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We show how the octonion Fourier transform can be embedded and studied in Clifford geometric algebra of three‐dimensional Euclidean space Cl(3,0)$$ Cl\left(3,0\right) $$. We apply a new form of dimensionally minimal embedding of octonions in geometric algebra that expresses octonion multiplication nonassociativity with a sum of up to four (individually associative) geometric algebra product terms. This approach leads to new polar representations of octonion analytic signals and signal reconstruction formulas. [ABSTRACT FROM AUTHOR]

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]

In this experimental study, we are presenting the ability of laser-induced plasmas with successive pulsation to identify combustion instabilities (CI) of a premixed lab-scale combustor. An acoustic disturbance equivalent to a shockwave perturbation is generated in the main air supply line of a swirled injector prior to the fuel addition by focusing nanosecond laser pulses of 1.6 W average power at 10 Hz. The shockwaves are attenuated to be strong pressure waves when reaching the combustor and impact the pressure field for short periods. After plasma breakdowns, the system returns back to its original state after 4 ms once the added acoustic energy has been fully dissipated. Given a set geometry, it is observed that the laser-induced breakdown amplifies the characteristic frequency peaks of the combustor system when actuated in cold flow. Furthermore, when applied to reacting flows, the pulsating acoustic perturbations impact the pressure fluctuation in the combustor, e.g., reducing the amplitude of the primary characteristic frequency peak at certain conditions. The identification of the main instability modes thanks to the plasma shockwave provides proof of the potential use of this novel diagnosis strategy in various and complex combustion systems. [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]

Using an asymptotic method, this article deals with flexural-gravity wave scattering with undulating bottom topography, including the effect of uniform currents. The interest in this problem lies in developing second-order solutions using the Fourier transform, which minimises the error gap between first and second-order solutions. The present method allows the physical processes involved in the sea-bed topography, uniform current, plate-covered surface, and wave interaction to be studied. Specifically, we observe Bragg resonance between the flexural-gravity waves and the bottom ripples, which are associated with the reflection of incident wave energy. We examine the effects of wave current and emphasise how crucial the asymptotic expansion method is to the emergence of the current response. We demonstrate that bottom topography dominates the effects of Bragg resonance for depth Froude numbers valued at 0.8 or less. Further, most reflected wave components have their frequencies shifted by the current, and wave action conservation causes reflected wave energy to be enhanced for following currents. Using the Joint North Sea Wave Observation Project spectrum and the discrete Fourier transform, the theory derived in the frequency domain is shown in the time domain to analyse wave propagation through the whole system. • A mathematical model to study flexural-gravity wave scattering with undulating bottom in the presence of uniform current. • The problem is studied using an asymptotic approach under linear water wave theory. • Develop the second-order solutions using the Fourier transform technique. • Identify the existence of Bragg resonance between the flexural-gravity waves and the bottom ripples. • The frequency domain analysis is illustrated in the time domain using JONSWAP. [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]

In this study, our primary emphasis lies in examining and exploring the decay rate of solutions of a Lord‐Shulman thermoelastic Timoshenko model with microtemperature impact. Through the application of the energy method within the Fourier space and subject to appropriate assumptions, we prove that the decay rate of the solution is of the form (1+t)−18$$ {\left(1+t\right)}^{-\frac{1}{8}} $$ in the case of absence of mechanical damping term. [ABSTRACT FROM AUTHOR]

Ocean observation has advanced rapidly in recent decades due to its crucial role in resource exploration and scientific research, with the Doppler factor being widely utilized. However, the precision of Doppler estimation is frequently constrained by frequency resolution. Traditional frequency estimation methods using single-tone signals face considerable challenges with low accuracy and poor robustness. In response, this paper introduces a novel Doppler-sensitive Acoustic Frequency Comb (AFC) for estimating the Doppler factor, enabling multiple measurements with a single transmission and reception of the signal. The proposed Combined Uneven Uncertainty (CUU) method based on AFC achieves a bias of less than 1.1x10-5, significantly surpassing the optimal result of 3.2x10-5 attained by other frequency estimation methods in the absence of noise. Compared to traditional single-tonemethods, the AFC approach improves spectral leakage performance and enhances estimation accuracy without increasing computational complexity. Experimental results demonstrate that the CUU method realizes a difference performance of less than 3.4x10-6, notably lower than that of 3.2x10-5 induced by coherent spectral leakage in fast Fourier Transform (FFT). [ABSTRACT FROM AUTHOR]

