A Criterion for Mapping Homogeneity
УДК 511.8
Abstract
The paper proposes a definition of a homogeneous mapping based on the concept of a group action. This definition can be used to describe generalizations of the concept of "homogeneous function," including positive, absolute, limited homogeneity, lambda-homogeneity, and homogeneous distributions. The main result is as follows: if a group acts on the set of assignments and values of the mapping, and acts commutatively on the set of values, then the mapping is homogeneous with respect to these actions if and only if the homogeneity condition is satisfied for the generating set of this group.
In most cases, the assignment sets and value sets of mappings are vector spaces, and the multiplicative group of the main field or its subgroup acts as an acting group. Therefore, it is important to know their generating sets. For example, for the multiplicative group R+ of positive real numbers, these are numerical intervals, possibly with excluded null sets. Thus, it follows that the homogeneity of a function, understood traditionally, is guaranteed by the fulfillment of the homogeneity condition for any numerical interval from R+. This fact was previously only established for differentiable functions using the well-known Euler identity for homogeneous functions.
Downloads
Metrics
References
Elghibi M., Othman H.A., A.-H.A. Al-Noshri. Homogeneous functions: New characterizations and applications // Transactions of a. Razmadze Mathematical Institute171.2017.
Фихтенгольц Г.М. Курс дифференциального и интегрального исчисления : в 3 т. М., 2009. Т. 1.
Shah V., Sharma J. Extension of Euler’s Theorem of homogeneous functions for finite variables and higher derivatives // Int J Eng Innovative Technol. 2014. Vol. 4. № 1.
Hiwarekar A.P. Extension of Euler’s Theorem of homogeneous functions to higher derivatives // Bull Marathwada Math.Soc. 2009.Vol. 10.№1.
Martinez F., Martinez-Vidal I., Paredes S., Wiley. Conformable Euler’s Theorem of homogeneous functions // A Comp and Math Methods. 2019.1:e 1048. https://doi.org/ 10.1002/cmm4.1048.
Adewumi M. Homogeneous Functions, Eulers Theorem and Partial Molar Quantities // сайт Pen State College of Earth and Mineral Sciences. https://www.e-education.psu. edu/png520/search/node/homogeneous. (дата обращения: 17.11.2022)
Однородная функция //Wikipedia.org:универсаль-ная интернет-энциклопедия со свободным контентом, русскоязычный раздел, 2001. https://ru.wikipedia.org/wiki/ Однородная функция (дата обращения: 17.11.2022).
Бугров Я.С., Никольский С.М. Высшая математика : учебник для вузов : в 3 т. T. 2. Дифференциальное и интегральное исчисление М., 2004.
Гельфанд И.М., Шапиро З.Я. Однородные функции и их приложения // УМН. 1955. Т. 10. Вып. 3 (65).
Поликанова И.В. Отображения, однородные относительно действия групп // Вестник АлтГПА. Серия: Ест. и точн. науки. 2014. № 20.
Поликанова И.В. Некоторые критерии однородности функции // Вестник БГПУ. Серия: Ест. и точн. науки. 2008. № 8.
Copyright (c) 2023 Ирина Викторовна Поликанова
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).