Mathematics modelling

Mathematical Modeling

---

1  Why mathematical modeling?

Mathematical modeling is the art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application.


Mathematical modeling

• is indispensable in many applications
• is successful in many further applications
• gives precision and direction for problem solution
• enables a thorough understanding of the system modeled
• prepares the way for better design or control of a system
• allows the efficient use of modern computing capabilities

Learning about mathematical modeling is an important step from a theoretical mathematical training to an application-oriented mathematical expertise, and makes the student fit for mastering the challenges of our modern technological culture.

---

2  A list of applications

In the following, I give a list of applications whose modeling I understand, at least in some detail. All areas mentioned have numerous mathematical challenges.
This list is based on my own experience; therefore it is very incomplete as a list of applications of mathematics in general. There are an almost endless number of other areas with interesting mathematical problems.
Indeed, mathematics is simply the language for posing problems precisely and unambiguously (so that even a stupid, pedantic computer can understand it).
Anthropology

• Modeling, classifying and reconstructing skulls

Archeology

• Reconstruction of objects from preserved fragments
• Classifying ancient artifices

Architecture

• Virtual reality

Artificial intelligence

• Computer vision
• Image interpretation
• Robotics
• Speech recognition
• Optical character recognition
• Reasoning under uncertainty

Arts

• Computer animation (Jurassic Park)

Astronomy

• Detection of planetary systems
• Correcting the Hubble telescope
• Origin of the universe
• Evolution of stars

Biology

• Protein folding
• Humane genome project
• Population dynamics
• Morphogenesis
• Evolutionary pedigrees
• Spreading of infectuous diseases (AIDS)
• Animal and plant breeding (genetic variability)

Chemical engineering

• Chemical equilibrium
• Planning of production units

Chemistry

• Chemical reaction dynamics
• Molecular modeling
• Electronic structure calculations

Computer science

• Image processing
• Realistic computer graphics (ray tracing)

Criminalistic science

• Finger print recognition
• Face recognition

Economics

• Labor data analysis

Electrical engineering

• Stability of electric curcuits
• Microchip analysis
• Power supply network optimization

Finance

• Risk analysis
• Value estimation of options

Fluid mechanics

• Wind channel
• Turbulence

Geosciences

• Prediction of oil or ore deposits
• Map production
• Earth quake prediction

Internet

• Web search
• Optimal routing

Linguistics

• Automatic translation

Materials Science

• Microchip production
• Microstructures
• Semiconductor modeling

Mechanical engineering

• Stability of structures (high rise buildings, bridges, air planes)
• Structural optimization
• Crash simulation

Medicine

• Radiation therapy planning
• Computer-aided tomography
• Blood circulation models

Meteorology

• Weather prediction
• Climate prediction (global warming, what caused the ozone hole?)

Music

• Analysis and synthesis of sounds

Neuroscience

• Neural networks
• Signal transmission in nerves

Pharmacology

• Docking of molecules to proteins
• Screening of new compounds

Physics

• Elementary particle tracking
• Quantum field theory predictions (baryon spectrum)
• Laser dynamics

Political Sciences

• Analysis of elections

Psychology

• Formalizing diaries of therapy sessions

Space Sciences

• Trajectory planning
• Flight simulation
• Shuttle reentry

Transport Science

• Air traffic scheduling
• Taxi for handicapped people
• Automatic pilot for cars and airplanes

---

3 Basic numerical tasks

The following is a list of categories containing the basic algorithmic toolkit needed for extracting numerical information from mathematical models.
Due to the breadth of the subject, this cannot be covered in a single course. For a thorough education one needs to attend courses (or read books) at least on numerical analysis (which usually covers some numerical linear algebra, too), optimization, and numerical methods for partial differential equations.
Unfortunately, there appear to be few good courses and books on (higher-dimensional) numerical data analysis.
Numerical linear algebra

• Linear systems of equations
• Eigenvalue problems
• Linear programming (linear optimization)
• Techniques for large, sparse problems

Numerical analysis

• Function evaluation
• Automatic and numerical differentiation
• Interpolation
• Approximation (Padé, least squares, radial basis functions)
• Integration (univariate, multivariate, Fourier transform)
• Special functions
• Nonlinear systems of equations
• Optimization = nonlinear programming
• Techniques for large, sparse problems

Numerical data analysis (= numerical statistics)

