Joint Embedding of Hierarchical Categories and Entities for Concept Categorization and Dataless Classification
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Vertex-Transplants on a Convex Polyhedron
CCCG 2020, Saskatoon, Canada, August 5{7, 2020 Vertex-Transplants on a Convex Polyhedron Joseph O'Rourke Abstract Regular Tetrahedron. Let the four vertices of a regu- lar tetrahedron of unit edge length be v1; v2; v3 forming Given any convex polyhedron P of sufficiently many ver- the base, and apex v0. Place a point x on the v3v0 edge, tices n, and with no vertex's curvature greater than π, close to v3. Then one can form a digon starting from x it is possible to cut out a vertex, and paste the excised and surrounding v0 with geodesics γ1 and γ2 to a point y portion elsewhere along a vertex-to-vertex geodesic, cre- on 4v1v2v0, with jγ1j = jγ2j = 1. See Fig. 1(a,b). This ating a new convex polyhedron P0 of n + 2 vertices. digon can then be cut out and its hole sutured closed. The removed digon surface can be folded to a doubly covered triangle, and pasted into edge v1v2. The re- 1 Introduction sulting convex polyhedron guaranteed by Alexandrov's Theorem is a 6-vertex irregular octahedron P0. The goal of this paper is to prove the following theorem: Theorem 1 For any convex polyhedron P of n > N vertices, none of which have curvature greater than π, there is a vertex v0 that can be cut out along a digon of geodesics, and the excised surface glued to a geodesic on P connecting two vertices v1; v2. The result is a new convex polyhedron P0 with n + 2 vertices. N = 16 suffices. -
Locating Diametral Points
Results Math (2020) 75:68 Online First c 2020 Springer Nature Switzerland AG Results in Mathematics https://doi.org/10.1007/s00025-020-01193-5 Locating Diametral Points Jin-ichi Itoh, Costin Vˆılcu, Liping Yuan , and Tudor Zamfirescu Abstract. Let K be a convex body in Rd,withd =2, 3. We determine sharp sufficient conditions for a set E composed of 1, 2, or 3 points of bdK, to contain at least one endpoint of a diameter of K.Weextend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in E. We also show that such criteria do not exist for n ≥ 4points. Mathematics Subject Classification. 52A10, 52A15, 53C45. Keywords. Convex body, diameter, geodesic diameter, diametral point. 1. Introduction The tangent cone at a point x in the boundary bdK of a convex body K can be defined using only neighborhoods of x in bdK. So, one doesn’t normally expect to get global information about K from the size of the tangent cones at one, two or three points. Nevertheless, in some cases this is what happens! A convex body K in Rd is a compact convex set with interior points; we shall consider only the cases d =2, 3. A convex surface in R3 is the boundary of a convex body in R3. Let S be a convex surface and x apointinS. Consider homothetic dilations of S with the centre at x and coefficients of homothety tending to infinity. The limit surface is called the tangent cone at x (see [1]), and is denoted by Tx. -
Designing a Rapid Deployment Emergency Services
Designing a Rapid Deployment Emergency Services (RapiDES) Vehicle for 2020: Addressing the problem of ever increasing traffic delaying the emergency services vehicles in big cities by Martin CS Tee A thesis submitted in partial fulfillment of the requirements for the degree of Master of Design in Industrial Design Department of Art and Design University of Alberta © Martin CS Tee, 2015 ABSTRACT This thesis/project is about the design process of designing a conceptual vehicle design that responds to a growing global phenomenon of ever increasing traffic congestion delaying emergency response times of various emergency services. This thesis aims at establishing the case for a smarter, smaller and more responsive emergency services vehicle design by first looking at the phenomenon of ever increasing traffic volume in big cities around the world particularly in North America, examining the causes and consequences of traffic congestion, and identifying the most critical impact it causes which is on the emergency response times of various emergency services which makes a difference between life and death with a minute’s delay to the scene. The thesis then attempts to offer a solution with the accompanying practical project focusing on designing a Rapid Deployment Emergency Services (RapiDES) vehicle for the near future (2020) to meet the challenges of saving lives and properties in a quick and responsive way. This design of RapiDES will serve as a baseline model for the North American context and can be modified to suit the needs and requirements for other markets. The design requirements and key features of RapiDES are generated and described in the Design Brief of RapiDES as a result of rigorous research mapping exercise and scenario studies of RapiDES in action for the three emergency services of EMS, fire rescue and law enforcement. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
