Abstract: Permafrost is a thick layer of soil that is frozen throughout the year and covers significant portions of the northern hemisphere. Currently, there is a large amount of carbon trapped in the permafrost, and as permafrost melts, a significant portion of this carbon will be released into the atmosphere as either carbon dioxide or methane. We use empirical data to estimate that, on average, permafrost currently extends from the arctic to latitude 61_N. We propose an adaption to the Budyko energy balance model to study the impacts of receding permafrost. We track the steady-state latitude of both the permafrost line and the snow line as greenhouse gas emissions, and consequently, global mean temperature increases. Using the change in permafrost surface area, we are able to quantify the total carbon feedback of melting permafrost. Focusing our analysis on scenarios described in recent IPCC reports and the Paris Climate Agreement, we use change in the permafrost line latitude to estimate the amount of carbon dioxide released by the melted permafrost. Similarly, we use the snow line to calculate the minimum average global temperature that would cause the ice caps to completely melt. We find that our adaption of the Budyko model produces estimates of carbon dioxide emissions within the range of projections of models with higher complexity.
Abstract: We exhibit an algorithm with continuous instructions for two robots moving without collisions on a track shaped as a wedge of three circles. We show that the topological complexity of the configuration space associated with this problem is 3. The topological complexity is a homotopy invariant that can be thought of as the minimum number of continuous instructions required to describe the movement of the robots between any initial configuration to any final one without collisions. The algorithm presented is optimal in the sense that it requires exactly 3 continuous instructions.
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Abstract: Quadratization is a transform of a system of ODEs with polynomial right-hand side into a system of ODEs with at most quadratic right-hand side via the introduction of new variables. It has been recently used as a preprocessing step for new model order reduction methods, so it is important to keep the number of new variables small. Several algorithms have been designed to search for a quadratization with the new variables being monomials in the original variables. To understand the limitations and potential ways of improving such algorithms, we study the following question: can quadratizations with not necessarily new monomial variables produce a model of substantially smaller dimension than quadratization with only new monomial variables?
Abstract: Full-waveform inversion (FWI) is a method used to determine properties of the Earth from information on the surface. We use the squared Wasserstein distance (squared W2 distance) as an objective function to invert for the velocity of seismic waves as a function of position in the Earth, and we discuss its convexity with respect to the velocity parameter. In one dimension, we consider constant, piecewise increasing, and linearly increasing velocity models as a function of position, and we show the convexity of the squared W2 distance with respect to the velocity parameter on the interval from zero to the true value of the velocity parameter when the source function is a probability measure. Furthermore, we consider a two-dimensional model where velocity is linearly increasing as a function of depth and prove the convexity of the squared W2 distance in the velocity parameter on large regions containing the true value. We discuss the convexity of the squared W2 distance compared with the convexity of the squared L2 norm, and we discuss the relationship between frequency and convexity of these respective distances. We also discuss multiple approaches to optimal transport for non-probability measures by first converting the wave data into probability measures.
Abstract: This paper compares the advantages, limitations, and computational considerations of using Finite-Time Lyapunov Exponents (FTLEs) and Lagrangian Descriptors (LDs) as tools for identifying barriers and mechanisms of fluid transport in two-dimensional time-periodic ows. These barriers and mechanisms of transport are often referred to as Lagrangian Coherent Structures," though this term often changes meaning depending on the author or context. This paper will specifically focus on using FTLEs and LDs to identify stable and unstable manifolds of hyperbolic stagnation points, and the Kolmogorov-Arnold-Moser (KAM) tori associated with elliptic stagnation points. The background and theory behind both methods and their associated phase space structures will be presented, and then examples of FTLEs and LDs will be shown based on a simple, periodic, time-dependent double-gyre toy model with varying parameters.
