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Міністерство освіти і науки, молоді та спорту України

Сумський державний університет

Збірник тематичних текстів та лексико-граматичних вправ з дисципліни «Англійська мова професійного спілкування»

для студентів спеціальності 080202 «Прикладна математика»

спеціалізації «Моделювання складних систем»

факультету ЕЛІТ

денної форми навчання


Сумський державний університет


Збірник тематичних текстів та лексико-граматичних вправ з дисципліни «Англійська мова професійного спілкування»

/ укладач А.М.Дядечко. – Суми : Сумський державний університет, 2011. – 64 с.

Кафедра іноземних мов

Unit 1

1 Can you remember any ideas and views of modern science that go back to the 19th century? Can you name the scientists who contributed to the development of science in the 19th century?

^ 2 Make sure that you know the words and word combinations below. Try to memorize them.

gene confident assumption

matter Euclidian geometry

outgrowth fundamental constituents

insight mutually

core ideas in terms of

heredity corpuscular

germ to raise a question

subjects and predicates to severe connection

deductive reasoning to trace

3 Read the text.


Twentieth-century science is an evolutionary outgrowth of the 19th-century science: in terms of the theories scientists created and the new ideas underlying them; in its organization and conduct as professional practice; and in its relationship to society.

From its 17th – century birth, modern science has had two mutually influential “sides”: an “inside”, intellectual dimension, and an “outside”, social relationship dimension. The inside, the outside and the relationship between them all changed character in the course of the 20th century. As 20th –century physical, life, and social science are built on the 19th –century science, identifying developments in the 19th –century science that played key roles in the 20th –century science is a precondition for appreciating the innovativeness of the 20th – century science. A preview of the major theories and the core ideas that cut across the scientific disciplines will help orient us as we begin tracing the rise of these innovate theories and ideas.

A In the 19th century, the atom came to represent the view that natural phenomena consisted of fundamental building blocks that could be configured to produce the vast number of forms we find in nature. The atom in physics and chemistry is an example of that conception, as are the gene in the theory of heredity and the germ in the germ theory of disease.

B In the 19th century, a science of energy was created, called thermodynamics, which identified energy as a new dimension of reality. This science recognized that energy was a phenomenon in nature parallel to matter.

C The 19th century also saw the development of the idea of fields of energy and force. A field is an immaterial phenomenon obeying natural laws and capable of exercising forces on material objects.

D Chemists in the 19th century discovered that structure is the feature that differentiates one substance from another, as opposed to the fundamental constituents of substances and the properties of these constituents.

E The fifth core idea of the 19th century that would influence 20th --century science was the discovery of non-Euclidean geometry. For approximately 2,300 years before the mid-19th century, Western philosophy, science and mathematics were based on the confident assumption that deductive reasoning was closely linked to truth. In the mid-19th century, mathematicians discovered deductively perfect geometries that contradict Euclidean geometry, raising the question of which form of geometry is true of space and severing the critical connection between reasoning and reality.

Another important development in mathematics in the 19th century was the invention of symbolic logic. From this development, we learned that notation can have a significant impact on our thinking. Simply replacing words with symbols can lead to new insights. Further, symbolic logic undermined the notion that subjects had priority over predicates, that is, that things were the ultimate reality and relationships were a secondary consequence of the organization of things. Through the use of symbolism, relationships were found to have properties of their own.

F The 19th century also saw the replacement of Newton’s particle,

or corpuscular, theory of light with the wave theory of light and the

subsequent expansion of this theory with James Clerk Maxwell’s electromagnetic theory of energy.

G Probability and statistics became important in the 19th century, specifically, the idea that natural processes exist that require probability to describe them.

H Finally, the 19th –century theory of evolution was a foundational idea of the 20th – century science.

^ 1 Answer the questions:

a) What foundation did 19th century science lay for 20th century science?

b) What two aspects of science development are considered in the text?

c) What sciences are mentioned in the text?

d) What do the terms “atom”, “gene” and “germ” have in common?

e) What science is known for presenting energy in a new way?

f) What is the impact of symbols on human thinking?

g) What does light consist of?

