VIII. Choose the correct answer to the questions.

1. What are ‘paper mills’?

1. Software programs that detect plagiarism.

2. Websites that offer ready-made essays.

2. Do you always have to pay to download an essay?

1. .Yes.

2. .No, not always.

3. When would a teacher start to suspect plagiarism?

1. If the level of the students’ work seemed too high.

2. If the level of the students’ work seemed too low.

4. When would a British teacher become suspicious?

1. If one of their students turned in an essay with lots of spelling mistakes.

2. If one of their students turned in an essay with lots of American spellings.

5. What can happen to students who are found guilty of plagiarism?

1. They can be thrown out of their school.

2. They can be taken to the police.

Part IV

The Development of Mathematical Science

Lesson 1

THE DEVELOPMENT OF MATHEMATICAL SCIENCE IN UKRAINE

Mathematics has been a teaching subject within the present boarders of Ukraine since the ancient times of Kiev Rus, although mainly restricted at that time to elements of arithmetic. In the 16^th –17^th centuries arithmetic and geometry were taught along with theological subjects at the Brotherhood seminaries in Lviv, Lutsk, Kiev, Vinnytsya and other Ukrainian cities.

With the foundation in the 19^th century of the universities in Kharkiv (1805), Kiev (1834) and Odessa(1865), which had departments of physics and mathematics, a new period was initiated in the development of mathematics in Ukraine. T.F.Osipovsky (1765-1832) was an eminent organizer of mathematical education in Ukraine during the first half of the 19^th century. Many of his students became professors of mathematics at the Universities of Kiev and Kharkiv. A. F. Pavlovsky(1789-1857) also participated in disseminating mathematical education. It was at that time that two Ukrainian mathematicians M. V.Ostrogradsky (1801-1862) and V.Y.Bunyakovsky(1804-1889), rose to distinguished positions and for several decades were the intellectual leaders of mathematical thought in Russia.

M.Y.Vashchenko-Zakharchenko (1825-1912) occupied a place of honour among the outstanding mathematicians of Ukraine. He taught mathematics for nearly half a century at Kiev University and wrote a number of monographs and textbooks. In his work Symbolic Calculus and its Application in Integration of Linear Differential Equations he anticipated some present-day approaches to operational calculus and to the theory of equations with a delayed argument. The scientists delivered a course of lectures on non-Euclidean geometry as early as the 1870’s

In the second half of the 19^th century the development of mathematical investigations in Ukraine was influenced by two remarkable scientific schools, those of St. Petersburg and Moscow. Scientific mathematical societies were formed at the Ukrainian universities: the Kharkiv Mathematical Society in 1879 and the Kiev Physics and Mathematics Society in 1890.

Mathematical science gained great success in Ukraine after the Great October Socialist Revolution. Many mathematicians were trained in Soviet times, whose contributions to mathematical science are of world significance.

The mathematics departments of Ukrainian universities play an important part in the development of mathematical research and in training scientists.

Mathematicians of Kiev University have been and are now elaborating the theory of differential equations: ordinary and partial, algebra, mathematical physics, the theory of complex variable functions and its application, the laws of probability and mathematical statistics.

Kharkiv University, where S. M. Bernstein (1880-1968) was a leading figure in the 1930’s, was and remains one of the mathematical centre in Ukraine. Important researches have been conducted at the Odessa and Dnipropetrovsk Universities in the field of functional analysis and its application, the theory of differential equations and a constructive theory of functions.

In the 1930’s investigations were started in Kiev on asymptotic methods in the theory of different equations, the theory of oscillations, theory of functions, etc. Proceeding from the general concepts and methods of non-linear mechanics, which were developed by M. M. Krylov(1879-1955) and M.M.Bogolyubov (1909), in the 1950’s Y.O. Mytropolsky (1917), Y.S. Shtokalo (1897) and others solved a number of problems pertaining to asymptotic approximations.

The Institute of Mathematics of the Ukrainian Academy of Sciences, organized in 1934, is the leading mathematical centre in Ukraine. The Institute of Mathematics has been closely associated with the work of many Ukrainian mathematicians.

In the development of mathematical science the Ukrainian scientists hold a place of honour. It was in the Ukrainian SSR that the study of many important trends in contemporary mathematics was initiated: constructive theory of functions, non linear mechanics, etc. Ukrainian mathematicians have published a number of monographs and textbooks which have been included in the treasure –house of the Ukrainian and world mathematical literature. Many Ukrainian scientists have been trained in Ukraine and are now working in various fields of modern mathematics,