Fourier Coefficients Research Articles

Fractional-Differential Models of the Time Series Evolution of Socio-Dynamic Processes with Possible Self-Organization and Memory

This article describes the solution of two problems. First, based on the fractional diffusion equation, a boundary problem with arbitrary values of derivative indicators was formulated and solved, describing more general cases than existing solutions. Secondly, from the consideration of the probability schemes of transitions between states of the process, which can be observed in complex systems, a fractional-differential equation of the telegraph type with multiples is obtained (in time: β, 2β, 3β, … and state: α, 2α, 3α, …) using orders of fractional derivatives and its analytical solution for one particular boundary problem is considered. In solving edge problems, the Fourier method was used. This makes it possible to represent the solution in the form of a nested time series (one in time t, the second in state x), each of which is a function of the Mittag-Leffler type. The eigenvalues of the Mittag-Leffler function for describing states can be found using boundary conditions and the Fourier coefficient based on the initial condition and orthogonality conditions of the eigenfunctions. An analysis of the characteristics of time series of changes in the emotional color of users’ comments on published news in online mass media and the electoral campaigns of the US presidential elections showed that for the mathematical expectation of amplitudes of deviations of series levels from the size of the amplitude calculation interval (“sliding window”), a root dependence of fractional degree was observed; for dispersion, a power law with a fractional index greater than 1.5 was observed; and the behavior of the excess showed the presence of so-called “heavy tails”. The obtained results indicate that time series have unsteady non-locality, both in time and state. This provides the rationale for using differential equations with partial fractional derivatives to describe time series dynamics.

Fourier coefficients of real analytic Eisenstein series at various cusps

Fourier coefficients of real analytic Eisenstein series at various cusps

On the Fourier Coefficients for General Product 𝐿-Functions

Let 𝑓 be a normalized primitive cusp form of even integral weight for the full modular group Γ = 𝑆𝐿(2, ℤ). In this paper, we investigate upper bounds for the error terms related to the average behavior of Fourier coefficients 𝜆𝑓 ⊗𝑓 ⊗⋯⊗𝑙𝑓 (𝑛) of 𝑙-fold product 𝐿-functions, where 𝑓 ⊗ 𝑓 ⊗ ⋯ ⊗𝑙 𝑓 denotes the 𝑙-fold product of 𝑓. These results improves and generalizes the recent developments of Venkatasubbareddy and Sankaranarayanan [41]. We also provide some other similar results related to the error terms of general product 𝐿-functions.

On the Fourier coefficients of Siegel Eisenstein series of degree 3 of an odd prime level with the quadratic character

In this paper we give an explicit formula of the Fourier coefficients of the Siegel Eisenstein series of degree 3, level $p$ with quadratic character. For the calculation we use the result of the generalized Gauss sum computed by Hiroshi Saito. After the tedious calculation we can get a rather simple result.

On certain kernel functions and shifted convolution sums of the Fourier coefficients

We study the behavior of the shifted convolution sum involving even power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the k-full kernel function for any fixed integer k≥2.

Otolith morphology and total length relationships in Schizothorax grahami

Otolith is important for studying fish populations and life histories. In this study, the dominant species of Schizothorax grahami in the source section of the Chishui River was taken to understand the relationships between otolith morphology and total length (TL). Results showed a large difference between the four TL groups (A/B/C/D), except group B is similar to group C. The combined discrimination success rate of linear discriminant analysis was 62.2%. Group A and D’s success rate is the highest, at around 75%. Meanwhile, the success rate for Group B and Group C is below 65%. The one-way ANOVA of the Shape Index and the Canonical analysis of Principal Coordinates with two coefficients (Fourier coefficients and Wavelet coefficients) showed that Group B is similar to Group C, with a large difference from the other two groups. When TL was greater than 100 mm (the pearl organs appearing), the otolith growth was lower changing. Otolith morphology still changes with growth after sexual maturity in fish, so the larger fish is more useful for conducting otolith morphology studies for accurate evaluation and management of local fishery resources.

A new method to characterize the nonlinear magneto-viscoelasticity behavior of magneto-active elastomers under large amplitude oscillatory axial (LAOA) loading

