Fourier Coefficients Research Articles

Oscillations of Fourier coefficients of product of L-functions at integers in a sparse set

Let [Formula: see text] be a normalized Hecke eigenform of weight [Formula: see text] for the full modular group [Formula: see text]. In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg [Formula: see text]-functions [Formula: see text], supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant [Formula: see text] We improve previous results in the case when the reduced form is given by [Formula: see text]

Thermal diffusity in copper benzene-1,3,5-tricarboxylate–reduced graphite oxide mechanical composites

AbstractMetal organic frameworks (MOFs) and particularly copper benzene-1,3,5-tricarboxylate (HKUST-1) are excellent materials for gas storage (e.g., CH4, N2, H2 adsorption) and gas separation. In this work, reduced graphene oxide (RGO)–HKUST-1 mechanical mixtures were studied in order to reveal the effect of RGO content on the pressure tolerance of the texture and heat conductivity. HKUST-1 was obtained by two different synthesis routes. Air-dried MOF and RGO were thoroughly mixed prior to the compression. Powder XRD and Raman spectroscopy were used to characterize the response of the crystal structure, while low-temperature nitrogen adsorption was used the follow the adsorption properties of the pellets. Finally, the "flash" heat pulse method was used to assess the thermal properties. The gas adsorption isotherms revealed that the adsorption capacity decreases when RGO is added. Based on Raman and XRD results, we found that the synthesis route has an effect on multiple scales. We experimentally confirmed that evaluation of the thermal diffusivity requires a model more complex than the simple Fourier equation, due to the inherent heterogeneous structure of the material. A good approximation of the Fourier coefficient of thermal diffusivity was obtained using the parameters of the Guyer–Krumhansl equation. The heat pulse experiments also revealed possible size-dependent behavior.

ESTIMATE FOR HIGHER MOMENTS OF CUSP FORM COEFFICIENTS OVER SUM OF TWO SQUARES COEFFICIENTS OVER SUM OF TWO SQUARES

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights k1 and k2 for the full modular group Γ = SL(2, Z), respectively. Denote by λf (n) and λg(n) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we consider the summatory function X n=a2+b2≤x λf (n)iλg(n)j , for x ≥ 2, where a, b ∈ Z and i, j ≥ 1 are positive integers.

From Pseudo to Real: Generalized Subspace Method for Power Spectrum Reconstruction

Subspace methods are widely used in statistical signal processing owing to their super-resolution. Conventional subspace methods, such as multiple signal classification (MUSIC) identify parameters containing frequency and direction of arrival by calculating pseudo-spectra without physical amplitudes. However, the real value of amplitude plays a significant role in fault diagnosis and condition monitoring. Additionally, due to hardware limitations and increased frequency, uniform (Nyquist) sampling becomes challenging. Nonuniform sampling, an alternative to high-rate sampling is common in practice. Consequently, Nonuniform (sub-Nyquist) signal processing methods are urgently required. In this paper, a generalized subspace method (GSM) is proposed to bridge the pseudo spectrum with the real spectrum. The proposed method is a supplement to the existing subspace methods and is applicable to both uniform and nonuniform sampling. In the GSM, the Fourier coefficient of the autocorrelation function is derived by signal subspace from the perspective of basis matching. Then, an alias-free real power spectrum is obtained by combining the power spectrum reconstructed by GSM with the classical pseudo spectrum. Finally, simulation and experiment demonstrate the performance of the proposed method in power spectrum reconstruction. Even under the dual influence of undersampled and nonuniform, the proposed method is still effective.

