Fourier Analysis Research Articles

Badly approximable grids and k$k$‐divergent lattices

AbstractLet be a matrix. In this paper, we investigate the set of badly approximable targets for , where is the ‐torus. It is well known that is a winning set for Schmidt's game and hence is a dense subset of full Hausdorff dimension. We investigate the relationship between the measure of and Diophantine properties of . On the one hand, we give the first examples of a nonsingular such that has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on that slightly strengthens nonsingularity, and show that under the assumption that satisfies this condition, is a null‐set with respect to any nontrivial algebraic measure on the torus. For this, we use naive homogeneous dynamics, harmonic analysis, and a novel concept that we refer to as mixing convergence of measures.

Exploring sperm cell motion dynamics: Insights from genetic algorithm-based analysis

Accurate analysis of sperm cell flagellar dynamics plays a crucial role in understanding sperm motility as flagella parameters determine cell behavior in the spatiotemporal domain. In this study, we introduce a novel approach by harnessing Genetic Algorithms (GA) to analyze sperm flagellar motion characteristics and compare the results with the traditional decomposition method based on Fourier analysis. Our analysis focuses on extracting key parameters of the equation approximating flagellar shape, including beating period time, bending amplitude, mean curvature, and wavelength. Additionally, we delve into the extraction of phase constants and initial swimming directions, vital for the comprehensive study of sperm cell pairs and bundling phenomena. One significant advantage of GA over Fourier analysis is its ability to integrate sperm cell motion data, enabling a more comprehensive analysis. In contrast, Fourier analysis neglects sperm cell motion by transitioning to a sperm-centered coordinate system (material system). In our comparative study, GA consistently outperform the Fourier analysis-based method, yielding a remarkable reduction in fitting error of up to 70% and on average by 45%. An in-depth exploration of the sperm cell motion becomes indispensable in a wide range of applications from complexities of reproductive biology and medicine, to developing soft flagellated microrobots.

The HANTS-fitted RSEI constructed in the vegetation growing season reveals the spatiotemporal patterns of ecological quality

Yuxi, located in China’s central plateau of Yunnan, is grappling with ecological and environmental challenges as it continues to develop its economy. While ecological quality assessment serves as the foundation for ecological protection, it is pivotal to have reliable and long-term methods for assessing the ecological status to support informed decision-making in ecological protection. Reliable and long-term methods for assessing ecological status in order to facilitate informed decision-making in ecological protection are applied. This study utilized Landsat data to reconstruct four indices (greenness, wetness, dryness, and heat) during the vegetation growth in Yuxi from 2000 to 2020 that employs Harmonic Analysis of Time Series (HANTS) method. Subsequently, the annual Remote Sensing Ecological Index (RSEI) was computed by using the reconstructed indices to evaluate ecological quality in Yuxi. Additionally, spatiotemporal patterns and determinants of Yuxi’s ecological quality are unveiled through Sen’s slope estimator and Mann–Kendall test (Sen + MK) trend analysis, spatial auto-correlation analysis, and geographical detectors applied to year-by-year RSEI data. The findings in the paper indicate that the accuracy of the RSEI is significantly influenced by the vegetation season, suggesting that constructing the RSEI model with data from the vegetation growth season is crucial. Moreover, the HANTS optimization method effectively enhances the ecological indices used in the RSEI model, leading to smoother and more continuous filling of missing data. The difference between the reconstructed RSEI and the original RSEI falls within the range of − 0.15 to 0.15. Yuxi has an average RSEI of 0.54 to emphasis a moderate level of comprehensive ecological quality. Compared with river valley plains, the ecological quality of mountainous areas is higher, and the ecological quality of Yuxi presents a distinct center-edge pattern. From 2000 to 2020, Yuxi’s ecological quality exhibited fluctuations, with a slight overall improvement. Land use patterns, particularly in forestry land and impervious surfaces, are identified as the main drivers of these changes. The research offers valuable insights for scientific decision-making related to sustainable development and ecological protection.

