Rouben Rostamian

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MATH 481: Mathematical Modeling

Spring 2017 Course information

Class Time/Place:	ThTh 5:30pm–6:45pm, SOND 109 MP 105
Office:	MP 406
Phone:	410–455–2405
Email:	rostamian@umbc.edu
Office hours:	TuTh 4:30–5:30, or by appointment

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 using analytical, computational and graphical tools, as appropriate.

Textbook

There is no textbook. You should attend every lecture and take notes.

Prerequisites

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

Writing Intensive designation

As of Fall 2009 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 LaTeX—the universal software for typesetting mathematics.
• Learn how to use Maple—a quite sophisticated software for symbolic computation and plotting.
• Learn how to interact with the Unix operating system.

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. There will be around 6 such reports in the semester. These will vary in complexity but a typical report is around 5–7 printed pages.

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 LaTeX

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

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

Once you convince yourself that LaTeX 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 LaTeX, is:

LaTeX: A Document Preparation System by Leslie Lamport.

After you have thoroughly mastered that manual, you may expand your knowledge by reading:

The LaTeX Companion (2nd Edition) by Mittelbach, Goossens, Braams, Carlisle and Rowley.

How to get LaTeX

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

UMBC's machines

LaTeX 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 LaTeX on the library machines, unfortunately.)

Your own Linux machine

All Linux distributions come with LaTeX. If you have your own Linux machine, you may install LaTeX with a few mouse clicks. Ask me if you don't know how.

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.

Windows and Mac platforms

The TeX Users Group's website has a wealth of information about LaTeX. In particular, in their Getting Started page they suggest MacTeX for MacOSX and proTeXt for Windows. I have used neither, but I consider that site's advice trustworthy.

About Maple

Maple™ is a computer software for symbolic computations. It factorizes polynomials and solves differential equations:

2x3 – 9x2 + x + 12	⇒	(2x – 3) (x – 4) (x+1),
y'' + y = tan x	⇒	y = c1 cos x + c2 sin x – cos x ln(sec x + tan x).

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

Maple is used as an analysis aid in the case studies. I will devote some class time to Maple tutorials. Probably you will need to use Maple or something equivalent for most of the homework assignments.

The complete and searchable documentation of Maple is available under the Help→Maple Help menu on its GUI window. You may download the same documentation as a stand-alone PDF file, if you so prefer, from Maple's Documentation Center. Be warned that it's a largish 56 GB file.

Buying Maple

Maple is available on all university machines, so you don't have to buy it. But, if you do wish to have a personal copy on your own computer, you may download the student version for $99 from Maplesoft's web store. According to their sales department, the student version is functionally equivalent to the regular version, the only difference is that you need to affirm that you are a student.

An additional 25% discount is available through the Maple Adoption Program. Ask me for the promotion code.

Exams and grading

There are are no exams in this course. Your work will be evaluated solely on the quality of your case study reports.

Writing style varies greatly among individuals, 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 a good writing when I see one". The following are some of the issues that come into consideration:

• Correct mathematics. Correct logic. Complete explanations. No mysterious gaps in the reasoning. Someone else with an education level comparable to yours should be able to follow the discourse and learn from it.
• Correct LaTeX markup:
  • No stray blank lines. In LaTeX 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.
  • In paragraph mode, put mathematical symbols and expressions within $...$. For instance:

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

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

Homework assignments

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

I won't take late reports; please don't ask for exceptions. However one lowest homework grade will be dropped to accommodate unanticipated events.
 

 

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.

For detailed policies on academic integrity consult:

Undergraduate students:

Student Academic Conduct Policy (a PDF file)

Graduate students:

Policy and Procedures for Student Academic Misconduct

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.