Fourier Series Research Articles

Response of semicircular canyons and movable cylindrical cavities to SH waves in anisotropic half-space geology

The development of tunnels or the laying of underground pipelines are essential engineering projects in modern society, and in canyon tunnels and underground pipeline projects, the surface motion and cavity edge motion have been topics of concern in ground vibration problems. We investigate the wave scattering problem in an elastic half-space anisotropic medium containing a semicircular canyon and a subsurface movable cylindrical cavity by using the wave function expansion method, the complex function method, and the mirror method. By deriving the governing equation and transforming it into the standard form of the Helmholtz equation satisfying the zero-stress boundary condition, we solve the corresponding displacement functions. Introducing a position correction coefficient, the scattered wavefield in a half-space anisotropic medium is constructed by the mirror method, which improves the problem of scattered wave source singularity in anisotropic half-space media. Then, combining the free boundary conditions with a Fourier series expansion method, we solve the unknown coefficients in the equations. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, using frequency- and time-domain analysis, we investigate the effects of the relevant parameters on the surface motion [Formula: see text], the dynamic stress concentration factor, and the displacement amplitude [Formula: see text]. The results indicate that rock anisotropy and the presence of semicircular canyons have a significant effect on the dynamic response of subsurface structures. This not only provides a theoretical basis for practical unlined tunnels or pipeline projects but also can provide a basis for seismic design of underground structures.

Long-range near-side correlation in e+e− collisions at 183-209 GeV with ALEPH archived data

The first measurement of two-particle angular correlations for charged particles with LEP-II data is presented. The study is performed using archived hadronic e+e− data collected by ALEPH at center-of-mass energies up to 209 GeV, above the W+W− production threshold, which provide access to unprecedented charged-particle multiplicities and more complex color-string configurations if compared to previous measurements at LEP-I energies. An intriguing long-range near-side excess is observed in the correlation function measured with respect to the thrust axis in the highest multiplicity interval (Ntrk≥50). Such a structure is not predicted by the Monte-Carlo simulation. The harmonic anisotropy coefficients vn, which result from the Fourier expansion of the two-particle correlation functions, were also measured for the first time in e+e− data, and compared to pythia 6 predictions and to the results obtained in proton-proton collisions. The results presented in the Letter provide novel experimental constraints on the formation of collective phenomena in point-like e+e− collisions.

Phase-Resolved Functional Lung MRI for Pulmonary Ventilation and Perfusion (V/Q) Assessment.

Fourier decomposition is a contrast agent-free 1H MRI method for lung perfusion (Q) and ventilation (V) assessment. After image registration, the time series of each voxel is analyzed with regard to the cardiac and breathing frequency components. Using a standard 2D spoiled gradient-echo sequence with a temporal resolution of ~300 ms, an image-sorting algorithm was developed to produce phase-resolved functional lung imaging (PREFUL) with an increased temporal resolution. Thus, it is feasible to evaluate regional flow volume loops (FVL) during tidal volume breathing and depict the propagation of the pulse wave during the cardiac cycle. This method can be applied at 1.5T or 3T with standard MR hardware without the necessity for sequence programming, as the described protocol can be implemented with the default SPGRE sequence on most systems. PREFUL ventilation MRI has been validated using 129Xe and 19F gas imaging with good regional agreement. Perfusion-weighted PREFUL MRI has been validated using SPECT as well as dynamic contrast enhanced (DCE) MRI. PREFUL has been tested in a dual center dual vendor setting and is currently applied in several ongoing multicenter trials. Furthermore, it is feasible across a range of field strengths (0.55T-3T) and different age groups, including newborns. Quantitative V/Q PREFUL MRI has been used in patients with cystic fibrosis, chronic obstructive pulmonary disease, chronic thromboembolic pulmonary hypertension, and corona virus disease-2019 to quantify disease and monitor treatment change after therapy. Furthermore, PREFUL V/Q imaging has been shown to predict transplant loss due to chronic lung allograft dysfunction in patients after lung transplantation. In summary, PREFUL MRI is a validated technique for quantitative ventilation and pulmonary pulse wave/perfusion imaging for regional pulmonary disease detection, quantification, and treatment monitoring with potential added value to the current clinical routine.

