Introduction to Mathematics and Computer Science: Учебное пособие

М И Н И СТ Е РСТ В О О БРА ЗО В А Н И Я И Н А У К И РФ Ф Е Д Е РА Л ЬН О Е А ГЕ Н СТ В О П О О БРА ЗО В А Н И Ю В О РО Н Е Ж СК И Й ГО СУ Д А РСТ В Е Н Н Ы Й У Н И В Е РСИ Т Е Т

INTRODUCTION TO MATHEMATICS AND COMPUTER SCIENCE У ч ебное п ос обие Г С Э .Ф .01

В О РО Н Е Ж 2005

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У тверж денонаучно-методическим советом ф-таРГФ (протокол № 5 от28 ию ня 2005 г.)

Составител и:

СтернинаМ .А . В оротниковаМ .И .

У чеб ное пособ ие подготовл ено на кафедре англ ийского язы ка факул ьтета романо-германской фил ол огии В оронеж скогогосударственногоуниверситета. П редназначенодл я студентов 1 курса дневногоотдел ения ф-та прикл адной математики, информатики и механики, об учаю щ ихся по специал ьности « П рикл адная математикаи информатика» .

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Н астоящ ее учеб ное пособ ие явл яется частью создаваемого на кафедре англ ийского язы ка учеб но-методического компл екса дл я студентов дневного отдел ения ф-та прикл адной математики, информатики и механики, об учаю щ ихся по специал ьности « П рикл адная математика и информатика» (010200) и предназначенодл я студентов 1 курса. П особ ие состоит из четы рех раздел ов, три из которы х посвящ ены соответственнотрем основны м математическим дисципл инам, читаемы м на 1 курсе факул ьтета: Э В М и программирование, математический анал из, ал геб раи геометрия, ачетверты й содерж иттексты дл я допол нител ьногочтения. А вторы ставят своей цел ью знакомствостудентов с л ексикой, характерной дл я каж дой из перечисл енны х математических дисципл ин, об учение их адекватному пониманию и переводу соответствую щ их текстов, а такж е привитие им навы ков аннотирования текстов поспециал ьности. Грамматический материал пособ ия охваты вает систему времен англ ийского гл агол а в действител ьном и страдател ьном зал огах, а такж е нел ичны е гл агол ьны е формы . В прил ож ении приводится список л ексико-грамматических кл ише дл я аннотирования и реферирования научны х текстов, а такж е об ъясняется чтение основны х математических формул .

И СП О Л ЬЗО В А Н Н А Я Л И Т Е РА Т У РА 1. Santiago Remacha Esteras. Infotech. – Cambridge University Press, 2002. 2. V.A. Zorich. Mathematical Analysis. – London: Springer, 2003. 3. R. Solomon. Abstract Algebra. – Brooks Cole Publishing Company, 2003.

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UNIT 1. COMPUTERS AND PROGRAMMING SECTION 1. 1. Read the international words and guess their meaning: term

fundamental

algorithm

operator

computer

result

instruction

problem

arithmetic

discrete

algebra

mathematics

procedure 2. Read the following words and memorize their meaning: appearance (n)

появл ение

version (n)

версия

branch (n)

отрасл ь

concept (n)

понятие, идея

notion (n)

понятие, представл ение

engineering (n)

техника

exact (a)

точны й

intelligibility (n)

понятность, доступность

intelligible (a)

понятны й

executor (n)

испол нител ь

property (n)

свойство

discreteness (n)

дискретность

vagueness (n)

неточнотсь, неясность

determinacy (n)

детерминированность

termination (n)

окончание

finite (a)

конечны й

to achieve (v)

достигать

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3. Memorize the following word combinations: sequence of operations

посл едовател ьность операций

distinguishing feature

отл ичител ьная черта

to carry out

вы пол нять

to take into account

принимать вовнимание

to have an influence

оказы вать внимание

due to

из-зачего-то, б л агодаря чему-то

both ... and

как, таки

to come into usage

войти в употреб л ение

4. Read the text. ALGORITHM The term “algorithm” has come into usage quite recently. Its appearance in our life is due to the rapid rise of computer science which has the study of algorithm as its focal point. The word “algorithm” originated in the Middle East. It comes from the Latin version of the last name of the Persian scholar Abu Jafar Mohammed ibn Musa al-Khowaresmi (Algorithmi), whose textbook on arithmetic, written in 825 A. D.*, gave birth to algebra as an independent branch of mathematics. In the 12th century this textbook was translated into Latin and it had a great influence for many centuries on the development of computing procedures. The name of the textbook’s author became associated with computation in general and used as a term “algorithm”. The concept of an algorithm is now one of the most fundamental notions, both in mathematics and engineering. An algorithm is defined as an exact and intelligible order for a certain executor to carry out a sequence of operations, aiming at getting certain results or solving a given problem. An algorithm has 5 properties of its own. The first of them is called discreteness. This property means that the process under description is to be separated into certain steps (instructions).

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The second property of an algorithm may be called its intelligibility. It means that an algorithm should take into account, what orders an executor can understand and carry out and what orders he or it cannot. The next distinguishing feature of an algorithm is that all vagueness must be eliminated – each instruction must have one single meaning. This property of an algorithm is called the property of determinacy. Another property of an algorithm is its mass character, which means that a given algorithm may be used for solving a certain class of problems. The last property of an algorithm is its effectiveness. It means that the exact carrying out of all orders of the algorithm should lead to termination of the process after a finite number of steps. * A.D. – anno Domini /л ат./ - нашей э ры 5. Answer the following questions: 1. When did the term “algorithm” come into usage? 2. Why is this term widely spread nowadays? 3. What is the origin of this term? 4. What is the definition of an algorithm? 5. How many properties does an algorithm have? Speak of each of them. 6. Who can be an executor of an algorithm, to your mind?

SECTION 2.

