MATH 4193.001/5093.001 - Introductory Mathematical Modeling / Mathematical Models - Spring 2017
TR 12:00--1:15 p.m., 359 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Monday 2:30-3:30, Wednesday 10:30-11:30, or by appointment, in 802 PHSC.

Course catalog description:
For 4193: Prerequisite: 3113 or 3413, 3333, 4733 or 4753, or permission of instructor. Mathematics models are formulated for problems arising in various areas where mathematics is applied. Techniques are developed for analyzing the problem and testing validity of proposed model. (F)
For 5103: Prerequisite: permission of instructor or admission to the M.S. program. May be repeated with change of content; maximum credit six hours. Mathematical models are formulated for problems arising in various areas in which mathematics has been applied. In each case, techniques are developed for analyzing the resulting mathematical problem, and this analysis is used to test the validity of the model. (Sp)

Text: We will use parts of the books below (all of them are feely available in pdf format for OU students from the OU library):
G. S. Layek. An Introduction to Dynamical Systems and Chaos, Springer, 2015.
M. W. Hirsch, S. Smale, R. L. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd edition, Academic Press, 2012.
J. D. Murray. Mathematical Biology, Vols. 1 and 2, 3rd edition, Springer, 2002, 2003.
A. C. King, J. Billingham, S. R. Otto. Differential Equations: Linear, Nonlinear, Ordinary, Partial. Cambridge University Press, 2003.
R. Seydel. Practical Bifurcation and Stability Analysis. 3rd edition, Springer, 2010.

A brief (tentative) list of topics to be covered:

• One-dimensional systems: flows on the line, bifurcations, flows on the circle.
• Two (and higher)-dimensional flows: linear systems, the phase plane, limit cycles, bifurcations.
• Chaos: the Lorenz equations, one-dimensional maps, fractals, strange attractors.

Homework:

• Homework 1, due on Thursday, February 2.
• Homework 2, due on Thursday, February 16.
• Homework 3, due on Thursday, March 2.
• Homework 4, due IN CLASS on Thursday, March 23.
• Homework 5, due on Tuesday, April 18.
• Homework 6, due on Thursday, May 4.

Content of the lectures:

