arXiv subjects

Julie Rowlett

Publications and source records attributed to Julie Rowlett.

At least 37 records · Page 2Linked to original sources

arXiv2020

Mathematics Indicates That an HIV-Style Strategy Could Be Applied to Manage the Coronavirus

We have learned to live with many potentially deadly viruses for which there is no vaccine, no immunity, and no cure. We do not live in constant fear of these viruses, instead, we have learned how to outsmart them and reduce the harm they cause. A new mathematical model that combines the spread of diseases that do not confer immunity together with the evolution of human behaviors indicates that we may be able to fight new diseases with the same type of strategy we use to fight viruses like HIV.

q-bio.PE↗

arXiv2020

Mathematical Models for Erosion and the Optimal Transportation of Sediment

We investigate a mathematical theory for the erosion of sediment which begins with the study of a non-linear, parabolic, weighted 4-Laplace equation on a rectangular domain corresponding to a base segment of an extended landscape. Imposing natural boundary conditions, we show that the equation admits entropy solutions and prove regularity and uniqueness of weak solutions when they exist. We then investigate a particular class of weak solutions studied in previous work of the first author and produce numerical simulations of these solutions. After introducing an optimal transportation problem for the sediment flow, we show that this class of weak solutions implements the optimal transportation of the sediment.

math.AP↗

arXiv2020

Eigenvalue Estimates on Bakry-Emery Manifolds

We demonstrate lower bounds for the eigenvalues of compact Bakry-Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalised maximum principle which allows gradient estimates in the Riemannian setting to be directly applied to the Bakry-Emery setting. Lower bounds for all eigenvalues are demonstrated using heat kernel estimates and a suitable Sobolev inequality.

math.SP↗

arXiv2020

The theory of games and microbe ecology

Using game theory we provide mathematical proof that if a species of asexually reproducing microbes does not possess maximum variability in competitive abilities amongst its individual organisms, then that species is vulnerable to replacement by competitors. Furthermore, we prove that such maximally variable species are neutral towards each other in competition for limited resources; they coexist. Our proof is constructive: given one species which does not possess maximum variability, we construct a species with the same (or lower) mean competitive ability which can invade, in the sense that its expected value in competition is positive whereas the expected value of the non-maximally variable species is negative. Our results point towards the mechanistic underpinnings for the frequent observations that (1) microbes are characterized by large intra-specific variability and that (2) the number of extant microbe species is very large.

q-bio.PE↗

arXiv2020

The level sets of typical games

In a non-cooperative game, players do not communicate with each other. Their only feedback is the payoff they receive resulting from the strategies they execute. It is important to note that within each level set of the total payoff function the payoff to each player is unchanging, and therefore understanding the structure of these level sets plays a key role in understanding non-cooperative games. This note, intended for both experts and non-experts, not only introduces non-cooperative game theory but also shows its fundamental connection to real algebraic geometry. We prove here a general result about the structure of the level sets, which although likely to be known by experts, has interesting implications, including our recent application to provide a new mathematical explanation for the "paradox of the plankton." We hope to encourage communication between these interrelated areas and stimulate further work in similar directions.

math.HO↗

arXiv2020

Many ways to stay in the game: Individual variability maintains high biodiversity in planktonic micro-organisms

In apparent contradiction to competition theory, the number of known, co-existing plankton species far exceeds their explicable biodiversity - a discrepancy termed the Paradox of the Plankton. We introduce a new game-theoretic model for competing micro-organisms in which one player consists of all organisms of one species. The stable points for the population dynamics in our model, known as strategic behavior distributions (SBDs), are probability distributions of behaviors across all organisms which imply a stable population of the species as a whole. We find that intra-specific variability is the key characteristic that ultimately allows co-existence because the outcomes of competitions between individuals with variable competitive abilities is unpredictable. Our simulations based on the theoretical model show that up to 100 species can coexist for at least 10000 generations, and that even small population sizes or species with inferior competitive ability can survive when there is intra-specific variability. In nature, this variability can be observed as niche differentiation, variability in environmental and ecological factors, and variability of individual behaviors or physiology. Therefore previous specific explanations of the paradox are consistent with and provide specific examples of our suggestion that individual variability is the mechanism which solves the paradox.

q-bio.PE↗

arXiv2020

One can hear the corners of a drum

We prove that the presence or absence of corners is spectrally determined in the following sense: any simply connected domain with piecewise smooth Lipschitz boundary cannot be isospectral to any connected domain, of any genus, which has smooth boundary. Moreover, we prove that amongst all domains with Lipschitz, piecewise smooth boundary and fixed genus, the presence or absence of corners is uniquely determined by the spectrum. This means that corners are an elementary geometric spectral invariant; one can hear corners.

