We study the blow-up of H-perimeter minimizing sets in the Heisenberg group H, n ≥ 2. We show that the Lipschitz approximations rescaled by the square root of excess converge to a limit function. Assuming a stronger notion of local minimality, we prove that this limit function is harmonic for the Kohn Laplacian in a lower dimensional Heisenberg group.

In the present paper, we study rotational surfaces in the three dimensional pseudo-Galilean space G3. Also, we characterize rotational surfaces in G3 in terms of the position vector field, Gauss map and Laplacian operator of the second fundamental form on the surface.

Several systems of differential operators are constructed and their study is commenced. These systems are generalizations, in a reasonable sense, of the Heisenberg Laplacian operators introduced by Folland and Stein. In particular, they admit large groups of conformal symmetries; various real form of the special linear groups, even special orthogonal groups, and the exceptional group of type E6...

The 1D discrete fractional Laplacian operator on a cyclically closed (periodic) linear chain with finite number N of identical particles is introduced. We suggest a ”fractional elastic harmonic potential”, and obtain the N -periodic fractional Laplacian operator in the form of a power law matrix function for the finite chain (N arbitrary not necessarily large) in explicit form. In the limiting ...

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-laplacian matrix, which consists of a dimensional lifting of the laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the st...

This work presents a new method for symmetrization of directed graphs that constructs an undirected graph with equivalent pairwise effective resistances as a given directed graph. Consequently a graph metric, square root of effective resistance, is preserved between the directed graph and its symmetrized version. It is shown that the preservation of this metric allows for interpretation of alge...

In this paper, we investigate the use of heat kernels as a means of embedding graphs in a pattern space. We commence by performing the spectral decomposition on the graph Laplacian. The heat kernel of the graph is found by exponentiating the resulting eigensystem over time. By equating the spectral heat kernel and its Gaussian form we are able to approximate the geodesic distance between nodes ...

The signless Laplacian separator of a graph is defined as the difference between the largest eigenvalue and the second largest eigenvalue of the associated signless Laplacian matrix. In this paper, we determine the maximum signless Laplacian separators of unicyclic, bicyclic and tricyclic graphs with given order.

This paper develops the necessary tools to understand the relationship between eigenvalues of the Laplacian matrix of a graph and the connectedness of the graph. First we prove that a graph has k connected components if and only if the algebraic multiplicity of eigenvalue 0 for the graph’s Laplacian matrix is k. We then prove Cheeger’s inequality (for dregular graphs) which bounds the number of...

In this article, we prove the existence of multiple solutions for following fractional Schrödinger-Poisson system with sign-changing potential (−∆)u+ V (x)u+ λφu = f(x, u), x ∈ R, (−∆)φ = u, x ∈ R, where (−∆)α denotes the fractional Laplacian of order α ∈ (0, 1), and the potential V is allowed to be sign-changing. Under certain assumptions on f , we obtain infinitely many solutions for this sys...