Search Results (93)

In this work, we consider a generalized form of the classical $(2+1)$-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results. Full article

This study presents an image inpainting model based on an energy functional that incorporates the $L^p$ norm of the fractional Laplacian operator as a regularization term and the $H^{-1}$ norm as a fidelity term. Using the properties of the fractional Laplacian operator, the $L^p$ norm is employed with an adjustable parameter p to enhance the operator’s ability to restore fine details in various types of images. The replacement of the conventional $L^2$ norm with the $H^{-1}$ norm enables better preservation of global structures in denoising and restoration tasks. This paper introduces a diffusion partial differential equation by adding an intermediate term and provides a theoretical proof of the existence and uniqueness of its solution in Sobolev spaces. Furthermore, it demonstrates that the solution converges to the minimizer of the energy functional as time approaches infinity. Numerical experiments that compare the proposed method with traditional and deep learning models validate its effectiveness in image inpainting tasks. Full article

This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly reducing the computational complexity of nonlinear terms. Temporal discretization is achieved using a second-order difference method, while spatial discretization utilizes a simple and easily implementable discrete approximation for the fractional Laplacian operator. The boundedness and convergence of the proposed numerical scheme under the maximum norm are rigorously analyzed, demonstrating its adherence to discrete energy conservation or dissipation laws. Numerical experiments validate the scheme’s effectiveness, structure-preserving properties, and capability for long-time simulations for both one- and two-dimensional problems. Additionally, the impact of the parameter $\epsilon$ on error dynamics is investigated. Full article

The 32-bit floating-point (FP32) format has many useful applications, particularly in computing and neural network systems. The classic 32-bit fixed-point (FXP32) format often introduces lower quality of representation (i.e., precision), making it unsuitable for real deployment, despite offering faster computations and reduced computational cost, which positively impacts energy efficiency. In this paper, we propose a switched FXP32 format able to compete with or surpass the widely used FP32 format across a wide variance range. It actually proposes switching between the possible values of key parameters according to the variance level of the data modeled with the Laplacian distribution. Precision analysis is achieved using the signal-to-quantization noise ratio (SQNR) as a performance metric, introduced based on the analogy between digital formats and quantization. Theoretical SQNR results provided in a wide range of variance confirm the design objectives. Experimental and simulation results obtained using neural network weights further support the approach. The strong agreement between the experiment, simulation, and theory indicates the efficiency of this proposal in encoding Laplacian data, as well as its potential applicability in neural networks. Full article

Background/Objectives: To compare the image quality of chest radiography with a dual-energy X-ray imaging system using AI technology (DE-AI) to that of conventional chest radiography with a standard protocol. Methods: In this prospective study, 52 healthy volunteers underwent dual-energy chest radiography. Images were obtained using two exposures at 60 kVp and 120 kVp, separated by a 150 ms interval. Four images were generated for each participant: a conventional image, an enhanced standard image, a soft-tissue-selective image, and a bone-selective image. A machine learning model optimized the cancellation parameters for generating soft-tissue and bone-selective images. To enhance image quality, motion artifacts were minimized using Laplacian pyramid diffeomorphic registration, while a wavelet directional cycle-consistent adversarial network (WavCycleGAN) reduced image noise. Four radiologists independently evaluated the visibility of thirteen anatomical regions (eight soft-tissue regions and five bone regions) and the overall image with a five-point scale of preference. Pooled mean values were calculated for each anatomic region through meta-analysis using a random-effects model. Results: Radiologists preferred DE-AI images to conventional chest radiographs in various anatomic regions. The enhanced standard image showed superior quality in 9 of 13 anatomic regions. Preference for the soft-tissue-selective image was statistically significant for three of eight anatomic regions. Preference for the bone-selective image was statistically significant for four of five anatomic regions. Conclusions: Images produced by DE-AI provide better visualization of thoracic structures. Full article