The teacher–student framework has been employed in unsupervised domain adaptation, which transfers knowledge learned from a labeled source domain to an unlabeled target domain. However, this framework suffers from two serious challenges: the domain gap, causing performance degradation, and noisy teacher pseudo-labels, which tend to mislead students. In this paper, we propose a Consistent and Contrastive Teacher with Fourier Transform (CCTF) method to address these challenges for high-performance cross-domain object detection. To mitigate the negative impact of domain shifts, we use the Fourier transform to exchange the low-frequency components of the source and target domain images, replacing the source domain inputs with the transformed image, thereby reducing domain gaps. In addition, we encourage the localization and classification branches of the teacher to make consistent predictions to minimize the noise in the generated pseudo-labels. Finally, contrastive learning is employed to resist the impact of residual noise in pseudo-labels. After extensive experiments, we show that our method achieves the best performance. For example, our model outperforms previous methods by 3.0% on FoggyCityscapes. [ABSTRACT FROM AUTHOR]

At present, computationally intensive numerical weather prediction systems based on physics equations are widely used for short-term weather forecasting. In this paper, we investigate the potential of accelerating the Weather Research and Forecasting (WRF-ARW) model using machine learning techniques. Two main approaches are considered. First, we assess the viability of complete replacing the numerical weather model with deep learning models, capable of predicting the full range forecast directly from basic initial data. Second, we consider a “super-resolution” technique involving low-resolution WRF computation and a machine learning based downscaling using coarse-grid forecast for conditioning. The process of downscaling is intrinsically an ill-posed problem. In both categories, several prominent and promising machine learning methods are evaluated and compared on real data from a variety of sources. for the Moscow region Namely, in addition to the ground truth WRF forecasts that were utilized for training, we compare the model predictions against ERA5 reanalysis and measurements from local weather stations. We show that deep learning approaches can be successfully applied to accelerate a numerical model and even produce more realistic forecasts in other aspects. As a practical outcome, this study offers empirically validated guidance for the selection and application of deep learning methods to accelerate the computation of detailed short-term atmospheric forecasts tailored to specific needs. [ABSTRACT FROM AUTHOR]

This work focuses on the asymptotic stability of nonlocal diffusion equations in N$$ N $$‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove L2$$ {L}^2 $$ and L∞$$ {L}^{\infty } $$‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if l>p$$ l>\left|p\right| $$, then the solution u(t,x)$$ u\left(t,x\right) $$ converges globally to the trivial equilibrium time‐exponentially. If l=p$$ l=\left|p\right| $$, then the solution u(t,x)$$ u\left(t,x\right) $$ converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when r>q$$ r>\left|q\right| $$, the solution u(t,x)$$ u\left(t,x\right) $$ converges globally to the positive equilibrium time‐exponentially, and when r=q$$ r=\left|q\right| $$, the solution u(t,x)$$ u\left(t,x\right) $$ converges globally to the positive equilibrium time‐algebraically. Here, l,p,r$$ l,p,r $$, and q$$ q $$ are the coefficients of each term contained in the linear part of the nonlinear term f$$ f $$. All convergence rates above are L2$$ {L}^2 $$ and L∞$$ {L}^{\infty } $$‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations. [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]

The weighted fractional Fourier transform (WFRFT) has always been considered a development of the discrete fractional Fourier transform (FRFT). This paper points out that the WFRFT is a discrete FRFT of eigenvalue decomposition, which will change the consistent understanding of the WFRFT. Extended definitions based on the WFRFT have been proposed and widely used in information processing. This paper proposes a unified framework for extended definitions, and existing extended definitions can serve as special cases of this unified framework. In further analysis, we find that the existing extended definitions are deficient. With the help of a unified framework, we systematically analyze the reasons for the deficiencies. This has great guiding significance for the application of the WFRFT and its extended definitions. [ABSTRACT FROM AUTHOR]

