Практикум для подготовки к экзамену по английскому языку (для студентов математического факультета I и II курсов)

Федеральное агентство по образованию Омский государственный университет им. Ф.М. Достоевского

УДК 802.0 ББК 81.2Англ.ж721 П69

Рекомендован к изданию редакционно-издательским советом ОмГУ Рецензент – ст. преподаватель кафедры иностранных языков Э.К. Сопелева

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Практикум для подготовки к экзамену по английскому языку (для студентов математического факультета I и II курсов)

Практикум для подготовки к экзамену по английскому языку (для студентов математического факультета I и II курсов) / Сост. Л.В. Жилина. – Омск: Изд-во ОмГУ, 2005. – 44 с. ISBN 5-7779-0600-1 Состоит из 5 разделов: I – тексты для письменного перевода со словарем; II – тексты и ключевые выражения для реферирования; III – разговорные темы; IV – чтение математических формул; V – лексический минимум, кроссворды для закрепления лексики и контрольные тесты. Для студентов математического факультета I и II курсов. УДК 802.0 ББК 81.2Англ.ж721

Изд-во ОмГУ

Омск 2005

ISBN 5-7779-0600-1

© Омский госуниверситет, 2005 2

ПРЕДИСЛОВИЕ Практикум для подготовки к экзамену по английскому языку предназначен для студентов I и II курсов математического факультета и включает в себя 5 разделов. Первый раздел содержит 3 текста, по сложности соответствующие экзаменационным текстам для письменного перевода с использованием словаря. Второй раздел посвящен реферированию текста без словаря. Для облегчения выполнения этого задания помимо текста дается план реферирования и фразы, помогающие грамотно изложить содержание статьи. Третий раздел – изложение разговорной темы. Он состоит из 6 текстов (ко второй и третьей теме даны дополнительные тексты), являющихся примерными разговорными темами, включенными в экзамен. Четвертый раздел поможет студентам научиться читать математические формулы, встречающиеся в текстах. Пятый раздел – приложение, в которое входят слова и словосочетания, часто встречающиеся в специализированных текстах, посвященных разным разделам математики, два кроссворда на знание математической лексики, три грамматических теста-задания и пример экзаменационного билета. Практикум предназначен для более эффективной подготовки студентов к экзамену по английскому языку.

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PART I Text 1 Translate the Text with dictionary in written form (time 45 minutes) About a Line and a Triangle Given ∆ ABC, extend the side AB beyond the vertices. Now, rotate the line AB around the vertex A until it falls on the side AC. Next rotate it (from its new position) around C until it falls on the side BC. Lastly, rotate it around B till it takes up its erstwhile position. It is virtually obvious that although the line now occupies exactly the same position as before, something has changed. After three rotations, the line turned around 180°. So, for example, the point A will now lie on a different side from B than before. We say that turning the line around the triangle changed its orientation. It appears that the line occupies the same position but not quite: points on the line did not preserve their locations. However, since there are just two possible orientations of the line, we come up with an interesting question: what happens to the line after it turns around the triangle twice? Will it occupy its original position exactly (point-for-point)? The answer is easily obtained from the following observation. After the first rotation the line occupies the same position but with a different orientation. Let's turn the line into coordinate axis. In other words, let's choose the origin – point O, the unit of measurements, and the positive direction. If, after the rotation, the point originally at the distance x from O will be now located at the position b-x. Therefore, there exists one point on the line that does not move even after a single rotation. This is the fixed point of the transformation. The fixed point solves the equation x = b-x. The rotation of the line around the triangle is simply equivalent to the rotation of the line around that point through 180°.