UHV AC/DC power grid geomagnetically induced currents harmonic characteristics analysis

The power grid’s geomagnetically induced currents (GIC) during geomagnetic storms, are associated with a broad and abrupt harmonic impact, with the potential for grid paralysis and extensive power outages in severe scenarios. Therefore, the harmonic characteristics of geomagnetically induced currents in China’s ultra-high voltage (UHV) AC/DC power grid are analyzed in this paper. Firstly, a harmonic equivalent model of transformer GIC is established, and the injection of harmonics into the UHV AC/DC power grid during GIC events is quantified.Secondly, a harmonic state-space model is established by combining the 12-pulse converter switching function theory with the exponential Fourier series. The harmonic transmission and dynamic coupling characteristics of GIC in the UHV AC/DC power grid are analyzed using the harmonic equivalent model of the main transformer GIC and the harmonic equivalent model of the converter transformer GIC. Finally, a harmonic simulation model of the UHV AC/DC power grid under GIC is constructed using PSCAD, and the distribution of harmonics in the UHV AC/DC power grid under GIC is calculated. The calculations indicate that with 30 A injected GIC in the sending end transformers, the 2nd harmonic current on the sending end network side alone is 61.45 A, and the 3rd harmonic voltage on the valve side is 8.57 kV. The characteristics of harmonic transmission in the power grid result in a 2nd harmonic current of 34.87 A on the receiving end network side. Simultaneously, the total harmonic distortion of the AC power grid at the sending end is 2.35 %, surpassing the permissible limit established by the national standard. Consequently, this issue should be prioritized by the relevant power authorities.

A Quantum Harmonic Analysis Approach to Segal Algebras

In this article, we study a commutative Banach algebra structure on the space L1(R2n)⊕T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\mathbb {R}^{2n})\\oplus {\\mathcal {T}}^1$$\\end{document}, where the T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}^1$$\\end{document} denotes the trace class operators on L2(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^{n})$$\\end{document}. The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.

Powering piezoelectric frequency up-converter with rotary magnetic forces for torque-sensing application

A piezoelectric frequency up-energy harvester (PFU-EH) utilizing an SECE circuit (synchronized electric charge extraction) is developed for torque-sensing application in rotating machinery. Development of an explicit theoretical model for the system with the PFU-EH, considering phase angle shifts and magnetic impulsive forces is proposed to formulate the frequency response equations for voltage and power outputs using Fourier analysis. In this study, harvesting electrical energy can be achieved through the magnetic force interaction between the cantilevered piezoelectric beam, which carries two attached magnets, and the compound rotating disk structure, also carrying two attached magnets. The compound rotating disk structure, composed of a metal ring (outer disk) and a flywheel (inner disk) connected by a spring system, can create a phase angle caused by an applied load. As a result, mechanical torque is formed for inducing magnetic excitation to the PFU-EH and subsequently generating an AC voltage signal. The functionality of the AC voltage harvested by the PFU-EH can be further leveraged for two primary benefits. First, it can be converted into DC voltage output using the SECE circuit, enhancing the effectiveness of the electric signal without requiring load matching under different modal vibrations. Second, the SECE harvesting voltage can simultaneously function as a sensing signal for detecting mechanical torque itself. The sensing signal detection process is further analyzed by applying feature extraction from the spectrogram and utilizing a 2D-CNN for auxiliary segment identification. Finally, the experimental results demonstrate reasonable agreement with the predicted outcomes.

New Methods for Old Questions: The Use of Elliptic Fourier Analysis for the Formal Study of Palaeolithic Art

New Methods for Old Questions: The Use of Elliptic Fourier Analysis for the Formal Study of Palaeolithic Art

A Comparison of Limited Information Estimation Methods for the Two-Parameter Normal-Ogive Model with Locally Dependent Items

The two-parameter normal-ogive (2PNO) model is one of the most popular item response theory (IRT) models for analyzing dichotomous items. Consistent parameter estimation of the 2PNO model using marginal maximum likelihood estimation relies on the local independence assumption. However, the assumption of local independence might be violated in practice. Likelihood-based estimation of the local dependence structure is often computationally demanding. Moreover, many IRT models that model local dependence do not have a marginal interpretation of item parameters. In this article, limited information estimation methods are reviewed that allow the convenient and straightforward handling of local dependence in estimating the 2PNO model. In detail, pairwise likelihood, weighted least squares, and normal-ogive harmonic analysis robust method (NOHARM) estimation are compared with marginal maximum likelihood estimation that ignores local dependence. A simulation study revealed that item parameters can be consistently estimated with limited information methods. At the same time, marginal maximum likelihood estimation resulted in biased item parameter estimates in the presence of local dependence. From a practical perspective, there were only minor differences regarding the statistical quality of item parameter estimates of the different estimation methods. Differences between the estimation methods are also compared for two empirical datasets.