1. Read the international words and guess their meaning: modern

machine

logic

electronic

mechanism

control

command

activities

section

2. Read the following words and memorize their meaning: digital (a)

цифровой

device (n)

устройство

to store (v)

хранить, записы вать

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to accept (v)

принимать, пол учать

to process (v)

об раб аты вать

software (n)

программное об еспечение

hardware (n)

аппаратная часть

equipment (n)

об орудование

unit (n)

единица, э л емент

byte (n)

б айт

bit (n)

б ит

input (n)

вход

output (n)

вы ход

data (n)

данны е

storage (n)

память

punched card (n)

перфокарта

punched tape (n)

перфол ента

keyboard

кл авиатура

printer (n)

печатаю щ ее устройство

peripheral (n,a)

периферийное устройство, периферийны й

buffer (n)

б уфер

speed (n)

скорость

to transmit (v)

передавать

capacity (n)

емкость, об ъем памяти

3. Memorize the following word combination: to store information

хранить информацию

internal storage

внутренняя память

external storage

внешняя память

access time

время об ращ ения кпамяти

to distinguish between

дел ать разл ичие меж ду

by means of

посредством, спомощ ью

as compared with

посравнению счем-то

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that is why

поэ тому

as well as

такж е как

to take place

происходить, иметь место

4. Read the text. COMPUTERS Modern electronic digital computers have become means of solving mathematical and other scientific problems. The sphere of their application is now practically boundless. An electronic computer is a device that can accept information, store it, process it and present the result of the processing in some acceptable form. When we speak about a computer, we must distinguish between its hardware and software. Hardware represents the material part of the machine, its equipment, while software is represented by computer instruction and programs. The computer is told what operations to perform by means of instructions. An instruction is a command to the computer. The basic unit of information for a computer is called a byte. A byte consists of eight bits. The word “bit” is formed from the letter “b” in the word “binary” and the letters “it” in the word “digit”. A bit is the smallest unit of information. The part of a computer that takes in information is called the input unit. It is the functional part of the computer that accepts the data to be operated and programs for operating. Input to the computer used to be provided by punched cards and punched tapes. Nowadays it is provided by a keyboard. The part of a computer that puts out the information is called the output unit. The computer can put out information in a form acceptable to people with the help of printer and display. Input and output device are usually called peripherals. These peripheral devices are rather slow as compared with the computer. That is why many computers have special buffers. The aim of the buffer is to provide a better match between the speed of internal electronic operations and input and output operations. Buffers thus are storage devices accepting information at a very high

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speed from the computer and releasing it at the proper speed for the peripheral equipment. The part of a computer which stores information is called storage of memory. It is the mechanism that can retain information during calculation and transmit it as needed to other parts of the machine. Storage is characterized by 2 main factors: an access time and capacity. Memory access time is the time required to transmit one computer word out of the memory to where it will be used. The capacity of a computer memory is the quantity of data and programs that the memory unit can hold. According to these two factors we can speak about 2 types of storage – internal, capable of quick access, and external storage, providing large capacity. The central processing unit (CPU) or central processor is the nerve center of any computer system. It coordinates and controls the activities of all other units and has 2 hardware section: an arithmetic and logic unit, and a central unit. The CPU has two functions: it must obtain instructions from the memory and interpret them as well as perform the actual operations. The first function is executed by the central unit. The second function of the CPU is performed by the arithmetic and logic unit. Thus the arithmetic and logic unit is that part of the CPU in which the actual computations take place. The central unit is that part of the CPU which obtains instructions from the memory, interprets them and generates the control signals.

5. Answer the following questions: 1. What is an electronic computer? 2. How can you define hardware? 3. What is software? 4. What is an instruction? 5. How do we call the basic unit of information for a computer? 6. How many bits does a byte consist of? 7. How can you define the input unit? 8. How can you define the output unit?

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9. How are input and output devices usually called together? 10. What is the aim of a buffer? 11. How can you define a buffer? 12. What is storage? 13. What is the synonym of the word “storage”? 14. What are the 2 main factors storage is characterized by? 15. What are the functions of the central processing unit? 16. How many parts is CPU composed of? 17. What is the general purpose of the central unit? 18. What is the arithmetic and logical unit responsible for? 19. What are the main parts of a computer?

6. Retell the text using the questions given in the previous exercise as a plan.

7. Read the summary of the text given below and write a summary of the text “Algorithm”. Use the expressions, given in supplement 1. The text given under the title “Computers” deals with the structure of modern computers. It carries material on the main parts of a computer: input, output, storage, central processor and defines them. The author also dwells upon such devices as buffers and speaks about the functions of the central processor. The text is concluded by the definitions of the arithmetic and central units of the CPU.

SECTION 3.

1. Read the international words and guess their meaning: process

code

program

popular

catalogue

commercial

2. Read the following words and memorize their meaning:

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subroutine (n)

подпрограмма

assembler (n)

ассемб л ер

to require (v)

треб овать

to constitute (v)

составл ять

available (adj)

доступны й, подходящ ий

advantage (n)

преимущ ество

familiar (adj)

знакомы й

acronym (n)

акроним

version (n)

вариант

3. Memorize the following word combinations: programming language

язы кпрограммирования

assembly language

язы кассемб л ер

machine-dependent language

машинно-зависимы й язы к

the so-called

такназы ваемы й

to stand for

об означать

to gain popularity

завоевы вать попул ярность

low-level language

язы книзкогоуровня

high-level language

язы квы сокогоуровня

4. Read the text. PPOGRAMMING Programming is the process by which a set of instructions is produced for a computer to make it perform a certain procedure. The word “program” means the sequence of instructions which a computer carries out. A program usually consists of subroutines or subprograms. A subroutine is part of a program which constitutes a logical section of the program. It is written only once in the program, but may be used many times during the computation.

