The Classification of Surfaces
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Note on 6-Regular Graphs on the Klein Bottle Michiko Kasai [email protected]
Theory and Applications of Graphs Volume 4 | Issue 1 Article 5 2017 Note On 6-regular Graphs On The Klein Bottle Michiko Kasai [email protected] Naoki Matsumoto Seikei University, [email protected] Atsuhiro Nakamoto Yokohama National University, [email protected] Takayuki Nozawa [email protected] Hiroki Seno [email protected] See next page for additional authors Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/tag Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Kasai, Michiko; Matsumoto, Naoki; Nakamoto, Atsuhiro; Nozawa, Takayuki; Seno, Hiroki; and Takiguchi, Yosuke (2017) "Note On 6-regular Graphs On The Klein Bottle," Theory and Applications of Graphs: Vol. 4 : Iss. 1 , Article 5. DOI: 10.20429/tag.2017.040105 Available at: https://digitalcommons.georgiasouthern.edu/tag/vol4/iss1/5 This article is brought to you for free and open access by the Journals at Digital Commons@Georgia Southern. It has been accepted for inclusion in Theory and Applications of Graphs by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. Note On 6-regular Graphs On The Klein Bottle Authors Michiko Kasai, Naoki Matsumoto, Atsuhiro Nakamoto, Takayuki Nozawa, Hiroki Seno, and Yosuke Takiguchi This article is available in Theory and Applications of Graphs: https://digitalcommons.georgiasouthern.edu/tag/vol4/iss1/5 Kasai et al.: 6-regular graphs on the Klein bottle Abstract Altshuler [1] classified 6-regular graphs on the torus, but Thomassen [11] and Negami [7] gave different classifications for 6-regular graphs on the Klein bottle. In this note, we unify those two classifications, pointing out their difference and similarity. -
Hydraulic Cylinder Replacement for Cylinders with 3/8” Hydraulic Hose Instructions
HYDRAULIC CYLINDER REPLACEMENT FOR CYLINDERS WITH 3/8” HYDRAULIC HOSE INSTRUCTIONS INSTALLATION TOOLS: INCLUDED HARDWARE: • Knife or Scizzors A. (1) Hydraulic Cylinder • 5/8” Wrench B. (1) Zip-Tie • 1/2” Wrench • 1/2” Socket & Ratchet IMPORTANT: Pressure MUST be relieved from hydraulic system before proceeding. PRESSURE RELIEF STEP 1: Lower the Power-Pole Anchor® down so the Everflex™ spike is touching the ground. WARNING: Do not touch spike with bare hands. STEP 2: Manually push the anchor into closed position, relieving pressure. Manually lower anchor back to the ground. REMOVAL STEP 1: Remove the Cylinder Bottom from the Upper U-Channel using a 1/2” socket & wrench. FIG 1 or 2 NOTE: For Blade models, push bottom of cylinder up into U-Channel and slide it forward past the Ram Spacers to remove it. Upper U-Channel Upper U-Channel Cylinder Bottom 1/2” Tools 1/2” Tools Ram 1/2” Tools Spacers Cylinder Bottom If Ram-Spacers fall out Older models have (1) long & (1) during removal, push short Ram Spacer. Ram Spacers 1/2” Tools bushings in to hold MUST be installed on the same side Figure 1 Figure 2 them in place. they were removed. Blade Models Pro/Spn Models Need help? Contact our Customer Service Team at 1 + 813.689.9932 Option 2 HYDRAULIC CYLINDER REPLACEMENT FOR CYLINDERS WITH 3/8” HYDRAULIC HOSE INSTRUCTIONS STEP 2: Remove the Cylinder Top from the Lower U-Channel using a 1/2” socket & wrench. FIG 3 or 4 Lower U-Channel Lower U-Channel 1/2” Tools 1/2” Tools Cylinder Top Cylinder Top Figure 3 Figure 4 Blade Models Pro/SPN Models STEP 3: Disconnect the UP Hose from the Hydraulic Cylinder Fitting by holding the 1/2” Wrench 5/8” Wrench Cylinder Fitting Base with a 1/2” wrench and turning the Hydraulic Hose Fitting Cylinder Fitting counter-clockwise with a 5/8” wrench. -
Surfaces and Fundamental Groups