Dancing with math: Using Kleins quartic for music generation

Studying music from a mathematical perspective is often based on the notions of symmetry and topological space. A group G acting on some set S of musical objects in a meaningful way is seen as a symmetry group in music. This can be used in analysing motif development and chord progressions. In neo-Riemannian analysis, one has three principal chord transformations that generate a group GD_12, acting on the set S of 24 major and minor triads. Moreover, this theory is visualized through a simplicial complex whose underlying space is a topological torus. In this paper, we first introduce various symmetry groups and graphic presentations in music theory. We then propose a way of doing neo-Riemannian analysis on Kleins quartic, which is a genus 3 surface instead of a torus, realizing an idea of John Baez. Finally, we use our theory to perform a harmonic analysis of The Imperial March from Star Wars.

Sea Level Tidal Attributes in Port Said Harbor, Eastern Egyptian Mediterranean Coast

Observed sea level data attained from a pressure sensor tide gauge, installed inside Port Said harbor in Egypt, was utilized to modernize tidal characteristics and surges in the study area. Five years of hourly sea level dataset (June 2010 to June 2015) were analyzed using the harmonic analysis approach of Delft-3D Tide suit. The results of the analysis revealed that the semi-diurnal constituents (M2, S2), together with the solar annual (Sa) dominate tide behavior in the area. Furthermore, a semidiurnal tidal regime by a ratio of 0.21 is based on the form factor (FF) equation. Moreover, tidal asymmetry (Ar.) shows a flood tide with a short period of a value equal to 0.01068 > 0.01, which reflects a tide wave distortion. From power spectral analysis, the sea-level change is controlled by both the significant tidal and non-tidal (storm surge) components by almost an equal percentage of 50.63 % and 49.37 % respectively. Additionally, the residual from sea level harmonic analysis was annually correlated with the meteorological parameters of wind, temperature, and atmospheric pressure to estimate their effect on Sea-Level Rise (SLR). A small positive trend line of SL was distinguished in the years from 2011 to 2015 by approximately (2 mm/yr.) due to the weak correlation of meteorological parameters, in conjunction with a conventional relationship with atmospheric pressure and temperature. In 2010, a slight negative linear tendency was noticed of 0.3 mm/yr., which can be related to the direct proportional relation of atmospheric pressure and surge component, besides an inversely proportional relation of air temperature and residual component, regardless of the weak correlation with wind vector. Overall, the research provides insights into the tidal characteristics, surges, and sea-level rise in Port Said Harbor. Understanding these attributes serves the objective of this paper to assess the impact of sea-level rise and develop appropriate adaptation strategies in coastal areas

An investigation of a self-powered low-frequency nonlinear vibration isolation system

High-static-low-dynamic stiffness vibration isolation are of utmost significance in low-frequency vibration isolation. However, their intricate mechanisms are always challenging for environmental vibration changes. In this study, a self-powered nonlinear vibration isolation system with two stages is studied, with the aiming of obtaining the broadband vibration isolation. Magnetic springs of negative stiffness combined with the coil could transform mechanical vibration into electrical energy in lower stage. The feedback electrical energy could power the piezoelectric actuator in the upper stage. Based on the governing equations measured in stable equilibrium, the harmonic balance analysis is used to derive the frequency response functions of the transmissibility and the output power in the mechatronic resonance. The frequency band of vibration isolation extends to the lower frequencies and two resonance peaks reduce to the lower levels. The numerical simulations results validate the analytical solutions. Then the shock isolation performance is illustrated under the round step displacement and rounded pulse displacement. The parameter study demonstrates numerically that introduction of the high-static-low-dynamic stiffness improves the shock isolation performance. Moreover, the lower stage mechanical energy could drive the piezoelectric stack, resulting in high-efficiency shock isolation. Finally, an experiment is employed to prove the high-efficiency vibration isolation.

