Chess
Published:
Wrote a GUI chess application in JavaFX using event driven programming.
Posts by Collection
projects
Trachtenberg Interactive Quiz
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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
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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
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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
talks
Polynomials that preserve nonnegative matrices
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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
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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
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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?
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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
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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
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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.