Rouben Rostamian

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MATH 481: Mathematical Modeling

Spring 2021 Course information

Class Time/Place:	MoWe 5:30pm–6:45pm (online on Blackboard Collaborate)
Email:	rostamian@umbc.edu

Course content

Mathematical modeling refers to the process of applying mathematical tools and reasoning to understand the world around us. In this course we will get a glimpse of such process in the context of several case studies. Here is a possible list of topics:

• The use of radioactive isotopes for finding an object's age
• Analyzing pollution in lakes
• Population dynamics of predators and prey
• Population dynamics of competing species
• Stable and unstable equilibria via linearization
• The effect of a predator in the stability of competing species
• Epidemics

All case studies lead to models involving differential equations. Each case study begins with a free-form description of an issue and a simple mathematical model. In most cases further analysis leads to more accurate but more complicated models. The models are explored through analytical, computational and graphical tools, as appropriate.

Textbook

There is no textbook but you will need to invest $75 in buying Maple. See below.

That also means that will need to attend the lectures and take notes.

Prerequisites

Math 221 (linear algebra), Math 225 (differential equations), Math 251 (multivariable calculus).

Writing Intensive designation

This course carries a Writing Intensive (WI) designation. As such, it meets the writing requirement of UMBC's General Education Program (GEP).

Course objectives

• Understand nonlinear systems of differential equations through phase space analysis.
• Learn how to set up, analyze and interpret mathematical models.
• Learn how to communicate mathematical ideas in writing.
• Learn how to use L^aT_eX—the universal software for typesetting mathematics.
• Learn how to use Maple—a quite sophisticated software for symbolic computation and plotting.

The writing component

This course differs from most mathematics courses in that writing is an essential part of the course. The “deliverable” for each homework assignment is a complete and self-contained report that describes the problem, the analysis, calculations, conclusions and citations, written in the style of a technical journal article. You will write five such reports in this semester. These will vary in complexity but a typical report is around 6–9 printed pages.

Writing style varies among individuals but good writing is not entirely arbitrary. Have a look at what these experts have to say about it.

Writing advice: No needless words

Vigorous writing is concise. A sentence should contain no unnecessary words, a paragraph no unnecessary sentences, for the same reason that a drawing should have no unnecessary lines and a machine no unnecessary parts. This requires not that the writer make all his sentences short, or that he avoid all detail and treat his subjects only in outline, but that every word tell.

— Strunk & White in The Elements of Style

Writing advice: How to write mathematics

The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order you want it said in, you must write it, rewrite it, and re-rewrite it several times, and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation. That’s all there is to it.

— P. R. Halmos in How to Write Mathematics

Help available at UMBC's Writing Center

UMBC's Writing Center provides help with writing. This may be a valuable resource if you need a little bit of hand holding with your writing assignments.

About L^aT_eX

Technical writing is easy with the right tool. L^aT_eX is the computer software of choice for technical writing, especially for articles that contain a lot of mathematics.

L^aT_eX is closer to being a programming language than a word processor. I will devote some class time to L^aT_eX tutorials and expect that you will write your assignments using L^aT_eX.

Once you convince yourself that L^aT_eX is for you, you should consider buying its manual and keeping it within an arm's reach at all times. The manual, written by the creator of L^aT_eX, is:

L^aT_eX: A Document Preparation System by Leslie Lamport.

How to get L^aT_eX

L^aT_eX is an open source software; it may be obtained freely and installed on any computer platform.

Overleaf: L^aT_eX in a web browser—the easy way

With Overleaf you won't need to install L^aT_eX on your machine at all! Go to https://www.overleaf.com and register for a free personal account. Then you will be able to access their fully functioning L^aT_eX remotely through your web browser. This works on all operating systems. Your files will reside on Overleaf's cloud servers by default but you can download them to your local computer when needed. In fact, you will need to do that in order to email them to me.

UMBC's machines

L^aT_eX is already installed on UMBC's Linux machines in the GL labs. You don't need to do anything special to use it. (There is no L^aT_eX on the library machines, unfortunately.)

Linux

All Linux distributions come with L^aT_eX. If you have your own Linux machine, you may install/activate L^aT_eX with a few mouse clicks. Ask me if you don't know how.
Aside: If you don't have Linux installed your computer, this course provides you with a good opportunity to install one and learn how to use it. Ask me if you don't know how.

Mac

Download and install MacTeX.

Windows

Download and install MiKTeX.

About Maple

Maple™ is a computer software system for doing symbolic computations.

