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Texture is the surface qualities and visual attributes of an object, determined by the arrangement, size, shape, density, and proportion of its fundamental components. In the manufacturing industry, products typically have uniform textures, allowing for automated visual inspections of the product surface to recognize defects. During this process, texture defect recognition techniques can be employed. In this paper, we propose a method that combines a convolutional autoencoder architecture with Fourier transform analysis. We employ a normal reconstructed template as defined in this study. Despite its simple structure and rapid training and inference capabilities, it offers recognition performance comparable to state-of-the-art methods. Fourier transform is a powerful tool for analyzing the frequency domain of images and signals, which is essential for effective defect recognition as texture defects often exhibit characteristic changes in specific frequency ranges. The experiment evaluates the recognition performance using the AUC metric, with the proposed method showing a score of 93.7%. To compare with existing approaches, we present experimental results from previous research, an ablation study of the proposed method, and results based on the high-pass filter used in the Fourier mask. [ABSTRACT FROM AUTHOR]

We obtain the exact order estimates for approximation of the functions of many variables from the generalized Nikol'skii–Besov classes B p , θ Ω R d by de la Vallée Poussin sums in the metrics of the spaces L ∞ R d and L 1 R d . These classes of functions for some given Ω coincide with the well-known classical isotropic Nikol'skii–Besov classes. [ABSTRACT FROM AUTHOR]

Long time series forecasting (LTSF), which involves modeling relationships within long time series to predict future values, has extensive applications in domains such as weather forecasting, financial analysis, and traffic prediction. Recently, numerous transformer-based models have been developed to address the challenges in LTSF. These models employ methods such as sparse attention to alleviate the inefficiencies associated with the attention mechanism and utilize decomposition architecture to enhance the predictability of the series. However, these complexity reduction methods necessitate additional calculations, and the series decomposition architecture overlooks the random components. To overcome these limitations, this paper proposes the Frequency Enhanced Decomposed Attention Free Transformer (FEDAF). FEDAF introduces two variants of the Frequency Enhanced Attention Free Mechanism (FEAFM), namely FEAFM-s and FEAFM-c, which seamlessly replace self-attention and cross-attention. Both variants perform calculations in the frequency domain without incurring additional costs, with the time and space complexity of FEAFM-s being O (L log L) . Additionally, FEDAF incorporates a time series decomposition architecture that considers random components. Unlike other models that solely decompose the series into trend and seasonal components, FEDAF also eliminates random terms by applying Fourier denoising. Our study quantifies data drift and validates that the proposed decomposition structure can mitigate the adverse effects caused by data shift. Overall, FEDAF demonstrates superior forecasting performance compared to state-of-the-art models across various domains, achieving a remarkable improvement of 19.49% for Traffic in particular. Furthermore, an efficiency analysis reveals that FEAFM enhances space efficiency by 12.8% compared to the vanilla attention mechanism and improves time efficiency by 43.63% compared to other attention mechanism variants. [ABSTRACT FROM AUTHOR]

A thermoelastic model for a solid cylinder consisting of two different isotropic thermoelastic homogeneous materials is created. Boundary conditions for the heat flow and stress tensors were discussed. A dual-phase lag model was applied to investigate its thermophysical properties. For their numerical evaluation, a two-layered structure with an interfacial thermal contact resistance and an integral elastic wave resistance, as well as some special cases, were considered. This study will be useful for theoretical modeling the thermoelasticity at the nanoscale and for designing nano and multilayered devices, plates, and surface coatings. [ABSTRACT FROM AUTHOR]

Gaussian processes are popular in spatial statistics, data mining and machine learning because of their versatility in quantifying spatial variability and in propagating uncertainty. Although there has been a prolific research activity about Gaussian processes over Euclidean domains, only recently this research has extended to non-Euclidean manifolds. This paper digs into vector-valued Gaussian processes defined over the product of a hypersphere and a Euclidean space of arbitrary dimension, which are of interest in various disciplines of the natural sciences and engineering. Under mild regularity conditions, we establish a surprising one-to-one correspondence between matrix-valued kernels associated with vector Gaussian processes over the product space, and what we term partial ultraspherical and Fourier transforms that are taken over either the sphere or the Euclidean subspace. The properties of our approach are illustrated in terms of new parametric classes of matrix-valued kernels for product spaces of a hypersphere crossed with a Euclidean space. We also provide two algorithms that allow for fast simulation of approximately Gaussian (in the sense of the central limit theorem) processes in such product spaces. [ABSTRACT FROM AUTHOR]

