**Instructor:** Hemanshu Kaul

**Office:** 125C, Engineering 1

**Phone:** (312) 567-3128

**E-mail:** kaul [at] iit.edu

**Time:** 10am, Monday and Wednesday.

**Place:** 121, Engg. 1 Bldg.

**Office Hours:** 1pm-2pm Monday and Wednesday; walk-ins; and by appointment. Emailed questions are also encouraged.

**TA Office Hours:** Chris Mitillos, cmitillo [at] hawk.iit.edu, 12noon-3pm, Friday, 129, E1 bldg.

|Course Information| |Advice| |Announcements| |Examinations| |Project| |Homework| |Class Log| |Books| |MATLAB/ Mathematica| |Useful Links|

The

Understanding Mathematics - a study guide

Excellent advice for math majors and graduate students, by Terry Tao, 2006 Fields medallist. Required reading.

*Thursday, 3/13*: HW # 7 has been uploaded. You have almost 2 weeks to do this HW based on last two weeks of classes.

Also, take a look at the project section below for important information about the project.*Wednesday, 2/26*: HW # 6 has been uploaded. It is due a week after the mid-term Exam#1.*Wednesday, 2/12*: Dates for Mid-term Exams #1 and #2 and Final Exam have been announced. Look below in the appropriate section.*Tuesday, 1/14*: Check this webpage regularly for homework assignments, announcements, etc.

*Exam # 1*: Wednesday, 2/26. Syllabus: Based on topics covered up to and including 2/12 [total 9 lectures].*Exam # 2*: Wednesday, 4/16. Syllabus: Based on topics covered from 2/17 up to 4/2 (including both dates [total 11 lectures].*Final Exam*: Thursday, May 8, 8am - 10am. Syllabus: All topics covered during the semester.

Read carefully through this

Look through the

3/5/2014: Email me with project team members (2-3 per team)

3/9/2014: Email me with your choice of project topic among the 14 project descriptions sent to you by email.

4/7/2014: Discuss the initial draft and model of your project with me in person. All members of your team must be present.

5/1/2014: Final submission of Project report and associated materials/programs/etc.

The homework problems listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.

If you have the 4th edition of the textbook, please compare the problem statements with those in 5th edition before solving them.

You are allowed the use of calculators /computational software, to aid in the basic computational work of the problems. In case of doubt, ask me for a clarification.

*Monday, 1/13*: Read and understand Examples from Section 1.1.*Wednesday, 1/15*: Read and understand Examples from Sections 1.2 and 1.3.**Homework #1:**Due Wednesday, 1/22: Math 486 students submit 5 problems, and Math 522 students submit all 6 problems.

Section 1.1: #3bc, #10, #13. Section 1.2: #2, #3, #9. Solutions distributed on 1/27.*Wednesday, 1/22*: Read and understand Examples from Sections 1.3 and 1.4.**Homework #2:**Due Wednesday, 1/29: Math 486 students submit all problems, and Math 522 students submit all problems.

Section 1.3: #1f, #2e, #3a, #6, #10. Section 1.4: #3 (read and understand example 2 first).*Monday, 1/27*: Read and understand Section 2.1.*Wednesday, 1/29*: Read and understand Examples from Sections 2.2, 2.3 and 2.4.**Homework #3:**Due Wednesday, 2/5: Math 486 students submit 5 out of 6 problems, and Math 522 students submit all problems.

Section 2.2: #3, #6. Section 2.3: #4, #9(read Example 2 first, do NOT solve part a or b). Section 3.1: #5, #7.*Wednesday, 2/5*: Read and understand Example 2 from Section 3.4.**Homework #4:**Due Wednesday, 2/12: Math 486 students submit 6 (note: 3.4.7ab and 3.4.8 are compulsory) out of 7 problems , and Math 522 students submit all problems.

Section 3.2: #2b, #3. Section 3.3: #2b, #4, #8. Section 3.4: #7ab, #8(compare all 4 models from #7 and #8 here).*Wednesday, 2/12*: Read and understand Example 2 from Section 7.1. Read and understand the examples in Sections 8.1 and 8.5.**Homework #5 [pdf].**Due Wednesday, 2/19. Solutions distributed on 2/19, Wednesday.*Monday, 2/24*: Read and understand Examples from Sections 7.2, 7.4, 6.1 and 6.2.**Homework #6:**Due Wednesday, 3/5: Math 486 students submit all problems, and Math 522 students submit all problems.

Section 7.2: #3, #13a. Section 7.4: #1. Section 6.1: #2, #3. Section 6.2: Project#1.*Monday, 3/12*: Read and understand Examples from Sections 6.3 and 5.3, and Example 1 from Section 5.5.**Homework #7:**Due Wednesday, 3/26: Math 486 students submit all problems, and Math 522 students submit all problems.

