Search Results (207)

This paper mainly studies a different infinite-point Caputo fractional differential equation, whose nonlinear term may be singular. Under some conditions, we first use spectral analysis and fixed-point index theorem to explore the existence of positive solutions of the equation, and then use Banach fixed-point theorem to prove the uniqueness of positive solutions. Finally, an interesting example is used to explain the main result. Full article

Frequency-modulated continuous-wave (FMCW) LiDAR systems frequently experience noise interference during multi-target measurements in real-world applications, resulting in target overlapping and diminished detection accuracy. Conventional denoising approaches—such as Empirical Mode Decomposition (EMD) and wavelet thresholding—are often constrained by challenges like mode mixing and the attenuation of weak target signals, which limits their detection precision. To address these limitations, this study presents a novel denoising framework that integrates an optimized Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) algorithm and singular value decomposition (SVD). The CEEMDAN algorithm’s two critical parameters—the noise standard deviation and the number of noise additions—are optimally determined using particle swarm optimization (PSO), with the envelope entropy of the intrinsic mode functions (IMFs) serving as the fitness criterion. IMFs are subsequently selected based on spectral and amplitude comparisons with the original signal to facilitate initial signal reconstruction. Following CEEMDAN-based decomposition, SVD is employed with a normalized soft thresholding technique to further suppress residual noise. Validation using both synthetic and experimental datasets demonstrates the superior performance of the proposed approach over existing methods in multi-target scenarios. Specifically, it reduces the root mean square error (RMSE) by 45% to 59% and the mean square error (MSE) by 34% to 69%, and improves the signal-to-noise ratio (SNR) by 1.85–4.38 dB and the peak signal-to-noise ratio (PSNR) by 1.18–6.94 dB. These results affirm the method’s effectiveness in enhancing signal quality and target distinction in noisy FMCW LiDAR measurements. Full article

This paper proposes a hierarchical Fourier extension framework for the accurate reconstruction of piecewise smooth functions with mixed-order singularities. To address key challenges in spectral approximation–namely boundary-induced artifacts, instability in edge detection, and loss of accuracy near discontinuities–the method integrates three main components: (1) boundary-focused Fourier extensions that isolate endpoint effects while preserving internal structures; (2) a multi-stage edge detection strategy combining spectral mollifiers and coordinate transformations to identify discontinuities in function values and their derivatives; (3) adaptive domain partitioning followed by localized Fourier extensions to retain spectral accuracy on smooth segments. Numerical results demonstrate near machine-precision accuracy (∼10⁻¹⁴–10⁻¹⁵) with significantly improved stability and performance over traditional global methods. Full article

Hyperspectral image (HSI) quality is generally degraded by diverse noise contamination during acquisition, which adversely impacts subsequent processing performance. Current techniques predominantly rely on nuclear norms and low-rank matrix approximation theory to model the inherent property that HSIs lie in a low-dimensional subspace. Recent research has demonstrated that HSI gradient maps also exhibit low-rank priors. The correlated total variation (CTV), which is defined as the nuclear norm of gradient maps, can simultaneously model low-rank and local smoothness priors, and shows better performance than the standard nuclear norm. However, similar to nuclear norms, CTV may excessively penalize large singular values. To overcome these constraints, this study introduces a non-convex correlated total variation (NCTV), which shows the potential to eliminate mixed noise (including Gaussian, impulse, stripe, and dead-line noise) while preserving critical textures and spatial–spectral details. Numerical experiments on both simulated and real HSI datasets demonstrate that the proposed NCTV method achieves better performance in detail retention compared with the state-of-the-art techniques. Full article

This paper concerns accurate spectral collocation solutions, more precisely Chebyshev collocation (ChC), to some third-order nonlinear and singular boundary value problems on unbounded domains. The problems model some draining or coating fluid flows. We use exclusively ChC, in the form of Chebfun, avoid any obsolete shooting-type method, and provide reliable information about the convergence and accuracy of the method, including the order of Newton’s method involved in solving the nonlinear algebraic systems. As a complete novelty, we combine a graphical representation of the convergence of the Newton method with a numerical estimate of its order of convergence for a more realistic value. We treat five challenging examples, some of which have only been solved by approximate methods. The found numerical results are judged in the context of existing ones; at least from a qualitative point of view, they look reasonable. Full article

