Dylan is a deep thinker with innovative ideas. His sales and marketing experience have been primarily internet based with profitable success. Passionate about the protection of the environment, Dylan continues to educate his peers on solution based ideas. He is a proven leader as captain of his basketball team and a team player in all aspects of his life. Also currently a USPA licensed skydiver and skydiving coach at Sky San Diego, Dylan has a passion for living by the motto, “live life to the fullest”. He recognizes entrepreneurship as a pathway to achieving his personal goals and making the world a better place for all. His goal for the future is to become a serial entrepreneur and positively impact the world through my endeavors.

]]>MS Applied Mathematics

THESIS DEFENSE

Monday, August 15, 2016

2:00pm GMCS 418

*Pseudo-Spectral Continuation Methods for the Planar Hamiltonian Korteweg-de Vries Equation*

Abstract:

The aim of this thesis was to develop a pseudo-spectral continuation scheme for periodic orbits of planar Hamiltonian dynamical systems. Traditionally, numerical continuation of Hamiltonian periodic orbits is performed by way of shooting methods. While we build our analysis on the more traditional approach, we remove any integration required by the shooting method in favor of solving the Hamiltonian dynamical system directly at every point along the continuation branch.

This thesis considers the Korteweg-de Vries (KdV) equation to build the pseudo-spectral continuation scheme. We first derive the standard KdV equation and build the corresponding Hamiltonian dynamical system. We then consider an unfolding parameter and a phase condition to ensure that we converge to unique solutions. Finally, we show that using the pseudo-spectral continuation method, we are able to solve for solutions along the continuation branch much faster and with a higher level of accuracy than shooting methods.

Thesis Committee:

Christopher Curtis, Thesis Chair, Department of Mathematics & Statistics

Peter Blomgren, Department of Mathematics & Statistics

Calvin Johnson, Department of Physics

THESIS DEFENSE

*The Inclusion-Exclusion and Mixing Ratio Conjectures From the Ingleton Inequality*

Abstract:

The Ingleton Inequality, which is based on a vector space theorem and adapted to entropy functionals by Matus, plays a role in describing the entropy region for four discrete random variables, depending on whether or not it is satisfied. The entropy regions that are defined by entropic vectors for discrete random variables are not known, in general. For , recent work has been accomplished by Mao and Boston in characterizing this space of -dimensional entropic vectors. They study how *group-characterizable* entropic vectors sometimes yield violations of the Ingleton Inequality/Ratio, thus expanding the understanding of this region. We discovered a decomposition of the Ingleton Ratio into what we dub the Mixing Ratio and the Inclusion-Exclusion Ratio, which resembles the inclusion-exclusion principle from set theory. We propose two conjectures based on the simpler Inclusion-Exclusion and Mixing Ratios, attempting to streamline the search for Ingleton Inequality-violators. Examples of these violators are reproduced in order to provide support for our conjectures. Characterizing the entropy region associated with discrete random variables is related to determining the maximum communication rate of a network coding system that uses discrete random variables. Characterizing this region for arbitrary remains an open problem.

Thesis Committee:

Michael O’Sullivan, Thesis Chair, Department of Mathematics & Statistics

Carmelo Interlando, Department of Mathematics & Statistics

Jean mark Gawron, Department of Linguistics

Turbulence mitigation algorithms have gained a lot of attention in the last decade and several algorithms are now available. Unfortunately, since this problem is generally involved in Defense applications, there is no common dataset to assess such algorithms. The purpose of Gilles’ grant is to create a free open dataset of observations acquired through atmospheric turbulence. A camera with different lenses will be purchased to run several scenarios of interest for the Defense community. The project involves an undergraduate student and we expect to use the proximity of the desert to make several observation by the end of the Summer. It is expected that this dataset will be released online by the end of 2016. Pictured: Nicholas Ferrante (SDSU), and Dr. Gilles.

For the next three years, the department will host a Research Experience for Undergraduates (REU) program. This will bring brilliant undergraduate students together with graduate students and faculty, to solve meaningful problems and learn about research careers, in eight summer weeks. The 2016 projects are in atmospheric imaging (run by Dr. Gilles) and in Markov chain Monte Carlo analysis (run by Dr. Roman). The students for these projects have been selected; they represent the whole country, from SDSU, to Georgetown, to the University of Massachusetts, to UCLA. For more information, see the program website http://www.sci.sdsu.edu/math-reu/index.html The grant PI is Vadim Ponomarenko.

]]>The project is based on a partnership between SDSU, three museums in Balboa Park: Mingei International Museum, Reuben H. Fleet Science Center, and the Museum of Photographic Arts, as well as the Boys’ and Girls’ Club of Southeast San Diego.

Current work and programs conducted by InforMath include:

1. A new exhibition at the Reuben H. Fleet Science Center entitled “Taking Shape” is open to the public. The exhibition brings together art and mathematics, the latter focused on topology concepts. The structure made out of packing tape is large enough for visitors to walk inside to explore its layout. It includes three areas inspired by the following surfaces of: 1) Torus, 2) Schwarz P, and 3) Pair-of-Pants. Working areas associated with these topologies are being setup.

2. A program on basket weaving and curvature has been conducted at the Mingei International Museum and the Boys’ and Girls’ Club of Southeast San Diego. Children from Chulavista who have volunteered to enroll worked three sessions at the Mingei and three at their after school facilities. They explored the current exhibition “Made in America,” created woven forms, and sewed fabric bowls.

3. A new exhibition at the Museum of Photographic Arts is in preparation. Its theme is 3D photography and the mathematics of depth perception.

4. A new program on Music and Mathematics is going to start in April. Children from the Boys’ and Girls’ Club of Southeast San Diego will be transported weekly to the Reuben H, Fleet Science Center. They will use a new exhibit called “Dance Math” to develop mathematical ways of representing dance and rhythm. The program will end with a public performance combining drumming, dance, and chanting.

]]>A significant contribution of the work will be connecting the research on mathematics learning generally with research on mathematics learning of English language learners. In addition to advancing theoretical understandings, the research will also contribute practical resources and guidance for mathematics teachers who teach English language learners.

The Faculty Early Career Development (CAREER) program is a National Science Foundation (NSF)-wide activity that offers awards in support of junior faculty who exemplify the role of teacher-scholars through outstanding research, excellent education, and the integration of education and research within the context of the mission of their organizations.

Zahner’s abstract is available on the NSF website below.

http://www.nsf.gov/awardsearch/showAward?AWD_ID=1553708&HistoricalAwards=false