Catalog Entry: MATH 641. Mathematical Analysis and Modeling (3).
Prerequisite: Undergraduate degree in mathematics or by instructor permission.
Examines mathematical models of real life phenomena and develops solution strategies for open-ended problems.
The models are based on Calculus, Differential Equations and Linear Algebra; they may include discrete and continuous population models, diffusion processes, business and economics models, continuous and discrete optimization problems with calculus and linear programming. Software may include Excel, Maple, Matlab or similar programs.
1.
Fibonacci Sequence I
(a) Definition
(b) Graph (logarithmic scale)
(c) Ratio of consecutive terms
2.
Fixed Point Analysis
(a) Definition
(b) Cobwebs
(c) Convergence Analysis
(i) Mean Value Theorem
(ii) Convergence Theorem
(iii) Attracting and Repelling Fixed Points
(d) Relation to Newton’s Method
3.
Fibonacci Sequence II
(a) Exact Calculation of the n-th Term
(b) Sum of Fibonacci Numbers (induction proof)
(c) Sum of Squares of Fibonacci Numbers (induction proof)
4.
Review of Eigenvalues
(a) Solutions of Linear Systems
(b) Singular and Non-Singular Matrices
(c) Determinants
(d) Definition of Eigenvalues/Eigenvectors
(e) Characteristic Polynomial
(f) Calculation of Eigenvectors
(g) Power Method
5.
Fibonacci Sequence III
Matrix Perspective
6.
Discrete Population Models I
(a) A first example (four age groups), growth rate, stable age distribution
(b) Leslie Matrix
(c) Age clusters
(d) State Diagrams
7.
Discrete Population Models II, Harvesting
Two dimensional model

(a) Survival Regions, graphical
(b) Survival regions, analysis (eigenvalues/vectors)
8.
Logistic Growth
(a) Fixed Points
(b) Behavior for r > 2 (bifurcations)
9.
Classical Bifurcation Model
(a) Change of Variables
(b) Two-cycles and the “every other term” function.
(c) General Bifurcation Diagram
10.
Logistic Growth with Harvesting
Constant and non-constant models
11.
Discrete Predator Prey Model I
(a) Derivation of the model
(b) Steady States
12.
Linearization
(a) The basic idea of linearization (drop higher order terms)
(b) Square Root computations. This is Newton’s Method without taking a derivative.
(c) Linearization and Tangent Lines
13.
Discrete Predator Prey Model II
(a) Linearization of the discrete predator prey model
(b) Classification of the steady states
14.
Continuous Models
(a) Transition process from discrete to continuous
(b) Fixed points and steady states
(c) Analytic Solutions, Separable Equations
(d) Direction fields and Euler’s method
(e) One dimensional State Diagrams
(f) The Phase Plane
This was one semester. More can be added if repetitions are avoided, and if students work on material on their own, rather than going over it in class.
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