging. 2D 1 H MRI series were segmented by enforcing the spatial similarity between adjacent images and the right-to-left lung volume-ratio. The segmented lung series were co-registered using a coarse-to-fine deformable registration framework that used dual optimization techniques. All pairwise registrations were implemented in parallel and FD was performed to generate 2D ventilation-weighted maps and ventilation-defect-percent (VDP). Lung segmentation and registration accuracy were evaluated by comparing algorithm and manual lung-masks, deformed manual lung-masks, and fiducials in the moving and fixed images using Dice-similarity-coefficient (DSC), mean-absolute-distance (MAD), and target-registration-error (TRE). The relationship of FD-VDP and 3 He-VDP was evaluated using the Pearson-correlation-coefficient (r) and Bland Altman analysis. Algorithm reproducibility was evaluated using the coefficient-of-variation (CoV) and intra-class-correlation-coefficient (ICC) for segmentation, registration, and FD-VDP components. RESULTS For lung segmentation, there was a DSC of 95 ± 1.5% and MAD of 2.3 ± 0.5 mm, and for registration there was a DSC of 97 ± 0.8%, MAD of 1.6 ± 0.4 mm and TRE of 3.6 ± 1.2 mm. Reproducibility for segmentation DSC (CoV/ICC = 0.5%/0.92), registration TRE (CoV/ICC = 0.4%/0.98), and FD-VDP (Cov/ICC = 3.9%/0.97) was high. The pipeline required 10 min/subject. FD-VDP was correlated with 3 He-VDP (r = 0.69, P < 0.001) although there was a bias toward lower FD-VDP (bias = -4.9%). CONCLUSIONS We developed and evaluated a pipeline that provides a rapid and precise method for FDMRI ventilation maps.

Atmospheric drag is one of the primary sources of error in the orbit determination and prediction of satellites in the low altitude LEO regime. Accurate modeling of the drag force is limited by unc...

A micro‐mechanically motivated phenomenological yield function, for polycrystalline cubic metals is presented. In the suggested yield function microstructure is taken into account by the crystallographic orientation distribution function in terms of tensorial Fourier coefficients. The yield function is presented in a polynomial form in powers of the stress state. Known group‐theoretic results are used to identify isotropic and anisotropic parts in the yield function, whereby anisotropic parts are characterized by tensorial Fourier coefficients. The form of the presented yield function is inspired by the classic, phenomenological von Mises ‐ Hill yield function first published in 1913. For a specific choice of material parameters, both functions coincide, thus a micro‐mechanically motivated generalization of the von Mises ‐ Hill yield function is presented. For the given yield function, two dimensional experimental results are sufficient, to identify a three dimensional anisotropic yield behavior. The work concludes with a treatment of the isotropic special case, i.e. a tension‐compression split in yield behavior as well as parameter ranges for convexity and shapes of the yield surface.

Ways to reduce the computational cost of periodic electronic structure calculations by using basis functions corresponding to linear combinations of planewaves have been examined recently. These contracted planewave (CPW) basis functions correspond to Fourier series representations of atom-centered basis functions, and thus provide access to some beneficial properties of planewave (PW) and localized basis functions. This study reports the development and assessment of a direct inversion of the iterative subspace (DIIS) method that employs unique properties of CPW basis functions to efficiently converge electronic wavefunctions. This method relies on access to a PW-based representation of the electronic structure to provide a means of efficiently evaluating matrix-vector products involving the application of the Fock matrix to the occupied molecular orbitals. These matrix-vector products are transformed into a form permitting the use of direct diagonalization techniques and DIIS methods typically employed with atom-centered basis sets. The abilities of this method are assessed through periodic Hartree-Fock calculations of a range of molecules and solid-state systems. The results show that the method reported in this study is approximately five times faster than CPW-based calculations in which the entire Fock matrix is calculated. This method is also found to be weakly dependent upon the size of the basis set, thus permitting the use of larger CPW basis sets to increase variational flexibility with a minor impact on computational performance. © 2018 Wiley Periodicals, Inc.

The η-phase field of the Cu–In system is unusually rich and shows wealth of phases that are all related to the B8 (NiAs/Ni2In) type. Previous electron diffraction work has revealed extensive super structure ordering; in this study, we report single crystal diffraction experiments on high-quality samples of two of the phases in the system, ht1-Cu5In3 and ht2-Cu5In3. Both these phases constitute super structures, caused by the ordering of interstitials. The structure ht2-Cu5In3 is a relatively simple structure with a bidimensional modulation but where only first order satellites are visible, indicating incomplete order whereas ht1-Cu5In3 displays many high order satellites and the refined structure exhibit nearly perfect order with step-function-like occupancy domains. The refinement of the structure of ht1-Cu5In3 is challenging for several reasons. It is difficult to integrate the data because of the unusual combination of very closely spaced satellites caused by q-vectors close to (1/3 1/3 0) and a paucity of reflections in total, caused by the small substructure. A second challenge is the refinement of what amounts to a two-dimensional step function. In effect, the shape of a two-dimensional occupation domain is defined by a Fourier series and the problem of defining that shape is non-trivial. (Less)