Application of the Fourier Modulation Analysis Method to the Problem of Derivatives Recovery

Abstract. In this work, formulas are obtained for finding higher derivatives of a function (dependence), expressed through its Fourier coefficients (harmonic amplitudes). These formulas were found by differentiating both a function with a harmonically modulated argument and its Fourier series. The expressions given make it possible to find higher derivatives of functions numerically with any accuracy by means of different methods. For example, numerical integration of the Euler-Fourier formulas may be used for harmonic amplitudes. Derivatives also can be found experimentally when studying a nonlinear physical process by digitally recording the harmonic amplitudes of the dependence under study under simultaneous static and harmonic influences. The problem of reconstructing derivatives from Fourier coefficients is posed and an analysis of its correctness is performed. Formulas for estimating recovery errors are defined and recommendations are given to reduce these errors. Examples are given with different smoothness of analytical functions and of functions used to explain experiments: 1) an analytical function used to explain the magnetic properties of superconductors, whose Fourier coefficients are determined numerically with the error of the computational algorithm in the Mathcad software environment; 2) current-voltage characteristic (CVC) of 2 back-to-back semiconductor diodes, whose harmonic amplitudes were determined experimentally with a given measurement error of the device. The resulting derivative of the current-voltage characteristic is compared with the derivative obtained using numerical differentiation formulas; 3) an analytical function whose derivative has a discontinuity of the first kind. Errors in measuring Fourier coefficients were added artificially using a random number generator.

Shifted convolution sums for GL(3)×GL(2)$GL(3)\times GL(2)$ averaged over weighted sets

AbstractLet be the Fourier coefficients of an Hecke–Maass cusp form and be those of an Hecke holomorphic or Hecke–Maass cusp form . Let and be a sequence. We show that if for some , for any , and a similar bound holds when . This improves Sun's bound and generalizes it to an average with arbitrary weights . Moreover, we demonstrate how one can recover the factorizable moduli structure given by Jutila's variant of the circle method via studying a shifted sum with weighted average. This allows us to recover Munshi's bound on the shifted sum with a fixed shift without using Jutila's circle method.

Optimal with Respect to Accuracy Recovery of Some Classes Functions by Fourier Series

Introduction. Function approximation (approximation or restoration) is widely used in data analysis, model building, and forecasting. The goal of function approximation is to find the function that best approximates the original function. This can be useful when the original function is too complex to analyze or when a model needs to be simplified for more efficient computation or interpretation. Function approximation is an important tool in science, engineering, economics, and other fields where data analysis and modeling are required. It allows you to simplify complex functions, identify patterns in the behavior of the object of study, and predict the value of a function beyond the available data. The purpose of the paper is consider the problems of approximation of a function, which on some interval is given by its values in some set of nodal points and belongs to some class of functions by trigonometric Fourier series with a given accuracy and at fulfillment of given constraints on its execution time. The main attention is paid to obtaining estimates of computational complexity (implementation time) and solving the problem of function approximation by Fourier series with a given or maximum possible accuracy using efficient algorithms for solving optimization problems. Results. The general formulation of the problem of approximation of functions by Fourier series in accordance with the technology of solving problems of computational and applied mathematics with specified values of quality characteristics is presented. Estimates of the error of the proposed approximation algorithms using for the computation of Fourier coefficients the optimal in accuracy and close to them quadrature formulas for the computation of integrals from rapidly oscillating functions of the classes of Helder and Lipschitz with given fixed values in the nodes of a fixed grid are given. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of the specified classes are given. Estimates of computational complexity of the given algorithms are obtained, which allow us to set real constraints on the time of algorithm implementation with a given or maximum possible accuracy. Conclusions. A comprehensive analysis of the quality of the considered algorithms for the approximation of functions by Fourier series using the accuracy-optimal (or close to them) quadrature formulas for the computation of Fourier coefficients for the computation of integrals from rapidly oscillating functions is presented. The estimates of their main characteristics – accuracy and computational complexity – are obtained. Keywords: function approximation, Fourier series, Fourier series coefficients, approximation error, computational complexity.

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for [Formula: see text]. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for [Formula: see text]. Furthermore, we present a conditional result regarding sign changes of these coefficients.

