# Research Opportunities

## Student Presentations, Projects, and Papers

### Hydrology Models for Quantifying Ecosystem Services

Post-baccaulaureate student Kate Meyer and Julia Signell (Engineering, '14) reviewed and developed hydrology models for quantifying ecosystem services. In addition to interpreting and comparing existing models, they investigated techniques for predicting direct surface water runoff using monthly rather than daily precipitation data, a capability that may inform environmental decisions in data-sparse regions. The timing and intensity of precipitation events can strongly influence how much flows quickly into surface waters: twelve inches of rain delivered as gentle mists over the course of a month typically generate less runoff than two six-inch storms. Therefore, accurate predictions of monthly runoff require augmenting knowledge of monthly precipitation with some knowledge of its distribution. Although results are still preliminary, it appears that integrating event-based runoff predictions across an exponential distribution of precipitation event depths allows a reasonable prediction of monthly runoff (faculty advisor Andrew Guswa, Picker Engineering Program).

### Pattern Formations by Social Interactions During Foraging

Weici Hu '12 investigated the effects of animal foraging strategies using partial differential equation models. When animals forage in groups, individuals may adapt different strategies. In particular, when prey is scarce and patchy, certain individuals may exploit others rather than search for prey on their own. Weici built and analyzed mathematical models which reflect different social interactions and found new spatio-temporal oscillatory patterns. This indicates that social interaction itself may contribute to the formation of patchy environment. She presented the result in a talk, "Pattern Formations by Social Interactions During Foraging," during MathFest 2012 in Madison, WI. (faculty advisor Nessy Tania).

### Randomization and bootstrapping down under

Prior to her Junior Semester Abroad at the University of Auckland, New Zealand, Kate Aloisio undertook a Praxis project analyzing qualitative data from pilot interviews for a new approach to teach Randomization and Bootstrapping methods using a simple data analysis system called iNZight. This initial analysis, finding common misconceptions and misinterpretations within the interviews, allowed the researchers to assess certain aspects of the course that needed improvements from the pilot to the main study, which included teaching these methods and software to over 2,500 students taking Introduction to Statistics at various high schools and universities in New Zealand.

### Latent class models for eating disorder research

Sarah Anoke and Zoe Zadworny have undertaken a series of statistical analyses of data from the Growing Up Today Study (GUTS) to better understand how to classify eating disorders for female adolescents. Latent class analysis was used to determine response patterns, with a variety of criteria used to determine the appropriate number of classes. (faculty advisor Nicholas Horton).

### Creating clueless puzzles

Colleen McGaughey, with Fred Henle and Jerry Butters, created a new class of numerical puzzles. Like *kenken* and *sudoku*, the answer is a Latin square (for a 6x6, for example, each row and column must contain one each of the digits, 1, 2, 3, 4, 5, and 6), but unlike *kenken* and *sudoku* there are no numerical clues. The only clue is that the sums of the numbers in each region are all the same (faculty advisor Jim Henle). Prof. Henle presented this work the Hudson River Undergraduate Mathematics Conference at Skidmore College in April 2011.

### Null models in ecological research

Kate Aloisio fit a series of null models for an ecological study led by Smith Professor Jesse Bellemare. The ecological relationship between angiosperm species richness and soil calcium content were considered using data from in a temperate deciduous forest region, in relation to other potentially important environmental factors. The use of null models, while computationally expensive, allow underlying assumptions to be loosened. (faculty advisor Nicholas Horton).

### A non-wellfounded model of arithmetic

Sarah Costrell, Elizabeth Cowdrey, Samantha Lowe, and Cora Waterman looked to see what standard theorems of arithmetic are true in the model and which fail. They found, for example, that ordinary induction fails but strong induction succeeds (faculty advisor Jim Henle). This work was presented at the Sectional Meeting of the American Mathematical Society at Holy Cross in April 2011.

### Sol Lewitt: Grid and arcs from the midpoints of four sides

In 2000, when SCMA had just closed for a major renovation and expansion, an anonymous donor from the class of 1947 gave SCMA Sol LeWitt’s Wall Drawing #139 (Grid and arcs from the midpoints of four sides) [1972]. This important early wall drawing, which is executed in black pencil, was installed at SCMA for the first time in January 2008 by Roland Lusk, a draftsman from LeWitt’s New York studio with the assistance of three Smith students: Isabel Barrios Cazali '10, Katherine Bessey '10, and Sophia LaCava-Bohanan '08. The Museum owns several of LeWitt’s sculptures and works on paper, but Wall Drawing #139 is different because it exists as a series of instructions until it was executed on a wall. A video and more details can be found here.

### Using Apple Xgrid to speed up statistical simulations

Sarah Anoke, Molly Johnson and Yuting Zhao researched the applications of parallel computing to large statistical simulations. Because such simulations involve certain tasks repeated a large number of times, they are well poised to be sped up by distributing repetitions to multiple systems for simultaneous processing. The Apple Xgrid system provides access to grids of computers that can be used to facilitate parallel processing. They used the Apple Xgrid system at Smith to assist with the documentation of grid use with R, performance of latent class methodology studies, development of an R package, helped draft a scientific manuscript on their work (published paper can be found here, faculty advisor Nicholas Horton).