Let n , m ≥ 1 and α > 0 . We denote by F α , m the m-analytic Bargmann–Segal–Fock space, i.e., the Hilbert space of all m-analytic functions defined on C n and square integrables with respect to the Gaussian weight exp (- α | z | 2) . We study the von Neumann algebra A of bounded linear operators acting in F α , m and commuting with all "horizontal" Weyl translations, i.e., Weyl unitary operators associated to the elements of R n . The reproducing kernel of F 1 , m was computed by Youssfi [Polyanalytic reproducing kernels in C n , Complex Anal. Synerg., 2021, 7, 28]. Multiplying the elements of F α , m by an appropriate weight, we transform this space into another reproducing kernel Hilbert space whose kernel K is invariant under horizontal translations. Using the well-known Fourier connection between Laguerre and Hermite functions, we compute the Fourier transform of K in the "horizontal direction" and decompose it into the sum of d products of Hermite functions, with d = n + m - 1 n . Finally, applying the scheme proposed by Herrera-Yañez, Maximenko, Ramos-Vazquez [Translation-invariant operators in reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31], we show that F α , m is isometrically isomorphic to the space of vector-functions L 2 (R n) d , and A is isometrically isomorphic to the algebra of matrix-functions L ∞ (R n) d × d . [ABSTRACT FROM AUTHOR]

Bent functions have a number of practical applications in cryptography, coding theory, and other fields. Fourier transform is a key tool to study bent functions on finite abelian groups. Using Fourier transforms, in this paper, we first present two necessary and sufficient conditions on the existence of bent functions via faithful actions of finite abelian groups and then show two constructions of sequences with ideal auto-correlation (SIACs). In addition, we construct a periodic complementary sequence set (PCSS) by rearranging a periodic multiple shift sequence (PMSS) corresponding to a bent function on a finite abelian group. Some concrete constructions of SIACs and PCSSs are provided to illustrate the efficiency of our methods. [ABSTRACT FROM AUTHOR]

The present study uses machine learning and time series spectral analysis to develop a novel technique to forecast the sunspot number (SN) in both hemispheres for the remainder of Solar Cycle 25 and Solar Cycle 26. This enables us to offer predictions for hemispheric SN until January 2038 (using the 13-month running average). For the Northern hemisphere, we find maximum peak values for Solar Cycles 25 and 26 of 58.5 in April 2023 and 51.5 in November 2033, respectively (root mean square error of 6.1). For the Southern hemisphere, the predicted maximum peak values for Solar Cycles 25 and 26 are 77.0 in September 2024 and 70.1 in November 2034, respectively (root mean square error of 6.8). In this sense, the results presented here predict a Southern hemisphere prevalence over the Northern hemisphere, in terms of SN, for Solar Cycles 25 and 26, thus continuing a trend that began around 1980, after the last period of Northern hemisphere prevalence (which, in turn, started around 1900). On the other hand, for both hemispheres, our findings predict lower maxima for Solar Cycles 25 and 26 than the preceding cycles. This fact implies that, when predicting the total SN as the sum of the two hemispheric forecasts, Solar Cycles 24 – 26 may be part of a centennial Gleissberg cycle's minimum, as was the case in the final years of the 19th century and the start of the 20th century (Solar Cycles 12, 13, and 14). [ABSTRACT FROM AUTHOR]

This study introduces a novel method for predicting the sunspot number ( S N ) of Solar Cycles 25 (the current cycle) and 26 using multivariate machine-learning techniques, the Sun's polar flux as a precursor parameter, and the fast Fourier transform to conduct a spectral analysis of the considered time series. Using the 13-month running average of the version 2 of the S N provided by the World Data Center—SILSO, we are thus able to present predictive results for the S N until January 2038, giving maximum peak values of 131.4 (in July 2024) and 121.2 (in September 2034) for Solar Cycles 25 and 26, respectively, with a root mean square error of 10.0. These predicted dates are similar to those estimated for the next two polar flux polarity reversals (April 2024 and August 2034). Furthermore, the values for the S N maxima of Solar Cycles 25 and 26 have also been forecasted based on the known correlation between the absolute value of the difference between the polar fluxes of both hemispheres at an S N minimum and the maximum S N of the subsequent cycle, obtaining similar values to those achieved with the previous method: 142.3 ± 34.2 and 126.9 ± 34.2 for Cycles 25 and 26, respectively. Our results suggest that Cycle 25 will have a maximum amplitude that lies below the average and Cycle 26 will reach an even lower peak. This suggests that Solar Cycles 24 (with a peak of 116.4), 25, and 26 would belong to a minimum of the centennial Gleissberg cycle, as was the case in the final years of the 19th and the early 20th centuries (Solar Cycles 12, 13, and 14). [ABSTRACT FROM AUTHOR]

We propose a novel methodology for forecasting spatio-temporal data using supervised semi-nonnegative matrix factorization (SSNMF) with frequency regularization. Matrix factorization is employed to decompose spatio-temporal data into spatial and temporal components. To improve clarity in the temporal patterns, we introduce a nonnegativity constraint on the time domain along with regularization in the frequency domain. Specifically, regularization in the frequency domain involves selecting features in the frequency space, making an interpretation in the frequency domain more convenient. We propose two methods in the frequency domain: soft and hard regularizations, and provide convergence guarantees to first-order stationary points of the corresponding constrained optimization problem. While our primary motivation stems from geophysical data analysis based on GRACE (Gravity Recovery and Climate Experiment) data, our methodology has the potential for wider application. Consequently, when applying our methodology to GRACE data, we find that the results with the proposed methodology are comparable to previous research in the field of geophysical sciences but offer clearer interpretability. [ABSTRACT FROM AUTHOR]

