Contents

PART A. INTRODUCTION TO MATHEMATICS ………………………………….6

Unit 1.Mathematics………………………………………………………7

Unit 2.Basic geometric concepts……………………………………...11

Unit 3.Texts for extracurricular work……………….………….……...14

Mathematics is the Queen of natural knowledge……….…....14

Fields of mathematics…………………………………….……..17

Poincare Conjecture……………………………………….…….21

Applied mathematics…………………………………….….…...26

Mathematical Methods for Business and Finance….….……..30

About a line and a triangle…………………………….…….…..34

Computer algebra……………………………………….……….35

Computer Software in Science and Mathematics….………...36

The main principles of axiomatic methods……….…………...37

Fields Medal………………………………….…………………..38

Mathematical economics…………………….………………….39

A modern view of geometry…………………..…………………41

Mathematical programming………………..……………………42

PART B. SCIENCE ITSELF…………………………………………………………44

Unit 4.Did Darwin’s finches do Math?............ .................................44

Unit 5.Introduction to computational complexity ………………….48

Unit 6.When you read an article………………………..…..…………50

Unit 7. Abstracts…………………………………………………………52

Unit 8. Сonclusion……………………………..……….………………58

Unit 9.Texts for extracurricular work…………….…………….……...58

Streaming Computation of Delaunay Triangulations..……….58

A proof of the Gibbs-Thomson formula. ................................67

Sublinear Time Bounds………………………………….………69

APPENDIX…………………………………………………………………………….70

Words and word combinations used in texts……………………………… 71

Wording of mathematics formulae……………………………………………78

Part a. Introduction to science Unit 1. Mathematics

Task 1. Read the names of the subjects and the object of their study. Match two parts of the sentences

1) Mathematical analysis... 2) Geometry... 3) Functional analysis... 4) Computer programming...

5) Discrete mathematics... 6) Differential equations... 7) Applied Statistics... 8) Algebra...

a) is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

b) is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions.

c) is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.

d) is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying «smoothly», the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.

e) is the study of mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.

f) refers to studies where the essence is the use of abstract methods to study problems where sets of numerical functions are involved. These «abstract methods» consist of applying theorems about functions between sets having an algebraic and a topological or a more general limit structure, i.e. one defining some kind of «limit process». One generally calls such structured sets spaces, and the functions between them are called functionals.

g) is the process of designing, writing, testing, debugging / troubleshooting, and maintaining the source code of computer programs.

h) is the science of the collection, organization, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.

Task 2. Match the words with their correct definitions

1) deduction a) a statement or proposition which is regarded as being established, accepted, or self-evidently true.

2) definition b) a judgement or decision reached by reasoning.

3) conjecture c) an opinion or conclusion formed on the basis of incomplete information;

an unproven mathematical or scientific theorem.

4) conclusion e) a statement of the exact meaning of a word, especially in a dictionary.

5) number f) an excellent example for others to follow.

6) patterns g) an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations;

a figure or group of figures used to identify someone or something.

7) axiom h) the inference of particular instances by reference to a general law or principle.