The five-membered fused six-membered nitrogen heteroaromatic ring system is a crucial skeleton in the design and synthesis of energetic compounds. Based on this skeleton, many high-performance energetic compounds have been synthesized. However, to date, no one has conducted a systematic study on the characteristics of this skeleton itself. To assess how the number and position of nitrogen atoms affect the energy and stability of this type of skeleton, one to four nitrogen-substituted skeleton molecules were analyzed using Density Functional Theory (DFT) calculations. Natural population analysis (NPA), Laplacian bond order (LBO) analysis, aromaticity studies, and enthalpy of formation calculations were performed. Patterns observed in the computational results were summarized, and their potential correlations were analyzed. Based on these findings, design recommendations for derivatives of these skeletons in energetic compounds were proposed to serve as a reference for energetic material chemists. Full article

We study the regularizing effect of the zero order term in some elliptic problems, which present in the principal part a nonlocal operator, i.e., the fractional Laplacian operator. We prove the existence of bounded energy solutions, even if the data are assumed to be very irregular. Full article

The aim of this paper is to demonstrate the existence of a unique positive solution to non-local fractional p-Laplacian equations of the Brézis–Oswald type involving Hardy potentials. The main feature of this paper is solving the difficulty that arises in the presence of a singular coefficient and in the lack of the semicontinuity property of an energy functional associated with the relevant problem. The main tool for overcoming this difficulty is the concentration–compactness principle in fractional Sobolev spaces. Also, the uniqueness result of the Brézis–Oswald type is obtained by exploiting the discrete Picone inequality. Full article

The analysis of land cover using deep learning techniques plays a pivotal role in understanding land use dynamics, which is crucial for land management, urban planning, and cartography. However, due to the complexity of remote sensing images, deep learning models face practical challenges in the preprocessing stage, such as incomplete extraction of large-scale geographic features, loss of fine details, and misalignment issues in image stitching. To address these issues, this paper introduces the Multi-Scale Modular Extraction Framework (MMS-EF) specifically designed to enhance deep learning models in remote sensing applications. The framework incorporates three key components: (1) a multiscale overlapping segmentation module that captures comprehensive geographical information through multi-channel and multiscale processing, ensuring the integrity of large-scale features; (2) a multiscale feature fusion module that integrates local and global features, facilitating seamless image stitching and improving classification accuracy; and (3) a detail enhancement module that refines the extraction of small-scale features, enriching the semantic information of the imagery. Extensive experiments were conducted across various deep learning models, and the framework was validated on two public datasets. The results demonstrate that the proposed approach effectively mitigates the limitations of traditional preprocessing methods, significantly improving feature extraction accuracy and exhibiting strong adaptability across different datasets. Full article

In recent years, the application of free-form surface spatial grid structures in large public buildings has become increasingly common. The layouts of grids are important factors that affect both the mechanical performance and aesthetic appeal of such structures. To achieve a triangular grid with good mechanical performance and uniformity on free-form surfaces, this study proposes a new method called the “strain energy gradient optimization method”. The grid topology is optimized to maximize the overall stiffness, by analyzing the sensitivity of nodal coordinates to the overall strain energy. The results indicate that the overall strain energy of the optimized grid has decreased, indicating an improvement in the structural stiffness. Specifically, compared to the initial grid, the optimized grid has a 30% decrease in strain energy and a 43.3% decrease in maximum nodal displacement. To optimize the smoothness of the grid, the study further applies the Laplacian grid smoothing method. Compared to the mechanically adjusted grid, the structural mechanical performance does not significantly change after smoothing, while the geometric indicators are noticeably improved, with smoother lines and regular shapes. On the other hand, compared to the initial grid, the smoothed grid has a 21.4% decrease in strain energy and a 28.3% decrease in maximum nodal displacement. Full article