• Lecture 1 (Tue, Jan 17): Introduction: Overview of the course, viewing of the NOVA documentary "The Strange New Science of Chaos" (31 January 1989) available on YouTube. On recent history, see J. Gleick's book Chaos: Making a new Science. (Penguin, 1987).
  The importance of being nonlinear: ODEs vs. PDEs; ODE + initial condition (IC) = initial value problem (IVP);
  example - harmonic oscillator equation (without damping or external forcing), x''+ω²x=0, general solution x(t)=C₁cos(ωt)+C₂sin(ωt) (with two arbitrary constants), particular solution for given initial conditions x(0)=x₀, x'(0)=v₀;
  complicating the harmonic oscillator equation in a non-significant way by adding a liner damping term γx'(0) (with γ=const>0) in the LHS and an external forcing term ƒ(t) in the RHS;
  complicating the harmonic oscillator equation in a significant way by adding a nonliner term βx³ in the LHS; another nonlinear example - pendulum equation, x''+(g/l)²sin(x)=0; nonlinear systems are so hard to solve because the principle of superposition fails (hence no normal modes, no Fourier transform, no Laplace transform...).
• Lecture 2 (Thu, Jan 19): The importance of being nonlinear: systems of autonomous first-order ODEs x'=f(x) for the unknown function x:[0,T]→Rⁿ (T is a positive number), where f:Rⁿ→Rⁿ is a given function; non-autonomous first-order systems: x'=f(x,t);
  converting a single higher-order ODE into a system of 1st-order ODEs; an example of an IVP without solution: x'=ƒ(x), where ƒ(x)=−1 for x≥0 and ƒ(x)=1 for x<0;
  an example of an IVP with infinitely many solutions: dV/dt=βV^2/3 (with β=const>0), V(0)=0, describing the volume of a water droplet in oversaturated vapor right at the moment of its formation.
  ODEs - basic geometric concepts: geometric representation of the solution of the IVP x'=f(x), x(0)=x⁽⁰⁾ as a parameterized curve in Rⁿ called flow and given by the equation φ_t(x⁽⁰⁾), t∈[0,T]; flow φ_t of an ODE; the flow satisfies dφ_t(x⁽⁰⁾)/dt=f(φ_t(x⁽⁰⁾)), φ₀(x⁽⁰⁾)=x⁽⁰⁾ (the latter can be written as φ₀=Id_Rⁿ (where Id_Rⁿ is the identity function in Rⁿ); semigroup property of the flow: φ_t(φ_s(x⁽⁰⁾))=φ_t+s(x⁽⁰⁾), i.e., φ_t∘φ_s=φ_t+s; example: flow of the ODE x'=−x, y'=x²+y see the complete solution of this problem, and check that the flow satisfies the semigroup property.
• Lecture 3 (Tue, Jan 24): ODEs - basic geometric concepts (cont): linear autonomous constant-coefficient ODEs: x'=Ax, where A is a constant square matrix; definition of an exponent of a matrix; the solution of the IVP x'=Ax, x(0)=x⁽⁰⁾ is given by φ_t(x⁽⁰⁾)=e^tAx⁽⁰⁾; example: exponentiating a diagonal matrix with distince diagonal elements; example: proof that if D=CAC⁻¹ is diagonal (where C is an invertible matrix) with distinct eigenvalues, then A=C⁻¹DC and e^tA=e^tC−1DC=C⁻¹e^tDC, which is easy to calculate; "differences" between 1-dimensional and 2-dimensional problems (the Intermediate Value Theorem holds in 1 dimension) and "differences" between 2-dimensional and 3-dimensional problems (in 2 dimensions, a non-self-intersecting line that "spirals in" must "spiral out" in order to leave the domain).
• Lecture 4 (Thu, Jan 26): ODEs - basic geometric concepts (cont): for a given flow φ_t of the ODE x'=f(x) in Rⁿ, a Poincare surface Σ is an (n−1)-dimensional surface in Rⁿ that is "transversal" to the integral lines of the ODE (i.e., at a point of intersection of an integral line and Σ, the tangent line to the integral line does not belong to the tangent plane to Σ at this point); Poincare map P:Σ→Σ for the ODE x'=f(x) (and a given choice of Σ); a fixed point (FP) or a periodic point of the Poincare map correspond to a periodic orbit of the flow of the ODE; example: simple harmonic oscillator.
• Lecture 5 (Tue, Jan 31): ODEs - basic geometric concepts (cont): a concrete example of Poincare map (Problem 3 of Homework 1).
  Maps - basic concepts: map - a function that is meant to be iterated; iterating a one-dimensional map ƒ:R→R graphically - a cobweb plot; fixed points on a cobweb plot - the x-coordinate of the intersection of the graph of ƒ with the diagonal y=x; stability of a fixed point; criterion for stability of a FP x_*: if |ƒ(x_*)|<1, then the FP x_* is stable, if |ƒ(x_*)|>1, then the FP x_* is unstable; observation: a periodic point of ƒ of period m is a fixed point of the iterated map ƒ^m; Intermediate Value Theorem (IVT); using the IVT to prove the existence of a fixed point of a continuous map ƒ:[a,b]→R if ƒ([a,b])⊆[a,b] - a graphical proof and a formal proof (apply the IVT to the function g(x)=ƒ(x)−x).
