Сборник материалов международной научной конференции студентов, магистрантов, аспирантов

Ю.М. Ермалович, И.И. Яцишкевич

Республика Беларусь, Брест, БрГУ имени А.С. Пушкина

Научный руководитель – П.Н. Резько

LEGENDRE NUMBER AND POLYNOMINALS THEORY

Adrien-Marie Legendre was a French mathematician. He invented mathematical theories of Legendre transformation and elliptic functions. He worked on quadratic reciprocity law in number theory. He made remarkable contribution to application of analysis to number theory and division of primes. His valuable research on elliptic functions has significant place in the arena of mathematics. Several theories of Legendre acted as a source of inspiration for many mathematical theories like Galois Theory, some parts of Gauss’ statistics and number theory and Abel’s work on elliptic functions. He developed the least squares method, which has wide ranging application in linear regression, signal processing, statistics, and curve fitting. He gave the concept of Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics. Legendre transformation is also used in thermodynamics to obtain the enthalpy and the Helmholtz and Gibbs (free) energies from the internal energy.

Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale from 1795. At the same time, he was associated with the Bureau des longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media. This treatise also brought him to the attention of Lagrange. The Académie des Sciences made Legendre an adjoin member in 1783. In 1789 he was elected a Fellow of the Royal Society [1].

Legendre lost his private fortune in 1793 during the French Revolution. That year, he also married Marguerite-Claudine Couhin, who helped him put his affairs in order. In 1795 Legendre became one of six members of the mathematics section of the reconstituted Académie des Sciences, renamed the Institut National des Sciences et des Arts. From 1799 to 1815, Legendre served as mathematics examiner for graduating artillery students at the École Militaire.

Most of his work was brought to perfection by others: his work on roots of polynomials inspired Galois theory; Abel’s work on elliptic functions was built on Legendre’s; some of Gauss' work in statistics and number theory completed that of Legendre. He developed the least squares method and firstly communicated it to his contemporaries before Gauss which has broad application in linear regression, signal processing, statistics, and curve fitting; this was published in 1806 as an appendix to his book on the paths of comets. Today, the term “least squares method” is used as a direct translation from the French “méthode des moindrescarrés” [2, p. 467].

In 1830 he gave a proof of Fermat’s last theorem for exponent n = 5, which was also proven by Lejeune Dirichlet in 1828. In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss; in connection to this, the Legendre symbol is named after him. He also did pioneering work on the distribution of primes, and on the application of analysis to number theory. His 1798 conjecture of the Prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896.

Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel’s stroke of genius to study the inverses of Jacobi’s functions and solve the problem completely.

He is known for the Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics. In thermodynamics it is also used to obtain the enthalpy and the Helmholtz and Gibbs (free) energies from the internal energy. He is also the namegiver of the Legendre polynomials, solutions to Legendre’s differential equation, which occur frequently in physics and engineering applications, e.g. electrostatics.

Legendre is best known as the author of Éléments de géométrie, which was published in 1794 and was the leading elementary text on the topic for around 100 years. This text greatly rearranged and simplified many of the propositions from Euclid’s Elements to create a more effective textbook.

Legendre died in Paris on 10 January 1833, after a long and painful illness, and Legendre’s widow carefully preserved his belongings to memorialize him. Upon her death in 1856, she was buried next to her husband in the village of Auteuil, where the couple had lived, and left their last country house to the village. Legendre’s name is one of the 72 names inscribed on the Eiffel Tower.

1. 
   Adrien-Marie Legendre Biography [Electronic resource]. – Mode of access: http://www.thefamouspeople.com/profiles/adrien-marie-legendre-591. php#yxWX427iSvkKEjFS.99. – Date of access: 04.03.2015.
   
2. 
   Stephen, M. Stigler Gauss and the Invention of Least Squares / M. Stephen. –1981. – P. 465-474.
   

Cтатья посвящена известному французскому математику Адриену Мари Лежа́ндру. Он первым привел доказательство, что сумма углов треугольника не может быть меньше 180°. Лежандр занимался эллиптическими функциями, сферическими функциями (многочлены Лежандра), вариационным исчислением (признак наличия экстремума интеграла Лежандра), вычислением орбит комет и проблемами простых чисел. Экспериментальным путем ему удалось найти формулу, позволившую с довольно большой точностью вычислить сумму простых чисел до одного миллиона.

