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Articles 931 - 960 of 1060

Full-Text Articles in Geometry and Topology

The Connective K-Theory Of Finite Groups, Robert R. Bruner, John Greenlees Jan 2003

The Connective K-Theory Of Finite Groups, Robert R. Bruner, John Greenlees

Mathematics Faculty Research Publications

This paper is devoted to the connective K homology and cohomology of finite groups G. We attempt to give a systematic account from several points of view. In Chapter 1, following Quillen [50, 51], we use the methods of algebraic geometry to study the ring ku^*(BG) where ku denotes connective complex K-theory. We describe the variety in terms of the category of abelian p-subgroups of G for primes p dividing the group order. As may be expected, the variety is obtained by splicing that of periodic complex K-theory and that of integral ordinary homology, however the way these parts fit …

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Partitioning Regular Polygons Into Circular Pieces I: Convex Partitions, Mirela Damian, Joseph O'Rourke Jan 2003

Partitioning Regular Polygons Into Circular Pieces I: Convex Partitions, Mirela Damian, Joseph O'Rourke

Computer Science: Faculty Publications

We explore an instance of the question of partitioning a polygon into pieces, each of which is as “circular” as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a …

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Computational Geometry Column 44, Joseph O'Rourke Jan 2003

Computational Geometry Column 44, Joseph O'Rourke

Computer Science: Faculty Publications

The open problem of whether or not every pair of equal-area polygons has a hinged dissection is discussed.

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On The Development Of The Intersection Of A Plane With A Polytope, Joseph O'Rourke Jan 2003

On The Development Of The Intersection Of A Plane With A Polytope, Joseph O'Rourke

Computer Science: Faculty Publications

Define a “slice” curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy's arm lemma to permit nonconvex “openings” of a planar convex chain.

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A Few Weight Systems Arising From Intersection Graphs, Blake Mellor Jan 2003

A Few Weight Systems Arising From Intersection Graphs, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

No abstract provided.

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A Geometric Interpretation Of Milnor's Triple Invariants, Blake Mellor, Paul Melvin Jan 2003

A Geometric Interpretation Of Milnor's Triple Invariants, Blake Mellor, Paul Melvin

Mathematics, Statistics and Data Science Faculty Works

Milnor's triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.

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Boundary Volume And Length Spectra Of Riemannian Manifolds: What The Middle Degree Hodge Spectrum Doesn't Reveal, Carolyn S. Gordon, Juan P. Rossetti Jan 2003

Boundary Volume And Length Spectra Of Riemannian Manifolds: What The Middle Degree Hodge Spectrum Doesn't Reveal, Carolyn S. Gordon, Juan P. Rossetti

Dartmouth Scholarship

No abstract provided.

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The Fine Structure Of The Kasparov Groups Ii: Topologizing The Uct, Claude Schochet Oct 2002

The Fine Structure Of The Kasparov Groups Ii: Topologizing The Uct, Claude Schochet

Mathematics Faculty Research Publications

The Kasparov Groups KK∗(A,B) have a natural structure as pseudopolonais groups. In this paper we analyze how this topology interacts with the terms of the Universal Coefficient Theorem (UCT) and the splitting sof the UCT constructed by J. Rosenberg and the author, as well as its canonical three term decomposition which exists under bootstrap hypotheses. We show that the various topologies on [cursive]Ext^{1}_{ℤ}(K∗(A),K∗(B)) and other related groups mostly coincide. Then we focus attention on the Milnor sequence and the fine structure subgroup of KK∗(A,B). …

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A Polytopal Generalization Of Sperner's Lemma, Jesus A. De Loera, Elisha Peterson '00, Francis E. Su Oct 2002

A Polytopal Generalization Of Sperner's Lemma, Jesus A. De Loera, Elisha Peterson '00, Francis E. Su

All HMC Faculty Publications and Research

We prove the following conjecture of Atanassov (Studia Sci. Math. Hungar.32 (1996), 71–74). Let T be a triangulation of a d-dimensional polytope P with n vertices v1, v2,…,vn. Label the vertices of T by 1,2,…,n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F. Then there are at least n−d full dimensional simplices of T, each labelled with d+1 different labels. We …

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Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil Aug 2002

Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil

Mathematical Sciences Technical Reports (MSTR)

In contribution to the classification of all tilings of low-genus surfaces, the kaleidoscopic and non-kaleidoscopic tilings by quadrilaterals are given up to genus 12. As part of their classification, the algebraic structure of the conformal tiling groups and the geometric structure of the tiles are specified. In addition, several infinite classes of tilings and tiling groups are presented.

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Computational Geometry Column 43, Joseph O'Rourke Jun 2002

Computational Geometry Column 43, Joseph O'Rourke

Computer Science: Faculty Publications

The concept of pointed pseudo-triangulations is defined and a few of its applications described.

