Laplacian Research Articles

Quantum state transfer between cells and binary tree model

Discussing quantum state transfer on a graph is a very difficult task, when it has a relatively large number of vertices. In this paper, we construct cells so that it contain a number of vertices by partitioning the vertices, so that a finite number of cells can be used to cover all vertices on the graph. Subsequently, we define the adjacency case between cells and the orthonormal basis on the complex Hilbert space. In this framework, the quantum state transfer based on binary tree model is studied by using Laplacian operator.

Continuum limit for Laplace and elliptic operators on lattices

Continuum limit for Laplace and elliptic operators on lattices

Least energy solutions for a class of (p1,p2)$(p_{1}, p_{2})$‐Kirchhoff‐type problems in RN$\mathbb {R}^{N}$ with general nonlinearities

AbstractWe examine the following ‐Kirchhoff‐type problem: where , , is the ‐Laplacian operator for , are two continuous Kirchhoff functions that satisfy suitable conditions, and is a general nonlinearity. Using appropriate variational arguments, we prove the existence of a radially symmetric least energy solution.

Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain

AbstractWe consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric.

A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

On a class of generalized capillarity system involving fractional ψ$$ \psi $$‐Hilfer derivative with p(·)$$ p\left(\cdotp \right) $$‐Laplacian operator

This research delves into a comprehensive investigation of a class of ‐Hilfer generalized fractional nonlinear differential system originated from a capillarity phenomena with Dirichlet boundary conditions, focusing on issues of existence and multiplicity of nonnegative solutions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz type condition. We use minimization arguments of Nehari manifold together with variational approach to show the existence and multiplicity of positive solutions of our problem with respect to the parameter in appropriate fractional ‐Hilfer spaces. Our main result is novel, and its investigation will enhance the scope of the literature on coupled systems of fractional ‐Hilfer generalized capillary phenomena.

Dynamics of a mistletoe-bird model on a weighted network.

Based on the patchy habitats of mistletoes and the mutualistic relationship between mistletoes and birds, we propose a mistletoe-bird model on a weighted network that is described by discrete Laplacian operators. Without considering mistletoes, the dynamics of a model of birds is investigated by a threshold parameter. Under the premise of the persistence of birds, the existence and uniqueness of solutions of a mistletoe-bird model are established, and the stability of solutions of the model is discussed by the ecological reproduction index , specifically, mistletoes go extinct when , and mistletoes coexist with birds when . Moreover, we show that network weights can induce changes of instantaneous dynamics of birds or mistletoes by the matrix perturbation method. By assuming that the weighted network is a river network and a star network, we simulate the extinction of mistletoes and the coexistence of mistletoes with birds, respectively.

On class of bi-nonlocal problems involving fractional p(.,.)-Laplacian operators

This work aims to obtain the existence and multiplicity of solutions for a bi-nonlocal problem involving the p(.,.)-Laplacian operator. Under specific assumptions, the paper uses the Mountain Pass Theorem combined with the Ekeland variational approach to establish the existence of two nontrivial solutions for the problem. The existence of infinitely many solutions is proved using the Symmetric Mountain Pass Theorem and the Ljusternik-Schnirelman Theorem. Finally, we study a p(.,.)-Laplacian problem with non-variational form.

Brownian motion in a vector space over a local field is a scaling limit

For any natural number d, the Vladimirov–Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a d-dimensional vector space V over a local field K. Just as the Laplace operator on L2(Rd) is the infinitesimal generator of Brownian motion with state space Rd, the Vladimirov–Taibleson operator on L2(V) is the infinitesimal generator of real-time Brownian motion with state space V. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted V to be the p-adic numbers.

High Spatiotemporal Precision Mapping of Optical Nanosensor Array Using Machine Learning.

