Search Results (29)

Remote sensing imaging technology is one of the safest and most effective tools for gas leakage monitoring in chemical parks, as it enables fast and accurate access to detailed information about the gas cloud (e.g., volume, distribution, diffusion, and location) in the case of gas leakage. While multi-spectral imaging systems are commonly used for hazardous gas leakage detection, efforts to realize the three-dimensional reconstruction of gas clouds through data obtained from multi-spectral imaging systems remain scarce. In this study, we propose a method for realizing the three-dimensional reconstruction of gas clouds with only two multi-spectral imaging systems; in particular, the two multi-spectral imaging systems are used to simultaneously observe the three-dimensional space with gas leakage and reconstruct gas cloud images in real time. A geometric method is used for the localization in the monitoring space and the construction of a three-dimensional spatial grid. The non-axisymmetric inverse Abel transform (IAT) is then applied to the extracted gas absorbance images in order to realize the reconstruction of each layer, and these are then stacked to form a 3D gas cloud. Through the above measurement, identification, and reconstruction processes, a 3D gas cloud with geometric information and concentration distribution characteristics is generated. The results of simulation experiments and external field tests prove that gas clouds can be localized under the premise that they are completely covered by the field of view of both scanning systems, and the 3D distribution of the leakage gas cloud can be reconstructed quickly and accurately with the proposed system. Full article

In this paper, we present a new integral representation for the Jacobi polynomials that follows from Koornwinder’s representation by introducing a suitable new form of Euler’s formula. From this representation, we obtain a fractional integral formula that expresses the Jacobi polynomials in terms of Gegenbauer polynomials, indicating a general procedure to extend Askey’s scheme of classical polynomials by one step. We can also formulate suitably normalized Fourier–Jacobi spectral coefficients of a function in terms of the Fourier cosine coefficients of a proper Abel-type transform involving a fractional integral of the function itself. This new means of representing the spectral coefficients can be beneficial for the numerical analysis of fractional differential and variational problems. Moreover, the symmetry properties made explicit by this representation lead us to identify the classes of Jacobi polynomials that naturally admit the extension of the definition to negative values of the index. Examples of the application of this representation, aiming to prove the properties of the Fourier–Jacobi spectral coefficients, are finally given. Full article

For studying the finite-size behavior of the Ising model under different boundary conditions, we propose an alternative to the standard transfer matrix technique approach based on Abelès theorem and Chebyshev polynomials. Using it, one can easily reproduce the known results for periodic boundary conditions concerning the Lee–Yang zeros, the exact position-space renormalization-group transformation, etc., and can extend them by deriving new results for antiperiodic boundary conditions. Note that in the latter case, one has a nontrivial order parameter profile, which we also calculate, where the average value of a given spin depends on the distance from the seam with the opposite bond in the system. It is interesting to note that under both boundary conditions, the one-dimensional case exhibits Schottky anomaly. Full article

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK $\alpha ,\gamma$-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique. Full article

Camellia oleifera (Camellia oleifera Abel.) is a key woody oilseed tree. In recent years, China’s Camellia oleifera industry has shifted from extensive to refined management, with an action plan launched to boost productivity and efficiency. This study utilized remote sensing technology to diagnose crop nutrient levels. Focusing on 240 Camellia oleifera trees from four varieties at the Dechang Cooperative in Shucheng County, Anhui Province, the study collected full-spectrum canopy reflectance data (350–2500 nm) across five growing stages: spring shoot, summer shoot, fruit expanding, fruit ripening, and full blooming. First-order derivative (FD) and second-order derivative (SD) transformations were used to preprocess the spectral data and analyze the relationships between leaf potassium concentration (LKC) and the raw spectra (R), FD, and SD. The VCPA-IRIV strategy was then applied to identify sensitive wavelengths and artificial neural network algorithms were used to construct LKC estimation models. The main conclusions are as follows. (1) In the spring shoot stage, LKC ranged from 1.93 to 8.06 g/kg, with an average of 3.70 g/kg; in the summer shoot stage, LKC ranged from 2.01 to 8.82 g/kg, with an average of 4.96 g/kg; in the fruit expanding stage, LKC ranged from 1.40 to 18.27 g/kg, with an average of 4.90 g/kg; in the fruit ripening stage, LKC ranged from 1.45 to 8.90 g/kg, with an average of 3.71 g/kg.; and in the full blooming stage, LKC ranged from 2.38 to 9.57 g/kg, with an average of 5.79 g/kg. Across the five growth stages, the LKC content of Camellia oleifera showed a pattern of initially increasing, then decreasing, and subsequently increasing again. (2) The optimal LKC model for the spring shoot stage was FD-[7,6,2], with ${\mathrm{R}}_{\mathrm{c}}^2$ = 0.6559, RMSEC = 0.1906 in the calibration set, ${\mathrm{R}}_{\mathrm{T}}^2$ = 0.4531, RMSET = 0.2009 in the test set. The optimal LKC model for the summer shoot stage was FD-[6,5,4], with ${\mathrm{R}}_{\mathrm{c}}^2$ = 0.7419, RMSEC = 0.2489 in the calibration set, and ${\mathrm{R}}_{\mathrm{T}}^2$ = 0.7536, RMSET = 0.2259 in the test set; the optimal LKC model for the fruit expanding stage was SD-[7,6,2], with ${\mathrm{R}}_{\mathrm{c}}^2$ = 0.3036, RMSEC = 0.2113 in the calibration set, and ${\mathrm{R}}_{\mathrm{T}}^2$ = 0.3314, RMSET = 0.1800 in the test set; the optimal LKC model for the fruit ripening stage was FD-[10,3,2], with ${\mathrm{R}}_{\mathrm{c}}^2$ = 0.4197, RMSEC = 0.2375 in the calibration set, and ${\mathrm{R}}_{\mathrm{T}}^2$ = 0.5649, RMSET = 0.1772 in the test set, and the optimal LKC model for the full blooming stage was SD-[10,3,2], with ${\mathrm{R}}_{\mathrm{c}}^2$ = 0.7013, RMSEC = 0.2322 in the calibration set, and ${\mathrm{R}}_{\mathrm{T}}^2$ = 0.5621, RMSET = 0.2507 in the test set. Full article