math.SP↗

arXiv2020

On the spectral theory and dynamics of asymptotically hyperbolic manifolds

We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.

math.SP↗

arXiv2020

La géométrie de Bakry-Émery et l'écart fondamental

This article is a brief presentation of results surrounding the fundamental gap. We begin by recalling Bakry-Emery geometry and demonstrate connections between eigenvalues of the Laplacian with the Dirichlet and Neumann boundary conditions. We then show a connection between the fundamental gap and Bakry-Emery geometry, concluding with a presentation of the key ideas in Andrews's and Clutterbuck's proof of the fundamental gap conjecture. We conclude with a presentation of results for the fundamental gap of triangles and simplices.

math.SP↗

arXiv2020

The sound of symmetry

This note begins with an introduction to the inverse isospectral problem popularized by M. Kac's 1966 article in the American Mathematical Monthly, "Can one hear the shape of a drum?" Although the answer has been known for some twenty years now, many open problems remain. Intended for general audiences, readers are challenged to complete exercises throughout this interactive introduction to inverse spectral theory. Following the introduction, the main techniques used in inverse isospectral problems are collected and discussed. These are then used to prove that one can hear the shape of: parallelograms, acute trapezoids, and the regular n-gon. Finally, we show that one can realistically hear the shape of the regular n-gon amongst all convex n-gons because it is uniquely determined by a finite number of eigenvalues; the sound of symmetry can really be heard!

math.SP↗

arXiv2020

How to hear the corners of a drum

We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary conditions, in the generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle fail, but can nonetheless be replaced by powerful microlocal analysis methods.

math.SP↗

arXiv2020

Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions

The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, Bérard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace eigenvalue equation are all trigonometric functions. In 2008, McCartin proved that in two dimensions, this special analytic property has both an equivalent algebraic formulation, as well as an equivalent geometric formulation. Here we generalize the results of Bérard and McCartin to all dimensions. We prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. Moreover, we prove that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. In conclusion, we connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach's conjecture.

math.SP↗

arXiv2020

The Dirichlet isospectral problem for trapezoids

We show that non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum. This extends our previous result, which was only concerned with the Neumann Laplace spectrum.

math.AP↗

arXiv2020

Decisions and disease: a mechanism for the evolution of cooperation

In numerous contexts, individuals may decide whether they take actions to mitigate the spread of disease, or not. Mitigating the spread of disease requires an individual to change their routine behaviours to benefit others, resulting in a 'disease dilemma' similar to the seminal prisoner's dilemma. In the classical prisoner's dilemma, evolutionary game dynamics predict that all individuals evolve to 'defect.' We have discovered that when the rate of cooperation within a population is directly linked to the rate of spread of the disease, cooperation evolves under certain conditions. For diseases which do not confer immunity to recovered individuals, if the time scale at which individuals receive information is sufficiently rapid compared to the time scale at which the disease spreads, then cooperation emerges. Moreover, in the limit as mitigation measures become increasingly effective, the disease can be controlled, and the rate of infections tends to zero. Our model is based on theoretical mathematics and therefore unconstrained to any single context. For example, the disease spreading model considered here could also be used to describe social and group dynamics. In this sense, we may have discovered a fundamental and novel mechanism for the evolution of cooperation in a broad sense.

physics.soc-ph↗

arXiv2019

Second variation of Selberg zeta functions and curvature asymptotics

We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichmüller space. We then use this formula to determine the asymptotic behavior as $\text{Re} (s) \to \infty$ of the second variation. As a consequence, for $m \in \mathbb{N}$, we obtain the complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t, \mathcal K_t)\to t\in \mathcal T$ of holomorphic m-differentials over the Teichmüller space $\mathcal T$, for $m$ large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $O(m^2 e^{-l_0 m}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.

math.SP↗

arXiv2018

Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

math.DG↗

arXiv2017

A Polyakov formula for sectors

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

math.SP↗

arXiv2017

The heat trace for the drifting Laplacian and Schrödinger operators on manifolds

We study the heat trace for both the drifting Laplacian as well as Schrödinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder estimate. We also demonstrate that the more precise asymptotic behavior of the remainder is determined by and conversely distinguishes higher (Sobolev) regularity on the potential or weight function. In the case of a smooth weight function, we determine the full asymptotic expansion of the heat trace for the drifting Laplacian for small times. We then use the heat trace to study the asymptotics of the eigenvalue counting function. In both cases the Weyl law coincides with the Weyl law for the Riemannian manifold with the standard Laplace-Beltrami operator. We conclude by demonstrating isospectrality results for the drifting Laplacian on compact manifolds.

math.DG↗