Related papers: On generalized divergence and Laplace operators as…
Taking as starting point the approach to the divergence operator on weighted graphs, we give a notion of divergence associated to transport coupling and coupled measures on locally compact Hausdorff spaces. We consider the induced Laplacian…
We study invariants under gauge transformations of linear partial differential operators on two variables. Using results of BK-factorization, we construct hierarchy of general invariants for operators of an arbitrary order. Properties of…
In this article we discuss the convergence of first order operators on a thickened graph (a graph-like space) towards a similar operator on the underlying metric graph. On the graph-like space, the first order operator is of the form…
We consider a generalization of the classical Laplace operator, which includes the Laplace-Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this…
We consider the normalized Laplace operator for directed graphs with positive and negative edge weights. This generalization of the normalized Laplace operator for undirected graphs is used to characterize directed acyclic graphs. Moreover,…
The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural…
The operators of fractional calculus come in many different types, which can be categorised into general classes according to their nature and properties. We conduct a formal study of the class known as weighted fractional calculus and its…
We extend to manifolds endowed with a general geometric structure, the classical notions of gradient as well as Laplace operator, and provide some of their natural properties.
In this note we study the application of generalized fractional operators to a particular class of nonstandard Lagrangians. These are typical of dissipative systems and the corresponding Euler-Lagrange and Hamilton equations are analyzed.…
This thesis generalizes the differential operators on standard oriented graphs and oriented hypergraphs introduced in 10.1137/15M1022793 and arXiv:2007.00325. The extended concepts of gradients, adjoints and $p$-Laplacians for vertices and…
Chemical hypergraphs and their associated normalized Laplace operators are generalized and studied in the case where each vertex--hyperedge incidence has a real coefficient. We systematically study the effect of symmetries of a hypergraph…
We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the sense of Sobolev embeddings. Our approach is of variational and…
The $p$-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex $p$-Laplacian and a hyperedge…
Pointwise estimates for the gradient of solutions to the $p$-Laplace system with right-hand side in divergence form are established. They enable us to develop a nonlinear counterpart of the classical Calder\'on-Zygmund theory in terms of…
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $\Psi$-fractional calculus. The operational calculus approach has proved useful…
Motivated by the problem of understanding theoretical bounds for the performance of the Belkin-Niyogi Laplacian eigencoordinate approach to dimension reduction in machine learning problems, we consider the convergence of random graph…
In this article we give a brief overview of some known results in the theory of obstacle-type problems associated with a class of fourth-order elliptic operators, and we highlight our recent work with collaborators in this direction.…
Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension $\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators $$…
Here we introduce connectivity operators, namely, diffusion operators, general Laplacian operators, and general adjacency operators for hypergraphs. These operators are generalisations of some conventional notions of apparently different…
We consider a generalization of the diffusion equation on graphs. This generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one-dimensional graphs. The generalization is based on the $k$-path…