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A list of all the posts and pages found on the site. For you robots out there, there is an XML version available for digesting as well.
Pages
Posts
Points and Boxes
Published:
Heawood Conjecture
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projects
Chess
Published:
Wrote a GUI chess application in JavaFX using event driven programming.
Trachtenberg Interactive Quiz
Published:
Text based interactive application that is designed to help people learn the Trachtenberg speed multiplication system. Information on Trachtenberg and his system are provided in the github repository.
Brain Grid
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BrainGrid is an open-source spiking neural network simulator that is intended to aid scientists and researchers by providing pre-built code that can be easily modified to fit different models.
Cribbage
Published:
Text based interactive cribbage game. There are several bot opponents coded up with varying skill levels.
Matrix Polynomial Analysis
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The goal of the project was to get a picture of what the space of polynomials that preserve the nonnegativity of matrices of a given size looks like.
Nonnegative matrix boundary optimizer
Published:
One of the difficult things about the nonnegative inverse eigenvalue problem (NIEP), is trying to reason about the matrices that lie on the boundary. This project aims to numerically solve for these matrices using a series of optimization steps. The idea is that once these matrices are found we can build conjectures on what the boundary of the NIEP looks like.
publications
Would gamers collaborate given the opportunity
Published in FDG 18: Proceedings of the 13th International Conference on the Foundations of Digital Games, 2018
Abstract: Understanding player preference and behavioral tendency in the presence of collaborative opportunities is fundamental to building cooperative video games.1 Such knowledge allows the integration and fine-tuning of features to further engage players. This paper presents an ongoing study that approaches the subject by posing a simple question: when opportunities are provided, would players choose to cooperate? The work analyzes existing well-established cooperative game design patterns and identifies effective attributes of game mechanics that are characterized by the patterns. Small multiplayer games that focused on each of the attributes are built where in each case the players have the options of collaborating or completing the tasks individually. In this way, the effects of each attribute can be analyzed independently to provide insights into players behavior under specific variations of conditions. While the testing is ongoing, our results will provide directions for future integration of collaborative features in games. Additionally, our approach of identifying implementable attributes based on the principles of cooperative game design patterns serves as a template for systematic approach to building cooperative video games.
Polynomials that preserve nonnegative matrices
Published in Journal of Linear Algebra and its Applications, 2022
Abstract: In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London ((1978/1979) [8]) posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If Pn denotes the set of all polynomials that preserve all n-by-n nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to Pn. However, it is known that Pn contains polynomials with negative entries. In this work, novel results for Pn with respect to the coefficients of the polynomials belonging to Pn. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed.
Polynomials that preserve nonnegative monomial matrices
Published in arXiv open access repository, 2022
Abstract: A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we provide a formula for computing an arbitrary power of a monomial matrix and a formula for computing the polynomial of a nonnegative monomial matrix.
Polynomials that preserve nonnegative matrices of order two
Published in Ball State Undergraduate Mathematics Exchange, 2022
Abstract: A known characterization for entire functions that preserve all nonnegative matrices of order two is shown to characterize polynomials that preserve nonnegative matrices of order two. Equivalent conditions are derived and used to prove that P3 ⊂ P2, which was previously unknown. A new characterization is given for polynomials that preserve nonnegative circulant matrices of order two.
The NIEP is solvable by reality and finitely many polynomial inequalities
Published in arXiv open access repository, 2024
Abstract: The nonnegative inverse eigenvalue problem (NIEP) is shown to be solvable by the reality condition, spectrum equal to its conjugate, as well as by a finite union and intersection of polynomial inequalities. It is also shown that the symmetric NIEP and real NIEP form semi-algebraic sets and can therefore be solved just by a finite union and intersection of polynomial inequalities. An overview of ideas are given in how tools from real algebraic geometry may be applied to the NIEP and related sub-problems.
talks
Polynomials that preserve nonnegative matrices
Published:
Abstract: In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83–90] posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If $\mathscr{P}_n$ denotes the set of all polynomials that preserve all $n$-by-$n$ nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to $\mathscr{P}_n$. However, it is known that $\mathscr{P}_n$ contains polynomials with negative entries. In this presentation, results for $\mathscr{P}_n$ with respect to the coefficients of the polynomials belonging to $\mathscr{P}_n$. Along the way, a generalization for the even and odd parts of a polynomial are given. This talk concludes with a characterization of $\mathscr{P}_2$.
Polynomials that preserve nonnegative matrices
Published:
Abstract: In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83–90] posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If $\mathscr{P}_n$ denotes the set of all polynomials that preserve all $n$-by-$n$ nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to $\mathscr{P}_n$. However, it is known that $\mathscr{P}_n$ contains polynomials with negative entries. In this presentation, results for $\mathscr{P}_n$ with respect to the coefficients of the polynomials belonging to $\mathscr{P}_n$. Along the way, a generalization for the even and odd parts of a polynomial are given. This talk concludes with a characterization of $\mathscr{P}_2$.
