Fourier Techniques Research Articles

Exploring the Influence of Generalized Kernels on Green’s Function in Fractional Differential Equations

The basic purpose of this article is to define the Green’s function in order to provide the solution of fractional differential equations in the presence of general analytic kernel. Using the technique of Laplace and Fourier transforms, we construct the Green’s function for ordinary and partial fractional differential equations. The presented results will provide the generalization of some models existing in the literature. Some examples are also provided to prove the results for some particular cases.

Fuzzy Fourier Approximations for Uncertain Data Theory and Applications

This study explores the integration of fuzzy logic with Fourier series and transforms to address the challenges posed by uncertainty and imprecision in real-world data. By representing uncertain data through fuzzy numbers and applying fuzzy Fourier approximations, this method enhances the accuracy and robustness of signal processing, image reconstruction, and time-series forecasting, particularly in noisy environments. The comparative analysis demonstrates that fuzzy Fourier methods outperform traditional Fourier techniques in handling uncertainty, while recognizing the computational complexity introduced by fuzzification. The study also explores future research directions, including multi-dimensional data processing, hybrid approaches with machine learning, and the use of fuzzy logic in quantum Fourier transforms. These advancements offer promising solutions for improving data analysis in fields like telecommunications, medical imaging, and financial forecasting, where uncertainty is a critical factor.

Experimental Investigation Using Robust Deep VMD-ICA and 1D-CNN for Condition Monitoring of Roller Element Bearing

Abstract A rotor-bearing system experiences numerous vibrations during the operation that frequently degrade performance and endanger operational safety. Roller bearing failure has significant consequences, leading to downtime or even a complete outage of rotating machinery. It is crucial to detect and diagnose incipient bearing defects promptly to ensure optimal operation of the machinery and minimize potential disruptions to the process. Deep Independent Component Analysis is a necessity used in modern condition monitoring to detect bearing failure earlier than it occurs. To address this issue the feasibility of utilizing Deep Independent Component Analysis (ICA) method based on Variational Modal Decomposition (VMD) with one-dimensional convolutional neural network (1D-CNN) to diagnose incipient bearing defect. Fast Fourier Techniques are utilized to extract the vibration signatures of artificially damaged bearings on a newly built test bed. VMD addresses to minimize data noise, allowing data to decompose into various sub-datasets for extraction of incipient defect features. With weak defect characteristic signal and noise interference, the Deep VMD-ICA model and 1D-CNN simplicity improved the accuracy of diagnosis corresponding to the experimental results. Moreover, Deep VMD-ICA with 1D-CNN has additionally demonstrated strong performance in comparison to experimental results and is useful for monitoring the condition of industrial machinery. The results reveal that this fault diagnosis approach is reliable, with a diagnostic accuracy of 98.93% for bearing faults.

Lake gravity anomalies from ICESat-2 laser altimetry and geodetic radar altimetry

In the current most accepted global geopotential model, EGM2008, there are often data gaps in the source data used to compute the model over inland water. As a result, EGM2008 may be less reliable over lakes. Satellite altimetry has the potential to estimate gravity anomalies and update EGM2008 over lakes. Here, we evaluate the first attempt to extract gravity anomalies from ICESat-2 laser altimetry over several medium (100–1000 km2) and large (>1000 km2) lakes and compare them with conventional radar altimetry from CryoSat-2 and SARAL to investigate the performance of ICESat-2 for gravity determination over lakes. Aerial gravimetry from the GRAV-D project over the United States are utilized as the best estimate of the gravity field over the lakes. Gravity determination from altimetry is done using Fast Fourier Techniques (FFT) within a remove-restore geoid-to-gravity approach. The resulting altimetry derived gravity anomalies are then compared to the EGM2008 geoid over each lake with respect to GRAV-D. 18 lakes with area ranging from 108 km2 to 82,220 km2 across the United States were considered. Overall, gravity determination from ICESat-2 provides more reliable estimates than the other two radar altimetry missions. For all considered lakes, the performance of ICESat-2, measured in terms of standard deviation with GRAV-D, is comparable or better than the EGM2008 field over the same lake. Lake Pend Orielle is the best performing case, in which the standard deviation of the ICESat-2 derived gravity field is 2.14 mGal and the standard deviation of the EGM2008 gravity field is 2.66 mGal with respect to the GRAV-D measurements. Over Lake Tahoe, which is surrounded by mountainous terrain, ICESat-2 performs comparably to EGM2008 and captures clear gravity signal related to the lake’s bathymetry, whereas CryoSat-2 produces very unstable results. The method presented here for deriving gravity anomalies from altimetry applied to ICESat-2 laser altimetry data produces results that validate in supplement to the GRAV-D project over medium to large lakes in the United States.