Abstract: The analytical solution for the large deformation of a cantilever beam under a point load, typically applied to the tip of the cantilever and perpendicularly to its axis, has been widely studied and published. However, the more complex case of two angled point loads applied to the cantilever has not been published. The current research delved into the following scenario: an upright cantilever, e.g. a pole, has point loads applied at two locations on the cantilever, where each point load is angled, i.e. the point load has both a horizontal component (which may result from wind loading) and downward vertical component of force (such as from weights). The aim of the research is to develop a methodology for finding,
At the two locations where the point loads are applied, the angle of deflection, the horizontal deflection, and the vertically deflected height. Ultimately, the research yielded a methodology based on the Complete and Incomplete Elliptic Integrals of the First Kind and Second Kind. The analytical solution developed in this research - specifically the method for calculating the angles of deflections - was compared against Finite Element Analysis and was found to produce nearly identical results. We conclude that the methodology shown can be extended to any number of point loads and will be a contribution to the field of non-linear mechanics.
Abstract: Since the end of 2019, COVID-19 has threatened human life around the globe. As the death toll continues to rise, development of vaccines and antiviral treatments have progressed at unprecedented speeds. This paper uses an SIR-type model, extended to include asymptomatic carrier and deceased populations as a basis for expansion to the effects of a time-dependent drug or vaccine. In our model, a drug is administered to symptomatically infected individuals, decreasing recovery time and death rate. Alternatively, a vaccine is administered to susceptible individuals and, if effective, will move them into the recovered population. We observe final mortality outcomes of these countermeasures by running simulations across different release times with differing effectivenesses. As expected, the earlier the drug or vaccine is released into the population, the smaller the death toll. We find that for earlier release dates, difference in the quality of either treatment has a large effect on total deaths. However as their release is delayed, these differences become smaller. Finally, we find that a vaccine is much more effective than a drug when released early in an epidemic. However, when released after the peak of infections, a drug is marginally more effective in total lives
Abstract: COVID-19 epidemics in many parts of the United States and the world have shown unexpected shifts from exponential to linear growth in the number of daily new cases. Epidemics on configuration model networks typically produce exponential growth, while epidemics on lattices produce linear growth. We explore a network-based epidemic model that interpolates between lattice-like and configuration model networks while keeping the degree distribution and basic reproduction number (R0) constant. This model starts with nodes assigned random locations in a unit square and connected to their nearest neighbors. A proportion p of the edges are disconnected and reconnected in a configuration model subnetwork. As p increases, we observe a shift from linear to exponential growth. Realistic human contact networks involve many local interactions and fewer long-distance interactions, so social distancing affects both the effective reproduction number Rt and the proportion of long-distance connections in the network. While the impact of changes in Rt is well-understood, far less is understood about the effect of more subtle changes in network structure. Our analysis indicates that the threshold between linear and exponential growth may occur even with a small percentage of reconfigured edges. Additionally, the number of total infected individuals in an epidemic substantially increases around this threshold even when R0 remains constant. This study reveals that implementing and relaxing social distancing restrictions can have more complex and dramatic effects on epidemic dynamics than previously thought.
Abstract: As the world becomes increasingly reliant on the internet, from online schooling to working from home, broader and higher quality access has never been more important. However, expanding internet infrastructure presents a unique challenge in terms of cost, economic efficiency, and capacity requirements. Our team aims to optimize the process of improving connectivity by predicting the price of bandwidth over the next 10 years, calculating band-width needs for a variety of household scenarios, and determining the best distribution of cellular nodes over a given region.
Abstract: Consider a "forest" of infinitely thin trees arranged on the lattice Z x Z. If you are standing at the origin, (0; 0), not all trees are visible despite the fact that they are infinitely thin. In particular, of the trees all lying on a line through (0; 0), only one such point is visible. In this article we conclusively classify all closest occurring invisible rectangular n x m blocks of points for 1 _ n;m _ 4. This (partially) resolves a question posed by Goins-Harris-Kubik-Mbirika. Furthermore, we compile statistics for all occurring arrangements up to size 4_4 and discuss interesting patterns that appear in that data. 152ee80cbc
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