# Mathematics Projects 2012

### Investigations in Combinatorics

Faculty Mentor - Dr. David Gove

Combinatorics is the theory and methodology of counting described sets. We will begin with the counting principles of list multiplication,permutations, combinations, and inclusion-exclusion. Typical questions include 1) how many poker hands are there and how many of each type? 2) how many 10-person committees could be made in the U.S. senate, assuming each committee member has to be from a different state? 3) how many ways could a secretary accidently stuff 20 envelopes with 20 letters so that no letter was in the correct envelope? We will move on the more powerful methods of generating functions and compare algebraic and combinatorial methods of proof. These will lead us to partition problems and Fibonacci numbers. Many open problems will be discussed and investigated.

### Statistical analysis of changes in Sierra Nevada snowpack levels

Faculty Mentor - Dr. Eduardo Montoya

Mountainous areas, particularly in the western United States, provide a large fraction of the fresh water supply. Water is stored in the snowpack and is released when the snow melts, which is generally believed to occur at the beginning of April. It is estimated that snowmelt derived from the Sierra Nevada of California provides 75% of the state's water supply for agriculture (Rosenthal & Dozier 1996). The impact of climate change on the snowpack could have important implications for California's hydrology. Therefore, it is important to understand how the snowpack in the Sierra Nevada may be affected by ongoing changes in climate. Snow Water Equivalent (SWE) is the most important snowpack measurement. The seasonal pattern in SWE measurements at a particular location across years contains both variation in the magnitude of the snowpack peak and variation in the start/end of the snow season and timing of the peak. Participants in this study will describe and analyze SWE data to determine spatial linear trend patterns in the magnitude of the Sierra Nevada snowpack over time, as well as its relationship to atmospheric and oceanic variability. Patterns of atmospheric and oceanic variability over the Pacific Ocean are known to influence the winter and spring climate of the Western USA. Time permitting, participants will then shift focus to the variation in the timing of the peak. Participants will use a popular open-source statistical software and possibly an open-source GIS software. In addition, project participants will learn skills used in a wide range of disciplines, such as statistics, computer programming, and snow hydrology. This research will provide students exposure to the type of research that will be conducted at the CSUB Center for Research Excellence in Science and Technology that is funded by the National Science Foundation.

### Explore the World of Chaos

Faculty Mentor - Dr. Maureen Rush

This workshop is for high school students who want to explore the mathematical phenomenon known as chaos. Chaos is what results when very small changes in the initial conditions of a system lead to very large discrepancies over time. This phenomenon can account for our inability to predict weather, spread of disease, among other things.

You will learn the basics of mathematical iteration, both linear and nonlinear, with only an algebra background. Much of our work will utilize graphing calculators and software spreadsheets. We'll create wonderful graphs and create movies!

## Disclaimer

These Web pages and any associated Adobe Acrobat Files are designed as supporting material for the respective projects. Please feel free to contact either of the program directors with any questions you might have.