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The aim of this work is to introduce a purely order-based Kantorovich–Wright type integration of scalar functions with respect to a vector measure defined on a δ-ring and taking values in a Dedekind σ-complete vector lattice and use it for obtaining general representation theorems for Dedekind complete vector lattices.

We give the first examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity. Extremally Ricci pinched G2-structures (introduced by Bryant) which are steady Laplacian solitons have also been found. All the examples are left-invariant G2-structures on solvable Lie groups., Comment: 14 pages

In this paper, we present the finite difference method for Caputo-type parabolic equation with fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0, 1) and the spatial derivative is the fractional Laplacian. The Caputo derivative is evaluated by the L 1 approximation, and the fractional Laplacian with respect to the space variable is approximated by the Caffarelli–Silvestre extension. The difference schemes are provided together with the convergence and error estimates. Finally, numerical experiments are displayed to verify the theoretical results.

An exact discretization of fractional-order Laplacian for N -dimensional space is suggested. Particular solutions of fractional-order partial difference equations with the proposed discrete Laplace operators are suggested.

As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [12]. Consequently, this result significantly improves the recent works [12, 38, 40]., Comment: This version fix several typos of the previous one. 25 pages

This paper proposes a new implicit definition of the fractional Laplacian. Compared with the existing explicit definitions in literature, this novel definition has clear physical significance and is mathematically simple and numerically easy to calculate for multidimensional problems. In stark contrast to a quick increasing and extensive applications of time-fractional derivative to diverse scientific and engineering problems, little has been reported on space-fractional derivative modeling. This is largely because the existing definitions are only feasible for one-dimensional case and become mathematically too complicated and computationally very expensive when applied to higher dimensional cases. In this study, we apply the newly-defined fractional Laplacian for modeling the power law behaviors of three-dimensional nonlocal heat conduction. The singular boundary method (SBM), a recent boundary-only collocation discretization method, is employed to numerically solve the proposed fractional Laplacian heat equation. And the computational costs are observed moderate owing to the proposed new definition of fractional Laplacian and the boundary-only discretization, meshfree, and integration-free natures of the SBM technique. Numerical experiments show the validity of the proposed definition of fractional Laplacian.

In this paper, we study the solutions of the following fractional Laplacian equation { ( − Δ ) s u = f ( x , u ) in Ω , u = 0 in Ω c . Under some assumptions on the nonlinearity, we will show that the boundedness of solutions is equivalent to the finiteness of their Morse indices. We extend the result of Bahri and Lions (1992) [1] , from the Laplacian equations to the fractional Laplacian equations.

From the basis of the Gauss divergence theorem applied on a circular control volume that was put forward by Isshiki (2011) in deriving the MPS-based differential operators, a more general Laplacian model is further deduced from the current work which involves the proposal of an altered kernel function. The Laplacians of several functions are evaluated and the accuracies of various MPS Laplacian models in solving the Poisson equation that is subjected to both Dirichlet and Neumann boundary conditions are assessed. For regular grids, the Laplacian model with smaller N is generally more accurate, owing to the reduction of leading errors due to those higher-order derivatives appearing in the modified equation. For irregular grids, an optimal N value does exist in ensuring better global accuracy, in which this optimal value of N will increase when cases employing highly irregular grids are computed. Finally, the accuracies of these MPS Laplacian models are assessed in an incompressible flow problem.

This paper presents an alternative means of deriving and discretizing spectral distances and kernels on a 3D shape by filtering its Laplacian spectrum. Through the selection of a filter map, we design new spectral kernels and distances, whose smoothness and encoding of both local and global properties depend on the convergence of the filtered Laplacian eigenvalues to zero. Approximating the discrete spectral distances through the Taylor approximation of the filter map, the proposed computation is independent of the evaluation of the Laplacian spectrum, bypasses the computational and storage limits of previous work, which requires the selection of a specific subset of eigenpairs, and guarantees a higher approximation accuracy and a lower computational cost.

Let G be a mixed graph with a nonzero Laplacian eigenvalue μ of multiplicity k. A line star set for μ in G is a set Y of k edges of G such that μ is not a Laplacian eigenvalue of G − Y . It is shown that line star set exists for any nonzero Laplacian eigenvalue of any mixed graph. Some basic properties for line star sets are given. We also obtain some results on line star sets in undirected graphs.