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Text 2

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Translate the Text with dictionary in written form (time 45 minutes)

Translate the Text with dictionary in written form (time 45 minutes)

Computer Algebra

Computer Software in Science and Mathematics

Symbols as well as numbers can be manipulated by a computer. New, general-purpose algorithms can undertake a wide variety of routine mathematical work and solve intractable problems by Richard Pavelle, Michael Rothstein and John Fitch

Computation offers a new means of describing and investigating scientific and mathematical systems. Simulation by computer may be the only way to predict how certain complicated systems evolve by Stephen Wolfram

Of all the tasks to which the computer can be applied none is more daunting than the manipulation of complex mathematical expressions. For numerical calculations the digital computer is now thoroughly established as a device that can greatly ease the human burden of work. It is less generally appreciated that there are computer programs equally well adapted to the manipulation of algebraic expressions. In other words, the computer can work not only with numbers themselves but also with more abstract symbols that represent numerical quantities. In order to understand the need for automatic systems of algebraic manipulation it must be appreciated that many concepts in science are embodied in mathematical statements where there is little point to numerical evaluation. Consider the simple expression 3π2/π. As any student of algebra knows, the fraction can be reduced by cancelling π from both the numerator and the denominator to obtain the simplified form 3π. The numerical value of 3π may be of interest, but it may also be sufficient, and perhaps of greater utility, to leave the expression in the symbolic, nonnumerical form. With a computer programmed to do only arithmetic, the expression 3 π2/π must be evaluated; when the calculation is done with a precision of 10 significant figures, the value obtained is 9,424777958. The number, besides being a rather uninformative string of digits, is not the same as the number obtained from the numerical evaluation (to 10 significant figured) of 3π. The latter number is 9,424777962; the discrepancy in the last two decimal places results from round-off errors introduced by the computer. The equivalence of 3 π2/π and 3π would probably not be recognized by a computer programmed in this way. 5

Scientific laws give algorithms, or procedures for determining how systems behave. The computer program is a medium in which the algorithms can be expressed and applied. Physical objects and mathematical structures can be represented as numbers and symbols in a computer, and a program can be written to manipulate them according to the algorithms. When the computer program is executed, it causes the numbers and symbols to be modified in the way specified by the scientific laws. It thereby allows the consequences of the laws to be deduced. Executing a computer program is much like performing an experiment. Unlike the physical objects in a conventional experiment, however, the objects in a computer experiment are not bound by the laws of nature. Instead they follow the laws embodied in the computer program, which can be of any consistent form. Computation thus extends the realm of experimental science: it allows experiments to be performed in a hypothetical universe. Computation also extends theoretical science. Scientific laws have conventionally been constructed in terms of a particular set of mathematical functions and constructs, and they have often been developed as much for their mathematical simplicity as for their capacity to model the salient features of a phenomenon. A scientific law specified by an algorithm, however, can have any consistent form. The study of many complex systems, which have resisted analysis by traditional mathematical methods, is consequently being made possible through computer experiments and computer models. Computation is emerging as a major new approach to the science, supplementing the long-standing methodologies of theory and experiment. 6

PART II Text Rendering of the text (15 min) Acta Mathematica Academiae Scientiarum Hungaricae Tomus 26 (1–2), (1975), 41–52.

FREE INVERSE SEMIGROUPS ARE NOT FINITELY PRESENTABLE

This paper contains two main results. The first one coincides with the title, the second consists in a description of free inverse semigroups (if a free inverse semigroup is presented as a quotient algebra of a free involuted semigroup, then each element of F£ is a class of equivalent words, we give a canonical form of the words). Certain corollaries with properties of free inverse semigroups follow. All results of the paper were reported by the author at a meeting of the semin-nar "Semigroups" in Saratov State University on October 21, 1971.

By В.М. SCHEIN (Saratov) In the memory of Professor A. Kertesz

Free inverse semigroups became a subject of intense studies in the last few years. Their existence was proved long ago: as algebras with two operations (binary multiplication and unary involution) inverse semigroups may be characterized by a finite system of identities, i.e. they form a variety of algebras. Therefore, free inverse semigroups do exist. A construction of a free algebra in a variety of algebras (as a quotient algebra of an absolutely free word algebra) is well known. Free inverse semigroups in such a form were considered by V.V. VAGNER who found certain properties of such semigroups. A monogenic free inverse semigroup (i.e. a free inverse semigroup with one generator) was described by L.M. GLUSKIN. Later this semigroup was described by H.E. SCHEIBLICH in a slightly different form. The most essential progress in this direction was made in a paper by H.E. SCHEIBLICH who described arbitrary free inverse semigroups. A relevant paper by C. EBERHART and J. SELDEN should be mentioned. There are papers on some special properties of free inverse semigroups. N.R. REILLY described free inverse subsemigroups of free inverse semigroups, results in this direction were obtained also by W.D. MUNN and L. O'CARROLL. Let F£x denote the free inverse semigroup with the set X of free generators. A monogenic free inverse semigroup will be denoted F£1. Time and then we will write F£ instead of F£x. We do not consider F£∅ a one-element inverse semigroup. 7

