Order Fourier Coefficients Research Articles

Second-Order Bounds on Correlations Between Increasing Families

Harris’s correlation inequality states that any two monotone functions on the Boolean hy-percube are positively correlated. Talagrand [11] started a line of works in search of quantitative versions of this fact by providing a lower bound on the correlation in terms of the influences of the functions. A famous conjecture of Chvátal [2] was found by Friedgut, Kahn, Kalai and Keller [5] to be equivalent to a certain strengthening of Talagrand’s bound, conjectured to hold true when one of the functions is antipodal (i.e., g(x) = 1 − g(−x). Motivated by this conjecture, we strengthen some of those bounds by giving estimates that also involve the second order Fourier coefficients of the functions. In particular, we show that the bounds due to Talagrand and due to Keller, Mossel and Sen [8] can be improved when one of the functions is antipodal. Our proofs follow a different route than the ones in the literature, and the analysis is carried out in the Gaussian setting.

Nonlocal interaction engineering of 2D roton-like dispersion relations in acoustic and mechanical metamaterials

The interior of the synthetic unit cells and their interactions determine the wave properties of metamaterials composed of periodic lattices of these cells. While local interactions with the nearest neighbors are well appreciated, nonlocal beyond-nearest-neighbor interactions are often considered as a nuisance. Here, by introducing a versatile effectively two-dimensional metamaterial platform for airborne sound and elastic waves, we exploit nonlocal effects as a powerful design tool. Within a simplified discrete model, we analytically show that the lowest band can be engineered by Fourier synthesis, where the N-th order Fourier coefficient is determined by the N-th nearest-neighbor interaction strength. Roton-like dispersion relations are an example. The results of the discrete model agree well with a refined model and with numerical calculations. In addition, we engineer the passage of waves from a local metamaterial into a nonlocal metamaterial by carefully tailoring the transition region between the two.

Reconstructing small perturbations of an obstacle for acoustic waves from boundary measurements on the perturbed shape itself

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 rigorous by using a 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.

Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions e λ e_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ → ∞ \lambda\to\infty if 0 < | ν | λ < 1 - δ 0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta for any fixed 0 < δ < 1 0<\delta<1 . Our result implies, for instance, that the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) .

Time domain simulations of piston-like resonant flow in the gap of an oscillating twin-hull using inviscid and viscous flow solvers

A numerical study of a semi-circular twin-hull section under heave oscillation is presented. Two different time domain simulation methods were used: Firstly, a boundary element method based on potential theory which incorporates fully nonlinear free surface boundary conditions by using a mixed Eulerian Lagrangian scheme. Secondly, a finite volume method in combination with a volume of fluid scheme for capturing the free surface. In order to evaluate the effects of viscosity, simulations with the finite volume method were carried out using inviscid as well as viscous flow assumptions. Hydrodynamic mass and damping coefficients were derived from first order Fourier coefficients and validated against results from linear theory and experiments. A detailed comparison of nonlinear results from both simulation methods was carried out in vicinity of the piston-mode resonance frequency and a very good agreement was found. The phase angles corresponding to the Fourier coefficients varied strongly with frequency, which went along with a phase shift of the fluid motion in the gap. Influence of viscosity on the flow was found to be present but had relatively low impact on the hydrodynamic forces.

Investigation of the linear and mode-coupled flow harmonics in Au+Au collisions at [formula omitted] = 200 GeV

Flow harmonics (vn) of the Fourier expansion for the azimuthal distributions of hadrons are commonly employed to quantify the azimuthal anisotropy of particle production relative to the collision symmetry planes. While lower order Fourier coefficients (v2 and v3) are more directly related to the corresponding eccentricities of the initial state, the higher-order flow harmonics (vn>3) can be induced by a mode-coupled response to the lower-order anisotropies, in addition to a linear response to the same-order anisotropies. These higher-order flow harmonics and their linear and mode-coupled contributions can be used to more precisely constrain the initial conditions and the transport properties of the medium in theoretical models. The multiparticle azimuthal cumulant method is used to measure the linear and mode-coupled contributions in the higher-order anisotropic flow, the mode-coupled response coefficients, and the correlations of the event plane angles for charged particles as functions of centrality and transverse momentum in Au+Au collisions at nucleon-nucleon center-of-mass energy sNN= 200 GeV. The results are compared to similar LHC measurements as well as to several viscous hydrodynamic calculations with varying initial conditions.