Maple can factorize polynomials:
----- factor($2x^3 - 9x^2 + x + 12$);	$\Rightarrow$ answer:	$(2x-3)(x-4)(x+1)$
Maple can solve algebraic equations:
----- solve($2x^3 - 9x^2 + x + 12 = 0$);	$\Rightarrow$ answer:	$3,\;\frac{3}{2},\; -1$
Maple can solve differential equations:
----- dsolve($y'' + y = \tan x$);	$\Rightarrow$ answer:	$y = c_1 \cos x + c_2 \sin x - \cos x \ln(\sec x + \tan x)$
Maple can evaluate integrals:
----- $\ds\int_0^\infty e^{-x^2} \,dx$	$\Rightarrow$ answer:	$\frac{1}{2} \sqrt{\pi}$

(Can you do that?) In fact, Maple knows just about all the undergraduate and some of the graduate material of the standard mathematics curriculum.

Additionally, Maple can plot functions in 2D or 3D that you may need for your reports:

----- plot($\ds[e^{-x}\sin(3x),\: 3/4e^{-x/2}\cos(3x)],\; x=0\,..\,2\pi\;$);
----- plot3d($\ds x e^{-x^2-y^2},\; x=-2\,..\,2,\; y=-2\,..\,2$ );

We will use Maple extensively in our case studies. For that reason I will devote some class time to Maple tutorials. You will need Maple or (something equivalent) to complete the homework assignments.

Buying Maple

Maple is available on all university machines, so you don't have to buy it, but you will have to go to a computer lab to use it, which can be inconvenient since you will be needing it for several hours every week. That is particularly challenging during this COVID pandemic—for all we know, the computer labs may not be accessible at all. It makes sense, therefore to buy a personal copy for use on your own computer. It costs $99 for students at Maplesoft's web store. A further 25% discount is available through the Maple Adoption Program. Ask me for the promotion code.

Maple's complete documentation is distributed with Maple and can be accessed from within it and therefore you won't need a separate manual, but if you really want, you may download it as a stand-alone PDF file from Maple's Documentation Center.

Exams and grading

This semester you will write and deliver reports on five projects. I will provide a general outline for each project and go over the mathematical tools needed to accomplish it. There will be no exams; your course evaluation will be determined solely on the quality of your reports.

Specifically, I will assign a grade of A, B, C, D, or F to each project. These correspond to numerical values of 4, 3, 2, 1, 0, respectively. At times, I may add +/− adjustments, such as “B+”.   A “+” raises the numerical value by 1/3.  A “−” reduces the numerical value by 1/3.  At the end of the semester I will calculate the average of your numerical scores and convert it back to a letter grade which will be your course grade. For instance, if your average is 3.4, it will be rounded to 3.0, which is a “B”. If your average is between 3.5 and 4.0, it will be rounded to 4.0, which is an “A”.

The first project is special—I will collect and grade it like all other projects but its grade won't count toward your course grade. There are at least two reasons for that.

1. I will do most of the writing of that project and pass it on to you as a semi-finished report. This is to provide you with a sense of what a report is expected to look like. You will add the missing parts and send the finished report back to me.
2. Even with a partially pre-written report, you will still have somewhat a steep hill to climb. That includes learning
   • the mathematics needed in this project;
   • how to interact with L^aT_eX and the L^aT_eX syntax;
   • how to interact with Maple and the Maple syntax;
   • how to transfer information from Maple to L^aT_eX;
   • how to express mathematical ideas in writing.

The purpose of removing Project #1 from the course grade calculation is to reduce the anxiety/pressure that may be associated with that learning process.

Writing style and evaluation rubric

Writing style is a personal matter, therefore I am not going to impose rigid writing rules. As a result, however, there is no conventional "grading rubric"—I will grade your work holistically in the sense that "I know good writing when I see one". The following are some points that come into consideration:

• Correct mathematics. Correct logic. Complete explanations. No mysterious gaps/leaps in the reasoning. Someone else with an education level comparable to yours should be able to follow the discourse and learn from it.
• Correct L^aT_eX markup:
  • No stray blank lines—in L^aT_eX a blank line signals a paragraph break;
  • Use \ref to refer to sections and figures;
  • Use \eqref to refer to equations;
  • Use \cite to refer to the bibliography;
  • Within a sentence, put mathematical symbols and expressions within $ signs. For instance:

        As $t$ gets larger, $f(t)$ get closer to $a$;

  • Use tilde to guard against undesired line breaks.
  See L^aT_eX Tips and Tricks for more advice.
• Nicely framed graphs to convey your ideas. Meaningful captions to tell the reader what the graphs are about.
• Overall impression:
  • Grammar, spelling, punctuation;
  • Exposition style;
  • No needless words.

Homework assignments

I will put homework assignments in the table below as we go along. You may study with others, however I expect that you will write the reports on your own; we don't want reports that are minor variations of each other.
 

 

The Official UMBC Honors Code

By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal.

Student Disability Services (SDS)

Services for students with disabilities are provided for all students qualified under the Americans with Disabilities Act of 1990, the ADAA of 2009, and Section 504 of the Rehabilitation Act who request and are eligible for accommodations. The Office of Student Disability Services is the UMBC department designated to coordinate accommodations that would allow for students to have equal access and inclusion in their courses.