CG, SYMMLQ, and MINRES are Krylóv subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ’s solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This underst...

The Arnoldi method is currently a very popular algorithm to solve large-scale eigenvalue problems. The main goal of this paper is to generalize the Arnoldi method to the characteristic equation of a delay-differential equation (DDE), here called a delay eigenvalue problem (DEP). The DDE can equivalently be expressed with a linear infinite dimensional operator whose eigenvalues are the solutions...

Abstract. Let φm be a polynomial satisfying some mild conditions. Given a set R ⊂ C, a continuous function f on R and its best approximation p n−1 from Πn−1 with respect to the maximum norm, we show that p ∗ n−1 ◦φm is a best approximation to f ◦ φm on the inverse polynomial image S of R, i.e. φm(S) = R, where the extremal signature is given explicitly. A similar result is presented for constra...

The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A cent...

In this paper, a new approach for solving the system of fractional integro-differential equation with weakly singular kernels is introduced. The method based on class symmetric orthogonal polynomials called shifted sixth-kind Chebyshev polynomials. First, operational matrices are constructed, and after that, described. This reduces equations (WSFIDEs) by collocation into algebraic equations. Th...

The Rayleigh–Ritz method is widely used for eigenvalue approximation. Given a matrix X with columns that form an orthonormal basis for a subspace X , and a Hermitian matrix A, the eigenvalues of XHAX are called Ritz values of A with respect to X . If the subspace X is A-invariant, then the Ritz values are some of the eigenvalues of A. If the A-invariant subspace X is perturbed to give rise to a...

Abstract We introduce an approach to treat localized correlated electronic states in the otherwise weakly host medium. Here, environment is dynamically downfolded on subspace. It captured via renormalization of one and two quasiparticle interaction terms which are evaluated using many-body perturbation theory. outline strategy how take dynamical effects into account by going beyond static limit...

This paper concerns the long-term behavior of population systems, and in particular of chemical reaction systems, modeled by deterministic mass-action kinetics. We approach two important open problems in the field of chemical reaction network theory: the Persistence Conjecture and the Global Attractor Conjecture. We study the persistence of a large class of networks called lower-endotactic and,...

Let H be an integral domain, and let Σ be a collection of integrally closed overrings of H. We show that if A is an overring of H such that H = ( T R∈Σ R)∩A, and if Σ is a Noetherian subspace of the space of all integrally closed overrings of H, then there exists a weakly Noetherian subspace Γ of integrally closed overrings of H such that H = ( T R∈Γ R) ∩ A, and no member of Γ can be omitted fr...