### The spring 2010 student presentations for MATH 495, Senior Seminar, are as follows:

## Multiplication Out of Order?

### Wednesday, March 31

### 12:00-1:00 pm

### Lindsay Wiard

If you are given two real numbers, a and b, can you answer these two simple questions?

- Which is bigger, a or b ? (order)
- What is the product of a and b? (multiplication)

Suppose a = 5 and b = 8. Of course you are able to answer the questions. But are the two answers related? No, there does not seem to be a connection between the concepts of order and multiplication. While this may be the case for real numbers, relationships can be found when working in different settings. In the study of matrices, Walter's Theorem establishes a remarkable connection between these two seemingly unrelated notions: order and multiplication. In this presentation, we will provide the background for understanding Walter's Theorem, state the theorem, and then outline a new elementary proof discovered here at Western State College!

## Non-Commutativity Exposed: Adventures Through Vector States

### Monday, April 4

### 12:00-1:00 pm

### Franklin Greener

Many mathematical operations are commutative, which means that the order in which the operation is applied does not affect the result. For instance, addition and multiplication of numbers are both commutative operations (i.e. a + b = b + a and ab = ba). A familiar non-commutative operation is division. In fact many mathematical operations are non-commutative, including subtraction and matrix multiplication. In the context of matrix multiplication, it is often the case that AB is not the same as BA, but one might wonder if relationships between AB and BA can be found. In this talk we will state a conjecture that establishes a relationship between the matrices AB and BA. Theoretical and numerical evidence supporting the conjecture will be presented as well as specific contexts in which the conjecture is known to be true.

## Math is Blowing in the Wind

### Friday, April 9

### 12:00-1:00 pm

### Joshua Krenz

What is wind? Wind is a driving force in most of the world's weather. Wind drives the hurricanes which destroy the Gulf Coast and the snow that covers our mountains. But most important of all, wind supplies power for developing countries, homes, and major cities. In 1919, Albert Betz discovered an elegant expression for the maximum power obtainable from wind. Using a simple wind tube, we will discuss how to model wind flow and derive equations which describe how a wind turbine converts the force of the wind into a useful power source. Interestingly, this result can be obtained from elementary calculus.

## Investigating Stream Incision Rates Over Time

### Monday, April 12

### 12:00-1:00 pm

### Maggie Michalowski

Geologists frequently use the ages of geologic formations to estimate dates of events in Earth's history. Rivers and streams carry loads of information about the earth's shape and features throughout time. But can the watersheds that contain them also give us reliable information about the age of what we see today? There is an unanswered question in geology on whether or not the rate that streams cut into the earth is constant with time. We will explore the factors that play into these incision rates to discover how depth and time are related.

## There Is Hope For Losers!

### Monday, April 19

### 12:00-1:00 pm

### Anthony Cook

If you played a game that consisted of randomly switching between two losing games you would expect to lose, right? Well, Parrondo's Paradox says otherwise; if you switch between two specific losing games you actually end up with a winning game. How can this work? We will answer this question by discussing Parrondo's Paradox and providing the mathematical background needed to show how the paradox works.

## The Game of Nim

### Friday, April 30

### 12:00-1:00 pm

### Steve Zehnder

Consider a two player game in which there are multiple heaps of tokens and players take turns taking tokens from each heap. The goal of the game is to take the last object, and though not obvious, there is a perfect winning strategy. In this talk we will explore the theory behind the game of Nim and cerain extensions of the game.

### Nicole Sul

will be looking at shadows in Euclidean, spherical, and hyperbolic geometry. A recurring related rates problem in calculus is one with a street lamp and a person walking away from the lamp at a certain rate. The goal is to find the rate at which the tip of the shadow is moving. Students usually use similar triangles to solve the problem. Unfortunately similar triangles do not exist in spherical or hyperbolic geometry. In order to generalize to these spaces, she will use the Pythagorean Theorem, because it has an analogous statement in each other geometry.

If you would like to give a talk or would like more information contact Dr. Bob Cohen