In this paper we mainly study large time behavior for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. The sharp L 2 decay rate was obtained on the co-rotational case. We prove that the optimal L 2 decay rate of the velocity of the general FENE dumbbell model is (1 + t) - d 4 with d ≥ 2 . Our obtained result is sharp and improves considerably the previous result in Luo and Yin (Arch Ration Mech Anal 224(1):209–231, 2017). [ABSTRACT FROM AUTHOR]

Let ρ be a rearrangement-invariant (r.i.) norm on the set M (R n) of Lebesgue-measurable functions on R n such that the space L ρ (R n) = f ∈ M (R n) : ρ (f) < ∞ is an interpolation space between L 2 (R n) and L ∞ (R n). The principal result of this paper asserts that given such a ρ , the inequality ρ (f ^) ≤ C σ (f) holds for any r.i. norm σ on M (R n) if and only if ρ ¯ U f ∗ ≤ C σ ¯ (f ∗). Here, ρ ¯ is the unique r.i. norm on M (R +) , R + = (0 , ∞) , satisfying ρ ¯ (f ∗) = ρ (f) and U f ∗ (t) = ∫ 0 1 / t f ∗ , in which f ∗ is the nonincreasing rearrangement of f on R + . Further, in this case the smallest r.i. norm σ for which ρ (f ^) ≤ C σ (f) holds is given by σ (f) = σ ¯ (f ∗) = ρ ¯ U f ∗ , where, necessarily, ρ ¯ ∫ 0 1 / t χ (0 , a) = ρ ¯ min { 1 / t , a } < ∞ , for all a > 0 . We further specialize and expand these results in the contexts of Orlicz and Lorentz Gamma spaces. [ABSTRACT FROM AUTHOR]

In an effort to estimate the number of true nulls in large scale multiplicity problems (the normal means problem), we generalize the current Fourier transform based oracle estimator with a Laplace transform based estimator. Our interest in this problem stems from the application of r-power which requires knowledge of the number of nulls (Dasgupta et al. in Sankhya B 78(1):96–118, 2016). We analytically show that our method is consistent and theoretically has lower mean squared error than the existing competitor (Jin in J R Stat Soc Ser B (Stat Methodol) 70(3):461–493, 2008). We follow up by a numerical example and a simulation study that ratifies our theoretical results. [ABSTRACT FROM AUTHOR]

Fourier PlenOctrees have shown to be an efficient representation for real-time rendering of dynamic neural radiance fields (NeRF). Despite its many advantages, this method suffers from artifacts introduced by the involved compression when combining it with recent state-of-the-art techniques for training the static per-frame NeRF models. In this paper, we perform an in-depth analysis of these artifacts and leverage the resulting insights to propose an improved representation. In particular, we present a novel density encoding that adapts the Fourier-based compression to the characteristics of the transfer function used by the underlying volume rendering procedure and leads to a substantial reduction of artifacts in the dynamic model. We demonstrate the effectiveness of our enhanced Fourier PlenOctrees in the scope of quantitative and qualitative evaluations on synthetic and real-world scenes. [ABSTRACT FROM AUTHOR]

In this study, we extend Beckner's seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform in several key aspects: it does not conform to the Plancherel theorem, alters the interplay between derivatives and multiplication, and the product of algebra elements does not necessarily maintain the magnitude relationships found in classical settings. To overcome these challenges, our approach involves constructing the Cayley–Dickson Fourier transform by sequentially applying classical Fourier transforms. A pivotal part of our strategy is the utilization of a theorem that facilitates the norm-preserving extension of linear operators between spaces L p and L q. Furthermore, our investigation brings new insights into the complexities surrounding the Beckner–Hirschman Entropic inequality in the context of non-associative algebras. [ABSTRACT FROM AUTHOR]

Let { (M n - 1 B n , C n) } n = 1 ∞ be a compatible tower on R d and let μ { M n } , { B n } be the Moran measure generated by infinite convolutions of discrete measures induced by them. In this paper, we first prove that under certain situations, the compatible tower condition can ensure that μ { M n } , { B n } is a spectral measure, that is the Hilbert space L 2 (μ { M n } , { B n } ) admits an exponential orthonormal basis. Furthermore, if we restrict { M n , B n } n = 1 ∞ to be a class of generalized Sierpinski-type family, then we obtain that the existence of compatible tower and the spectrality of μ { M n } , { B n } are equivalent. [ABSTRACT FROM AUTHOR]

This paper is concerned with the valuation of single and double barrier knock-out call options in a Markovian regime switching model with specific rebates. The integral formulas of the rebates are derived via matrix Wiener-Hopf factorizations and Fourier transform techniques, also, the integral representations of the option prices are constructed. Moreover, the first-passage time density functions in two-state regime model are derived. As applications, several numerical algorithms and numerical examples are presented. [ABSTRACT FROM AUTHOR]