The nonlinear viscoelasticity of magneto-active elastomers (MAEs) under large amplitude oscillatory shear (LAOS) loading has been extensively characterized. A reliable and effective methodology, however, is lacking for such characterizations under large amplitude oscillatory axial (LAOA) loading. This is partly due to complexities associated with experimental compression mode characterizations of MAEs and in-part due to their asymmetric stress–strain behavior leading to different elastic moduli during extension and compression. This study proposes a set of new nonlinear measures to characterize nonlinear and asymmetric behavior of MAEs subject to LAOA loading. These include differential large/zero strain moduli and large/zero strain-rate viscosity, which could also facilitate physical interpretations of the inter- and intra-cycle nonlinearities observed in asymmetric and hysteretic stress–strain responses. The compression mode stress–strain behavior of MAEs was experimentally characterized under different magnitudes of axial strain (0.025 to 0.20), strain rate (frequency up to 30 Hz) and magnetic flux density (0 to 750mT). The measured stress–strain responses were decomposed into elastic, viscous and viscoelastic stress components using Chebyshev polynomials and Fourier series. The stress decomposition based on Chebyshev polynomials permitted determination of equivalent nonlinear elastic and viscous stress components, upon which the proposed measures were obtained. An equivalent set of Fourier coefficients was also obtained for estimating equivalent elastic/viscous stress, thereby facilitating faster calculation of the proposed material measures. The proposed methodology is considered to serve as an effective tool for deriving constitutive models for describing nonlinear and asymmetric characteristics of MAEs.

Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions

Abstract High-dimensional partial differential equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell ^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.

Fault classification of the rolling bearing based on minimax entropy domain adaption augmented with signal generation algorithm

Rolling bearing faults have been capturing substantial research attention as they are the root causes of malfunctions in mechatronics systems than any other factors. The detection of rolling bearing faults in the early stage is therefore a mandatory requirement demanded by reliable industrial plants. To release the dependence of diagnostic methods on human expertise and system’s understanding, this work proposes a fault classification method for rolling bearings that is based on a deep learning framework. The framework consists of a minimax entropy domain adaptation algorithm augmented with a signal generalization algorithm. The function of the signal generalization algorithm is to reduce the domain shift between training and testing datasets that are often obtained experimentally from different working conditions. The generalized signal is then represented in the form of Fourier series whose coefficients contain intrinsic information that associated with different types of bearing faults. A convolutional neural network extracts the hidden information of bearing faults buried in the Fourier coefficients and then categorises the working condition of the bearing under test. By combining the advantages of both signal processing techniques in the frequency domain and the minimax entropy domain adaptation, the novel diagnostic framework is able to detect bearing faults from different working conditions. The effectiveness of the proposed diagnostic algorithm is experimentally verified by two case studies that were prepared with different types and levels of bearing faults.

Octonionic Magical Supergravity, Niemeier Lattices, and Exceptional & Hilbert Modular Forms

AbstractThe quantum degeneracies of Bogomolny‐Prasad‐Sommerfield (BPS) black holes of octonionic magical supergravity in five dimensions are studied. Quantum degeneracy is defined purely number theoretically as the number of distinct states in charge space with a given set of invariant labels. Quantum degeneracies of spherically symmetric stationary BPS black holes are given by the Fourier coefficients of modular forms of exceptional group . Their charges take values in the lattice defined by the exceptional Jordan algebra over integral octonions. The quantum degeneracies of rank 1 and rank 2 BPS black holes are given by the Fourier coefficients of singular modular forms and . The rank 3 (large) BPS black holes will be studied elsewhere. Following the work of N. Elkies and B. Gross on embeddings of cubic rings A into the exceptional Jordan algebra we show that the quantum degeneracies of rank 1 black holes described by such embeddings are given by the Fourier coefficients of the Hilbert modular forms (HMFs) of . If the discriminant of the cubic ring A is with p a prime number then the isotropic lines in the 24 dimensional orthogonal complement of A define a pair of Niemeier lattices which can be taken as charge lattices of some BPS black holes. The current status of the searches for the M/superstring theoretic origins of the octonionic magical supergravity is also reviewed.

