Search Results (23)

We study the Sneddon $\mathcal{R}$-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel $xcos\left(xt\right)+hsin\left(xt\right)$, the $\mathcal{R}$-transform acts as a unifying operator for sine- and cosine-type integral transforms. Boundedness, continuity, and weighted $L^p$-estimates are established in an appropriate Banach space framework, together with Parseval–Goldstein type identities. Initial and final value theorems are derived for generalized functions in Zemanian-type spaces, yielding precise asymptotic behaviour at the origin and at infinity. A finite-interval theory is also developed, leading to polynomial growth estimates and final value theorems for the finite $\mathcal{R}$-transform. Full article

The paper concerns the solution of the ordinary differential equation $y^{″}\pm x^{m}y=0$, which may be designated the generalized Airy equation, since the original Airy equation corresponds to the particular case $m=1$ with the + sign. The solutions may be designated generalized circular (hyperbolic) sines and cosines for the + (−) sign, since the particular case $m=0$ corresponds to the elementary circular (hyperbolic) sines and cosines. There are 3 cases of solution of the generalized Airy equation, depending on the parameter m: (I) for m a non-negative integer, the coefficient $x^m$ is an analytic function, and the solutions are also analytic series; (II) for m complex other than an integer, the coefficient $x^m$ has a branch point at the origin, and the solutions also have a branch point multiplied by an analytic series; (III) for m a negative integer, the coefficient $x^m$ has a pole of order m, and the generalized Airy equation is singular. Case III has four subcases: (III-A) for $m=-1$, the coefficient $x^{-1}$ is a simple pole, and the solutions are Frobenius–Fuchs series of two kinds; (III-B) for $m=-2$, the coefficient is a double pole, and the solutions are a combination of elementary functions, namely exponential, logarithmic, and circular (hyperbolic) sine and cosine for the + (−) sign; (III-C,D) for $m=-3,-4,\dots$, the coefficient is a pole of multiplicity $\left|m\right|$, and the generalized Airy differential equation has an irregular singularity of degree $\left|m\right|-2$ at the origin. In the sub-cases (III-C,D), the solutions can be obtained by inversion as asymptotic series of descending powers specified by (III-C) Frobenius–Fuchs series of two kinds for a triple pole $m=-3$; (III-D) for higher-order poles $m=-4,-5,\dots$ by generalized circular (hyperbolic) sines and cosines of $1/x$. It is shown that in all cases the ascending and descending series are absolutely and uniformly convergent with the n-th term decaying like $O\left(n^2\right)$. This enables the use of a few terms of the series to obtain tables and plot graphs of the solutions of the generalized Airy differential equation as generalized circular and hyperbolic sines and cosines for several values of the parameter m. As a physical application, it is shown that the generalized circular (hyperbolic) cosines and sines specify the motion of a linear oscillator with natural frequency a power of time in the oscillatory (monotonic) case when the origin is an attractor (repeller). Full article

Multi-angular remote sensing plays a critical role in the study domains of ecological monitoring, climate change, and energy balance. The successful retrieval of the surface Bidirectional Reflectance Distribution Function (BRDF) and albedo from multi-angular remote sensing observations for various applications relies on the sensitivity of an appropriate BRDF model to both the number and the sampling distribution of multi-angular observations. In this study, based on selected high-quality multi-angular datasets, we designed three representative angular sampling schemes to systematically analyze different illuminating–viewing configurations of the retrieval results in a kernel-driven BRDF model framework. We first proposed an angular information index (AII) by incorporating a weighting mechanism and information effectiveness to quantify the angular information content for the angular sampling distribution schemes. In accordance with the principle that observations on the principal plane (PP) provide the most representative anisotropic scattering features, the assigned weight gradually decreases from the PP towards the cross-principal plane (CPP). The information effectiveness is determined based on the cosine similarity between the observations, effectively reducing the information redundancy. With such a method, we assess the AII of the different sampling schemes and further analyze the impact of angular distribution on both BRDF inversion and the estimation of snow surface albedo, including White-Sky Albedo (WSA) and Black-Sky Albedo (BSA) based on the RossThick-LiSparseReciprocal-Snow (RTLSRS) BRDF model. The main conclusions are as follows: (1) The AII approach can serve as a robust indicator of the efficiency of different sampling schemes in BRDF retrieval, which indicates that the RTLSRS model can provide a robust inversion when the AII value exceeds a threshold of −2. (2) When the AII value reaches such a reliable level, different sampling schemes can reproduce the BRDF shapes of snow across different bands to somehow varying degrees. Specifically, observations with smaller view zenith angle (VZA) ranges can reconstruct a BRDF shape that amplifies the anisotropic effect of snow; in addition, the forward scattering tends to be more pronounced at larger solar zenith angles (SZAs), while the variations in BRDF shape reconstructed from off-PP observations depend on both wavelength and SZAs. (3) The relative differences in both BSA and WSA grow with increasing wavelength for all these sampling schemes, mostly within 5% for short bands but up to 30% for longer wavelengths. With this novel AII method to quantify the information contribution of multi-angular sampling distributions, this study offers valuable insights into several main multi-angular BRDF sampling strategies in satellite sensor missions, which relate to most of the fields of multi-angular remote sensing applications in engineering. Full article