^ 2 Guess the words from the text by their definitions:

disease-producing microorganism –

physical matter or material –

nearly identically –

letter, figure or other conventional mark –

supposition –

basic unit of heredity –

understanding -

^ 3 Find in the text the words that are synonyms of the words below:

non-material to break (2 words)

emergence characteristic

product, result influence

link to direct

4 Derive nouns from the following verbs:

to assume – to replace- to contradict-

to deduce- to expand to identify-

to produce- to influence- to differentiate –

^ 5 Add prefixes, suffixes or both to the words below to form their derivatives from the text. Then sort out the newly formed words into nouns and adjectives:

build, Euclid, cent, evolution, concept, intellect, relation, real, second, connect, chemist, symbol, foundation, approximate, material, place, view, condition, innovative.

^ 6 Translate into Ukrainian/Russian paying special attention to the Infinitive constructions.

a) … . . . . the atom came to represent the view that natural phenomena consisted of fundamental building blocks…

b) … . relations were found to have properties of their own.

c) … natural processes require probability to describe them.

^ 7 Fill in the blanks with the appropriate prepositions:

a) Structure is a feature that differentiates one substance ____ another , as opposed ____ the fundamental constituents of substances and properties of these substances.

b) We learned that notation can have a significant impact ____ our thinking.

c) The relationship ______ them all changed character _____ the course of the 20th century.

d) Simply replacing words _____ symbols can lead _____ new insights.

e) A field is capable _____ exercising forces on material objects.

^ Unit 2

1 What knowledge needed for your future speciality are you expected to gain at SSU?

2 What do you know about the history of applied mathematics? Try to remember facts, dates and names.

3 What fields does applied mathematics include today? Which of them have you studied? Share your learning experience.

4 Make sure that you can read and understand the words and word combinations below. Point out the terms related directly to applied mathematics study. Memorize them.

legacy fluid mechanics

emergence numerical analysis

simulation data set

cryptology neural networks

domain to spawn

advent to change over time

probability to spawn

lattice theory to arise out of

differential equations to cover

approximation theory applicable

5 Read the text


Applied mathematics is a branch of mathematics that concerns itself with the mathematical technique typically used in the application of mathematical knowledge to other domains.

There is no consensus of what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees.

Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory

(broadly construed to include representations, asymptotic methods, variational methods, and numerical analysis), and applied probability. These areas of mathematics were intimately tied to the development of Newtonian Physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left legacy as well; until the early 20th century subjects such as classical mechanics were often taught in applied mechanics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments.

Today, the term applied mathematics is used in a broader sense. It includes the classical areas above, as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptology), though they are not generally considered to be a part of the fields of applied mathematics. Sometimes the term applicable mathematics is used to distinguish between the traditional field of applied mathematics and the many more areas of mathematics that are applicable to real world problems.

Mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and the applications of mathematics within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it. Industrial mathematics is used to solve industrial problems. It is sometimes split in two branches: techno-mathematics (covering problems coming from technology) and econo-mathematics (for problems in economy and finance).

The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics, computational science, and computational engineering, which use high performance computing for the simulation of phenomena and solution of problems in the science and engineering.

Historically mathematics was most important in the natural sciences and engineering. However, after World War II, fields outside of the physical sciences have spawned the creations of new areas of mathematics, such as game theory, which grew out of economic considerations, or neural networks, which arose out of the study of the brain in neuroscience, or bioinformatics, from the importance of analyzing large data sets in biology.

The advent of the computer has created new applications, both in studying and using the new computer technology itself (computer science, which uses combinatorics, formal logic, and lattice theory), as well as using computers to study problems arising in other areas of science (computational science), and of course studying the mathematics of computation (numerical analysis). Statistics is probably the most widespread application of mathematics in the social sciences, but other areas of mathematics are proving increasingly useful in these disciplines, especially in economics and management sciences.

^ 1 Answer the questions:

a) Do you agree with the definition of applied mathematics given in the text? If not, give the one you consider more appropriate.

b) Is it easy to distinguish between physics and applied mathematics?

c) What was the role of Newtonian physics in the development of applied mathematics?

d) Can we consider applied mathematics as a purely theoretical science?

e) In what way did the development of computer software broadened the applied mathematics area?

f) What areas of applied mathematics appeared in the 20th century?

^ 2 What do we call people who study or deal with

mathematics, history, chemistry, management, biology,

mechanics, physics, nervous system ?