A Supermassive Binary Black Hole Candidate in Mrk 501

Using multifrequency observations, from radio to γ-rays of the blazar Mrk 501, we constructed their corresponding light curves and built periodograms using RobPer and Lomb–Scargle algorithms. Long-term variability was also studied using the power density spectrum and the detrended function analysis. Using the software VARTOOLS Version 1.40, we also computed the analysis of variance, box-least squares and discrete fourier transform. The result of these techniques showed an achromatic periodicity ≲229d. This, combined with the result of pink-color noise in the spectra, led us to propose that the periodicity was produced via a secondary eclipsing supermassive binary black hole orbiting the primary one locked inside the central engine of Mrk 501. We built a relativistic eclipsing model of this phenomenon using Jacobi elliptical functions, finding a periodic relativistic eclipse occurring every ∼224d in all the studied wavebands. This implies that the frequency of the emitted gravitational waves falls slightly above 0.1 mHz, well within the operational range of the upcoming LISA space-based interferometer, and as such, these gravitational waves must be considered as a prime science target for future LISA observations.

Isotropic optimizations of finite difference discretization

The isotropy is a fundamental requirement for solving partial differential equations by the finite difference scheme. In this work, we first introduce the concept of the virtual nodes to construct the finite difference scheme, and propose several stencils of finite difference discretization to optimize the isotropy of the gradient and Laplacian operators. The isotropic error of the optimized gradient stencil is reduced to 4.1% and 7.5% of the conventional scheme and the typical 4th-order isotropic stencil that is widely used in the multiphase lattice Boltzmann method, while the optimized Laplacian stencil displays the best performance in the Fourier analysis. Then, the optimized stencils are applied to suppress the spurious currents in the multiphase lattice Boltzmann method. The spurious current is reduced to only 7.2% of that calculated by the typical 4th-order isotropic stencil at the reduced temperature 0.6, at which the liquid/gas density ratio is near to 1000. Furthermore, they are adopted to rectify the dendrite directions in the alloy solidification simulations by the phase field method. The angle deviation of dendrite growth is reduced to 15.7% of the conventional scheme.

Some asymptotic profiles for the viscous Moore-Gibson-Thompson equation in the Lq framework

This manuscript studies some qualitative properties of solutions to the Cauchy problem for the viscous Moore-Gibson-Thompson (MGT) equation. For one thing, by applying the WKB analysis and diagonalization procedure, we derive some Lp−Lq decay estimates and the large time asymptotic profile for a suitable energy term, which provides a new way to treat higher order MGT-type coupled systems. For another, we obtain the global (in time) singular limits in the Lq framework and the higher order asymptotic profile with respect to small thermal relaxation via the multi-scale analysis and the Fourier analysis. Especially, provided the incompatible initial condition between the viscous MGT equation and the strongly damped wave equation, the formation of initial layer is rigorously justified.

Quantum Fisher kernel for mitigating the vanishing similarity issue

Quantum kernel (QK) methods exploit quantum computers to calculate QKs for the use of kernel-based learning models. Despite a potential quantum advantage of the method, the commonly used fidelity-based QK suffers from a detrimental issue, which we call the vanishing similarity issue; the exponential decay of the expectation value and the variance of the QK deteriorates implementation feasibility and trainability of the model with the increase of the number of qubits. This implies the need to design QKs alternative to the fidelity-based one. In this work, we propose a new class of QKs called the quantum Fisher kernels (QFKs) that take into account the geometric structure of the data source. We analytically and numerically demonstrate that the QFK can avoid the issue when shallow alternating layered ansatzes are used. In addition, the Fourier analysis numerically elucidates that the QFK can have the expressivity comparable to the fidelity-based QK. Moreover, we demonstrate synthetic classification tasks where QFK outperforms the fidelity-based QK in performance due to the absence of vanishing similarity. These results indicate that QFK paves the way for practical applications of quantum machine learning toward possible quantum advantages.

