7.3 Движение декабристов (стр. 4 )

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.

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.

The mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher Leibniz considered recursive self-similarity.

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 his papers. 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 were based mostly on recursion.

Four common techniques for generating fractals are:

Escape-time 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.

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 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. 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. 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.

Fractal patterns have been found in the ch patterns can also be found in African art, architecture, textiles, sculpture, and even cornrow hairstyles.

1 Say if the following statements are TRUE or FALSE.

1 If translated from Latin, “fractal” means “breaking”.

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

3 Any fractal can be characterized by 4 features.

4 All semi-similar objects are fractals.

5 Today lots of fractal examples are introduced artificially.

6 Recurrence self-similarity was investigated by Leibniz.

7 Koch is known for constructing his triangle.

8 Iterated function systems are based on a fixed geometric rule.

9 Self-similarity ranges from the strongest to the weakest type.

2 Answer the questions.

1 How can you define a fractal?

2 What are 5 typical characteristics of fractals?

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

4 What algorithm can be used to build up fractals by computer?

5 What are the techniques used to generate fractals?

6 What are typical examples of fractals in nature?

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

origin _______________ ______________

dimension _______________ xxxxxxxxxxxxxx

imagination _______________ ______________

curve xxxxxxxxxxxxxxx ______________

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

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.

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.

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.

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 and later explicitly applied to biology in the 1970s. 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.