Degree of approximation of functions conjugate to the functions belonging to Lip((𝜉1,𝜉2)𝑟)-class through double Nörlund means

Abstract In this paper, we determine the error of approximation of functions, conjugate to the functions of two variables ( 2 ⁢ π 2\pi -periodic in both variables) belonging to the Lipschitz class Lip ( ( ξ 1 , ξ 2 ) ; r \operatorname{Lip}((\xi_{1},\xi_{2});r ) ( r ≥ 1 r\geq 1 ), through the double Nörlund means of its conjugate double Fourier series. Additionally, in the form of corollaries, we present some particular cases of the theorem proved here.

Semi-analytical vibration modeling of complex axisymmetric shells using shifted Legendre series

This paper presents a semi-analytical approach based on the shifted Legendre series for vibration modeling of complex axisymmetric shells. This method divides the structure along its generator into complex axisymmetric shells comprised solely of conical and spherical shell segments. A global cylindrical coordinate system is established to unify the coordinate system of complex axisymmetric shells. Coupling between segments and the simulation of arbitrary boundary conditions are achieved using artificial spring technology. Based on the first-order shear deformation theory (FSDT), the energy functional of each segment is derived. The circumferential displacement is expanded using Fourier series, while the meridional displacement is expanded using the shifted Legendre series at shifted Gauss-Lobatto nodes. Utilizing the inner product matrix and differential matrix of the shifted Legendre series, the integration and differentiation processes in the vibration characteristic equation are simplified into a product form, significantly enhancing computational efficiency. Through convergence analysis and comparison of the computational results of the present method with numerical simulation results, experimental results and literature results demonstrate the advantages of high convergence, high accuracy and fast solution speed of this solution method. On this basis, the vibration characteristics of ring-stiffened double-layer hemispherical shells are investigated. The results demonstrate the great applicability of this method for vibration modeling of complex axisymmetric shells in engineering.

A method for evaluating the design scheme of continuous measuring bridge of the instrumented wheelset

This paper proposes a method for evaluating the design scheme of continuous measuring bridge of the instrumented wheelset, based on the theory of continuous measurement of wheel-rail forces, Finite Element Analysis, and Levenberg-Marquardt algorithm. The elastic strain at each placement point of the strain gauge on the web surface of the wheel is expanded into Fourier series, and the general mathematical expressions of the output signals for each symmetric component, each bridge, and the synthesized bridge are provided. Nine typical measuring bridge schemes of strain gauge arrangement on the wheel web surface and seven typical measuring bridge schemes of strain gauge arrangement inside the measuring holes are given. The finite element model of the wheel is established using software ABAQUS, and the moving force is applied to the rolling surface of the wheel for simulation calculations. The output signals of 16 typical measuring bridges are then obtained. By utilizing the mathematical formulas of the bridge outputs and the simulation results, a matrix expression describing the relationship between amplitudes of all harmonics and the bridge output is given, and the amplitudes of all the former 20 orders harmonics of the bridge signal are obtained by applying Levenberg-Marquardt algorithm. Based on the analysis results of the harmonic amplitudes, a method is proposed to quantitatively evaluate the output signals of a single bridge (which can be either harmonic or triangular wave) and the synthesized bridge. The mathematical expressions of the evaluation indicators are provided within the range of 0 to 1. Finally, the analysis results of the evaluation indicators for 16 typical measuring bridges are presented and compared. The proposed method is not only applicable to the bridge design of the instrumented wheelset but also to the evaluation of signals from channels of the data collector when the instrumented wheelset is being calibrated on the test rig.

