Instructor: Hemanshu Kaul
Office: 125C Rettaliata Engg Center.
E-mail: kaul [at] iit.edu

Time: 1:50-3:05pm Tuesday and Thursday.
Place: 121 Rettaliata Engg Center

Discussion Forums: Math 380 at Campuswire.

Office Hours: Tuesday and Thursday at 3:05-4pm. And by appointment in-person or through Zoom (send email to setup appointment).
Questions through Campuswire Discussion Forums are strongly encouraged.

TA Office Hours: Minshen Xu, Wednesday 12-3pm, at RE 129 or at Math Tutoring Center.

• Thursday, 1/12 : Dates for Mid-term Exams #1 and #2 have been announced. Look below in the appropriate section.



• Tuesday, 1/10 : Check this webpage regularly for homework assignments, announcements, etc.

• Exam #1 : Thursday, February 23rd. Syllabus: Based on topics, examples, applications corresponding to HWs #1-#5.
• Exam #2 : Thursday, April 13th. Syllabus: Based on topics, examples, applications corresponding to HWs #6-#10.
• Final Exam : 5/3, Wednesday, 2-4pm in RE 122. Syllabus: All topics covered during the semester.

• Week #1 : 2 Lectures.
  
  • Topics:Discussion of class structure and purpose. The process of math modeling - discussion with examples, Principle of proportionality, difference equations - examples from accounting/ finance/ science, discrete time vs. continuous time, limiting behavior of DDS(discrete dynamical system) and example of modeling births/deaths/resources through non-linear discrete dynamical systems. (From Sections 1.0, 1.1, 1.2, and elsewhere)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#1.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand all Examples from Sections 1.1 and 1.2.
    • HW#1 for Submission. Due Thursday, 1/19, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Section 1.1: #3bc, #10, (#12a and #13a).
      Section 1.2: Submit any two of the following three sets of problems: #2, (#6 and #7), #9.
      Questions? Ask on Campuswire. Or, send email.
      HW Solutions distributed in class.


• Week #2 : 2 Lectures.
  
  • Topics:Drug dosage model and its analysis, (stable and unstable) Equilibrium values/fixed points and solutions of DDS, Solutions methods and stability of equilibrium values of homogenous and nonhomogenous linear DDS, Interacting discrete dynamical systems via a interacting species population model (Competitive Hunter model). (From Sections 1.3, 1.4)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#2.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand Examples from Sections 1.3 and 1.4.
    • HW#2 for Submission. Due Thursday, 1/26, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Section 1.3: #1f, #2e, #3a, #6, #14ad.
      Section 1.4: #2, #4.
      Questions? Ask on Campuswire. Or, send email.
      HW Solutions distributed in class.


• Week #3 : 2 Lectures.
  
  • Topics: 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; Geometric similarity - Examples from physics (raindrops) and biology (fish); short discussion of SIR epidemic model; Strategy analysis using System of DDS in an Astronaut spacecraft docking procedure and its analysis with interpretation. Model fitting vs. data fitting/ Interpolation. (From Sections 2.2, 2.3, 1.4 and elsewhere)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#3.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand Examples from Sections 2.2, 2.3, 2.4, and 3.1. Think about Section 2.3 Problem #9 (main question, not parts ab).
    • HW#3 for Submission. Due Thursday, 2/2, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Section 2.2: submit one of (#6, #12).
      Section 2.3: #4, Project#2.
      Section 3.1: submit one of (#5, #7). [Comment: Look at Table 3.1 and 3.2 and the discussion inbetween them (we will also discuss this on Tuesday). Explain the use of appropriate data transformation, and just roughly estimate (using your computational software if you wish) the values of the parameters from the plots.]
      Questions? Ask on Campuswire. Or, send email.
      HW Solutions distributed in class.<


• Week #4 : 2 Lectures.
  
  • Topics: Model fitting vs. data fitting/ Interpolation; Sources of qualitative and quantitative errors in the modeling process, Transforming data to fit linear systems. What is error for a collection of data points vs. a model? Fitting a model to data as minimizing appropriate l_p distance between vector of observations and vector of predictions. Chebyshev Approximation Criterion - min max absolute deviation as a linear optimization problem. Minimizing average deviation. Least squares criterion - min sum of squares of deviations, Applying calculus to find best fit model in LSC, Normal equations and critical points in LSC plus application of second derivative test, Using LSC for fitting a straight line, Fitting a power curve with fixed exponent, fitting a power curve with unknown exponent, Transforming non-linear data. Comparing various models for the same set of observed data with LSC and with CAC, Error calculations. (From Sections 3.1, 3.2, 3.3, 3.4, and elsewhere)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#4.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand Examples 1 and 2 from Section 3.4.
    • HW#4 for Submission. Due Friday, 2/10, by 10am in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Section 3.2: #2b, #3.
      Section 3.3: Submit one of (#4, #8).
      Section 3.4: #7ab, #8(compare all 4 models from #7 and #8 here, as we did in the lecture).
      [Comment: When solving the problems for Chebyshev Approximation Criterion, it is expected that your solution explicitly includes the linear optimization problem that has to be solved to apply CAC. After writing the linear optimization problem, you should solve it using any solver. For example, you can use: MATLAB : LinProg; MATHEMATICA - I; MATHEMATICA - II; Python; OTHER SOLVERS.]
      Questions? Ask on Campuswire. Or, send email.
      