Joint inversion of gravity and gravity gradient data are of paramount importance in geophysical exploration, as the integration of these datasets enhances subsurface resolution and facilitates the accurate delineation of ore body shapes and boundaries. Conventional regularization methods, such as the L₂-norm, frequently yield excessively smooth solutions, which complicates the recovery of sharp boundaries. Furthermore, disparities in data units, magnitudes, and noise levels introduce additional complexities in selecting appropriate weighting functions and inversion parameters. To address these challenges, this study proposes a greedy inversion method based on cosine similarity, which identifies the most relevant cells and reduces the complexity involved in data weighting and parameter selection. Additionally, it incorporates prior information on density limits to achieve a high-resolution and sparse solution. To further enhance the stability and accuracy of the inversion process, a pruning mechanism is introduced to dynamically detect and remove erroneously selected cells, thereby suppressing error propagation. Synthetic model experiments demonstrate that incorporating the pruning mechanism significantly improves inversion accuracy. The method not only accurately resolves models of varying volumes while avoiding local convergence issues in the presence of major anomalies, but also exhibits strong robustness against noise, successfully delineating clear boundaries even when applied to complex composite models contaminated with 10% Gaussian noise. Finally, when applied to the joint inversion of measured gravity and gravity gradient tensor data from the Vinton salt dome, the results closely align with previous studies and actual geological observations. Full article

A new text matching model based on dynamic gated sparse attention feature distillation with a faithful semantic preservation strategy is proposed to address the fact that text matching models are susceptible to interference from weakly relevant information and that they find it difficult to obtain key features that are faithful to the original semantics, resulting in a decrease in accuracy. Compared to the traditional attention mechanism, with its high computational complexity and difficulty in discarding weakly relevant features, this study designs a new dynamic gated sparse attention feature distillation method based on dynamic gated sparse attention, aiming to obtain key features. Weakly relevant features are obtained through the synergy of dynamic gated sparse attention, a gradient inversion layer, a SoftMax function, and projection theorem literacy. Among these, sparse attention enhances weakly correlated feature capture through multimodal dynamic fusion with adaptive compression. Then, the projection theorem is used to identify and discard the noisy features in the hidden layer information to obtain the key features. This feature distillation strategy, in which the semantic information of the original text is decomposed into key features and noise features, forms an orthogonal decomposition symmetry in the semantic space. A new variety of faithful semantic preservation strategies is designed to make the key features faithful to the original semantic information. This strategy introduces an interval loss function and calculates the angle between the key features and the original hidden layer information with the help of cosine similarity in order to ensure that the features reflect the semantics of the original text. This can further update the iterative key features and thus improve the accuracy. The strategy builds a feature fidelity verification mechanism with a symmetric core of bidirectional considerations of semantic accuracy and correspondence to the original text. The experimental results show that the accuracies are 89.10% and 95.01% in the English datasets MRPC and Scitail, respectively; 87.8% in the Chinese dataset PAWX; and 80.32% and 80.27% in the Ant Gold dataset, respectively. Meanwhile, the accuracies in the KUAKE-QTR dataset and Macro-F1 are 70.10% and 68.08%, respectively, which are better than other methods. Full article

