GIT Compactifications of M0,nM_{0,n} from Conics

We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$ admits a morphism to each such GIT quotient, analogous to the well-known result of Kapranov for the simpler $(\mathbb{P}^1)^n$ quotients. Moreover, these morphisms factor through Hassett's moduli spaces of weighted pointed rational curves, where the weight data comes from the GIT linearization data.Comment: 15 pages, 5 figures; corrected inequality in Lemma 5.1, Int. Math. Res. Notices Vol. 201

Accurate and efficient calculation of response times for groundwater flow

We study measures of the amount of time required for transient flow in heterogeneous porous media to effectively reach steady state, also known as the response time. Here, we develop a new approach that extends the concept of mean action time. Previous applications of the theory of mean action time to estimate the response time use the first two central moments of the probability density function associated with the transition from the initial condition, at $t=0$, to the steady state condition that arises in the long time limit, as $t \to \infty$. This previous approach leads to a computationally convenient estimation of the response time, but the accuracy can be poor. Here, we outline a powerful extension using the first $k$ raw moments, showing how to produce an extremely accurate estimate by making use of asymptotic properties of the cumulative distribution function. Results are validated using an existing laboratory-scale data set describing flow in a homogeneous porous medium. In addition, we demonstrate how the results also apply to flow in heterogeneous porous media. Overall, the new method is: (i) extremely accurate; and (ii) computationally inexpensive. In fact, the computational cost of the new method is orders of magnitude less than the computational effort required to study the response time by solving the transient flow equation. Furthermore, the approach provides a rigorous mathematical connection with the heuristic argument that the response time for flow in a homogeneous porous medium is proportional to $L^2/D$, where $L$ is a relevant length scale, and $D$ is the aquifer diffusivity. Here, we extend such heuristic arguments by providing a clear mathematical definition of the proportionality constant.Comment: 22 pages, 3 figures, accepted version of paper published in Journal of Hydrolog

New homogenization approaches for stochastic transport through heterogeneous media

The diffusion of molecules in complex intracellular environments can be strongly influenced by spatial heterogeneity and stochasticity. A key challenge when modelling such processes using stochastic random walk frameworks is that negative jump coefficients can arise when transport operators are discretized on heterogeneous domains. Often this is dealt with through homogenization approximations by replacing the heterogeneous medium with an $\textit{effective}$ homogeneous medium. In this work, we present a new class of homogenization approximations by considering a stochastic diffusive transport model on a one-dimensional domain containing an arbitrary number of layers with different jump rates. We derive closed form solutions for the $k$th moment of particle lifetime, carefully explaining how to deal with the internal interfaces between layers. These general tools allow us to derive simple formulae for the effective transport coefficients, leading to significant generalisations of previous homogenization approaches. Here, we find that different jump rates in the layers gives rise to a net bias, leading to a non-zero advection, for the entire homogenized system. Example calculations show that our generalized approach can lead to very different outcomes than traditional approaches, thereby having the potential to significantly affect simulation studies that use homogenization approximations.Comment: 9 pages, 2 figures, accepted version of paper published in The Journal of Chemical Physic

Atomic force microscopy measurements of topography and friction on dotriacontane films absorbed on a SiO2 surface [abstract]

Faculty Mentor: Dr. Haskell Taub, Physics & AstronomyAbstract only availableWe report comprehensive atomic force microscopy (AFM) measurements of the nanoscale topography and lateral friction on the surface of thin films of an intermediate-length normal alkane, dotriacontane, at room temperature. The aim of these studies is twofold: (i) to study the film structure by a direct imaging technique and compare it with that inferred from previous x-ray scattering results on the same films; and (ii) to determine the lateral frictional force exerted on the tip of the microscope cantilever and correlate this with the topography. As a prototypical system, we chose dotriacontane (n-C32H66) films adsorbed from a heptane (n-C7H16) solution onto SiO2-coated Si(100) single-crystal substrates whose structure we had previously characterized. Our topographic and frictional images simultaneously recorded in the contact mode reveal a multilayer structure of the dotriacontane films. Nearest the SiO2 surface, we observe one or two layers of molecules that are oriented with their long axis parallel to the interface. Above this “parallel film,” solid layers adsorb in which the molecules are oriented perpendicular to the surface. Regardless of the concentration of the initial dotriacontane-heptane solution, all of the perpendicular layers are incomplete with higher layers having progressively smaller area. The thickness of a perpendicular layer that we measure with AFM agrees with that inferred from previous x-ray specular reflectivity measurements. We also observe bulk dotriacontane particles and, in contrast to our previous measurements, are able to determine their location relative to the parallel and perpendicular layers. The particles have a terrace structure on top that indicates their molecules are oriented nearly perpendicular to the SiO2 surface. We find that the lateral friction is sensitive to the molecular orientation in the underlying crystalline film and can be used effectively with topographic measurements to resolve uncertainties in the film structure.MU Undergraduate Research Scholars Progra

