Features of representations of exponential functions in HNS of high dimensions by software of hypercomplex calculations
By one of the creators of hyper-complex numbers V. Hamilton was the first proposed a constructive definition of nonlinear transcendental functions from hypercomplex variables. In particular, the researcher proposed to define the exponential function of a quaternion variable as the sum of a power ser...
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Інститут проблем реєстрації інформації НАН України
2021
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Data Recording, Storage & Processing| id |
drspiprikievua-article-239191 |
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Data Recording, Storage & Processing |
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OJS |
| language |
Ukrainian |
| topic |
гіперкомплексна числова система представлення функцій експонента характеристичне число системи комп’ютерної алгебри алгебраїчна операція таблиця Келі hypercomplex number system (HNS) function representation exponent characteristic number computer algebra systems algebraic operation Cayley table |
| spellingShingle |
гіперкомплексна числова система представлення функцій експонента характеристичне число системи комп’ютерної алгебри алгебраїчна операція таблиця Келі hypercomplex number system (HNS) function representation exponent characteristic number computer algebra systems algebraic operation Cayley table Боярінова, Ю. Є. Каліновський, Я. О. Features of representations of exponential functions in HNS of high dimensions by software of hypercomplex calculations |
| topic_facet |
гіперкомплексна числова система представлення функцій експонента характеристичне число системи комп’ютерної алгебри алгебраїчна операція таблиця Келі hypercomplex number system (HNS) function representation exponent characteristic number computer algebra systems algebraic operation Cayley table |
| format |
Article |
| author |
Боярінова, Ю. Є. Каліновський, Я. О. |
| author_facet |
Боярінова, Ю. Є. Каліновський, Я. О. |
| author_sort |
Боярінова, Ю. Є. |
| title |
Features of representations of exponential functions in HNS of high dimensions by software of hypercomplex calculations |
| title_short |
Features of representations of exponential functions in HNS of high dimensions by software of hypercomplex calculations |
| title_full |
Features of representations of exponential functions in HNS of high dimensions by software of hypercomplex calculations |
| title_fullStr |
Features of representations of exponential functions in HNS of high dimensions by software of hypercomplex calculations |
| title_full_unstemmed |
Features of representations of exponential functions in HNS of high dimensions by software of hypercomplex calculations |
| title_sort |
features of representations of exponential functions in hns of high dimensions by software of hypercomplex calculations |
| title_alt |
Особливості побудови представлень експоненціальних функцій у гіперкомплексних числових системах високих вимірностей засобами пакету гіперкомплексних обчислень |
| description |
By one of the creators of hyper-complex numbers V. Hamilton was the first proposed a constructive definition of nonlinear transcendental functions from hypercomplex variables. In particular, the researcher proposed to define the exponential function of a quaternion variable as the sum of a power series similar to the real case.
Over time, this approach has been generalized to other transcendental functions of the hypercomplex variable: trigonometric sine and cosine, hyperbolic sine and cosine, and others, and is now generally accepted. After Hamilton, many works have been used to construct an image of an exponential function from a quaternion. For this purpose, various methods are used, which are based on the symmetric properties of quaternions. Images of other transcendental functions of the quaternion are also constructed: logarithmic function, trigonometric sine and cosine.
These functions have found important use not only in scientific applications: physics, mechanics, but also in technical ones: orientation of a solid body in space, gyroscopy, robotics, etc. The problem of constructing images of nonlinear functions from a hypercomplex variable is reduced to their definition from the point of view of the structure of calculations over the hyper-complex argument and their representation in the form of a hyper-complex function.
Knowledge of the methods of performing algebraic operations in hypercomplex number systems (HNS) allows us to construct form of them in the form of a hypercomplex function when determining linear or nonlinear functions. The construction of the form of transcendental functions from the hypercomplex argument is reduced to the image of power series of type in the form of hypercomplex functions.
In some simple cases, this can be done directly. But in the general case requires the development of special methods. The aim of this work is to solve some algorithmic problems that arise when modeling representations of exponential functions in high-dimensional HNS.
