The fundamental relationship between matrices over the rational numbers and a newly defined type of graph, a Kirchhoff graph, is discussed. For a given matrix, a Kirchhoff graph for that matrix represents the orthogonal complementarity of the null and row spaces of the matrix. A number of basic results are proven, and then a relatively complicated Kirchhoff graph is constructed for a matrix tha...

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear comp...

In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coeff...

In this paper, we prove the existence of at least three solutions to the following Kirchhoff nonlocal fractional equation:  M (∫ Rn×Rn |u(x)− u(y)| K(x− y)dxdy − ∫ Ω |u(x)|dx ) ((−∆)u− λu) ∈ θ(∂j(x, u(x)) + μ∂k(x, u(x))), in Ω, u = 0, in R \ Ω, where (−∆) is the fractional Laplace operator, s ∈ (0, 1) is a fix, λ, θ, μ are real parameters and Ω is an open bounded subset of R, n > 2s, with L...