Last time, we talked about a very simple solver for Laplacian-based systems of linear equations, i.e., systems of linear equations of the form Ax = b, where the constraint matrix A is the Laplacian of a graph. This is not fully-general—Laplacians are SPSD matrices of a particular form—but equations of this form arise in many applications, certain other SPSD problems such as those based on SDD m...

Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical numerical approaches. The purpose of this work is study the effects spectral Laplacian on bifurcation structure reaction-diffusion systems bounded domains. In order do we use advanced continuation tech...

In this paper, we study the following fractional Schrödinger-poisson systems involving fractional Laplacian operator { (−∆)su+ V (|x|)u+ φ(|x|, u) = f(|x|, u), x ∈ R3, (−∆)tφ = u2, x ∈ R3, (1) where (−∆)s(s ∈ (0, 1)) and (−∆)t(t ∈ (0, 1)) denotes the fractional Laplacian. By variational methods, we obtain the existence of a sequence of radial solutions. c ©2016 All rights reserved.

Boman and Hendrickson [BH01] observed that one can solve linear systems in Laplacian matrices in time O ( m ln(1/ǫ) ) by preconditioning with the Laplacian of a low-stretch spanning tree. By examining the distribution of eigenvalues of the preconditioned linear system, we prove that the preconditioned conjugate gradient will actually solve the linear system in time Õ ( m ln(1/ǫ) ) .

The p-Laplacian problem is one of the typical examples of degenerate nonlinear systems arising from nonlinear diffusion and filtration, powerlaw materials and quasi-Newtonian flows. In this article we give a survey of quasi-norm techniques to establish optimal error estimates for finite element approximation of p-Laplacian.

let $a = (a_{i,j})_{1 leq i,j leq n}$ be an $n times n$ matrixwhere $n geq 2$. let $dt(a)$, its second immanant be the immanant corresponding to the partition $lambda_2 = 2,1^{n-2}$. let $g$ be a connected graph with blocks $b_1, b_2, ldots b_p$ and with$q$-exponential distance matrix $ed_g$. we given an explicitformula for $dt(ed_g)$ which shows that $dt(ed_g)$ is independent of the manner in ...

In this paper, we investigate the existence of multiple periodic solutions for two classes of nonlinear difference systems involving (φ1,φ2)-Laplacian. First, by using an important critical point theorem due to B. Ricceri, we establish an existence theorem of three periodic solutions for the first nonlinear difference system with (φ1,φ2)-Laplacian and two parameters. Moreover, for the second no...