ACCELERATIONS RECORDED BY LOW-FREQUENCY WEARABLE SENSORS AS EFFECTIVE DISCRIMINATORS OF KNEE AND HIP OSTEOARTHRITIS

Wearable inertial sensors can detect abnormal gait associated with knee or hip osteoarthritis (OA). However, few studies have compared sensor-derived gait parameters between patients with hip and knee OA or evaluated the efficacy of sensors suitable for remote monitoring in distinguishing between the two. Hence, our study seeks to examine the differences in accelerations captured by low-frequency wearable sensors in patients with knee and hip OA and classify their gait patterns.We included patients with unilateral hip and knee OA. Gait analysis was conducted using an accelerometer ipsilateral with the affected joint on the lateral distal thighs. Statistical parametric mapping (SPM) was used to compare acceleration signals. The k-Nearest Neighbor (k-NN) algorithm was trained on 80% of the signals' Fourier coefficients and validated on the remaining 20% using 10-fold cross-validation to classify the gait patterns into hip and knee OA.We included 42 hip OA patients (19 females, age 70 [63–78], BMI of 28.3 [24.8–30.9]) and 59 knee OA patients (31 females, age 68 [62–74], BMI of 29.7 [26.3–32.6]). The SPM results indicated that one cluster (12–20%) along the vertical axis had accelerations exceeding the critical threshold of 2.956 (p=0.024). For the anteroposterior axis, three clusters were observed exceeding the threshold of 3.031 at 5–19% (p = 0.0001), 39–54% (p=0.00005), and 88–96% (p = 0.01). Regarding the mediolateral axis, four clusters were identified exceeding the threshold of 2.875 at 0–9% (p = 0.02), 14–20% (p=0.04), 28–68% (p < 0.00001), and 84–100% (p = 0.004). The k-NN model achieved an AUC of 0.79, an accuracy of 80%, and a precision of 85%.In conclusion, the Fourier coefficients of the signals recorded by wearable sensors can effectively discriminate the gait patterns of knee and hip OA. In addition, the most remarkable differences in the time domain were observed along the mediolateral axis.

Data analysis methods and applications of the eddy current diagnostic system in the Keda Torus eXperiment device

Since the establishment of the eddy current diagnostic system within the Keda Torus eXperiment (KTX) device, it has unveiled many applications. Recent developments have introduced innovative data analysis techniques alongside compelling experimental results, underscoring the necessity for a comprehensive summary of the system's data analysis approaches and broad applications. Notable features of the system encompass exceptional precision, the ability to encompass shell currents on the entirety of the closed boundary, vector detection of shell currents, and measurement of diverse physical quantities. In terms of data analysis methodologies, meticulous scrutiny of the null field region is conducted, and we reveal a distinctive characteristic within the complex shell current signals, namely the asymmetry of the amplitudes of ±n Fourier coefficients. Moreover, the Hodge decomposition emerges as a pivotal technique, allowing for the distinctive separation of shell currents into three orthogonal components based on their distinct spatial topological properties. With regard to practical applications, an in-depth examination of the vector potential and magnetic helicity flux densities are presented in detail, further highlighting the far-reaching utility of the system's capabilities.

Fourier Series Weight in Quantum Machine Learning

In this work, we aim to confirm the impact of the Fourier series on the quantum machine learning model. We will propose models, tests, and demonstrations to achieve this objective. We designed a quantum machine learning leveraged on the Hamiltonian encoding. With a subtle change, we performed the trigonometric interpolation, binary and multiclass classifier, and a quantum signal processing application. We also proposed a block diagram of determining approximately the Fourier coefficient based on quantum machine learning. We performed and tested all the proposed models using the Pennylane framework.

Attentional Modulation in Early Visual Cortex: A Focused Reanalysis of Steady-state Visual Evoked Potential Studies.

Steady-state visual evoked potentials (SSVEPs) are a powerful tool for investigating selective attention. Here, we conducted a combined reanalysis of multiple studies employing this technique in a variety of attentional experiments to, first, establish benchmark effect sizes of attention on amplitude and phase of SSVEPs and, second, harness the power of a large data set to test more specific hypotheses. Data of eight published SSVEP studies were combined, in which human participants (n = 135 in total) attended to flickering random dot stimuli based on their defining features (e.g., location, color, luminance, or orientation) or feature conjunctions. The reanalysis established that, in all the studies, attention reliably enhanced amplitudes, with color-based attention providing the strongest effect. In addition, the latency of SSVEPs elicited by attended stimuli was reduced by ∼4 msec. Next, we investigated the modulation of SSVEP amplitudes in a subset of studies where two different features were attended concurrently. Although most models assume that attentional effects of multiple features are combined additively, our results suggest that neuronal enhancement provided by concurrent attention is better described by multiplicative integration. Finally, we used the combined data set to demonstrate that the increase in trial-averaged SSVEP amplitudes with attention cannot be explained by increased synchronization of single-trial phases. Contrary to the prediction of the phase-locking account, the variance across trials of complex Fourier coefficients increases with attention, which is more consistent with boosting of a largely phase-locked signal embedded in non-phase-locked noise.

Moments of ideal class counting functions

We consider the counting function of ideals in a given ideal class of a number field of degree d. This describes, at least conjecturally, the Fourier coefficients of an automorphic form on GL(d), typically not a Hecke eigenform and not cuspidal. We compute its moments, and also investigate the moments of the corresponding cuspidal projection.

Ramanujan type congruences for quotients of Klein forms

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p>19. Canonical bases for modular forms of level 5≤p≤19 are constructed by summing weight one Hecke Eisenstein series of levels 5≤p≤19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ1(p) and Γ(p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ1(p) as a direct sum of subspaces of modular forms for Γ(p) of the form qr/pZ[[q]]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences.