For ν ∈ R , we consider the invariant Laplacians Δ ν in the unit complex ball B n = (S U (n , 1) / S (U (n) × U (1)) Δ ν = 4 (1 - | z | 2) { ∑ i , j = 1 n (δ ij - z i z j ¯) ∂ 2 ∂ z i ∂ z j ¯ - ν 2 ∑ j = 1 n z j ∂ ∂ z j + ν 2 ∑ j = 1 n z j ¯ ∂ ∂ z j ¯ + ν 2 4 } and the spectral projectors Q λ , ν associated to Δ ν defined by Q λ , ν f = | c ν (λ) | - 2 f ∗ φ λ , ν (z) , where φ λ , ν is the S (U (n) × U (1)) -invariant eigenfunction of Δ ν and c ν (λ) the Harish-Chandra function. The goal of this paper is to give an image characterization of Q λ , ν of C c ∞ (B n) and L 2 ( B n , (1 - | z | 2 ) - n - 1 d m (z)) . [ABSTRACT FROM AUTHOR]

In this paper, we establish some of the deformed calculus for R-Minkowski space-time. We start by defining the deformed derivative in R-Minkowski spacetime, the deformed exponential and trigonometric functions, deformed Fourier transform and its inverse transformation, and then we apply all that to the study of the one-dimensional box problem in a position representation. Thereafter, we study the one-dimensional Dirac harmonic oscillator, and as a consequence we obtain the energy spectrum of the latter in R-Minkowski space-time. [ABSTRACT FROM AUTHOR]

In this article, the nonclassical transient heat propagation process in a cracked strip is investigated by Guyer–Krumhansl (G–K) model, which incorporates both the time lagging behavior and the spatially nonlocal effect. The impulsive thermal loading as well as cyclic loading exerted on the top bounding surface are examined to explore the non-Fourier thermal characteristics. By means of the Laplace transform and Fourier transform, the governing partial differential equations subjected to mixed boundary conditions are converted to a group of singular integral equations. With the aid of numerical Laplace inversion, the transient temperatures are calculated to make comparisons of thermal responses determined by Fourier's law, Cattaneo–Vernotte (C–V) equation, and G–K model. The numerical results display the specific thermal behaviors of G–K model in the cracked medium and demonstrate the G–K model's capabilities in eliminating the unrealistic phenomena accompanied by C–V equation. Our research would contribute to achieving a better understanding of the transient heat conduction in small-sized systems or composites at the macroscopic scale. [ABSTRACT FROM AUTHOR]

We construct a crystalline measure on the real line that is not a Fourier quasicrystal. [ABSTRACT FROM AUTHOR]

Universal style transfer (UST) infuses styles from arbitrary reference images into content images. Existing methods, while enjoying many practical successes, are unable of explaining experimental observations, including different performances of UST algorithms in preserving the spatial structure of content images. In addition, methods are limited to cumbersome global controls on stylization, so that they require additional spatial masks for desired stylization. In this work, we first provide a systematic Fourier analysis on a general framework for UST. We present an equivalent form of the framework in the frequency domain. The form implies that existing algorithms treat all frequency components and pixels of feature maps equally, except for the zero-frequency component. We connect Fourier amplitude and phase with a widely used style loss and a well-known content reconstruction loss in style transfer, respectively. Based on such equivalence and connections, we can thus interpret different structure preservation behaviors between algorithms with Fourier phase. Given the interpretations, we propose two plug-and-play manipulations upon style transfer methods for better structure preservation and desired stylization. Both qualitative and quantitative experiments demonstrate the improved performance of our manipulations upon mainstreaming methods without any additional training. Specifically, the metrics are improved by 6% in average on the content images from MS-COCO dataset and the style images from WikiArt dataset. We also conduct experiments to demonstrate (1) the abovementioned equivalence, (2) the interpretability based on Fourier amplitude and phase and (3) the controllability associated with frequency components. [ABSTRACT FROM AUTHOR]

The main result of this note is the strengthening of a quite arbitrary a priori Fourier restriction estimate to a multi-parameter maximal estimate of the same type. This allows us to discuss a certain multi-parameter Lebesgue point property of Fourier transforms, which replaces Euclidean balls by standard ellipsoids or axes-parallel rectangles. Along the lines of the same proof, we also establish a d-parameter Menshov–Paley–Zygmund-type theorem for the Fourier transform on R d . Such a result is interesting for d ⩾ 2 because, in a sharp contrast with the one-dimensional case, the corresponding endpoint L 2 estimate (i.e., a Carleson-type theorem) is known to fail since the work of C. Fefferman in 1970. Finally, we show that a Strichartz estimate for a given homogeneous constant-coefficient linear dispersive PDE can sometimes be strengthened to a certain pseudo-differential version. [ABSTRACT FROM AUTHOR]

The thermoreflectance method, which can measure thermal diffusivity in the cross-plane direction of thin films, mainly has two possible configurations; rear-heat front-detect (RF) and front-heat front-detect (FF) configuration. FF configuration is applicable to a wide variety of thin films including thin films deposited on opaque substrates, but this configuration has some problems in determination of the thermal diffusivity. One of the main problems is the effect of the penetration of pump beam and probe beam in thin film, which affects the initial temperature distribution near the sample's surface after pulse heating. Several studies have tried to analyze the effect but there have been no practical analytical solutions which can solve this problem in FF configuration. In this paper, we propose a new analytical solution which considers the penetration of pump beam and probe beam into thin film, and by applying Fourier expansion analysis which we developed in a previous study to thermoreflectance signals, we have determined the thermal diffusivity of thin film in the thermoreflectance method under FF configuration. We measured platinum thin films with different thickness under both FF and RF configuration and obtained consistent thermal diffusivity values from both configurations. [ABSTRACT FROM AUTHOR]