The Internet of Things (IoT) and Wireless Sensor Networks (WSNs) heavily rely on the lifetime of sensor nodes, which is inversely proportional to transmission power. Nodes with greater separation demand higher transmission power, while those closer together require less power. In practice, node placement varies significantly due to diverse terrain and contours, making power transmission configuration a critical and challenging issue in WSNs. This paper introduces an Enhanced Grey Wolf Optimization (EGWO) algorithm designed to optimize power transmission in WSN environments. Traditional Grey Wolf Optimization (GWO) employs a parameter that decreases linearly with iterations to regulate exploitation. In contrast, the proposed EGWO adopts a concave decline in the exploitation rate, allowing for more precise optimization in areas under exploration. The enhancement utilizes a cosine function that gradually decreases from 1 to 0, providing a smoother and more controlled transition. The experimental results demonstrate that EGWO outperforms other optimization algorithms. The proposed method achieves the lowest fitness value of −4.21, compared to 1.22 for standard GWO, −2.81 for PSO, and 2.86 for BESO, indicating its superiority in optimizing power transmission in WSNs. Full article

This article investigates the transmission characteristics of Laguerre–Gaussian (LG) beams under cosine modulation, power function modulation and linear modulation based on the variable coefficient fractional Schrödinger equation (FSE), respectively. In the absence of modulation, the LG beam undergoes diffraction-induced expansion as the transmission distance increases, with the degree of spreading increasing with a rising Lévy index. Under the cosine modulation, the evolution of the beam exhibits a periodic inversion, where the higher modulation frequency leads to a shorter oscillation period. The oscillation amplitude enlarges with a higher Lévy index and lower modulation frequency. When taking a power function modulation into account, the beam gradually evolves into a stable structure over propagation, with its width broadening with a growing Lévy index and modulation coefficient. In a linear modulation, the propagation of the LG beam forms a “trumpet-like” structure due to an accelerated diffraction effect. Notably, the transmission of the beam is not affected by the radial and azimuthal indices, but its ring number and phase singularity are changed correspondingly. The beam behaves in a similar evolutionary law under different modulations when the Lévy index is below 1. These findings offer valuable insights for applications in optical manipulation and communication. Full article

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant $\pi$ and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind. Full article

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media. Full article

Numerous analytical radar-scattering laws have been published through the past decades to interpret planetary radar observations, such as Hagfors’ law, which has been commonly used for the Moon, and the cosine law, which is commonly used in the shape modeling of asteroids. Many of the laws have not been numerically validated in terms of their interpretation and limitations. This paper evaluates radar-scattering laws for self-affine fractal surfaces using a numerical approach. Traditionally, the autocorrelation function and, more recently, the Hurst exponent, which describes the self-affinity, have been used to quantify the height correlation. Here, hundreds of three-dimensional synthetic surfaces parameterized using a root-mean-square (rms) height and a Hurst exponent were generated, and their backscattering coefficient functions were computed to evaluate their consistency with selected analytical models. The numerical results were also compared to empirical models for roughness and radar-scattering measurements of Hawaii lava flows and found consistent. The Gaussian law performed best at predicting the rms slope regardless of the Hurst exponent. Consistent with the literature, it was found to be the most reliable radar-scattering law for the inverse modeling of the rms slopes and the Fresnel reflection coefficient from the quasi-specular backscattering peak, when homogeneous statistical properties and a ray-optics approach can be assumed. The contribution of multiple scattering in the backscattered power increases as a function of rms slope up to about 20% of the backscattered power at normal incidence when the rms slope angle is 46°. Full article

