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University of Richmond

Department of Math & Statistics Faculty Publications

Series

2020

Articles 1 - 6 of 6

Full-Text Articles in Physical Sciences and Mathematics

The Surface Diffusion And The Willmore Flow For Uniformly Regular Hypersurfaces, Jeremy Lecrone, Yuanzhen Shao, Gieri Simonett Dec 2020

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 …

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Topology-Guided Roadmap Construction With Dynamic Region Sampling, Read Sandström, Diane Uwacu, Jory Denny, Nancy M. Amato Oct 2020

Topology-Guided Roadmap Construction With Dynamic Region Sampling, Read Sandström, Diane Uwacu, Jory Denny, Nancy M. Amato

Department of Math & Statistics Faculty Publications

Many types of planning problems require discovery of multiple pathways through the environment, such as multi-robot coordination or protein ligand binding. The Probabilistic Roadmap (PRM) algorithm is a powerful tool for this case, but often cannot efficiently connect the roadmap in the presence of narrow passages. In this letter, we present a guidance mechanism that encourages the rapid construction of well-connected roadmaps with PRM methods. We leverage a topological skeleton of the workspace to track the algorithm's progress in both covering and connecting distinct neighborhoods, and employ this information to focus computation on the uncovered and unconnected regions. We demonstrate …

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Asymptotically-Optimal Topological Nearest-Neighbor Filtering, Read Sandström, Jory Denny, Nancy M. Amato Oct 2020

Asymptotically-Optimal Topological Nearest-Neighbor Filtering, Read Sandström, Jory Denny, Nancy M. Amato

Department of Math & Statistics Faculty Publications

Nearest-neighbor finding is a major bottleneck for sampling-based motion planning algorithms. The cost of finding nearest neighbors grows with the size of the roadmap, leading to a significant computational bottleneck for problems which require many configurations to find a solution. In this work, we develop a method of mapping configurations of a jointed robot to neighborhoods in the workspace that supports fast search for configurations in nearby neighborhoods. This expedites nearest-neighbor search by locating a small set of the most likely candidates for connecting to the query with a local plan. We show that this filtering technique can preserve asymptotically-optimal …

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On Quasilinear Parabolic Equations And Continuous Maximal Regularity, Jeremy Lecrone, Gieri Simonett Mar 2020

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.

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A Template For Success: Celebrating The Work Of Judith Grabiner, Della Dumbaugh, Adrian Rice Mar 2020

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 …

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Score Following With Hidden Tempo Using A Switching State-Space Model, Yucong Jiang, Chris Raphael Jan 2020

Score Following With Hidden Tempo Using A Switching State-Space Model, Yucong Jiang, Chris Raphael

Department of Math & Statistics Faculty Publications

A score-following program traces the notes in a musical score during a performance. This capability is essential to many meaningful applications that synchronize audio with a score in an on-line fashion. Existing algorithms often stumble on certain difficult cases, one of which is piano music. This paper presents a new method to tackle such cases. The method treats tempo as a variable rather than a constant (with constraints), allowing the program to adapt to live performance variations. This is first expressed by a Kalman filter model at the note level, and then by an almost equivalent switching state-space model at …

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