### Exponential domination in grid graphs

Emily Hale-Sills, Kim Lockrow, Emily Merrill, and Samantha Lowe considered a variation on the domination number of a graph. They defined *G* as a graph, *S* a subset of the set of vertices *V* and *d(u,v)* the distance between vertices *u, v*. They also defined *w _{S}(v)* as

*S*is an

*exponential dominating set*if for all vertices

*v*,

*w*is greater than or equal to 1. The exponential dominating number is a function of

_{S}(v)*G*and is defined as the least number of vertices in an exponential dominating set. They research this exponential dominating number for various classes of grid graphs (faculty advisor Ruth Haas). This work was presented at the Joint Mathematics Meetings of the American Mathematical Society in New Orleans in January 2011.

### Solid-coloring objects built from rectangular bricks

Lily Du, Jessica Lord, Micaela Mendlow, Emily Merrill, Viktoria Pardey, Rawia Salih, and Stephanie Wang considered a *brick* as a rectangle in 2D, a rectangular box in 3D, and the natural generalization in R^{d}. An *object built from bricks* is a connected collection of bricks glued together whole-face-to-whole face. A *solid-coloring* of such an object colors each brick so that no two bricks that share a face have the same color. In R^{2} objects built from square bricks are 2-colorable, and objects built from rectangle bricks are 3-colorable. In R^{3}, objects built from cube bricks are again 2-colorable, but they had only proved that objects built from rectangular-box bricks are 4-colorable, although they had no example that needs more than 3 colors. They reported on progress proving that special classes of 3D objects built from bricks are 3-colorable, and on generalizations to other brick shapes and to higher dimensions (faculty advisor Joseph O'Rourke). This work was presented at the Sectional Meeting of the American Mathematical Society at Holy Cross in April 2011.

### Experiments in monotone kinetic visibility

Lily Du, Stefanie Wang, and Yonit Bousany asked "Given a simple planar polygon, is it possible to move its vertices in such a way that the internal visibility graph increases monotonically?" Mathematica experiments led to possible approaches for answering this question, some of which can be proven to work for special subclasses (faculty advisor Ileana Streinu). This work was presented at the Joint Mathematics Meetings of the American Mathemtical Society in New Orleans in January 2011.

### A paracomplete logic and the "sortanatural" numbers

Cora Waterman, Samantha Lowe, Elizabeth Cowdery, and Sarah Costrell studied a logic with infinitely many truth values in which statements may be neither true nor false. They derived properties of the logic, such as the failure of the Law of the Excluded Middle, and showed that it resolves a number of paradoxes. They then applied the principles of the logic to arithmetic to construct a number system with many curious properties and a provably strong but provably limited resemblance to the natural numbers (faculty advisor Jim Henle). This work was presented at the Sectional Meeting of the American Mathematical Society at Holy Cross in April 2011.

### Geometry and dynamics of an ecological arms race

Abigail Fisher, Elizabeth Cowdery, Michelle Winerip, Allison Reed-Harris, and Jayna Resman examined the geometry and dynamics of a predator-prey arms race by comparing growth modes, environmentally induced responses, and foraging effectiveness of two introduced crab species and defense effectiveness of a snail species along the New England coast. One component of the research concentrates on the dynamical aspects of the predator-prey and competitions systems. The other aspect develops geometric models for the snail shape and explores the influence of shell thickness on overall morphogenesis. The geometric part of the research feeds in the dynamical one by determining the relevant morphogenetic parameters that may influence the population dynamics (faculty advisor Chris Golé). This work was presented at the Joint Mathematics Meetings of the American Mathemtical Society in New Orleans in January 2011.

### Determining the extent of defensive response to an invading crab predator

Sarah Anoke, Margaret Nyamumbo, and Sigma Shams analyzed data from a biological invasion study performed by Prof. L. David Smith in the Biological Sciences department. It was hypothesized that snails could vary their shell morphology in the presence of predator effluent, and Prof. Smith tested this hypothesis by conducting a longitudinal study. In this study, he measured the shell thickness of snails exposed to varying amounts of crab effluent. Sarah, Margaret, and Sigma selected an appropriate model for analysis of these data and were able to determine that the presence of crabs appeared to increase snail shell thickness (faculty advisor Katherine Halvorsen).

### Risk factors for pFLUTD in cats

Kate Aloisio undertook a series of statistical analyses for a veterinarian studying an observational cohort of male cats. The goal of the project was to determine risk factors for prolonged feline lower urinary tract disease (pFLUTD) signs and short-term recurrent urethral obstruction (rUO) in cats treated with in-dwelling urinary catheterization and hospitalization for urethral obstruction. The project results were presented at the International Veterinary Emergency and Critical Care Symposium (faculty advisor Nicholas Horton).

### Geometry of the boundary configurations in a phyllotaxis model

Jay Griffiths, Allison Reed-Harris, and Rebecca Terry investigated phyllotaxis, the study of the spiral arrangements in plants. These spirals most often come in two sets, whose numbers are consecutive Fibonacci numbers. They explored the set of initial conditions in a simple two-postulate dynamic-geometric model of phyllotaxis. They focused on the stability question and in the geometry of the boundary between Fibonacci and Lucas configurations (faculty advisor Pau Atela). This work was presented at the Sectional Meeting of the American Mathematical Society at Holy Cross in April 2011.