\section*{What is Applied mathematics?}

\begin{definition}
Applied mathematics problems are mathematical problems which
importance is self-evident.
\end{definition}


Applied math is a group of methods aimed for solution of problems in
sciences, engineering, economics, or medicine. These methods are
originated by Newton, Euler, Lagrange, Gauss and other giants.
 Modern areas of applied math
 include Mathematical physics, Mathematical biology, 
Control theory, Aero-space engineering,
optimal planning, math finance. 
There is a fuzzy boundary between applied math and engineering
and at the other side, between applied and pure mathematics. 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). 
 
Study of applied math requires expertise in many areas of 
mathematics and science, physical intuition and common sense, 
and collaboration skills. 
Applied math  allows for many approaches to the problem, a choice
 of objectives, and variety of methods.
\begin{definition}
Success in applied math 
is a new elegant 
model that catches the phenomenon and novel clever methods to solve
it.
\end{definition}
% \newpage
\section*{Approaches of applied math}

In the majority of applied problems, one need to build mathematical
model of a phenomenon, solve this model, and develop the recommendations for
improvement of the performance.

\paragraph{Math modeling}
 Typically we deal with large systems (thousands of variables,
equations and inequalities).  A
clever model separates the main phenomenon from the noise 
and allows for analytic treatment of the problem, 
followed by extensive numerical development. There may be several
models suggested for the same phenomena with various level of
details:  No one need a one-to-one geographical map.
The art of simplification is to make the problem solvable but not
trivial. All models have ranges of applicability (Take for example the
ideal gas model, compare with the black hole model).

 %\newpage
 \paragraph{Method of solutions}

The 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. These will be
intensively discussed in the course. 
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. 
Examples: Weather forecast; conflict situation: Worst case
scenario or average outcome.

Numerical models allow for solution of classical problems and address
novel classes of problems (like math finance, math genetics,
weather prediction). They also call for new theoretical approaches: 
Development of the theory of free boundary
problems was caused by the possibility of numerical solution the
problem in an arbitrary domain.

 \paragraph{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 that will be discussed in the class.

\end{document}

%\newpage

\section*{Example of an applied math field: Optimal design}



\paragraph{Intuitive design}
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.

\paragraph{Math models}
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.

\paragraph{Common-sense improvement, educated guess}
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.
% \newpage

\paragraph{Preconditions}

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.  

This progress has stimulated an interest
in the mathematical
foundations of structural optimization.
 A design problem asks for  the
best geometry of layouts of different materials in a given domain.
Of course, this approach simplifies (or, as a mathematician would
say, idealizes) the real engineering problem, because questions
such as convenience or cost of manufacturing are not considered.

Analysis of optimal structures allows us to formulate  general
principles of an optimally designed construction.  In particular,
we can extend the intuitive principle of equally stressed
construction to a multidimensional situation and find optimal
structures that are, in a sense, hybrids of  simple mechanisms.
The construction that adapts to the varying load has some common
features of living tissues and provides a bridge to math biology.