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THE PLAN FOR RENDERING THE ARTICLE 1. The headline of the article, the author of the article, where and when the article was published The article is headlined... The headline of the article I have read... The article is entitled... The headline of the article is... The author of the article is... The article is written by... It was published in ...(newspaper) on...(date) (on May 23d, 2003) It was printed in... The article under the title... was published in ... on ... 2. The main idea of the article The main idea of the article is... Basically, the article is about... The article is devoted to the problem of... The article touches upon... The article dwells upon... The article tells the readers about... The author discusses an important problem of... The purpose of the article is to give the reader some information on... The aim of the article is to provide the reader with some material (data) on... 3. The contents of the article (some facts, figures) The author starts by telling the reader that... The author writes (states, stresses, thinks, points out) that... The author highlights the fact that... The author focuses on the fact that... According to the text... Further the author reports (says) that... The article goes on to say that... In conclusion... 9

Finally... The author comes to the conclusion that... 4. Your opinion on the article, your attitude towards it I find the article interesting, important, useful, informative, upto-date, disputable, dull, of no value, too hard to understand... On reading the article I realize the fact that… Additional task Reproduce the text in your own words Differential equations give adequate models for the overall properties of physical processes such as chemical reactions. They describe, for example, the changes in the total concentration of molecules: they do not, however, account for the motions of individual molecules. These motions can be modelled as random walks: the path of each molecule is like the path that might be taken by a person in a milling crowd. In the simplest version of the model the molecule is assumed to travel in a straight line until it collides with another molecule; it then recoils in a random direction. All the straight-line steps are assumed to be of equal length. It turns out that if a large number of molecules are following random walks, the average change in the concentration of molecules with time can in fact be described by a differential equation called the diffusion equation. There are many physical processes, however, for which no such average description seems possible. In such cases differential equations are not available and one must resort to direct simulation. The motions of many individual molecules or components must be followed explicitly; the overall behavior of the system is estimated by finding the average properties of the results. The only feasible way to carry out such simulations is by a computer experiment: essentially no analysis of the systems for which analysis is necessary could be made without the computer.

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PART III Speak on the topic 1. Read and translate the text Topic 1 A modern view of geometry For a long time geometry was intimately tied to physical space, actually beginning as a gradual accumulation of subconscious notions about physical space and about forms, content, and spatial relations of specific objects in that space. We call this very early geometry "subconscious geometry". Later, human intelligence evolved to the point where it became possible to consolidate some of the early geometrical notions into a collection of somewhat general laws or rules. We call this laboratory phase in the development of geometry "scientific geometry". About 600 B.C. the Greeks began to inject deduction into geometry giving rise to what we call "demonstrative geometry". In time demonstrative geometry becomes a material-axiomatic study of idealized physical space and of the shapes, sizes, and relations of idealized physical objects in that space. The Greeks had only one space and one geometry; these were absolute concepts. The space was not thought of as a collection of points but rather as a realm or locus, in which objects could be freely moved about and compared with one another. From this point of view, the basic relation in geometry was that of congruence With the elaboration of analytic geometry in the first half of the seventeenth century, space came to be regarded as a collection of points; and with the invention, about two hundred years later of the classical non-Euclidean geometries. But space was still regarded as a locus in which figures could be compared with one another. Geometry came to be rather far removed from its former intimate connection with physical space, and it became a relatively simple matter to invent new and even bizarre geometries. At the end of the last century, Hilbert and others formulated the concept of formal axiomatics. There developed the idea of branch of mathematics as an abstract body of theorems deduced from a set of 11