3 Find in the text and make a list of 6 adverbs that end in –ly and 12 nouns that end in –tion.

4 Complete the sentences

a) High performance computers make it possible to simulate ……………………………

b) ……………… is important in cryptology.

c) …………………covers problems coming from technology.

d) The problems in economy and finance are covered by ………………………..

e) Statistics is widely used in ………………………………………

^ 5 Fill in the blanks with the proper Passive forms:

a) Classical mechanics _______________( teach) in applied mathematics departments.

b) Industrial mathematics _______________( split) in two branches: techno-mathematics and econo-mathematics.

c) These areas of mathematics ___________(tie) to the development of Newtonian Physics.

d) Such categorizations ________________( make) difficult by the way mathematics and science change over time.

^ Unit 3

1 What are typical features of a complex system?

2 Give examples of complex systems from different spheres of human activities.

3 Make sure that you can read and understand the words and word combinations below. Try to memorize them.

consensus to encompass

attribute to seek

neuroscience to constrain

hallmark to exhibit

aspiration to brandish

chaos spontaneous

a weak fit dispersed

coupling rules ominous

pattern prediction essentially

precise prediction notably

linear equations

reciprocal dependence

4 Read the text


Complex systems is a scientific field which studies the common properties of systems that are considered fundamentally complex. Such systems are used to model processes in biology, economics, physics and many other fields. The scientific field under discussion is also called complex systems theory, complexity science, study of complex systems, sciences of complexity, non-equilibrium physics, and historical physics. The key problems of complex systems are difficulties with their formal modeling and simulation.

At present, the consensus related to one universal definition of complex systems does not exist yet. In different research contexts complex systems are defined on the base of their different attributes.

The study of complex systems is bringing an old approach to many scientific questions that are a weak fit for the usual mechanistic view of reality present in science. Complex systems encompass an approach in many diverse disciplines including anthropology, artificial life, chemistry, computer science, economics, evolutionary computation, earthquake prediction, meteorology, molecular biology, neuroscience, physics, psychology and sociology.

Scientists often seek simple non-linear coupling rules which lead to complex phenomena. Human societies and probably human brains are complex systems in which neither the components nor the couplings are simple. Nevertheless, they exhibit many of the hallmarks of complex systems. It is worth remarking that non-linearity is not a necessary feature of complex systems modeling: macro-analyses that concern unstable equilibrium and evolution processes of certain biological/social/economic systems can usefully be carried out also by sets of linear equations, which do nevertheless entail reciprocal dependence between variable parameters.

Complex systems is a new approach to science that studies how relationships between parts give rise to the collective behaviors of a system and how the system interacts and forms relationships with its environment.

The earliest precursor to modern complex systems theory can be found in the classical political economy of the Scottish Enlightenment, late developed by the Austrian school of economics, which says that order in market systems is spontaneous and that it is the result of human action, but not the execution of any human design. Upon this the Austrian school developed from the 19th to the early 20th century the economic calculation problem, along with the concept of dispersed knowledge, which were to fuel debates against the then-dominant Keynesian economics. This debate would lead economists, politicians and others to explore the question of computational complexity.

A pioneer in the field, and inspired by Karl Popper’s and Warren Weaver’s works, Novel prize economist and philosopher Friedrich Hayek dedicated much of his work, from early to the late 20th century, to the study of complex phenomena, not constraining his work to human economies but to other fields such as psychology, biology and cybernetics.

Further Steven Strogatz stated that “every decade or so, a grandiose theory comes along, bearing similar aspirations and often brandishing an ominous-sounding C-name. In the 1960s it was cybernetics. In the 70s it was catastrophe theory. Then came chaos theory in the 80s and complexity theory in the 90s.”

One of Hayek’s main contributions to early complexity theory is his distinction between the human capacity to predict the behavior of simple systems and its capacity to predict the behavior of complex systems through modeling. He believed that economics and the sciences of complex phenomena in general (biology, psychology and so on) could not be modeled after the sciences that deal with essentially simple phenomena like physics. Hayek would notably explain that complex phenomena, through modeling, can only allow pattern predictions, compared with the precise predictions that can be made out of non-complex phenomena.

^ 1 Are statements below TRUE or FALSE?

a) Complex systems theory development needed new approaches.

b) Complex systems are not easy to simulate.

c) Complex systems approach fits weather forecasting.

d) People can design market system order.

e) Warren Weaver was awarded a Nobel prize in economics.

f) Complexity theory came after cybernetics but prior to catastrophe theory.