This paper presents a new method for estimating the eigenvalues of the Laplacian matrix associated with the graph describing the network topology of a multi-agent system. Given an approximate value of the average of the initial condition of the network state and some intermediate values of the network state when performing a Laplacian-based average consensus, the estimation of the Laplacian eigenvalues is obtained by solving the factorization of the averaging matrix. For this purpose, in contrast to the state of the art, we formulate a convex optimization problem that is solved in a distributed way by means of the Alternating Direction Method of Multipliers (ADMM). The main variables in the optimization problem are the coefficients of a polynomial whose roots are precisely the inverse of the distinct nonzero Laplacian eigenvalues. The performance of the proposed method is evaluated by means of simulation results.

Through an Euler–Hopf-type formula, we establish existence result to a Nirenberg-type problem involving the square root of the Laplacian in sphere S 2 .

The multi-dimensional scalar Spherical Harmonics Discrete Ordinate Method (SHDOM) is extended to the vector case. The vector model uses complex and real generalized spherical harmonics in the energetic representation of the Stokes vector, and retains some powerful features of the scalar model, as for example, the combination of the generalized spherical harmonic and the discrete ordinate representations of the radiance field, the use of a linear short characteristic method for computing the corner-point values of the Stokes vector, and the application of the adaptive grid technique. Results illustrating the accuracy of the vector model are shown for realistic simulated clouds.

In this paper, we are concerned with the existence and non-existence of global solutions of a semi-linear heat equation with fractional Laplacian. We obtain some extension of results of Weissler who considered the case α = 1, and h ≡ 1.

In a three-dimensional bounded possibly multiply-connected domain, we prove the existence and uniqueness of vector potentials in L p -theory, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, various Sobolev-type inequalities are given.

C-grid discretizations based on a hexagonal or triangular mesh can be investigated with the help of a planar trivariate coordinate system, where the vector components are either defined tangentially (hexagonal C-grid) or perpendicularly (triangular C-grid) to the coordinate lines. Inspecting the Helmholtz decomposition of a vector in case of the linearly dependent trivariate coordinate description reveals insights into the structure and stencil of the discretized divergence and vorticity on such grids. From a vector Laplacian, which is consistent with the Helmholtz decomposition, a general formulation for the inner product operator at grid edges can be derived. Thus, the vector reconstruction of the tangential wind for the Coriolis term in the shallow water equations can be given even for a slightly distorted tesselation as present on icosahedral spherical grids. Furthermore, a rigorous comparison of the triangular and the hexagonal C-grid linear shallow water equations is performed from a theoretical and also from an experimental viewpoint. It turns out that the additional degree of freedom in the height field in the triangular C-grid case compared to the hexagonal C-grid is responsible for the decoupling of divergence values on upward and downward directed triangles. This problem occurs especially for small Rossby deformation radii, and practically requires additional explicit diffusion. In contrast, the hexagonal C-grid discretization has remarkable similarity to the quadrilateral C-grid case in the wave dispersion properties and eigenvector structure. Numerical experiments performed with that option proved resilent.

The notion of a scalar function that is l -stable at a point is introduced in [D. Bednařik, K. Pastor, On second-order conditions in unconstrained optimization, Math. Program. Ser. A 113 (2008) 283–298]. In the present paper a characterization of the l -stable functions is obtained. Further, the notion of an l -stable function is generalized from scalar to vector functions. In an application, optimality conditions for constrained vector problems with l -stable data are established.

We propose a modification of the generalized gradient vector flow field techniques based on a continuous force field analysis. At every iteration the generalized gradient vector flow method obtains a new, improved vector field. However, the numerical procedure always employs the original image to calculate the gradients used in the source term. The basic idea developed in this paper is to use the resulting vector field to obtain an improved edge map and use it to calculate a new gradient based source term. The improved edge map is evaluated by new continuous force field analysis techniques inspired by a preceding discrete version. The approach leads to a better convergence and better segmentation accuracy as compared to several conventional gradient vector flow type methods.

In this paper, we consider the existence, nonexistence and multiplicity of a positive solution for a Gelfand type generalized Laplacian system with a singular indefinite weight and a vector parameter. By using the upper and lower solution method and fixed point index theory, we obtain a global multiplicity result with respect to the parameter.

In this paper we consider nonlinear periodic systems driven by the nonhomogeneous vector p -Laplacian differential operator and with a nonsmooth potential. Using variational techniques, based on the nonsmooth critical point theory, we prove existence and multiplicity results. Finally, we obtain an existence result for systems driven by the ordinary vector p -Laplacian and we show that it is still true in the more general framework of the previous results.