Uniform Convergence of Fourier Series for Hölder Continuous Functions and Functions of Bounded Variation

This article discusses two significant results about the uniform convergence of Fourier series. One is the uniform convergence of the Fourier series for Hölder-α(α>0) continuous functions, while the other is the uniform convergence of the Fourier series for functions that can be expressed as the sum of finitely many continuous monotonic functions. The two results are particularly useful. Because Hölder continuity condition ensures a specific regularity that facilitates the analysis and provides a robust framework for proving uniform convergence and many practical functions can be decomposed into monotonic components, making the theorem broadly applicable. By providing detailed proofs and applications, this study demonstrates the significant implications of uniform convergence in both theoretical and practical contexts. Moreover, this work paves the way for future research in higher-dimensional Fourier analysis, non-periodic functions, and adaptive Fourier techniques, underscoring the ongoing relevance and versatility of Fourier series in modern mathematics. Several well-known results emerge as corollaries of these findings, highlighting the robustness and applicability of these theorems in Fourier analysis.

A comparison between Hankel and Fourier methods for photothermal radiometry analysis

AbstractPhotothermal radiometry has recently been investigated for use in the multidimensional thermal characterization of anisotropic samples. In application, there are two principal thermal models available for such characterization: a Cartesian model for the heat equation, which requires the application of three Fourier transforms to arrive at a solution (dubbed the Fourier technique), and a cylindrical model for the heat equation, which requires the application of a Hankel transform and a single Fourier transform (dubbed the Hankel technique). The Fourier technique allows for three‐dimensional characterization, while the Hankel technique is expected to greatly reduce the computational time required. As these models can be very computationally expensive, the potential to reduce this cost is of great interest. In this work, these multidimensional models are presented after which they are compared for accuracy, computational time, and assumption limitations. It was found that both the Fourier and Hankel techniques could accurately arrive at desired thermal properties, but that the Hankel Technique reduced the computational time by between 100× and 250× depending upon mesh spacings. Accuracy limitations were found as the eccentricity of the heating laser was increased with a less than 13% error being induced from a beam with a 3–1 axis ratio. The Hankel technique shows ideal application in computationally expensive models which employ a relatively circular beam shape.

CMB polarization signal demodulation with a rotating half-wave plate

ABSTRACT Several forthcoming Cosmic Microwave Background polarization experiments will employ a Continuously Rotating Half-Wave Plate (CRHWP), the primary purpose of which is to mitigate instrumental systematic effects. The use of a CRHWP necessitates demodulating the time-ordered data during the early stages of data processing. The standard approach is to “lock in” on the polarization signal using the known polarization modulation frequency and use Fourier techniques to filter out the remaining unwanted components. However, an alternative, less well-studied option is to incorporate the demodulation directly into the map-making step. Using simulations, we compare the performance of these two approaches to determine which is most effective for B-mode signal recovery. Testing the two techniques in multiple experimental scenarios, we find that the lock-in technique performs best over the full multipole range explored. However, for the recovery of the largest angular scales (multipoles, $\ell \lt 100$) we find essentially no difference in the recovery of the signal between the lock-in and map-making approaches, suggesting that a parallel analysis based on the latter approach could represent a powerful consistency check for primordial B-mode experiments employing a CRHWP. We also investigate the impact of a detector-differencing step, implemented prior to demodulation, finding that, in most scenarios, it makes no difference whether differencing is used or not. However, analysing detectors individually allows the point at which information from multiple detectors is combined to be moved to later stages in the analysis pipeline. This presents alternative options for dealing with additional instrumental systematic effects that are not mitigated by the CRHWP.