Complementary N-Gon Waves and Shuffled Samples Noise
Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx2020),(DAFx-20), Vienna, Vienna, Austria, Austria, September September 8–12, 2020-21 2020 COMPLEMENTARY N-GON WAVES AND SHUFFLED SAMPLES NOISE Dominik Chapman ABSTRACT A characteristic of an n-gon wave is that it retains angles of This paper introduces complementary n-gon waves and the shuf- the regular polygon or star polygon of which the waveform was fled samples noise effect. N-gon waves retain angles of the regular derived from. As such, some parts of the polygon can be recog- polygons and star polygons of which they are derived from in the nised when the waveform is visualised, as can be seen from fig- waveform itself. N-gon waves are researched by the author since ure 1. This characteristic distinguishes n-gon waves from other 2000 and were introduced to the public at ICMC|SMC in 2014. Complementary n-gon waves consist of an n-gon wave and a com- plementary angular wave. The complementary angular wave in- troduced in this paper complements an n-gon wave so that the two waveforms can be used to reconstruct the polygon of which the Figure 1: An n-gon wave is derived from a pentagon. The internal waveforms were derived from. If it is derived from a star polygon, angle of 3π=5 rad of the pentagon is retained in the shape of the it is not an n-gon wave and has its own characteristics. Investiga- wave and in the visualisation of the resulting pentagon wave on a tions into how geometry, audio, visual and perception are related cathode-ray oscilloscope. -
Draining a Polygon –Or– Rolling a Ball out of a Polygon
CCCG 2008, Montr´eal, Qu´ebec, August 13–15, 2008 Draining a Polygon –or– Rolling a Ball out of a Polygon Greg Aloupis∗ Jean Cardinal∗ S´ebastienCollette†∗ Ferran Hurtado‡ Stefan Langerman§∗ Joseph O’Rourke¶ Abstract are labeled counterclockwise (ccw). Let us assume that vi is a local minimum with respect to y, i.e., both vi−1 We introduce the problem of draining water (or balls repre- and vi+1 are above vi. Now we are permitted to ro- senting water drops) out of a punctured polygon (or a poly- tate P in the vertical plane (or equivalently, alter the hedron) by rotating the shape. For 2D polygons, we obtain gravity vector). In the Rotation model, B does not combinatorial bounds on the number of holes needed, both move from vi until one of the two adjacent edges, say for arbitrary polygons and for special classes of polygons. ei = vivi+1, turns infinitesimally beyond the horizontal, 2 We detail an O(n log n) algorithm that finds the minimum at which time B rolls down ei and falls under the in- number of holes needed for a given polygon, and argue that fluence of gravity until it settles at some other convex the complexity remains polynomial for polyhedra in 3D. We vertex vj. For example, in Fig. 1, B at v4 rolls ccw make a start at characterizing the 1-drainable shapes, those v2 that only need one hole. v1 v13 1 Introduction v3 v11 Imagine a closed polyhedral container P partially filled v v0 v12 with water. How many surface point-holes are needed 5 v4 to entirely drain it under the action of gentle rotations v15 of P ? It may seem that one hole suffices, but we will show that in fact sometimes Ω(n) holes are needed for v14 v9 a polyhedron of n vertices. -
Four Colours Suffice? Transcript
Four Colours Suffice? Transcript Date: Monday, 21 October 2002 - 12:00AM Location: Barnard's Inn Hall FOUR COLOURS SUFFICE? Robin Wilson, former Visiting Gresham Professor in the History of Mathematics [Introduction] In October 1852, Francis Guthrie, a former student of Augustus De Morgan (professor of mathematics at University College London), was colouring a map of England. He noticed that if neighbouring countries had to be differently coloured, then only four colours were needed. Do four colours suffice for colouring all maps, however complicated?, he wondered. On 23 October of that year, 150 years ago this week, Guthrie's brother Frederick asked De Morgan, who immediately became fascinated with the problem and communicated it to his friends. De Morgan's famous letter of 23 October 1852 to the Irish mathematical physicist Sir William Rowan Hamilton included the following extract. A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. [He then described the problem, and gave a simple example of a map for which four colours are needed.] Query: cannot a necessity for five or more be invented? De Morgan also wrote about the problem to the philosopher William Whewell, Master of Trinity College Cambridge, and others. The problem first appeared in print in the middle of an unsigned book review (actually by De Morgan) of Whewell's Philosophy of Discovery. This review contained the following very strange passage: Now, it must have been always known to map-colourers that four different colours are enough. -