In cancer therapy, magnetic drug targeting is considered as an effective treatment to reduce chemotherapy's side effects. The accurate design and shaping of magnetic fields are crucial for healthy cells to be immune from chemotherapeutics. In this paper, arbitrary 2-dimensional spatial patterns of magnetic fields from DC to megahertz are represented in terms of spatial Fourier spectra with sinusoidal eigenfunctions. Realization of each spatial frequency was investigated by a set of elliptical coils. Therefore, it is shown that the desired pattern was synthesized by simultaneous use of coil sets. Currents running on each set were obtained via fast and straightforward analytical Fourier series calculation. Experimentally scanned sample patterns were in close agreement with full wave analysis. Discussions include the evaluation of the Fourier series approximation error and cross-polarization of produced magnetic fields. It was observed that by employing the controlled magnetic field produced by the proposed setup, we were able to steer therapeutic particles toward the right or left half-spheres of the breast, with an efficiency of 90%. Such a pattern synthesizer may be employed in numerous human arteries as well as other bioelectromagnetic patterning applications, e.g., wireless power transfer, magnetic innervation, and tomography. Bioelectromagnetics. 39:325-338, 2018. © 2018 Wiley Periodicals, Inc.

A key limitation of the most constitutive models that reproduce a Degradation of quasi-brittle materials is that they generally do not address issues related to fatigue. One reason is the huge computational costs to resolve each load cycle on the structural level. The goal of this paper is the development of a temporal Integration scheme, which significantly increases the computational efficiency of the finite element method in comparison to conventional temporal integrations. The essential constituent of the fatigue model is an implicit gradient-enhanced formulation of the damage rate. The evolution of the field variables is computed as amultiscale Fourier series in time.On a microchronological scale attributed to single cycles, the initial boundary value problem is approximated by linear BVPs with respect to the Fourier coefficients. Using the adaptive cycle jump concept, the obtained damage rates are transferred to a coarsermacrochronological scale associated with the duration of material deterioration. The performance of the developedmethod is hence improved due to an efficient numerical treatment of the microchronological problem in combination with the cycle jump technique on the macrochronological scale. Validation examples demonstrate the convergence of the obtained solutions to the reference simulations while significantly reducing the computational costs.

A transversely isotropic thermoelastic half-space in both mechanical and thermal points of view is considered as the domain of the initial boundary value problem involved in this paper. The governing partial differential equations of thermoelasticity in a cylindrical coordinate system are uncoupled with the aid of a complete set of displacement-potential and temperature-potential functions, which with the help of Fourier series decomposition and Hankel-Laplace integral transforms, are reduced to ordinary differential equations in terms of depth. Then, the general solutions due to an arbitrary patch-load and surface heat flux are investigated for the case of a point load varying with time as Heaviside step function and a point heat flux varying with time as Dirac delta function in order to compute the related Green's functions. The governing equations for the potential functions are in such a way that different longitudinal and transverse waves are recognized and the transport properties can be discovered from the governing equations. Some numerical illustrations are also presented to depict the dependency of response on the thermal properties as well as the anisotropy of the medium.

Purpose To develop an algorithm for the Fourier analysis of posterior corneal videokeratographic data and to evaluate the derived parameters in the diagnosis of Subclinical Keratoconus (SKC) and Keratoconus (KC). Methods This was a cross-sectional, observational study that took place in the Eye Institute of Thrace, Democritus University, Greece. Eighty eyes formed the KC group, 55 eyes formed the SKC group while 50 normal eyes populated the control group. A self-developed algorithm in visual basic for Microsoft Excel performed a Fourier series harmonic analysis for the posterior corneal sagittal curvature data. The algorithm decomposed the obtained curvatures into a spherical component, regular astigmatism, asymmetry and higher order irregularities for averaged central 4 mm and for each individual ring separately (1, 2, 3 and 4 mm). The obtained values were evaluated for their diagnostic capacity using receiver operating curves (ROC). Logistic regression was attempted for the identification of a combined diagnostic model. Results Significant differences were detected in regular astigmatism, asymmetry and higher order irregularities among groups. For the SKC group, the parameters with high diagnostic ability (AUC > 90%) were the higher order irregularities, the asymmetry and the regular astigmatism, mainly in the corneal periphery. Higher predictive accuracy was identified using diagnostic models that combined the asymmetry, regular astigmatism and higher order irregularities in averaged 3and 4 mm area (AUC: 98.4%, Sensitivity: 91.7% and Specificity:100%). Conclusions Fourier decomposition of posterior Keratometric data provides parameters with high accuracy in differentiating SKC from normal corneas and should be included in the prompt diagnosis of KC.