^ 2 Answer the questions:

a) Where can complex systems be applied? Give examples.

b) What doesn’t mechanical approach fit today?

c) Are complex systems easy to define and model?

d) Can you consider human brain and a society as a complex system?

e) Do complex systems need linearity or non-linearity?

f) What is the main concept of a complex system?

g) When and where were the complex system applied first?

h) What are “c-theories” that the humanity gained in the 2nd half of the 20th century?

i) What was the role of Friedrich Hayek in the complexity theory development?

j) Can complex systems be predicted?

^ 3 Try to define the nouns below:

simulation, definition, computation, prediction, equation, execution, distinction, action.

4 Fill in the gaps with the proper prepositions:

a) The scientific field ______ discussion is also called complex systems theory.

b) The consensus related ______ one universal definition of complex systems doe not exist yet.

c) Scientists often seek simple non-linear rules which lead ____ complex phenomena.

d) The evolution processes of certain economic systems can be carried

_____ also _____ sets of linear equations.

e) Complex systems is a new approach ____ science that studies how relationships ____ parts give rise _____ the collective behaviors of the system.

f) People can predict the behavior of complex systems _____ modeling.

^ 5 Enlarge on the topic covered in the text by giving brief presentations about

a) Keynesian economics

b) Karl Popper’s and Warren Weaver’s main ideas and views

c) cybernetics

d) catastrophe theory

e) chaos theory.

^ Unit 4

1What basic principles of synergetics do you remember? Give some examples of synergetic approach .

2 Make sure that you can read and understand the words and word combinations below. Memorize them.

cybernetics qualitative changes bifurcation theory

emergence macroscopic scales chaos theory

neuron strong interplay catastrophe theory

hormone cooperative behavior stochastic process

fluctuation paradigmic example instability point

variable theoretical treatment to obey

cognition macroscopic state to elaborate

interdisciplinary order parameters macroscopically

spatial slaving principle irrespective of

temporal circular causality

spontaneous phase transition

3 Read the text


Synergetics (Greek: ‘working together”) is an interdisciplinary field of research originated by Hermann Haken in 1969.The term “synergetics”is widely used, but still not generally accepted. Instead, “theory of self-organization” is used as a more traditional term.

Synergetics can be considered as a modern stage of development within the traditions of cybernetics (N.Weiner, W.R.Ashby) and system-structural analysis (attempts to elaborate the general theory of systems). However, while cybernetics investigates the functioning of complex using an abstract model of “black box”, synergetics studies some physical mechanisms of the complex structures formation, i.e. it tries to look into the “black box”.

Synergetics deals with material or immaterial systems, composed of, in general, many individual parts. It focuses its attention on the spontaneous, i.e. self-organized emergence of new qualities which may be structures, processes or functions. The basic question dealt with by Synergetics is: are there general principles of self-organization irrespective of the nature of the individual parts of a system? In spite of the great variety of the individual parts, which may be atoms, molecules, neurons (nerve cells), up to individuals in a society, this question could not be answered in the positive for large classes of systems, provided attention is focused on qualitative changes on macroscopic scales. Here “macroscopic scales” means spatial and temporal scales that are large compared to those of the elements. “Working together” may take place between parts of a system, between systems or even between scientific disciplines. Characteristic of synergetics is the strong interplay between experiment and theory.

The development of the theory of cooperative behaviour, made by H.Haken and called by himself synergetics proper, originated from the investigation of the coherent radiation of lasers. Lasers became a paradigmic example of synergetics.

The systems under experimental or theoretical treatment are subject to control parameters which may be fixed from the outside or may be generated by part of the system considered. An example for an external control parameter is the power input into a gas laser by an electric current. An example for an internally generated control parameter is hormones in the human body or neurotransmitters in the brain.

When control parameters reach specific critical values the system may become unstable and adopt a new macroscopic state. Close to such instability points, a new set of collective variables can be identified: the order parameters. They obey, at least in general, low dimensional dynamics and characterize the system macroscopically. According to the slaving principle, the order parameters determine the behavior of the individual parts which may be still be subject to fluctuations. Their origin may be internal or external. Because the cooperation of the individual parts enables the existence of order parameters that in turn determine the behavior of the individual parts, one speaks of circular causality. At a critical point, a single order parameter may undergo a non-equilibrium phase transition with symmetry breaking, critical slowing down and critical fluctuations.

