The eigenvalue shift technique is the most well-known and fundamental tool for matrix computations. Applications include the search of eigeninformation, the acceleration of numerical algorithms, the study of Google’s PageRank. The shift strategy arises from the concept investigated by Brauer [1] for changing the value of an eigenvalue of a matrix to the desired one, while keeping the remaining ...

1. Endomorphisms of vector spaces Throughout these notes, k is an algebraically closed field, varieties are reduced and irreducible k-schemes of finite type, and morphisms of varieties are k-morphisms. A moduli problem for a class of algebraic objects consists in two parts: finding the equivalence classes of the objects under a suitable equivalence relation (usually isomorphism), and parametriz...

We consider discretization of the ‘geometric cover problem’ in the plane: Given a set P of n points in the plane and a compact planar object T0, find a minimum cardinality collection of planar translates of T0 such that the union of the translates in the collection contains all the points in P . We show that the geometric cover problem can be converted to a form of the geometric set cover, whic...

An approach of design of stable of control systems with ultimately wide ranges of uncertainly disturbed parameters is offered. The method relies on using of nonlinear structurally stable functions from catastrophe theory as controllers. Theoretical part presents an analysis of designed nonlinear second-order control systems. As more important the integrators in series, canonical controllable fo...

We study log canonical thresholds (also called global canonicalthreshold or $\alpha$-invariant) of $\mathbb{R}$-linear systems. prove existence positive lower bounds in different settings, particular, proving a conjecture Ambro. then show that the Borisov-Alexeev-Borisov holds; is, given natural number $d$ and real $\epsilon$, set Fano varieties dimension with $\epsilon$-log singularities forms...

Matrix functions are used in many areas of linear algebra and arise in numerical applications in science and engineering. In this paper, we introduce an effective approach for determining matrix function f(A)=g(q(A)) of a square matrix A, where q is a polynomial function from a degree of m and also function g can be a transcendental function. Computing a matrix function f(A) will be time- consu...

Transfer matrices are used widely for the dynamic analysis of engineering structures, increasingly so for static analysis, and are particularly useful in the treatment of repetitive structures for which, in general, the behaviour of a complete structure can be determined through the analysis of a single repeating cell, together with boundary conditions if the structure is not of infinite extent...

We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, selfadjoint extensions of s...