• Week #5 : 2 Lectures.
  
  • Topics: Optimization based Decision making, general form of an optimization problem, Introduction to Linear Optimization through variants of a model, Integer, Mixed Integer and binary optimization problems, multiobjective problems and how to convert them into single objective through three approaches, Binary programs and using 0-1 variables to make decisions with various constraints, Knapsack Problem and budgeting problems, modeling dependent "yes/no" decisions, approximating a non-linear function by a piecewise linear function using 0-1 variables. Geometric solutions of Linear programs, Geometric and linear algebraic intuition behind simplex algorithm, Local search algorithms - underlying concepts like neighborhood and step-size; thinking of simplex algorithm, optimization from Calculus as local search algorithms. (From Sections 7.1, 7.2, 7.3, and elsewhere)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#5.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand Examples 1 and 3 from Section 7.1. And Examples 1 and 2 from Section 7.2.
    • Homework #5 for submission [PDF]. Due Thursday, 2/16, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Questions? Ask on Campuswire. Or, send email.
      


• Weeks #6 and #7 : 3 Lectures and 1 Mid-term Exam.
  
  • Topics: Local search algorithms - underlying concepts like neighborhood and step-size; thinking of simplex algorithm, optimization from Calculus as local search algorithms. Graphs and networks - basic concepts such as visual and algebraic representations, examples of vertices and edge relations from Math, Computer Science, Operations Research, Network Science, Data Science, Biology, Epidemiology, etc. Conflict-free allocation of scarce resources as Graph Coloring and related examples of scheduling/ allocation of resources, Proper coloring and chromatic number of a graph, Chromatic number of a graph as 0-1 Linear optimization problem, Greedy Algorithms, Greedy coloring for proper coloring of a graph, Monitoring a network model as a vertex cover problem, Greedy algorithm for vertex cover, Vertex Cover problem modeled as a 0-1 linear optimization problem.
    Maximum Flow problem on networks as a linear optimization problem, Modeling Matching problem as a max flow problem, Min cost transportation problem for logistics as a linear program, Eulerian graphs and Eulerian tours as a model for Konigsberg bridge problem. (From Sections 8.1, 8.2, 8.3, 8.5 and elsewhere)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#6 and Notes#7 and Notes#8.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand the examples in Sections 8.1, 8.3, and 8.5.
    • Homework #6 for submission[PDF]. [Based on 3+ lectures]. Due Friday, 3/3, by 10am in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Questions? Ask on Campuswire. Or, send email.
      


• Week #8 : 2 Lectures.
  
  • Topics:Maximum Flow problem on networks as a linear optimization problem, Min cost transportation problem for logistics as a linear program, Eulerian graphs and Eulerian tours as a model for Konigsberg bridge problem. (From Sections 8.1, 8.3, 8.4, 8.5, and elsewhere)
    Monte Carlo algorithm for calculating area under a curve, volume under a surface, etc., the concept of bounding box and its importance. Idea of using randomness in computation - MCMC, volume of high dimensional body, etc. Random point generation - middle square method, linear congruence method. (Monte Carlo) Modeling probabilistic behavior using random numbers - flipping fair/ unfair coin, roll of fair/unfair die, flipping pairs of coins, etc. Monte Carlo simulation modeling of rainy days and probability of three consecutive rainy days, Comparison with simulating a "rainy week", Simulating "nice week" with at least 4 days that have no rain and low winds, multiple weather conditions, etc. Underlying assumptions and criticism of these models. (From Section 5.1, 5.2, 5.3, 5.5 and elsewhere)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#8 and Notes#9.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand the Example from Sections 5.5.
      [Try these two problems but do not submit] Section 5.2: #1b, #2c.
    • HW#7 for Submission. Due Thursday, 3/9, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      
      • Section 5.1: Submit 2 out of the 3 problems listed below for this Section.
        [For each of these problems, first carefully write all steps of the algorithm as applied to the problem. Remember you have to find the appropriate bounding rectangle/box. Write the solution in the way I did such problems in class.]
        #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 5.3: Submit both problems listed below for this Section.
        [First write the step-by-step algorithm you will be using and then run it on computer for different values of n (number of trials). Write the solution in the way I did such problems in class.]
        #2,
        #4.
      • Final Problem. Use Monte Carlo Simulation to predict the outcome of the following game of chance over n rounds:
        In each round of the game, two fair coins are flipped simultaneously and Player 1 calls "Evens" or "Odds". "Evens" means that coins must match (both heads or both tails), and "Odds" means that coins will not match. If Player 1 calls correctly then he/she wins $1 from Player 2. If Player 1 calls incorrectly then Player 2 wins $1 from Player 1. (When you run the algorithm on computer, if you wish you may assume that player 1 always has a fixed strategy, e.g. always calls "even", or always calls "odd", etc., and compare the outcomes based on each strategy.)
      Questions? Ask on Campuswire. Or, send email.
      