• Lecture 6 (Thu, Feb 2): Flows on the line - a geometric way of thinking: geometric meaning of solving an ODE x'=ƒ(x) for a function x:[0,T]→R - drawing the integral lines of the vector field ƒ(x) in the (t,x)-plane; goal - to find the asymptotic behavior of the solution of an IVP, i.e., the behavior of x(t) as t→∞, without solving the ODE, directly from the "phase portrait", i.e., from the ODE x'=ƒ(x) of ƒ considered as a graph of the function ƒ in the "phase plane" (i.e., the (x',x)-plane); Euler method for approximate solution of an ODE:
  x(t_i+1)=x(t_i+Δt)=x(t_i)+x'(t_i)Δt+(1/2!)x''(t_i)(Δt)²+...≈x(t_i)+x'(t_i)Δt=x(t_i)+ƒ(x(t_i))Δt;
  if x^*∈R is a value such that ƒ(x_*)=0, then the (unique under appropriate condition on ƒ) solution of the IVP x'=ƒ(x), x(0)=x_* is x(t)=x_* for all t≥0; studying the stability of a fixed point x_* directly from the graph of ƒ in the (x,x')-plane in the examples x'=x−1, x'=sin(x), and x'=(x−1)(x−3)²(x−5)³; stable FPs (atracting FPs, attractors, sinks), unstable FPs (repelling FPs, repellers, sources); semi-stable fixed points, like the fixed point x_* in the ODE x'=(x−1)(x−3)²(x−5)³;
  idea of a generic situation (when the number and/or type of fixed points does not change dramatically under arbitrarily small perturbations), and of a non-generic situation (when dramatic changes may occur due to arbitrarily small perturbations); example of a non-generic situation: the quadratic equation x²−2x+1=0, i.e., (x−1)²=0 - the equation has exactly one root, but for arbitrarily small ε>0, the equation (x−1)²+ε=0 has no real roots, while the equation (x−1)²−ε=0 has two real roots;
  non-generic situation in an ODE : the ODE x'=(x−1)(x−3)²(x−5)³ has three fixed points: an attracting FP at 1, a semi-stable FP at 3, and a repelling one at 5; the semi-stable fixed point x_*=3 in the ODE x'=(x−1)(x−3)²(x−5)³ can be destroyed easily by adding or subtracting an arbitrarily small number to the right-hand side: for arbitrarily small ε>0, the ODE x'=(x−1)(x−3)²(x−5)³−ε has only two FPs (a stable FP slightly smaller than 1, and an unstable FP slightly bigger than 5); on the other hand, for arbitrarily small ε>0, the ODE x'=(x−1)(x−3)²(x−5)³+ε has four fixed points (a stable FP slightly bigger than 1, an unstable FP slightly smaller than 3, a stable FP slightly bigger than 3, and an unstable FP slightly smaller than 5) - all this is obvious from the graph of ƒ in the phase plane; in fact, the FP at 5 of the original equation (which comes from the factor (x−5)sup>3 in the right-hand side of the ODE) is also non-generic (this is not so obvious, we will study this soon).
  Suggested reading (if the discussion in class was not enough): Sec. 1.7 of Layek (try to understand the pictures), Sec. 1.2 and 1.3 of Hirsch-Smale-Devaney (again, the most important thing for us are the pictures).
• Lecture 7 (Tue, Feb 7): Saddle-node (tangent, blue sky) bifurcation in a 1-parameter family of ODEs on R: x'=ƒ_μ(x)=μ+x²: birth of two fixed points of opposite stability when the parameter value passes through the critical value μ_c (which in this case is 0); behavior of solutions for μ>μ_c and for μ<μ_c; a more complicated example - the 1-parameter family of ODEs x'=ƒ_μ(x)=μ−x−e^−x: writing ƒ_μ(x) as a difference of two functions: ƒ_μ(x)=φ_μ(x)−ψ(x), with φ_μ(x)=μ−x, ψ(x)=e^−x, and finding the critical value μ_c as the value for which the graphs of φ_μ(x) and ψ(x) are tangent.
• Lecture 8 (Thu, Feb 9): Saddle-node (tangent, blue sky) bifurcation in a 1-parameter family of ODEs on R (cont.): finishing the calculations from the complicated example from Lecture 7: computing the critical value μ_c=1 and the fixed point x^*_c=0 for μ=μ_c; Taylor series of a function of two variables, expansion of ƒ(x) in a Taylor series near the point (μ,x)=(1,0); computing the approximate values of the fixed points for μ slightly above μ_c, so that only the lowest-order terms in the Taylor expansion can be retained while the rest of the terms are ignored; bifurcation diagram - positions of the fixed points as functions of the parameter μ.
  Supercritical pitchfork bifurcation in a 1-parameter family of ODEs on R: studying the bifurcation in the 1-parameter family x'=ƒ_μ(x)=−x³+μx: when μ<0, there is only one stable FP, namely 0, while for μ>0, the FP 0 becomes unstable while two stable FPs are born (one positive, one negative); bifurcation diagram of the supercritical pitchfork bifurcation; a physical example of a system that undergoes a supercritical pitchfork bifurcaition - a ferromagnet.
• Lecture 9 (Tue, Feb 14): More examples of bifurcations: studies of the bifurcations in the 2-parameter family x'=ƒ_a,b(x)=−x³+3ax+b, which exhibits two saddle-node bifurcations.
  Insect outbreak: derivation of a system modeling the dynamics of the budworm population accounting for the limited resources and the predation:
  
