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Elements of differential equations in the mathematics course for students of economics

100%
Biernacki M.
Didactics of Mathematics
|
2015
|
issue 12(16)
5-10
XX
The article presents a proposal of one lecture with elements of differential equations included in a 30-hour course in mathematics for students of economics at the University of Economics in Wrocław. The author puts forward a presentation of some basic methods for solving first order differential equations exemplified by two macroeconomic growth models: the Domar model and the Solow model.
2
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Prace z równań różniczkowych w „Pamiętniku Akademii Umiejętności w Krakowie”

88%
Koroński J.
Zagadnienia Filozoficzne w Nauce
|
2013
|
issue 53
199-230
PL
This paper concerns the general characteristics of the Academy of Arts and Sciences in Cracow and the Memoirs of the Academy of Arts and Sciences in Cracow. Moreover, in the context of the global development of the theory of differential equations we present in this paper the articles of Alojzy Jan Stodółkiewicz (1856-1934), Władysław Zajączkowski (1837-1898), Jan Rajewski (1857-1906), Wawrzyniec Żmurko (1824-1889) and Edward Władysław Skiba (1843-1911) on differential equations, which were published in the Memoirs of the Academy of Arts and Sciences in Cracow.
3
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Prace z równań różniczkowych w „Pamiętniku Towarzystwa Nauk Ścisłych w Paryżu”

88%
Koroński J.
Zagadnienia Filozoficzne w Nauce
|
2013
|
issue 53
231-262
PL
This paper concerns the general characteristics of the Natural Science Society in Paris and the Memoirs of the Natural Science Society in Paris. Moreover, in the context of the development of the theory of differential equations in the world, we present in this paper the articles of Y. Villarceau (1813), W. Zajączkowski (1837–1898) and W. Folkierski (1842–1904) on differential equations, which were published in Memoirs of the Natural Science Society in Paris.
4

Analiza dynamiki modeli wzrostu gospodarczego za pomocą środowiska obliczeniowego Mathematica

63%
Zawadzki H.
Zeszyty Naukowe Uniwersytetu Ekonomicznego w Krakowie
|
2015
|
issue 4(940)
59-69
PL
Podstawowy aparat matematyczny teorii wzrostu gospodarczego (w przypadku modeli z czasem ciągłym) stanowią równania różniczkowe oraz sterowanie optymalne, czyli wyznaczanie rozwiązań równań różniczkowych spełniających określone warunki i optymalizujących pewien funkcjonał całkowy. W artykule przedstawiono (na przykładzie modeli wzrostu Mankiwa-Romera-Weila oraz Lucasa-Uzawy) wybrane możliwości programu Mathematica w zakresie: numerycznego rozwiązywania równań różniczkowych (równań ruchu) i wyznaczania optymalnych wartości funkcjonałów (całkowitej zdyskontowanej użyteczności) w przypadku, gdy rozwiązania optymalne wyznaczone są jedynie w sposób numeryczny, symbolicznego różniczkowania rozwiązań równań różniczkowych, wyrażonych za pomocą pewnej klasy funkcji specjalnych – funkcji hipergeometrycznych Gaussa oraz graficznej prezentacji rozwiązań równań różniczkowych.
EN
Differential equations and optimal control theory are the basic tools of the mathematical theory of economic growth. The aim of this paper is to show (using the example of the Mankiw-Romer-Weil and Lucas-Uzawa growth models) some capabilities of Mathematica in the symbolic and numerical solving of differential equations, calculating optimal values of integral functional (discounted lifetime utility) when only numerical optimal solutions are known, the symbolic differentiation of solutions given in terms of hypergeometric Gauss functions and the graphical presentation of solutions to differential equations.
5
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Governmental combat of migration between competing terrorist organisations

51%
HAUSKEN K.
Operations Research and Decisions
|
2020
|
vol. 30
|
issue 3
21-46
EN
Each terrorist organisation is modelled with four coupled differential time equations for the evo-lution of ideologues, criminal mercenaries, captive participants, and capital sponsoring. Emigration of ideologues may cause unbounded growth of the organisation receiving ideologues. The organisation losing ideologues may reach a stationary state where ideologues are supported by capital sponsors and mercenaries. Emigration of mercenaries may cause the organisation losing mercenaries to experience growth. The organisation receiving mercenaries may lose capital sponsors permanently, allowing for the presence of mercenaries, or capital sponsors may rebound deterring mercenaries. Emigration of ideologues from one organisation to another requires more government intervention into the latter to ensure termination. Emigration of mercenaries from one organisation to another may require more gov-ernment intervention into the latter, since mercenaries support ideologues. Competing terrorist organi-sations may facilitate their mutual extinction. Various intervention strategies are considered: the most threatening organisation is eliminated first, aided by competition from the least threatening, after which the remaining organisation is eliminated. The government’s instantaneous and accumulated utilities are analysed through time and compared, depending on emigration, competition, and government interven-tion strategies.
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