Estimating Directional Wave Spectra Properties in Non-Breaking Waves from a UAS-Mounted Multi-beam Lidar

Abstract Wave spectra and directional moment measurements are of scientific and engineering interest and are routinely estimated with wave buoys. Recently, both fixed-location and Uncrewed Aircraft System (UAS)-mounted lidar have estimated surfzone wave spectra. However, nearshore wave statistics seaward of the surfzone have not been measured with lidar due to low return number and nearshore directional moments have not been measured at all. We use a multi-beam scanning lidar mounted on a gasoline-powered UAS to estimate wave spectra, wave slope spectra, and directional moments on the inner shelf in ≈ 10 m water depth from an 11-min hover and compare to a co-located wave buoy. Lidar returns within circular sampling regions with varying radius R are fit to a plane and a 2D parabola, providing sea-surface and slope timeseries. Wave spectra across the sea-swell (0.04–0.4 Hz) are robustly estimated for R ≥ 0.8 m. Estimating slope spectra is more challenging. Large R works well in the swell band and smaller R work well at higher frequencies, in good agreement with a wave buoy inferred slope spectrum. Directional Fourier coefficients, estimated from wave and slope spectra and cross-spectra, are compared to a wave buoy in the sea-swell band. Larger R and the 2D parabola-fit yield better comparison to the wave buoy. Mean wave angles and directional spreads, functions of the directional Fourier coefficients, are well reproduced at R = 2.4 m and the 2D parabola-fit, within the uncertainties of the wave buoy. The internal consistency of the UAS-lidar-derived results and their good comparison to the Spotter wave buoy demonstrate the effectiveness of this tool for estimating wave statistics.

Effects of high-low frequency electromagnetic radiation on vibrational resonance in Hodgkin–Huxley neuronal system

In this paper, based on the Hodgkin–Huxley (H–H) neuron model, the effects of high-low frequency (HLF) electromagnetic radiation on vibrational resonance (VR) in a single neuron is investigated. It is found that VR can be observed in a single H–H neuron model with or without considering HLF electromagnetic radiation. However, HLF electromagnetic radiation can cause changes in the structure of the resonance peaks and make the maximum region and intensity of the response amplitude smaller, which weakens the VR in the neuronal system. This is because the HLF electromagnetic radiation makes the neuronal electrical activities less regular, resulting in weaker synchronization between input low-frequency signal and output firing. In addition, the effects of the system parameters such as the HF electromagnetic radiation amplitude on the Fourier coefficients are investigated, and it is found that these parameters can also cause changes in the number of resonance peaks, resulting in vibrational mono-resonance and vibrational bi-resonances. When the frequency of the HLF electromagnetic radiation is between 0.014 and 0.041, local anti-resonance can be also observed. This has practical significance in reducing some of the unexpected vibration in aerospace engineering and mechanical fields. The HLF electromagnetic radiation plays an important role in weakening the VR of neuronal systems. Therefore, this study could provide new insights to further control the abnormal electrical activities and information transmission of neurons, which can lead to better design and development of therapeutic programs for neurological disorders and neuroplasticity therapies, and advance neuroscience.