SAH in Paris 5 October 11, 2012 Fall Board Meeting
Journal The Society of Automotive Historians, Inc. Issue 255 Electronic March - April 2012 Diamonds Aren’t Forever, Page 12. Inside Date Reminders President’s Message 3 Retromobile 6 June 11, 2012 Scharburg Award Nominations Due SAH Chapter News 3 Pilgrimage to Alsace 8 July 31, 2012 SAH News 4 Book & Media Reviews 10 Valentine Award Nominations Due. Letters 4 Inside the Fisher Body Craftsmen’s Guild August 1, 2012 Bradley Award Nominations Due. The Fisher Body Craftsman’s Guild recap SAH in Paris 5 October 11, 2012 Fall Board Meeting. Cover Vehicle: 1901 Sunbeam-Mabley at the October 12, 2012 www.autohistory.org Louwman Museum. Photo: Lincoln Sarmanian Annual MeetingMarch and- April Awards 2012 Banquet. Journal The Society of Automotive Historians, Inc. Issue 255 March - April 2012 Offi cers SAH Annual Awards J. Douglas Leighton President Benz Award, Chair: Don Keefe, [email protected] John Heitmann Vice President The Carl Benz Award is presented each year for the best article published in the previous calendar year. SAH Robert R. Ebert Secretary Awards of Distinction are awarded for exemplary articles not receiving the Benz Award. Patrick D. Bisson Treasurer 2011 Terry V. Boyce, “1951 Buick XP-300: Mr. Chayne Builds His Dream Car,”in Collectible Automobile 2010 John L. Baeke, M.D, “The Lebarons: Heir Apparent to the Throne,” in The Reunion Board of Directors 2009 Jim Chase, “Packard and Winton: The Transcontinental Rivalry,” in The Packard Cormorant Through October 2012 Thomas S. Jakups, Leslie Kendall, Bradley Award, Chair: Judith Endelman, [email protected] Steve Wilson The James J. Bradley Distinguished Service Award is presented to deserving archives and libraries for exemplary efforts in preserving motor vehicle resource materials. -
Vertex-Transplants on a Convex Polyhedron
CCCG 2020, Saskatoon, Canada, August 5{7, 2020 Vertex-Transplants on a Convex Polyhedron Joseph O'Rourke Abstract except when P0 has only a few vertices or special sym- metries. Given any convex polyhedron P of sufficiently many ver- In the examples below, we use some notation that will tices n, and with no vertex's curvature greater than π, not be fully explained until Sec. 3. it is possible to cut out a vertex, and paste the excised portion elsewhere along a vertex-to-vertex geodesic, cre- Cube. Fig. 1 shows excising a unit-cube corner v0 with ating a new convex polyhedron P0. Although P0 could geodesics γ1 and γ2, each of length 1, and then sutur- have, in degenerate situations, as many as 2 fewer ver- ing this digon into the edge v v . Although a paper 0 1 2 tices, the generic situation is that P has n + 2 ver- model reveals a clear 10-vertex polyhedron (points x 0 tices. P has the same surface area as P, and the same and y become vertices of P0), I have not constructed it total curvature but with some of that curvature redis- numerically. tributed. v2 v0 v y 2 γ1 y v1 1 Introduction v0 v1 γ2 The goal of this paper is to prove the following theorem: x x Theorem 1 For any convex polyhedron P of n > N vertices, none of which have curvature greater than π, there is a vertex v0 that can be cut out along a digon of geodesics, and the excised surface glued to a geodesic Figure 1: Left: Digon xy surrounding v0. -
Tiling with Penalties and Isoperimetry with Density
Rose-Hulman Undergraduate Mathematics Journal Volume 13 Issue 1 Article 6 Tiling with Penalties and Isoperimetry with Density Yifei Li Berea College, [email protected] Michael Mara Williams College, [email protected] Isamar Rosa Plata University of Puerto Rico, [email protected] Elena Wikner Williams College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Li, Yifei; Mara, Michael; Plata, Isamar Rosa; and Wikner, Elena (2012) "Tiling with Penalties and Isoperimetry with Density," Rose-Hulman Undergraduate Mathematics Journal: Vol. 13 : Iss. 1 , Article 6. Available at: https://scholar.rose-hulman.edu/rhumj/vol13/iss1/6 Rose- Hulman Undergraduate Mathematics Journal Tiling with Penalties and Isoperimetry with Density Yifei Lia