**Math 139A, Fall 2019 **

**Applications of Mathematics **

**Instructor
**John E. M^{c}Carthy

**Class **
Tu 10.00-11.00 in Cupples I, Rm 218

**JM Office **
105 Cupples I

**JM Office Hours** M 4.00-5.00,
Tu 11.00-12.00,
Th. 2.30-3.30, and by appointment

**Phone **
935-6753

**Prerequisites**

Math 131, taken concurrently

**Description**

Mathematics can often seem intimidatingly abstract.

"Why do we need to know this?" and "What is this good for?" are
common

questions, which sometimes are not adequately answered.

It is all very well to say that mathematics is needed for cell-phone
design, or

to make ultrasound images, or for Google to calculate page-rank; but
explaining

exactly how it is used in any of these applications takes a great deal
ot time.

The purpose of this course is to give examples of how mathematics can
be used to

understand real world problems. It is aimed at students who are also
enrolled in Calculus I, Math 131,

so we will start with problems that only need pre-calculus to solve,
and work up to ones that use

calculus.

**Do I need to be a math wiz to take this course?**

No. The course is for students who are curious about how mathematics is
used,

and want some inkling of its scope.

**Content **

Here is a tentative schedule. The first seven do not use calculus,
the next six do.

We may change some of these topics.

- Dimensional Analysis. How to guess plausible formulas.
- The mathematics of convoys. Are they a good idea? What are
the pros and cons? Reading
for class on Sep 3

- Fibonacci Numbers. See this site for pictures
- The golden ratio
- Fractals I : Coastlines
- Fractals II : what is dimension?
- Linear regression I. Application: the Gutenberg-Richter law of earthquake magnitudes. How can we speak of a twenty thousand year event? Earthquakes
- Linear Regression II: How to find the best line. Regression to the mean Application: Metabolic rate versus animal size Metabolic Rates
- Network capacities. Braess's paradox - building extra roads can increase congestion.
- Fractals III: Blood flow.
- Should we all have the same mitochondrial DNA? Galton's approach to surnames.
- How genes spread through populations.
- Sigmoid curves for populations, and the logistic equation.
- SIR model of infectious diseases.

Basis for Grading

Grading will be based on classrooom participation, and a term paper.

The topic of the term paper should be chosen in consultation with the
Instructor.

A preliminary draft should be handed in October 29th. This will be
graded and returned,

and a final version should be handed in December 3rd.

Classroom Participation (contribution to discussion): 20%

Term Paper - first draft: 30%

Term Paper - final draft: 50%

**Class**

I do expect you to come to class every day, and to participate in
class discussions.

I expect you to read the corresponding section in Korner's book.
I may call on you at

any time to answer a question.

**Texts ** The Pleasures of Counting,
by T.W. Korner (Cambridge, 1996).

This is a lovely book. If you get
bogged down in some section, it is okay to move on.