An Examination of Learning Using Fourier Analysis of Mathematical Models of Consciousness

The concept of consciousness remains quite possibly the ultimate mystery for humanity. It is somehow linked to the ability to learn and experience the external environment. The process of learning is of utmost importance to any species and yet remains largely not understood. While many experts from varying fields throughout history have attempted to quantify and measure learning, all failed to capture the process with a mathematical explanation. The purpose of this paper is to present three different models of consciousness and their related Fourier transform and evaluate the ability of each to capture some of the behavior that is understood about the conscious experience. There is some evidence that the experience of consciousness as a function of time can be measured in an expression of phase. With this hypothesis, the technique of Fourier transform becomes a useful tool to examine a mathematical model of consciousness that can be transformed from the phase plane to the linearization of time. This paper summarizes some current research on consciousness and learning and perception of time. This paper then presents three different models of consciousness represented as a pulse function, Dirac Delta function, and exponential decay function and examines each model utilizing Fourier transform. Finally, this paper concludes that the intersection between psychology, neuroscience and cognitive science can be made by utilizing analysis tools in the field of applied mathematics.

Impact of oil price, economic globalization, and inflation on economic output: Evidence from Latin American oil-producing countries using the quantile-on-quantile approach

This research examines the impact of oil prices, economic globalization, and inflation on the economic output in oil-producing countries in Latin America. We employ advanced time series techniques to achieve precision in findings and reliability of policy inferences. We use cointegration techniques with and without structural breaks because oil prices are highly volatile, and the economies analyzed are unstable in the face of political and social changes. Additionally, we performed a sensitivity analysis using standard quantile and the newly developed quantile-on-quantile regressions to measure the impacts of the covariates on real per capita output. Finally, we use causality Fourier techniques to identify the direction of causal relationships between the series. We found that the oil price, economic globalization, inflation, and output have a long-term equilibrium relationship in the presence of structural breaks. Likewise, quantile models show that the impacts of inflation, economic globalization, and oil prices on economic output are extensively non-linear across the quantile distribution. We found a negative connection between inflation and economic output in the surveyed territories. We propose that those responsible for energy policy in the countries studied should rethink crude oil policies to maximize the benefits of oil exploitation under stagnant globalization and growing inflation.

High-contrast, High-angular-resolution Optical Speckle Imaging: Uncovering Hidden Stellar Companions

We explore the possibility of detecting very faint, very close-in stellar companions using large aperture ground-based telescopes and the technique of optical speckle imaging. We examine the state of high-angular-resolution speckle imaging and contrast levels being achieved using current speckle cameras on the Gemini 8 m telescope. We then explore the use of the modern image reconstruction technique—multiframe blind deconvolution (MFBD)—applied to speckle imaging from the Gemini 8 m telescope. We show that MFBD allows us to measure the flux ratio of the imaged stars to high accuracy and the reconstructed images yield higher precision astrometry. Both of these advances provide a large refinement in the derived astrophysical parameters compared with current Fourier techniques. MFBD image reconstructions reach contrast levels of ∼5 × 10−3, near the diffraction limit, to ∼10−4 about 1.″0 away. At these deep contrast levels with angular limits starting near the 8 m diffraction limit (∼20 mas), most stellar companions to a solar-like stars can be imaged in the optical to near-IR bandpass (320–1000 nm). “To Xanadu we go...”—adapted from S. T. Coleridge.

Two-Dimensional Fourier Transform Ion Cyclotron Resonance Mass Spectrometry of N-Linked Glycopeptides.