Bladder Boundary Estimation by Gravitational Search Algorithm Using Electrical Impedance Tomography

The noninvasive method for monitoring urine collection in the bladder is highly recommended for paraplegia patients. These patients are unable to discharge urine at the right time due to a weaker sensation for bladder volume. If the urine is not discharged in time, then the bladder size will increase and affect the neighboring organs and tissues. Electrical impedance tomography (EIT) is a noninvasive alternative method that can be used for bladder status monitoring. Size estimation of the bladder with EIT can clarify the bladder status. The bladder is a nonuniform structure with complex shape; therefore, higher order Fourier series is needed to represent the true shape. In estimating the higher order Fourier coefficients, the performance of a conventional modified Newton-Raphson (mNR) algorithm does not give the desired performance. In this article, the gravitational search algorithm (GSA) that is known for solving optimization problems in high-dimensional search space is proposed for the shape estimation of the bladder. GSA is a global optimization algorithm that has fast convergence and does not require the computation of Jacobian. Numerical experiments and phantom studies are performed to estimate the bladder size, and the results showed good performance.

Linear quasi-randomness of subsets of abelian groups and hypergraphs

We establish an equivalence between the two seemingly distant notions of quasi-randomness: small linear bias of subsets of abelian groups and uniform edge distribution for uniform hypergraphs.For a subset A⊂G of an abelian group G consider the k-uniform Cayley (sum) hypergraph H(k)(A). The vertex set of H(k)(A) is G and the edges are k-element sets {x1,…,xk}∈Gk with x1+⋯+xk∈A. For d∈(0,1) we show that sets A⊂G of density d+o(1) have all non-trivial Fourier coefficients of order o(|G|) if and only if e(U)=d|U|k+o(|G|k) for all U⊂V(H(k)(A)).This connects the work of Chung and Graham on quasi-random subsets of the integers and that of Conlon–Hàn–Person–Schacht on weak/linear quasi-random hypergraphs. Further, it extends the work of Chung and Graham who established the corresponding result for k=2 and G=Zn.

Azimuthally variation of elastic impedances for fracture estimation

Fracture parameters including fracture orientation and fracture density, work as significant information in the fields of the engineering and exploration geophysics. However, it nowadays still remains challenging to concisely and stably estimate those fracture parameters from seismic data due to the limited signal to noise ratio of the azimuthal seismic data. A method based on azimuthally anisotropic elastic impedance (azimuthal AEI) in terms of Fourier series variation with azimuth is proposed to characterize the fracture distribution. Firstly, the azimuthal AEI equation in terms of truncated Fourier series is derived to reduce the number of unknown parameters in the fracture parameters estimation. In addition, KL (Karhunen-Loève) transformation is used to extract the primary feature of reflection amplitude variation with azimuth and weaken the random noise for azimuthal pre-stack seismic data. And then the azimuthal AEI is estimated from KL transformed azimuthal pre-stack seismic data with the constrained sparse spike inversion method. The fracture orientation without 90° ambiguity can be estimated by an innovative azimuthal AEI equation incorporating the azimuthal AEI equation in terms of Fourier series and the fracture orientation prior information. And the biased estimate for the anisotropic gradient of the azimuthal AEI can be estimated from the second order Fourier coefficient to forecast the fracture density. Besides, we introduce a more convenient way, which is employing the standard deviation of the nature logarithm (SDNL) of the azimuthal AEI in the azimuth domain, than that based on the anisotropic gradient in the fracture density estimation. The synthetic cases demonstrate the rationality and robustness of the innovative primary wave azimuthal AEI equation and the fracture parameters estimation method even with sparse 30° sampling interval in the azimuth. A field data example demonstrates that the proposed method is effective and efficient in the fracture parameters prediction.