Study on the tidal variability related to flooding and hydroelectric operations in the Qiantang river estuary

The Qiantang River estuary, located in southeastern China, displays abnormally large tidal ranges due to its funnel shape. Although numerous studies have discussed river-tide interplay in the Qiantang River estuary, these studies usually treat tidal waves with dramatically different periods as a whole and use ZhaKou as the upstream boundary. To date, the area upstream from ZhaKou to the Fuchun River Power Station (FRPS) remains unexplored, which motives the current study. This research systematically explores the spatiotemporal variations of major tidal constituents in the Qiantang River estuary using classical harmonic analysis (T_TIDE), non-stationary harmonic analysis (NS_TIDE), and variational mode decomposition (VMD). The hindcast of NS_TIDE explains 72.9%–93.9% of original signal variances, better than the T_TIDE. However, the improvement is less from ZhaKou to CangQian, with the hindcast explaining less than 80% of the original signal variance, possibly due to the assumption of a static riverbed topography in NS_TIDE misaligned with the Qiantang River's conditions. Long-period constituents such as MSf, Mf, and Mm have large amplitudes near FRPS. Furthermore, diurnal S1 and P1 tides show notable amplitude increases at TongLu due to hydroelectric operations. The VMD method successfully captured the seasonal changes of tidal parameters caused by nonlinear tide-river-morphology interplay. The amplitudes of semi-diurnal and quarter-diurnal tides increase significantly between July and September due to decreased bottom friction caused by runoff erosion; thereafter, they gradually decrease due to increased bottom friction induced by sediment accumulation. This study has implications for similar fluvial estuaries worldwide with significant variations in water depth.

Quantum signal processing and nonlinear Fourier analysis

Elucidating a connection with nonlinear Fourier analysis (NLFA), we extend a well known algorithm in quantum signal processing (QSP) to represent measurable signals by square summable sequences. Each coefficient of the sequence is Lipschitz continuous as a function of the signal.

Directional sensitivity of bone conduction stimulation on the otic capsule in a finite element model of the human temporal bone

Sound transmission to the human inner ear by bone conduction pathway with an implant attached to the otic capsule is a specific case where the cochlear response depends on the direction of the stimulating force. A finite element model of the temporal bone with the inner ear, no middle and outer ear structures, and an immobilized stapes footplate was used to assess the directional sensitivity of the cochlea. A concentrated mass represented the bone conduction implant. The harmonic analysis included seventeen frequencies within the hearing range and a full range of excitation directions. Two assessment criteria included: (1) bone vibrations of the round window edge in the direction perpendicular to its surface and (2) the fluid volume displacement of the round window membrane. The direction of maximum bone vibration at the round window edge was perpendicular to the round window. The maximum fluid volume displacement direction was nearly perpendicular to the modiolus axis, almost tangent to the stapes footplate, and inclined slightly to the round window. The direction perpendicular to the stapes footplate resulted in small cochlear responses for both criteria. A key factor responsible for directional sensitivity was the small distance of the excitation point from the cochlea.

Theta blocks

We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta function and show that they give a new powerful method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers to this question.

The history of short-tailed whip scorpions: changes in body size and flagellum shape in Schizomida

Representatives of Schizomida, known as short-tailed whip scorpions, are an understudied group within Megoperculata. They are found subterraneanly in tropical and sub-tropical regions. They lack eyes, have superior mobility, and possess a flagellum which is relevant to their mating, but in general little is known about their general biology. Fossil representatives of Schizomida from a variety of time periods are available for study. Using 23 fossil specimens (14 of which are described here for the first time) and 86 extant individuals from the literature, changes in both body size and flagella shape in Schizomida were compared over time. Measurements of prosoma length and leg length were used as a proxy for body size, and a comparative size analysis was carried out. Individuals from the Cretaceous period were found to be significantly smaller than extant individuals, contrary to our expectations. For flagellum shape, images were used to create reconstructions, which were then analysed using an elliptic Fourier analysis followed by a principal component analysis. The morphological diversity of the male flagellum shape was found to have decreased between the Cretaceous and modern fauna. We discuss potential explanations for our findings, although a greater understanding of the general biology of Schizomida is required to properly interpret our results.