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Articles 1 - 17 of 17
Full-Text Articles in Physical Sciences and Mathematics
A Note On Solid Coloring Of Pure Simplicial Complexes, Joseph O'Rourke
A Note On Solid Coloring Of Pure Simplicial Complexes, Joseph O'Rourke
Computer Science: Faculty Publications
We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in Rd may be colored with d+1 colors so that no two simplices that share a (d-1)-facet have the same color. In R2 this says that any planar map all of whose faces are triangles may be 3-colored, and in R3 it says that tetrahedra in a collection may be "solid 4-colored" so that no two glued face-to-face receive the same color.
Slider-Pinning Rigidity: A Maxwell-Laman-Type Theorem, Ileana Streinu, Louis Theran
Slider-Pinning Rigidity: A Maxwell-Laman-Type Theorem, Ileana Streinu, Louis Theran
Computer Science: Faculty Publications
We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley’s direction networks.
Smoothness Of Lipschitz Minimal Intrinsic Graphs In Heisenberg Groups ℍN, N > 1, Luca Capogna, Giovanna Citti, Maria Manfredini
Smoothness Of Lipschitz Minimal Intrinsic Graphs In Heisenberg Groups ℍN, N > 1, Luca Capogna, Giovanna Citti, Maria Manfredini
Mathematics Sciences: Faculty Publications
We prove that Lipschitz intrinsic graphs in the Heisenberg groups ℍn, with n > 1, which are vanishing viscosity solutions of the minimal surface equation, are smooth and satisfy the PDE in a strong sense.
Flat Zipper-Unfolding Pairs For Platonic Solids, Joseph O'Rourke
Flat Zipper-Unfolding Pairs For Platonic Solids, Joseph O'Rourke
Computer Science: Faculty Publications
We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.
Curvedland: An Applet For Illustrating Curved Geometry Without Embedding, Gary Felder, Stephanie Erickson
Curvedland: An Applet For Illustrating Curved Geometry Without Embedding, Gary Felder, Stephanie Erickson
Physics: Faculty Publications
We have written a Java applet to illustrate the meaning of curved geometry. The applet provides a mapping interface similar to MapQuest or Google Maps; features include the ability to navigate through a space and place permanent point objects and/or shapes at arbitrary positions. The underlying two-dimensional space has a constant, positive curvature, which causes the apparent paths and shapes of the objects in the map to appear distorted in ways that change as you view them from different relative angles and distances.
Convexity And Horizontal Second Fundamental Forms For Hypersurfaces In Carnot Groups, Luca Capogna, Scott D. Pauls, Jeremy T. Tyson
Convexity And Horizontal Second Fundamental Forms For Hypersurfaces In Carnot Groups, Luca Capogna, Scott D. Pauls, Jeremy T. Tyson
Mathematics Sciences: Faculty Publications
We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli-Garofalo- Nhieu or Lu-Manfredi-Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph.
On Folding A Polygon To A Polyhedron, Joseph O'Rourke
On Folding A Polygon To A Polyhedron, Joseph O'Rourke
Computer Science: Faculty Publications
We show that the open problem presented in "Geometric Folding Algorithms: Linkages, Origami, Polyhedra" [DO07] is solved by a theorem of Burago and Zalgaller [BZ96] from more than a decade earlier.
Poset Pinball, Gkm-Compatible Subspaces, And Hessenberg Varieties, Megumi Harada, Julianna Tymoczko
Poset Pinball, Gkm-Compatible Subspaces, And Hessenberg Varieties, Megumi Harada, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a …
On Flat Polyhedra Deriving From Alexandrov's Theorem, Joseph O'Rourke
On Flat Polyhedra Deriving From Alexandrov's Theorem, Joseph O'Rourke
Computer Science: Faculty Publications
We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov's gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n3) time for polygons whose gluings are specified by n labels.