• Lecture 10 (Thu, Feb 16): Insect outbreak (cont.): non-dimensionalizing the problem:
  
  pictorial analysis of the location of the fixed points as functions of the parameters r and k.
• Lecture 11 (Tue, Feb 21): Insect outbreak (cont.): complete analysis of the dynamics of the budworm population - bistability, bifurcation diagrams.
• Lecture 12 (Thu, Feb 23): Definition and examples of two-dimensional linear systems: systems of two ODEs; converting second-order ODEs into systems of two first order ODEs; example - harmonic oscillator, Hooke's force, conservative (potential) forces, minimum of potential energy as a fixed point of the system, small oscillations around a non-degenerate equilibrium point are approximately harmonic oscillations (in the generic case), fixed points, closed orbits, derivation of a conserved quantity "energy" by multiplying the equation of motion by and integrating to obtain , obtaining the shape of the periodic orbits in the phase plane by using conservation of energy; fixed points of a 2-dimensional system; Taylor series of a function of two variables; linearization around a fixed point of the 2-dimensional system by using Taylor series.
• Lecture 13 (Tue, Feb 28): Definition and examples of two-dimensional linear systems (cont.): detailed analysis of the linear system , a stable node (a<0, a≠−1), a star (a=−1), a line of fixed points (a=0), saddle points (a>0).
• Lecture 14 (Thu, Mar 2): Classification of linear systems: eigenvectors, eigenvalues, and their role in constructing the general solution of a linear system of ODEs; characteristic equation; trace and determinant of a matrix; proof of the invariance of the trace and determinant with respect of similarity transformations, i.e., , ; proof that the roots of a polynomial equation with real coefficients are either real or they come as pairs of complex conjugate numbers; corollary: the eigenvalues of a matrix with real entries are either real or come in pairs of complex conjugate numbers; classification of the possibilities for a linear system of two ODEs: ,
  • real eigenvalues:
  • λ₁<λ₂<0: stable node;
  • λ₁<0<λ₂: saddle point;
  • 0<λ₁<λ₂: unstable node;
  • λ₁=0, λ₂≠0: a line of attracting (if λ₂<0) or repelling (if λ₂>0) fixed points;
  • λ₁=λ₂=0: all points are fixed;
  • λ₁=λ₂:
    • A diagonalizable, i.e., : in this case A has two distinct eigenvectors: the phase portrait is a stable star (λ<0) or an unstable star (λ>0);
    • A non-diagonalizable, in which case : in this case A has only one eigenvector: the phase portrait is a stable degenerate node (λ<0) or an unstable degenerate node (λ>0);
  • non-real eigenvalues:
  • λ_1,2=α±iβ (α,β∈R, α≠0): stable spiral (α<0) or unstable spiral (α>0);
  • λ_1,2=±iβ (β∈R): center.
• Lecture 15 (Tue, Mar 7): Classification of linear systems (cont.): the trace-determinant plane.
  Assembling the global picture from local information: example: rabbits versus sheep - linear analysis of the fixed points, basins of attraction of the stable fixed points, stable and unstable manifold of the saddle point.
• Lecture 16 (Thu, Mar 9): On the importance of the hyperbolicity of a fixed point: example: x'=x, y'=−y+x² - exact solution, linearization at (0,0), comparison of the local behavior of the nonlinear system and the linearized system in a small neighborhood of the fixed point (0,0), change of variables that transforms the nonlinear system into its linearization near the fixed point, computation of the stable and unstable manifolds of the fixed point (0,0);
  Hartman-Grobman Theorem: If x^* is a hyperbolic fixed point of x'=f(x), then there exists a change of variables in a small neighborhood of x^* that transforms the original system into its linearization, u'=Df(x^*)⋅u, at x^*;
  example: the system x'=x, y'=y² has a non-hyperbolic fixed point at (0,0), the solutions of the original system and of its linearization at (0,0) behave very differently;
  nullclines of a nonlinear system in R².
• Lecture 17 (Tue, Mar 21): Assembling the global picture from local information: rabbits vs. sheep - fixed points, stable and unstable manifolds, basins of attraction.
• Lecture 18 (Thu, Mar 23): Limit cycles: definition and simple examples; ruling out limit cycles by using Lyapunov functions.
• Lecture 19 (Tue, Mar 28): Midterm exam (on the material covered in Lectures 1-17).
