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Full-Text Articles in Physical Sciences and Mathematics
Regularity Of Non-Characteristic Minimal Graphs In The Heisenberg Group ℍ1, Luca Capogna, Giovanna Citti, Maria Manfredini
Regularity Of Non-Characteristic Minimal Graphs In The Heisenberg Group ℍ1, Luca Capogna, Giovanna Citti, Maria Manfredini
Mathematics Sciences: Faculty Publications
Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are apriori estimates on the solutions of the approximating Riemannian PDE and the ensuing C∞ regularity of the sub-Riemannian minimal surface along its Legendrian foliation.
Using Labeled Data To Evaluate Change Detectors In A Multivariate Streaming Environment, Albert Y. Kim, Caren Marzban, Donald B. Percival, Werner Stuetzle
Using Labeled Data To Evaluate Change Detectors In A Multivariate Streaming Environment, Albert Y. Kim, Caren Marzban, Donald B. Percival, Werner Stuetzle
Statistical and Data Sciences: Faculty Publications
We consider the problem of detecting changes in a multivariate data stream. A change detector is defined by a detection algorithm and an alarm threshold. A detection algorithm maps the stream of input vectors into a univariate detection stream. The detector signals a change when the detection stream exceeds the chosen alarm threshold. We consider two aspects of the problem: (1) setting the alarm threshold and (2) measuring/comparing the performance of detection algorithms. We assume we are given a segment of the stream where changes of interest are marked. We present evidence that, without such marked training data, it might …
Sparse Hypergraphs And Pebble Game Algorithms, Ileana Streinu, Louis Theran
Sparse Hypergraphs And Pebble Game Algorithms, Ileana Streinu, Louis Theran
Computer Science: Faculty Publications
A hypergraph G=(V,E) is (k,ℓ)-sparse if no subset V′⊂V spans more than k|V′|−ℓ hyperedges. We characterize (k,ℓ)-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behavior in terms of the sparsity parameters k and ℓ. Our constructions extend the pebble games of Lee and Streinu [A. Lee, I. Streinu, Pebble game algorithms …
Q-Groupoids And Their Cohomology, Rajan Amit Mehta
Q-Groupoids And Their Cohomology, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
We approach Mackenzie's L{script}A{script}-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of L{script}A{script}-groupoids to the category of Q-groupoids. We associate to every Qgroupoid a double complex that provides a model for the Q-cohomology of the classifying space. As examples, we obtain models for equivariant Q-and orbifold Q-cohomology, and for equivariant Lie algebroid and orbifold Lie algebroid cohomology. We obtain double complexes associated to Poisson groupoids and groupoid-algebroid "matched pairs".
Transition To Mixing And Oscillations In A Stokesian Viscoelastic Flow, Becca Thomases, Michael Shelley
Transition To Mixing And Oscillations In A Stokesian Viscoelastic Flow, Becca Thomases, Michael Shelley
Mathematics Sciences: Faculty Publications
In seeking to understand experiments on low-Reynolds-number mixing and flow transitions in viscoelastic fluids, we simulate the dynamics of the Oldroyd-B model, with a simple background force driving the flow. We find that at small Weissenberg number, flows are "slaved" to the extensional geometry imposed by forcing. For large Weissenberg number, such solutions become unstable and transit to a structurally dissimilar state dominated by a single large vortex. This new state can show persistent oscillatory behavior with the production and destruction of smaller-scale vortices that drive mixing.
Generalized Mean Curvature Flow In Carnot Groups, Luca Capogna, Giovanna Citti
Generalized Mean Curvature Flow In Carnot Groups, Luca Capogna, Giovanna Citti
Mathematics Sciences: Faculty Publications
In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4] and [12]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.
Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke
Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke
Computer Science: Faculty Publications
The Yao graph for k = 4, Y4, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners.
Sparsity-Certifying Graph Decompositions, Ileana Streinu, Louis Theran
Sparsity-Certifying Graph Decompositions, Ileana Streinu, Louis Theran
Computer Science: Faculty Publications
We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k …
Grünbaum Colorings Of Toroidal Triangulations, Michael O. Albertson, Hannah Alpert, Sarah-Marie Belcastro, Ruth Haas
Grünbaum Colorings Of Toroidal Triangulations, Michael O. Albertson, Hannah Alpert, Sarah-Marie Belcastro, Ruth Haas
Mathematics Sciences: Faculty Publications
We prove that if G is a triangulation of the torus and χ(G) 6 ≠ 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors.
Q-Algebroids And Their Cohomology, Rajan Amit Mehta
Q-Algebroids And Their Cohomology, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
A Q-algebroid is a graded Lie algebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a Q-groupoid. We associate to every Q-algebroid a double complex. As a special case, we define the Becchi-Rouet-Stora-Tyutin (BRST) model of a Lie algebroid, which generalizes the BRST model for equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman isomorphism of the BRST and Weil models, and we suggest a definition of a basic subcomplex which, however, requires a choice of a connection. Other examples include Roytenberg's homological double of a Lie bialgebroid, Ginzburg's model of equivariant Lie algebroid …