The Math Major Vol. 2, No. 12

The Math Major

CSU Fresno Mathematics Department

Vol 2. No. 12

Math Department Colloquia

Date: Thursday, April 2, 4:10 pm
Speaker: Dr. Roger Alperin, San Jose State University Department of Mathematics
Title: "Irrationals, Rationals and the Modular Group"
Location: Science 143
Abstract: The modular group PSL₂( Z) is the group of integer matrices with determinant 1 under a simple equivalence relation. This group has a rich structure. I will discuss the group theoretic structure and its action on the projective line.

Math Field Day News

The math department's annual Math Field Day will be held on Saturday April 18. Dr. Della Duncan would like all volunteers to stop by her office, or send email, to confirm assignments.

Wolfram's Mathematica Empowerment Tourtruck (dubbed the MathMobile) will be on campus Friday and Saturday (April 17-18) for demonstrations of their powerful program Mathematica.

Actuary Exam Deadline

If you are planning to take the May 12, 1998 Actuary Exam, then you must register by April 1, 1998. For information on actuarial science, you can look at the web pages of the Society of Actuaries and the American Academy of Actuaries.

Science Center News

The School of Natural Sciences is planning an innovative new Science Center, which will house programs in all sciences, including mathematics. The math department is planning to have some components to serve students, secondary school teachers, and the community. If you have some ideas about what you would like to see mathematically in the museum, please send them to Dr. Sean Cleary.

Calculus Help

Mechanical Engineering students are offering tutorial help for calculus (Math 75, 76, 77 and 81). The lab is in EW 120 at the times: Mon (1-2), Tues (9-11, 3:30-4:30), Wed (1-2) and Thur (10-11, 3:30-4:30).

Problem Corner

Problem 2.11: The position of a ball on a circular billiard board is given. Describe the triangular path the ball follows in order to pass through its original position after touching the cushion twice.

Solution to Problem 2.11: Let P be the original position of the ball, O the center of the circular board with radius r, and a is the distance from O to P. Obviously the path must be an isosceles triangle. The only question is how must theta = angle OPB depend upon a and r?

We use the fact that at the point where the ball bounces off the circular boundary, the angle of incidence must be equal to the angle of reflection. This, coupled with the angle sum for a triangle = 180 degrees (look at triangle ABP), gives us

angle PBO = 45degrees - theta/2

Now apply the law of sines to triangle POB to get

sin(theta)/r = sin(45 degrees - theta/2)/a.

We use the half angle formula for sin to get

sin(45 degrees - theta/2) = sqrt{(1 - sin(theta))/2}

when 0 <=theta <= 90 degrees. And so with a little help from algebra, we get the equation (quadratic in sin(theta))

2 a² sin² (theta) + r² sin(theta)- r² = 0.

Use the quadratic formula (choose positive root) to solve for sin(theta) to get

theta = arcsin (-r² + r sqrt{r² + 8a²}/(4 a²).

There was only one solution turned in, and it was incorrect.

New Problem

Problem 2.11: (Due Thursday April 16 by 4 pm) Prove that if a, b and c are distinct (no two the same) positive real numbers then a³ + b³ + c³> 3 abc.

Solutions may be delivered to the math department office (for Dr. Cusick) or by e-mail at larryc@csufresno.edu. There is a $75 dollar first prize and a $50 second prize to be awarded at the end of the semester to the student(s) who submit the most correct solutions.