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. Full article

A new numerical method for solving Volterra linear convolution integral equations (CVIEs) of the second kind is presented in this work. This new approach uses first-order infinite impulse response digital filters method (IIRFM). Three convolutive kernels were analyzed, the unit kernel and two singular kernels: the logarithmic and generalized Abel kernels. The IIRFM is based on the combined use of the Laplace transformation, a first-order decomposition, and a bilinear transformation. This approach often leads to simple analytical expressions of the approximate solutions, enabling efficient numerical calculation, even using single-precision floating-point numbers. When compared with the method of homotopic perturbations with Laplace transformation (HPM-L), the IIRFM approach does not present, in linear cases, the convergence difficulties inherent to iterative approaches. Unlike most solution methods based on the Laplace transform, the IIRFM has the dual advantage of not requiring the calculation of the Laplace transform of the source function, and of not requiring the systematic calculation of inverse Laplace transforms. Full article

Soil nutrient transformation and the microbial metabolism are primarily regulated by soil microorganisms, including fungi and bacteria, which exhibit distinct growth patterns, energy substrate utilization, and survival strategies. Despite their significance, our understanding of the key microorganisms governing the soil microbial metabolism and multifunctionality in subtropical woodlands remains limited. To address this knowledge gap, we conducted a large-scale investigation and assessment of the soil microbial metabolic limitation and soil multifunctionality in Camellia oleifera Abel and Pinus massoniana Lamb. woodlands in subtropical China. Our results reveal that the microbial phosphorus limitation was more severe in C. oleifera compared to P. massoniana woodlands. Nonetheless, the pattern of carbon metabolic limitation for microbes and soil multifunctionality was similar in both types of woodland. Specifically, the microbial carbon limitation was positively associated with both bacterial and fungal richness, while the microbial phosphorus limitation was significantly correlated with fungi including the richness and community structure in the P. massoniana woodland. By contrast, we did not observe significant correlations between microbial metabolic limitation indices and microbial parameters in C. oleifera woodlands. Regarding soil multifunctionality, the results reveal a strong positive correlation between the soil multifunctionality and fungal community in both P. massoniana and C. oleifera woodlands. Furthermore, our structural equation modeling revealed that the soil fungal community, rather than the bacterial community, had a significant effect on the microbial metabolic limitation and soil multifunctionality. Overall, our study provides profound insights into the relative importance of bacterial and fungal communities in shaping the soil microbial metabolic limitation and soil multifunctionality in subtropical woodlands. The findings of our study have important implications for the management and conservation of subtropical woodlands. Full article

The temperature field and chemiluminescence measurements of axisymmetric flame are obtained simultaneously in only one image. Digital Laser Speckle Displacement measures temperature fields, and direct image flame determines chemiluminescence values. Applying the Abel transform of axisymmetric objects for volume visualization requires smooth intensity profiles. Due to the nature of the experimental setup, direct image flame is corrupted with speckle noise and a crosstalk effect. These undesirable effects deteriorate the measurement results. Then, experimental data need crosstalk correction and speckle noise reduction to improve the measurements. This work aims to implement a methodology to reduce the speckle noise of highly noisy data intensity profiles to create smooth profiles appropriate to applying the Abel transform. The method uses a Four-Order Partial Differential Equation to reduce speckle noise and a Curve fitting utilizing a set of Gaussian functions to decrease residual undesirable effects. After this, correction of crosstalk is necessary to avoid this effect. The methodology is applied to premixed flames generated with Liquid Petroleum Gas for different mixes. Full article