In the pursuit of advancing COVID-19 diagnosis through imaging, this paper introduces a novel approach utilizing adaptive Fourier neural operators (AFNO) for the analysis of high-resolution computed tomography (HRCT) chest images. The study population comprised 395 patients with 181,106 labeled high-resolution COVID-19 CT images from the HRCTCov19 dataset, categorized into four classes: ground glass opacity (GGO), crazy paving, air space consolidation, and negative for COVID-19. The methods included image preprocessing, involving resizing and normalization, followed by the application of the AFNO model, which enables efficient token mixing in the Fourier domain independent of input resolution. The model was trained using the Adam optimizer with a learning rate of 1 × 10−⁴ and evaluated using metrics such as accuracy, precision, recall, and F1 score. The results demonstrate AFNO's superior performance in few-shot segmentation tasks over traditional self-attention mechanisms, achieving an overall accuracy of 94%. Specifically, the model showed high precision and recall for the GGO and negative classes, indicating its robustness and effectiveness. This research has significant implications for the development of AI-powered diagnostic tools, particularly in environments with limited access to high-quality imaging data and those where computational efficiency is critical. Our findings suggest that AFNO could serve as a powerful model for analyzing HRCT images, potentially leading to improved diagnosis and understanding of COVID-19, representing a critical step in combating the pandemic. [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]

The analysis of ground-penetrating radar (GPR) data is of vital importance for detecting various subsurface features that might manifest as hyperbolic peaks, which are indicators of a buried object or grayscale variation in the case of contrast in the soil texture. This method focuses on identifying exaggerated patterns through a series of image-processing steps. Two GPR images are initially read and preprocessed by extracting channels, flipping, and resizing. Then, specific regions of interest (ROIs) are cropped, and the Fourier transform is further applied to turn them into the frequency domain. With the help of their frequency signatures, these patterns are extracted from the images, and binary masks are constructed to obtain features of interest. These masked images were reconstructed and merged to make hyperbolic features visible. Finally, Local Binary Pattern (LBP) analysis is used to emphasize these hyperbolic peaks, thereby facilitating their recognition across the whole image. The proposed approach improves the detection of performance subsurface features in GPR data; hence, it is an important tool for geophysical surveys and other related applications. The results prove the high performance of the proposed procedure in improving GPR image characteristics. [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]

This paper introduces a groundbreaking method, Homotopy-based Fourier transform, integrating Fourier transform and Homotopy perturbation for refined nonlinear problem-solving. The modification enhances solution technique efficiency, notably accelerating convergence, particularly in solving the Korteweg–de Vries equation. Demonstrating versatility, the method effectively addresses ordinary and partial differential equations, showcasing its applicability across diverse mathematical scenarios. Moreover, the approach is extended to nonlinear dynamical systems, illustrating its robustness in handling complex dynamic behaviors. This method proves especially suitable for highly nonlinear differential equations, offering an efficient and effective tool for scientists and engineers dealing with intricate mathematical models. By significantly improving convergence rates, the Homotopy-based Fourier transform stands out as a valuable asset in unraveling the complexities of nonlinear systems across various scientific and engineering applications. [ABSTRACT FROM AUTHOR]

Modern electricity networks are facing significant challenges in terms of power quality due to the increasing integration of power electronics. Even though low order harmonic control has largely been achieved, the emergence of supraharmonics is becoming a new cause for concern. This topic has gained interest in the past decade since power quality issues have become important due to the proliferation of highly sensitive electrical and electronic equipment. This special case of harmonics is mainly due to the power electronic converters utilized in industrial as well as residential applications, including electromobility, motor drive systems, and photovoltaic installations. While the adverse effects of supraharmonics have been pointed out in numerous studies and intensive research is underway on the crucial subject of supraharmonic measurements, the parameters affecting their levels have not received adequate attention. We attempted to shed more light on this important issue in the specific case of a small grid-connected PV system. In particular, the supraharmonic emission levels of the system were investigated through experimental measurements, and useful conclusions on the impact of specific environmental factors were derived. [ABSTRACT FROM AUTHOR]