Michael Marab Isamar Rosa Platac Elena Wiknerd Volume 13, No. 1, Spring 2012 aDepartment of Mathematics and Computer Science, Berea College, Berea, KY 40404 [email protected] bDepartment of Mathematics and Statistics, Williams College, Williamstown, MA 01267 [email protected] cDepartment of Mathematical Sciences, University of Puerto Rico at Sponsored by Mayagez, Mayagez, PR 00680 [email protected] dDepartment of Mathematics and Statistics, Williams College, Rose-Hulman Institute of Technology Williamstown, MA 01267 [email protected] Department of Mathematics Terre Haute, IN 47803 Email: [email protected] http://www.rose-hulman.edu/mathjournal Rose-Hulman Undergraduate Mathematics Journal Volume 13, No. 1, Spring 2012 Tiling with Penalties and Isoperimetry with Density Yifei Li Michael Mara Isamar Rosa Plata Elena Wikner Abstract. We prove optimality of tilings of the flat torus by regular hexagons, squares, and equilateral triangles when minimizing weighted combinations of perime- ter and number of vertices. -
Arxiv:2007.03355V2 [Math.GT] 23 Jul 2020
GEOMETRIC DECOMPOSITIONS OF SURFACES WITH SPHERICAL METRIC AND CONICAL SINGULARITIES GUILLAUME TAHAR Abstract. We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical analog of the geometric triangulations for flat surfaces with conical singularities. The irreducible components include not only spherical triangles but also other interesting spherical poly- gons. In particular, we present the class of half-spherical concave polygons that are spherical polygons without diagonals and that can be arbitrarily complicated. Finally, we introduce the notion of core as a geometric invariant in the settings of spherical sur- faces. We use it to prove a reducibily result for spherical surfaces with a total conical angle at least (10g − 10 + 5n)2π. Contents 1. Introduction1 2. Basic notions3 3. Irreducible polygons5 4. Core of a spherical surface 12 References 18 1. Introduction A natural generalization of uniformization theorem concerns surfaces with prescribed sin- Pn gularities. For any compact Riemann surface S of genus g and any real divisor i=1 αiPi formed by positive real numbers αi > 0 and points Pi 2 S, we may ask if there is a metric with conical singularities of angle 2παi at each point Pi and constant curvature elsewhere. If there is a metric with constant curvature K on S n fP1;:::;Png, then angle defect at the conical singularities is interpreted as singular curvature in a generalized Gauss-Bonnet formula: n K:Area(S) X = 2 − 2g − n + α 2π i arXiv:2007.03355v2 [math.GT] 23 Jul 2020 i=1 The sign of this latter quantity, that depends only on the genus of the surface and the prescribed angles, determines if the constant curvature is negative, zero or positive. -
Geometry in Design Geometrical Construction in 3D Forms by Prof
D’source 1 Digital Learning Environment for Design - www.dsource.in Design Course Geometry in Design Geometrical Construction in 3D Forms by Prof. Ravi Mokashi Punekar and Prof. Avinash Shide DoD, IIT Guwahati Source: http://www.dsource.in/course/geometry-design 1. Introduction 2. Golden Ratio 3. Polygon - Classification - 2D 4. Concepts - 3 Dimensional 5. Family of 3 Dimensional 6. References 7. Contact Details D’source 2 Digital Learning Environment for Design - www.dsource.in Design Course Introduction Geometry in Design Geometrical Construction in 3D Forms Geometry is a science that deals with the study of inherent properties of form and space through examining and by understanding relationships of lines, surfaces and solids. These relationships are of several kinds and are seen in Prof. Ravi Mokashi Punekar and forms both natural and man-made. The relationships amongst pure geometric forms possess special properties Prof. Avinash Shide or a certain geometric order by virtue of the inherent configuration of elements that results in various forms DoD, IIT Guwahati of symmetry, proportional systems etc. These configurations have properties that hold irrespective of scale or medium used to express them and can also be arranged in a hierarchy from the totally regular to the amorphous where formal characteristics are lost. The objectives of this course are to study these inherent properties of form and space through understanding relationships of lines, surfaces and solids. This course will enable understanding basic geometric relationships, Source: both 2D and 3D, through a process of exploration and analysis. Concepts are supported with 3Dim visualization http://www.dsource.in/course/geometry-design/in- of models to understand the construction of the family of geometric forms and space interrelationships.