Synergetics has a number of connections with to other disciplines, such as complexity theory, dynamic systems theory, bifurcation theory, chaos theory, catastrophe theory, the theory of stochastic processes. The connection with chaos theory and catastrophe theory is in particular established by the concept of order parameters and the slaving principle, according to which close to instabilities the dynamics even of complex systems is governed by few variables only. Synergetics goes beyond the framework of strict mathematical models and narrow applications. Scholars are trying to apply synergetics not only to new fields of natural sciences, but also to humanities. There are some attempts to use synergetic models in understanding human artistic and scientific creativity, cognition, health, education, communication, etc.

^ 1 Say if the statements below are TRUE or FALSE.

a) The term “synergetics” is still being discussed today.

b) Synergetics is being developed and applied by mathematicians.

c) Lasers happened to become quite important for synergetics development.

d) Fluctuations are typically caused from outside.

e) Synergetics can help investigate human abilities and behavior.

^ 2 Answer the questions.

a) What is the origin of the term “synergetics”?

b) How is synergetics related to cybernetics?

c) What is synergetics focused on?

d) What meaning does the term “macroscopic scales” acquire in the synergetic context?

e) What systems can be investigated by means of synergetics?

f) What is the contribution of Hermann Haken into synergetics development?

g) What is the difference between types of control parameters?

h) What are order parameters?

i) When does circular causality take place?

j) What are connections and newest trends of synergetics?

^ 3 Comment on the terms from the text:

interdisciplinary field phase transition

coherent radiation spatial/temporal scales

instability point stochastic process

4 Complete the chart by forming nouns and adverbs from the adjectives enlisted:

Noun Adjective Adverb

__________ qualitative ____________

___________ traditional ____________

___________ stochastic ____________

___________ critical ____________

___________ artistic ____________

___________ creative ____________

___________ natural ____________

___________ cooperative ____________

___________ paradigmatic ____________

^ 5 Add negative prefixes to the words below and sort them out into three columns as in the examples:

in- im- ir-

instability immaterial irrespective

regular, possible, moral, finite, migrant, dependent, capable, movable, mobile, rational, adequate, accurate, resistible, responsive

^ 6 Fill in proper prepositions.

a) Synergetics tries to look _______ the black box”.

b) Synergetics focuses its attention ______ the spontaneous emergence of new qualities.

c) Synergetics proper originated _______ the investigation of the coherent radiation of lasers.

d) According _____ the slaving principle, the order parameters determine the behavior of the individual parts.

e) _____ a critical point, a single order parameter may undergo a non-equilibrium phase transition.

f) Close ____ instabilities the dynamics even of complex systems is governed by few variables only.

g) Scholars are trying to apply synergetics not only _____ new fields of natural sciences, but also ____ humanities.

^ Unit 5

1 Say what you remember about fractals. Give examples of fractals.

2 What scientists investigated fractals? Try to remember names and countries.

3 Make sure that you can read and understand the words and the word combinations below. Memorize them:

magnification reduced copy Brownian motion

snowflake Hausdoff dimension Levy flight

coastline mountain range fractal landscape

artifact lightning bolt aggregation cluster

triangle true fractals blood vessel

set iterated functions pulmonary vessel

sponge recurrence relation to derive

aggregation complex plane to exhibit

broccoli replacement rule to denote

replica deterministic process to take shape

to coin

4 Read the text


A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced copy of the whole,” a property called self-similarity. The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractus meaning “broken”. A fractal often has the following features:

It has a fine structure at arbitrary small scales.

It is too irregular to be easily described in traditional Euclidian geometric language.

It is self-similar (at least approximately or stochastically).

It has a Hausdorff dimension which is greater than its topological dimensions.

It has a simple and recursive definition.

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex. Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snowflakes. However, not all self-similar objects are fractals – for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics.

Images of fractals can be created using fractal generating software. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, such as when it is possible to zoom into a region of the fractal that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.

The mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).