We study a new approach to eigen-decomposition of symmetric matrices, called the vector field method. In contrast to the methods currently in use, the vector field method computes an eigenvector by computing a discrete approximation to the integral curve of a special tangent vector field on the unit sphere. The optimization problems embedded in each iteration of the vector field algorithms admit closed-form solutions making the vector field approach promisingly efficient. Besides establishing the general convergence results about the vector field method and the complexity bounds for computing ϵ -approximations to eigenvectors, we also computationally compare the performance of a family of algorithms called the recursive vector field algorithms with that of the power method. The vector field method, like the power method, offers a significant advantage in that it can exploit sparseness to speed up computation even further. The convergence analysis, that we present, also suggests that one could speed up the vector field algorithms through preprocessing that contracts the spectral radius of the given matrix.

The behavior of the finite-difference time-domain method (FDTD) is investigated with respect to the approximation of the two-dimensional Laplacian, associated with the curl-curl operator. Our analysis begins from the observation that in a two-dimensional space the Yee algorithm approximates the Laplacian operator via a strongly anisotropic 5-point approximation. It is demonstrated that with the aid of a transversely extended-curl operator any 9-point Laplacian can be mapped onto FDTD update equations. Our analysis shows that the mapping of an isotropic Laplacian approximation results in an isotropic and less dispersive FDTD scheme. The properties of the extended curl are further explored and it is proved that a unity Courant number can be achieved without the resulting scheme suffering from grid decoupling. Additionally, the case of a 25-point isotropic Laplacian is examined and it is shown that the corresponding scheme is fourth order accurate in space and exhibits isotropy up to sixth order. Representative numerical simulations are performed that validate the theoretically derived results.

We show that the Watson and Cramer–von Mises statistics are related to Green's function of the Laplacian on a circle. A generalization leads to a new U-statistic whose kernel is the Green function of the Laplacian on the sphere.

It is well known that the resistance distance between two arbitrary vertices in an electrical network can be obtained in terms of the eigenvalues and eigenvectors of the combinatorial Laplacian matrix associated with the network. By studying this matrix, people have proved many properties of resistance distances. But in recent years, the other kind of matrix, named the normalized Laplacian, which is consistent with the matrix in spectral geometry and random walks [Chung, F.R.K., Spectral Graph Theory, American Mathematical Society: Providence, RI, 1997], has engendered people's attention. For many people think the quantities based on this matrix may more faithfully reflect the structure and properties of a graph. In this paper, we not only show the resistance distance can be naturally expressed in terms of the normalized Laplacian eigenvalues and eigenvectors of G, but also introduce a new index which is closely related to the spectrum of the normalized Laplacian. Finally we find a non-trivial relation between the well-known Kirchhoff index and the new index.

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.

We show that the action of conformal vector fields on functions on the sphere determines the spectrum of the Laplacian (or the conformal Laplacian), without further input of information. The spectra of intertwining operators (both differential and non-local) with principal part a power of the Laplacian follows as a corollary. An application of the method is the sharp form of Gross' entropy inequality on the sphere. The same method gives the spectrum of the Dirac operator on the sphere, as well as of a continuous family of nonlocal intertwinors, and an infinite family of odd-order differential intertwinors., 19 pages, LaTeX

In this paper, two geometric processing problems are considered: (1) point on a surface nearest to an external point, and (2) silhouette curve of a surface when observed from a given point. Problem (1) is solved by constructing gradient curves on the surface associated with a distance scalar field. Problem (2) appears as the intersection of surfaces (implicit case), or as tracing a plane curve (parametric case). Formulations are geometric-differential, and lead to explicit, first-order systems of ordinary differential equations (ODEs), with initial conditions that can be efficiently integrated by standard numerical methods. The methodology allows us to deal with both implicit and parametric representations, these having any functional structure for which the differential statements are meaningful.

The central aim of this paper is the study of the spectrum of the Hodge Laplacian on differential forms of any order k in L p . The underlying space is a C ∞ -smooth open manifold M N with Ricci Curvature bounded below and uniformly subexponential volume growth. It will be demonstrated that on such manifolds the L p spectrum of the Hodge Laplacian on differential k -forms is independent of p for 1 ⩽ p ⩽ ∞ , whenever the Weitzenbock Tensor on k -forms is also bounded below. It follows as a corollary that the isolated eigenvalues of finite multiplicity are L p independent. The proof relies on the existence of a Gaussian upper bound for the Heat kernel of the Hodge Laplacian. By considering the L p spectra on the Hyperbolic space H N + 1 we conclude that the subexponential volume growth condition is necessary in the case of one-forms. As an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on certain manifolds with a pole or that are in a warped product form. This will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the L 1 spectrum of the Laplacian.