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Many common subroutines are used quite often. That is why it is necessary to have a library of subroutines stored in some part of the computer memory and available to the programmer. In order to make a computer perform certain instructions or commands, people use the so-called programming languages. The most primitive type of programming language is known as an assembly language. It provides commands that are very similar to the machine language of the computer. The assembly language is the most machine-dependent one. The advantage of this language is that it is easy for the computer to understand it. The assembly language is a low level language which is oriented to the machine code of a computer. But there are also the so-called high level languages which allow users to write in a notation which they are more familiar with. High level languages are usually aimed at a certain problem. There are several types of high level languages. One of them is FORTRAN. FORTRAN is an acronym for FORmula TRANslation. It is a problem oriented high level programming language for scientific and mathematical use. FORTRAN was the first high level programming language. It was introduced in 1954. There were several versions of FORTRAN. Another high level language is PL/1. It was introduced in 1964. PL/1 stands for Programming Language 1. It is a general-purpose high level programming language for scientific and commercial applications. ALGOL is another high level language. It was introduced in the early 1960s and gained popularity in Europe more than in the United States. ALGOL is an acronym for ALGOrithmic Language. It is a problem oriented high level programming language for mathematical and scientific use. COBOL is an acronym for COmmon Business Oriented Language. It is internationally accepted programming language developed for general commercial use.

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BASIC is a high level programming language designed by solving mathematical and business problems. It was developed in 1965 and stands for Beginners Allpurpose Symbolic Instruction Code. One more high level programming language is PASCAL. It is a general-purpose high level language named after the French mathematician Blaise Pascal. C is a modern high level programming language, designed in the 1970s for usage with UNIX operating system. It replaced the programming language B which had been intended for UNIX. C is much more flexible than other high level languages and due to this it has now become a widely used professional language for various reasons. There are two modifications of this language – C+ and C++. One more modern high level programming language is Java. It looks a lot like C++, but its strength lies in a slightly different area than that of C++. A well-written Java program is generally far simpler and much easier to understand than the equivalent C program. Java’s error handling is a big improvement over most other languages which lead to greater programming productivity.

5. Answer the following questions: 1. What is programming? 2. How can you define a program? 3. What is a subroutine? 4. What are advantages of the assembly language? 5. What high level programming languages do you know? 6. What do the names of these programming languages stand for? 7. What are the most modern programming languages? 8. What are the modern programming languages characterized by?

6. Write a summary of the text. Use expressions given in Supplement 1.

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LEXICAL EXERCISES TO UNIT 1. 1. Find equivalents: 1. device

1. вы вод

2. punched card

2. устройство

3. storage capacity

3. хранить информацию

4. calculating procedure

4. матоб еспечение

5. branch of mathematics

5. ввод

6. to store information

6. перфокарта

7. software

7. память компью тера

8. input

8. об ъем памяти

9. programming language

9. язы кпрограммирования

10. output

10. об л асть математики

11. computer storage

11. процессвы числ ения

2. Insert the proper word: 1. The concept of ............... is now one of the most fundamental notions both in mathematics and engineering. 2. The next distinguishing feature of an algorithm is that all ............... must be eliminated. 3. An electronic computer is ............... that can accept information, store it, process it and present the result of processing in some acceptable form. 4. ............... represents the material part of the machine, its equipment. 5. ............... is represented by computer instructions and programs. 6. The basic unit of information for a computer is called ............... 7. The part of a computer that takes in information is called ............... 8. The part of a computer that puts out information is called ............... 9. Input and output devices are usually called ............... 3. Give English equivalents: устройствовводаи вы вода цифровы е компью теры

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принимать, хранить и об раб аты вать информацию материал ьная часть компью тера периферийное устройство наб оринструкций подпрограмма отл ичител ьная черта основная единицаинформации

UNIT 2.

MATHEMATICAL ANALYSIS

SECTION 1 1. Read the international words and guess their meaning: limit

analysis

element

mathematics

function

2. Read the following words and word combinations, memorize their meaning: mapping (n)

отоб раж ение

set (n)

множ ество

quantity (n)

кол ичество, вел ичина

infinite (adj)

б есконечны й

definition (n)

определ ение

members of sequence

чл ены посл едовател ьности

as follows

сл едую щ им об разом, сл едую щ ее

to converge (v)

сходится

numerical sequence

числ овая посл едовател ьность

ideal numbers

несоб ственны е числ а

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3. Read the text. While reading use the formulae in Supplement 2. LIMITS OF SEQUENCES AND FUNCTIONS The mapping f : N → M of the set of real numbers N into a certain set M is called the sequence of elements of the set M. If we define the number f (n) by xn , the sequence will be written as follows: x1 , x 2 ,..., x n ,.... or

{x n } . The elements

x1 , x 2 ,..., x n ,.... are called the members of the

sequence. Definition 1. The number a is called the limit of the sequence {x n } , if for any number ε >0 there will be n0 such that xn − a < ε when n ≥ n0 . If a is the limit of sequence {xn } , it is written as lim xn = a . They say that in this case the sequence {x n } converges to a, that is xn → a . As it can be seen from definition 1, if a = lim xn then for any number ε > 0 , beginning with a certain number, depending on ε , all the members of the sequence differ from a by quantity less than ε , however small this number may be. The limit of a numerical sequence is a particular case of a more general notion – the notion of the limit of a function. This notion as well as that of the limit of a sequence, is one of the most important in mathematical analysis. Definition 2. Let the function f be given on the set X ⊂ R , having the limit point a. The number l is called the limit of the function f(x) at x → a if for any ε >0 there exists such δ > 0 , that from x ∈ U δ (a) I ( X \ a) follows f ( x) ⊂ U ε (l ) . In this definition a and l may also be infinite ideal numbers. 4. Answer the following questions: 1. What is called the sequence of elements? 2. What is called the limit of the sequence? 3. What are the mathematical relations between the notion of the limit of a numerical sequence and that of the limit of a function? 4. How is the limit of a function defined?

5. Give a summary of the text.

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SECTION 2.

