Search Results (13)

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

In this paper, we focus on the multiplicity of positive solutions for a singular tempered fractional initial-boundary value problem with a p-Laplacian operator and a changing-sign perturbation term. By introducing a truncation function and combing with the properties of the solution of isomorphic linear equations, we transform the changing-sign tempered fractional initial-boundary value problem into a positive problem, and then the existence results of multiple positive solutions are established by the fixed point theorem in a cone. It is worth noting that the changing-sign perturbation term only satisfies the weaker Carathèodory conditions, which implies that the perturbation term ℏ can be allowed to have an infinite number of singular points; moreover, the value of the changing-sign perturbation term can tend to negative infinity in some singular points. Full article

Globally positive unbounded solutions, with zero derivative at infinity, are here considered for ordinary differential equations involving the generalized Euclidean mean curvature operator. When $p\ge 2$, the results highlight an analogy with an auxiliary equation with the p-Laplacian operator. The results are obtained using some comparison criteria for the principal solutions of a class of associated half-linear equations. Full article

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with $\varphi$-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that $\varphi$ be a homeomorphism, using Schauder’s fixed-point theorem, coupled with concepts of $L^1$-Carathéodory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth. Full article

In this paper, three sorts of network indices for the weighted three-layered graph are studied through the methods of graph spectra theory combined with analysis methods. The concept of union of graphs are applied to design two sorts of weighted layered multi-fan composed graphs, and the accurate mathematical expressions of the network indices are obtained through the derivations of Laplacian spectra; furthermore, the asymptotic properties are also derived. We find that when the cardinalities of the vertices on a sector-edge-link tend to infinity, the indices of FONC and EMFPT are irrelevant with the number of copies of the fan-substructure based on the considered graph framework. Full article

Depth map estimation is crucial for a wide range of applications. Unfortunately, it often presents missing or unreliable data. The objective of depth completion is to fill in the “holes” in a depth map by propagating the depth information using guidance from other sources of information, such as color. Nowadays, classical image processing methods have been outperformed by deep learning techniques. Nevertheless, these approaches require a significantly large number of images and enormous computing power for training. This fact limits their usability and makes them not the best solution in some resource-constrained environments. Therefore, this paper investigates three simple hybrid models for depth completion. We explore a hybrid pipeline that combines a very efficient and powerful interpolator (infinity Laplacian or AMLE) and a series of convolutional stages. The contributions of this article are (i) the use a Texture+Structuredecomposition as a pre-filter stage; (ii) an objective evaluation with three different approaches using KITTI and NYU_V2 data sets; (iii) the use of an anisotropic metric as a mechanism to improve interpolation; and iv) the inclusion of an ablation test. The main conclusions of this work are that using an anisotropic metric improves model performance, and the ablation test demonstrates that the model’s final stage is a critical component in the pipeline; its suppression leads to an approximate 4% increase in $MSE$. We also show that our model outperforms state-of-the-art alternatives with similar levels of complexity. Full article

This article mainly studies the double index logarithmic nonlinear fractional $g\text{-}$Laplacian parabolic equations with the Marchaud fractional time derivatives ${\displaystyle {\partial }_t^{\alpha }}$. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional $g\text{-}$Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional $g\text{-}$Laplacian parabolic equations is studied. Full article

In this article, a robust index named first-order network coherence (FONC) for the multi-agent systems (MASs) with layered lattice-like structure is studied via the angle of the graph spectra theory. The union operation of graphs is utilized to construct two pairs of non-isomorphic layered lattice-like structures, and the expression of the index is acquired by the approach of Laplacian spectra, then the corresponding asymptotic results are obtained. It is found that when the cardinality of the node sets of coronary substructures with better connectedness tends to infinity, the FONC of the whole network will have the same asymptotic behavior with the central lattice-like structure in the considered classic graph frameworks. The indices of the networks were simulated to illustrate the the asymptotic results, as described in the last section. Full article

By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers: ${(-\Delta )}^{\frac{\alpha }{2}}u\left(x\right)+u^{-\gamma }\left(x\right)+v^{-q}\left(x\right)=0,x\in R^N,$ ${(-\Delta )}^{\frac{\beta }{2}}v\left(x\right)+v^{-\sigma }\left(x\right)+u^{-p}\left(x\right)=0,x\in R^N,$ $u\left(x\right)\gtrsim {\left|x\right|}^a,v\left(x\right)\gtrsim {\left|x\right|}^b\mathrm{as}\left|x\right|\to \infty ,$ where $\alpha ,\beta \in (0,2)$, and $a,b>0$ are constants. We study the decay at infinity and narrow region principle for the fractional Laplacian system with different negative powers. The same results hold for nonlinear Hénon-type fractional Laplacian systems with different negative powers. Full article

In this article, the consensus-related performances of the triplex multi-agent systems with star-related structures, which can be measured by the algebraic connectivity and network coherence, have been studied by the characterization of Laplacian spectra. Some notions of graph operations are introduced to construct several triplex networks with star substructures. The methods of graph spectra are applied to derive the network coherence, and some asymptotic behaviors of the indices have been derived. It is found that the operations of adhering star topologies will make the first-order coherence increase a constant value under the triplex structures as parameters tend to infinity, and the second-order coherence have some equality relations as the node related parameters tend to infinity. Finally, the consensus related indices of the triplex systems with the same number of nodes but non-isomorphic graph structures have been compared and simulated to verify the results. Full article

We consider the exterior as well as the interior free-boundary Bernoulli problem associated with the infinity-Laplacian under a non-autonomous boundary condition. Recall that the Bernoulli problem involves two domains: one is given, the other is unknown. Concerning the exterior problem we assume that the given domain has a positive reach, and prove an existence and uniqueness result together with an explicit representation of the solution. Concerning the interior problem, we obtain a similar result under the assumption that the complement of the given domain has a positive reach. In particular, for the interior problem we show that uniqueness holds in contrast to the usual problem associated to the Laplace operator. Full article

In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016). Full article