We represent processes in highly anisotropic, nonhomogeneous, either ferromagnetic or ferrimagnetic materials, by coupling the Maxwell equations with a vector extension of the standard rectangular hysteresis loop. We then derive a two-length-scale vector model via a homogenization procedure. Here, the fine-scale magnetization M depends on the fine-scale magnetic field H , at variance with the corresponding scalar problem. This amends the usual formulation of the vector Preisach model, which is not consistent with the law ∇ · B =0 .

A spherical-harmonics ( P N ) method that incorporates and couples existent scalar and vector versions of the method is developed. The scalar (group-by-group) version of method is applied to the slowing-down energy range and the vector version to the thermal energy range. The link of coupling between these methods is the slowing-down source to the thermal groups which becomes available once the problems for all of the slowing-down groups are solved with the scalar method. The advantage of using the coupled P N method over the vector method when solving transport problems defined in extended energy range is demonstarted in this work.

In this paper, we introduce and study a class of Generalized Vector Quasi-Variational-Like Inequality Problem (GVQVLIP) involving set-valued mappings with certain monotonicity. By employing the scalarization technique, several existence results for solutions of the (GVQVLIP) are established under noncompact setting in topological vector spaces. These new existence results improve, unify, and generalize many known results for scalar and vector variational inequalities in recent literature.

One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of spanning trees in a graph (see [1, 17, 11, 15]). The usual notion of the combinatorial Laplacian for a graph involves edge weights. Namely, a Laplacian L forGis a matrix with rows and columns indexed by the vertex setVofG, and the (u,v)-entry of L, foru,vinG,u≠v, is associated with the edge-weight of the edge (u,v). It is not so obvious to consider Laplacians with vertex weights (except for using some symmetric combinations of the vertex weights to define edge-weights). In this note, we consider a vertex weighted Laplacian which is motivated by a problem arising in the study of algebro-geometric aspects of the Bethe Ansatz [18]. The usual Laplacian can be regarded as a special case with all vertex-weights equal. We will generalize the matrix-tree theorem to matrix-tree theorems of counting “rooted” directed spanning trees. In addition, the characteristic polynomial of the vertex-weighted Laplacian has coefficients with similar interpretations. We also consider subgraphs with non-trial boundary. We will shown that the Laplacian with Dirichlet boundary condition has its determinant equal to the number of rooted spanning forests. The usual matrix-tree theorem is a special case with the boundary consisting of one single vertex.

The probability density function for survivals, that is for transitions without hitting a barrier, for a collection of particles driven by correlated Brownian motions is analyzed. The analysis is known to lead to a study of the spectrum of the Laplacian on domains on the sphere in higher dimensions. The rst eigenvalue of the Laplacian governs the large time behavior of the probability density function and the asymptotics of the hitting time distribution. It is found that the solution leads naturally to a spectral function, a ‘generating function’ for the eigenvalues and multiplicities of the Laplacian. Analytical properties of the spectral function suggest a simple scaling procedure for determining the eigenvalues, readily applicable for a homogeneous collection of correlated particles. Comparison of the rst eigenvalue with the available theoretical and numerical results for some specic domains shows remarkable agreement.

We present comparison theorems for the eigenvalues of the covariant Laplacian acting on the sections of vector bundle with the structure group U ( N ).

Different approximating expressions for the surface Laplacian operator on a triangulated surface are derived. They are evaluated on a triangulated spherical surface for which the analytical expression of the surface Laplacian is known. It is shown that in order to obtain accurate results, due care has to be taken of irregularities present in the triangulation grid. If this is done, the approximation will equal the performance of an expression based on least squares which can be derived. Next the different approximations obtained are used as a regularization operator in the solution of an ill-posed inverse problem in electrical volume conduction. It is shown that in this application a crude approximation to the surface Laplacian suffices.