Using the data from weather observation stations, an assessment of statistical significance of the first harmonic amplitude of the long-term average daily precipitation series in Russia in 1961‒2020 and their changes in 1991‒2020 compared to 1961‒1990 is made. It is shown that at most stations the first harmonic has a reliable value different from the noise, except for eleven stations in the southern regions of the European part of Russia. It is revealed that on average there is a significant decrease in the first harmonic amplitudes of the daily precipitation series across Russia. An analysis of the spatial distribution of the studied quantities is performed and the presence of large areas with their homogeneous character is demonstrated. [ABSTRACT FROM AUTHOR]

In order to analyze the ability of STA/LTA, AIC, STA/LTA + AIC, and W-AIC methods to recognize interference signals during P wave pickup, we selected a large number of KiK-Net records for P wave arrival time recognition. The results indicate that a single method has limited recognition ability for interference signals. The STA/LTA combined with AIC method adds limiting conditions at the trigger point, achieving recognition accuracy of 93.0% and 91.7% for spike signals and drift signals. This method has good stability and accuracy for picking up P wave arrival. For regular interference signals, the accuracy of picking up after wavelet decomposition and reconstruction reaches 87.8%. In order to more accurately identify interference signals and improve pickup accuracy, different methods can be combined. On the other hand, there is a linear relationship between the amplitude and magnitude of the P band Fourier spectrum. The variances for P waves in the first 5 s in calculating earthquake magnitudes are 0.602, 0.462, and 0.423 for acceleration, velocity, and displacement, respectively. The results can provide reference for earthquake early warning algorithms based on artificial intelligence technology. [ABSTRACT FROM AUTHOR]

The present article motivates great interests on the constitutive modeling of thermoelastic coupling behavior in a transversely isotropic thick plate in the context of a new theory of ultrafast heating condition, known as Moore–Gibson–Thompson (MGT) theory. Corresponding to this new theory, the heat transport law is formulated in an integral form of a common derivative within a slipping interval by assimilating the memory-dependent heat transfer. Upper surface of the plate is stress-free subjected to prescribed surface temperature, whereas the lower surface rests in a rigid foundation and is thermally insulated and under the action of the gravitational field. Applying the Laplace–Fourier transforms successively, the solutions of the governing equations have been achieved in the transformed domain and the corresponding solutions in the space-time domain have been obtained using suitable numerical technique based on the expansion of Fourier series. According to the graphical representations corresponding to the numerical results, conclusions about the new theory are constructed. Excellent predictive capability is demonstrated due to the presence of memory-dependent derivative, gravitational field and kernel functions. A comparative study between isotropic and transversely isotropic material is also demonstrated. [ABSTRACT FROM AUTHOR]

We define the wavelet and the continuous wavelet transform in the framework of the Lebedev–Skalskaya transform (LSCWT) and discuss some operational results. Using these results, we obtain the Parseval's type relation and a reconstruction formula for LSCWT. Moreover, we investigate the composition of LSCWTs and derive its Parseval's type relation and reconstruction formula. [ABSTRACT FROM AUTHOR]

Let ξ ∈ R , and ρ i ∈ R with 0 < | ρ i | < 1 for 1 ≤ i ≤ 3 . For an expanding real matrix M = ρ 1 - 1 0 ξ 0 ρ 2 - 1 - ξ 0 0 ρ 3 - 1 ∈ M 3 (R) and an integer digit set D = { (0 , 0 , 0) t , (1 , 0 , 0) t , (0 , 1 , 0) t , (0 , 0 , 1) t } ⊂ Z 3 , let μ M , D be the self-affine measure defined by μ M , D (·) = 1 | D | ∑ d ∈ D μ M , D (M (·) - d) . In this paper, we prove that if ρ 1 = ρ 2 , then L 2 (μ M , D) admits an infinite orthogonal set of exponential functions if and only if | ρ i | = (p i / q i) 1 r i for some p i , q i , r i ∈ N + with gcd (p i , q i) = 1 and 2 | q i , i = 1 , 2 . In particular, if ρ 1 , ρ 2 , ρ 3 ∈ { p q : p , q ∈ 2 Z + 1 } and ρ 1 = ρ 2 , then there exist at most 4 mutually orthogonal exponential functions in L 2 (μ M , D) , and the number 4 is the best. [ABSTRACT FROM AUTHOR]