Titchmarsh’s-type theorem for two-sided quaternion Fourier transform and sharp Hausdorff–Young inequality for quaternion linear canonical transform

In this work, we first introduce the two-sided quaternion Fourier transform and demonstrate its essential properties. We generalize Titchmarsh’s-type theorem in the framework of the two-sided quaternion Fourier transform. Based on the interaction between the quaternion Fourier transform and quaternion linear canonical transform we explore sharp Hausdorff–Young inequality for the quaternion linear canonical transform. The obtained result can be considered as a generalized version of sharp Hausdorff–Young inequality for the two-dimensional quaternion Fourier transformation in the literature.

A local and hierarchical Koopman spectral analysis of fluid dynamics

AbstractA local and hierarchical Koopman spectral analysis is proposed to extend Koopman spectral analysis typically used in a linear system or an ergodic process to its application in general nonlinear dynamics. The continuous and analytic Koopman eigenfunctions and eigenvalues, derived from operator perturbation theory, are capable of dealing with a nonlinear transition process with mathematical rigorousness. A proliferation rule is identified to derive high‐order eigenvalues and eigenfunctions from lower‐order ones, thus various spectral patterns may be generated through recursive proliferations. The locally linear map around each state constructs base local Koopman eigenvalues and eigenfunctions from an algebraic eigenvalue problem, and high‐order ones are generated via the proliferation rule to express the systematic nonlinearity. The aforementioned hierarchy simplifies the Koopman spectral analysis and is verified by studying the development of Kármán vortex streets. The triangular chain and the lattice distribution of Koopman eigenvalues confirm the critical role of the proliferation rule and the hierarchy structure of Koopman eigenvalues. The local spectral analysis on the transition process shows that the periodic flow forms as the growth rates of the critical Koopman modes reduce to zero, and meanwhile, the Koopman modes at the same frequency superpose on each other to form the well‐known Fourier or Floquet modes, where the latter are the enhanced nonlinear motions due to the alignment of Koopman eigenvalues with the critical ones.

Leak detection in water distribution networks based on deep learning and kriging interpolation method

ABSTRACT The burgeoning growth of urban areas has escalated the necessity for efficient and precise leak detection in water distribution networks. Automatic detection methods based on deep learning are a state-of-the-art research topic. In this paper, a methodology that combines deep learning and data imaging is proposed. The framework employs pressure monitoring data and is anchored on the following three pillars: (1) the generation of a comprehensive dataset, encompassing one year of leak-free demand data derived from Fourier Series analysis and monitoring pressure under normal and leak conditions, (2) the transformation of pressure time series into images using kriging interpolation, (3) establishing convolution neural network (CNN) and evaluating its performance of abnormal identification. The effectiveness of the proposed methodology is assessed in different image sets under various leak conditions. The findings reveal that this method meets dependable and effective outputs for leak detection, with the deep learning model achieving a high true positive rate (TPR) of 98% and an area under the curve (AUC) of 94%. This study provides invaluable information for strategic action planning and the enhancement of water loss management protocols, especially in situations where water utilities and regulatory authorities grappling with limited budgets and diminishing revenues.

MRI-based ventilation and perfusion imaging to predict radiation-induced pneumonitis in lung tumor patients at a 0.35 T MR-Linac

Background and purpose:Radiation-induced pneumonitis (RP), diagnosed 6–12 weeks after treatment, is a complication of lung tumor radiotherapy. So far, clinical and dosimetric parameters have not been reliable in predicting RP. We propose using non-contrast enhanced magnetic resonance imaging (MRI) based functional parameters acquired over the treatment course for patient stratification for improved follow-up. Materials and methods:23 lung tumor patients received MR-guided hypofractionated stereotactic body radiation therapy at a 0.35T MR-Linac. Ventilation- and perfusion-maps were generated from 2D-cine MRI-scans acquired after the first and last treatment fraction (Fx) using non-uniform Fourier decomposition. The relative differences in ventilation and perfusion between last and first Fx in three regions (planning target volume (PTV), lung volume receiving more than 20Gy (V20) excluding PTV, whole tumor-bearing lung excluding PTV) and three dosimetric parameters (mean lung dose, V20, mean dose to the gross tumor volume) were investigated. Univariate receiver operating characteristic curve - area under the curve (ROC–AUC) analysis was performed (endpoint RP grade≥1) using 5000 bootstrapping samples. Differences between RP and non-RP patients were tested for statistical significance with the non-parametric Mann–Whitney U test (α=0.05). Results:14/23 patients developed RP of grade≥1 within 3 months. The dosimetric parameters showed no significant differences between RP and non-RP patients. In contrast, the functional parameters, especially the relative ventilation difference in the PTV, achieved a p-value<0.05 and an AUC value of 0.84. Conclusion:MRI-based functional parameters extracted from 2D-cine MRI-scans were found to be predictive of RP development in lung tumor patients.