An off-axis digital holographic interferometry technique integrated with a Mach–Zehnder interferometer based setup is demonstrated for measuring the temperature and temperature profile of a transparent medium. This technique offers several advantages: it does not require precise optomechanical adjustments or accurate definition of the frequency carrier mask, making it simple and cost-effective. Additionally, high-quality optics are not necessary. The methodology relies on measuring the phase difference between two digitally reconstructed complex wave fields and utilizing the temperature coefficient of the refractive index. In this way, we presented an equation of the temperature as a function of phase changes and the temperature coefficient of refractive index. This approach simplifies the calculation process and avoids the burden of complicated mathematical inversions, such as the inverse Abel transformation. It also eliminates the need for additional work with the Lorentz–Lorentz equation and Gladstone–Dale relation and can be extend for 3D measurements. Full article

Legendre coefficients of an integrable function $f\left(x\right)$ are proved to coincide with the Fourier coefficients with a nonnegative index of a suitable Abel-type transform of the function itself. The numerical computation of N Legendre coefficients can thus be carried out efficiently in $\mathcal{O}\left(NlogN\right)$ operations by means of a single fast Fourier transform of the Abel-type transform of $f\left(x\right)$. Symmetries associated with the Abel-type transform are exploited to further reduce the computational complexity. The dual problem of calculating the sum of Legendre expansions at a prescribed set of points is also considered. We prove that a Legendre series can be written as the Abel transform of a suitable Fourier series. This fact allows us to state an efficient algorithm for the evaluation of Legendre expansions. Finally, some numerical tests are illustrated to exemplify and confirm the theoretical results. Full article

Optical Calorimetry (OC) is a 2D Digital Holographic Interferometry (DHI)-based measurement technique with potential applications for the 3D dosimetry of ultra-high dose rate (FLASH) radiation therapy beams through tomographic reconstruction. This application requires accurate measurements of DHI signals in environments with low signal-to-noise ratios (SNRs) in order to accurately measure absorbed energy to a medium per unit mass (Dose). However, tomographic reconstruction accuracy is sensitive to noise in the measurements. In this study, a virtual model of an OC dosimeter was used to characterize and model major sources of noise within a DHI setup, allowing for the modelled noise sources to be selectively reduced. The tomographic reconstruction of the 3D dose distribution was achieved using the inverse Abel transform. Reducing the noise contribution from atmospheric turbulence and mechanical vibration by one half improved the central axis reconstruction error from 6.5% to 1.3% and 1.1%, respectively, and the mean dose difference from 2.9% to 0.4% and 0.3%, respectively. This indicates the potential of the tomographic DHI-based 3D OC dosimeter to reconstruct accurate 3D dose distributions from a single projection if the specified sources of noise can be reduced to acceptable levels. The used methodology is applicable to any application of tomographic DHI where reconstruction quality is highly sensitive to noise. Full article

We generate the Einstein–Gauss–Bonnet field equations in higher dimensions for a spherically symmetric static spacetime. The matter distribution is a neutral fluid with isotropic pressure. The condition of isotropic pressure, an Abel differential equation of the second kind, is transformed to a first order nonlinear canonical differential equation. This provides a mechanism to generate exact solutions systematically in higher dimensions. Our solution generating algorithm is a different approach from those considered earlier. We show that a specific choice of one potential leads to a new solution for the second potential for all spacetime dimensions. Several other families of exact solutions to the condition of pressure isotropy are found for all spacetime dimensions. Earlier results are regained from our treatments. The difference with general relativity is highlighted in our study. Full article

The conversion of airglow intensity to volume emission rate (VER) is a common method for studying the ionosphere, but the contribution of the intensity conversion process to the uncertainty in estimated electron or ion density is significant. The Abel inversion is a commonly used method for retrieving VERs from vertical profiles of airglow intensities accumulated along the rays horizontally at the tangent point, but it requires that the intensities converge to zero at their uppermost height, which is often not the case due to observational limitations. In this study, we present a method for optimizing the retrieval of VER from satellite-measured airglow intensities using the techniques of deep learning and Abel inversion. This method can be applied to fill in unobserved or discontinuous observations in airglow intensity profiles with the Chapman function, allowing them to be used with the Abel inversion to determine VERs. We validate the method using limb 135.6 nm airglow emission intensity data from the NASA Global-scale Observations of the Limb and Disk (GOLD) mission. Our training process involves using three hidden layers with varying numbers of neurons, and we compare the performance of the best-performing deep learning models to Abel-transformed results from real-time observations. The combination of Abel inversion and deep learning has the potential to optimize the process of converting intensity to VER and improve the capacity for analyzing ionospheric observations. Full article

Abel’s integral equation is an efficient singular integral equation that plays an important role in diverse fields of science. This paper aims to investigate Abel’s integral equation and its solution using $G_{\alpha }$-transform, which is a symmetric relation between Laplace and Sumudu transforms. $G_{\alpha }$-transform, as defined via distribution space, is employed to establish a solution to Abel’s integral equation, interpreted in the sense of distributions. As an application to the given theory, certain examples are given to demonstrate the efficiency and suitability of using the $G_{\alpha }$-transform method in solving integral equations. Full article