• Visualization (2D and 3D computational geometry)
• Parameter estimation (least squares, maximum likelihood)
• Prediction
• Classification
• Time series analysis (signal processing, filtering, time correlations, spectral analysis)
• Categorical time series (hidden Markov models)
• Random numbers and Monte Carlo methods
• Techniques for large, sparse problems

Numerical functional analysis

• Ordinary differential equations (initial value problems, boundary value problems, eigenvalue problems, stability)
• Techniques for large problems
• Partial differential equations (finite differences, finite elements, boundary elements, mesh generation, adaptive meshes)
• Stochastic differential equations
• Integral equations (and regularization)

Non-numerical algorithms

• Symbolic methods (computer algebra)
• Sorting
• Compression
• Cryptography
• Error correcting codes

---

4 The modeling diagram

The nodes of the following diagram represent information to be collected, sorted, evaluated, and organized.



The edges of the diagram represent activities of two-way communication (flow of relevant information) between the nodes and the corresponding sources of information.
S. Problem Statement

• Interests of customer/boss
• Often ambiguous/incomplete
• Wishes are sometimes incompatible

M. Mathematical Model

• Concepts/Variables
• Relations
• Restrictions
• Goals
• Priorities/Quality assignments

T. Theory

• of Application
• of Mathematics
• Literature search

N. Numerical Methods

• Software libraries
• Free software from WWW
• Background information

P. Programs

• Flow diagrams
• Implementation
• User interface
• Documentation

R. Report

• Description
• Analysis
• Results
• Validation
• Visualization
• Limitations
• Recommendations

Using the modeling diagram

• The modeling diagram breaks the modeling task into 16=6+10 different processes.
• Each of the 6 nodes and each of the 10 edges deserve repeated attention, usually at every stage of the modeling process.
• The modeling is complete only when the 'traffic' along all edges becomes insignificant.
• Generally, working on an edge enriches both participating nodes.
• If stuck along one edge, move to another one! Use the general rules below as a check list!
• Frequently, the problem changes during modeling, in the light of the understanding gained by the modeling process. At the end, even a vague or contradictory initial problem description should have developed into a reasonably well-defined description, with an associated precisely defined (though perhaps inaccurate) mathematical model.

---

5  General rules

• Look at how others model similar situations; adapt their models to the present situation.
• Collect/ask for background information needed to understand the problem.
• Start with simple models; add details as they become known and useful or necessary.
• Find all relevant quantities and make them precise.
• Find all relevant relationships between quantities ([differential] equations, inequalities, case distinctions).
• Locate/collect/select the data needed to specify these relationships.
• Find all restrictions that the quantities must obey (sign, limits, forbidden overlaps, etc.). Which restrictions are hard, which soft? How soft?
• Try to incorporate qualitative constraints that rule out otherwise feasible results (usually from inadequate previous versions).
• Find all goals (including conflicting ones)
• Play the devil's advocate to find out and formulate the weak spots of your model.
• Sort available information by the degree of impact expected/hoped for.
• Create a hierarchy of models: from coarse, highly simplifying models to models with all known details. Are there useful toy models with simpler data? Are there limiting cases where the model simplifies? Are there interesting extreme cases that help discover difficulties?
• First solve the coarser models (cheap but inaccurate) to get good starting points for the finer models (expensive to solve but realistic)
• Try to have a simple working model (with report) after 1/3 of the total time planned for the task. Use the remaining time for improving or expanding the model based on your experience, for making the programs more versatile and speeding them up, for polishing documentation, etc.
• Good communication is essential for good applied work.
• The responsibility for understanding, for asking the questions that lead to it, for recognizing misunderstanding (mismatch between answers expected and answers received), and for overcoming them lies with the mathematician. You cannot usually assume your customer to understand your scientific jargon.
• Be not discouraged. Failures inform you about important missing details in your understanding of the problem (or the customer/boss) - utilize this information!
• There are rarely perfect solutions. Modeling is the art of finding a satisfying compromise. Start with the highest standards, and lower them as the deadline approaches. If you have results early, raise your standards again.
• Finish your work in time.

Lao Tse: ''People often fail on the verge of success; take care at the end as at the beginning, so that you may avoid failure.''

---

6  Conflicts

Most modeling situations involve a number of tensions between conflicting requirements that cannot be reconciled easily.