1. Read the following words and memorize their meaning: meaning (n)

значение

segment (n)

сегмент, отрезок

continuous (adj)

непреры вны й

neighborhood (n)

окрестность

corollary (n)

сл едствие (из теоремы )

bound (n)

граница

discontinuity (n)

разры в

closed set

замкнутое множ ество

2. Read the text. While reading use the formulae in Supplement 2.

CONTINUOUS FUNCTIONS Definition1. Let f : X → R , where X ⊂ R . The function f is called continuous at the point x0 of the set X if for any ε > 0 there exists such δ > 0 , that for every x ∈ X and x − x0 < δ , f ( x) − f ( x 0 ) < ε . We may

say that the function f : X → R is called

continuous at the point x0 ∈ X if for any neighborhood U ε ( f ( x0 )) of the point f ( x0 ) there exists such neighborhood U δ ( x0 ) of the point x0 that f : U δ ( x0 ) I X → U ε ( f ( x0 )) . The function f continuous at every point of the set X is called continuous on this set. If the function f is not continuous at the point x0 , x0 is called the point of function discontinuity. Theorem 1. /Weierstrass /. Let X ⊂ R be a limited and closed set and the function f : X → R be continuous on X. Then f(X) is also a limited and closed set.

Corollary. In case X is a segment [a, b] of a numerical straight line, we get the classical Weierstrass theorems: if the function f(x) is continuous on [a, b] , it

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1. is limited at this segment; 2. reaches its exact lower and upper bounds at the segment.

3. Answer the following questions: 1. What function is called continuous? 2. What point is called the point of function discontinuity? 3. How can the Weierstrass theorem be stated? 4. How many corollaries does this theorem have? 5. What are they? 4. Write a summary of this text.

SECTION 3.

1. Read the international words and guess their meaning: interval

differential

argument

function

2. Read the following words and word combinations and memorize their meaning: value (n)

значение, вел ичина

increment (n)

приращ ение

variable (n)

переменная (вел ичина)

homogeneous (adj)

однородны й

remainder (n)

остаток

ratio (n)

отношение

condition (n)

усл овие

derivative (n)

производная

differential (n)

дифференциал

differentiable (adj)

дифференцируемы й

it is necessary and sufficient

необ ходимои достаточно

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3. Read the text. While reading use the formulae in Supplement 2. THE DEFINITION OF DERIVATIVE AND DIFFERENTIAL Let the function y = f (x) be defined in an open interval X, x ∈ X be the inner point of this interval and the value h >< 0 be such that the point x + h belongs to X. Then if we go over from the value of the argument x to a new value x + h the function will change by the value ∆y = ∆f ( x) = f ( x + h) − f ( x) , which is called the increment of the function f(x), corresponding to the increment h of an independent variable. The increment of the independent variable is also denoted by ∆x . Definition 1. Function f is called differentiable at the point x0 ∈ X if there exists such a homogeneous function l x (h) that for all h ∈ R , for which x0 + h ∈ X we have 0

f ( x 0 + h) − f ( x 0 ) = l x0 (h) + ω ( x0 , h) , where

ω ( x 0 , h) h → 0 . →0 h

The linear function l x (h) is called the differential of the given function f at point 0

x0 and is denoted α y or α f ( x 0 ) , as far as ω ( x0 , h) is concerned it is called the

remainder of the function increment. Definition 2. The ratio limit of the function increment ∆y to the increment h of an independent variable, as h tends to zero (if it is finite) is called the derivative of the function y = f (x) at the point x0 . Theorem 1. For the function f : X → R to be differentiable at the point x0 ∈ X it is necessary and sufficient for it to have a derivative at this point.

4. Answer the following questions: 1. What is called the increment of the function? 2. What function is called differentiable? 3. What is called the derivative of the function y = f (x) at the point x0 ? 4. What theorem is stated in the text?

5. Give a summary of the text.

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LEXICAL EXERCISES TO UNIT 2

1. Give Russian equivalents: Value, interval, linear function, remainder, increment, independent variable, differentiable function, remainder of the function increment, derivative of the function, finite limit.

2. Give English equivalents: В нутренняя точка промеж утка, независимая переменная,

приращ ение

независимой переменной, дифференцируемая функция, л инейная функция, производная функция.

UNIT 3.

ALGEBRA AND GEOMETRY

SECTION 1. 1. Read the international words and guess their meaning: coordinate

correlation

vector

equivalent

perpendicular

collinear

2. Read the following words and memorize their meaning: plane (n)

пл оскость

arbitrary (adj)

произвол ьны й

correlation (n)

соотношение

rectangular (adj)

прямоугол ьны й

conclusion (n)

вы вод, закл ю чение

to satisfy (v)

удовл етворять

to denote (v)

об означать

consequently

сл едовател ьно

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solution (n)

решение

to subtract (v)

вы читать

to determine (v)

определ ять

identity (n)

тож дество

degree (n)

степень

3. Memorize the following word combinations: frame of reference

системакоординат

arbitrary point

произвол ьная точка

to come to the conclusion

сдел ать вы вод

to satisfy the equation

удовл етворять уравнению

with respect to

относител ьночего-л иб о

the equation of the first order

уравнение первогопорядка

4. Read the text. While reading use the formulae in Supplement 2.

THE EQUATION OF STRAIGHT LINE AND PLANE If a certain frame of reference is given, the coordinates of points, lying on the straight line or on the plane cannot be arbitrary but should meet certain correlations. Let Cartesian rectangular frame of reference Oxy be fixed on the plane and the straight line L be given. Let us consider a non-zero vector n = ( A, B) (1) perpendicular to L. It is clear that all other vectors perpendicular to this straight line will be collinear to n. Let us take an arbitrary point M 0 ( x0 , y0 ) on a straight line. A point M(x,y) lies on the straight line L if and only if the vectors M 0 M and n are perpendicular, that is ( M 0 M , n) = 0 (2)

As M 0 M = ( x − x0 , y − y0 ) , it follows from 1 and 2, that A( x − x0 ) + B( y − y 0 ) = 0 .

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Denoting − Ax0 − By 0 = C we come to the conclusion that in the given frame of reference Oxy the coordinates of the points of the straight line L and only they satisfy the equation Ax + By + C = 0

(3)

Among the numbers A, B there is one that does not equal zero. That is why equation (3) will be called the equation of the first degree with respect to variables x, y. Let us prove that any equation of the first degree (3) determines a certain straight line with respect to a fixed frame of reference Oxy. As equation (3) is the equation of the first order, at least one of the constants A, B is different from zero. Consequently equation (3) has at least one solution ( x0 , y0 ) , for example, x0 = −

in addition

AC , A + B2

y0 = −

2

BC , A + B2 2

Ax0 + By 0 + C = 0.