• Week #9 and #10 : 2 Lectures and 2 Spring Break holidays.
  
  • Topics:Discussion of Exam 1 grades and solutions.
    Models derived from Markov chains, Estimating Stationary distribution of a MC, Different ways of representing and analyzing Markov Chains - State Digraph, Transition probabilities and Matrix, Eigenvector for eigenvalue 1, Examples - Snakes-Ladders game, weather, Car Rental company with two locations, Election voting tendencies, Markov model for "oblivious random web surfer" and the page rank formula as the stationary distribution. (From Sections 6.1, 6.2, 6.3, and elsewhere).
    Component reliability - series system and parallel system, Combinations of series and parallel components. Linear regression and what it means, Its calculation using SSE, SST, SSR, properties, Plots of residuals. (From Sections 6.2, 6.3, and elsewhere).
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#9 and Notes#10.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand Example 2 from Section 6.1, and Example 3 from Section 6.2, and Example 1 from Section 6.3.
    • HW#8 for Submission. Due Thursday, 3/23, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Section 6.1: #2, Project#1.
      Section 6.2: Project#1.
      Final Problem: Find the simplified pagerank of all the webpages in the 5 webpage example (you can add either a loop on W5 or an arc from W5 to W1 with probability 1) in the lecture notes (Notes#10 above).
      Questions? Ask on Campuswire. Or, send email.
      


• Week #11 : 2 Lectures.
  
  • Topics: Ordinary Differential Equations for instantaneous rate of change for continuous problems and as approximate average rate of change in discrete problems. Population growth models - Malthusian equation, solving the linear ODE, Population growth under unlimited resources, Population growth under limited resources - logistic growth model and its solution & properties - how to estimate max population capacity; underlying assumptions. Solutions curves for an ODE and its relation to initial value, Autonomous DE, phase diagram using the behavior of first and second derivatives, Using phase diagram to sketch solutions curves and solutions fields, Qualitative vs Quantitative Analysis of a mathematical model. Equilibrium values - stable and unstable. Solution field for the logistic model. Continuous dynamical system - system of differential equations, Solutions curves as parametric system, stable/asymptotically stable/unstable equilibrium points and related concepts for a dynamical system, Phase plane and using the direction of derivatives of x and y w.r.t. t to plot the solution curve (x(t), y(t)). (From Sections 11.1, 11.4, 11.5, 12.1, and elsewhere)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#11.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand Examples from Sections 11.1, 11.2, and 11.5 not covered in class.
    • HW#9 for Submission. Due Thursday, 3/30, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Section 11.1: #4, #6.
      Section 11.4: #4.
      Section 12.1: #6.
      Questions? Ask on Campuswire. Or, send email.
      


• Week #12 : 2 Lectures.
  
  • Topics:A competitive-hunter model as a simplification of competing species model with underlying exponential rather than logistic growth, Equilibrium points, graphical analysis and interpretation of the model. A predator-prey model as a variation of the competitive-hunter model, equilibrium points, graphical analysis, and interpretation of the model and the periodic fluctuation with time lag of the relative populations of the predator and the prey, "Harvesting" and Volterra's principle, Volterra-Lotka model. Modeling and interpreting the Economic aspects of an Arms Race between two coountries. 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 under various scenarios of the competition factor.
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#12 and Notes#13
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand Examples from Section 12.4 not covered in class.
    • HW#10 for Submission. Due Thursday, 4/6, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Section 12.2: #3, #6.
      Section 12.3: #5abcdf.
      Questions? Ask on Campuswire. Or, send email.
      


• Weeks #13 and #14 : 3 Lectures and 1 Exam.
  