It took until 1872 before a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass’s very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake. In 1915,Waclaw Sierpiski constructed his triangle and, one year later, his carpet. Originally these geometric fractals were described as curves rather than 2D shapes that they are known as in their modern constructions. The idea of self-similar curves was taken further by Paul Pierrre Levy, who, in his 1938 paper “Space Curves and Surfaces Consisting of Parts Similar to the Whole” described a new fractal curve, the Levy C curve. George Cantor also gave examples of subsets of the real line with unusual properties – these Cantor sets are also now recognized as fractals.

Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincare, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of many of the objects that they had discovered.

In the 1960s, Benoit Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which was built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word “fractal” to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term “fractal”.

Four common techniques for generating fractals are:

^ Escape-time fractals (also known as “orbits” fractals ) are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal, the Nova fractal and the Lyapunov fractal. The 2D vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data ) are passed through this field repeatedly.

^ Iterated function systems have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Peano curve, Koch snowflake, Harter-Heighway dragon curve, Menger sponge, are some examples of such fractals.

Random fractals are generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Levy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.

^ Strange attractors are generated by iteration of the solution of a system of initial-value differential equations that exhibit chaos.

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

Exact self-similarity is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity.

Quasi-self-similarity is a loose form of self-similarity; the fractal appears approximately (but not exactly ) identical at different scales. Quasi-self-similarity fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar.

^ Statistical self-similarity is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of “fractal” trivially imply some form of statistical self-similarity.(Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.

Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snowflakes, crystals, mountain ranges, lightning, river networks, broccoli, and systems of blood vessels and pulmonary vessels. Coastlines may be loosely considered fractal in nature.

Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature.

Fractal patterns have been found in the paintings. They are also prevalent in African art and architecture. Such patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.

^ 1 Say if the following statements are TRUE or FALSE.

a) If translated from Latin, “fractal” means “breaking”.

b) The term “fractal” was introduced by B. Mandelbrot in the 20th century.

c) Any fractal can be characterized by 4 features.

d) Fractals are often viewed as very complex.

e) Today lots of fractal examples are introduced artificially.

f) Leibniz thought that only the straight line was self-similar.

g) Koch is known for constructing his triangle.

h) By “orbits” we mean random fractals.

i) Self-similarity ranges from the strongest to the weakest type.

^ 2 Answer the questions.

a) How can you define a fractal?

b) What are 5 typical characteristics of fractals?

c) What were the roles of Leibniz and Mandelbrot in the fractal theory development?

d) Whose views did Mandelbrot follow?

e) What helps fractals capture everyone’s imagination?

f) What are the techniques used to generate fractals?

^ 3 Guess the words from the text through their definitions:

a) plane figure with three straight sides;

b) line of which no part is straight and which changes direction without angles;

c) crystals of frozen vapour falling from the sky;

d) coming, happening again, repeated;

e) number of things of the same kind that belong together.

4 Complete the table with the derivatives.

Noun Adjective Verb

­­­­­­-­­­­­­­­­­­­­­­­­­­­­­­__­_________ visual ______________

relation ______________ ______________

____________ ______________ function

____________ xxxxxxxxxxxxxx exhibit

____________ topological xxxxxxxxxxxxxx

____________ ______________ include

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^ Unit 5

1 Tell as much as you remember about game theory. Give names, facts, main concepts.

2 What film deals closely with game theory?

3 Make sure that you can read and understand the words and word combinations from the text:

equilibrium to attempt explicitly

payoff to capture unilaterally

move to expand

outcome to mitigate

insight to come into being

Nash equilibrium solution concept current set

equilibrium strategy strategic interaction hostile situation

traffic flow game semantics multi-agent system

4 Read the text


Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology, engineering, political science, international relations, computer science, and philosophy. Game theory attempts to mathematically capture behavior in strategic situations, in which an individual’s success in making choices depends on the choices of others. While initially developed to analyze competitions in which one individual does better at another’s expense (zero sum games), it has been expanded to treat a wide class of interactions, which are classified according to several criteria.

Well-defined mathematically, a game consists of a set of players, a set of moves (or strategies ) available to those players, and a specification of payoffs for each combination of strategies. Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

Traditional applications of game theory attempt to find equilibria in these games. In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide.

Although some developments occurred before it, the field of game theory came into being with the 1944 book “Theory of Games and Economic Behavior” by John von Neumann and Oskar Morgenstern. This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game theorists have won Nobel prizes in economics, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.