postulates. Each geometry became, from this point of view, a particular branch of mathematics. In the twentieth century the study of abstract spaces was inaugurated and some very general studies came into being. A space became merely a set of objects together with a set of relations in which the objects are involved, and a geometry became the theory of such a space. The boundary lines between geometry and other areas of mathematics became very blurred, if not entirely obliterated. There are many areas of mathematics where the introduction of geometrical terminology and procedure greatly simplifies both the understanding, and the presentation of some concept or development. The best way to describe geometry today is not as some separate and prescribed body of knowledge but as a point of view – a particular way of looking at a subject. Not only is the language of geometry often much simpler and more elegant than the language of algebra and analysis, but it is at times possible to carry through rigorous trains of reasoning in geometrical terms without translating them into algebra or analysis. A great deal of modern analysis becomes singularly compact and unified through the employment of geometrical language and imagery. 2. Retell the text

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1. Read and translate this text Topic 2 N.L. Lobachevsky

Non-Euclidean geometry has developed into an extremely useful instrument for application in the physical world. After 1840 Lobachevsky published a number of papers on convergence of infinite series and the solution of defined integrals. In modern books on defined integrals about 200 integrals was solved by Lobachevsky.

N.I. Lobachevsky was born in 1792 in Nizhny Novgorod. After his father death in 1797 the family moved to Kazan where Lobachevsky graduated from the University. He stayed in Kazan all his life occupying the position of dean of the faculty of Physics and Mathematics and rector of the University. He lectured on mathematics, physics and astronomy. Lobachevsky is the creator of non-Euclidean geometry. His first book appeared in 1829. Few people took notice of it. Non-Euclidean geometry remained for several decades an obscure field of science. Most mathematicians ignored it. The first leading scientist who realized its full importance was Riemann. There is one axiom of Euclidean geometry. This is the famous postulate of the unique parallel, which states that through any point not on a given line one and only one line can be drawn parallel to the given line. It goes without saying that there are many lines through a point, which do not intersect a given line within any distance, however large. So, this axiom can never be verified by experiment. All the other axioms of Euclidean geometry have a finite character since they deal with finite portions of lines and with plane figures of finite extent. The fact that the parallel axiom is not experimentally verifiable raises the question of whether or not it is independent of the other axioms. If it were a necessary logical consequence of the other, then "it would be possible not to regard it as an axiom and to give a proof of it in terms of the other Euclidean axioms. For centuries mathematicians have tried to find such a proof and in the long run were appeared that the parallel postulate is really independent of the others. What does the independence of the parallel postulate mean? Simply that it is possible to construct a consistent system of geometrical statements dealing with points and lines, by deduction from a set of axioms in which the parallel postulate is replaced by a contrary postulate. Such a system is called non-Euclidean geometry. Lobachevsky settled the question by constructing in all detail a geometry in which the parallel postulate does not hold.

An extraordinary woman, Sofia Kovalevskaya was not only a great mathematician, but also a writer and advocate of women's rights in the 19th century. It was her struggle to obtain the best education available which began to open doors at universities to women. In addition, her ground-breaking work in mathematics made her male counterparts reconsider their archaic notions of women's inferiority to men in such scientific arenas. Sofia Kovalevskaya was born in 1850. As the child of a Russian family of minor nobility, Sofia was raised in plush surroundings. She was not a typically happy child, though. She felt very neglected as the middle child in the family of a well admired, first-born daughter, Anya, and of the younger male heir, Fedya. For much of her childhood she was also under the care of a very strict governess who made it her personal duty to turn Sofia into a young lady. As a result, Sofia became fairly nervous and withdrawn – traits which were evident throughout her lifetime. Sofia's exposure to mathematics began at a very young age. She claims to have studied her father's old calculus notes that were papered on her nursery wall in replacement for a shortage of wallpaper. Sofia

