Let G be a simple graph and L = L(G) the Laplacian matrix of G. G is called L-integral if all its Laplacian eigenvalues are integer numbers. It is known that every cograph, a graph free of P4, is L-integral. The class of P4-sparse graphs and the class of P4-extendible graphs contain the cographs. It seems natural to investigate if the graphs in these classes are still L-integral. In this paper ...

We prove an integrability criterion of order 3 for a homogeneous potential of degree −1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the ...

In this paper, necessary and sufficient conditions for the existence of the positive definite solutions of the nonlinear matrix equation X + A*X ! A =Q are presented, when A is nonsingular and s an integer, as well as some new properties and bounds for the eigenvalues of matrices related to A,Q are discussed. An algebraic method for the computation of the solutions is proposed, based on the alg...

We consider the stability problem for the difference system xn = Axn−1 +Bxn−k, where A, B are real matrixes and the delay k is a positive integer. In the case A=−I , the equation is asymptotically stable if and only if all eigenvalues of the matrix B lie inside a special stability oval in the complex plane. If k is odd, then the oval is in the right half-plane, otherwise, in the left half-plane...

In a previous paper (Corrigan-Sasaki), many remarkable properties of classical Calogero and Sutherland systems at equilibrium are reported. For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all “integer valued”. The equilibrium positions of Calogero and Sutherland systems for the classical root systems (Ar, Br, Cr an...

We derive an exchange energy functional of generalized gradient form with a corresponding potential that changes discontinuously at integer particle numbers. The functional is semilocal, yet incorporates key features that are connected to the derivative discontinuity of Kohn-Sham density-functional theory. We validate our construction for several paradigm systems and explain how it addresses ce...

On a compact connected 2m-dimensional Kähler manifold with Kähler form ω, given a smooth function f : M → R and an integer 1 < k < m, we want to solve uniquely in ω the equation ω̃ ∧ωm−k eω, relying on the notion of k-positivity for ω̃ ∈ ω the extreme cases are solved: k m by Yau in 1978 , and k 1 trivially . We solve by the continuity method the corresponding complex elliptic kthHessian equation...

The purpose of this note is to recall the theory of the (homogenized) spectral problem for Schrödinger equation with a polynomial potential and its relation with quadratic differentials. We derive from results of this theory that the accumulation rays of the eigenvalues of the latter problem are in 1 − 1-correspondence with the short geodesics of the singular planar metrics induced by the corre...

Considering the linear delay difference system x n 1 ax n Bx n − k , where a ∈ 0, 1 , B is a p × p real matrix, and k is a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrix B. It is also shown that the stability domain becomes smaller as the delay increases. These results may be successfully applied in the stability...

We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. If the walk is periodic, all eigenvalues of time evolution matrices must algebraic integers. Focusing on this, we explore which ring coefficients characteristic polynomials should belong to. On other hand, $\eta$-Hermitian adjacency have implications. From these, can find...