In the game, Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

The Nash equilibrium concept is used to analyze the outcome of the strategic interaction of several decision makers. In other words, it is a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the decision of each one depends on the decision of the others. The simple insight underlying John Nash’s idea is that we cannot predict the result of the choices of multiple decision makers if we analyze those decisions in isolation. Instead, we must ask what each player would do, taking into account the decision-making of the others.

Nash equilibrium has been used to analyze hostile situations like war and arms races (see Prisoner’s dilemma), and also how conflict may be mitigated by repeated interaction (see Tit-for-tat). It has also been used to study to what extent people with different preferences can cooperate (see Battle of the sexes), and whether they will take risks to achieve a cooperative outcome (see Stag hunt). It has been used to study the adoption of technical standards, and also the occurrence of bank runs and currency crises (see Coordination game). Other applications include traffic flow (see Wardrop’s principle), how to organize auctions (see Auction theory), and even penalty kicks in soccer (see Matching pennies).

Game theory has been used to study a wide variety of human or animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The use of game theory in the social sciences has expanded, and the game theory has been applied to political, sociological, and psychological behaviors as well.

Game theory analysis was initially used to study animal behavior by Ronald Fisher in the 1930s (although even Charles Darwin makes a few informal game theoretic statements). The developments in economics were later applied to biology largely by John Maynard Smith in his book “Evolution and the Theory of Games”.

Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations. Also, game theory provides a theoretical basis to the field of multi-agent systems.

The field of algorithmic game theory combines computer science concepts of complexity and algorithm design with game theory and economic theory. The emergence of the internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets.

^ 1 Say if the statements below are TRUE or FALSE:

a) Any player depends on other players’ choices.

b) The equilibrium concept differs from one field of science to another.

c) Game theory came into being when the book by J. F. Nash appeared.

d) Game theorists have been welcomed by the Nobel prize committee.

e) Nash equilibrium cannot be applied to the military.

f) Both humans and animals can be described by game theory.

^ 2 Answer the questions:

a) What is a game from a mathematical and logical point of view?

b) What do you know about the history of game theory development?

c) What is the idea of the Nash equilibrium concept?

d) Who may benefit from the Nash equilibrium concept?

e) In what way did Charles Darwin contribute to the game theory evolution?

f) In what field the demand for game theory is increasingly high today?

^ 3 Complete the list of Latin origin nouns by giving their singular or plural form:

___________ data

equilibrium ___________

formulae ___________

___________ criteria

___________ axes

___________ quanta

^ 4 Give antonyms of the following words:

hostile explicitly cooperative normal

informal extensive to expand to mitigate

5 Use synonyms to explain the following words:

outcome insight standard scholar

currency concept to occur unilaterally

^ 6 Fill in the blanks in the sentences below with the proper prepositions:

a) An individual success ____ making choices depends ____ the choices ____ others.

b) Game theory has been widely recognized ____ an important tool in many fields.

c) No player can benefit ____ changing his or her strategy.

d) Eight game theorists have won Nobel prizes ____ economics.

e) The developments ___ economics were later applied ____ biology largely ____ John Maynard Smith.

^ Unit 6

1 Do you often wait in a queue? When and where do you queue? How do you feel when waiting in a queue?

2 What is the main idea of queuing theory? Where can it be applied?

3 Make sure that you can read and understand the following words and word combinations from the text. Memorize them.

iteration finite to abandon

switchboard infinite to grant

entity discrete to exceed

customer viable to balk

server excessive to jockey

congestion manually

simulation randomly

4 Read the text


Queuing theory deals with problems which involve queuing (or waiting). Typical examples might be: banks/supermarkets (waiting for service), computers (waiting for a response), failure situations (waiting for a failure to occur e.g. in a piece of machinery), public transport (waiting for a train or a bus). As we know queues are a common everyday experience. They form because resources are limited. It makes economic sense to have queues just to know for example how many supermarket tills you would need to avoid queuing or how many buses or trains would be needed to avoid queues.