Dynamic characteristics of composite drive shaft in complex environment based on Carrera unified formulation

This article uses Carrera unified formulation (CUF) to analyze the dynamics characteristics of composite shaft with consideration of thermal effects, axial loads, and rotational speeds. The shaft radial displacement is described by Taylor expansion terms, and the axial displacement is described by improved Fourier series. Hamilton’s principle is used to obtain the control equations. The calculation results indicate that: The increase of temperature can reduce composite shaft natural frequency, and the influence is not obvious when the layer angle is small; The effect of axial load on composite shaft natural frequency is roughly linear, but the influence varies under different boundaries, length-radius ratio, layer orders; Due to the gyroscopic effect, the natural frequency of the composite shaft exhibits two states under the action of rotation speed: forward and backward mode; In forward mode, the influence of temperature and axial load on the composite shaft still follows the previous conclusion; In the backward mode, when the rotational speed exceeds the critical speed, the influence of temperature and axial load on the composite shaft shows the opposite conclusion. These findings are of great significance for the practical application of composite shaft in engineering.

Deret Fourier dengan Pendekatan Numerik Menggunakan Metode Trapezium

Fourier series has the ability to decompose a periodic function into a collection of sine and cosine wave sums. Another way to express a function into an infinite series is with the Maclaurin series or Taylor series but both of these series require that the function must have the first derivative, the second derivative and so on of the function is defined at its expansion point and there must be no discontinuity value at that point. However, the advantage of the Fourier series is that we can decompose the function into an infinite series containing the sum of sine and cosine terms even though the function cannot be differentiated and has discontinuity at some points. The Fourier series equation of an analytical function is obtained by first finding the values ​​of the Fourier coefficients using analytical integrals. In the case of functions formed from numerical data that are assumed to have a periodic pattern, the Fourier series formula that uses analytical integrals to find its Fourier coefficients can no longer be used so the numerical approach using the Trapezium method is an alternative that allows us to find the Fourier series equation for a periodic function expressed numerically. We use 12 points as the basis for the calculation in the Trapezium method to approximate the Fourier series numerically. This paper presents a numerical approach to solving the problem.

A two-dimensional vertex model for curvy cell-cell interfaces at the subcellular scale.

Cross-sections of cell shapes in a tissue monolayer typically resemble a tiling of convex polygons. Yet, examples exist where the polygons are not convex with curved cell-cell interfaces, as seen in the adaxial epidermis. To date, two-dimensional vertex models predicting the structure and mechanics of cell monolayers have been mostly limited to convex polygons. To overcome this limitation, we introduce a framework to study curvy cell-cell interfaces at the subcellular scale within vertex models by using a parametrized curve between vertices that is expanded in a Fourier series and whose coefficients represent additional degrees of freedom. This extension to non-convex polygons allows for cells with the same shape index, or dimensionless perimeter, to be, for example, either elongated or globular with lobes. In the presence of applied, anisotropic stresses, we find that local, subcellular curvature or buckling can be energetically more favourable than larger scale deformations involving groups of cells. Inspired by recent experiments, we also find that local, subcellular curvature at cell-cell interfaces emerges in a group of cells in response to the swelling of additional cells surrounding the group. Our framework, therefore, can account for a wider array of multicellular responses to constraints in the tissue environment.