Subtracting the given identity from equation (3) we obtain the equation A( x − x0 ) + B( y − y 0 ) = 0 ,

which is equivalent to equation (3). But it means that any point M(x,y), the coordinates of which satisfy the given equation lies on the straight line, passing through the point M 0 ( x0 , y0 ) and perpendicular to vector (1). Thus, under the fixed frame of reference on the plane any equation of the first degree determines the straight line and the coordinates of the points of any straight line satisfy the equation of the first degree. Equation (3) is called the general equation of the straight line on the plane, vector n (1) is called the normal vector of the straight line.

5. Answer the following questions: 1. Can the coordinates of points be arbitrary if a certain frame of reference is given? 2. Why is equation (3) called the equation of the first degree? 3. What kind of vector is called the normal vector of the straight line?

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4. What kind of equation do we call the general equation of the straight line on the plane?

6. Give a summary of this text.

SECTION 2.

1. Read the international words and guess their meaning: operator

operation

symbolically

element

prototype

2. Read the following words and word combinations and memorize their meaning: non-empty (adj)

непустой

to correspond (v)

соответствовать

subspace (n)

подпространство

to map (v)

отоб раж ать

dimension (n)

размерность

domain (n)

об л асть определ ения

rank (n)

ранг, категория

image (n)

об раз

kernel (n)

ядро

totality (n)

совокупность

hence (adv)

отсю да, сл едовател ьно

deliberately (adv)

преднамеренно, заведомо

one-to-one correspondence

взаимнооднозначное соответствие

range of values

об л асть значений

alongside with

нарядус

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3. Read the text. While reading use the formulae in Supplement 2.

OPERATORS The rule, according to which every element x of a certain non-empty set X corresponds to the only element Y, is called an operator. The result y of applying the operator A to the element x is denoted symbolically: y = A(x),

y = Ax

(1)

They say that the operator A maps X into Y. The set X is called the domain of the operator A. The element y of (1) is called the image of the element x, and the element x itself is called the prototype of the element y. The totality TA of all images is called the range of values of the operator A. In case each element

y ∈ Y has

only one prototype, rule (1) is called one-to-one

correspondence. Operator is also called mapping or operation. Let linear spaces X, Y be given over the same field P. Let us consider the operator A, the domain of which is the subspace Y and the range of values is a certain subset of Y. The operator A is called linear, if A(αu + βv) = αAu + βAv (2)

for any vectors u, v ∈ X and any numbers α, β ∈ P . The range of values T A of the linear operator A is the subspace of the space Y. If z = Au, w = Av , the vector αz + βw will deliberately be the image of vector αu + βv at

all numbers α, β . Hence, the vector αz + βw belongs to the range of values of the operator A. The dimension of subspace TA is called the rank of the operator and is denoted rA . Alongside with TA we shall consider the set N A of vectors x ∈ X , satisfying the equation Ax = 0 . This set is also a subspace and is called the kernel of the operator A.

4. Answer the following questions: 1. What is an operator?

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2. What does an operator do? 3. What algebraic notions are defined in the text? 4. How are they defined?

5. Give a summary of the text.

SECTION 3.

1. Read the following words and word combinations and memorize their meaning: eigen value

соб ственное значение

eigen vector

соб ственны й вектор

range of values

об л асть значений

to occur (v)

оказаться, иметь место

to simplify (v)

упрощ ать

operational research

иссл едование операций

to extend (v)

расширять

to add (v)

приб авл ять, скл ады вать

mutually (adv)

взаимно, об ою дно

2. Read the text. While reading use the formulae in Supplement 2.

EIGEN-VALUES AND EIGEN-VECTORS Let the linear operator A function in space X. It means that each vector x ∈ X corresponds to a certain vector y ∈ Ax of the same space X. It may occur that for a certain non-zero vector x its image and its prototype are collinear. This fact greatly simplifies operational research. A number λ is called the eigen-value and a non-zero vector x the eigen-vector of the linear operator A. If they are related by means of the correlation, Ax = λx.

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Note, that if x is the eigen-vector corresponding to the eigen-value of λ , each collinear vector αx at α ≠ 0 will also be an eigen-vector. If the eigen-value λ corresponds to 2 eigen-vector x, y, then each non-zero vector of the type αx + βy will also be an eigen-vector. The zero vector is not an eigen-vector by definition. That is why the set X λ of all eigen-vectors, being linear combinations of any number of the given eigen-vectors, corresponding to one and the same eigen-value λ is not a subspace. If we extend X λ , adding to it the zero vector, X λ will become a space. This subspace is called eigen subspace of the operator A, corresponding to the eigen-value of λ . It is easy to understand that all non-zero vectors of the space X will be eigenvectors of the operators O, E , αE . Eigen-vectors have the following properties: 1. The system of the eigen-vectors x1 , x2 ,..., xm of the operator A, mutually corresponding to different eigen-values λ1 , λ 2 ,..., λ m is linear independent. 2. A linear operator, acting in m-dimensional space has no more than m mutually different eigen-values.

3. Answer the following questions: 1. How can you define an eigen-vector and eigen-value? 2. Can the zero vector be considered an eigen-vector? 3. What subspace is called an eigen subspace? 4. What 2 properties do eigen-vectors have?

4. Give a summary of the text.

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LEXICAL EXERCISES TO UNIT 3

1. Give the Russian equivalents: Frame of reference, arbitrary, non-zero vector, collinear, equivalent to, normal vector, to satisfy the equation, eigen-value, eigen-vector, non-empty set, one-to-one correspondence, kernel of the operator.

2. Give the English equivalents: Соотношение; прямоугол ьная системакоординат; перпендикул ярны й вектор; относител ьно; уравнение первогопорядка; сл едовател ьно; решение уравнения; тож дество; л иния, проходящ ая через точку; об щ ее уравнение прямой л инии на пл оскости; перпендикул ярны й вектор прямой л инии; об л асть значений ; отоб раж ение; соб ственное значение.

