## 02. Students

### Current

#### Christopher Bresten

- Numerical convergence of the logistic map
- RBF-Gegenbauer Reconstruction Method for Discontinuous Solutions to PDEs. Chris L. Bresten, Sigal Gottlieb, Jae-Hun Jung. 14 th Sigma Xi Research Exhibition, University of Massachusetts Dartmouth, April, 2008

Abstract: Approximations of PDEs with discontinuous solutions using high order methods suffer from a form of oscillatory error known as the Gibbs phenmenon. Radial Basis Functions provide a cheap and simple way to approximate the solutions to such PDEs, but are not immune to the Gibbs oscillations.The Gegenbauer polynomial reconstruction method provides an effective way to remove such oscillations. This work ties Radial Basis Functions and the Gegenbauer post-processing technique together to produce a non-oscillatory solution to a discontinuous problem. Oscillatory error is detected and removed as needed in real time for stable solutions to time-dependent PDEs.

#### Adam Callahan

- Power laws and heavy-tailed distributions of words in literary texts
- Quantitative aspects of literary texts Adam J. Callahan, Gary E. Davis; Department of Mathematics. 14 th Sigma Xi Research Exhibition, University of Massachusetts Dartmouth, April, 2008

Abstract: We examine a wide range of literary texts, both in English and other languages, and show that the ratio of the number of new words in the first n words of the text to n – has a heavy tailed distribution, meaning that except for the early parts of the text, the ratio of the number of new words to total words decays as a power function of the number of words. This indicates the type-token ratio is scale-invariant – the same statistical properties on almost all scales. This had previously been demonstrated across particular authors’ works but not within a given text. We also examine the Shannon entropy of the first n words of a text, from the word frequencies, and show that in all texts we examined the entropy increases logarithmically with the number of words. We apply this to the Voynich manuscript to show an unusual decrease of entropy in that text. Finally, we borrow an idea from mathematical finance and show that the logarithm of the returns of word frequencies is a symmetric distribution with significantly broad-shoulders, quite distinct from a normal distribution. This is part of an ongoing project to examine text from a dynamical systems viewpoint.

#### Daniel Higgs

- Hybrid RBF-WENO methods for nonlinear hyperbolic conservation laws (OCTAVE)
- Population Dynamics with Variable Non-Chronlogically Based Aging. Daniel L. Higgs, Jae-Hun Jung, Sigal Gottlieb. 14 th Sigma Xi Research Exhibition, University of Massachusetts, April, 2008

Abstract: Population dynamics is the mathematical study of population changes in reaction to biological and environmental processes. One specific problem in population dynamics is modeling the size and age of a given population over time. Traditionally, these models apply a static rate of aging to all individuals within a population and that rate is directly equated with the passage of time. While this is useful for investigating the chronological age of a population, it abstracts the processes which govern age, and doesn’t allow for any variability of those processes between individuals. In this project, we will investigate other, more biological measures of age and construct population models based on an aging rate which varies between individuals and changes over time. Special attention will be given to models which yield a discontinuous solution. These models will be solved numerically with a multidomain hybrid WENO-RBF technique concurrently being researched.

### Past

#### Andrew Correia

- Relationship Between The Derivatives of The Mean and The Moments of The Distribution. Andrew W. Correia, Jae-Hun Jung. 14 th Sigma Xi Research Exhibition, University of Massachusetts Dartmouth, April 2008.

Abstract: In a paper recently published by Dr. Tim J. Cole, et al, he and his group did research on how the skewness and kurtosis of children’s height change with age during puberty. As part of the research, the group found that during puberty, the curves of the standard deviation, skewness, and kurtosis closely resemble those of the first, second, and third derivatives, respectively, of the mean of the distribution. Here we look to see what the specific conditions are that make this true in the general case. We will then generate random data, fit the data using Chebyshev polynomials, and do the appropriate calculations to test our conclusion.

#### Vincent Durante

#### Adam Nishri

- Level set methods for speech segmentation based on polynomial reconstructions. Adam Nishri, Jae-Hun Jung (Mathematics, UMass Dartmouth), Chang-Yeol Jung (Mathematics, Indiana). 14 th Sigma Xi Research Exhibition, University of Massachusetts Dartmouth, April 2008.

Abstract: Speech segmentation is an important process of the signal processing for many areas of applications such as machine translation, speech recognition, etc. Speech segmentation is to detect the local discontinuities or boundaries between words or consonants and vowels in the speech data. Several methods have been actively developed for the speech segmentation in the last decades including the level set method. In this work, we adopt the level set approach using the energy spectrum of the speech data. The level set method has been widely used for the image reconstruction. The same methodology is applied to the signal data. Using the Euler-Lagrange equations, the level set function is evaluated with several iterations. The main development of this work is to use the orthogonal polynomials instead of the trigonometric functions. Several preliminary examples will be presented