In this paper, we focus on how to better represent video data in the field of action recognition. We propose a new image feature named phase spectrum reconstruction map to facilitate action recognition, which extracts the contour features of RGB frame images from a video clip beneficial for action recognition. We demonstrate the effectiveness of such a feature with ablation experiments using the channel-based feature fusion method and two-stream method. Also, we verify that the reconstructed map does contain motion-related features and can be learned by convolutional neural networks only using the reconstructed map as input features. Our method is trained and evaluated using the benchmark datasets HMDB-51 and UCF-101, and both show significant improvements over other methods without adding the reconstructed map features. [ABSTRACT FROM AUTHOR]

Accurately predicting the future trend of a time series holds immense importance for decision-making and planning across various domains, including energy planning, weather forecasting, traffic warning, and other practical applications. Recently, deep learning methods based on transformers and time convolution networks (TCN) have achieved a surprising performance in long-term sequence prediction. However, the attention mechanism for calculating global correlation is highly complex, and TCN methods do not fully consider the characteristics of time-series data. To address these challenges, we introduce a new learning model named wavelet-based Fourier-enhanced network model decomposition (W-FENet). Specifically, we have used trend decomposition and wavelet transform to decompose the original data. This processed time-series data can then be more effectively analyzed by the model and mined for different components in the series, as well as capture the local details and overall trendiness of the series. An efficient feature extraction method, Fourier enhancement-based feature extraction (FEMEX), is introduced in our model. The mechanism converts time-domain information into frequency-domain information through a Fourier enhancement module, and the obtained frequency-domain information is better captured by the model than the original time-domain information in terms of periodicity, trend, and frequency features. Experiments on multiple benchmark datasets show that, compared with the state-of-the-art methods, the MSE and MAE of our model are improved by 11.1 and 6.36% on average, respectively, covering three applications (i.e. ETT, Exchange, and Weather). [ABSTRACT FROM AUTHOR]

In the multidimensional Euclidean space, except the classical real Hardy space, there are numerous product ones. We associate with each of them a class of functions related to the known variations and new ones. Such a characterization is fulfilled by means of the integrability of the Fourier transform [ABSTRACT FROM AUTHOR]

Let G be a compact group and G/H a homogeneous space where H is a closed subgroup of G. Define an operator T H : C (G) → C (G / H) by T H f (t H) = ∫ H f (t h) d h for each t H ∈ G / H . In this paper, we extend T H to a norm-decreasing operator between L p -spaces with a vector measure for each 1 ≤ p < ∞ . This extension will be used to derive properties of invariant vector measures on G/H. Moreover, a definition of the Fourier transform for L p -functions with a vector measure is introduced on G/H. We also prove the uniqueness theorem and the Riemann–Lebesgue lemma. [ABSTRACT FROM AUTHOR]

The aim of this work is to derive a symbol calculus on L 2 (R n) for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be positive definite. What relates these two parts is the powerful method of diagonalization of a normal Hausdorff operator elaborated earlier by the second named author. [ABSTRACT FROM AUTHOR]

Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2, which is used in the Walsh analysis. The role of harmonics is played by some analogues of the classical Walsh functions. The concept of Fourier transform is introduced, and the Poisson summation formula, Plancherel theorem, Vilenkin-Chrestenson formulas and so on are proved. A kind of analogue of the Schwartz class is studied. This class consists of functions such that both the function itself and its Fourier transform have compact support. [ABSTRACT FROM AUTHOR]

The Dirichlet problem for a multidimensional elliptic system in the half-space is considered. With the help of the Fourier transform, the problem of the solvability of the problem is reduced to the study of a second-order partial differential equation. [ABSTRACT FROM AUTHOR]

In this work, we propose a morphologic analysis by means of the construction of 3D models of the SARS-CoV-2 VP (viral particles) with algorithms in Python and Matlab based on the processing of frames. To this aim, we simulate the Takeda method to induce periodicity and apply the Fourier transform to obtain the phase of objects under analysis. To this aim, we analyze several research works focused on infected tissues by SARS-CoV-2 virus culture cells, highlighting the obtained medical images of the virus from microscopy and tomography. We optimize the results by performing image processing (segmentation and periodic noise removal) in order to obtain an accurate ROI (Region of Interest) segmentation containing only information on SARS-CoV-2 cells. We apply our algorithm to these images (3D tomographic medical images) to simulate the Takeda method (which also filters the image), considering the periodicity induced by us in the image to carry out a phase unwrapping process. Finally, we use the image phase to focus on the body, center (RNA, Protein M-N), and spikes (Protein S) of the SARS-CoV-2 cells to identify them as characteristic elements of the SARS-CoV-2 virion morphology to build a 3D model based only in the metadata of clinical studies on cell cultures. The latter results in the construction of a mathematical, physical, biological, and numerical model of the SARS-CoV-2 virion, a tool with volumes, or 3D non-speculative or animated models, based only on medical images (3D tomography) in clinical tests, faithful to the virus. [ABSTRACT FROM AUTHOR]

For the fourth-order partial differential equation proposed by Fushchich for mathematical description of heat and mass transfer processes, the class of uniqueness and the class of correctness of the Cauchy problem have been established. Using the Fourier transform, a solution to the Cauchy problem is found in an explicit analytical form. It is shown that the new equation does not improve the properties of the classical heat conduction equation, but it allows one to describe heat transfer processes taking into account relaxation effects within the framework of the Galileo-invariant mathematical theory of transfer. [ABSTRACT FROM AUTHOR]