In this paper we study the topological classification problem for homogeneous polynomial vector fields of degree k > 2 in three variables. In 1960 Markus [7] (see also [6]) classified topologically the quadratic homogeneous vector fields in two variables. Subsequent works relative to the differentiable classification of these vector fields can be found in Date [4], Newton [lo], Samardzija [13], Vulpe and Sibirskii [14]. Let us now consider a homogeneous vector field Q = (Q,, Q2, Q,), of degree k > 2 in R3 with an isolated singularity at 0 E R3, i.e., all the Qi, i = 1, 2, 3, are homogeneous polynomials of degree k, with an isolated common zero at 0 E R3. The vector field defined as Qr(.?) = Q(i) (X, Qi(.%) + X2Q2(X) +X3 Q&f)). X, X = (2, ,X2, X3) E S2, is tangent to the 2-sphere S2, and for any regular orbit y of Qr the set

Three two-dimensional plasma simulation models have been compared using an electrostatic two-stream instability as a test case. In one model the usual five-point Laplacian was used in Poisson's equation, and the field was smoothed by interpolation with a bilinear spline function. In another model the difference approximation to the Laplacian was the nine-point operator prescribed by Hamilton's variational principle, and the electric field was computed as the exact negative gradient of the potential. In the third model the nine-point Laplacian was used with the smoothed field. The test problem was an unstable double-streaming plasma that was initially cold in the streaming direction. The major result is that the evolution of electric field energy did not depend strongly on the choice of model. There was a much stronger dependence on the random numbers that were selected to represent the initial velocity distribution.

A survey is made of possible methods to verify features of the nuclear single-particle model in which the nucleon is assumed to move relativistically in a field of scalar character plus one given as the fourth component of a four-vector (σ + ω model). Electromagnetic and weak probes are considered, and sensitivity to the nucleon effective mass—considerably changed in the presence of the scalar plus vector fields—is sought. A number of direct effects are in large measure lost because terms involving the mass usually involve gradient operators as well, and these tend to give little contribution when acting on the potentials in the nuclear interior where they are large but flat. A promising candidate as a probe is the ratio of induced pseudoscalar coupling constant to axial vector coupling constant, which is proportional to the effective mass.

A MACSYMA program is described which performs symbolic vector algebra and vector calculus. The program can combine and simplify symbolic expressions including dot products and cross products, together with the gradient, divergence, curl and Laplacian operators. The distribution of these operators over sums or products is under user control, as are various other expansions, including expansion into components in any specific orthogonal curvilinear co-ordinate system. There is also a capability for deriving the scalar or vector potential of a vector field. Examples include derivation of the partial differential equations describing fluid flow and magnetohydrodynamics, for 12 different classic orthogonal curvilinear co-ordinate systems.

A general theory is developed for TEM waves guided by any form of cylindrical conductors. The vector Laplacian in general curvilinear coordinates leads to very simple equations for E and H . With electromagnetic waves, a scalar potential does not exist, since curl E ≠ 0. But if E ·curl E = 0, a scalar quasipotential φ can be introduced, from which the E -vector can be obtained. The use of a quasipotential constitutes a second method for handling TEM waves. The paper produces no new practical results; but it does develop general equations that are free from the ambiguities existing in previous treatments.

The structure of Wick-rotated spinor-spinor Bethe-Salpeter equations is studied in coordinate space. Investigations are extended for nonzero external mass differences, the interaction includes ladder-like contributions of scalar, pseudoscalar, vector and axialvector mesons. After eliminating the γ matrices, a set of coupled partial differential equations is obtained which is written in terms of scalar and three-vector amplitudes. By decomposition of the amplitudes in terms of four-dimensional scalar and three-vector spherical harmonics, two systems of coupled ordinary differential equations are derived for the radial wavefunctions. These equations involve a set of matrix elements which are summarized for arbitrary values of angular momenta in a transparent form. Some important symmetry properties of the radial Bethe-Salpeter equations are discussed. The method is feasible for computations including numerical calculations of Regge trajectories. As a by-product, useful relations for four-dimensional scalar and three-vector spherical harmonics are derived.

The increasing use of Maxwell's equations necessitates a careful consideration of the best formulation for electromagnetic problems. Advance in the past has been hindered by an unfortunate notation that employs the same symbol ▿2 for two entirely different operators—the vector Laplacian and the scalar Laplacian. A general equation is developed in this paper for the vector Laplacian in any orthogonal, curvilinear coordinate system; and this equation allows the simple formulation of all electrodynamic problems by means of the vector Helmholtz equation, ★E ∗ +γ 2 E ∗ =0 .

By means of the asymptotic condition it is shown that starting from a vector field non-vanishing divergence one can introduce a spin-one vector field without changing the four-momentum operator nor the S -matrix.

The a3 coefficient is derived for the scalar field for arbitrary coupling constant ξ by the point separation method.

A proposal has been made recently that a special notation be adopted for the Laplacian of a vector. It is shown that such a notation is unjustified since the usual scalar Laplacian gives the correct result when operating on a vector.