UNIT 4. ADDITIONAL TEXTS FOR READING Mathematical analysis Analysis is the branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in general settings. Historically, analysis originated in the 17th century, with the invention of calculus by Newton and Leibniz. In the 17th and 18th centuries, analysis topics such as the calculus of variations, differential and partial differential equations, Fourier analysis and generating functions were developed mostly in applied work. Calculus techniques were applied successfully to approximate discrete problems by continuous ones. All through the 18th century the definition of the concept function was a subject of debate among mathematicians. In the 19th century, Cauchy was the first to put calculus on a firm logical foundation by introducing the concept of Cauchy sequence. He also started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. In the middle of the century Riemann introduced his theory of integration. The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the ε-δ definition of limit. Then, mathematicians started worrying that they were assuming

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the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the discontinuity sets of real functions. Jordan developed his theory of measure, Cantor developed what is now called naïve set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert space to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

Algebra Algebra (from the Arabic "al-jabr" meaning "reunion", "connection" or "completion") is a branch of mathematics which may be defined as a generalization and extension of arithmetic. The process of "balancing and restoration" is important in algebra. It is the process of equaling both sides of any given equation. Basically, "whatever you do to one side of the equation, you do to the other". Also in Algebra the distributive law is used to properly factor out and solve equations. The field of algebra may be roughly divided in • elementary algebra, where the properties of operations on the real number system are recorded, symbols are used as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied; •

abstract algebra, where algebraic structures such as groups, rings and fields are axiomatically defined and investigated; •

linear algebra which studies the specific properties of vector spaces;

•

universal algebra, where those properties common to all algebraic structures are studied; •

computer algebra, where algorithms for the symbolic manipulation of mathematical objects are collected.

Geometry Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible of proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. Because of its immediate practical applications, geometry was one of the first branches of mathematics to be developed. Likewise, it was the first field to be put on

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an axiomatic basis by Euclid. The Greeks were interested in many questions about ruler-and-compass constructions. The next most significant development had to wait until a millennium later, and that was analytic geometry, in which coordinate systems are introduced and points are represented as ordered pairs or triples of numbers. This sort of representation has since then allowed us to construct new geometries other than the standard Euclidean version. The central notion in geometry is that of congruence. In Euclidean geometry, two figures are said to be congruent if they are related by a series of reflections, rotations, and translations. Other geometries can be constructed by choosing a new underlying space to work with (Euclidean geometry uses Euclidean space, Rn) or by choosing a new group of transformations to work with (Euclidean geometry uses the inhomogeneous orthogonal transformations, E(n)). The latter point of view is called the Erlangen program. In general, the more congruences we have, the fewer invariants there are. As an example, in affine geometry any linear transformation is allowed, and so the first three figures are all congruent; distances and angles are no longer invariants, but linearity is.

Infinite series The sum of an infinite series is a limit of partial sums of infinitely many terms. Such a limit can have a finite value; if it has, the series is said to converge; if it does not, it is said to diverge. The simplest convergent infinite series is perhaps

It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2 — in other words, the series has an upper bound. This series is a geometric series and mathematicians usually write it as:

An infinite series is formally written as

where the elements an are real (or complex) numbers. We say that this series converges towards S, or that its value is S, if the limit

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exists and is equal to S. If there is no such number, then the series is said to diverge. The sequence of partial sums is defined as the sequence

indexed by N. Then, the definition of series convergence simply says that the sequence of partial sums has limit S, as N → ∞ . Formal definition of series Indeed, mathematicians usually define a series as the above sequence of partial sums. The notation represents then a priori this sequence, which is always well defined, but which may or may not converge. Only in the latter case, i.e. if this sequence has a limit, the notation is also used to denote the limit of this sequence. To make a distinction between these two completely different objects (sequence vs. numerical value), one may omit the limits (atop and below the sum's symbol) in the former case. History of the theory of infinite series The investigation of the validity of infinite series is considered to begin with Gauss. Euler had already considered the hypergeometric series

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence. Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long before by Gregory (1668). Euler and Gauss had given various criteria, and Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form. Abel (1826) corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m and x. He showed the necessity of considering the subject of continuity in questions of convergence. Cauchy's methods led to special rather than general criteria. General criteria began with Kummer, and were studied by Eisenstein, Weierstrass in his various contributions to the theory of functions and many others.

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Notations for differentiation The simplest notation for differentiation that is in current use is due to Lagrange and uses the prime, ′. To take derivatives of f(x) at the point a, we write: f ′(a) for the first derivative, f ″(a) for the second derivative, f ″′(a) for the third derivative and then f(n)(a) for the nth derivative (n > 3). For the function whose value at each x is the derivative of f(x), we write f ′(x). Similarly, for the second derivative of f we write f ″(x), and so on. The other common notation for differentiation is due to Leibniz. For the function whose value at x is the derivative of f at x, we write:

We can write the derivative of f at the point a in two different ways:

If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as:

Higher derivatives are expressed as or for the n-th derivative of f(x) or y respectively. Leibniz's notation is versatile in that it allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. Newton's notation for differentiation was to place a dot over the function name:

and so on. Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration. It is usually used for first and second derivatives.