Glycosylation is a common modification across living organisms and plays a central role in understanding biological systems and disease. Our ability to probe the gylcome has grown exponentially in the past several decades. However, further improvements to the analytical toolbox available to researchers would allow for increased capabilities to probe structure and function of biological systems and to improve disease treatment. This article applies the developing technique of two-dimensional Fourier transform ion cyclotron resonance mass spectrometry to a glycoproteomic workflow for the standard glycoproteins coral tree lectin (CTL) and bovine ribonuclease B (BRB) to demonstrate its feasibility as a tool for glycoproteomic workflows. 2D infrared multiphoton dissociation and electron capture dissociation spectra of CTL reveal comparable structural information to their 1D counterparts, confirming the site of glycosylation and monosaccharide composition of the glycan. Spectra collected in 2D of BRB reveal correlation lines of fragment ion scans and vertical precursor ion scans for data collected using infrared multiphoton dissociation and diagonal cleavage lines for data collected by electron capture dissociation. The use of similar techniques for glycoproteomic analysis may prove valuable in instances where chromatographic separation is undesirable or quadrupole isolation is insufficient.

A nonlocal photoacoustic effect with variable thermal conductivity of semiconductor material subjected to laser heat source

This study explores the behavior of laser-exposed photo-excited carriers, investigating the propagation of photoacoustic waves in the thermoelastic domain. It also delves into the theoretical generation of surface acoustic waves in semiconductors through photo-thermoelastic processes. The research considers the interaction between thermomechanical and acoustic waves in a nonlocal medium with temperature-dependent thermal conductivity. Unlike relying on electron–phonon or electron-hole thermalization processes, photoacoustic waves here result from thermoelastic stress induced by the laser-induced temperature increase. The investigation accounts for the optical, mechanical, and thermoelastic properties of nanoscale semiconductor materials. Predictions of photoacoustic signals are derived by solving a combined thermal diffusion issue and a thermoelastic problem, using Laplace and Fourier transforms in the mathematical model. Numerical solutions encompass various physical fields within the time domain using the inversion technique of Laplace and Fourier transforms, such as thermal, acoustic, mechanical, and carrier density diffusion. The study evaluates and compares the influences of thermal memory and thermal conductivity presenting visual representations.

Simulation of time-intensity curve based on k-space filling in breast dynamic contrast-enhanced three-dimensional magnetic resonance imaging.

This study elucidated the effects of a three-dimensional k-space trajectory incorporating the partial Fourier (PF) technique on a time-intensity curve (TIC) in a dynamic contrast-enhanced magnetic resonance imaging of a typical malignant breast tumor using a digital phantom. Images were obtained from the Cancer Imaging Archive Open Data for Breast Cancer, and 1-min scans with high temporal resolution were analyzed. The order of the k-space trajectory was set as Linear (sequential), Low-High (centric), PF (62.5%; Z-, Y-, and both directions), and Low-High Radial. k0 (center of the k-space) timing and TIC shape were affected by the chosen k-space trajectory and implementation of the PF technique. A small TIC gradient was obtained using a Low-High Radial order. If the k-space filling method (particularly the radial method) produces a gentle TIC gradient, misinterpretation could arise during the assessment of tumor malignancy status.

As seis pinturas de Massimo Campigli do MAC USP discutidas sob o olhar das artes plásticas e dos resultados de análises com técnicas de imagens e espectroscópicas

This article focuses on the discussion of the results obtained with non-invasive image and spectroscopic analyzes (to determine chemical elements and compounds) on the six paintings by the German artist, naturalized Italian, Massimo Campigli, that belong to the collection of the Museum of Contemporary Art of the University of São Paulo (MAC USP). Aspects of the artist’s trajectory, the plastic solutions of the artworks, along with the analyzes made with the spectroscopic techniques of energy dispersive X-ray fluorescence (ED-XRF), Raman, and Fourier transform infrared (FTIR), and the imaging techniques of visible photography (VIS), raking light (RAK), transmitted light (TRANS), visible ultraviolet fluorescence (UVF), infrared reflectography (IRR), and radiography (RAD) offer important information on the artist and on his work. These non-invasive analyzes were carried out in the museum itself with portable equipment from the Laboratory of Archaeometry and Applied Sciences to Cultural Heritage (LACAPC) of the Physics Institute (IF) of USP. Overall, the results obtained corroborate with Campigli’s testimonies about his artistic practice, as well as those of people who saw him working in his studio. His palette choice, composed by natural and synthetic pigments, mostly whites and paints with earthy tones, and his creative steps of creating and recreating the paintings, reusing canvases many times, were proven to be true with the analytical investigations.

