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The Surface Diffusion And The Willmore Flow For Uniformly Regular Hypersurfaces, Jeremy Lecrone, Yuanzhen Shao, Gieri Simonett
The Surface Diffusion And The Willmore Flow For Uniformly Regular Hypersurfaces, Jeremy Lecrone, Yuanzhen Shao, Gieri Simonett
Department of Math & Statistics Faculty Publications
We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are C1+α–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are C1+α–close to a sphere, and we prove …
A Template For Success: Celebrating The Work Of Judith Grabiner, Della Dumbaugh, Adrian Rice
A Template For Success: Celebrating The Work Of Judith Grabiner, Della Dumbaugh, Adrian Rice
Department of Math & Statistics Faculty Publications
Judith Grabiner is a mathematician who specializes in the history of mathematics. She is currently the Flora Sanborn Pitzer Professor Emerita of Mathematics at Pitzer College, one of the Claremont Colleges in Claremont, California. She has authored more than forty articles, as well as three books: The Origins of Cauchy’s Rigorous Calculus (1981), The Calculus as Algebra: J.-L. Lagrange, 1736–1813 (1990), and A Historian Looks Back: The Calculus as Algebra and Selected Writings (2010), which won the Beckenbach Prize from the Mathematical Association of America in 2014. She deliv- ered an invited address titled “The Centrality of Mathemat- ics in …
On Quasilinear Parabolic Equations And Continuous Maximal Regularity, Jeremy Lecrone, Gieri Simonett
On Quasilinear Parabolic Equations And Continuous Maximal Regularity, Jeremy Lecrone, Gieri Simonett
Department of Math & Statistics Faculty Publications
We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.
Almost Difference Sets In 2-Groups, Xin Yutong
Almost Difference Sets In 2-Groups, Xin Yutong
Honors Theses
Difference sets have been studied for decades due to their applications in digital communication, cryptography, algebra, and number theory. More recently, mathematicians have expanded their focus to the field of almost difference sets. Almost difference sets have similar functionalities with difference sets, yet with more potential of finding new constructions. In this paper I will introduce the definitions, properties, and applications of difference sets and almost difference sets, and discuss our effort and results in the exploration of almost difference sets in cyclic and non-cyclic groups.
Internal Migration Of Foreign-Born In Us: Impacts Of Population Concentration And Risk Aversion, Thin Yee Mon Su
Internal Migration Of Foreign-Born In Us: Impacts Of Population Concentration And Risk Aversion, Thin Yee Mon Su
Honors Theses
Internal migration in the US has been declining since the 1990s and research has mostly focused on labor market dynamics and aging population to explain the migration trends. This paper analyzes migration patterns of foreign-born groups in the US from 2000 to 2019. Along with the migration determinants such as education and employment, the paper focuses on population concentration as a factor that shapes foreign-born decisions to relocate in the US. Population concertation is defined to be a measure of how geographically concentrated each foreign-born group is across the US. I find that the likelihood of migrating to another state …
Biasing Medial Axis Rapidly-Exploring Random Trees With Safe Hyperspheres, David Qin
Biasing Medial Axis Rapidly-Exploring Random Trees With Safe Hyperspheres, David Qin
Honors Theses
Motion planning is a challenging and widely researched problem in robotics. Motion planning algorithms aim to not only nd unobstructed paths, but also to construct paths with certain qualities, such as maximally avoiding obstacles to improve path safety. One such solution is a Rapidly-Exploring Random Tree (RRT) variant called Medial Axis RRT that generates the safest possible paths, but does so slowly. This paper introduces a RRT variant called Medial Axis Ball RRT (MABallRRT) that uses the concept of clearance -- a robot's distance from its nearest obstacle -- to efficiently construct a roadmap with safe paths. The safety of …
Fast Medial Axis Sampling For Use In Motion Planning, Hanglin Zhou
Fast Medial Axis Sampling For Use In Motion Planning, Hanglin Zhou
Honors Theses
Motion planning is a difficult but important problem in robotics. Research has tended toward approximations and randomized algorithms, like sampling-based planning. Probabilistic RoadMaps (PRMs) are one common sampling-based planning approach, but they lack safety guarantees. One main approach, Medial Axis PRM (MAPRM) addressed this deficiency by generating robot configurations as far away from the obstacles as possible, but it introduced an extensive computational burden. We present two techniques, Medial Axis Bridge and Medial Axis Spherical Step, to reduce the computational cost of sampling in MAPRM and additionally propose recycling previously computed clearance information to reduce the cost of connection in …
Estimating Value-At-Risk Of An Unconventional Portfolio, Elizabeth N. Mejía-Ricart
Estimating Value-At-Risk Of An Unconventional Portfolio, Elizabeth N. Mejía-Ricart
Honors Theses
Since the 2008 financial crisis, interest rates and bond yields have been low all through the recovery and expansion that followed, and they are still low. As a result, more investors have been attracted to US equities, a space of possibly higher returns. However, these returns come with a potential downside: risk of loss. One of the methods to assess this potential downside is value-at-risk (VaR), which gained momentum in the late 1990s. At the time, the market risk amendment to the 1988 Basle Capital Accord required commercial banks with significant trading activities to put aside capital to cover market …
Computer-Assisted Coloring-Graph Generation And Structural Analysis, Wesley Su
Computer-Assisted Coloring-Graph Generation And Structural Analysis, Wesley Su
Honors Theses
Graphs are a well studied construction in discrete math, with one of the most common areas of study being graph coloring. The graph coloring problem asks for a color to be assigned to each vertex in a graph such that no two adjacent vertices share a color. An assignment of k colors that meets these criteria is called a k-coloring. The coloring graph Ck(G) is defined as the graph where every vertex represents a valid k-coloring of graph G and edges exist between colorings that di↵er by one vertex. We call graph G the base graph of the k-coloring graph …