In the paper, we introduce the signless Laplacian $ABC$-matrix $\tilde{\mathrm{Q}}(G)=\overline{D}(G)+\tilde{\mathrm{A}}(G)$, where $\overline{D}(G)$ is the diagonal matrix of $ABC$-degrees and $\tilde{\mathrm{A}}(G)$ is the $ABC$-matrix of $G.$ The eigenvalues of the matrix $\tilde{\mathrm{Q}}(G)$ are the signless Laplacian $ABC$-eigenvalues of G. We give some basic properties of the matrix $\tilde{\mathrm{Q}}(G)$, which includes relating independence number and clique number with signless Laplacian $ABC$-eigenvalues. For bipartite graphs, we show that the signless Laplacian $ABC$-spectrum and the Laplacian $ABC$-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian $ABC$-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian $ABC$-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix $\tilde{\mathrm{Q}}(G)-{\displaystyle \frac{tr(\tilde{\mathrm{Q}}(G))}{n}}I$, called the signless Laplacian $ABC$-energy of G. We obtain some upper and lower bounds for signless Laplacian $ABC$-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the $ABC$-energy with the signless Laplacian $ABC$-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian $ABC$-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same $ABC$-energy. Full article

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the $L^{\infty }$-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for weak solutions to nonlinear elliptic problems involving a singular nonlinear term such as Hardy potentials has not been studied extensively. To overcome this, we utilize a truncated energy technique and the De Giorgi iteration method. As its application, we demonstrate that the problem above has at least two distinct nontrivial solutions by exploiting a variant of Ekeland’s variational principle and the classical mountain pass theorem as the key tools. Furthermore, we prove the existence of a sequence of infinitely many weak solutions that converges to zero in the $L^{\infty }$-norm. To derive this result, we employ the modified functional method and the dual fountain theorem. Full article

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation ${(-\Delta )}^{s}u+V\left(ϵx\right)u=\lambda u+ulog{u}^2\mathrm{in}{\mathbb{R}}^N,$ under the mass constraint ${\int }_{{\mathbb{R}}^N}{\left|u\right|}^{2}dx=a.$ Here, $N\ge 2$, $a,ϵ>0$, $\lambda \in \mathbb{R}$ is an unknown parameter, ${(-\Delta )}^s$ is the fractional Laplacian and $s\in (0,1)$. We introduce a function space where the energy functional associated with the problem is of class ${\mathcal{C}}^1$. Then, under some assumptions on the potential V and using the Lusternik–Schnirelmann category, we show that the number of normalized solutions depends on the topology of the set for which the potential V reaches its minimum. Full article

The pulse-coupled neural network (PCNN), due to its effectiveness in simulating the mammalian visual system to perceive and understand visual information, has been widely applied in the fields of image segmentation and image fusion. To address the issues of low contrast and the loss of detail information in infrared and visible light image fusion, this paper proposes a novel image fusion method based on an improved adaptive dual-channel PCNN model in the non-subsampled shearlet transform (NSST) domain. Firstly, NSST is used to decompose the infrared and visible light images into a series of high-pass sub-bands and a low-pass sub-band, respectively. Next, the PCNN models are stimulated using the weighted sum of the eight-neighborhood Laplacian of the high-pass sub-bands and the energy activity of the low-pass sub-band. The high-pass sub-bands are fused using local structural information as the basis for the linking strength for the PCNN, while the low-pass sub-band is fused using a linking strength based on multiscale morphological gradients. Finally, the fused high-pass and low-pass sub-bands are reconstructed to obtain the fused image. Comparative experiments demonstrate that, subjectively, this method effectively enhances the contrast of scenes and targets while preserving the detail information of the source images. Compared to the best mean values of the objective evaluation metrics of the compared methods, the proposed method shows improvements of 2.35%, 3.49%, and 11.60% in information entropy, mutual information, and standard deviation, respectively. Full article

In this paper, we will study a singular problem involving the fractional $(q_1(x,.)$-$q_2(x,.))$-Laplacian operator in the whole space ${\mathbb{R}}^N,(N\ge 2)$. More precisely, we combine the variational method with monotonicity arguments to prove that the associated functional energy admits a critical point, which is a weak solution for such a problem. Full article