Open Access. Powered by Scholars. Published by Universities.®
- Institution
- Keyword
-
- Tiling (2)
- 4-dimensional manifold (1)
- Absolute neighborhood retract (1)
- Bar framework (1)
- Bochner's theorem (1)
-
- Calibrated geometry (1)
- Cell-like mapping (1)
- Codimension 3 (1)
- Fine structure subgroup (1)
- Geodesic Tiling (1)
- Geometric permutation (1)
- Hyperbolic geometry (1)
- Kaleidoscopic Tiling (1)
- Kasparov KK-groups (1)
- Knots (1)
- Line transversal (1)
- Locally simply co-connected (1)
- Locked chains (1)
- Pfaffian systems (1)
- Polonais group (1)
- Polygonal chain (1)
- Pontryagin form (1)
- Positive definite functions (1)
- Pseudopolonais group (1)
- Quasi-unital (1)
- Quasihomomorphism (1)
- Shrinking theorem (1)
- Stabbing (1)
- Systole (1)
- Tiling Group (1)
Articles 1 - 11 of 11
Full-Text Articles in Geometry and Topology
Polygonal Chains Cannot Lock In 4d, Roxana Cocan, Joseph O'Rourke
Polygonal Chains Cannot Lock In 4d, Roxana Cocan, Joseph O'Rourke
Computer Science: Faculty Publications
We prove that, in all dimensions d ≥ 4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polynomial number of “moves.” These results contrast to those known for d = 2, where trees can “lock,” and for d = 3, where open and closed chains can lock.
Computational Geometry Column 42, Joseph S. B. Mitchell, Joseph O'Rourke
Computational Geometry Column 42, Joseph S. B. Mitchell, Joseph O'Rourke
Computer Science: Faculty Publications
A compendium of thirty previously published open problems in computational geometry is presented.
The Fine Structure Of The Kasparov Groups I: Continuity Of The Kk-Pairing, Claude Schochet
The Fine Structure Of The Kasparov Groups I: Continuity Of The Kk-Pairing, Claude Schochet
Mathematics Faculty Research Publications
In this paper it is demonstrated that the Kasparov pairing is continuous with respect to the natural topology on the Kasparov groups, so that a KK-equivalence is an isomorphism of topological groups. In addition, we demonstrate that the groups have a natural pseudopolonais structure, and we prove that various KK-structural maps are continuous.
Uniqueness Of Volume-Minimizing Submanifolds Calibrated By The First Pontryagin Form, Daniel A. Grossman, Weiqing Gu
Uniqueness Of Volume-Minimizing Submanifolds Calibrated By The First Pontryagin Form, Daniel A. Grossman, Weiqing Gu
All HMC Faculty Publications and Research
One way to understand the geometry of the real Grassmann manifold Gk(Rk+n) parameterizing oriented k-dimensional subspaces of Rk+n is to understand the volume-minimizing subvarieties in each homology class. Some of these subvarieties can be determined by using a calibration. In previous work, one of the authors calculated the set of 4-planes calibrated by the first Pontryagin form p1 on Gk(Rk+n) for all k,n ≥4, and identified a family of mutually congruent round 4-spheres which are consequently homologically volume-minimizing. In the present work, we associate to the family of calibrated …
Computational Geometry Column 41, Joseph O'Rourke
Computational Geometry Column 41, Joseph O'Rourke
Computer Science: Faculty Publications
The recent result that n congruent balls in Rd have at most 4 distinct geometric permutations is described.
Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot
Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot
Mathematical Sciences Technical Reports (MSTR)
Consider a surface, S, with a kaleidoscopic tiling by non-obtuse triangles (tiles), i.e., each local reflection in a side of a triangle extends to an isometry of the surface, preserving the tiling. The tiling is geodesic if the side of each triangle extends to a closed geodesic on the surface consisting of edges of tiles. The reflection group G*, generated by these reflections, is called the tiling group of the surface. This paper classifies, up to isometry, all geodesic, kaleidoscopic tilings by triangles, of hyperbolic surfaces of genus up to 13. As a part of this classification the tiling groups …
Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods
Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods
Mathematical Sciences Technical Reports (MSTR)
The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. We determine if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, we determine if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles π/2 , π/3 , and π/7 whose tiling groups are all PSL2(13). These tilings can be distinguished by the lengths of their systoles.
Extended Powers Of Manifolds And The Adams Spectral Sequence, Robert R. Bruner
Extended Powers Of Manifolds And The Adams Spectral Sequence, Robert R. Bruner
Mathematics Faculty Research Publications
The extended power construction can be used to create new framed manifolds out of old. We show here how to compute the effect of such operations in the Adams spectral sequence, extending partial results of Milgram and the author. This gives the simplest method of proving that Jones’ 30-manifold has Kervaire invariant one, and allows the construction of manifolds representing Mahowald’s classes η4 and η5, among others.
Strictly Positive Definite Functions On A Compact Group, Mohamed Allali, Tomasz Przebinda
Strictly Positive Definite Functions On A Compact Group, Mohamed Allali, Tomasz Przebinda
Mathematics, Physics, and Computer Science Faculty Articles and Research
We recognize a result of Schreiner, concerning strictly positive definite functions on a sphere in an Euclidean space, as a generalization of Bochner's theorem for compact groups.
A 4-Dimensional 1-Lcc Shrinking Theorem, Mladen Bestvina, Robert J. Daverman, Gerard A. Venema
A 4-Dimensional 1-Lcc Shrinking Theorem, Mladen Bestvina, Robert J. Daverman, Gerard A. Venema
University Faculty Publications and Creative Works
This paper contains several shrinking theorems for decompositions of 4-dimensional manifolds. Let f : M → X be a closed, cell-like mapping of a 4-manifold M onto a metric space X and let Y be a closed subset of X such that X - Y is a 4-manifold and Y is locally simply co-connected in X. The main result states that f can be approximated by homeomorphisms if Y is a 1-dimensional ANR. The techniques of the proof also show that f can be approximated by homeomorphisms in case Y is an arbitrary 0-dimensional closed subset. Combining the two results …
On The Existence Of Finite Type Link Homotopy Invariants, Blake Mellor, Dylan Thurston
On The Existence Of Finite Type Link Homotopy Invariants, Blake Mellor, Dylan Thurston
Mathematics, Statistics and Data Science Faculty Works
We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of the first author.