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2. Retell the text Additional text (Topic 2) Read and translate this text Sofia Kovalevskaya January 15, 1850 – February 10, 1891 Kovalevskaya Stamps issued in 1951 and 1996. Written by Becky Wilson, Class of 1997 (Agnes Scott College)

credits her uncle Peter for first sparking her curiosity in mathematics. He took an interest in Sofia and found time to discuss numerous abstractions and mathematical concepts with her. When she was fourteen years old she taught herself trigonometry in order to understand the optics section of physics book that she was reading. The author of the book and also her neighbor, Professor Tyrtov, was extremely impressed with her capabilities and convinced her father to allow her to go off to school in St. Petersburg to continue her studies. After concluding her secondary schooling, Sofia was determined to continue her education at the university level. However, the closest universities open to women were in Switzerland, and young, unmarried women were not permitted to travel alone. To resolve the problem Sofia entered into a marriage of convenience to Vladimir Kovalevsky in September 1868. The couple remained in Petersburg for the first few months of their marriage and then travelled to Heidelburg where Sofia gained a small fame. People were enthralled by the quiet Russian girl with an outstanding academic reputation. In 1870, Sofia decided that she wanted to pursue studies under Karl Weierstrass at the University of Berlin. Weierstrass was considered one of the most renowned mathematicians of his time, and at first he did not take Sofia seriously. Only after evaluating a problem set he had given her did he realize the genius at his hand. He immediately set to work privately tutoring her because the university still would not permit women to attend. Sofia studied under Weierstrass for four years. She is quoted as having said, "These studies had the deepest possible influence on my entire career in mathematics. They determined finally and irrevocably the direction I was to follow in my later scientific work: all my work has been done precisely in the spirit of Weierstrass". At the end of her four years she had produced three papers in the hopes of being awarded a degree. The first of these, "On the Theory of Partial Differential Equations," was even published in Crelle's journal, a tremendous honor for an unknown mathematician. In July of 1874, Sofia Kovalevskaya was granted a Ph.D. from the university of Gottingen. Yet even with such a prestigious degree and the help of Weierstrass, who had grown quite fond of his pupil, she was not able to find employment. She and Vladimir decided to return to her family in Palobino. Shortly after her return home, her father died

unexpectedly. It was during this period of sorrow that Sofia and Vladimir fell in love. Their marriage produced one daughter. While at home, Sofia neglected her work in mathematics but instead developed her literary skills. She tried her hand at fiction, theater reviews, and science articles for a newspaper. In 1880, Sofia returned to work in mathematics with a new fervor. She presented a paper on Abelian integrals at a scientific conference and was very well received. Once again she was faced with the dilemma of finding employment doing what she loved most – mathematics. She decided to return to Berlin, also home to Weierstrass. She was not there long before she learned of Vladimir's death. He had committed suicide when all of his business ventures had collapsed. Sofia's grief threw her into her work more passionately than ever. Then, in 1883, Sofia's luck took a turn for the better. She received an invitation from an acquaintance and former student of Weierstrass, Gosta Mittag-Leffler, to lecture at the University of Stockholm. In the beginning it was only a temporary position, but at the and of a five year period, Sofia had more than proved her value to the university. Then came a series of great accomplishments. She gained a tenured position at the university, was appointed an editor for a mathematics journal, published her first paper on crystals, and in 1885, was also appointed Chair of Mechanics. At the same time, she co-wrote a play, "The Struggle for Happiness," with her friend, Anna Leffler. In 1887, Sofia again received devastating news. The death of her sister, Anya, was particularly hard on Sofia because the two had always been very close. Fortunately, it was not long afterward that Sofia achieved "her greatest personal triumph". In 1888, she entered her paper, "On the Rotation of a Solid Body about a Fixed point," in a competition for the Prix Bordin by the French Academy of Science and won. "Prior to Sofia Kovalevsky's [Sofia Kovalevskaya] work the only solutions to the motion of a rigid body about fixed point had been developed for the two cases where the body is symmetric". In her paper, Sofia developed the theory for an unsymmetrical body where the center of its mass is not on an axis in the body. The paper was highly regarded that the prize money was increased from 3000 to 5000 francs. Also at this time, a new man entered her life. Maxim Kovalesky came to Stockholm for a series of lectures. There he met Sofia, and the