Achieving high-order accuracy in finite difference/spectral methods for space-time fractional differential equations often relies on very restrictive and usually unrealistic smoothness assumptions in the spatial and/or temporal domains. For spatial discretization, spectral methods using smooth basis functions are commonly employed. However, spatial–fractional derivatives pose challenges, as they often lack guaranteed spatial smoothness, requiring non-smooth basis functions. In the temporal domain, finite difference schemes on uniformly graded meshes are commonly employed; however, achieving accuracy remains challenging for non-smooth solutions. In this paper, an efficient algorithm is adopted to improve the accuracy of finite difference/Pertrov–Galerkin spectral schemes for a time-space fractional reaction–diffusion equation, with a hyper-singular integral fractional Laplacian and non-smooth solutions in both time and space domains. The Pertrov–Galerkin spectral method is adapted using non-smooth generalized basis functions to discretize the spatial variable, and the L1 scheme on a non-uniform graded mesh is used to approximate the Caputo fractional derivative. The unconditional stability and convergence are established. The rate of convergence is $O\left(N^{\mu -\gamma }+K^{-min\{\rho \beta ,2-\beta \}}\right),$ achieved without requiring additional regularity assumptions on the solution. Finally, numerical results are provided to validate our theoretical findings. Full article

An estimate of the trigger effect of the proton flux on seismicity was obtained. The proton flux time series with a time step of 5 min, 2000–2024, was analyzed. In each time interval of 5 days, statistics of the proton flux time series were calculated: mean values, logarithm of kurtosis, spectral slope, singularities spectrum support width, wavelet-based entropy, and the Donoho–Johnston wavelet-based index. For each of the used statistics, time points of local extrema were found, and for each pair of time sequences of proton flux statistics and earthquakes with a magnitude of at least 6.5 in sliding time windows, the “advance measures” of each time sequence relative to the other were estimated using a model of the intensity of interacting point processes. The difference between the “direct” measure of the advance of time points of local extrema of proton flux statistics relative to the time moments of earthquakes and the “inverse” measure of the advance was calculated. The maximum proportion of the intensity of seismic events for which the proton flux was a trigger was estimated as 0.28 for using the points of the local minima of the singularities spectrum support width. Full article

Comprehensive assessment of pore structure and multiphase water distribution is critical to the flow and transport process in coalbed methane (CBM) reservoirs. In this study, nuclear magnetic resonance (NMR) and multifractal analysis were integrated to quantify the multiscale heterogeneity of nine medium- and high-rank coals under water-saturated and dry conditions. By applying the box-counting method to transverse relaxation time (T₂) spectra, multifractal parameters were derived to characterize pore heterogeneity and residual water distribution. The influencing factors of pore heterogeneity were also discussed. The results show that pore structures in high-rank coals (HCs) exhibit a broader multifractal spectrum and stronger rightward spectrum than those of medium-rank coals, reflecting micropore-dominated heterogeneity and the complexity induced by aromatization in HCs. The vitrinite content enhances micropore development, increasing the heterogeneity and complexity of pore structure and residual water distribution. Inertinite content shows opposite trends compared to vitrinite content for the effect on pore structure and water distribution. Volatile yield reflects coal metamorphism and thermal maturity, which inversely correlates with pore heterogeneity and complexity. Residual water mainly distributes to adsorption pores and pore throats, shortening T₂ relaxation (bound water effect) and reducing spectral asymmetry. The equivalence of the multifractal dimension and singularity spectrum validates their joint utility in characterizing pore structure. Minerals enhance pore connectivity but suppress complexity, while moisture and ash contents show negligible impacts. These findings provide a theoretical reference for CBM exploration, especially in optimizing fluid transportation and CBM production strategies and identifying CBM sweet spots. Full article

This study presents a novel method for dynamically tuning singular states in one-dimensional (1D) photonic lattices (PLs) using air-slit-based structural modifications. Singular states, arising from symmetry-breaking-induced resonance radiation, generate diverse spectral features through interactions between resonance modes and background radiation. By strategically incorporating air slits to break symmetry in 1D PLs, we demonstrated effective control of resonance positions, enabling dual functionalities including narrowband band pass and notch filtering. These singular states originate from asymmetric guided-mode resonances (aGMRs), which can be interpreted by analytical modeling of the equivalent slab waveguide. Moreover, the introduction of multiple air slits significantly enhances spectral tunability by inducing multiple folding behaviors in the resonance bands. This approach allows for effective manipulation of optical properties through simple adjustments of air-slit displacements. This work provides great potential for designing multifunctional photonic devices with advanced metamaterial technologies. Full article

The precoder obtained using the traditional singular value decomposition (SVD) method for legitimate user’s channel, while achieving the highest spectral efficiency for the legitimate user, cannot defend against eavesdropping attacks, thus posing a security vulnerability. This paper investigates the millimeter wave (mmWave) secure beam hybrid precoding technology and proposes a generalized singular value decomposition (GSVD)-based secure beam hybrid precoding algorithm, termed GSVD-Sparsity, leveraging the sparsity of the mmWave beamspace channel. The algorithm selects the most powerful paths from the legitimate user’s beamspace channel representation and utilizes their corresponding angle information to construct a radio frequency (RF) precoder. It then constructs a hybrid precoder that closely approximates the optimal digital precoder derived from the GSVD-based scheme in a fully digital system. The simulation results indicate that, compared to the SVD-based scheme that focuses on spectral efficiency, the GSVD-based precoding scheme can form secure beams in a fully digital system. Under the condition that the legitimate user experiences a certain loss in the received signal-to-noise ratio (SNR), the eavesdropper is unable to correctly reconstruct the original constellation diagram, ensuring the scheme has strong anti-eavesdropping capabilities. In a hybrid precoding system, the low-complexity GSVD-Sparsity algorithm can achieve a spectral efficiency close to that of the GSVD-based scheme in a fully digital system while maintaining anti-eavesdropping capabilities. Full article

