Instructor: Hemanshu Kaul

Office: 125C, REC (Engineering 1)
Phone: (312) 567-3128
E-mail: kaul [at] iit.edu

Time: 11:25am, Tuesday and Thursday.
Place: 025, REC (Engineering 1)

Office Hours: 1:50-2:50pm Tuesday and Thursday; and by appointment. Emailed questions are also encouraged.
As well as the discussion forum at Piazza
Math TA Office Hours: Check the schedule at 129, Retalliata Engg.

• Thursday, 1/18 : Dates for Mid-term Exams #1 and #2 have been announced. Look below in the appropriate section.


• Thursday, 1/14 : Check this webpage regularly for homework assignments, announcements, etc.

• Exam # 1 : Thursday, February 22nd. Syllabus: Based on topics, examples, applications corresponding to HWs #1-#5.
• Exam # 2 : Tuesday, April 17th. Syllabus: Based on topics, examples, applications corresponding to HWs #6-#10.
• Final Exam : 8am-10am, Wednesday, May 2nd. Syllabus: All topics covered during the semester.

• Tuesday, 1/9 : Read and understand Examples from Section 1.1.


• Thursday, 1/11 : Read and understand Examples from Sections 1.2 and 1.3.


• Homework #1: Due Thursday, 1/18. Solutions distributed on 1/18.
  Section 1.1: #3bc, #10, (#12a and #13a). Section 1.2: #2, #3, #9.


• Tuesday, 1/16 : Read and understand Examples from Sections 1.3 and 1.4.


• Thursday, 1/18 : Read and understand Section 2.1.


• Homework #2: Due Thursday, 1/25. Solutions distributed on 1/30
  Section 1.3: #1f, #2e, #3a, #6, #10[Instead of Example 4 of Section 1.2, Use the DDS a(n+1) = (0.69) a(n) + 0.1; a(0)=0.5]. Section 1.4: #2, #4, [Also try but do not submit #3].



• Tuesday, 1/23 : Read and understand Examples from Sections 2.2, 2.3 and 2.4.


• Thursday, 1/25 : Read and understand Section 3.1.


• Homework #3: Due Thursday, 2/1. Solutions distributed on 2/1
  Section 2.2: #6. Section 2.3: #4, #9(read Example 2 first, do NOT solve part a or b: you just have to explain the strengths and weaknesses of each model based on their underlying assumptions), Project#2. Section 3.1: #5, #7 [Comment: just roughly estimate (using your computational software if you wish) the values of the parameters from the plots; do NOT apply any theory from later sections. Use the appropriate log transformation].



• Thursday, 2/1 : Read and understand Example 2 from Section 3.4.


• Homework #4: Due Thursday, 2/8: Submit 5 out of 6 problems [Note: 3.4.7ab and 3.4.8 are compulsory; choose 3 out of the 4 problems in Sections 3.2 and 3.3]
  Section 3.2: #2b, #3. Section 3.3: #4, #8. Section 3.4: #7ab, #8(compare all 4 models from #7 and #8 here). Solutions distributed on 2/13.
  
  [Comment: When solving the problems for Chebyshev Approximation Criterion, it is expected that your solution explicitly includes the linear program that has to be solved to apply CAC.
  After writing the linear program, you should solve it using any solver/ calculator. For example, you can use:
  MATLAB : LinProg; MATHEMATICA - I; MATHEMATICA - II; OTHER SOLVERS]
  



• Thursday, 2/8 : Read and understand Example 2 from Section 7.1. Read and understand Examples from Sections 7.2, 7.3.


• Homework #5: Due Thursday, 2/15 Solutions distributed on 2/15.
  Section 7.1: #4.
  Section 7.2: #3, #13a (Set each as a Linear program and then solve it graphically by considering the family of lines corresponding to the objective function and seeing where it is maximized/minimized in the geometric feasible region which you should sketch).
  Section 7.3:#3, #7a (Set each as a Linear program and then solve it graphically by finding the corner points (intersections of the lines) of the geometric feasible region and checking which corner gives the max or min value for the objective function.).
  



• Tuesday, 2/21 : Read and understand the examples in Sections 8.1 and 8.5.


• Homework #6 [pdf]. Due Thursday, 3/1. Solutions distributed on 3/6.


• Thursday, 3/1 : Read and understand Examples from Sections 6.1, 6.2, and 6.3.