  • Topics:Discussion and Review for Mid-term Exam#2.
    Completed the discussion 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 under various scenarios of the competition factor. Euler's method for approximate solution of a system of Differential equations. Discretization of the Dynamical system for Blue and Fin whale populations with time step of 1 year and computing the time needed by populations of the two species to achieve the equilibrium populations. Sensitivity analysis of competing species model with varying values of competition factor using Euler's method, discretization with increasing values of step size (delta t) to illustrate the phenomenon of chaos arising out of discrete Dynamical system.
    Unconstrained multivariable optimization - fixed rate of change vs. varying rate of change. Submodels within a "Profit = Revenue - Cost" model and their criticism. Analysis using calculus; Sensitivity analysis and its use in understanding a model and its robustness. All illustrated through a simple pig selling problem. 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. Using the gradient (with Newton's method), or the gradient method of steepest ascent/descent to solve unconstrained multivariable optimization, relation between stepsize and gradient, choice of stepsize. (From Sections 12.5, 13.1, 13.2, and elsewhere)
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#13. Notes#14.
  • Homework:
    The homeworks listed below are from the course textbook, Giordano, Fox, Horton, A First Course in Mathematical Modeling, 5th edition.
    Follow the detailed instructions and rules for HWs given in the Course Information Handout and through Campuswire and email comments.
    
    • Reading HW: Read and understand Examples from Sections 12.5, 13.1, 13.2, 13.3, and 13.4 not covered in class.
    • HW#11 for Submission. Due Thursday, 4/20, by 11pm in Blackboard. Submit a single PDF file through Blackboard Assignment.
      Submit 3 out of the following 4 problems.
      Section 12.5: #7 [first write down Euler's method applied to this problem and then solve it using Matlab, etc.].
      Section 13.1: #7.
      Section 13.2: #6 (Use Gradient = 0 method as well as method of steepest ascent/descent).
      Section 13.3: #3.
      
      Questions? Ask on Campuswire. Or, send email.
      


• Weeks #15 and #16 : 4 Lectures.
  
  • Topics:Discussion of Exam#2 scores and problems.
    
    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 and its underlying principle (sensitivity and economic interpretation of Lagrange multipliers). Examples and models. Model for optimal harvesting of a renewable resource like fish. (From Sections 13.2, 13.3, 13.4, and elsewhere)
    
    Review of Mathematical modeling process, intuitive vs formal mathematics.
    What is ethical decision-making? How do we make ethical decisions? How are ethics relevant to the mathematical modeling process? Ethical concerns in each step of the modeling process - algorithmic bias, bioethics, statistical bias, etc. Impact of the bias in assumptions underlying the model formulation, Love Canal controversy, Wales Radioactivity model, Precautionary principle, etc. How can we modify a transportation model to incorporate equitable accessibility to social resources while maintaining overall service levels, Example from CTA bus services in Chicago, etc. (From elsewhere).
    
  • Lecture Notes: Outlines of lectures without all the details as discussed in the classroom. Notes#15. Notes#16.

• M.M.Meerschaert, Mathematical Modeling, Fourth Edition.
• H.P. Williams, Model Building in Mathematical Programming, Fifth Edition.
• Hillier and Lieberman, Introduction to Operations Research, 7th edition onwards.


• Wikipedia on Math Models
• OR Models


• For a bit of insight into what randomness means and one way of characterizing a random sequence of numbers, read through this short essay: Probability vs Randomness.


• Chasing the Elusive Numbers That Define Epidemics
• Hard Lessons of Modeling the Coronavirus Pandemic
• Mathematicians and Blue Crabs
• Ecologists nightmare is a mathematicians dream
• Universal Geometry of Geology
• Modeling Brains: Ignore the Right Details
• Equation-Free Prediction in Ecological models
• Nature's Critical Warning System
• A Mathematical Model Unlocks the Secrets of Vision
• How Nature Defies Math in Keeping Ecosystems Stable
• Evidence is only as good as the model, and modeling can be dangerous business. So how much evidence is enough?
• Universal Law of Turbulence
• A Statistical Search for Genomic Truths
• All Change Is a Mix of Order and Randomness

• You can access MATLAB through the IIT's Virtual Computing Lab (VCL) which is accessible through your my.iit.edu account.
  See FAQs regarding VCL.
  Refer to VCL Instructions here.
• MATLAB Overview
• Getting Started with MATLAB
• MATLAB Development Environment
• Preface of Moler's Textbook
• Chapter 1 of Moler's Textbook
• Single Page summary of essential MATLAB commands
• Twelve Page summary of essential MATLAB commands
• Another Introduction to MATLAB from a more advanced book of Moler.

• Mathematica at IIT
• Mathematica Tutorials - Videos.
• Mathematica Tutorials.
• A fast introduction to Mathematica for Programmers.
• An elementary introduction to Mathematica.
• Introduction to Mathematica.