On the information complexity for integration in subspaces of the Wiener algebra

Recently, Goda proved the polynomial tractability of integration on the following function subspace of the Wiener algebraFd:={f∈C(Td)|‖f‖Fd:=∑k∈Zd|fˆ(k)|max⁡(1,minj∈supp(k)⁡log⁡|kj|)<∞}, where T:=R/Z=[0,1), fˆ(k) is the k-th Fourier coefficient of f and supp(k):={j∈{1,…,d}|kj≠0}. Goda raised an open question as to whether the upper bound of the information complexity for integration in Fd can be improved. In this note, we give a positive answer. By establishing a Monte Carlo sampling method and using Rademacher complexity to estimate the uniform convergence rate, the upper bound can be improved to Θ(d/ϵ3), where ϵ∈(0,1/2) is the target accuracy. We also use the same technique to estimate the information complexity for a Hölder continuous subspace of Wiener algebra. Compared to the previous upper bound Θ(max⁡(d2ϵ2,d1/qϵ1/α)), we present a new upper bound Θ((dlog⁡dq+dlog⁡(1/ϵ)α)/ϵ2), where q∈[1,∞),α∈(0,1] are the parameters of Hölder continuity. Ignoring the logarithmic factors, the order of our upper bound is superior to the previous result, especially for the case where the Hölder exponent α is small.

Rademacher Expansion of a Siegel Modular Form for $${{\mathcal {N}}}= 4$$ Counting

AbstractThe degeneracies of 1/4 BPS states with unit torsion in heterotic string theory compactified on a six torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form $$\Phi _{10}$$ Φ 10 of weight 10. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher-type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of $$1/\Phi _{10}$$ 1 / Φ 10 . The construction uses two distinct $$\textrm{SL}(2, {\mathbb {Z}})$$ SL ( 2 , Z ) subgroups of $$\textrm{Sp}(2, {\mathbb {Z}})$$ Sp ( 2 , Z ) which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of $$1/\eta ^{24}$$ 1 / η 24 by means of a continued fraction structure.

Symmetries and spectral statistics in chaotic conformal field theories. Part II. Maass cusp forms and arithmetic chaos

We continue the study of random matrix universality in two-dimensional conformal field theories. This is facilitated by expanding the spectral form factor in a basis of modular invariant eigenfunctions of the Laplacian on the fundamental domain. The focus of this paper is on the discrete part of the spectrum, which consists of the Maass cusp forms. Both their eigenvalues and Fourier coefficients are sporadic discrete numbers with interesting statistical properties and relations to analytic number theory; this is referred to as ‘arithmetic chaos’. We show that the near-extremal spectral form factor at late times is only sensitive to a statistical average over these erratic features. Nevertheless, complete information about their statistical distributions is encoded in the spectral form factor if all its spin sectors exhibit universal random matrix eigenvalue repulsion (a ‘linear ramp’). We ‘bootstrap’ the spectral correlations between the cusp form basis functions that correspond to a universal linear ramp and show that they are unique up to theory-dependent subleading corrections. The statistical treatment of cusp forms provides a natural avenue to fix the subleading corrections in a minimal way, which we observe leads to the same correlations as those described by the [torus]×[interval] wormhole amplitude in AdS3 gravity.

Spatio-temporal data prediction of multiple air pollutants in multi-cities based on 4D digraph convolutional neural network.

In response to the problem that current multi-city multi-pollutant prediction methods based on one-dimensional undirected graph neural network models cannot accurately reflect the two-dimensional spatial correlations and directedness, this study proposes a four-dimensional directed graph model that can capture the two-dimensional spatial directed information and node correlation information related to multiple factors, as well as extract temporal correlation information at different times. Firstly, A four-dimensional directed GCN model with directed information graph in two-dimensional space was established based on the geographical location of the city. Secondly, Spectral decomposition and tensor operations were then applied to the two-dimensional directed information graph to obtain the graph Fourier coefficients and graph Fourier basis. Thirdly, the graph filter of the four-dimensional directed GCN model was further improved and optimized. Finally, an LSTM network architecture was introduced to construct the four-dimensional directed GCN-LSTM model for synchronous extraction of spatio-temporal information and prediction of atmospheric pollutant concentrations. The study uses the 2020 atmospheric six-parameter data of the Taihu Lake city cluster and applies canonical correlation analysis to confirm the data's temporal, spatial, and multi-factor correlations. Through experimentation, it is verified that the proposed 4D-DGCN-LSTM model achieves a MAE reduction of 1.12%, 4.91%, 5.62%, and 11.67% compared with the 4D-DGCN, GCN-LSTM, GCN, and LSTM models, respectively, indicating the good performance of the 4D-DGCN-LSTM model in predicting multiple types of atmospheric pollutants in various cities.