Given a one-parameter family of continuous linear operators , with , we consider the optimal recovery of the values of on the whole space by approximate information on the values of , where runs over a compact set and . We find a family of optimal methods for recovering the values of . Each of these methods uses approximate measurements at no more than two points in and depends linearly on these measurements. As a corollary, we provide some families of optimal methods for recovering the solution of the heat equation at a given moment of time from inaccurate measurements on other time intervals and for solving the Dirichlet problem for a half-space on a hyperplane by inaccurate measurements on other hyperplanes. The optimal recovery of the values of from the indicated information reduces to finding the value of an extremal problem for the maximum with continuum many inequality-type constraints, i.e., to finding the exact upper bound of the maximized functional under these constraints. This rather complicated task reduces to the infinite-dimensional problem of linear programming on the vector space of all finite real measures on the -algebra of Lebesgue measurable sets in . This problem can be solved by some generalization of the Karush–Kuhn–Tucker theorem, and its significance coincides with the significance of the original problem. [ABSTRACT FROM AUTHOR]

We find explicit expressions for optimal recovery methods in the problem of recovering the values of continuous linear operators on a Sobolev function class from the following information: The Fourier transform of functions is known approximately on some measurable subset of the finite-dimensional space on which the functions are defined. As corollaries, we obtain optimal methods for recovering the solution to the heat equation and solving the Dirichlet problem for a half-space. [ABSTRACT FROM AUTHOR]

In this note we study character sheaves for graded Lie algebras arising from inner automorphisms of special linear groups and Vinberg's type II classical graded Lie algebras. [ABSTRACT FROM AUTHOR]

Many generalizations of the Fourier transform (FT) have been introduced in the literature, including fractional Fourier transform (FrFT) and linear canonical transform (LCT). This paper aims to extend the discrete combined Fourier transform (DCFT) to the FrFT domains and investigate its applications in image enhancement. The DCFT of a signal is a linear combination of the forward and inverse DFTs of the signal. The discrete combined fractional Fourier transform (DCFrFT) introduced in this work can be seen as a generalization of the of DCFT to the FrFT domains. Simulation results of the proposed DCFrFT and the original signal recovery are given to validate the proposed approach. To present an example of application of the proposed combined transform, we propose a methodology for image enhancement in which the root filtering (RF) technique is extended to the DCFrFT domains. Here the proper selection of the angle parameter of the proposed DCFrFT and the RF parameter are crucial in the enhancement process. For this purpose, simulation results for different values of the RF parameter are carried out and its appropriate value is chosen based on the subjective evaluation. It is seen here that the enhancement approach in the proposed transform domain can improve degraded images when an appropriate RF parameter is used. [ABSTRACT FROM AUTHOR]

A Fourier multiplier related to Mittag-Leffler function is introduced. We prove that our multiplier is radial on R n and generalizes the Bessel function. Furthermore, we study the L 2 boundedness of the related Mittag-Leffler maximal function, the Littlewood–Paley g-function, and the discrete singular integral operator. We prove that the three operators are bounded on L 2 (R n) . In addition, our formulation of the introduced Mittag-Leffler maximal function is a solution of a diffusion equation. Our results generalize previously known results. [ABSTRACT FROM AUTHOR]

Image segmentation is one of the most challenging and difficult tasks in digital image processing. It has many medical applications such as cancerous tumors segmentation, organ segmentation, or abnormalities segmentation. Recent techniques combining convolution-based models and transformers are proposed for automatic medical segmentation tasks. These techniques achieve good results but require much time and resources. In this paper, we propose a new model to segment medical images which combines CNNs and frequency transformers in a parallel way to minimize the number of parameters and to reduce computation time. This work presents a powerful model, composed of two main branches, able to learn global-local feature interactions which are currently in a medical image. The first branch based on Frequency Transformer (FT) employs Fourier Transform instead of multi-head attention to capture global dependencies. While a no-deeper convolutional neural network (CNN) is employed to get rich local information. With a small number of parameters, the proposed model was tested on many public medical image databases and achieves state-of-the-art results for lesion/tumor segmentation tasks. [ABSTRACT FROM AUTHOR]