Thermodynamics of a gas of hadrons with attractive and repulsive interactions within an S -matrix formalism

We report the effect of including repulsive interactions on various thermodynamic observables calculated using a S-matrix based Hadron Resonance Gas (HRG) model to already available corresponding results with only attractive interactions [A. Dash, S. Samanta, and B. Mohanty, Phys. Rev. C 97, 055208 (2018)]. The attractive part of the interaction is calculated by parameterizing the two body phase shifts using K-matrix formalism while the repulsive part is included by fitting to the experimental phase shifts which carry the information about the nature of the interaction. We find that the bulk thermodynamic variables for a gas of hadrons such as energy density, pressure, entropy density, speed of sound and specific heat are suppressed by the inclusion of repulsive interactions and are more pronounced for second and higher order correlations and fluctuations, particularly for the observables $\chi^2_Q$, $\chi^2_B-\chi^4_B$ and $C_{BS}$ in the present model. We find a good agreement between lattice QCD simulations and the present model for $C_{BS}$. We have also computed two leading order Fourier coefficients of the imaginary part of the first order baryonic susceptibility at imaginary baryon chemical potential within this model and compared them with the corresponding results from lattice. Additionally, assuming that the value of interacting pressure versus temperature for a gas of hadrons calculated in S-matrix formalism is same as that from a van der Waals HRG (VDWHRG) model, we have quantified the attractive and repulsive interactions in our model in terms of attractive and repulsive parameters used in the VDWHRG model. The values of parameters thus obtained are $a=1.54\pm 0.064$ GeV $\text{fm}^{3}$ and $r=0.81\pm 0.014$ fm.

Preferred Alignments of Angular Momentum Vectors of Galaxies in the SDSS Supercluster S[202-001+0084

We present a study of spin vector orientation of 1331 SDSS (Sloan Digital Sky Survey) galaxies in SDSS Supercluster S[202-001+0084] having redshift 0.02 to 0.03. The data are provided by our collaborator, Institute of Astro-particle Physics, Innsbruck University, Austria. We have converted two dimensional data to three dimensional by Godlowskian Transformation. Our aim is to find out non-random effects in the preferred alignments of angular momentum vectors of galaxies. In general, no preferred alignments angular momentum vectors of galaxies are noticed, supporting Hierarchy model of galaxy formation. In almost all sub samples the 1st order Fourier coefficient suggests that the angular momentum vectors are randomly oriented but other statistical tests suggest anisotropic distribution probably due to the binning effect and gravitational shearing effect.

Measurements of Fourier Harmonics of Azimuthal Anisotropy in Pb+Pb and Xe+Xe Collisions

The high-statistics data sets collected by the LHC experiments during the heavy-ion runs allow for a detailed study of the azimuthal anisotropy of charged particles. This report presents the selected results of $v_{n}$ measured in Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$ TeV and Xe+Xe collisions at $\sqrt{s_\mathrm{NN}} = 5.44$ TeV, with emphasis on the higher order Fourier coefficients, $v_{n}$, sensitive to initial state fluctuations of the medium created in ultra-relativistic heavy-ion collisions.