The study of animal sounds in biology and ecology relies heavily upon time–frequency (TF) visualisation, most commonly using the short‐time Fourier transform (STFT) spectrogram. This method, however, has inherent bias towards either temporal or spectral details that can lead to misinterpretation of complex animal sounds. An ideal TF visualisation should accurately convey the structure of the sound in terms of both frequency and time, however, the STFT often cannot meet this requirement. We evaluate the accuracy of four TF visualisation methods (superlet transform [SLT], continuous wavelet transform [CWT] and two STFTs) using a synthetic test signal. We then apply these methods to visualise sounds of the Chagos blue whale, Asian elephant, southern cassowary, eastern whipbird, mulloway fish and the American crocodile. We show that the SLT visualises the test signal with 18.48%–28.08% less error than the other methods. A comparison between our visualisations of animal sounds and their literature descriptions indicates that the STFT's bias may have caused misinterpretations in describing pygmy blue whale songs and elephant rumbles. We suggest that use of the SLT to visualise low‐frequency animal sounds may prevent such misinterpretations. Finally, we employ the SLT to develop 'BASSA', an open‐source, GUI software application that offers a no‐code, user‐friendly tool for analysing short‐duration recordings of low‐frequency animal sounds for the Windows platform. The SLT visualises low‐frequency animal sounds with improved accuracy, in a user‐friendly format, minimising the risk of misinterpretation while requiring less technical expertise than the STFT. Using this method could propel advances in acoustics‐driven studies of animal communication, vocal production methods, phonation and species identification. [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]

In this paper, we study the structural properties of Nevanlinna measures, that is, Borel measures that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually performed via the polydisc. [ABSTRACT FROM AUTHOR]

For the octonion Fourier transform, we establish the real Paley‐Wiener theorem. It relates the mean of derivatives of a function with the support of its octonion Fourier transform via limm→∞‖∂mαf‖L2(ℝ3,핆)1/m=(2π)α‖wα‖L∞(suppF핆{f},핆)for any octonion‐valued Sobolev function f∈H∞(ℝ3,핆). [ABSTRACT FROM AUTHOR]

Time series forecasting has played an important role in different industries, including economics, energy, weather, and healthcare. RNN-based methods have shown promising potential due to their strong ability to model the interaction of time and variables. However, they are prone to gradient issues like gradient explosion and vanishing gradients. And the prediction accuracy is not high. To address the above issues, this paper proposes a Fractional-order Lipschitz Recurrent Neural Network with a Frequency-domain Gated Attention mechanism (FLRNN-FGA). There are three major components: the Fractional-order Lipschitz Recurrent Neural Network (FLRNN), frequency module, and gated attention mechanism. In the FLRNN, fractional-order integration is employed to describe the dynamic systems accurately. It can capture long-term dependencies and improve prediction accuracy. Lipschitz weight matrices are applied to alleviate the gradient issues. In the frequency module, temporal data are transformed into the frequency domain by Fourier transform. Frequency domain processing can reduce the computational complexity of the model. In the gated attention mechanism, the gated structure can regulate attention information transmission to reduce the number of model parameters. Extensive experimental results on five real-world benchmark datasets demonstrate the effectiveness of FLRNN-FGA compared with the state-of-the-art methods. [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]

This paper presents the creation and realization of a precision time base (PTB) for mitigating jitter and measuring time interval error (TIE) in digital sampling oscilloscopes (DSO), achieved through the utilization of a field programmable gate array (FPGA). Our proposed method focuses on mitigating jitter through PTB, which involves the sampling of two reference channels having a phase shift of approximately 90∘ (at quadrature) for an accurate timebase correction. Through extensive experimentation and analysis, we were able to achieve a reduction in root mean square (RMS) jitter, minimizing it to around 260 fs. This outcome demonstrates the effectiveness of our approach in enhancing the accuracy and reliability of DSO measurements. Expanding on the application of our previously proposed technique, we demonstrate that the same sampling error correction approach can be utilized for TIE measurement. By sampling a large amount of data at either the rising or falling edge of a sinusoidal signal, TIE can be accurately evaluated. This extension enhances the versatility of our FPGA-based implementation by enabling comprehensive TIE analysis alongside PTB compensation in DSOs. [ABSTRACT FROM AUTHOR]