Section 6.3: #2. Section 5.1: #3(run the simulation on Mathematica/Matlab for n=100, 200, 300, etc. to get approximate values of pi), #5, #7(compare your answer to actual value of the volume). Section 5.2: #1b, #2c. Section 5.3: #2 (First write the algorithm you will be using and then run it on computer).*Wednesday, 3/26*: Read and understand Examples from Sections 11.1, 11.2, and 11.5 not covered in class.**Homework #8:**Due Wednesday, 4/2: Math 486 students submit all problems, and Math 522 students submit all problems. As always, explain each step of your solution, simply asking the software to give you the final answer is not acceptable.

Section 11.1: #1, #4, One of {6, 7}. Section 11.4: #4, #7. Section 11.5: #4.*Wednesday, 4/2*: Read and understand Examples from Sections 12.1, 12.2, and 12.5 not covered in class.**Homework #9:**Due Wednesday, 4/9: Math 486 students can skip one problem from Section 12.1, and Math 522 students submit all problems. As always, explain each step of your solution, simply asking the software to give you the final answer is not acceptable.

Section 12.1: #2, #5, #6. Section 12.2: #3, #6. Section 12.3: #5. Section 12.5: #7 (first write down Euler's method applied to this problem).*Monday, 4/14*: Read and understand Examples 1 and 2 from Section 13.3.**Homework #10:**Due Wednesday, 4/23: Math 486 and Math 522 students submit all problems. As always, explain each step of your solution, simply asking the software to give you the final answer is not acceptable.

Section 13.1: #7. Section 13.2: #6 (Use Calculus as well as method of steepest ascent/descent). Section 13.3: #3, #6.