Abstract 6224: Drop-like deformation of nuclei in cancer patient tissue

Abstract In this study, we investigated nuclear deformations in cancer tissues and their clinical relevance to diagnostics. Introduction: Cancers are graded by pathologists examining biopsy slides stained with hematoxylin and eosin for factors including nuclear atypia, which can include features like lobules, invaginations, large nuclear size, and prominent nucleoli. While the importance of nuclear atypia has been known for centuries, its cause remains unclear. Nuclear shaping has been studied extensively in vitro, leading to the liquid drop model of nuclear shaping, which states that nuclear shape changes can occur due to excess area in the form of wrinkles or folds in the lamina. When the lamina becomes fully taut, the nucleus reaches a limiting steady shape. Wrinkles allow the nuclear shape to mirror the cell shape such as in invading cancer cells. The applicability of this model to nuclei in cancer tissues from human patients is untested. Here, tissues were stained for nuclear lamina proteins to test for the presence of excess area in the lamina. Diagnostic relevance of the drop model was assessed by quantitative Fourier analysis of high-resolution images of lamin-stained nuclei. Methods: Deidentified patient tissues from breast, colon, head and neck, and skin were immunostained for lamin B1 and imaged at high resolution (60X, 1.5 NA). Nuclei were then roughly segmented by the deep learning program Cellpose, followed by segmentation with a custom approach that sensitively detects micron-scale variations in the laminar contour. An Elliptical Fourier Analysis was performed to quantify nuclear irregularity by describing nuclear shape with a series of elliptical harmonics. More irregular nuclei are described by larger Fourier coefficients at higher frequencies. Results: Folds and wrinkles in the nuclear lamina were observed in healthy tissue from all four tissue types studied, indicating the presence of excess area in the lamina. The presence of irregular nuclear shape including wrinkling/folds in the lamina was quantified with the Elliptical Fourier Coefficient (EFC) ratio. Posterior probability analysis related EFC ratio to tumor grade to assess the clinical value of the excess area found in the nuclear lamina. This revealed cancer-type dependent trends between EFC ratio and tumor grade. Conclusions: Observations of wrinkling/folds in the nuclear lamina in healthy and cancerous tissue are consistent with the liquid drop model of nuclear deformation. Quantification of nuclear lamina wrinkling with the EFC ratio, along with other nuclear shape parameters, can be a powerful approach for computer-based, explainable, clinical grading. Our findings reveal a fundamental principle of nuclear deformation in cancer tissues and suggest a new approach for improving the accuracy in cancer grading. Citation Format: Christina R. Dubell, Ting-Ching Wang, Ishita Singh, Daniel G. Rosen, Saptarshi Chakraborty, Tanmay P. Lele. Drop-like deformation of nuclei in cancer patient tissue [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 6224.

On products of Fourier coefficients of half-integral weight cusp forms

On products of Fourier coefficients of half-integral weight cusp forms

Fourier coefficients of cusp forms on special sequences

In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms [Formula: see text] and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.

Shifted convolution sums of divisor functions with Fourier coefficients

Let f(z) be a holomorphic cusp form of weight κ for the full modular group SL2(Z). Denote its n-th normalized Fourier coefficient by λf(n). Let τk(n) denote that k-th divisor function with k≥4. In this paper, we consider the shifted convolution sum∑n≤Xτk(n)λf(n+h). We succeed in obtaining a non-trivial upper bound, which is uniform in the shift parameter h.

Conditional simulation of stationary non-Gaussian processes based on unified hermite polynomial model

The conditional simulation of non-Gaussian excitations utilizing records from the monitoring system is of great significance for hazard mitigation. To this end, this paper proposes a novel conditional non-Gaussian simulation method. In this method, the Unified Hermite Polynomial Model (UHPM) is used to describe the transformation relationship between recorded and unrecorded non-Gaussian processes and their underlying Gaussian counterparts. Meanwhile, an explicit transformation model between their correlation functions is also provided. Then, the covariance matrix of Fourier coefficients of the underlying Gaussian processes is constructed. Based on this covariance matrix, the conditional samples of Fourier coefficients are generated and substituted into the Spectral Representation Method (SRM) to perform the conditional simulation of the underlying Gaussian processes. Finally, the conditionally simulated samples of the underlying Gaussian processes are transformed into the non-Gaussian samples by the UHPM. To showcase the precision and efficacy of the proposed method, two numerical examples involving the conditional simulations of non-Gaussian ground motions and non-Gaussian wind pressure coefficients are provided.