• Homework #7: Due Thursday, 3/8: Submit 6 out of the 7 problems. Solutions distributed on 3/20.
  Section 8.1: #3, #6, #7. Section 8.5: #1ab. Section 6.1: #2, Project#1. Section 6.2: Project#1.


• Thursday, 3/8 : Read and understand Examples from Sections 5.1 and 5.3..


• Homework #8: Due Thursday, 3/22. Solutions to be distributed on 3/27.
  Section 5.1:[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.]
  #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: [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)]
  #2 .
  



• Thursday, 3/22 : Read and understand Examples from Sections 11.1, 11.2, and 11.4 not covered in class.


• Homework #9: Due Thursday, 3/29: As always, explain each step of your solution, simply asking the software to give you the final answer is not acceptable.
  Solutions distributed on 4/3.
  Section 5.3: [For this problem, first write the step-by-step algorithm you will be using and then run it on computer for different number of trials]
  Use Monte Carlo Simulation to predict the outcome of the following game of chance:
  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. (n is the number of rounds in one game.) (When you run the algorithm on computer, if you wish you may assume that player 1 always calls "even".)
  Section 11.1: #4, #6, #7. Section 11.4: #4, #7.



• Thursday, 3/29 : Read and understand Examples from Sections 12.1, 12.2, and 12.5 not covered in class.


• Homework #10: Due Tuesday, 4/10: [This HW is due in 12 days. NOTE the DUE DATE and the EXAM#2 date] As always, explain each step of your solution, simply asking the software to give you the final answer is not acceptable.
  Solutions distributed on 4/10.
  Section 12.1: [Read Example 1 first] #6. Section 12.2:[Read and understand the context of equation 12.6 first] #3, #6. Section 12.3: #5abc. Section 12.5: #7 [first write down Euler's method applied to this problem].



• Thursday, 4/12 : Read and understand Examples from Section 13.2 and 13.3 that were not covered in class.


• Homework #11: Due Tuesday, 4/24: [NOTE the DATE] 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. Solutions to be distributed on 4/24.