In this paper, we give an explicit new formulation for the three-dimensional mode I weight function of Oore–Burns in the case where the crack border agrees with a star domain. Analysis in the complex field allows us to establish the asymptotic behaviour of the Riemann sums of the Oore–Burns integral in terms of the Fourier expansion of the crack border. The new approach gives remarkable accuracy in the computation of the Oore–Burns integral with the advantage of reducing the size of the mesh. Furthermore, the asymptotic behaviour of the stress intensity factor at the tip of an elliptical crack subjected to uniform tensile stress is carefully evaluated. The obtained analytical equation shows that the error of the Oore–Burns integral tends to zero when the ratio between the ellipse axes tends to zero as further confirmation of its goodness of fit.

We introduce a moving Fourier transformation for locally stationary time series, which captures the time-varying spectral density in a similar manner as the classical Fourier transform does for stationary time series. In particular, the resulting Fourier coefficients as well as moving local periodograms are shown to be (almost all) asymptotically uncorrelated. The moving local periodogram is obtained by thinning the local periodogram to avoid multiple information present at different but close points in time. We obtain consistent estimators for the local spectral density at each point in time by smoothing the moving local periodogram. Furthermore, the moving Fourier coefficients, respectively periodograms, are well suited to adapt stationary frequency domain bootstrap methods to the locally stationary case. For the wild time frequency toggle bootstrap, it is shown that the corresponding bootstrap covariance of a global locally stationary bootstrap samples captures the time-varying covariance structure of the underlying locally stationary time series correctly. Furthermore, this bootstrap in addition to adaptations of other frequency domain bootstrap methods is used in a simulation study to obtain uniform confidence bands for the time-varying autocorrelation at lag 1. Finally, this methodology is applied to a wind data set.

We give a Rankin–Selberg integral representation for the Spin (degree eight) $L$-function on $\operatorname{PGSp}_{6}$ that applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\unicode[STIX]{x1D70B}$ corresponds to a level-one Siegel modular form $f$ of even weight, and if $f$ has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$ of $\unicode[STIX]{x1D70B}$.

This paper proposes a new test for the presence of a nonlinear deterministic trend approximated by a Fourier expansion in a univariate time series for which there is no prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. Our approach builds on the work of Perron and Yabu (2009a) and is based on a Feasible Generalized Least Squares procedure that uses a super

The problem of plane deformation of the elastic ring with an inner radial cut is considered. On the inner edge of the ring hydrostatic pressure affected. The outer boundary is reinforced with a thin flexible coating, the outer boundary of it is free. As a coating model used special boundary conditions are formulated on the basis of the asymptotic analysis of the exact solution of the elasticity problem for the ring. It is difficult to find a range of geometrical and physical parameters of the problem analytically. That's why a series of numerical experiments using FEM package carried out. In particular, it was found that the error of the accepted model increases with the coating hardness and thickness. Discontinuous solution method technology in the Fourier series implemented. The boundary conditions on the crack faces satisfied. As a result of summation of series, the problem reduces to the solution of a singular integral equation for the derivative crack expanding function. The singular part is a Cauchy kernel and corresponds to the limiting classical case. A regular part of the kernel depends on the geometrical and physical parameters of the problem. The convergence of the obtained series investigated. The solution of the integral equation is constructed by collocation method as a linear combination of basis functions. Of course we taking into account the characteristic peaks in the vicinity of the crack. A first kind Chebyshev polynomials are used. Reduced stress intensity factor as an influence factor established. Qualitative and quantitative features of effect of the material and coating thickness on the intensity of the stresses in the vicinity of the crack tip established.

Summary A new, reliable algorithm for nonnegative, stabilizing solutions for the periodic Riccati differential equation (PRDE) is proposed based on Fourier series expansion and the precise integration method (PIM). Taking full advantages of periodicity, we expand coefficient matrices of the underlying linear time-varying periodic Hamiltonian system associated with the PRDE in Fourier series, and a novel extended PIM for the transition matrix of linear time-varying periodic systems is developed by combining the doubling algorithm with the increment-storage technique. This method needs to compute the matrix exponential and its related integrals only once for all evenly divided subintervals, which greatly improves the computational efficiency. Further, by introducing the Riccati transformation, a fast recursive formula for the PRDE is derived based on the block form of the transition matrix computed by the extended PIM. Finally, two numerical examples are presented to verify the numerical accuracy and efficiency of the proposed algorithm with compared results.

Temperature variations in the hot roll during a hot rolling process were analysed by solving heat conduction equations for boundary conditions using an analytical method. The analysis was conducted in a steady-state regime, taking into account the effects of process parameters such as the contact surface, roll velocity and various cooling boundary conditions. Assuming the periodicity of the process, the development of a solution in the Fourier series was employed to solve the governing equations. The temperature and its gradient distributions in the roll depth were analytically expressed according to the process parameters. The accuracy of the predicted results was examined through comparison with predictions presented in the literature (finite element solutions and measurements). Results showed that an increase in the rolling speed leads to a shorter contact time, which decreases the temperature field in the work-roll.