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On rational functions related to algorithms for a computation of roots. I

Baran Mirosław

Science, Technology and Innovation

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2019

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Vol. 7, no. 4

17--25

EN

We discuss a less known but surprising fact: a very old algorithm for computing square root known as the Bhaskara-Brouncker algorithm contains another and faster algorithms. A similar approach was obtained earlier by A.K. Yeyios [8] in 1992. By the way, we shall present a few useful facts as an essential completion of [8]. In particular, we present a direct proof that k – th Yeyios iterative algorithm is of order k. We also observe that Chebyshev polynomials Tn and Un are a special case of a more general construction. The most valuable idea followed this paper is contained in applications of a simple rational function Φ(w; z) = z-w/z+w.

2

On a recurrence for permanents of a sequence of 3-tridiagonal matrices

Trojovský Pavel, Zvoníková Iva

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 4

95--100

EN

This is a corrigendum of the paper: Küçük, A. Z. & Düz, M. (2017). Relationships between the permanents of a certain type of k-tridiagonal symmetric Toeplitz and the Chebyshev polynomials. Journal of Applied Mathematics and Computational Mechanics, 16, 75-86. We will show that Remark 9, on page 84, does not hold, what is the consequence of the incorrect proof, which authors formulated there.

3

Relationships between the permanents of a certain type of k-tridiagonal symmetric Toeplitz matrix and the Chebyshev polynomials

Kucuk A. Z., Duz M.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 1

75--86

EN

In this study, the recursive relations between the permanents of a certain type of the k-tridiagonal symmetric Toeplitz matrix with complex entries and the Chebyshev polynomials of the second kind are presented.

4

A note on the order of polynomial-like iterative equations

Draga S.

Commentationes Mathematicae

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2016

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Vol 56, No. 2

243--249

EN

We show that, under reasonable assumptions, two negative roots can be eliminated from the characteristic equation of a polynomial-like iterative equation. This result gives a new case where we may lower the order of such an equation.

5

Orthogonal polynomials on ellipses and their recurrence relations

Lauric V.

Demonstratio Mathematica

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2014

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Vol. 47, nr 1

137--145

EN

In this note we study the connection between orthogonal polynomials on an ellipse and orthogonal Laurent polynomials on the unit circle relative to some multiplicative measures and then establish the recurrence relations for orthogonal polynomials on an ellipse. The matrix representation of the operator of multiplication by coordinate function is obtained.

6

On a new type Bernstein-Stancu operators

Taşdelen, Başcanbaz-Tunca G., Erençin

Fasciculi Mathematici

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2012

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Nr 48

119--128

EN

In the present paper, firstly we obtain a functional differential equation corresponding to new type Bernstein-Stancu operators defined in [8]. Next we introduce some properties of these new type operators. In the end k-th order generalization of such operators is established.

7

Asymptotic Properties of the Factors of Words Over a Finite Alphabet

Tomescu I.

Fundamenta Informaticae

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2005

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Vol. 64, nr 1-4

463-470

EN

Let A be an alphabet of cardinality m, kn be a sequence of positive integers and w e A* (|w| = kkn). In this paper it is shown that if lim sup n→∞knn <1/lnm, then almost all words of length n over A contain the factor w, but if lim sup n→∞kn/lnn > 1/lnm, then this property is not true. Also, if lim inf...kn/lnn > 1/lnm , then almost all words of length n over A do not contain the factor w. Moreover, if lim...(ln n - knln m) = a e IR, then lim sup...|W(n,kn,w,A)|/m^n < 1-exp(-exp(a)) and liminf n→∞|W(n,kn,w,A)|/m^n >1-exp(-(1-1/m)exp(a)), where W(n,kn,w,A) denotes the set of words of length n over A containing the factor w of length kn.