Ben Clark

Ben Clark

Questionable software engineer and pretend mathematician

Publications

Journal Articles

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.

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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.

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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.

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Unpublished reports

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.

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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.

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