• fast - slow
• cheap - expensive
• short term - long term
• simplicity - complexity
• low quality - high quality
• approximate - accurate
• superficial - in depth
• sketchy - comprehensive
• concise - detailed
• short description - long description

Einstein: ''A good theory'' (or model) ''should be as simple as possible, but not simpler.''

• perfecting a program - need for quick results
• collecting the theory - producing a solution
• doing research - writing up
• quality standards - deadlines
• dreams - actual results

The conflicts described are creative and constructive, if one does not give in too easily. As a good material can handle more physical stress, so a good scientist can handle more stress created by conflict.
''We shall overcome'' - a successful motto of the black liberation movement, created by a strong trust in God. This generalizes to other situations where one has to face difficulties, too.
Among other qualities it has, university education is not least a long term stress test - if you got your degree, this is a proof that you could overcome significant barriers. The job market pays for the ability to persist.

---

7 Attitudes

• Do whatever you do with love. Love (even in difficult circumstances) can be learnt; it noticeably improves the quality of your work and the satisfaction you derive from it.
• Do whatever you do as a service to others. This will improve your attention, the feedback you'll get, and the impact you'll have.
• Take responsibility; ask if in doubt; read to confirm your understanding. This will remove many impasses that otherwise would delay your work.

Jesus: ''Ask, and you will receive. Search, and you will find. Knock, and the door will be opened for you.''

---

8  References

For more information about mathematics, software, and applications, see, e.g., my home page, athttp://www.mat.univie.ac.at/~neum/

---

Bounds (बंध )

Upper Bound and Bounded Above:-
Let S is a subset of R. And a∈R. Then a is called upper bound of S if x≤a for all x∈S.
And then S is called bounded above.

Lower Bound and Bounded Below:- 
Let S is a subset of R. And a∈R. Then a is called lower bound of S if a≤x for all x∈S.
And then S is called bounded below.

Least Upper Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) x≤a for all x∈S.
2) there are exist a Upper Bound t such that a≤t.
I.e. lub is less or equal all other upper bounds.


Greatest lower Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) a≤x for all x∈S.
2) there are exist a lower Bound w such that w≤a.
I.e. glb is greater or equal all other lower bounds. 

Mathspics

Mathspics

Complex numbers

A complex number is a number that can be expressed in the form a + bi, where a and bare real numbers and i is the imaginary unit, that satisfies the equation i² = −1.In this expression, a is the real part and b is the imaginarypart of the complex number.

   
  Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.

As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics,electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.

Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation

has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where i² = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are−3 + 3i and −3 − 3i, as can be verified using the fact that i² = −1:

According to the fundamental theorem of algebra, all polynomial equational with real or complex coefficients in a single variable have a solution in complex numbers.

Definition:-

A complex number is a number of the form a+ bi, where a and b are real numbers and i is the imaginary unit, satisfying i² = −1. For example, −3.5 + 2i is a complex number.

The real number a is called the real part of the complex number a + bi; the real numberb is called the imaginary part of a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part.The real part of a complex number z is denoted by Re(z) orℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). For example,

Hence, in terms of its real and imaginary parts, a complex number z is equal to . This expression is sometimes known as the Cartesian form of z.

A real number a can be regarded as a complex number a + 0i whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi whose real part is zero. It is common to write a for a + 0i and bifor 0 + bi. Moreover, when the imaginary part is negative, it is common to write a − bi withb > 0 instead of a + (−b)i, for example 3 − 4iinstead of 3 + (−4)i.

The set of all complex numbers is denoted by ℂ, 

Addition:-

To add two complex numbers we add each part separately:

(a+bi) + (c+di) = (a+c) + (b+d)i

Example: (3 + 2i) + (1 + 7i) = (4 + 9i)

Multiplying

To multiply complex numbers:

Each part of the first complex number gets multiplied by
each part of the second complex number

Just use "FOIL", which stands for "Firsts, Outers,Inners, Lasts"

Firsts:a × c Outers:a × di Inners:bi × c Lasts:bi × di

(a+bi)(c+di) = ac + adi + bci + bdi2

Like this:

Example: (3 + 2i)(1 + 7i)

(3 + 2i)(1 + 7i)		= 3×1 + 3×7i + 2i×1+ 2i×7i
		= 3 + 21i + 2i + 14i2
		= 3 + 21i + 2i − 14	(because i2 = −1)
		= −11 + 23i

But There is a Quicker Way!