Türkiye’den Salmo cinsine ait iki türün karşılaştırmalı otolit morfolojisi

In this study, the morphology of the sagittal otolith of Salmo coruhensis Turan, Kottelat & Engin, 2010 and Salmo fahrettini Turan, Kalayci, Bektaş, Kaya & Bayçelebi, 2020 from Çam Stream (Artvin) and Terme Stream (Samsun) was described by images of scanning electron microscopy. Its shape and contour were also analyzed with shape indices, elliptic Fourier coefficients and wavelet transforms. As the study material, a total of 30 S. coruhensis sample and 20 S. fahrettini sample were obtained. Interspecies differences in otolith shape and morphometry were evaluated by principal components analysis, canonical discrimination analysis, and permutational multivariate analysis of variance. The two salmonid species studied were distinguished by both morphometric and shape analysis methods. However, wavelet transform was found to be more effective than shape indices and elliptic Fourier coefficients in species discrimination, with an overall classification success rate of 80%. Our results showed that saccular otolith morphology could be an additional diagnostic character for trout species differentiation.

Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework.

FCSN: Global Context Aware Segmentation by Learning the Fourier Coefficients of Objects in Medical Images.

The encoder-decoder model is a commonly used Deep learning (DL) model for medical image segmentation. Encoder-decoder models make pixel-wise predictions that focus heavily on local patterns. As a result, the predicted mask often fails to preserve the object's shape and topology, which requires an understanding of the image's global context. In this work, we propose a Fourier Coefficient Segmentation Network (FCSN)---a novel global context-aware DL model that segments an object by learning the complex Fourier Coefficients of the object's masks. The Fourier coefficients are calculated by integrating over the mask's contour. Hence, FCSN is naturally motivated to incorporate a broader image context when estimating the coefficients. The global context awareness of FCSN helps produce more accurate segmentation and is more robust to local perturbations, such as additive noise or motion blur. We compare FCSN on other state-of-the-art global context-aware models (UNet++, DeepLabV3+, UNETR) on 5 medical image segmentation tasks (ISIC_2018, RIM_CUP, RIM_DISC, PROSTATE, FETAL). When compared with UNETR, FCSN attains significantly lower Hausdorff scores with 19.14 (6%), 17.42 (6%), 9.16 (14%), 11.18 (22%), and 5.98 (6%) for ISIC_2018, RIM_CUP, RIM_DISC, PROSTATE, and FETAL tasks respectively. Moreover, FCSN is lightweight by discarding the decoder module. FCSN only requires 29.7 M parameters which are 75.6 M and 9.9 M fewer than UNETR and DeepLabV3+, respectively. FCSN attains inference/training speeds of 1.6 ms/img and 6.3 ms/img, which is 8 and 3 faster than UNet and UNETR. Our work is available at https://github.com/nus-morninlab/FCSN.

Classification of the machine state in turning processes by using the acoustic emission

Processing digital information stands as a crucial foundation of Industry 4.0, facilitating a spectrum of activities from monitoring processes to their understanding and optimization. The application of data processing techniques, including feature extraction and classification, coupled with the identification of the most suitable features for specific purposes, continues to pose a significant challenge in the manufacturing sector. This research investigates the suitability of classification methods for machine and tool state classification by employing acoustic emission (AE) sensors during the dry turning of Ti6Al4V. Features such as quantiles, Fourier coefficients, and mel-frequency cepstral coefficients are extracted from the AE signals to facilitate classification. From this features the 20 best are selected for the classification to reduce the dimension of the feature space and redundancy. Algorithms including decision tree, k-nearest-neighbors (KNN), and quadratic discriminant analysis (QDA) are tested for the classification of machine states. Of these, QDA exhibits the highest accuracy at 98.6 %. Nonetheless, an examination of the confusion matrix reveals that certain classes, influenced by imbalanced training data, exhibit a lower prediction accuracy. In summary, the study affirms the potential of AE sensors for machine state recognition and tool condition monitoring. Although QDA emerges as the most acurate classifier, there remains an avenue for refinement, particularly in training data optimization and decision-making processes, to augment accuracy.