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two had a scandalous, rocky affair. The basic problem was that they were both too passionate about their work to give it up for the other. Maxim's work took him away from Stockholm and he wanted Sofia to give up her hard-earned positions to simply be his wife. Sofia flatly rejected such an idea but still could not bear the loss of him. She remained in France with him for the summer and fell into another one of her frequent depressions. Again, she turned to her writing. While she was in France, she finished Recollections of Childhood. In the fall of 1889, she returned to Stockholm. She was still miserable at the loss of Maxim even though she frequently travelled to France to visit him. She eventually became ill with depression and pneumonia. On February 10, 1891, Sofia Kovaleskaya died and the scientific world mourned her loss. During her career she published ten papers in mathematics and mathematical physics and also several literary works. Many these scientific papers were ground-breaking theories or the impetus for future discoveries. There is no question that Sofia Krukovsky Kovalevskaya was an incredible person. The President of the Academy of Sciences, which awarded Sofia the Prix Bordin, once said: "Our co-members have found that her works bear witness not only to profound and broad knowledge, but to mind of great inventiveness".

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1. Read and translate the text Topic 3 The Nature of Mathematics Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems. As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth. The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics – theorems and theories – are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty. In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Experience with mathematical modes of thought builds mathematical power – a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the informationladen world in which we live. During the first half of the twentieth century, mathematical growth was stimulated primarily by the power of abstraction and deduction, climaxing more than two centuries of effort to extract full benefit from the mathematical principles of physical science formulated by Isaac Newton. Now, as the century closes, the historic alliances of mathematics with science are expanding rapidly; the highly

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developed legacy of classical mathematical theory is being put to broad and often stunning use in a vast mathematical landscape. Several particular events triggered periods of explosive growth. The Second World War forced development of many new and powerful methods of applied mathematics. Postwar government investment in mathematics, fueled by Sputnik, accelerated growth in both education and research. Then the development of electronic computing moved mathematics toward an algorithmic perspective even as it provided mathematicians with a powerful tool for exploring patterns and testing conjectures. At the end of the nineteenth century, the axiomatization of mathematics on a foundation of logic and sets made possible grand theories of algebra, analysis, and topology whose synthesis dominated mathematics research and teaching for the first two thirds of the twentieth century. These traditional areas have now been supplemented by major developments in other mathematical sciences – in number theory, logic, statistics, operations research, probability, computation, geometry, and combinatorics. In each of these subdisciplines, applications parallel theory. Even the most esoteric and abstract parts of mathematics – number theory and logic, for example – are now used routinely in applications (for example, in computer science and cryptography). Fifty years ago, the leading British mathematician G.H. Hardy could boast that number theory was the most pure and least useful part of mathematics. Today, Hardy's mathematics is studied as an essential prerequisite to many applications, including control of automated systems, data transmission from remote satellites, protection of financial records, and efficient algorithms for computation. In 1960, at a time when theoretical physics was the central jewel in the crown of applied mathematics, Eugene Wigner wrote about the "unreasonable effectiveness" of mathematics in the natural sciences: "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." Theoretical physics has continued to adopt (and occasionally invent) increasingly abstract mathematical models as the foundation for current theories. For example, Lie groups

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and gauge theories – exotic expressions of symmetry – are fundamental tools in the physicist's search for a unified theory of force. During this same period, however, striking applications of mathematics have emerged across the entire landscape of natural, behavioral, and social sciences. All advances in design, control, and efficiency of modern airliners depend on sophisticated mathematical models that simulate performance before prototypes are built. From medical technology (CAT scanners) to economic planning (input/output models of economic behavior), from genetics (decoding of DNA) to geology (locating oil reserves), mathematics has made an indelible imprint on every part of modern science, even as science itself has stimulated the growth of many branches of mathematics. Applications of one part of mathematics to another – of geometry to analysis, of probability to number theory – provide renewed evidence of the fundamental unity of mathematics. Despite frequent connections among problems in science and mathematics, the constant discovery of new alliances retains a surprising degree of unpredictability and serendipity. Whether planned or unplanned, the cross-fertilization between science and mathematics in problems, theories, and concepts has rarely been greater than it is now, in this last quarter of the twentieth century. 2. Retell the text Additional text (Topic 3) Read and translate the text MATHEMATICS Mathematics is queen of natural knowledge

