Related papers: Homotopy Methods for Eigenvector-Dependent Nonline…
Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers,…
In the paper we suggest the homotopy method for solving of the non linear evolution equation. This method consists of two steps. First is the analytical solution for the linearized version of the non-linear evolution deep in the saturation…
The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or…
We consider the following problem for a fixed graph H: given a graph G and two H-colorings of G, i.e. homomorphisms from G to H, can one be transformed (reconfigured) into the other by changing one color at a time, maintaining an H-coloring…
We developed a new and powerful algorithm by which numerical solutions for excited states in a gravito optical surface trap have been obtained. They represent solutions in the regime of strong nonlinearities of the Gross--Pitaevskii…
We obtain global and local theorems on the existence of invariant manifolds for perturbations of non autonomous linear difference equations assuming a very general form of dichotomic behavior for the linear equation. The results obtained…
In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another…
A highly nonlinear eigenvalue problem is studied in a Sobolev space with variable exponent. The Euler-Lagrange equation for the minimization of a Rayleigh quotient of two Luxemburg norms is derived. The asymptotic case with a "variable…
We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be…
Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be…
We consider the homotopical dynamics on compact orientable surfaces of positive genus g. We establish a sufficient and necessary algebraic criterion for homotopy classes with infinitely many periodic points of maps on such surfaces in terms…
We consider two classes of nonlinear eigenvalue problems with double-phase energy and lack of compactness. We establish existence and non-existence results and related properties of solutions. Our analysis combines variational methods with…
We present previously unknown solutions to the 3D Gross--Pitaevskii equation describing atomic Bose-Einstein condensates. This model supports elaborate patterns, including excited states bearing vorticity. The discovered coherent structures…
In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulted from the Legendre dual-Petrov-Galerkin (LDPG) method for the $m$th-order initial value problem (IVP): $u^{(m)}(t)=\sigma u(t),\,…
An exact, number-conserving solution to the generalized, orbit-dependent pairing problem is derived by introducing an infinite-dimensional algebra. A method for obtaining eigenvalues and eigenvectors of the corresponding Hamiltonian is also…
We consider the most general class of linear boundary-value problems for higher-order ordinary differential systems whose solutions and right-hand sides belong to the corresponding Sobolev spaces. For parameter-dependent problems from this…
We present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by its signed multiindex. The method has the interpretation of a semismooth Newton method applied to certain functions…
Second order nonlinear eigenvalue problems are considered for which the spectrum is an interval. The boundary conditions are of Robin and Dirichlet type. The shape and the number of solutions are discussed by means of a phase plane…
In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential…
In this article we study homotopes of finite-dimensional algebras (not necessarily, associative). In the case of associative algebras we study homotopes by methods of Category theory and give description of so-called well-tempered elements…