Optical nanosensors, including single-walled carbon nanotubes (SWCNTs), provide real-time spatiotemporal reporting at the single-molecule level within a nanometer-scale area. However, their superior sensitivity also makes them susceptible to slight environmental influences such as reference analytes in media, external fluid flow, and mechanical modulations. Consequently, they often fail to achieve the optimal limit of detection (LOD) and frequently convey misinformation spatiotemporally. To address this challenge, we developed a single-pixel mapping technique for optical nanosensor arrays that operates with high spatiotemporal precision using machine learning. We systematically measured the spatial sensing images of various analyte concentrations below the LOD by using a near-infrared (nIR) fluorescent SWCNT nanosensor array. For dopamine (DA) as an example analyte, we extracted single-pixel level sensing features such as entropy, the Laplacian operator, and neighboring values under noise levels. We then trained the artificial intelligence (AI) model to accurately identify specific reaction pixels of the nanosensor array, even below the LOD region. Additionally, our method can distinguish subtle noise caused by fluid in the media or mechanical modulation of the array substrate. As a result, our approach significantly improved the detection sensitivity of the nanosensor array, achieving a 13-fold increase over the original LOD and halving the detection time of the reporter pixels, with F1 scores exceeding 0.9. This method not only lowers the LOD of optical nanosensors but also isolates sensor responses specific to the analyte, providing accurate spatiotemporal information to the user, even in noisy conditions. It can be universally applied to various optical nanosensor materials and analytes, maximizing the sensitivity and accuracy of the nanosensors used in diagnostics and analysis.

Existence of positive solutions for generalized fractional Brézis-Nirenberg problem

In this article, we study the fractional Brézis-Nirenberg type problem on whole domain $\R^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation} \label{Pr1} (-\Delta_{p})^{s}u - \lambda w |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in } \mathcal{D}^{s,p}\big(\mathbb{R}^{N}\big) , \tag{P} \end{equation} where $s\in (0,1), p\in (1, {N}/{s})$, $p_{s}^{*}={Np}/({N-sp})$ and the operator $(-\Delta_{p})^{s}$ is the fractional $p$-Laplace operator. The space $\mathcal{D}^{s,p}\big(\mathbb{R}^{N}\big)$ is the completion of $C_c^\infty\big(\R^N\big)$ with respect to the Gagliardo semi-norm. In this article, we prove the existence of a positive solution to problem \eqref{Pr1} by allowing the Hardy weight function $w$ to change its sign.

Existence of Solutions for a Class of Dirichlet Problems Involving Fractional (p, q)-Laplacian Operators

Existence of Solutions for a Class of Dirichlet Problems Involving Fractional (p, q)-Laplacian Operators

Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise.

Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn-Knopp (SK) iterations. This paper proves the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian with rates, when [Formula: see text] data points are i.i.d. sampled from a general [Formula: see text]-dimensional manifold embedded in a possibly high-dimensional space. Under certain joint limit of [Formula: see text] and kernel bandwidth [Formula: see text], the point-wise convergence rate of the graph Laplacian operator (under 2-norm) is proved to be [Formula: see text] at finite large [Formula: see text] up to log factors, achieved at the scaling of [Formula: see text]. When the manifold data are corrupted by outlier noise, we theoretically prove the graph Laplacian point-wise consistency which matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors among themselves and with data vectors. Motivated by our analysis, which suggests that not exact bi-stochastic normalization but an approximate one will achieve the same consistency rate, we propose an approximate and constrained matrix scaling problem that can be solved by SK iterations with early termination. Numerical experiments support our theoretical results and show the robustness of bi-stochastically normalized graph Laplacian to high-dimensional outlier noise.

Entanglement in Lifshitz fermion theories

We study the static entanglement structure in (1+1)-dimensional free Dirac-fermion theory with Lifshitz symmetry and arbitrary integer dynamical critical exponent. This model is different from the one introduced in [Hartmann et al., SciPost Phys.11 (2021) 031] due to a proper treatment of the square Laplace operator. Dirac fermion Lifshitz theory is local as opposed to its scalar counterpart which strongly affects its entanglement structure. We show that there is quantum entanglement across arbitrary subregions in various pure (including the vacuum) and mixed states of this theory for arbitrary integer values of the dynamical critical exponent. Our numerical investigations show that quantum entanglement in this theory is tightly bounded from above. Such a bound and other physical properties of quantum entanglement are carefully explained from the correlation structure in these theories. A generalization to (2+1)-dimensions where the entanglement structure is seriously different is addressed.

Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds

Abstract The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.

Some useful tools in the study of nonlinear elliptic problems

This paper gives an overview of some basic aspects concerning the qualitative analysis of nonlinear, nonhomogeneous elliptic problems. We are concerned with two classes of elliptic equations with Dirichlet boundary condition. The first problem is driven by a general nonhomogeneous differential operator, which includes several usual operators (such as the (p,q)-Laplace operator introduced by P. Marcellini). Next, we focus on differential operators with unbalanced growth in the nonautonomous case. Our analysis will point out some relevant differences between balanced and unbalanced growth problems. The presentation is done in the context of Dirichlet problems but a similar analysis can be developed for other boundary conditions, such as Neumann or Robin.

Treatment of 3D diffusion problems with discontinuous coefficients and Dirac curvilinear sources

Three-dimensional diffusion problems with discontinuous coefficients and unidimensional Dirac sources arise in a number of fields. The statement we pursue is a singular-regular expansion where the singularity, capturing the stiff behavior of the potential, is expressed by a convolution formula using the Green kernel of the Laplace operator. The correction term, aimed at restoring the boundary conditions, fulfills a variational Poisson equation set in the Sobolev space H1, which can be approximated using finite element methods. The mathematical justification of the proposed expansion is the main focus, particularly when the variable diffusion coefficients are continuous, or have jumps. A computational study concludes the paper with some numerical examples. The potential is approximated by a combined method: (singularity, by integral formulas, correction, by linear finite elements). The convergence is discussed to highlight the practical benefits brought by different expansions, for continuous and discontinuous coefficients.

Hermitian random walk graph Fourier transform for directed graphs and its applications

Signal processing on directed graphs present additional challenges since a complete set of eigenvectors is unavailable generally. To solve this problem, in this paper, a novel graph Fourier transform is constructed for representing and processing signals on directed graphs. Firstly, we introduce a Hermitian random walk Laplacian operator and derive that it is Hermitian positive semi-definite. Hence, the obtained Laplacian operator is diagonalizable and yields orthogonal eigenvectors as graph Fourier basis. Secondly, we propose the Hermitian random walk graph Fourier transform (HRWGFT) with good properties including unitary and preserving inner products. Furthermore, HRWGFT records the directionality of edges without sacrificing the information about the graph signal. Then, using these favorable properties, we derive spectral convolution to define the graph filter which is the core tool for processing graph signals. Finally, based on the proposed Laplacian matrix and HRWGFT, we present several applications on synthetic and real-world networks, including signal denoising, data classification. The rationality and validity of our work are verified by simulations.

Bayesian inversion of a fractional elliptic system derived from seismic exploration

In this paper, we concentrate on the Bayesian inversion of a dispersion‐dominated fractional Helmholtz (DDFH) equation, which has been introduced in studies concerning seismic exploration. To establish the inversion theory, we meticulously examine the DDFH equation. We transform it into a system comprising both fractional‐ and integer‐order elliptic equations, extending the conventional definition of the spectral fractional Laplace operator to accommodate non‐homogeneous boundary conditions. Subsequently, we establish the well‐posedness theory for scenarios involving both small and large wavenumbers. Our proof hinges upon the regularity attributes of select fractional elliptic equations and capitalizes fully on the structural peculiarities of the elliptic system, which distinguish it from classical cases. Thereafter, we focus on the inverse medium scattering problem pertinent to the DDFH equation, framed within the Bayesian statistical framework. We address two scenarios: one devoid of model reduction errors and another characterized by such errors—arising from the implementation of certain absorbing boundary conditions. More precisely, based on the properties of the forward operator, well‐posedness of the posterior measures have been proved in both cases, which provide an opportunity to quantify the uncertainties of this problem.

The Bessel–Clifford Function Associated to the Cayley–Laplace Operator

The Bessel–Clifford Function Associated to the Cayley–Laplace Operator