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1

Comparison of main geometric characteristics of deformed sphere and standard spheroid

Kovalchuk Vasyl, Mladenov Ivaïlo M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 5

art. no. e147058

EN

In the paper we compare the geometric descriptions of the deformed sphere (i.e., the so-called λ-sphere) and the standard spheroid (namely, World Geodetic System 1984’s reference ellipsoid of revolution). Among the main geometric characteristics of those two surfaces of revolution embedded into the three-dimensional Euclidean space we consider the semi-major (equatorial) and semi-minor (polar) axes, quartermeridian length, surface area, volume, sphericity index, and tipping (bifurcation) point for geodesics. Next, the RMS (Root Mean Square) error is defined as the square-rooted arithmetic mean of the squared relative errors for the individual pairs of the discussed six main geometric characteristics. As a result of the process of minimization of the RMS error, we have obtained the proposition of the optimized value of the deformation parameter of the λ-sphere, for which we have calculated the absolute and relative errors for the individual pairs of the discussed main geometric characteristics of λ-sphere and standard spheroid (the relative errors are of the order of 10−6 – 10−9). Among others, it turns out that the value of the,sup> flattening factor of the spheroid is quite a good approximation for the corresponding value of the deformation parameter of the λ-sphere (the relative error is of the order of 10−4).

2

Numerical detection of bifurcation point in the curve

Gulgowski J.

Mathematica Applicanda

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2015

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Vol. 43, No. 1

3--18

EN

We are presenting a numerical method which detects the presence and position of a bifurcation simplex, the regular -dimensional simplex, which may be considered as "fat bifurcation point", in the curve of zeroes of the C1 map f : Rk+1 → Rk. On the other hand the bifurcation simplex appears in the neighbourhood of the bifurcation point, meaning that we have the method to locate the bifurcation point as well. The method does not require any estimation of the derivative of the function and refers to the values of the map only in the vertices of certain triangulation. The bifurcation simplex is detected by change of the Brouwer degree value of the restriction of the map to the appropriate -simplex.

PL

W pracy podany jest opis metody numerycznej wykrywającej sympleksy bifurkacji, foremne (k + 1)-wymiarowe sympleksy, które można uważać za ”pogrubione punkty bifurkacji”, leżące na krzywej zawierającej zera odwzorowania klasy C1 f : Rk+1 → Rk. Sympleksy bifurkacji znajdują się (zwykle) w pobliżu punktów bifurkacji, co oznacza, że podana metoda pozwala zlokalizować przybliżone położenie punktów bifurkacji leżących na krzywej w zbiorze zer odwzorowania f. Podana metoda nie wymaga żadnych oszacowań na pochodną odwzorowania f – odwołuje się jedynie do wartości odwzorowania f w wierzchołkach sympleksów pewnej triangulacji przestrzeni Rk+1. Sympleks bifurkacji wykrywany jest poprzez zmianę wartości stopnia topologicznego (stopnia Brouwera) na odpowiednich obcięciach f do k-wymiarowych sympleksów zawartych w przestrzeni Rk+1.

3

Phenomenon of solutions branching in the problems of the objects placement and experimental design

Grigoriev Y. D.

Metody Informatyki Stosowanej

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2011

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nr 5

5-20

EN

The examples of solutions branching are presented in the construction of D- and A-optimal placement of beacons in distance measurement problem of navigation and in the construction of Bayesian and maximin D-efficient designs of experiment..

4

Some theorems of Rabinowitz type for nonlinearizable eigenvalue problems

Przybycin J.

Opuscula Mathematica

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2004

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Vol. 24, no. 1

115-121

EN

We discuss the structure of the solution set for nonlinearizable eigenvalue problems in a Hilbert space.

5

Behaviour of Solutions to Marchuk's Model Depending on a Time Delay

Bodnar M., Foryś U.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

97-112

EN

Marchuk's model of an immune reaction is a system of differential equations with a time delay. The aim of this paper is to study the behaviour of solutions to Marchuk's model depending upon the delay of immune reaction and the history of an illness. We study Marchuk's model without delays, with aconstant delay and with an infinite delay. A continuous dependence on thedelay is considered. Bifurcation points are found using computer simulations.