Star Unfolding Convex Polyhedra Via Quasigeodesic Loops, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu
Star Unfolding Convex Polyhedra Via Quasigeodesic Loops, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu
Computer Science: Faculty Publications
We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron ℘ to a simple (nonoverlapping) planar polygon: cut along one shortest path from each vertex of ℘ toQ, and cut all but one segment of Q.
The Yao Graph Y6 Is A Spanner, Joseph O'Rourke
The Yao Graph Y6 Is A Spanner, Joseph O'Rourke
Computer Science: Faculty Publications
We prove that Y6 is a spanner. Y6 is the Yao graph on a set of planar points, which has an edge from each point x to a closest point y within each of the six angular cones of 60◦ surrounding x .
Schubert Polynomials And Classes Of Hessenberg Varieties, Dave Anderson, Julianna Tymoczko
Schubert Polynomials And Classes Of Hessenberg Varieties, Dave Anderson, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a " Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, …
Lie Algebroid Structures On Double Vector Bundles And Representation Theory Of Lie Algebroids, Alfonso Gracia-Saz, Rajan Amit Mehta
Lie Algebroid Structures On Double Vector Bundles And Representation Theory Of Lie Algebroids, Alfonso Gracia-Saz, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.
A Sharp Diameter Bound For Unipotent Groups Of Classical Type Overℤ /Pℤ, Jordan S. Ellenberg, Julianna Tymoczko
A Sharp Diameter Bound For Unipotent Groups Of Classical Type Overℤ /Pℤ, Jordan S. Ellenberg, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
The unipotent subgroup of a finite group of Lie type over a prime field Fp comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n2 log p, where n is the rank of the associated Lie group. This generalizes the result of Ellenberg, A sharp diameter bound for an upper triangular matrix group, Harvard University, 1993, which treated the case of …
Highway Hull Revisited, Greg Aloupis, Jean Cardinal, Sébastien Collette, Ferran Hurtado, Stefan Langerman, Joseph O'Rourke, Belén Palop
Highway Hull Revisited, Greg Aloupis, Jean Cardinal, Sébastien Collette, Ferran Hurtado, Stefan Langerman, Joseph O'Rourke, Belén Palop
Computer Science: Faculty Publications
A highway H is a line in the plane on which one can travel at a greater speed than in the remaining plane. One can choose to enter and exit H at any point. The highway time distance between a pair of points is the minimum time required to move from one point to the other, with optional use of H. The highway hull H(S,H) of a point set S is the minimal set containing S as well as the shortest paths between all pairs of points in H(S,H), using the highway time distance. We provide a Θ(nlogn) worst-case …
On Baryon Number Non-Conservation In Two-Dimensional O(2n+1) Qcd, Tamar Friedmann
On Baryon Number Non-Conservation In Two-Dimensional O(2n+1) Qcd, Tamar Friedmann
Mathematics Sciences: Faculty Publications
We construct a classical dynamical system whose phase space is a certain infinite dimensional Grassmannian manifold, and propose that it is equivalent to the large N limit of two-dimensional QCD with an O(2N + 1) gauge group. In this theory, we find that baryon number is a topological quantity that is conserved only modulo 2. We also relate this theory to the master field approach to matrix models.
Morphing Of Triangular Meshes In Shape Space, Stefanie Wuhrer, Prosenjit Bose, Chang Shu, Joseph O'Rourke, Alan Brunton
Morphing Of Triangular Meshes In Shape Space, Stefanie Wuhrer, Prosenjit Bose, Chang Shu, Joseph O'Rourke, Alan Brunton
Computer Science: Faculty Publications
We present a novel approach to morph between two isometric poses of the same non-rigid object given as triangular meshes. We model the morphs as linear interpolations in a suitable shape space S. For triangulated 3D polygons, we prove that interpolating linearly in this shape space corresponds to the most isometric morph in R3 . We then extend this shape space to arbitrary triangulations in 3D using a heuristic approach and show the practical use of the approach using experiments. Furthermore, we discuss a modified shape space that is useful for isometric skeleton morphing. All of the newly presented …