Published in Journal 1, 2009
This paper is about the number 1. The number 2 is left for future work.
Recommended citation: Your Name, You. (2009). "Paper Title Number 1." Journal 1. 1(1).
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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 $\mathscr{P}_2$.
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 $\mathscr{P}_2$.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.