Defending Truthmaker Non-Maximalism

This is the final published version, originally published by Wiley at http://onlinelibrary.wiley.com/journal/10.1002/%28ISSN%292161-2234.Jago (2012) argues that truthmaker non-maximalism, the view that some but not all truths require truthmakers, is vulnerable to a challenge from truths which ascribe knowledge of propositions about things which don’t exist. Such truths, Jago argues, can only be dealt with using maximalist resources. I argue that Jago’s point relies on the claim that the relevant truths require truthmakers, a point that non-maximalists can coherently and plausibly deny. Moreover, I argue that by making use of a safety account of knowledge, non-maximalists can fully answer Jago’s challenge.AHR

O man do not scribble on the book : print and counter-print in a Scottish Englightenment university

The University of St. Andrews in the latter half of the eighteenth century was a small and under-funded institution. Nevertheless, it shared in the changes and accomplishments that made Scottish intellect in that period a European phenomenon. Some evidence of the university's intellectual vitality at that time can be seen in the very full records that survive from its contemporary Library: records of the Library's administration and of its everyday business, and also such records as the books themselves represent. It is with this last sort of evidence that I will be mainly concerned here. The books at St. Andrews unwittingly preserve a remarkable corpus of marginalia added by the students. In this article, I hope to relate these student writings to their educational context. Making use of the distinctions that Walter Ong has so instructively drawn between print, manuscript, and oral habits of mind, I will suggest that the marginalia oppose the Enlightenment ideology of their university with the values of an older style of discourse.Note, quotation marks removed from title to ensure alphabetical order. Difference as follows; "O man do not scribble on the book": Print and Counter-print in a Scottish Enlightenment Universit

Reversible signal transmission in an active mechanical metamaterial

Mechanical metamaterials are designed to enable unique functionalities, but are typically limited by an initial energy state and require an independent energy input to function repeatedly. Our study introduces a theoretical active mechanical metamaterial that incorporates a biological reaction mechanism to overcome this key limitation of passive metamaterials. Our material allows for reversible mechanical signal transmission, where energy is reintroduced by the biologically motivated reaction mechanism. By analysing a coarse grained continuous analogue of the discrete model, we find that signals can be propagated through the material by a travelling wave. Analysis of the continuum model provides the region of the parameter space that allows signal transmission, and reveals similarities with the well-known FitzHugh-Nagumo system. We also find explicit formulae that approximate the effect of the timescale of the reaction mechanism on the signal transmission speed, which is essential for controlling the material.Comment: 20 pages, 7 figure

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterising stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealisations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with MATLAB implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community

A sharp-front moving boundary model for malignant invasion

We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells in both populations undergo diffusive migration and logistic proliferation. The interface between the two populations moves according to a two-phase Stefan condition. Unlike many reaction-diffusion models of malignant invasion, the moving boundary model explicitly describes the motion of the sharp front between the cancer and surrounding tissues without needing to introduce degenerate nonlinear diffusion. Numerical simulations suggest the model gives rise to very interesting travelling wave solutions that move with speed $c$, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative $x$-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed $c \ge c^* > 0$, the moving boundary model leads to travelling wave solutions with $|c| < c^{**}$. We interpret these travelling wave solutions in the phase plane and show that they are associated with several features of the classical Fisher-Kolmogorov phase plane that are often disregarded as being nonphysical. We show, numerically, that the phase plane analysis compares well with long time solutions from the full partial differential equation model as well as providing accurate perturbation approximations for the shape of the travelling waves.Comment: 48 pages, 21 figure