The recent advancements in vision technology have had a significant impact on our ability to identify multiple objects and understand complex scenes. Various technologies, such as augmented reality-driven scene integration, robotic navigation, autonomous driving, and guided tour systems, heavily rely on this type of scene comprehension. This paper presents a novel segmentation approach based on the UNet network model, aimed at recognizing multiple objects within an image. The methodology begins with the acquisition and preprocessing of the image, followed by segmentation using the fine-tuned UNet architecture. Afterward, we use an annotation tool to accurately label the segmented regions. Upon labeling, significant features are extracted from these segmented objects, encompassing KAZE (Accelerated Segmentation and Extraction) features, energy-based edge detection, frequency-based, and blob characteristics. For the classification stage, a convolution neural network (CNN) is employed. This comprehensive methodology demonstrates a robust framework for achieving accurate and efficient recognition of multiple objects in images. The experimental results, which include complex object datasets like MSRC-v2 and PASCAL-VOC12, have been documented. After analyzing the experimental results, it was found that the PASCAL-VOC12 dataset achieved an accuracy rate of 95%, while the MSRC-v2 dataset achieved an accuracy of 89%. The evaluation performed on these diverse datasets highlights a notably impressive level of performance. [ABSTRACT FROM AUTHOR]

The experimental IR spectra of (S)-4′-(1-methylheptyloxycarbonyl) biphenyl-4-yl 4-[2-(2,2,3,3,4,4,4-heptafluorobutoxy) ethyl-1-oxy]-2-fluorobenzoate in the crystal phase are analyzed with the help of dispersion-corrected density functional theory (DFT+D3) calculations for isolated molecular monomer and dimer models, and a periodic model computed at the extended density functional tight-binding (xTB) level of theory. It is found that the frequency scaling coefficients obtained with the results of the molecular calculations are good matches for the crystal phase, being close to 1. The molecular and periodic models both confirm that varied intra- and intermolecular interactions are crucial in order to reproduce the broadened shape of the experimental band related to C=O stretching; the key factors are the conjugation of the ester groups with the aromatic rings and the varied intermolecular chemical environments, wherein the C=O group that bridges the biphenyl and F-substituted phenyl groups seems particularly sensitive. The C=O stretching vibrations are investigated as a function of temperature, covering the range of the crystal, smectic CA*, smectic C* and isotropic liquid phases. The structure changes are followed based on the X-ray diffraction patterns collected in the same temperatures as the IR spectra. The experimental and computational results taken together indicate that the amount of weak C=O...H-C hydrogen bonds between the molecules in the smectic layers decreases with increasing temperature. [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]

Abstract Reconstructing signals which are embedding spatial patterns such as Electrical resistivity tomography, is a process that should require to reconstruct first the spatial correlation of the damaged signals. This paper proposes an approach that implements an Unstationary Kriging (UNK) to reconstruct the experimental variogram of a damaged synthetic pseudo section within a set of pseudo sections coming from the same survey. We used and compared 02 other simple methods which are Linear Regression (LR) and Ordinary Kriging (OK), to test the hypothesis we formulate to link the experimental variograms coming from the same ERT survey. We implemented the UNK using Discrete Fourier Transforms (DFT) for trend modeling. After an implementation of the hybrid process (UNK) on 02 sets of data which are synthetics, we observed that the LR and the UNK methods present an interest. They both reconstruct signals with a +90% rate of accuracy, but when there is no structure or spatial correlation within the data, the LR is unstable. DFT was also tested alone for reconstruction but was mainly used in this study to help in computing the trends for each set of variographic signals. In the end, we conclude on an evidence that is: the proposed hybrid process is a promising way to reconstruct variographic signals, since we can improve it after more time invested to dig deep into the modeling of each of his components.

In this article, we apply the Fourier transform to prove the Hyers-Ulam and Hyers-Ulam-Rassias stability for the first- and second-order nonlinear differential equations with initial conditions. Additionally, we extend the results to investigate the Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability of these differential equations using the proposed method.