Study of Consistency between Two-Dimensional, sin<sup>2</sup><i>ψ</i>, cos<i>α</i> and Fourier Transformation Methods for X-Ray Residual Stress Measurement by Debye Ring Simulation

We investigated the consistency of four methods by analyzing the distorted Debye rings with various stress states. Firstly, the distorted Debye rings were generated under various uni-, bi-and tri-axial stress states by using the fundamental stress-strain relationship in two dimensional X-ray stress analysis theory. Secondly, we performed reverse analysis of stress states in the frameworks of sin2ψ and cosα methods from the simulated Debye rings. Moreover, the profiles of simulated Debye rings were Fourier transformed and the stress states were calculated in accordance with the obtained 1st and 2nd order Fourier coefficients. As a result, it was confirmed that all these methods provide the same stress values. That is to say, all these methods are consistent. Therefore, we can guarantee the validity of choosing any appropriate method considering the geometrical restrictions of experiment.

A note on Maass forms of icosahedral type

Using ideas of Ramakrishnan, we consider the icosahedral analogue of the theorems of Sarnak and Brumley on Hecke–Maass newforms with Fourier coefficients in a quadratic order. Although we are unable to conclude the existence of an associated Galois representation in this case, we show that one can deduce some implications of such an association, including weak automorphy of all symmetric powers and the value distribution of Fourier coefficients predicted by the Chebotarev density theorem.

Inner Product of Eigenfunctions over Curves and Generalized Periods for Compact Riemannian Surfaces

We show that for a smooth closed curve $$\gamma $$ on a compact Riemannian surface without boundary, the inner product of two eigenfunctions $$e_\lambda $$ and $$e_\mu $$ restricted to $$\gamma $$ , $$|\int e_\lambda \overline{e_\mu }\,\text {d}s|$$ , is bounded by $$\min \{\lambda ^\frac{1}{2},\mu ^\frac{1}{2}\}$$ . Furthermore, given $$0<c<1$$ , if $$0<\mu <c\lambda $$ , we prove that $$\int e_\lambda \overline{e_\mu }\,\text {d}s=O(\mu ^\frac{1}{4})$$ , which is sharp on the sphere $$S^2$$ . These bounds unify the period integral estimates and the $$L^2$$ -restriction estimates in an explicit way. Using a similar argument, we also show that the $$\nu $$ th order Fourier coefficient of $$e_\lambda $$ over $$\gamma $$ is uniformly bounded if $$0<\nu <c\lambda $$ , which generalizes a result of Reznikov for compact hyperbolic surfaces, and is sharp on both $$S^2$$ and the flat torus $$\mathbb T^2$$ . Moreover, we show that the analogs of our results also hold in higher dimensions for the inner product of eigenfunctions over hypersurfaces.

Predictions of ship maneuverability based on virtual captive model tests

Maneuverability is an important hydrodynamic performance of a ship, and should be taken into account during the ship design stage. The present study of Computational Fluid Dynamic (CFD) calculations aims to offer a numerical tool for maneuvering prediction with high accuracy. The virtual captive model tests for a model scale KCS container ship are conducted using unsteady Reynolds-averaged Navier–Stokes (RANS) computation to obtain the full set of linear and nonlinear hydrodynamic derivatives in the 3rd-order Abkowitz model. The numerical uncertainty analysis is carried out for the pure sway and yaw–drift tests to verify the numerical accuracy. It is concluded that the lower order Fourier coefficients are preferred in the computation of the hydrodynamic derivatives. Moreover, part of the computed hydrodynamic forces and moments are compared with the available captive model test data, and good agreement is obtained. By substituting the computed hydrodynamic derivatives into the mathematical model, the standard turning and zigzag maneuvers are predicted. By comparing the predicted maneuvering results with the available experimental data and the prediction results by others, it is demonstrated that acceptable prediction accuracy can be achieved with the present method, which shows the effectiveness of the present method in predicting ship maneuverability.

