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Articles 1 - 28 of 28
Full-Text Articles in Mathematics
Coplanar Constant Mean Curvature Surfaces, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan
Coplanar Constant Mean Curvature Surfaces, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan
Robert Kusner
We consider constant mean curvature surfaces with finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane: we construct and classify the entire family of these genus-zero, coplanar constant mean curvature surfaces.
Band Unfoldings And Prismatoids: A Counterexample, Joseph O'Rourke
Band Unfoldings And Prismatoids: A Counterexample, Joseph O'Rourke
Computer Science: Faculty Publications
This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding.
Unfolding Restricted Convex Caps, Joseph O'Rourke
Unfolding Restricted Convex Caps, Joseph O'Rourke
Computer Science: Faculty Publications
This paper details an algorithm for unfolding a class of convex polyhedra, where each polyhedron in the class consists of a convex cap over a rectangular base, with several restrictions: the cap’s faces are quadrilaterals, with vertices over an underlying integer lattice, and such that the cap convexity is "radially monotone," a type of smoothness constraint. Extensions of Cauchy’s arm lemma are used in the proof of non-overlap.
A Fixed Point Theorem For The Infinite-Dimensional Simplex, Douglas Rizzolo '08, Francis E. Su
A Fixed Point Theorem For The Infinite-Dimensional Simplex, Douglas Rizzolo '08, Francis E. Su
All HMC Faculty Publications and Research
We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.
A Comparison Of The Deck Group And The Fundamental Group On Uniform Spaces Obtained By Gluing, Raymond David Phillippi
A Comparison Of The Deck Group And The Fundamental Group On Uniform Spaces Obtained By Gluing, Raymond David Phillippi
Doctoral Dissertations
We de…ne a uniformity on a glued space under uniformly continuous attachment maps. If the component spaces are uniform coverable then the resulting glued space is uniform coverable. We consider examples including the glued uniformity on a …nite dimensional CW complex which is shown to be uniformly coverable. For one dimensional CWcomplexes, the resulting deck group is equivalent to the fundamental group. Other properties of the deck group are explored.
A Complexification Of Rolle’S Theorem, J. P. Pemba, A. R. Davies, N. K. Muoneke
A Complexification Of Rolle’S Theorem, J. P. Pemba, A. R. Davies, N. K. Muoneke
Applications and Applied Mathematics: An International Journal (AAM)
A new version of the classical Rolle’s theorem is proved for any complex-valued differentiable function of the complex variable on an open connected convex subset of the complex field. The associated Mean-Value theorem follows naturally. A few explicit illustrative examples are provided in the closing section of the paper.
Epsilon-Unfolding Orthogonal Polyhedra, Mirela Damian, Robin Flatland, Joseph O'Rourke
Epsilon-Unfolding Orthogonal Polyhedra, Mirela Damian, Robin Flatland, Joseph O'Rourke
Computer Science: Faculty Publications
An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, …
The Lsb Theorem Implies The Kkm Lemma, Gwen Spencer '05, Francis E. Su
The Lsb Theorem Implies The Kkm Lemma, Gwen Spencer '05, Francis E. Su
All HMC Faculty Publications and Research
No abstract provided in this article.
The Center Of Some Braid Groups And The Farrell Cohomology Of Certain Pure Mapping Class Groups, Craig A. Jensen, Yu Qing Chen, Henry H. Glover
The Center Of Some Braid Groups And The Farrell Cohomology Of Certain Pure Mapping Class Groups, Craig A. Jensen, Yu Qing Chen, Henry H. Glover
Mathematics Faculty Publications
In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the primary part of the Farrell cohomology of a few pure mapping class groups.
The Euler Characteristic Of The Whitehead Automorphism Group Of A Free Product, Craig A. Jensen, Jon Mccammond, John Meier
The Euler Characteristic Of The Whitehead Automorphism Group Of A Free Product, Craig A. Jensen, Jon Mccammond, John Meier
Mathematics Faculty Publications
A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms...