This paper analyzes the current state of water quality detection equipment and, based on the demand for portable water quality detection systems that are on-site, rapid, accurate, cost-effective, and capable of multi-parameter measurements using spectral analysis, represents the future development direction of water quality detection. By focusing on indicators of heavy metal ion water pollution, this study aims to achieve the “rapid and accurate detection of water quality using spectral analysis” and emphasizes key technologies such as “visible absorption spectroscopy in photoelectric detection technology and spectral analysis”, “spectral denoising methods”, and “Convolutional Neural Network (CNN) modeling and deployment”. A novel combined denoising method integrating Ensemble Empirical Mode Decomposition (EEMD) and Singular Value Decomposition (SVD) is developed and applied for the first time in spectral water quality detection to improve accuracy. The system uses a ZYNQ-based spectral analysis platform to detect heavy metal ion concentrations, enhancing detection speed. Comparative tests with copper ion standard solutions against Chinese national standards show good accuracy and reproducibility. The developed EEMD-SVD method demonstrates superior denoising effectiveness in processing actual spectral data within the water quality detection system. Full article

In this work, we analyze the vanishing cycles of Feynman loop integrals by the means of the Mayer–Vietoris spectral sequence. A complete classification of possible vanishing geometries is obtained. We use this result for establishing an asymptotic expansion for the loop integrals near their singularity locus and then give explicit formulas for the coefficients of such an expansion. Further development of this framework may potentially lead to exact calculations of one- and two-loop Feynman diagrams, as well as other next-to-leading and higher-order diagrams, in studies of radiative corrections for upcoming lepton–hadron scattering experiments. Full article

Pansharpening is essential for remote sensing applications requiring high spatial and spectral resolution. In this paper, we propose a novel Singular Spectrum Analysis-Enhanced Generative Adversarial Network (SSA-GAN) for multispectral pansharpening. We designed SSA modules within the generator, enabling more effective extraction and utilization of spectral features. Additionally, we introduce Pareto optimization to the nonreference loss function to improve the overall performance. We conducted comparative experiments on two representative datasets, QuickBird and Gaofen-2 (GF-2). On the GF-2 dataset, the Peak Signal-to-Noise Ratio (PSNR) reached 30.045 and Quality with No Reference (QNR) achieved 0.920, while on the QuickBird dataset, PSNR and QNR were 24.262 and 0.817, respectively. These results indicate that the proposed method can generate high-quality pansharpened images with enhanced spatial and spectral resolution. Full article

In this article, we study singular fractional-order differential equations with a variable coefficient, namely the linear operator of the differential equation containing a linear term with a variable coefficient. The coefficient $a\left(s\right)$ permits singularity at $s=0, 1$, and the nonlinearity $\mathfrak{f}(s,\chi )$ may be singular at $s=0, 1$ and $\chi =0$. By utilizing the fixed-point index theory, the existence of positive solutions are derived under sharp conditions concerning spectral radius. Full article

The emergence of 5G technology promises remarkable advancements in wireless communication, particularly in the realm of mmWave (millimeter-wave) massive multiple input multiple output (m-MIMO) systems. However, the realization of its full potential is hindered by the challenge of pilot overhead, which compromises system efficiency. The efficient usage of pilot signals is crucial for precise channel estimation and interference reduction to maintain data integrity. Nevertheless, this requirement brings up the challenge of pilot overhead, which utilizes precious spectrum space, thus reducing spectral efficiency (SE). To address this obstacle, researchers have progressively turned to artificial intelligence (AI) and machine learning (ML) methods to design hybrid beam-forming systems that enhance SE while reducing changes to the bit error rate (BER). This study addresses the challenge of pilot overhead in hybrid beamforming for 5G mmWave m-MIMO systems by leveraging advanced artificial intelligence (AI) techniques. We propose a framework integrating k-clustering, linear regression, random forest regression, and neural networks with singular value decomposition (NN-SVD) to optimize pilot placement and hybrid beamforming strategies. The results demonstrate an 82% reduction in pilot overhead, a 250% improvement in spectral efficiency, and a tenfold enhancement in bit error rate at low SNR conditions, surpassing state-of-the-art methods. These findings validate the efficacy of the proposed system in advancing next-generation wireless networks. Full article