Linear quasi-randomness of subsets of abelian groups and hypergraphs

We establish an equivalence between the two seemingly distant notions of quasi-randomness: small linear bias of subsets of abelian groups and uniform edge distribution for uniform hypergraphs.For a subset A⊂G of an abelian group G consider the k-uniform Cayley (sum) hypergraph H(k)(A). The vertex set of H(k)A is G and the edges are k-element sets {x1,…,xk}∈(Gk) with x1+…+xk∈A. For d∈(0,1) we show that sets A⊂G of density d+o(1) have all non-trivial Fourier coefficients of order o(|G|) if and only if e(U)=d(|U|k)+o(|G|k) for all U⊂V(H(k)(A)).This connects the work of Chung and Graham on quasi-random subsets of the integers and that of Conlon-Hàn-Person-Schacht on weak/linear quasi-random hypergraphs. Further, it extends the work of Chung and Graham who established the corresponding result for k=2 and G=Zn.

Predicting Apple Firmness and Soluble Solids Content Based on Hyperspectral Scattering Imaging Using Fourier Series Expansion

Abstract. This article reports on using a Fourier series expansion method to extract features from hyperspectral scattering profiles for apple fruit firmness and soluble solids content (SSC) prediction. Hyperspectral scattering images of ‘Golden Delicious’ (GD), ‘Jonagold’ (JG), and ‘Delicious’ (RD) apples, harvested in 2009 and 2010, were acquired using an online hyperspectral imaging system over the wavelength region of 500 to 1000 nm. The moment method and Fourier series expansion method were used to analyze the scattering profiles of apples. The zeroth-first order moment (Z-FOM) spectra and Fourier coefficients were extracted from each apple, which were then used for developing fruit firmness and SSC prediction models using partial least squares (PLS) and least squares support vector machine (LSSVM). The PLS models based on the Fourier coefficients improved the standard errors of prediction (SEP) by 4.8% to 19.9% for firmness and by 2.4% to 13.5% for SSC, compared with the PLS models using the Z-FOM spectra. The LSSVM models for the prediction set of Fourier coefficients achieved better SEP results, with improvements of 4.4% to 11.3% for firmness and 2.8% to 16.5% for SSC over the LSSVM models for the Z-FOM spectra data and 3.7% to 12.6% for firmness and 5.4% to 8.6% for SSC over the PLS models for the Fourier coefficients. Experiments showed that Fourier series expansion provides a simple, fast, and effective means for improving Keywords: Apples, Firmness, Fourier series expansion, Hyperspectral scattering imaging, Least squares support vector machine, Partial least squares, Soluble solids content.

Azimuthally anisotropic emission of low-momentum direct photons in Au + Au collisions at sNN=200 GeV

The PHENIX experiment at the Relativistic Heavy Ion Collider has measured 2nd and 3rd order Fourier coefficients of the azimuthal distributions of direct photons emitted at midrapidity in Au$+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV for various collision centralities. Combining two different analysis techniques, results were obtained in the transverse momentum range of $0.4<p_{T}<4.0$ GeV/$c$. At low $p_T$ the second-order coefficients, $v_2$, are similar to the ones observed in hadrons. Third order coefficients, $v_3$, are nonzero and almost independent of centrality. These new results on $v_2$ and $v_3$, combined with previously published results on yields, are compared to model calculations that provide yields and asymmetries in the same framework. Those models are challenged to explain simultaneously the observed large yield and large azimuthal anisotropies.

PHENIX results on collectivity in high-multiplicity p + Au, d + Au and 3He + Au collisions

We present recent PHENIX results for long-range two-particle correlation functions across a large pseudorapidity(η) gap in high-multiplicity p+Au, d+Au and 3He+Au collisions. Enhanced correlations are observed for particles with small azimuthal separation. Measurements of the second order Fourier coefficients (v2) in p/d/3He+Au collisions, and the third order Fourier coefficient (v3) in 3He+Au collision are performed using the event plane method. Scaling of v2 with the initial eccentricity (ɛ2) is performed in each system to investigate the role of the initial geometry in the development of a final-state anisotropy in the particle emission. The v2 coefficients for identified π±, K± and (anti-)proton are measured as a function of transverse kinetic energy (KET) and their magnitudes are found to scale approximately with the number of constituent quarks in the hadron.