An Algorithm for Finding the Exact Value of the Argument for the Modulus of Continuity in Estimate of Approximation of a Continuous Periodic Function by a Partial Sum of Its Fourier Series

An Algorithm for Finding the Exact Value of the Argument for the Modulus of Continuity in Estimate of Approximation of a Continuous Periodic Function by a Partial Sum of Its Fourier Series

Approximation by linear means of Vilenkin–Fourier series in Stepanets spaces

Approximation by linear means of Vilenkin–Fourier series in Stepanets spaces

Control of electroactive shear waves in a piezoelectric waveguide by different surface actions

Abstract In various cases of electromechanical surface actions, the problems of surface control of the propagation of an electroactive transverse wave in a piezoelectric waveguide (piezoelectric class 6 mm of hexagonal symmetry) are considered. In both problems, one of the surfaces of the piezoelectric waveguide is mechanically free, and an electrical displacement normal to the surface acts. The second surface of the waveguide is firmly clamped and grounded, in one task, or there is a drift of electrons over the surface and there are no mechanical effects, in another task. The formulated initial boundary value problems are solved by the method of Fourier series. True electromechanical fields are built in the form of series of eigenmodes of electroacoustic oscillations in accordance with the harmonics of surface effects. Analytical and numerical calculations of the control process in cases of surface action are given.

Estimate of the Convergence Rate in the Riemann Localization Principle for Trigonometric Fourier Series of Continuous Functions

Estimate of the Convergence Rate in the Riemann Localization Principle for Trigonometric Fourier Series of Continuous Functions

A Generative adversarial network model for estimating temporal frequency variation of vehicle-bridge interaction using modified Stockwell transform

The presence of massive and directional vehicles moving on a railroad bridge causes temporal fluctuations in the fundamental frequencies of both bridge and vehicle systems, making traditional structural inspections difficult to employ effectively. Thus, this paper proposes an image-to-image Generative Adversarial Network (GAN) that estimates temporal frequency variation for the vehicle-bridge interaction system. To address the challenges associated with conventional approaches, eigenvalue analysis, numerical substitution, and Fourier series approximations are examined using a simple degree of freedom and a more complex model. Thus, the GAN model is proposed to overcome those limitations. In the framework, based on the properties of a real bridge and vehicle model, the modified Stockwell transform of bridge acceleration from the dynamic simulation is used as input data with the paired data from the numerical substitution approach. Then, the model is validated through quantitative measures and laboratory scale experiments. Results showed that the model has strong performances across the various scenarios, demonstrating the potential for bridge condition assessment under operational conditions.

Heat Properties for Groups

We revisit Fourier’s approach to solve the heat equation on the circle in the context of (twisted) reduced group C*-algebras, convergence of Fourier series and semigroups associated to negative definite functions. We introduce some heat properties for countably infinite groups and investigate when they are satisfied. Kazhdan’s property (T) is an obstruction to the weakest property, and our findings leave open the possibility that this might be the only one. On the other hand, many groups with the Haagerup property satisfy the strongest version. We show that this heat property implies that the associated heat problem has a unique solution regardless of the choice of the initial datum.

Causal analysis among azimuthal Fourier modes in linear plasma based on multivariate time series models

This study investigates the causal relationships among azimuthal Fourier modes in linear plasma turbulence using multivariate time series models. We elucidate the dynamics of mode interactions in magnetized plasmas by employing the vector autoregressive model and Granger causality analysis. Our analysis, based on data from the plasma assembly for nonlinear turbulence analysis, reveals significant variations in causality with changing pressure conditions. Modes form weakly coupled clusters at lower pressures, while higher pressures lead to stronger coupling and larger clusters. The impulse response function further provides insights into the temporal propagation and nature of influences between modes. These findings enhance the understanding of spatial pattern formation in magnetized plasmas and offer a quantitative framework for analyzing plasma turbulence dynamics.