Use this rule:

(a+bi)(c+di) = (ac−bd) + (ad+bc)i

Example: (3 + 2i)(1 + 7i) = (3×1 − 2×7) + (3×7 + 2×1)i = −11 + 23i

1. Add: (7 + 5i) + (8 - 3i)



2. Add: (2 + 3i) + (-8 - 6i)



3. Express the sum of and
in the form .



4. Add and .


Example: i²

i can also be written with a real and imaginary part as 0 + i

Complex Plane

We can also put complex numbers on a Complex Plane.

• The Real part goes left-right
• The Imaginary part goes up-down

i2 = (0 + i)2		= (0 + i)(0 + i)
		= (0×0 − 1×1) + (0×1 + 1×0)i
		= −1 + 0i
		= −1

Prove that  for any integer n.

Corollary 1.2.15: DeMoivre's Formula
	For any integer n and any real number twe have (cos(t) + i sin(t))n = cos(nt) + i sin(nt) Proof

DeMoivre's Formula is quite something. It says that if you take a number on the unit circle (i.e. with lenght 1) with initial argument (angle) t and multiply it by itself, it simply rotates around the unit circle by that angle t. Each time you multiply the number by itself, the vector rotates another tdegrees. In other words, in this case the power operator results in a simple rotation.

Powers of a vector z with |z|=1

Two interesting questions related to this rotation, taken from the field of Complex Dynamics, are: suppose z is a complex number with |z|=1. Then:

• find conditions for Arg(z) such that zⁿ = z for some integer n. Such a point, incidentally, is called periodic of order n.
• if Arg(z)/ is irrational, what can you say about the sequence {z, z², z³, z⁴, ...}? Does it, for example, converge? Such a sequence, incidentally, is called the orbit of z.

Polar coordinates can be especially helpful for finding roots, in particular for complex numbers of lenght 1.

Proposition 1.2.16: Finding Roots
	For any positive integer n and any non-zero complex number a = r cis(t) the equation zn = a has exactly n distinct roots given by: z = where k = 0, 1, 2, ... n-1. Proof

In a previous example we found the two square roots of i, which turned out to be a fair amount of work. The above proposition allows us to dispense with such a question quickly. For example, the two solutions for

z² = i = cis(/2)

are:

z₁ = cis(/2/2) = cis(/4)
z₂ = cis((/2 + 2)/2) = cis(5/4)

which you can quickly check using DeMoivre's Formula. Here is a geometric interpretation of this proposition: let's find, for example, the three third-roots of i, i.e. we want to find all solutions toz³ = i.

1: Draw the vector i	2: Divide angle by 3 for first root	3: Draw 3 equally spaced segments, starting at the first root

Note, in particular, that the third root of i turned out to be -i, which indeed checks out:

(-i)(-i)(-i) = i*i*(-i) = (-1)*(-i) = i

This proposition is very satisfying: it says that at least every simple polynomial equation of degreen has n solutions. Later we will see that this is true in general: every n-th degree polynomial hasn roots, no "if's" and "but's". This, in fact, is a sign of things to come: many theorems in complex analysis will turn out to be very "satisfying" and nicely structured, which is one reason that the study of complex analysis is a lot of fun (I think -:). But first a few more 'profane' examples.

Example 1.2.17: Finding roots geometrically
	Find the cube roots of 8 and iand draw them. Find all 4 fourth-roots of -1 and draw them geometrically Find all 5 fifth-roots of 1 and draw them geometrically Find both square-roots of 3i-2 by (a) using polar coordinates and (b) using rectangular coordinates and aformula from the previous section. Confirm that both methods result in the same answers.

Let's conclude this chapter with a result that illustrates what a 'nicely structured' theorem in complex analysis can look like.

Proposition 1.2.18: Roots of Unity
	The n n-th roots of unity are given bywnk, where k = 0, 1, 2, ... n-1 and wn = cis(2/n) They form the vertices of a regular polygon and add up to zero, i.e. they satisfy the equation: 1 + wn + wn2 + ... + wnn-1 = 0 Proof

This is neat: not only does the equation zⁿ = 1have exactly n solutions (one of which is, of course, z=1), but the solutions have this really pretty geometric structure of forming a regular polygon, which implies algebraically that they add up to zero as vectors. Here are, for example, the eight roots of z⁸=1:

You can see their regular structure. When you add them all as vectors, you indeed get the zero vector.


(.Ref from Wikipedia and Mathisfun private.Shu.edu)