INFO-I
690
Mathematical Methods for
Informatics
Instructor
Office: Eigenmann Hall 906
Telephone: 821-856-1833
Email: schnell@indiana.edu
Website: www.informatics.indiana.edu/schnell
Classroom
Monday and Wednesday 9.30-10:45 am
Course description
This
course, an introduction to applied mathematics, is concerned with the
construction, analysis, and interpretation of mathematical models that shed
light on significant scientific problems.
It is intended to provide material of interest to student in informatics
and the natural and social sciences at graduate level.
Most
of the applied mathematics courses present collections of useful mathematical
techniques and illustrate the various techniques by solving classical problems
of mathematical physics. Our approach is
different. Typically, we select an
important scientific problem whose solution will involve some useful
mathematics. After briefly discussing
the required scientific background, we formulate a relevant mathematical
problem with some care. The formulation
step is often difficult. Not many
courses or textbooks actually demonstrate this, but we try to give due weight
to the challenges involved in constructing our mathematical models. A new technique may be then introduced to
solve the mathematical problem, or a technique known in simpler contexts may be
generalized. In most instances we take
care to determine what the mathematical results tells us about the scientific
process that motivated the problem in the first place.
Aims
The students
will learn the three steps of making applied mathematics:
(i)
The formulation of the scientific problem in mathematical terms,
(ii)
The solution of the mathematical problem thus created, and
(iii)
The interpretation of the solution and its empirical verification in
scientific terms.
Prerequisites
We
have assumed that the potential student has had an introductory college course
in physics or chemistry and is familiar with calculus, differential equations,
statistics and probabilities. Potential
students who feel inadequacies in calculus and physical reasoning would profit
from consulting:
K. F.
Riley, M. P. Hobson and S. J. Bence, Mathematical Methods in Physics and
Engineering: A comprenhesive Guide, 9th
edition. Wiley:
J. L.
Hodges Jr., E. L. Lehmann, Basic concepts
of probability and statistics, 2nd edition, Classic in Applied
Mathematics, No. 48, SIAM: Philadelphia, 2005.
Syllabus
A
preliminary assessment will be made to select several of the topics below. The instructor will make a quiz the first day
of class and discuss the students’ research aims to focus on certain topics.
1.
What is applied mathematics?
1.1.
The scope, purpose and practice of applied mathematics
1.2.
Applied mathematics contrasted with pure mathematics
1.3.
Applied mathematics contrasted with theoretical science
1.4.
Applied mathematics in natural science and engineering
2.
Deterministic systems and ordinary differential equations
2.1.
Ordinary differential equations in sciences
2.2.
Some fundamental procedures illustrated on ordinary differential equations
2.3.
Simplification, dimensional analysis and scaling
2.3.1.
The basic simplification procedure
2.3.2.
Dimensional analysis
2.3.2.1.Putting differential equations
into dimensionless form
2.3.2.2.Nondimensionalization of a functional relationship
2.3.2.3.Use of scale models
2.3.3.
Scaling
2.3.3.1.Definition of scaling
2.3.3.2.Order of magnitude
2.3.3.3.Scaling functions
2.3.3.4.Scaling and perturbation theory
2.4.
A systems of ordinary differential equations
2.4.1.
Nonlinear systems of two ordinary differential equations
2.4.2.
Phase plane methods and qualitative analysis
2.4.2.1.Curves in the phase
2.4.2.2.Nullclines and separatrices
2.4.2.3.Critical points
2.4.2.4.Linear stability analysis
2.4.2.4.1. Behavior or trajectories near
critical points
2.4.2.4.2. Limit cycles
2.4.2.5.Elements of bifurcation theory
3.
Random processes
3.1.
Random walk in one dimension: Langevin’s equation
3.1.1.
The one-dimensional random walk model
3.1.2.
Explicit solution
3.1.3.
Mean, variance, and the generating function
3.1.4.
Use of stochastic differential equation to obtain Boltzmann’s constant from
observations of Brownian motion
3.2.
From Langevin to Fokker-Plank
3.2.1.
Probability distributions
3.2.2.
Fokker-Plank equations
3.3.
Some random models in science
4.
Discrete models
4.1.
Scalar discrete-time models
4.1.1.
Cobwebbing, fixed points and linear stability analysis
4.1.2.
Chaos theory and the logistic model
4.2.
Systems of discrete time equations
4.2.1.
Fixed points and linear stability analysis
5.
Stochastic Discrete Models
5.1.
Common roots between deterministic and stochastic models
5.1.1.
Event, probability, and transition probability
5.2.
Master equation
5.3.
Elements of Markov processes
5.3.1.1.Modeling reactions as a Markov
Process
5.3.1.2.The transition probability
matrix
5.3.1.3.Dwell times
5.4.
Life and death processes
5.4.1.
Difference differential equation
5.4.2.
Poisson process
5.4.3.
The Master Equation
5.5.
Fundamentals of queuing theory
5.5.1.
Markov chains and queuing
5.5.2.
Life and death process and queuing
5.6.
Cellular automata
Textbook
Marc T. Figge, Michael Meyer-Hermann and Santiago Schnell, Random Models in Biology, Oxford
University Press, to be published in 2008.
The
students can also consult:
D.
Kaplan and L. Glass, Understanding nonlinear dynamics, Springer:
S. H. Strogatz, Nonlinear dynamics and chaos, Westview
Press:
G. de Vries, T Hillen, M. Lewis, J. Müller
and B. Schönfisch, A course in mathematical biology,
E. Parzen, Stochastic
Process, Classics in Applied Mathematics No. 24,
Grading
The course grade will be calculated by weighing your homework, and Final Project Paper in the proportions 60% and 40% respectively. Homework problem sets will be assigned every other week, they are due one week from the date of assignment.
Course Policies
Attendance:
We expect
that students will approach the course as they should a professional job –
attend every class. If you cannot attend class we would appreciate your
notifying the instructor that you will not be present and why. An email is sufficient.
Assignments:
Assignments
will be turned in by the beginning of class on the date they are due. Assignments which are late will not be graded
unless you have requested an extension at least three days before the due
date. Each student may be granted only
one extension. Not turning your assignment in the due date will mean that you
will fail the assignment. The final
paper must be turned in on time.
Academic Integrity:
As with other aspects of professionalism in
this course, you are expected to abide by the proper standards of professional
ethics and personal conduct. This includes the usual standards on acknowledgment
of joint work and other aspects of the
Incomplete Grade:
An incomplete (‘I’) final grade will be given only by prior arrangement in exceptional circumstances conforming to university and departmental policy which requires, among other things, that the student must have completed the bulk of the work required for the course with a passing grade, and that the remaining work can be made up within 30 days after the end of the semester
Use
of laptops in Class:
If the purpose of use is related to this
class, students may use their laptops during class and discussion
sections. However, they should not be
used if they distract your attention from what is going on in class, and use
should be minimized, since it is distracting to other students.
Accommodation:
We would like to hear from anyone who has a disability that may require some modifications of seating, or other class requirements so that appropriate arrangements can be made. Please see the instructor after class or during office hours.
We would like to know early in the semester of any possible conflicts between course requirements/deadlines and religious holy days or holidays (http://www.indiana.edu/~deanfac/holidays.html), so that accommodations can be made. Please see the instructor after class or during office hours.
We welcome feedback on the class
organization, material, lectures, assignments and exams. Please share your
comments and suggestions so that we can improve the class.
Further
information
Please contact the instructor, Santiago
Schnell, for more information.