The cone of polynomials that preserve nonnegative matrices
Published:
Abstract: The set of polynomials that preserve nonnegative matrices of a given order form a convex cone. In this talk, I will introduce some tools from convex analysis and nonnegative matrix theory. Then apply those tools to study this cone and present recent findings that the cone is not-polyhedral even when the degree of the polynomials is fixed to twice the size of the matrices.
Polynomials that preserve nonnegative matrices
Published:
Abstract: In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83–90] posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If $\mathscr{P}_n$ denotes the set of all polynomials that preserve all $n$-by-$n$ nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to $\mathscr{P}_n$. However, it is known that $\mathscr{P}_n$ contains polynomials with negative entries. In this presentation, results for $\mathscr{P}_n$ with respect to the coefficients of the polynomials belonging to $\mathscr{P}_n$. This talk concludes with a characterization of $\mathscr{P}_2$ and places to purse further research.
NIEP, SNIEP, RNIEP, and other problems in spectral nonnegative matrix theory
Published:
Abstract: Eigenvalues play a central role in linear algebra. For a given matrix, the process of finding these eigenvalues is straightforward. But what if, instead of a single matrix, I had a set of matrices? What shared behaviors do the corresponding eigenvalues have? Can we say where those eigenvalues are located? The talk will start with some background to known nonnegative matrix theory results and with an overview of some of the foundational questions in inverse eigenvalues problems. We will then discuss why we know these problems are solvable and what it would mean to solve them. Finally, I will end with some of my current ideas on approaching these problems.
The nonnegative inverse eigenvalue problem is solvable and the algorithm to solve it exists. So why is the problem unsolved?
Published:
Abstract: The nonnegative inverse eigenvalue problem (NIEP) asks what are the necessary and sufficient conditions such that a list of complex numbers forms the spectra of a nonnegative matrix. In this talk, I will give some background into the NIEP and its related subproblems. Next, I will discuss how the NIEP forms a semialgebraic set and why it can then be solved by polynomial inequalities. Finally, I will give an overview of some algorithms that can give a solution to the NIEP and outline why we can’t directly use them.
An experimental approach to the NIEP using algebraic geometry
Published:
Abstract: The nonnegative inverse eigenvalue problem (NIEP) asks for the necessary and sufficient conditions for a list of complex numbers to be the spectra of a nonnegative matrix. Using tools from algebraic geometry, it is shown that the NIEP is solvable by polynomial inequalities and the reality condition. An experimental approach is then presented for forming the desired feasibility region of the NIEP for small matrices. We conclude by proving that under this approach, solving the real NIEP solves the NIEP.
An experimental approach to the S-SNIEP using algebraic geometry
Published:
Abstract: The stochastic symmetric nonnegative inverse eigenvalue problem (S-SNIEP) asks for the necessary and sufficient conditions for a list of real numbers to be the spectra of a stochastic symmetric nonnegative matrix. Using tools from algebraic geometry, it is shown that the S-SNIEP is solvable by polynomial inequalities and the reality condition. An experimental approach is then presented for forming the desired feasibility region of the S-SNIEP for small matrices. This approach is then used to generate a conjectured solution to the n=4 case. We conclude by giving short comings to this approach and potential future ideas.
teaching
Calculus 1 lab
Running Lab, Washington State University, Department of Mathematics and Statistics, 2022
For this course, I ran a calculus 1 lab. This entailed running the lab twice a week, making a couple of quizzes, and helping grade.
Calculus 2 lab
Running Lab, Washington State University, Department of Mathematics and Statistics, 2023
For this course, I ran a calculus 2 lab. This entailed running the lab twice a week, making a couple of quizzes, and helping grade.
Discrete Mathematics
Instructor, Washington State University, Department of Mathematics and Statistics, 2023
For this course, I taught one section of Discrete Mathematics. This entailed writing and presenting lecture notes, giving/grading homeworks and exams, and being the instructor of record for my class.
Calculus 2 lab
Running Lab, Washington State University, Department of Mathematics and Statistics, 2024
For this course, I ran a calculus 2 lab. This entailed running the lab twice a week, making a couple of quizzes, and helping grade.
Discrete Mathematics
Instructor, Washington State University, Department of Mathematics and Statistics, 2024
For this course, I taught one section of Discrete Mathematics. This entailed writing and presenting lecture notes, giving/grading homeworks and exams, and being the instructor of record for my class.
Abstract Algebra
Instructor, Washington State University, Department of Mathematics and Statistics, 2025
For this course, I taught one section of Abstract algebra. This entailed writing and presenting lecture notes, giving/grading homeworks and exams, and being the instructor of record for my class.