7.3 Движение декабристов (стр. 2 )

Method of solutions

A large variety of applied math problem requires a variety of methods. However, there are several commonly used methods in the applied math workshop. These methods include linear algebra, differential and integral equations, approximation theory and asymptotics, variational principles, and numerics. The same physical problem can be approached differently, using either statistics, of differential equations, or variational methods, or a combination of them. Typically, there is no single ``correct'' approach to a complicated problem, each approach highlights different sides of it.

Improvement and recommendation

When people develop a model of a process, they usually want to improve it. A number of theories answer the question: How to maximize an objective. These are Optimization, variational methods, control theory, game theory. Optimization is the ultimate objective of study of an engineering problem. Sometimes the improvement is achieved by varying the parameters, but generally it is a serious math problem.

Intuitive design and Math models

The process of design always includes a mysterious element: The designer chooses the shape and materials for the construction using intuition and experience. Since ancient times this approach has proved effective. For centuries, engineering landmarks such as aqueducts, cathedrals, and ships were all built without mathematical or mechanical theories.

From the time of Galileo and Hooke, engineers and mathematicians have developed theories to determine stresses, deflections, currents and temperature inside structures. This information helps in the selection of a rational choice of structural elements.

Certain principles of optimality are rooted in common sense. For example, one wants to equalize the stresses in a designed elastic construction by a proper choice of the layout of materials. The overstressed parts need more reinforcement, and the understressed parts can be lightened. These simple principles form a basis for rational construction of amazingly complicated mechanical structures, like bridges, skyscrapers, and cars. Still, knowledge of the stresses in a body is mostly used as a checking tool, parallel with the design proper, which remains the responsibility of the design engineer.

In the past few decades, it has become possible to turn the design process into algorithms thanks to advances in computer technology. Large contemporary projects require the use of computer-aided design systems. These systems often incorporate algorithms that gradually improve the initial design by a suitable variation of design variables, namely, the materials' cost and layout. Optimization techniques are used to effect changes in a design to make it stronger, lighter, or more reliable.

1 Answer the questions:

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

2 Is it easy to differentiate applied math from other fields of science?

3 What does the study of applied mathematics require?

4 What does the solution of applied problems typically include?

5 What are the components of large systems that applied mathematics deals with?

6 Why is simplification so important in applied mathematics?

7 What are common methods of applied math?

8 In what way are optimization and improvement related in applied math?

9 Did people always base their work on exact calculations?

10 Is common sense important in engineering and research?

11 What kind of improvement can computer-aided design systems provide?

2 What do we call people who study or deal with

mathematics, history, chemistry, management, biology,

mechanics, physics, nervous system?

3 Look at the list of words from the text. Sort them out into nouns, adjectives and adverbs.

fuzzy, boundary, elasticity, theory, majority, typically, inequality, applicability, variety, commonly, differently, usually, optimally, amazingly, respectively, contemporary

4 Complete the sentences from the text.

1 Applied math is a group of methods aimed for solution of problems in sciences, engineering, economics, or medicine.

2 Applied math discovers new problems which could become subjects of pure math (like geodesics), or develop to become a new engineering discipline (like elasticity theory).

3 A large variety of applied math problem requires a variety of methods.

4 Sometimes the improvement is achieved by varying the parameters,

5 The designer chooses the shape and materials for the construction using intuition and experience.

6 The overstressed parts need more reinforcement, and the understressed parts can be lightened.

5 Fill in the blanks with the proper Passive forms:

1These methods _______________ by Newton, Euler, Lagrange, Gauss and other giants.

2 The same physical problem can ______________ differently.

3 Aqueducts, cathedrals, and ships _________ without mathematical or mechanical theories.

3 Certain principles of optimality ___________in common sense.

4 Optimization techniques ____________to effect changes in a design to make it stronger, lighter, or more reliable.

( to use, to approach, to originate, to root, to build)

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

Complex systems is a scientific field which studies the common properties of systems that are considered fundamentally ch 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 plex 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.