Lecturer: Jorge Viveros
office: CIMA 1 (MF1 bldg)
email: jviveros[at]uaeh.edu.mx
Dates I will be away: TBA
Classroom: MF3-4
Lectures: Mon 11-13, Tue and Wed 11-12, Thr 15-16
Office hours: Wed 15-21 (or by appointment)
As I must set up appointments through the institutional tutoring and counseling webpage, students need to first send me an email to schedule a meeting (no walk-ins).
PREREQUISITES: differential calculus in one variable, linear algebra 1. Please do not register for this class if you have not taken and passed those classes.
TOPICS BY UNIT
We will try to adhere to the list of topics below, however we may omit some topics and cover others more in depth. The choice of topics reflects the fact that this is a class for applied mathematics majors, therefore we will have to balance method, technique, and analysis, with concrete applications. Also, we need to cope with the limitations of this being a 3rd-semester class, thus we will have to take time to introduce a few necessary but basic concepts from probability theory (at the expense of cutting down to a minimum or removing entirely topics you will luckily cover, for instance, in a linear programming class and in a statistics class). Main references for each unit are indicated in brackets (see bibliography below).
UNIT I: Deterministic Models of change.
1. Difference equations and compartmental analysis. 2. Closed-form solutions and mathematical analysis. 3. Variable growth rates and the logistic model. 4. Systems of recurrence relations. 5. Scaling and proportionality in modeling. 6. Geometric similarity and dimensional analysis in modeling. 7. Analytic methods for model fitting: Chebyshev’s method, least absolute deviation sum, least squares. 8. Measure of fit, and statistics of simple regression. References: [GFH13] and [MSw07] for 1-4, [GFH13] and [Ben00] for 5-6, [GFH13] for 7, [MSw07] for 8.
UNIT II: Probabilistic and Stochastic Modeling.
1. Measures of center and spread of stochastic data: mean, variance, standard deviation, histograms, boxplots and five-point summary. 2. Some probability background: random variable, probability density function, cumulative distribution function, mean and variance of a random variable. 3. Uniform distribution, normal distribution, Bernoulli variables and binomial distribution. 4. A stochastic demographic model for the Florida sandhill crane. 5. Model validation: chi-square test statistic. 6. Simple examples of Monte Carlo simulation. 7. Examples of deterministic state models and their eigenvalue analysis. 8. Fundamentals of Markov chains and applications. References: [MSw07] for 1-5 and 7, [GFH13] and [Ben00] for 6, [MSw07] and [GFH13] for 8.
UNIT III: Modeling with graphs.
1. Examples. 2. Connectedness. 3. Digraphs and matrices. 4. Tournaments. 5. Traffic flow and vulnerability. 6. Garbage trucks and colorability. 7. Dijkstra’s shortest-path algorithm. 8. Maximum-flow algorithm.
References: [Rob76] for 1-6, [GFH13] for 7-8.
UNIT IV: Principles of Decision and Game theories.
1. Decision trees. 2. Sequential decisions and conditional probabilities, and alternative criteria. 3. Total conflict and linear programming models. 4. Games against nature. 5. Alternative methods for determining pure strategy solutions. 6. Partial conflict games. 7. Modeling examples.
References: [GFH13].
BIBLIOGRAPHY
No textbook will be required for this class. The books in this reference list contain a wide range of applications of math modeling to real-life problems, and are directed to a broad audience from actuarial to mathematical sciences. Several of these books also treat continuous-time models, see the list of topics above to know which books you may want to focus on while taking this class.
| [Bar16] | BARTON, J.T. (2016) Models for Life: An Introduction to Discrete Mathematical Modeling with Microsot Office Excel Office. Wiley. 978-1119039754. |
| [Ben00] | BENDER, E.A. (2000) An Introduction to Mathematical Modeling. Dover Books on Computer Science. 978-0486411804. |
| [Bil11] | BILLEY, S.; BURKE, J.; CHARTIER, T.; GREENBAUM, A.; LEVEQUE, R. (2011) Discrete Mathematical Modeling: Math 381 Course Notes. University of Washington. |
| [GFH13] | GIORDANO, F.R.; FOX, W.P.; HORTON, S.B. (2013) A First Course in Mathematical Modeling, 5th ed. Cengage Learning. 978-1285050904. |
| [LRT83] | LUCAS, W.F.; ROBERTS, F.S.; THRALL, R.M. (eds.) (1983) Discrete and System Models. Springer. 978-0387907246. |
| [Mee13] | MEERSCHAERT, M.M. (2013) Mathematical Modeling, 4th ed. Academic Press. 978-0123869128. |
| [Mes07] | MESTERTON-GIBBONS, M. (2007) A Concrete Approach to Mathematical Modelling. Wiley-Interscience. 978-0470171073. |
| [MSw07] | MOONEY, D.; SWIFT, R. (2007) A Course in Mathematical Modeling. AMS. 978-0883857120. |
| [Rob76] | ROBERTS, F.S. (1983) Discrete Mathematical Models with Applications to Social, Biological, and Environmental Problems. Pearson. 978-0132141710. |
| [Tun07] | TUNG, K.K. (2007) Topics in Mathematical Modeling. Princeton University Press. 978-0691116426. |
Evaluation
| 4 midterm exams | (25% each) | 29 Ago, | 26 Sep, | 24 Oct, | 21 or 28 Nov |
CLASS MATERIALS
For test 1: ordered sets & induction, difference/recurrence eqs., systems I, systems II.
LibreOffice calc sheets: fleets, hawks vs owls, traveler’s tendencies.
Octave code to iterate 2d map X(j+1)=JX(j) and sample presentation.
Cobweb diagrams.
practice test A, practice test B, test 1.
For test 2: least-squares, Chebyshev’s, and mean absolute deviation sum methods for fitting functions to data: (presentation), Octave code for demos: linear_fit.m, nonlin_fit.m, nonlinf_fit2.m, spline_demo.m, test 2-I.
For test 3:
For test 4:
SOFTWARE AND ONLINE RESOURCES
LibreOffice, Octave (Linux & Ubuntu OS: install Octave directly via system admin), Octave-online, Geogebra, Cobweb diagrams.
Jorge Viveros · Centro de Investigación en Matemáticas · Instituto de Ciencias Básicas e Ingeniería · Universidad Autónoma del Estado de Hidalgo · jviveros[at]uaeh.edu.mx
Jorge Viveros / CIMA-UAEH 