*Monday, 1/13*: The process of math modeling - discussion with examples, Principle of proportionality, difference equations - examples from accounting/ finance/ science, discrete time vs. continuous time. (From Section 1.1 and elsewhere)*Wednesday, 1/15*: Approximating change via difference equations -examples, modeling births/deaths/resources through non-linear discrete dynamical systems. (From Sections 1.2, 1.3, and elsewhere)*Monday, 1/20*: MLK Jr. Day*Wednesday, 1/22*: Systems of discrete dynamical systems via astronaut docking procedure, limiting behavior of DDS(discrete dynamical system) and example of drug dosage, Equilibrium values and solutions of DDS. (From Sections 1.3, 1.4, and elsewhere)*Monday, 1/27*: Solutions methods and stability of equilibrium values of homogenous and nonhomogenous linear DDS, More nonlinear DDS via a hunter-prey model. (From Section 1.4, 2.2, and elsewhere)*Wednesday, 1/29*: Properties of proportionality, proportionality in non-linear or translated linear systems -examples from physics, Geometric similarity - relationship between geometric notions like volume, surface area, etc. in terms of a fundamental dimension, Ideas underlying model fitting - visual model fitting. (From Sections 2.2, 2.3, 3.1, and elsewhere)*Monday, 2/3*: Four sources of error in the modeling process, Transforming data to fit linear systems, Chebyshev Approximation Criterion - min max deviation, Writing Chebyshev AC as a linear optimization problem. (From Sections 3.1, 3.2 and 3.3 )*Wednesday, 2/5*: Minimizing sum of absolute deviations, Least squares criterion - min sum of squares of deviations, Normal equations and critical points in LSC, Fitting a straight line, Fitting a power curve with fixed exponent, fitting a power curve with unknown exponent, Transforming non-linear data, Comparing the various LSC models derived using transformations. (From Sections 3.3 and 3.4)*Monday, 2/10*: Comparing different models, Relationship between CAC and LSC models, Introduction to Linear Optimization - various forms, standard form (how to convert into standard form), integer programs, geometric properties. (From Sections 3.4 and 7.1, and elsewhere)*Wednesday, 2/12*: Graphs and networks - basic concepts and examples from applications, Network models, Network flow - max flow, min cost flow, Expressing Graph models using linear programs, Using 0-1 variables to make decisions, Graph Coloring and its relation to scheduling. (From Sections 8.1, 8.3, 8.5, and elsewhere)*Monday, 2/17*: Graph Coloring and its relation to maps, Eulerian trails in graphs, Vertex covers and expression as linear program, History and uses of Graph theory in various fields. (From Sections 8.1, 8.5, and elsewhere)*Wednesday, 2/19*: Uses of Linear programs in solving hard optimization problems, Geometric solutions of Linear programs, Algebraic solution of Linear Program using simplex algorithm, Geometric intuition behind simplex algorithm, Local search algorithms - underlying concepts and thinking of simplex as a local search algorithm, a local search algorithm for traveling salesman problem. (From Sections 7.2, 7.4, and elsewhere)*Monday, 2/24*: Markov chains - properties and examples, Models derived from Markov chains, Estimating Stationary distribution of a MC, Component reliability - series system and parallel system, Combinations of series and parallel components. (From Sections 6.1, 6.2, and elsewhere)*Wednesday, 2/26*: Mid-term Exam#1.*Monday, 3/3*: Fundamentals of Finite, Discrete time Markov Chains - irreducible, a periodic, recurrent, stationary distribution and it calculation as a limit and as an eigenvalue problem. (From elsewhere).*Wednesday, 3/5*: Linear regression and what it means, Its calculation using SSE, SST, SSR, properties, Plots of residuals, relation between linear regression and correlation, Calculating area under a curve using random points. (From Sections 6.3 and 5.1)*Monday, 3/10*: Monte Carlo algorithm for calculating area under a curve, volume under a surface, etc., Random point generation - middle square method, linear congruence method; (Monte Carlo) Modeling probabilistic behavior using random numbers - fair/ unfair coin, roll of fair/unfair die, etc. (From Sections 5.2 and 5.3)*Wednesday, 3/12*: Discussion of Mid-term Exam#1.*Monday, 3/24*: Finish discussion of Mid-term Exam #1. Ordinary Differential Equations for instantaneous rate of change for continuous problems and as approximate average rate of change in discrete problems, Population growth models, solving the linear ODE, Population growth under limited resources - logistic growth model. (From Section 11.1 and elsewhere)*Wednesday, 3/26*: Population growth under limited resources - logistic growth model and its solution, Autonomous DE, Equilibrium values - stable and unstable, phase diagram using the behavior of first and second derivatives, Using phase diagram to sketch solutions curves and solutions fields, solution field for the logistic model, Euler's method for approximate solution to 1st order DE. (From Sections 11.1, 11.4 and 11.5)*Monday, 3/31*: Continuous dynamical system - system of differential equations, Euler's method for approximate solution of a system of Differential equations, Ecological modeling problem with competing species of Blue and Fin whales - modified logistic model with competition factor, equilibrium points including the assumption on competition factor to get a non-extinction equilibrium point, stable vs. unstable equilibrium points, discretization of the Dynamical system with time step of 1 year and computing the time needed by populations of the two species to achieve the equilibrium populations, Sensitivity analysis with varying values of competition factor. (From Sections 12.1, 12.2, 12.5, and elsewhere)*Wednesday, 4/2*: Continuing the example of Blue and Fin whales - Sensitivity analysis with varying values of competition factor, discretization with increasing values of h(=delta t) to illustrate the phenomenon of chaos arising out of discrete Dynamical system. Basic terminology and definitions of Autonomous system of Differential equations, phase plane, solution, trajectory, unstable and stable equilibrium, etc. A predator-prey model for Baleen whales and Krill - derivation of a modified logistic model, equilibrium points and its analytic solution using separable differential equation. (From Sections 12.1, 12.3 and elsewhere)*Monday, 4/7*: Analyzing the trajectories of the predator-prey model, and using them to interpret the model and the cyclic fluctuation with time lag of the relative populations of the predator and the prey, Solutions of linear dynamical system (linear systems of Differential equations) using eigenvalues and eigenvectors and matrix exponentials. (From Sections 12.3 and elsewhere)*Wednesday, 4/9*: Unconstrained multivariable optimization - using calculus. (From Section 13.2)*Monday, 4/14*: Using the gradient method of steepest ascent/descent to solve unconstrained multivariable optimization, relation between stepsize and gradient, choice of stepsize. Constrained multivariable optimization - using Lagrange multipliers; economic meaning of lagrange multipliers. (From Sections 13.2, 13.3, and elsewhere)*Wednesday, 4/16*: Mid-term Exam#2.*Monday, 4/21*: Newton's Method for finding root of single equation, or common root of multiple equations, the principle of global method to find approximate region of solution and local method like Newton's to find the exact solution with required accuracy, example for sensitivity analysis using NM. (From elsewhere)*Wednesday, 4/23*: Discussion of Mid-term Exam#2.*Monday, 4/28*: Complex Networks, examples, fundamental properties - degree distribution, average distance between vertices, clustering coefficient; Erdos-Renyi Random graph model, Stanley Milgram's six degrees of separation, Watts-Strogatz small world network model. (From elsewhere)*Wednesday, 4/30*: Dynamics on small-world networks - SI, SIR models; Degree distributions in real-world networks, Power law distribution and fat tail, Scale-free networks , Robustness via removal of arbitrary nodes or high degree nodes, Barabasi-Albert's preferential attachment model, modification of PA model. (From elsewhere)

For an alternate point-of-view and for additional applications, refer to the following books:

M.M.Meerschaert, Mathematical Modeling, Fourth Edition.

H.P. Williams, Model Building in Mathematical Programming, Fifth Edition.

MATLAB - getting started at IIT, by Greg Fasshauer

A crash course

MATLAB Tutorial

Older Guide I

Older Guide II

Mathematica Tutorials.

A Beginner's guide to Mathematica.

Introduction to Mathematica.

Wikipedia on Math Models

OR Models