With the rapid development of technologies like artificial intelligence, high-performance computing chips are playing an increasingly vital role. The inverse hyperbolic sine and inverse hyperbolic cosine functions are of utmost importance in fields such as image blur and robot joint control. Therefore, there is an urgent need for research into high-precision, high-performance hardware Intellectual Property (IP) for arcsinh and arccosh functions. To address this issue, this paper introduces a novel 128-bit low-latency floating-point hardware IP for arcsinh and arccosh functions, employing an enhanced Coordinate Rotation Digital Computer (CORDIC) algorithm, achieving a computation precision of 113 bits in just 32 computation cycles. This significantly enhances computational efficiency while reducing hardware implementation latency. The results indicate that, when compared to Python standard results, the calculated error of the proposed hardware IP does not exceed $8\times {10}^{-34}$. Furthermore, this paper synthesizes the completed IP using the TSMC 65 nm process, with a total IP area of 2.1056 mm${}^2$. Operating at a frequency of 300 MHz, its power is 22.4 mW. Finally, hardware implementation and resource analysis are conducted and compared on an Field Programmable Gate Array (FPGA). The results show that the improved algorithm trades a slight area increase for lower latency and higher accuracy. The designed hardware IP is expected to provide a more accurate and efficient computational tool for applications like image processing, thereby advancing technological development. Full article

Contrasting the dilemma that the traditional time-domain least mean square (LMS) algorithm in the existing satellite navigation receiver anti-interference system cannot satisfy the convergence time is short and maintain a low level of steady-state error at the same time, an inverse cosine variable step size LMS algorithm (ICVS-LMS) is proposed. To begin with, the LMS algorithm, with a fixed step size focuses on its effectiveness in attenuating and suppressing interference signals, is analyzed, and then the proposed ICVS-LMS algorithm is analyzed. In conclusion, both the ICVS-LMS algorithm and the traditional algorithm are simulated and compared in terms of their effectiveness in suppressing interference in satellite navigation signals. The experimental results demonstrate that the improved algorithm significantly reduces convergence time while maintaining a small steady-state error. The improved algorithm demonstrates high robustness and an obvious suppression effect on interference signals. The anti-interference performance is 8.41–12.22% higher than that of the proposed algorithm. Full article

Sound rendering is the process of determining the sound propagation path from an audio source to a listener and generating 3D sound based on it. This task demands complex calculations, including trigonometric functions. This paper presents hardware-based inverse cosine function calculations using the table method and linear approximation. This approach maintains a high accuracy while limiting hardware size for suitability in sound rendering applications. Consequently, our proposed hardware-based inverse cosine calculation method is a valuable tool for achieving high efficiency and accuracy in 3D sound rendering. Full article

Timing-mismatch errors among channels in time-interleaved analog-to-digital converters (TIADCs) greatly degrade the whole performance of the system. Therefore, techniques for calibrating timing mismatch are indispensable, and a new fully-digital calibration technique is presented in this article. Based on a Hilbert filter, modified moving averagers (MMAs) and inverse cosine functions, the proposed estimation algorithm is fast (within 1200 sample points) and accurate. Meanwhile, the coordinate rotational digital computer (CORDIC) algorithm, which is used to implement inverse cosine functions, is also improved, giving it higher precision. In addition, a compensation method based on second-order Taylor series approximation with less hardware resource consumption is provided. Through analyses and simulations, this calibration technique proved to be suitable for TIADCs with an arbitrary number of channels, in which the signal-to-noise and distortion ratio (SNDR) and the spurious-free dynamic range (SFDR) were, respectively, improved from 24.06 dB and 24.57 dB to 67.96 dB and 85.69 dB. Full article

When producing orthomosaic from aerial images of a forested area, challenges arise when the forest canopy is closed, and tie points are hard to find between images. The recent development in deep leaning has shed some light in tackling this problem with an algorithm that examines each image pixel-by-pixel. The scale-invariant feature transform (SIFT) algorithm and its many variants are widely used in feature-based image stitching, which is ideal for orthomosaic production. However, although feature-based image registration can find many feature points in forest image stitching, the similarity between images is too high, resulting in a low correct matching rate and long splicing time. To counter this problem by considering the characteristics of forest images, the inverse cosine function ratio of the unit vector dot product (arccos) is introduced into the SIFT-OCT (SIFT skipping the first scale-space octave) algorithm to overcome the shortfalls of too long a matching time caused by too many feature points for matching. Then, the fast sample consensus (FSC) algorithm was introduced to realize the deletion of mismatched point pairs and improve the matching accuracy. This optimized method was tested on three sets of forest images, representing the forest core, edge, and road areas of a loblolly pine plantation. The same process was repeated by using the regular SIFT and SIFT-OCT algorithms for comparison. The results showed the optimized SIFT-OCT algorithm not only greatly reduced the splicing time, but also increased the correct matching rate. Full article