Analyzing India's coal, natural gas, and biomass energy consumption: Evidence from a Fourier technique to promote sustainable development

AbstractInalignment with the United Nations Sustainable Development Goals (UN‐SDGs),which pinpoint vital aspects such as SDG‐7 and SDG‐8, this study delves intothe impact of biomass energy on economic growth in the Indian context. It alsoconsiders the roles of fossil fuels, specifically natural gas and coal. Toaccomplish this, a comprehensive analysis is initiated, making use of annualdata spanning from 1975 to 2022. A unique Fourier ARDL test serves as a centralanalytical tool alongside the Fourier Toda Yamamoto test. The empirical resultsunveil the presence of co‐integration among the variables scrutinized. Moreover,it becomes evident that coal and biomass make positive contributions toeconomic growth, whereas natural gas appears to exert a negative influence oneconomic growth. To validate these findings, a Fourier‐based Toda‐Yamamoto testis applied, demonstrating that natural gas, coal, and biomass possesspredictive capabilities concerning economic growth. These discoveries carrysubstantial implications for comprehending the factors influencing economicgrowth in India.

On the use of machine learning and data-transformation methods to predict hydration kinetics and strength of alkali-activated mine tailings-based binders

The escalating production of mine tailings (MT), a byproduct of the mining industry, constitutes significant environmental and health hazards, thereby requiring a cost-effective and sustainable solution for its disposal or reuse. This study proposes the use of MT as the primary ingredient (≥70%mass) in binders for construction applications, thereby ensuring their efficient upcycling as well as drastic reduction of environmental impacts associated with the use of ordinary Portland cement (OPC). The early-age hydration kinetics and compressive strength of MT-based binders are evaluated with an emphasis on elucidating the influence of alkali activation parameters and the amount of slag or cement that are used as minor constituents. This study reveals correlations between cumulative heat release and compressive strengths at different ages; these correlations can be leveraged to estimate the compressive strength based on hydration kinetics. Furthermore, this study presents a random forest (RF) model—in conjunction with fast Fourier and direct cosine transformation techniques to overcome the limitations associated with limited volume and diversity of the database—to enable high-fidelity predictions of time-dependent hydration kinetics and compressive strength of MT-based binders in relation to mixture design. Overall, this study demonstrates a sustainable approach to upcycle mine tailings as the primary component in low-carbon construction binders; and presents both analytical and machine learning-based approaches for accurate a priori predictions of hydration kinetics and compressive strength of these binders.

Fast MRI Techniques for Abdominal Imaging Simplified: Part 1—Making Sense of It All

Motion is a major limiting factor in abdominal MRI. Scan time is related to MR signal acquisition and k-space filling. K-space is the raw data matrix that stores digitized MR signal, which is Fourier transformed into an image. So rapid signal acquisition, reduced k-space filling, and partial Fourier techniques can reduce scan time. This article aims to provide a simplified explanation of the physics behind various techniques for fast abdominal MRI.