• Tuesday, 1/9 : The process of math modeling - discussion with examples, Principle of proportionality. (From Section 1.1 and elsewhere)
• Thursday, 1/11 : difference equations - examples from accounting/ finance/ science, discrete time vs. continuous time, Approximating change via difference equations - examples, limiting behavior of DDS(discrete dynamical system) and example of modeling births/deaths/resources through non-linear discrete dynamical systems. (From Sections 1.2, 1.3, and elsewhere)
• Tuesday, 1/16 : Equilibrium values and solutions of DDS, Solutions methods and stability of equilibrium values of homogenous and nonhomogenous linear DDS. (From Sections 1.3, 1.4)
• Thursday, 1/18 : Systems of discrete dynamical equations via a interacting species population model and its analysis, Strategy analysis using System of DDS via Astronaut spacecraft docking procedure and its analysis. (From Section 1.4, and elsewhere)
• Tuesday, 1/23 : Completion of Strategy analysis using System of DDS via Astronaut spacecraft docking procedure and its analysis. 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. (From Sections 2.2, 2.3, and elsewhere)
• Thursday, 1/25 : Geometric similarity - Examples from physics (raindrops) and biology (fish). Model fitting vs. data fitting. Four sources of error in the modeling process, Transforming data to fit linear systems, What is error for a collection of data points vs. a model? Chebyshev Approximation Criterion - min max deviation as a linear optimization problem. (From Sections 3.1, 3.2 and 3.3)
• Tuesday, 1/30 : Writing Chebyshev AC as a linear optimization problem, Minimizing sum of absolute deviations, Least squares criterion - min sum of squares of deviations, Mathematical ideas for measuring distance - l_p distances, Examples for CAC and LSC, Applying calculus to find best fit model in LSC. (From Section 3.3 and elsewhere)
• Thursday, 2/1 : Least squares criterion - 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 the various LSC models derived using transformations. (From Sections 3.4 and elsewhere)
• Tuesday, 2/6 : Comparing the various LSC models derived using transformations along with model derived from CAC - how to compare any given models. Detailed discussion of HW#3 project using geometric similarity and proportionality. (From Sections 3.4 and elsewhere)
• Thursday, 2/8 : Relation between errors of CAC and LSC models. Optimization Problems - how to convert multiobjective problems into single objective through two approaches, Introduction to Linear Optimization - example; Geometric solutions of Linear programs. (From Sections 3.3, 7.1, 7.2, 7.3, and elsewhere)
• Tuesday, 2/13 : Geometric 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; local search algorithm for Traveling Salesman Problem; How to modify local search to allow "bad" moves - Simulated Annealing. (From Sections 7.3, and elsewhere)
• Thursday, 2/15 : Graphs and networks - how to express a logistics problem on a distribution network/ Transportation problem as a linear program. Binary programs and using 0-1 variables to make decisions. (From Section 8.1, and elsewhere)
• Tuesday, 2/20 : Review for mid-term exam. Mixed integer linear programs, Binary programs and using 0-1 variables to make decisions, Knapsack Problem and budgeting problems. (From Section 8.1, 8.4, and elsewhere) <\li>
• Thursday, 2/22 : Mid-term Exam#1.
• Tuesday, 2/27 : Graphs and Networks- basic concepts such as visual and algebraic representations and comparative notations for undirected and directed graphs. Graph Coloring and its relation to scheduling and its relation to coloring maps/ 4-color theorem. Network flow - max flow problem and expressing it using linear program. Markov chains - properties and examples, (From Sections Sections 8.1, 8.3, 8.5 and elsewhere)
• Thursday, 3/1 : Vertex cover, applications, and expression as linear program, Eulerian trails in graphs, Models derived from Markov chains, Estimating Stationary distribution of a MC, Different ways of representing and analyzing Markov Chains. (From Sections 8.1, 8.3, 8.5, and 6.1, and elsewhere.
• Tuesday, 3/6 : 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. Discussion of Exam 1 grades, score distribution, and solutions. (From Section 6.3)
• Thursday, 3/8 : Discussion of some HW#6 0-1 variable problems. Monte Carlo algorithm for calculating area under a curve, volume under a surface, etc., the concept of bounding box and its importance. Random point generation - middle square method, linear congruence method. (From Sections 5.1 and 5.2)
• Tuesday, 3/20 : (Monte Carlo) Modeling probabilistic behavior using random numbers - fair/ unfair coin, roll of fair/unfair die, etc. Monte Carlo modeling of rainy days and probability of three consecutive rainy days. Idea underlying Markov Chain Monte Carlo. Ordinary Differential Equations for instantaneous rate of change for continuous problems and as approximate average rate of change in discrete problems. (From Sections 5.2 and 5.3, and elsewhere)
• Thursday, 3/22 : Population growth models - Malthusian equation, solving the linear ODE, 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, Equilibrium values - stable and unstable, phase diagram using the behavior of first and second derivatives. (From Section 11.1 and elsewhere)
• Tuesday, 3/27 : Qualitative vs Quantitative Analysis of a mathematical model. 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. Continuous dynamical system - system of differential equations, Solutions curves as parametric system, stable/asymptotically stable/unstable equillibrium points and related concepts for a dynamical system. (From Sections 11.1, 11.4, 12.1.)
• Thursday, 3/29 : 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)); A competitive-hunter model as a simplification of competing species model with underlying exponential rather than logistic growth, Equillibrium points, graphical analysis and interpretation of the model. 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. (From Section 12.2 and elsewhere)
• Tuesday, 4/3 : Motivation and mathematics for Euler's method for approximate solution to 1st order DE. Continuous dynamical system - system of differential equations, 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 Eulers method, discretization with increasing values of delta t to illustrate the phenomenon of chaos arising out of discrete Dynamical system. (From Section 12.5 and elsewhere)
• Thursday, 4/5 : 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. (From Section 12.3 and elsewhere)
• Tuesday, 4/10 : 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. (From elsewhere)
• Thursday, 4/12 : 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. (From Section 13.1, 13.2, and elsewhere)
• Tuesday, 4/17 : Mid-term Exam#2.
• Thursday, 4/19 : Constrained multivariable optimization - using Lagrange multipliers and its underlying principle (and economic interpretation of Lagrage multipliers). Using the gradient method of steepest ascent/descent to solve unconstrained multivariable optimization, relation between stepsize and gradient, choice of stepsize. (From Sections 13.2, 13.3, and elsewhere)
• Tuesday, 4/24 : 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, Degree distributions in real-world networks, Power law distribution and fat tail, Scale-free networks . (From elsewhere. For more see textbook: Mark Newman, Networks: an introduction.)
• Thursday, 4/26 : Discussion of Mid-term Exam#2 and what to expect in the final.