The Math Major Vol. 3, No. 5

The Math Major

CSU Fresno Mathematics Department

Vol 3. No. 5 (October 27, 1998)

Math Department Colloquium & Video Series

The department colloquium & video series is Thursdays at 1 p.m. (refreshments included). Every one is cordially invited. We have a colloquium coming up on October 29 and a video on November 5.

Colloquium:

Date & Location: Thursday, October 29 in Science 139
Speaker:Dr. Kurt Kreith, from the UC Davis Department of Mathematics
Title: "Iterative Algebra and Dynamic Modeling"
Description: What if computers had been invented 500 years earlier? How might the study of mathematics have been affected by such an event? And now that computer technology has become so commonplace, what are its implications for the study and application of mathematics?
These questions underlie a book project and a course I am offering at Davis High School on "Iterative Algebra and Dynamic Modeling." Starting with two familiar problems,
- solving quadratic equations, and
- solving a system of 2 linear equations in 2 unknowns.
The computer's capacity for cheap and easy iteration is used to develop a curriculum very different from that which is commonly taught under the banner of high school algebra.

Video:

Date & Location: Thursday, November 5 in PB 390.
Title: "Not Knot"
Description: Not Knot is a guided tour into computer-animated hyperbolic space. It proceeds from the world of knots to their complementary spaces -- what's not a knot. Profound theorems of recent mathematics show that most known complements carry the structure of hyperbolic geometry, a geometry in which the sum of three angles of a triangle always is less than 180 degrees.

From Not Knot

University Scholarships Deadline

California State University, Fresno administers over three hundred and twenty-seven donor-sponsored scholarships that range from $200 to $4000. The application filing period for the 1999-2000 academic year scholarships will end after November 13, 1998.

Also, beginning with the 1999-2000 academic year, the university will introduce a new four-year renewable scholarship. All students accepted into the Smittcamp Family Honors College will be designated President's Scholars and guaranteed a scholarship/grant award that covers the full cost of fees ($1794), the full costs of a room in University Housing ($2800) and a $200 book allowance. Additional details are available upon request. Inquiries should be addressed to the E-mail address: honors@csufresno.edu. Applications are now available in the Academic Departments and Financial Aid/Scholarship Office in Joyal 296. For more information visit the web site .

Math in the News

(From the MAA Online page. ) The auction house Christie's will be auctioning off the famous Archimedes palimpsest, discovered in the early 1900s, which contains the text of the long-lost "Method". Its rediscovery was of great importance because it shows that Archimedes used a kind of "method of indivisibles" to determine the areas and volumes which he wanted to compute. He did not consider, however, such computations to be proofs, and gave proofs using the method of exhaustion. The manuscript was apparently sold to a private collector, and is now up for sale again. Historians are excited about the opportunity to once again examine the manuscript, but also worried that the buyer may be another private collector.

A Mathematical Quote

"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country."

--David Hilbert (1862-1943)

David Hilbert

Problem Corner

Problem 3.4: Find all real solutions (x, y) to the equation

5^x-1 + 5^3-x = 2(8 cos(xy) sin(xy) + 1). --From Dr. Tuska

Solution to Problem 3.4: (Solution by Anar Ahmedov.) First look at the left hand side 5^x-1 + 5^3-x. We claim that this is at least 10, and 10 is realized only when x = 2. To see this, factor out a 5, 5(5^x-2 + 5^2-x). This is in the form 5(a + 1/a). It is easy to see that a + 1/a >=2 is equivalent to (a - 1)² >= 0 and equality is achieved only when a = 1, that is x = 2. This proves the claim.

Now, look at the right hand side

2(8cos(xy) sin(xy) + 1) = 2(4 sin(2xy) + 1)

(double angle formula). Clearly the maximum value for this function is 10.

It follows from above that the only possible solution to the original equation occurs when both left and right hand sides are 10. The former when x = 2, and hence the latter when 10 = 2(4 sin(4y) + 1). The solution to this is y = pi/8 + pi k/2. So all solutions are in the form

(x, y) = (2, pi/8 + pi k/2) where k is an integer.

Correct solutions were received from Anar Ahmedov, Bryan Chaffe, John Jamisom and Garth Kinnier and Amy Peterson (jointly).

New Problem

Problem 3.5: (Due Thursday November 5, 3 p.m.) We define a function f:[0, infinity ) -> [0, infinity) using the recursive formula:

f(x) = x if 0 <= x < 1 and 3 f(x/2) - 1/2 otherwise.

Prove that f is a continuous increasing function and find the unique positive solution x to the equation f(x) = 9.

Solutions may be delivered to the math department office (for Dr. Cusick) or by e-mail.

CSU Fresno Math Department Home Page

California State University, Fresno