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Groups In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms. For example, the set of integers is a group under the operation of addition. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste Galois (1830), concerning the problem of when an algebraic equation is soluble by radicals. Previous to this work, groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms. A great many of the objects investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, each under multiplication. Another important example is given by non-singular matrices under multiplication, and more generally, invertible functions under composition. Group theory allows for the properties of these systems and many others to be investigated in a more general setting, and its results are widely applicable. Group theory is also a rich source of theorems. Groups underlie many other algebraic structures such as fields and vector spaces and are also important tools for studying symmetry in all its forms. For these reasons, group theory is considered to be an important area in modern mathematics, and it has many applications to mathematical physics (for example, in particle theory). Basic definitions A group (G, * ) is a nonempty set G together with a binary operation * : G x G -> G, satisfying the group axioms. "a * b" represents the result of applying the operation * to the ordered pair (a, b) of elements of G. The group axioms are the following: * Associativity: For all a, b and c in G, (a * b) * c = a * (b * c). * Identity element: There is an element e in G such that for all a in G, e * a = a * e = a. * Inverse element: For all a in G, there is an element b in G such that a * b = b * a = e, where e is the identity element from the previous axiom. You will often also see the axiom * Closure: For all a and b in G, a * b belongs to G. The way that the definition above is phrased, this axiom is not necessary, since binary operations are already required to satisfy closure. When determining if * is a group operation, however, it is nonetheless necessary to verify that * satisfies closure; this is part of verifying that it is in fact a binary operation. The above axioms are not strictly minimal from a logical viewpoint; they contain a small amount of redundancy. However, the difference is slight and in practice one usually just checks the above axioms. It should be noted that there is no requirement that the group operation be commutative, that is there may exist elements such that a * b is not equal to b * a. A

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group G is said to be abelian (after the mathematician Niels Abel) (or commutative) if for every a, b in G, a * b = b * a. Groups lacking this property are called nonabelian. The order of a group G, denoted by |G| or o(G), is the number of elements of the set G. A group is called finite if it has finitely many elements, that is if the set G is a finite set. Note that we often refer to the group (G, * ) as simply "G", leaving the operation * unmentioned. But to be perfectly precise, different operations on the same set define different groups. Notation for groups Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively. That is: * We write "a · b" or even "ab" for a * b and call it the product of a and b; * We write "1" for the identity element and call it the unit element; * We write "a −1 " for the inverse of a and call it the reciprocal of a. However, sometimes the group operation is thought of as analogous to addition and written additively: * We write "a + b" for a * b and call it the sum of a and b; * We write "0" for the identity element and call it the zero element; * We write "-a" for the inverse of a and call it the opposite of a. Usually, only abelian groups are written additively, although abelian groups may also be written multiplicatively. When being noncommittal, one can use the notation (with "*") and terminology that was introduced in the definition, using the notation a −1 for the inverse of a. If S is a subset of G, and x an element of G then in multiplicative notation, xS is the set of all products {xs : s in S}; similarly the notation Sx = {sx : s in S}; and for two subsets S and T of G, we write ST for {st : s in S, t in T}. In additive notation, we write x + S, S + x, and S + T for the respective sets. An abelian group: the integers under addition A group that we are introduced to in elementary school is the integers under addition. For this example, let Z be the set of integers, {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}, and let the symbol "+" indicate the operation of addition. Then (Z,+) is a group (written additively). Proof: * If a and b are integers then a + b is an integer. (Closure; + really is a binary operation) * If a, b, and c are integers, then (a + b) + c = a + (b + c). (Associativity) * 0 is an integer and for any integer a, 0 + a = a + 0 = a. (Identity element) * If a is an integer, then there is an integer b := -a, such that a + b = b + a = 0. (Inverse element)

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This group is also abelian: a + b = b + a. The integers with both addition and multiplication together form the more complicated algebraic structure of a ring. In fact, the elements of any ring form an abelian group under addition, called the additive group of the ring. Not a group: the integers under multiplication On the other hand, if we consider the operation of multiplication, denoted by "·", then (Z,·) is not a group: * If a and b are integers then a · b is an integer. (Closure; · really is a binary operation) * If a, b, and c are integers, then (a · b) · c = a · (b · c). (Associativity) * 1 is an integer and for any integer a, 1 · a = a · 1 = a. (Identity element) * But, it is not true that whenever a is a non-zero integer, there is a non-zero integer b such that ab = ba = 1. For example, a = 2 is a non-zero integer, but no matter what non-zero integer b we choose, |ab| = |2b| >= 2 > 1. Note that we cannot choose b = 1/2 because 1/2 is not an integer. (Inverse element fails) . Since not every element of (Z,·) has an inverse, (Z,·) is not a group. The most we can say is that it is a commutative monoid. An abelian group: the nonzero rational numbers under multiplication Consider the set of rational numbers Q, that is the set of numbers a/b such that a and b are integers and b is nonzero, and the operation multiplication, denoted by "·". Since the rational number 0 does not have a multiplicative inverse, (Q,·), like (Z,·), is not a group. However, if we use the set Q \ {0} instead of Q, that is we include every rational number except zero, then (Q \ {0},·) does form an abelian group (written multiplicatively). The inverse of a/b is b/a, and the other group axioms are simple to check. We don't lose closure by removing zero, because the product of two nonzero rationals is never zero. Just as the integers form a ring, so the rational numbers form the algebraic structure of a field. In fact, the nonzero elements of any given field form a group under multiplication, called the multiplicative group of the field.

A finite nonabelian group: permutations of a set For a more abstract example, consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block". In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB > RBG > BRG, i.e., "take the last block and move it to the front". If we write e for "leave the blocks as they are" (the identity

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action), then we can write the six permutations of the set of three blocks as the following actions: * e : RGB > RGB * a : RGB > GRB * b : RGB > RBG * ab : RGB > BRG * ba : RGB > GBR * aba : RGB > BGR Note that the action aa has the effect RGB > GRB > RGB, leaving the blocks as they were; so we can write aa = e. Similarly, * bb = e, * (aba)(aba) = e, and * (ab)(ba) = (ba)(ab) = e; so each of the above actions has an inverse. By inspection, we can also determine associativity and closure; note for example that * (ab)a = a(ba) = aba, and * (ba)b = b(ab) = aba. This group is called the symmetric group on 3 letters, or S3. It has order 6 (or 3 factorial), and is non-abelian (since, for example, ab is not equal to ba). Since S3 is built up from the basic actions a and b, we say that the set {a,b} generates it. Every group can be expressed in terms of permutation groups like S3; this result is Cayley's theorem and is studied as part of the subject of group actions. Simple theorems * A group has exactly one identity element. * Every element has exactly one inverse. * You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x * a = b and exactly one solution y in G to the equation a * y = b. * The expression "a1 * a2 * ··· * an" is unambiguous, because the result will be the same no matter where we place parentheses. * (Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a * b) −1 = b −1 * a −1 . These and other basic facts that hold for all individual groups form the field of elementary group theory. Constructing new groups from given ones 1. If a subset H of a group (G,*) together with the operation * restricted on H is itself a group, then it is called a subgroup of (G,*).

