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1

On graduation of enrolment size in a multi-echelon educational system

Ekhosuehi V. U.

Mathematica Applicanda

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2017

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Vol. 45, No. 2

119--126

EN

This paper focuses on an educational system wherein demotion and double promotion are not allowed. The total enrolment in such a system is modelled as a linear model within the context of factor analysis. The goal is to represent the total enrolment in terms of latent factors which generate the flows in the system. The notion of the matrix spectrum and the spectral radius are used to benchmark the specific variances and to approximate the factor loading vector, respectively. Two main theorems are propounded alongside with their proofs. A numerical illustration is given.

PL

Artykuł koncentruje się na systemie edukacyjnym, w którym nie jest możliwe powtarzanie i podwójna promocja są niedozwolone. Całkowita liczba uczestników w takim systemie może być badana w ramach modelu liniowego z zastosowaniem analizy czynnikowej. Celem jest przedstawienie całkowitej liczby rekrutowanych w zależności od ukrytych czynników, które generują przepływy w systemie. Pojęcie spektrum macierzy i promienia spektralnego są używane dla porównania wariancji i do przybliżenia odpowiednio wektora wag czynników. Sformułowano dwa twierdzenia wraz z dowodami. Podano numeryczną ilustrację.

2

Numerical radius inequalities for finite sums of operators

Mirmostafaee A. K., Rahpeyma O. P., Omidvar M. E.

Demonstratio Mathematica

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2014

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Vol. 47, nr 4

963--970

EN

In this paper, we obtain some sharp inequalities for numerical radius of finite sums of operators. Moreover, we give some applications of our result in estimation of spectral radius. We also compare our results with some known results.

3

Some properties of the spectral radius of a set of matrices

Czornik A., Jurgaś P.

International Journal of Applied Mathematics and Computer Science

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2006

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Vol. 16, no 2

183-188

EN

In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.

4

Universal symbols on locally compact abelian groups

Gorin E. A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 2

199-204

EN

Let G be a locally compact abelian group, let X be its dual, let M(G) be the convolution algebra of all regular Borel measures of bounded variation on G, and let M (X) = {my | my is an element of M (G)}. A complex-valued function q on X is called symbol, if (the restriction) q|E is an element of M(X)|E for any compact subset E is a subset of X. If A is a Banach algebra then the group of invertible elements of A is denoted by A* and the spectral radius of a is an element of A is denoted by |a| . We consider continuous representations G is an element of g --> Tg is an element of A* such that ||Tg|| = 1 for all g is an element of G. In this case put my(T) = integral of GTg[my]dg). Obviously, we obtain a representation of M (G) (this representation is denoted by the same symbol T). It can easily be checked that hull(Ker(T)) is a subset of X. It leads to the correct definition of q(T) for all symbols q. Symbol q is called universal, if ||q(T)|| = |q(T)| for all T. The aim of the paper is to give a function-theoretic description of universal symbols.