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Articles 1 - 17 of 17
Full-Text Articles in Mathematics
Rental Harmony: Sperner's Lemma In Fair Division, Francis E. Su
Rental Harmony: Sperner's Lemma In Fair Division, Francis E. Su
All HMC Faculty Publications and Research
No abstract provided in this article.
Computational Geometry Column 36, Joseph O'Rourke
Computational Geometry Column 36, Joseph O'Rourke
Computer Science: Faculty Publications
Two results in "computational origami" are illustrated.
Pushpush Is Np-Hard In 3d, Joseph O'Rourke, The Smith Problem Solving Group
Pushpush Is Np-Hard In 3d, Joseph O'Rourke, The Smith Problem Solving Group
Computer Science: Faculty Publications
We prove that a particular pushing-blocks puzzle is intractable in 3D. The puzzle, inspired by the game PushPush, consists of unit square blocks on an integer lattice. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover, each block, when pushed, slides the maximal extent of its free range. We prove this version is NP-hard in 3D by reduction from SAT. The corresponding problem in 2D remains open.
Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton
Splitting Tiled Surfaces With Abelian Conformal Tiling Group, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Let p be a reflection on a closed Riemann Surface S, i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sp denote the fixed point subset of p, which is also called the mirror of p. If S −Sp has two components, then p is called separating and we say that S splits at the mirror Sp. Otherwise p is called non-separating. We assume that the system of mirrors, Sq, as q varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines …
Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith
Divisible Tilings In The Hyperbolic Plane, Sean A. Broughton, Dawn M. Haney, Lori T. Mckeough, Brandy M. Smith
Mathematical Sciences Technical Reports (MSTR)
We consider triangle-quadrilateral pairs in the hyperbolic plane which "kaleidoscopically" tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1,2, and 3 parameter families as well as a finite number of exceptional cases.
Art And Geometry: Proportion And Similarity, Catherine A. Gorini
Art And Geometry: Proportion And Similarity, Catherine A. Gorini
Humanistic Mathematics Network Journal
No abstract provided.
Zero-Parity Stabbing Information, Joseph O'Rourke, Irena Pashchenko
Zero-Parity Stabbing Information, Joseph O'Rourke, Irena Pashchenko
Computer Science: Faculty Publications
Everett et al. [EHN96, EHN97] introduced several varieties of stabbing information for the lines determined by pairs of vertices of a simple polygon P, and established their relationships to vertex visibility and other combinatorial data. In the same spirit, we define the “zero-parity (ZP) stabbing information” to be a natural weakening of their “weak stabbing information,” retaining only the distinction among {zero, odd, even > 0} in the number of polygon edges stabbed. Whereas the weak stabbing information’s relation to visibility remains an open problem, we completely settle the analogous questions for zero parity information, with three results: (1) ZP information …
Tilings Which Split A Mirror, Jim Belk
Tilings Which Split A Mirror, Jim Belk
Mathematical Sciences Technical Reports (MSTR)
We consider the mirror of a reflection which consists of its subset of fixed points. We investigate a number of conditions on the tiling that guarantee that the surface splits at a mirror.
Ossa's Theorem And Adams Covers, Robert R. Bruner
Ossa's Theorem And Adams Covers, Robert R. Bruner
Mathematics Faculty Research Publications
We show that Ossa’s theorem splitting ku ∧ BV for elementary abelian groups V follows from general facts about ku ∧ BZ/2 and Adams covers. For completeness, we also provide the analogous results for ko ∧ BV .
Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton
Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
We have all seen many of the beautiful patterns obtained by tiling the hyperbolic plane H by repeated reflection in the sides of a "kaleidoscopic" polygon. Though there are such patterns on the sphere and the euclidean plane, these positively curved and fiat geometries lack the richness we see in the hyperbolic plane. Many of these patterns have been popularized by the beautiful art of M.C. Escher. For a list of references and a more complete discussion on the construction of artistic tilings see [6].
Locked And Unlocked Polygonal Chains In 3d, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Mark Overmars, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides
Locked And Unlocked Polygonal Chains In 3d, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Mark Overmars, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides
Computer Science: Faculty Publications
In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a …
The Topological Snake Lemma And Corona Algebras, Claude Schochet
The Topological Snake Lemma And Corona Algebras, Claude Schochet
Mathematics Faculty Research Publications
We establish versions of the Snake Lemma from homological algebra in the context of topological groups, Banach spaces, and operator algebras. We apply this tool to demonstrate that if ƒ : B → B′ is a quasi-unital C*-map of separable C*-algebras, so that it induces a map of Corona algebras ƒ̄ : QB → QB′, and if ƒ is mono, then the induced map ƒ̄ is also mono.
Intersection Homology Of Toric Varieties And A Conjecture Of Kalai, Tom Braden, D. Macpherson
Intersection Homology Of Toric Varieties And A Conjecture Of Kalai, Tom Braden, D. Macpherson
Tom Braden
No abstract provided.
Geometrical Models For Grain Dynamics, Giovani L. Vasconcelos, J. J. P. Veerman
Geometrical Models For Grain Dynamics, Giovani L. Vasconcelos, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
We study models for the gravity-driven, dissipative motion of a single grain on an inclined rough surface. Imposing some conditions on the momentum loss due to the collisions between the particle and the surface, we arrive at a class of models in which the grain dynamics is described by one-dimensional maps. The dynamics of these maps is studied in detail. We prove the existence of various dynamical phases and show that the presence of these phases is independent of the restitution law (within the class considered).
Knot Theory And Wild Knots, Cherie Annette Reardon
Knot Theory And Wild Knots, Cherie Annette Reardon
Theses Digitization Project
No abstract provided.
Finite Type Link Homotopy Invariants, Blake Mellor
Finite Type Link Homotopy Invariants, Blake Mellor
Mathematics Faculty Works
Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the pairwise linking numbers of the components.
Computational Geometry Column 35, Joseph O'Rourke
Computational Geometry Column 35, Joseph O'Rourke
Computer Science: Faculty Publications
The subquadratic algorithm of Kapoor for finding shortest paths on a polyhedron is described.