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2. The product of two groups (G,*) and (H,•) is the set GxH together with the operation (g1,h1)(g2,h2) = (g1*g2,h1•h2). The product can also be defined with an infinite number of terms. 3. The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non zero terms. If the family is finite the direct sum and the product are of course the same. 4. Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.

Fermat's last theorem Fermat's last theorem (also called Fermat's great theorem) is one of the most famous theorems in the history of mathematics. It states that:

There are no positive natural numbers a, b, and c such that an + bn = cn in which n is a natural number greater than 2. The 17th-century mathematician Pierre de Fermat wrote about this in 1637 in his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus': "I have discovered a truly remarkable proof but this margin is too small to contain it". This statement is significant because all the other theorems proposed by Fermat were settled, either by proofs he supplied, or by rigorous proofs found afterwards. Mathematicians were long baffled, for they were unable either to prove or to disprove it. The theorem was therefore not the last that Fermat conjectured, but the last to be proved. The theorem is generally thought to be the mathematical result that has provoked the largest number of incorrect proofs. For various special exponents n, the theorem had been proved over the years, but the general case remained elusive. In 1983 Gerd Faltings proved the Mordell conjecture, which implies that for any n > 2, there are at most finitely many coprime integers a, b and c with an + bn = cn. Using sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the English mathematician Andrew Wiles, from Princeton University, with help from his former student Richard Taylor, devised a proof of Fermat's last theorem that was published in 1995 in the journal Annals of Mathematics. Ken Ribet had proved in 1986 Gerhard Frey's epsilon conjecture that every counterexample an + bn = cn to Fermat's last theorem would yield an elliptic curve y2 = x(x - an)(x + bn), which would provide a counterexample to the Taniyama-Shimura conjecture.

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This latter conjecture proposes a deep connection between elliptic curves and modular forms. Wiles and Taylor were able to establish a special case of the Taniyama-Shimura conjecture sufficient to exclude such counterexamples arising from Fermat's last theorem. The story of the proof is almost as remarkable as the mystery of the theorem itself. Wiles spent seven years in isolation working out nearly all the details. When he announced his proof in June 1993, he amazed his audience with the number of ideas and constructions used in his proof. Unfortunately, upon closer inspection a serious error was discovered: it seemed to lead to the breakdown of this original proof. Wiles and Taylor then spent about a year trying to revive the proof. In September 1994, they were able to resurrect the proof with some different, discarded techniques that Wiles had used in his earlier attempts. There is considerable doubt over whether Fermat's "truly remarkable proof" was correct. The methods used by Wiles were unknown when Fermat was writing, and it seems inconceivable that Fermat managed to derive all the necessary mathematics to demonstrate the same solution (in the words of Andrew Wiles, "it's impossible; this is a 20th century proof"). The alternatives are that there is a simpler proof that all other mathematicians up until this point have missed, or that Fermat was mistaken. In fact, a plausible faulty proof that might have been accessible to Fermat has been suggested. It is based on the mistaken assumption that unique factorization works in all rings of integral elements of algebraic number fields. The fact that Fermat never published an attempted proof, or even publicly announced that he had one, suggests that he may have found his own error and simply neglected to cross out his marginal note. An interesting conjecture related to Fermat's Last Theorem was proposed by Leonhard Euler in 1769. It states that for every integer n greater than 2, the sum of n1 n-th powers of positive integers cannot itself be an n-th power. The conjecture was disproved by L. J. Lander and T. R. Parkin in 1966 when they found the following counterexample for n = 5: 275 + 845 + 1105 + 1335 = 1445. In 1988, Noam Elkies found a method to construct counterexamples for the n = 4 case. His smallest counterexample was the following: 26824404 + 153656394 + 187967604 = 206156734. Roger Frye subsequently found the smallest possible n = 4 counterexample by a direct computer search using techniques suggested by Elkies: 958004 + 2175194 + 4145604 = 4224814. No counterexamples for n > 5 are currently known.

SUPPLEMENT 1. CLICHES FOR SUMMARIZING The text (article, paper) under consideration is entitled – рассматриваемы й текст (статья, раб ота) озагл авл ен(а)

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The text is devoted to (is concerned with, studies, describes) – текст посвящ ен (касается, изучает, описы вает) The text begins (opens, starts) with – текстначинается с The author touches upon (considers, supposes) – автор касается, (рассматривает, предпол агает) The author introduces the notions of . . . and defines them. – автор вводит понятия . . . и определ яетих The text is concluded by . . . – Т екстзаканчивается . . .

SUPPLEMENT 2. READING OF THE MAIN MATHEMATICAL FORMULAE N→M

N tends to M

f :N →M

f transfers N into M

f (n)

f of n

x1 , x 2 ,..., x m

x sub one, x sub two, ... , x sub m

xn = 0

x sub n equals 0, x sub n is equal to 0

a>b

a is greater than b

a
a is less than b

a≥b

a is greater than or equals b

xn

the module (the absolute value) of x sub n

X ⊂R

the set X is contained in the set R

X ⊃R

the set X contains the set R

x ∈U δ

x belong to U capital sub δ

AI B

the intersection of A and B

AU B

the union of A and B

∆x

the increment of x

a b

a divided by b

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CONTENTS Unit 1. Computers and programming

p.4

Unit 2. Mathematical analysis

p.15

Unit 3. Algebra and geometry

p.20

Unit 4. Additional texts for reading

p.27

Supplement 1. Cliches for summarizing

p.37

Supplement 2. Reading of the main mathematical formulae

p.38

Составител и: СтернинаМ аринаА б рамовна В оротниковаМ аринаИ вановна Редактор – БунинаТ .Д .