• Lecture 20 (Thu, Mar 30): Limit cycles (cont.): Poincaré-Bendixson Theorem for proving existence of limit cycles; trapping region; example: proving the existence of a limit cycle for the system r'=r(1−r²)+μrcos(θ), θ'=1 for small values of μ.
• Lecture 21 (Tue, Apr 4): Limit cycles (cont.): oscillatory biochemical processes: glycolysis; Sel'kov system; constructing the outside boundary of a trapping region for the Sel'kov system.
• Lecture 22 (Thu, Apr 6): Limit cycles (cont.): constructing the inside boundary of a trapping region for the Sel'kov system, computing the conditions on the values of the parameters (a,b)) for which the Sel'kov system has a limit cycle, resp. an attracting fixed point; discussion of the meaning of systems related to chemical kinetics: the concentration x(t)=[X](t) of a substance X that is pumped in the system at a rate b, decays spontaneously at a rate λ, and disappears due to a chemical reaction 2X+Y=Z satisfies the ODE dx/dt=db−λx−αx²y, where y(t)=[Y](t) is the concentration of the substance Y.
  Bifurcations in 2-dimensional systems: refresher: saddle-node bifurcation on R; an example of a saddle-node bifurcation in the system x'=μ−x², y'=−y.
• Lecture 23 (Tue, Apr 11): Bifurcations in 2-dimensional systems (cont.): finishing the analysis of the saddle-node bifurcation in the system x'=μ−x², y'=−y: a "ghost" remaining after the disappearance of the fixed points in a saddle-node bifurcation, estimating the time needed to pass through the "ghost": T∼(μ-μ_c)^1/2; occurrence of a generic saddle-node bifurcation at intersection points of the nullclines;
  supercritical pitchfork bifurcation in two dimensions in a prototypical example x'=μx−x³, y'=−y.
  Hopf bifurcations: behavior of the eigenvalues of a 2-dimensional matrix with real entries; supercritical Hopf bifurcation in the model system r'=μr−r³, θ'=ω+br²: linearization around the fixed point, computing the eigenvalues; generic supercritical Hopf bifurcation - the behavior of the amplitude and the frequency is amplitude = O((μ-μ_c)^1/2), frequency = O(1) (i.e., the frequence does not depend strongly on the parameter μ near the critical value μ_c).
• Lecture 24 (Thu, Apr 13): Hopf bifurcations (cont.): subcritical Hopf bifurcation in the model system r'=μr+r³−r⁵, θ'=ω+br²: when the parameter μ goes from negative to positive values, the repelling unstable limit cycle around the stable fixed point (0,0) shrinks to an unstable fixed point, and the only stable trajectory is a stable limit cycle with a large radius; hysteresis in the transition of μ back and forth through μ_c;
  viewing a movie of the Tacoma Narrows bridge collapse (likely caused by a subcritical Hopf bifurcation);
  a different kind of disaster: viewing a movie of the London Millenium Bridge oscillations;
  general culture: resonance in a harmonic oscillator with a periodic external driving force, plot of the amplitude as a function of the external driving frequency; soldiers do not march on a bridge (because some eigenfrequency of the bridge may be close to the frequency of the marching).
• Lecture 25 (Tue, Apr 18): Basic partial differential equations: examples of elliptic (Laplace), hyperbolic (wave), and parabolic (heat/diffustion) PDEs; elementary examples: u_x(x,y)=xy³, u_xxy(x,y,z)=xy³e^z; arbitrariness in the general solution of a PDE (the general solution of a PDE of order k for a function of n variables has k arbitrary functions of (n−1) variables; this works for ODEs as well, considering a constant as a function of 0 variables); why is a function "equivalent" to infinitely many constants;
  physical meaning and behavior of solutions of the heat/diffusion equation;
  physical meaning and behavior of solutions of the wave equation on R, general solution: u(x,t)=ƒ(x−ct)+g(x+ct), physical meaning of each term.
• Lecture 26 (Tue, Apr 20): Basic partial differential equations (cont.): physical interpretation of more complicated equations, like u_t(x,t)=α²u_xx+au_t+φ(u) - a diffusion term α²u_xx, a term au_t corresponding to a constant profile propagation to the right, and a term φ(u) reflecting a change in the amount of the substance due to a chemical reaction (or a similar effect); biological interpretation os such equations; Fisher's equation: u_t(x,t)=α²u_xx+ru(1−u/K).
  