Compass errors can be regarded as a deviation of the vessel from the expected heading. Gyrocompass errors are randomly oscillating in nature, and it is difficult to describe the behaviour of a gyrocompass sufficiently accurately using mathematical relationships. Fibre-optic gyroscopes have no mechanical components, so the variability in their indications has a different nature; the computational processes and inertial sensors used cause certain types of errors. Thus far, compass studies have focused on presenting absolute errors in the time domain. However, compasses exhibit specific characteristics in the frequency domain that affect the amplitude of their deviation. This leads to the issue of identifying the oscillatory spectrum of errors in the operation of such compasses, and how this spectrum is impacted by the dynamic movement of the vessel. We attempt to assess this phenomenon by means of measurements taken on board the training and research vessel M/S NAWIGATOR XXI. The application of a fast Fourier transform allows for calculation of the absolute compass errors in the frequency domain, meaning that the frequency of occurrence of errors can be observed as noise against the background of the useful signal. Our results confirm the value of applying a finite impulse response filter, which is used to filter out noise in the form of absolute compass errors from the useful signal background. The convolution function proposed here considerably extends the possibilities for analysing the signal spectrum in the frequency domain when testing for the accuracy of compass device indications, and enables the elimination of random errors with a low frequency of occurrence..

Image watermarking often involves the use of handheld devices under non-structured conditions for authentication purposes, particularly in the print-cam process where smartphone cameras are used to capture watermarked printed images. However, these images frequently suffer from perspective distortions, making them unsuitable for automated information detection. To address this issue, Cam-Unet, an end-to-end neural network architecture, is presented to predict the mapping from distorted images to rectified ones, specifically tailored for print-cam challenges applied to ID images. Given the limited availability of large-scale real datasets containing ground truth distortions, we created an extensive synthetic dataset by subjecting undistorted images to print-cam attacks. The proposed network is trained on this dataset, using various data augmentation techniques to improve its generalization capabilities. Accordingly, this paper presents an image watermarking system for the print-cam process. The approach combines Fourier transform-based watermarking with Cam-Unet as perspective distortion correction. Results show that the proposed method outperforms existing watermarking approaches typically employed to counter print-cam attacks and achieves an optimal balance between efficiency and cost-effectiveness. [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]

The first part of this work provides explicit solutions for two integral equations; both are solved by means of Fourier transform. In the second part of this paper, sufficient conditions for the existence and uniqueness of the solutions satisfying sandwich constraints for two types of full moment problems are provided. The only given data are the moments of all positive integer orders of the solution and two other linear, not necessarily positive, constraints on it. Under natural assumptions, all the linear solutions are continuous. With their value in the subspace of polynomials being given by the moment conditions, the uniqueness follows. When the involved linear solutions and constraints are positive, the sufficient conditions mentioned above are also necessary. This is achieved in the third part of the paper. All these conditions are written in terms of quadratic expressions. [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]

In this article, we study the recent development of the qualitative uncertainty principle on certain Lie groups. In particular, we consider that if the Weyl transform on certain step-two nilpotent Lie groups is of finite rank, then the function has to be zero almost everywhere as long as the nonvanishing set for the function has finite measure. Further, we consider that if the Weyl transform of each Fourier–Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the nonvanishing set for each Fourier–Wigner piece has finite measure. [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]

This article is devoted to the problem of signal decomposition into periodic and aperiodic components. According to the proposed approach, there is no need to evaluate the aperiodic component as a difference between the total signal of its periodic components. This research aims to create a general analytical approach that combines the Fourier and Maclaurin series methodologies into a single comprehensive series. As a result, analytical expressions for determining deposition coefficients were established for an aperiodic signal with a monoharmonic overlay. Recurrence relations were established to determine the coefficients of this series. These relations allow direct integrations of the obtained values of integrals to be avoided. The evaluated numerical values of the coefficients are also presented graphically and tabulated. It was proven that the values of these coefficients are universal numbers since they do not depend on the period/frequency of oscillations. The reliability of the proposed approach was confirmed by the fact that the evaluated coefficients are equal to the Fourier series coefficients in the case of a periodic signal. Also, for an aperiodic signal, these coefficients were reduced to the coefficients of the Maclaurin series. The usability of the proposed generalized analytical approach for signal decomposition is for control and monitoring systems of processes. [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]