Analysis of Aerodynamic Loads on Heliostats at Operation Position Using Large Eddy Simulation and the Consistent Discrete Random Flow Generation Method

Abstract This study utilizes the large eddy simulation model (LES) and a synthetic method based on the Fourier technique called consistent discrete random flow generation (CDRFG) to analyze the peak aerodynamic loads on heliostats due to the atmospheric boundary layer. With the CDRFG technique, key flow parameters, including mean velocity profile, turbulent intensities, integral length scales, and turbulent spectra generated in wind tunnels, can be replicated while also satisfying the divergence-free condition. A three-facet heliostat with an elevation angle of α = 45 deg and the rear aligned to the inflow was analyzed. The heliostat behaves like a lifting surface in this orientation, accentuating the aerodynamic effect. The methodology proposed in this study can accurately reproduce flow statistics and predict the peak loads. Compared to experimental data, differences of 2.62% for drag, 7.43% for lift, and 11.0% for overturning were observed. Furthermore, the simulation reveals the generation of wingtip vortices on the sides of the heliostat, which contribute to the aerodynamic load. Overall, this technique has been demonstrated to be effective in replicating the atmospheric boundary layer and predicting the aerodynamic coefficients of heliostats.

Quantitative photothermal analysis and multispectral imaging of dental structures: insights into optical and thermal properties of carious and healthy teeth.

In the analysis of two-layered turbid dental tissues, the outer finite-thickness layer is modeled by an optical transport coefficient distinct from its underlying semi-infinite substrate layer. The optical and thermophysical parameters of healthy and carious teeth across the various wavelengths were measured leading to the determination of the degree of reliability of each of the fitted parameters, with most reliable being thermal diffusivity and conductivity, enamel thickness, and optical transport coefficient of the enamel layer. Quantitative pixel-by-pixel images of the key reliable optical and thermophysical parameters were constructed. We introduced a theoretical model of pulsed photothermal radiometry based on conduction-radiation theory and applied to quantitative photothermal detection and imaging of biomaterials. The theoretical model integrates a combination of inverse Fourier transformation techniques, avoiding the conventional cumbersome analytical Laplace transform method. Two dental samples were selected for analysis: the first sample featured controlled, artificially induced early caries on a healthy tooth surface, while the second sample exhibited natural defects along with an internal filling. Using an Nd:YAG laser and specific optical parametric oscillator (OPO) wavelengths (675, 700, 750, and 808nm), photothermal transient signals were captured from different points on these teeth and analyzed as a function of OPO wavelength. Measurements were also performed with an 808-nm laser diode for comparison with the same OPO wavelength excitation, particularly for the second sample with natural defects. The findings demonstrated that the photothermal transient signals exhibit a fast-decaying pattern at shorter wavelengths due to their higher scattering nature, while increased scattering and absorption in the carious regions masked conductive and radiative contributions from the underlayer. These observations were cross-validated using micro-computed tomography, which also enabled the examination of signal patterns at different tooth locations. The results of our study showed the impact of optical and thermal characteristics of two-layered turbid dental tissues via an inverse Fourier technique, as well as the interactions between these layers, on the patterns observed in depth profiles.

One-dimensional deep convolutional autoencoder active infrared thermography: Enhanced visualization of internal defects in FRP composites

Fiber-reinforced polymer (FRP) composites have been widely applied in different industrial fields, thereby necessitating the employment of non-destructive testing (NDT) methods to ensure structural integrity and safety. Active infrared thermography (AIRT) is a fast and cost-efficient NDT technique for inspecting FRP composites. However, this method is easily affected by factors such as inhomogeneous heating, leading to a low level of visualization of defects. To address this issue, this study proposes a novel method called one-dimensional deep convolutional autoencoder active infrared thermography (1D-DCAE-AIRT) to enhance the visualization of internal defects in FRP composites. This method first preprocesses the thermal image sequence acquired by AIRT inspections. Subsequently, high-level thermal features at the pixel level are extracted from the aforementioned preprocessed thermal image sequence using a designed one-dimensional deep convolutional autoencoder (1D-DCAE) model. Finally, the extracted high-level thermal features are employed to generate enhanced visualization results that exhibit improved defect visibility. The results of three kinds of AIRT (eddy current pulsed thermography, flash thermography, and vibrothermography) experiments on FRP composite specimens with artificially introduced defects show that 1D-DCAE-AIRT can effectively enhance the visualization of internal defects. The enhancement effect is better than the conventional techniques of fast Fourier transform (FFT), principal component analysis (PCA), independent component analysis (ICA), and partial least-squares regression (PLSR).