The first to develop a viable queuing theory was the French mathematician S. D. Poisson (1781-1840). He created a distribution function to describe the probability of a prescribed outcome after repeated iterations of independent trials. Poisson used a statistical approach. His distributions could be applied to any situation where excessive demands are made on a limited resource. The most important application of queuing theory occurred during the late 1800s, when telephone companies were faced with the problem of how many operators to place on duty at a given time. At the time, all calls were switched manually by an operator who physically connected a wire to a switchboard. A Danish mathematician named A.K Erlang developed a different approach to traffic engineering based on Poisson’s work. He established formulas for calls that are abandoned (called Erlang-B) and for those that are held until service is granted (Erlang-C).

All queuing systems can be broken into individual subsystems consisting of entities queuing for some activity. To analyze these subsystems we need information related to:

arrival process:

- how many customers arrive singly or in groups;

- how the arrivals are distributed in time (what is the probability distribution of time between successive arrivals – the interarrival time distribution);

- whether there is a finite population of customers or an infinite number.

service mechanism:

- a description of the resources needed for service to begin;

- how long the service will take (the service time distribution);

- the number of severs available;

- whether the servers are in series (each server has a separate queue) or in parallel (on queue for all servers);

- whether preemtion is allowed (a server can stop processing a customer to deal with another “emergency” customer).

queue characteristics:

- how, from a set of customers waiting for service, do we choose the one to be served next (FIFO- first-in first-out also known as FCFS –first-come first-served, LIFO – last-in first out, randomly);

- do we have balking (customers deciding not to join the queue if it is too long),

reneging (customers leave the queue if they have waited too long for a service),

jockeying (customers switch between queues if they think they will get faster by doing so);

a queue of finite capacity or of infinite capacity.

Changing the queue discipline (the rule by which we select the next customer to be served) can often reduce congestion.

In terms of the analysis of queuing situations the types of questions typically concerned with measures of system performance include:

- How long does a customer expect to wait in the queue before they are served, and how long will they have to wait before the service is complete?

- What is the probability of a customer having to wait longer than a given time interval before they are served?

- What is the average length of the queue?

- What is the probability that the queue will exceed a certain length? –

- What is the expected utilization of the server and the expected time period during which he will be fully occupied (servers cost money so we need them to keep busy).

In order to get answers to the above questions there are two basic approaches: analytic methods or formula based queuing theory and simulation or computer based queuing theory. Analytic methods are only available for relatively simple queuing systems. Complex queuing systems are almost always analyzed using simulation (more technically known as discrete-event simulation).

^ 1 Say if the statements below are TRUE or FALSE:

a) Queuing is still a problem today.

b) The French scientist made the theoretical basis for queuing theory.

c) A. K. Erlang’s approach helped many telephone stations to solve their problems.

d) To solve a queuing problem one must build up its simplified model.

e) The customers are often served randomly.

f) Congestion is the result of an effective management.

g) Computation of queuing problems means their simulation.

^ 2 Answer the questions.

a) Why is queuing theory in such demand today?

b) What did S. D. Poisson create?

c) What big application did queuing theory occur in 1800s?

d) What aspects must be considered when analyzing a queuing subsystem?

e) What parameters can arrival process include?

f) What does service mechanism consist of?

h) What do abbreviations FIFO and LIFO stand for?

i) What is the difference between balking, reneging and jockeying?

j) What is a queue discipline?

k) What points are especially important for any service manager?

l) What are two basic approaches in queuing theory?

^ 3 Match the words with their definitions:

a) the person who receives a service entity

b) done with hands viable

c) pretence, imitation customer

d) the state of being too full, overcrowded distribution

e) to go away from, not intending

to return to congestion

f) to do again and again

h) the device for making or breaking

connection infinite

i) endless, without limits

j) putting, giving or sending out parts of a to iterate

set of things

k) money-register to abandon

l) able to exist manually

m) something that really exists simulation

^ 4 Give synonyms or explain the meaning of the noun from the text:

iteration, distribution, application, population, simulation,


5 Find in the text the phrases or sentences that make the context for the following adverbs:

relatively, technically, physically, randomly, manually, typically

6 Practice prepositions:

a) All calls were switched manually _____ an operator who connected a wire _____ a switchboard.

b) Telephone companies were faced _____ a problem of how many operators to place ____ duty _____ a given time.

c) Poisson’s distributions could be applied ____ any situation.

d) The queue discipline is the rule _____ which we select the next customer to be served.

e) All queuing systems can be broken _____ individual systems consisting _____ entities queuing _____ some activity.

7 Speak about your own queuing experiences.


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