The structure of the algorithm for constructing the representation of an exponential function in hypercomplex number systems of high dimensionality by the method of an associated system of linear differential equations is considered in the paper. The necessary brief information about the software of hypercomplex calculations (SHCC) is given, by means of which the necessary cumbersome operations on symbolic expressions at construction of representation of an exponent in HNS of the fifth dimension are carried out. The work is accompanied by fragments of programs in the environment of SHCC and the results of symbolic calculations. Refs: 14 titles. |
| publisher |
Інститут проблем реєстрації інформації НАН України |
| publishDate |
2021 |
| url |
http://drsp.ipri.kiev.ua/article/view/239191 |
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drspiprikievua-article-2391912021-08-31T12:26:55Z Features of representations of exponential functions in HNS of high dimensions by software of hypercomplex calculations Особливості побудови представлень експоненціальних функцій у гіперкомплексних числових системах високих вимірностей засобами пакету гіперкомплексних обчислень Боярінова, Ю. Є. Каліновський, Я. О. гіперкомплексна числова система, представлення функцій, експонента, характеристичне число, системи комп’ютерної алгебри, алгебраїчна операція, таблиця Келі hypercomplex number system (HNS), function representation, exponent, characteristic number, computer algebra systems, algebraic operation, Cayley table By one of the creators of hyper-complex numbers V. Hamilton was the first proposed a constructive definition of nonlinear transcendental functions from hypercomplex variables. In particular, the researcher proposed to define the exponential function of a quaternion variable as the sum of a power series similar to the real case. Over time, this approach has been generalized to other transcendental functions of the hypercomplex variable: trigonometric sine and cosine, hyperbolic sine and cosine, and others, and is now generally accepted. After Hamilton, many works have been used to construct an image of an exponential function from a quaternion. For this purpose, various methods are used, which are based on the symmetric properties of quaternions. Images of other transcendental functions of the quaternion are also constructed: logarithmic function, trigonometric sine and cosine. These functions have found important use not only in scientific applications: physics, mechanics, but also in technical ones: orientation of a solid body in space, gyroscopy, robotics, etc. The problem of constructing images of nonlinear functions from a hypercomplex variable is reduced to their definition from the point of view of the structure of calculations over the hyper-complex argument and their representation in the form of a hyper-complex function. Knowledge of the methods of performing algebraic operations in hypercomplex number systems (HNS) allows us to construct form of them in the form of a hypercomplex function when determining linear or nonlinear functions. The construction of the form of transcendental functions from the hypercomplex argument is reduced to the image of power series of type in the form of hypercomplex functions. In some simple cases, this can be done directly. But in the general case requires the development of special methods. The aim of this work is to solve some algorithmic problems that arise when modeling representations of exponential functions in high-dimensional HNS. The structure of the algorithm for constructing the representation of an exponential function in hypercomplex number systems of high dimensionality by the method of an associated system of linear differential equations is considered in the paper. The necessary brief information about the software of hypercomplex calculations (SHCC) is given, by means of which the necessary cumbersome operations on symbolic expressions at construction of representation of an exponent in HNS of the fifth dimension are carried out. The work is accompanied by fragments of programs in the environment of SHCC and the results of symbolic calculations. Refs: 14 titles. Розглянуто структуру алгоритму побудови представлення експонен-ціальної функції у гіперкомплексних числових системах (ГЧС) високої вимірності методом асоційованої системи лінійних диференціальних рівнянь. Наведено необхідні короткі відомості про програмний комп-лекс гіперкомплексних обчислень (ПКГО), за допомогою якого проведено необхідні громіздкі операції над символьними виразами при побудові представлення експоненти в ГЧС п’ятої вимірності. Робота супроводжується фрагментами програм у середовищі ПКГО і результатами символьних обчислень. Інститут проблем реєстрації інформації НАН України 2021-06-29 Article Article application/pdf http://drsp.ipri.kiev.ua/article/view/239191 10.35681/1560-9189.2021.23.2.239191 Data Recording, Storage & Processing; Vol. 23 No. 2 (2021); 12-26 Регистрация, хранение и обработка данных; Том 23 № 2 (2021); 12-26 Реєстрація, зберігання і обробка даних; Том 23 № 2 (2021); 12-26 1560-9189 uk http://drsp.ipri.kiev.ua/article/view/239191/237859 Авторське право (c) 2021 Реєстрація, зберігання і обробка даних |