А.А. Ефременко
Республика Беларусь, Минск, БГПУ имени М. Танка
Научный руководитель – Д.Ч. Кочерго

LANGUAGE AND CULTURE. INTERCULTURAL COMMUNICATION

In the world everything is interconnected from the smallest microorganisms to global processes in the universe. Culture and language ­– these are two components that cannot exist without each other. Mastering the language of a country, we are immersed in its culture, traditions, customs, learn a lot about their mother tongue, about their attitudes towards certain things and their world as a whole. It would seem that people are essentially the same in its internal structure, and hence the behavior, attitude to things to be the same. But man is unique in everything: in thoughts, actions and their own culture. For example, the English communication requires mandatory use of the word “please”: “Give me the salt, please”, “Please pour me some tea”. In Russian, too, there is the word “please” but for us it is not so commonplace as to the British.

Language is an essential part of any culture, as all the other elements of culture, such as attitude, can be passed through. Also, people through the language express what they think. Language can be defined as a means of communication of people exercising with signs and symbols to help people acquire knowledge and share them with others, share experience from generation to generation.

Over time, in terms of systemic transformation of society, the culture of people, and at the same time and language changed. How often, in everyday communication, do we use the literary word? Increasingly in our speech began to appear the words invented to facilitate specific concepts borrowed from other languages ​​or formed as a result of cuts. It makes our speech less clear for large groups of people, thereby dividing society into different categories.

The possibility of cultural and intercultural communication today is no accident attracts attention. Communication is included today in everyday life in a new and broader sense of the word associated with the term “communication”, but not identical to it.

Image of the world, which is composed of representatives from the various ethnic groups in the process of understanding of diversity in the world, affects the language, as well as determines the specific communicative behaviour of the inhabitants of a country. For example, in English there are two semantically divorced concepts: “house” and “home”, which translated into the Russian language the same way: “house”. The first of these refers to the house as a structure, a building that is more emphasizes the physical characteristics and the second – means personal property: “my home”. This suggests that in the minds of the people, the antithesis of his (own) group is important, that is why there was a division of these concepts.

The differences in the reflection of reality manifested not only by contact of different nationalities, but even within the same ethnic group may appear crops other than “basic”. And this phenomenon can be explained by the emergence of various kinds of slang (youth, professional, etc.).

Cultures differ because contacts between the cultures are weaker than inside them. And if people communicate with other cultures as well as within their own, the culture as such would disappear very quickly. Nevertheless, one cannot deny the fact that the representatives of each culture expressed in their own language is their own vision of the world around them.

Intercultural communications teach people tolerance and understanding. Linguistic aspect is no less important in the issue of cross-cultural communication. Human communication is never interrupted and the exchange of information continues to take its course. To do this, we need a language as a means of communication, and to achieve this, we need to learn to treat other cultures with respect. After starting to respect other cultures, we will be able to communicate on equal terms.

Almost all of the current conflicts in the world were formed on the linguistic and cultural grounds. This happens because people simply do not want to understand other peoples, ethnic groups, social groups. Adhering only to its point of view, we do not give ourselves to develop, to learn to think differently, do not give other people a chance to speak, to share his idea with the world.

We must always remain humans, at any stage of intercultural communication, understand the importance of human relationships, respect other cultures and share other people’s worldview.

1. 
   Язык и культура [Электронный ресурс]. – Режим доступа: http://www.referatbank.ru/referat/preview/8566/referat-yazyk-kultura.html. – Дата доступа: 6.03.2015.
   
2. 
   Межкультурная коммуникация [Электронный ресурс]. – Режим доступа: http://otherreferats.allbest.ru/psychology/00103274_0.html. – Дата доступа: 9.03.2015.
   

Статья посвящена осмыслению важности взаимосвязи между языком и культурой, их влиянию друг на друга и существованию в условиях системной трансформации общества. Показано, как межкультурные коммуникации влияют на нашу речь и культуру. Различия между социальными, этническими группами говорят нам об их разном мировоззрении, о разном уровне культуры и, как следствие, различном языке, искусственно дополненном для придания определенной группе характерных черт, отличающих ее от других. Главным вопросом является воспитание в людях терпимости и уважения к другим культурам и языкам.