Relationships Between Braid Length And The Number Of Braid Strands, Cornelia A. Van Cott
Relationships Between Braid Length And The Number Of Braid Strands, Cornelia A. Van Cott
Mathematics
For a knot K, let ℓ(K,n) be the minimum length of an n–stranded braid representative of K. Fixing a knot K, ℓ(K,n) can be viewed as a function of n, which we denote by ℓK(n). Examples of knots exist for which ℓK(n) is a nonincreasing function. We investigate the behavior of ℓK(n), developing bounds on the function in terms of the genus of K. The bounds lead to the conclusion that for any knot K the function ℓK(n) is eventually stable. We study the stable behavior of ℓK(n), with stronger results for homogeneous knots. For knots of nine or fewer …
Centers And Shore Points In Λ-Dendroids, Van C. Nall
Centers And Shore Points In Λ-Dendroids, Van C. Nall
Department of Math & Statistics Faculty Publications
A dendroid is the disjoint union of the set of centers and the set of shore points. We show this is also true for λ-dendroids and use this fact to show that the finite union of shore continua in a λ-dendroid is a shore set.
Minimal Surfaces, Maria Guadalupe Chaparro
Minimal Surfaces, Maria Guadalupe Chaparro
Theses Digitization Project
The focus of this project consists of investigating when a ruled surface is a minimal surface. A minimal surface is a surface with zero mean curvature. In this project the basic terminology of differential geometry will be discussed including examples where the terminology will be applied to the different subjects of differential geometry. In addition the focus will be on a classical theorem of minimal surfaces referred to as the Plateau's Problem.
On Groups Of Homological Dimension One, Jonathan Cornick
On Groups Of Homological Dimension One, Jonathan Cornick
Publications and Research
It has been conjectured that the groups of homological dimension one are precisely the nontrivial locally free groups. Some algebraic, geometric and analytic properties of any potential counter example to the conjecture are discussed.
Conics In The Hyperbolic Plane, Trent Phillip Naeve
Conics In The Hyperbolic Plane, Trent Phillip Naeve
Theses Digitization Project
An affine transformation such as T(P)=Q is a locus of an affine conic. Any affine conic can be produced from this incidence construction. The affine type of conic (ellipse, parabola, hyperbola) is determined by the invariants of T, the determinant and trace of its linear part. The purpose of this thesis is to obtain a corresponding classification in the hyperbolic plane of conics defined by this construction.
Tutte Polynomial In Knot Theory, David Alan Petersen
Tutte Polynomial In Knot Theory, David Alan Petersen
Theses Digitization Project
This thesis reviews the history of knot theory with an emphasis on the diagrammatic approach to studying knots. Also covered are the basic concepts and notions of graph theory and how these two fields are related with an example of a knot diagram and how to associate it to a graph.
An Upperbound On The Ropelength Of Arborescent Links, Larry Andrew Mullins
An Upperbound On The Ropelength Of Arborescent Links, Larry Andrew Mullins
Theses Digitization Project
This thesis covers improvements on the upperbounds for ropelength of a specific class of algebraic knots.
Derived Categories And The Analytic Approach To General Reciprocity Laws. Part Ii, Michael Berg
Derived Categories And The Analytic Approach To General Reciprocity Laws. Part Ii, Michael Berg
Mathematics Faculty Works
Building on the topological foundations constructed in Part I, we now go on to address the homological algebra preparatory to the projected final arithmetical phase of our attack on the analytic proof of general reciprocity for a number field. In the present work, we develop two algebraic frameworks corresponding to two interpretations of Kubota's n-Hilbert reciprocity formalism, presented in a quasi-dualized topological form in Part I, delineating two sheaf-theoretic routes toward resolving the aforementioned (open) problem. The first approach centers on factoring sheaf morphisms eventually to yield a splitting homomorphism for Kubota's n-fold cover of the adelized special linear group …
Removing Sets From Connected Spaces While Preserving Connectedness, Melvin Henriksen, Amir Nikou
Removing Sets From Connected Spaces While Preserving Connectedness, Melvin Henriksen, Amir Nikou
All HMC Faculty Publications and Research
As per the title, the nature of sets that can be removed from a product of more than one connected, arcwise connected, or point arcwise connected spaces while preserving the appropriate kind of connectedness is studied. This can depend on the cardinality of the set being removed or sometimes just on the cardinality of what is removed from one or two factor spaces. Sometimes it can depend on topological properties of the set being removed or its trace on various factor spaces. Some of the results are complicated to prove while being easy to state. Sometimes proofs for different kinds …
A Model Of Dna Knotting And Linking, Erica Flapan, Dorothy Buck
A Model Of Dna Knotting And Linking, Erica Flapan, Dorothy Buck
Pomona Faculty Publications and Research
We present a model of how DNA knots and links are formed as a result of a single recombination event, or multiple rounds of (processive) recombination events, starting with an unknotted, unlinked, or a (2,m)-torus knot or link substrate. Given these substrates, according to our model all DNA products of a single recombination event or processive recombination fall into a single family of knots and links.
