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Published: August 14, 2015

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Published: August 14, 2012

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Nonstandard existence proofs for reaction diffusion equations

Published in involve, 2019

Abstract: We give an existence proof for distribution solutions to a scalar reaction diffusion equation, with the aim of illustrating both the differences and the common ingredients of the nonstandard and standard approaches. In particular, our proof shows how the operation of taking the standard part of a nonstandard real number can replace several different compactness theorems, such as Ascoli’s theorem and the Banach–Alaoglu theorem on weak∗-compactness of the unit ball in the dual of a Banach space.

Recommended citation: Olson, C., Mueller, M., & Angenent, S. B. (2019). Nonstandard existence proofs for reaction diffusion equations. Involve, 12(6), 1015–1034.
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On the translates of general dyadic systems on R

Published in Mathematische Annalen, 2020

Abstract: Many techniques in harmonic analysis use the fact that a continuous object can be written as a sum (or an intersection) of dyadic counterparts, as long as those counterparts belong to an adjacent dyadic system. Here we generalize the notion of adjacent dyadic system and explore when it occurs, leading to some new and perhaps surprising classifications. In particular, we show that every dyadic grid is determined by two parameters, the shift and the location; moreover two dyadic grids form an adjacent dyadic system if and only if their shifts and locations satisfy readily verifiable conditions.

Recommended citation: Anderson, T.C., Hu, B., Jiang, L. et al. On the translates of general dyadic systems on R. Math. Ann. 377, 911–933 (2020).
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Nonconvex ancient solutions to curve shortening flow

Published in Tran. Amer. Math. Soc., 2023

Abstract: We construct an ancient solution to planar curve shortening. The solution is at all times compact and embedded. For $t\ll0$ it is approximated by the rotating Yin-Yang soliton, truncated at a finite angle $\alpha(t)=−t$, and closed off by a small copy of the Grim Reaper translating soliton.

Recommended citation: Zhang, Y., Olson, C., Khan, I., & Angenent, S. (2023). Nonconvex ancient solutions to curve shortening flow. Trans. Am. Math. Soc.
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Community formation in wealth-mediated thermodynamic strategy evolution

Published in Chaos, 2023

Abstract: We study a dynamical system defined by a repeated game on a 1D lattice, in which the players keep track of their gross payoffs over time in a bank. Strategy updates are governed by a Boltzmann distribution, which depends on the neighborhood bank values associated with each strategy, relative to a temperature scale, which defines the random fluctuations. Players with higher bank values are, thus, less likely to change strategy than players with a lower bank value. For a parameterized rock–paper–scissors game, we derive a condition under which communities of a given strategy form with either fixed or drifting boundaries. We show the effect of a temperature increase on the underlying system and identify surprising properties of this model through numerical simulations.

Recommended citation: Olson, C., Belmonte, A., & Griffin, C. (2022). Community formation in wealth-mediated thermodynamic strategy evolution. Chaos, 32(10), 103103.
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John Cross, epidemic theory, and mathematically modeling the Norwich smallpox epidemic of 1819

Published in PLOS One, 2024

Abstract: In this paper, we reintroduce Dr. John Cross’ neglected and unusually complete historical data set describing a smallpox epidemic occurring in Norwich, England in 1819. We analyze this epidemic data in the context of early models of epidemic spread including the Farr–Evans–Brownlee Normal law, the Kermack–McKendrick square Hyperbolic Secant and SIR laws, along with the modern Volz–Miller random-network law. We show that Cross’ hypothesis of susceptible pool limitation is sufficient to explain the data under the SIR law, but requires parameter estimates differing from the modern understanding of smallpox epidemiology or large errors in Cross’ data collection. We hypothesize that these discrepancies are due to the mass-action hypothesis in SIR theory, rather than significant errors by Cross, and use Volz–Miller theory to support this. Our analysis demonstrates the difficulties arising in inference of attributes of the disease from death incidence data and how model hypotheses impact these inferences. Our study finds that, combined with Volz–Miller modeling theory, Cross’ death incidence data and population observations give smallpox attributes which largely cohere to those used in modern smallpox models.

Recommended citation: Olson CD, Reluga TC (2024) John Cross, epidemic theory, and mathematically modeling the Norwich smallpox epidemic of 1819. PLoS ONE 19(11): e0312744. https://doi.org/10.1371/journal.pone.0312744
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Social Distancing Equilibria in Games under Conventional SI Dynamics

Published in arXiv, 2026

Abstract: The mathematical characterization of social-distancing games in classical epidemic theory remains an important question, for their applications to both infectious-disease theory and memetic theory. We consider a special case of the dynamic finite-duration SI social-distancing game where payoffs are accounted using Markov decision theory with zero-discounting, while distancing is constrained by threshold-linear running-costs, and the running-cost of perfect-distancing is finite. In this special case, we are able construct strategic equilibria satisfying the Nash best-response condition explicitly by integration. Our constructions are obtained using a new change of variables which simplifies the geometry and analysis. As it turns out, there are no singular solutions, and a time-dependent bang-bang strategy consisting of a wait-and-see phase followed by a lock-down phase is always the unique strategic equilibrium. We also show that in a restricted strategy space the bang-bang Nash equilibrium is an ESS, and that the optimal public policy exactly corresponds with the equilibrium strategy.

Recommended citation: Olson, C. D., & Reluga, T. C. (2026). Social distancing equilibria in games under conventional SI dynamics. arXiv [Cs.GT]. Retrieved from http://arxiv.org/abs/2603.12107
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Connor D. Olson

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