Mathematics grew up with civilization as man’s quantitative needs increased. It arose out of practical and man’s needs. As soon as man began to count even on his fingers mathematics began. It was the first of sciences to develop formally. It is growing faster today than in its early beginnings . New questions are always arising partly from 20

practical problems and partly from pure theoretical problems. In each generation men have developed new methods and ideas to solve these problems. Why has mathematics become so impotent in recent years? Can the new electronic brains solve mathematical problems faster and more accurately than a person and eliminate the needs for mathematicians? To answer these questions we need to know what mathematics is and how it is used. Mathematics is much more than the arithmetic, which is the science of number and computation. It is algebra, which is the language of symbols, operations and relations. It is much more than the geometry, which is the study of shape, size and space. It is much more than the statistics, which is the science of interpreting data and graphs. It is much more than the calculus, which is the study of change, limits, and infinity. Mathematics is all these and more. Mathematics is a way of thinking and a way of reasoning, where new ideas are being discovered every day. It is way of thinking that is used to solve all kinds of problems in the government and industry. Mathematics method is reasoning of the highest level known to man and every field of investigation – be it law, politics, psychology, medicine or anthropology – has felt its influence. There are various ways in which mathematics serves scientific investigation: 1. Mathematics supplies a language for the treatment of the quantitative problems of the physical and social sciences. 2. Mathematics supplies science with numerous methods and conclusions. 3. Mathematics enables the science to make predictions. 4. Mathematics supplies science with ideas to describe phenomena. The language of mathematics is precise and concise, it is a language of symbols, that is understood in all civilized nations of the world. Mathematics style aims at brevity and formal perfections. The student must always remember that the understanding of any subject in mathematics a clear and define knowledge of what precedes. This is the reason why the study of mathematics is discouraging to weak minds.

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1. Read and translate the text Topic 4 Students studies at the faculty of Mathematics Last year I passed my entrance exams successfully, and entered Omsk State University, which was founded in 1974. Now I am a second year student at the faculty of mathematics. The faculty of mathematics is one of the largest in our university, It trains research workers, school teachers and instructors for higher school, and technical schools. So the graduates of the faculty work in different branches of industry, research laboratories, computer centers and educational establishments. There are 7 chairs in the faculty: the chair of mathematical analysis, the chair of mathematical modeling, the chair of methods of teaching, the chair of algebra, the chair of logic and logical programming, the chair of applied mathematics and the new chair of informational systems. The teaching staff of our faculty is highly qualified. 8 doctors of sciences and many candidates of sciences work there. The academic year is divided into 2 terms. At the end of each term students have their credit tests and take their terminal exams. State scholarship is paid to the advanced students. At the end of the course of studies students are to submit a graduation paper and pass their final state examinations. The full university course lasts 5 years. During this period of time the maths students take 3 years of general courses followed by 2 year of specialized training in some special fields of mathematics. The main aim of the first stage of the mathematics program is to provide a broad and solid foundation for professional knowledge. The curriculum is rather wide and versatile. The math students study general education subjects, such as philosophy, political economy, foreign languages and so on. All the students study modern computers and the way of using them in different kinds of calculations. They spend a lot of time in the well-equipped labs of the faculty operating with computers and compiling programs. The syllabus also offers a wide range of specialized courses. Acquiring skills in research is the major goal of the final stage of the mathematics program. 22

The student does research mainly for his graduation paper, which reflects the knowledge and practical skills; he has gained in his special field. It is, as a rule, a small research project carried out by the student under the guidance of a supervisor. Then the student submits his graduation paper and defends it before an examination board. If he does this with honors he may be recommended to take postgraduate course. 2. Retell the text