From the beginning of COVID-19 pandemic, numerous mutants of SARS-CoV-2 have since been evolved owing to high transmissibility and virulence. Due to the limited effectiveness of previously imposed vaccines and preventive therapies, these strains are still causing concern. This paper proposes comparative evaluation of three novel genomic signal processing-based methods employing discrete wavelet decomposition with lifting (DWT), discrete Fourier transform (DFT), and singular value decomposition (SVD) for the classification of emerging SARS-CoV-2 variants utilizing feature extraction from collected SARS-CoV-2 variants acquired from the NCBI virus database. The efficiency and accuracy of the proposed alignment-free algorithms have been tested using three Coronavirus datasets including human Coronavirus (HCoV), SARS-CoV-2 variants (CoV-Variants and Omicron). The viral nucleotide sequences which are converted into numerical representation leveraging purine-pyrimidine mapping, DNA walk & Z-curve are fed into DWT, SVD, & DFT processors, respectively. In the approach with DWT, the second-generation wavelet transform employs two best wavelet bases Daubechies (Db) and Biorthogonal (Bior) based on the validation of the HCoV dataset for the feature extraction of the CoV-Variants dataset. Various machine learning algorithms, such as Support Vector Machine, K-nearest neighbors, and ensemble, are used to classify the virus strains and evaluate the efficacy of the algorithm. Finally, hyper-parametric tuning is done utilizing the Bayesian optimization technique to select the best fit model for KNN and SVM. The proposed algorithm has successfully classified the CoV-Variants dataset with an average accuracy of 98.76% utilizing the DWT, DFT, and SVD, while the best-achieved accuracy for this dataset is 98.9% using the DWT technique employing purine–pyrimidine mapping. The best-achieved accuracy rate for predicting Omicron is 99.8% using SVD-based technique. The best-obtained accuracy for HCoV dataset is 100% resulted in all three methods. [ABSTRACT FROM AUTHOR]

Fault diagnosis of analog circuits is a classical problem, and its difficulty lies in the similarity between fault features. To address the issue, an end-to-end mutually exclusive autoencoder (EEMEAE) fault diagnosis method for analog circuits is proposed. In order to make full use of the advantages of Fourier transform(FT) and wavelet packet transform(WPT) for extracting signal features, the original signals processed by FT and WPT are fed into two autoencoders respectively. The hidden layers of the autoencoders are mutually exclusive by Euclidean distance restriction. And the reconstruction layer is replaced by a softmax layer and 1-norm combined with cross-entropy that can effectively enhance the discriminability of features. Finally, the learning rate is adjusted adaptively by the difference of loss function to further improve the convergence speed and diagnostic performance of the model. The proposed method is verified by the simulation circuit and actual circuit and the experimental results illustrate that it is effective. [ABSTRACT FROM AUTHOR]

We consider estimates for the Fourier transform of measures concentrated on smooth surfaces given by the graph of a smooth function with simple Arnold singularities such that both principal curvatures of the surface vanish at some point. We prove that if the multiplicity of the critical point of the function does not exceed , then the Fourier transforms of the corresponding surface measures belong to for any . Note that for any smooth surface the Fourier transform of a nontrivial surface measure with compact support does not belong to ; i.e., the -estimate obtained is sharp. Moreover, there exists a function with an singularity (the multiplicity of the critical point of the function is equal to ) such that the Fourier transform of the corresponding surface measure does not belong to , which shows the sharpness of the results for the multiplicity of the critical point. [ABSTRACT FROM AUTHOR]

A major application of the inverse scattering and tomography methods is imaging all types of structural, physical, chemical and biological features of matter. The term vortex beam refers to a beam of electromagnetic radiation, electrons, photons or others—whose phase changes in corkscrew-like manner along the direction of propagation. The paper is devoted to the use of scalar Bessel beams of integer and fractional mode for the reconstruction of scattering potential. In practical applications, one naturally deals with Bessel beams truncated in the radial direction. The inversion formula for truncated Bessel beams is also obtained. Instead of the conventional Fourier diffraction theorem (Kak and Slaney in Principles of computerized tomographic imaging, SIAM, New York, 2001), the relations connecting the scattered field and the scattering potential in the Fourier space are obtained in the explicit form. [ABSTRACT FROM AUTHOR]

The thermoreflectance technique is one of the few methods which can measure thermal diffusivity of thin films as thin as 100 nm or thinner in the cross-plane direction. The thermoreflectance method under rear-heat front-detect configuration is sometimes called ultrafast laser flash method because of its similarity to laser flash method. Up to now it has typically only been possible to attempt to evaluate the interfacial thermal resistance between the thin films by preparing and measuring several samples with different thicknesses. In this study, a method to directly determine interfacial thermal resistance by a single measurement of a thin film on substrate is represented, by analyzing the shape of thermoreflectance signals with analytical solutions in frequency domain and time domain. Thermoreflectance signals observed from metallic thin films on sapphire substrate with different thickness steps were analyzed by Fourier analysis and fitted by analytical equations with four parameters: heat diffusion time across the first layer, ratio of virtual heat sources, characteristic time of cooling determined by interfacial thermal resistance and relative amplitude of the signal. Interface thermal resistance between the thin film and substrate was able to be determined reliably with smaller uncertainty. [ABSTRACT FROM AUTHOR]

Influenced by PT operators (parity and time reversal operators) the general conjugations acting in the L 2 space on the real line with the Lebesgue measure are studied. The behaviour of conjugations with respect to multiplication operators and translations is investigated. We consider conjugations defined as the convolution of the Fourier transform of a given unimodular (even) function and test functions. We characterize all conjugations commuting (intertwining) with all translation operators on the real line. The conjugations leaving kernels of Toeplitz operators (a generalization of model spaces) on H 2 space on the upper half plane invariant are also characterized. Analogous results are obtained for conjugations and kernels of Wiener–Hopf operators. [ABSTRACT FROM AUTHOR]