A New Lower Bound On Guard Placement For Wireless Localization, Mirela Damian, Robin Flatland, Joseph O'Rourke, Suneeta Ramswami
A New Lower Bound On Guard Placement For Wireless Localization, Mirela Damian, Robin Flatland, Joseph O'Rourke, Suneeta Ramswami
Computer Science: Faculty Publications
The problem of wireless localization asks to place and orient stations in the plane, each of which broadcasts a unique key within a fixed angular range, so that each point in the plane can determine whether it is inside or outside a given polygonal region. The primary goal is to minimize the number of stations. In this paper we establish a lower bound of ⌊2n/3⌋−1 stations for polygons in general position, for the case in which the placement of stations is restricted to polygon vertices, improving upon the existing ⌈n/2⌉ lower bound.
Virtual Spatial Graphs, Thomas Fleming, Blake Mellor
Virtual Spatial Graphs, Thomas Fleming, Blake Mellor
Mathematics Faculty Works
Two natural generalizations of knot theory are t he study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spat ial graphs.
Exotic Statistics For Strings In 4d Bf Theory, John C. Baez, Derek K. Wise, Alissa S. Crans
Exotic Statistics For Strings In 4d Bf Theory, John C. Baez, Derek K. Wise, Alissa S. Crans
Mathematics Faculty Works
After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can …
Connecting Polygonizations Via Stretches And Twangs, Mirela Damian, Robin Flatland, Joseph O'Rourke, Suneeta Ramswami
Connecting Polygonizations Via Stretches And Twangs, Mirela Damian, Robin Flatland, Joseph O'Rourke, Suneeta Ramswami
Computer Science: Faculty Publications
We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n2 ) “moves” between simple polygons. Each move is composed of a sequence of atomic moves called “stretches” and "twangs". These atomic moves walk between weakly simple "polygonal wraps" of S. These moves show promise to serve as a basis for generating random polygons.
Intrinsic Linking And Knotting In Virtual Spatial Graphs, Thomas Fleming, Blake Mellor
Intrinsic Linking And Knotting In Virtual Spatial Graphs, Thomas Fleming, Blake Mellor
Mathematics Faculty Works
We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and nonterminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the virtual unknotting number of a knot, and show that any knot with nontrivial Jones polynomial has virtual unknotting number at least 2.
Boundary Slopes Of 2-Bridge Links Determine The Crossing Number, Jim Hoste, Patrick D. Shanahan
Boundary Slopes Of 2-Bridge Links Determine The Crossing Number, Jim Hoste, Patrick D. Shanahan
Mathematics Faculty Works
A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and the same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of …
The Topology Of Surface Mediatrices, James Bernhard, J. J. P. Veerman
The Topology Of Surface Mediatrices, James Bernhard, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
Given a pair of distinct points p and q in a metric space with distance d, the mediatrix is the set of points x such that d(x,p)=d(x,q). In this paper, we examine the topological structure of mediatrices in connected, compact, closed 2-manifolds whose distance function is inherited from a Riemannian metric. We determine that such mediatrices are, up to homeomorphism, finite, closed simplicial 1-complexes with an even number of incipient edges emanating from each vertex. Using this and results from [J.J.P. Veerman, J. Bernhard, Minimally separating sets, mediatrices and Brillouin spaces, Topology Appl., in press], we give the classification …
Clarifications Of Rule 2 In Teaching Geometric Dimensioning And Tolerancing, Cheng Lin, Alok Verma
Clarifications Of Rule 2 In Teaching Geometric Dimensioning And Tolerancing, Cheng Lin, Alok Verma
Engineering Technology Faculty Publications
Geometric dimensioning and tolerancing is a symbolic language used on engineering drawings and computer generated three-dimensional solid models for explicitly describing nominal geometry and its allowable variation. Application cases using the concept of Rule 2 in the Geometric Dimensioning and Tolerancing (GD&T) are presented. The rule affects all fourteen geometric characteristics. Depending on the nature and location where each feature control frame is specified, interpretation on the applicability of Rule 2 is quite inconsistent. This paper focuses on identifying the characteristics of a feature control frame to remove this inconsistency. A table is created to clarify the confusions for students …