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1. Read and translate the text Topic 5 20th century mathematics Twentieth-century mathematics is highly specialized and abstract. The advance of set theory and discoveries involving infinite sets, transfinite numbers, and purely logical paradoxes caused much concern as to the foundations of mathematics. In addition to purely theoretical developments, devices such as high-speed computers influenced both the content and the teaching of mathematics. Among the areas of mathematical research that have been developed in the 20th century are abstract algebra, non-Euclidean geometry, abstract analysis, mathematical logic, and the foundations of mathematics. Modern abstract algebra includes the study of groups, rings, algebras, lattices, and a host of other subjects developed from a formal, abstract point of view. This approach formed the cornerstone of the work of a group of mathematicians called Bourbaki. Bourbaki uses abstract algebra in an axiomatic framework to develop virtually all branches of higher mathematics, including set theory, algebra, and general topology. The significance of non-Euclidean geometry was realized early in the 20th century when the geometry was applied in mathematical physics. It has come to play an essential role in the theory of relativity. Another area of mathematics, abstract analysis, has produced theories of the derivatives and integrals in abstract and infinitedimensional spaces. There are many areas of special interest in the field of abstract analysis, including functional analysis, harmonic analysis, families of functions and so on. The most notable development in the area of logic began in the 20th century with the work of two English logicians and philosophers, Bertrand Russell and Alfred North Whitehead. The object of their three-volume publication, "Prinseipia Mathematical” was to show that mathematics can be deduced from a very small number of logical principles. The foundations of mathematics have many "schools". At the turn of the century, David Hilbert was determined to preserve the powerful methods of transfinite set theory and the use of the infinite in 24

mathematics, despite apparent paradoxes and numerous objections (see Hilbert). His program was virtually abandoned in the 1930s when Kurt Godel demonstrated that for any general axiomatic system there are always theorems that cannot be proved or disproved (see Godel). Hilbert's followers, known as formalists, view mathematics in terms of abstract structures. The oldest philosophy of mathematics is usually ascribed to Plato. Platonism asserts the existence of eternal truths, independent of the human mind. In this philosophy the truths of mathematics arise from an abstract, ideal reality. 2. Retell the text

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1. Read and translate the text Topic 6 Algebra Algebra from arabic means the reunion of broken parts. The branch of mathematics, which deals in the most general way with properties and relations of numbers. The first known use of the word is in the title of a book by Muhammad ibn-Musa al-Khwarizmi, one of the most important Arab mathematicians of the 9-th century. The story of algebra begins from 1800 B.C. when any problem that we should now solve by algebra was solved by guessing or by some cumbersome arithmetic process. Then Alexandrian school had appeared and geometric method was in use. In the 3rd century mathematicians began to use symbols to write an equation. And in 17th century algebra already allows us the equation ax2+bx+c=0. Algebra is a generalization of arithmetic. Each state of arithmetic deals with particular numbers. The square of the sum of any two numbers a and b can be computed by the rule (a+b)2 = a +2ab+b. This is a general rule which remains true no matter what particular numbers may replace the symbols a and b. A rule of this kind is often called a formula. Algebra is the system of rules concerning the operations with numbers. The operations of addition, subtraction, multiplication, division, raising to a power are called algebraic expressions. Algebraic expressions consisting of more than one term are called multinomials. Algebra is the base of modern science, so formulas and expressions are widely used not only in mathematics but also in physics, chemistry, biology and others. Many branches of science need to handle formulas and expressions automatically. That's why computer algebra is the rapidly developed branch of algebra For numerical calculations the digital computer is now thoroughly established as a device that can greatly ease the human burden of work. It is less generally appreciated that there are computer programs equally well adapted to the manipulation of algebraic expressions. Algebraic programs have three advantages over purely numerical ones. First it is frequently more economical of computer time to simplify an expression algebraically before evaluating it numerically. Sec26

ond, unlike the numerical approximations generated by a computer, algebraic answers are exact. The third and perhaps the most important advantage is that the goals of scientific investigation are often better served by a result in algebraic form. The algebra developing history shows that algebra is gradually transforming in many other branches of science that have their own direction. 2. Retell the text

PART IV Wording of mathematics formulae ½ 0 + – * () a` a`` F1 F2 ab` x2 y3 z-10 C x>0 x