• Lecture 27 (Tue, Apr 25): Fisher's equation: looking for solutions of u_t(x,t)=u_xx+u(1−u) of the form of a traveling front with constant shape: u(x,t)=U(x−ct), where U(z)→const as z→−∞, U(z)→0 as z→∞, U(z)≥0 for all z∈R; deriving a 2nd order ODE for U(z): U''+cU'+U(1−U)=0, rewriting the 2nd order ODE for U(z) as a first-order system: U'=V, V'=−U(1−U)+cV; linearizing the system for U and V at the fixed points (0,0) and (1,0); the fixed point (1,0) is always a saddle, while the fixed point (0,0) is a stable spiral for c<2 and a stable node for c>2; impossibility of physically meaningfull solutions for c<2 (due to the condition that U(z)≥0 which comes from the condition u(x,t)≥0).
• Lecture 28 (Thu, Apr 27): Pendulum: derivaiton of the equation of a pendulum of mass m of length l swinging in a vertical plane in the gravity field with free-fall acceleration g; one degree-of-freedom, with coordinate θ∈S¹; cofiguration space S¹, phase space S¹×R which can be interpreted as a cylinder of infinite length with coordinates θ∈S¹ and v∈R; phase portrait of the pendulum in S¹×R separatrix.

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also expected. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Homework: Homework will be assigned regularly and will be posted on the this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. The lowest homework grade will be dropped. Giving just an answer to a problem is not worthy any credit - you have to write a complete solution which gives your step-by-step reasoning and is written in grammatically correct English. Although good exposition takes time and effort, writing your thoughts carefully will greatly increase your understanding and retention of the material.
You are allowed to discuss the homework problems with the other students in the class. However, each of you will need to prepare individual solutions written in your own words - this is the only way to achieve real understanding!
Your homework should have your name clearly written on it, and should be stapled. Please write the problems in the same order in which they are given in the assignment. No late homework will be accepted!

Exams: There will be one in-class midterm and a (comprehensive) final.
A tentative date for the midterm is March 23 (Thursday).
The final exam is scheduled from 1:30 to 3:30 p.m. on May 8 (Monday).
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking any of the exams at the scheduled time.

Grading: Your grade will be determined by your performance on the following coursework:

Technology: We will frequently engage in computations to illustrate the mathematical results that we discuss. You will have to do some elementary programming in Mathematica and MATLAB which are avaliable on the computers in the University's computer labs. The amount of programming you will need to do will be small, and previous programming experience is not assumed.

Policy on W/I Grades : You can withdraw from the course with an automatic "W" no later than March 31 (Friday) for undergraduate students and February 24 (Friday) for graduate students. Dropping after April 3 (Monday) requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates in the academic calendar!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Code.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

MATLAB tutorials:

• A 33-page Practical Introduction to Matlab, by Mark Gockenbach (Michigan Technological University); the same Introduction, in html format.
• A 16-page MATLAB Tutorial by Peter Blossey (University of Washington).
• A 8-page Introduction to Plotting with Matlab from the University of Washington.
• A very detailed (125 pages) MATLAB Overview by Ed Overman (Ohio State University).
• A 35-page MATLAB Primer by Kermit Sigmon (University of Florida).

Good to know:

• The greek_alphabet.
• Some useful notations.