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The Mathematics Of T. Benny Rushing, James Keesling, Luis Montejano, Gerard A. Venema May 2002

The Mathematics Of T. Benny Rushing, James Keesling, Luis Montejano, Gerard A. Venema

University Faculty Publications and Creative Works

No abstract provided.

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Representing Homology Classes Of Simply Connected 4-Manifolds, Vo Thanh Liem, Gerard A. Venema May 2002

Representing Homology Classes Of Simply Connected 4-Manifolds, Vo Thanh Liem, Gerard A. Venema

University Faculty Publications and Creative Works

The main theorem asserts that every 2-dimensional homology class of a compact simply connected PL 4-manifold can be represented by a codimension-0 submanifold consisting of a contractible manifold with a single 2-handle attached. One consequence of the theorem is the fact that every map of S2 into a simply connected, compact PL 4-manifold is homotopic to an embedding if and only if the same is true for every homotopy equivalence. The theorem is also the main ingredient in the proof of the following result: If W is a compact, simply connected, PL submanifold of S4, then each element of H2(W;ℤ) …

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Nonorthogonal Polyhedra Built From Rectangles, Melody Donoso, Joseph O'Rourke May 2002

Nonorthogonal Polyhedra Built From Rectangles, Melody Donoso, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that any polyhedron of genus zero or genus one built out of rectangular faces must be an orthogonal polyhedron, but that there are nonorthogonal polyhedra of genus seven all of whose faces are rectangles. This leads to a resolution of a question posed by Biedl, Lubiw, and Sun [BLS99].

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Enumerating Foldings And Unfoldings Between Polygons And Polytopes, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke Mar 2002

Enumerating Foldings And Unfoldings Between Polygons And Polytopes, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke

Computer Science: Faculty Publications

We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are, roughly: exponentially many, or nondenumerably infinite.

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Applications Of Graph Theory To Separability, Stephen Young Jan 2002

Applications Of Graph Theory To Separability, Stephen Young

Mathematical Sciences Technical Reports (MSTR)

Let S be a surface with a triangular tiling T. Let R be a reflection a side of one of the triangles; so that R is an orientation reversing isometry of the surface. Define M = {s in S |S : Rs = s}. We then say that the surface S separates along the reflection R if S-R has two components. This paper considers the applications of graph theoretic methods to determining whether a reflection is separating or not and compares the algorithmic efficiency of these methods to the current known methods.

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Intrinsic Knotting And Linking Of Complete Graphs, Erica Flapan Jan 2002

Intrinsic Knotting And Linking Of Complete Graphs, Erica Flapan

Pomona Faculty Publications and Research

We show that for every m∈N, there exists an n∈N such that every embedding of the complete graph Kn in R3 contains a link of two components whose linking number is at least m. Furthermore, there exists an r∈N such that every embedding of Kr in R3 contains a knot Q with |a2(Q)| ≥ m, where a2(Q) denotes the second coefficient of the Conway polynomial of Q.

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A Density Property Of The Tori And Duality, Peter Loth Jan 2002

A Density Property Of The Tori And Duality, Peter Loth

Mathematics Faculty Publications

In this note, a short proof of a recent theorem of D. Dikranjan and M. Tkachenko is given, and their result is extended.

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Vertex-Unfoldings Of Simplicial Manifolds, Erik D. Demaine, David Eppstein, Jeff Erickson, George W. Hart, Joseph O'Rourke Jan 2002

Vertex-Unfoldings Of Simplicial Manifolds, Erik D. Demaine, David Eppstein, Jeff Erickson, George W. Hart, Joseph O'Rourke

Computer Science: Faculty Publications

We present an algorithm to unfold any triangulated 2-manifold (in particular, any simplicial polyhedron) into a non-overlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension.

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Polygonal Chains Cannot Lock In 4d, Roxana Cocan, Joseph O'Rourke Nov 2001

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.

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Computational Geometry Column 42, Joseph S. B. Mitchell, Joseph O'Rourke Oct 2001

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.

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The Fine Structure Of The Kasparov Groups I: Continuity Of The Kk-Pairing, Claude Schochet Oct 2001

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.

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Uniqueness Of Volume-Minimizing Submanifolds Calibrated By The First Pontryagin Form, Daniel A. Grossman, Weiqing Gu Jun 2001

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 …

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Computational Geometry Column 41, Joseph O'Rourke Apr 2001

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.

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Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot Feb 2001

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 …

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Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods Feb 2001

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.

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Extended Powers Of Manifolds And The Adams Spectral Sequence, Robert R. Bruner Jan 2001

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.

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Strictly Positive Definite Functions On A Compact Group, Mohamed Allali, Tomasz Przebinda Jan 2001

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.

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A 4-Dimensional 1-Lcc Shrinking Theorem, Mladen Bestvina, Robert J. Daverman, Gerard A. Venema Jan 2001

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 …

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On The Existence Of Finite Type Link Homotopy Invariants, Blake Mellor, Dylan Thurston Jan 2001

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.

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