HOMEWORK 5 — SURFACES AND FUNDAMENTAL GROUPS DANNY CALEGARI This homework is due Wednesday November 22nd at the start of class. Remember that the notation e1; e2; : : : ; en w1; w2; : : : ; wm h j i denotes the group whose generators are equivalence classes of words in the generators ei 1 and ei− , where two words are equivalent if one can be obtained from the other by inserting 1 1 or deleting eiei− or ei− ei or by inserting or deleting a word wj or its inverse. Problem 1. Let Sn be a surface obtained from a regular 4n–gon by identifying opposite sides by a translation. What is the Euler characteristic of Sn? What surface is it? Find a presentation for its fundamental group. Problem 2. Let S be the surface obtained from a hexagon by identifying opposite faces by translation. Then the fundamental group of S should be given by the generators a; b; c 1 1 1 corresponding to the three pairs of edges, with the relation abca− b− c− coming from the boundary of the polygonal region. What is wrong with this argument? Write down an actual presentation for π1(S; p). What surface is S? Problem 3. The four–color theorem of Appel and Haken says that any map in the plane can be colored with at most four distinct colors so that two regions which share a com- mon boundary segment have distinct colors. Find a cell decomposition of the torus into 7 polygons such that each two polygons share at least one side in common. Remark. -
An Introduction to Topology the Classification Theorem for Surfaces by E
An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5. -
Volumes of Prisms and Cylinders 625
11-4 11-4 Volumes of Prisms and 11-4 Cylinders 1. Plan Objectives What You’ll Learn Check Skills You’ll Need GO for Help Lessons 1-9 and 10-1 1 To find the volume of a prism 2 To find the volume of • To find the volume of a Find the area of each figure. For answers that are not whole numbers, round to prism a cylinder the nearest tenth. • To find the volume of a 2 Examples cylinder 1. a square with side length 7 cm 49 cm 1 Finding Volume of a 2. a circle with diameter 15 in. 176.7 in.2 . And Why Rectangular Prism 3. a circle with radius 10 mm 314.2 mm2 2 Finding Volume of a To estimate the volume of a 4. a rectangle with length 3 ft and width 1 ft 3 ft2 Triangular Prism backpack, as in Example 4 2 3 Finding Volume of a Cylinder 5. a rectangle with base 14 in. and height 11 in. 154 in. 4 Finding Volume of a 6. a triangle with base 11 cm and height 5 cm 27.5 cm2 Composite Figure 7. an equilateral triangle that is 8 in. on each side 27.7 in.2 New Vocabulary • volume • composite space figure Math Background Integral calculus considers the area under a curve, which leads to computation of volumes of 1 Finding Volume of a Prism solids of revolution. Cavalieri’s Principle is a forerunner of ideas formalized by Newton and Leibniz in calculus. Hands-On Activity: Finding Volume Explore the volume of a prism with unit cubes. -
Quick Reference Guide - Ansi Z80.1-2015
QUICK REFERENCE GUIDE - ANSI Z80.1-2015 1. Tolerance on Distance Refractive Power (Single Vision & Multifocal Lenses) Cylinder Cylinder Sphere Meridian Power Tolerance on Sphere Cylinder Meridian Power ≥ 0.00 D > - 2.00 D (minus cylinder convention) > -4.50 D (minus cylinder convention) ≤ -2.00 D ≤ -4.50 D From - 6.50 D to + 6.50 D ± 0.13 D ± 0.13 D ± 0.15 D ± 4% Stronger than ± 6.50 D ± 2% ± 0.13 D ± 0.15 D ± 4% 2. Tolerance on Distance Refractive Power (Progressive Addition Lenses) Cylinder Cylinder Sphere Meridian Power Tolerance on Sphere Cylinder Meridian Power ≥ 0.00 D > - 2.00 D (minus cylinder convention) > -3.50 D (minus cylinder convention) ≤ -2.00 D ≤ -3.50 D From -8.00 D to +8.00 D ± 0.16 D ± 0.16 D ± 0.18 D ± 5% Stronger than ±8.00 D ± 2% ± 0.16 D ± 0.18 D ± 5% 3. Tolerance on the direction of cylinder axis Nominal value of the ≥ 0.12 D > 0.25 D > 0.50 D > 0.75 D < 0.12 D > 1.50 D cylinder power (D) ≤ 0.25 D ≤ 0.50 D ≤ 0.75 D ≤ 1.50 D Tolerance of the axis Not Defined ° ° ° ° ° (degrees) ± 14 ± 7 ± 5 ± 3 ± 2 4. Tolerance on addition power for multifocal and progressive addition lenses Nominal value of addition power (D) ≤ 4.00 D > 4.00 D Nominal value of the tolerance on the addition power (D) ± 0.12 D ± 0.18 D 5. Tolerance on Prism Reference Point Location and Prismatic Power • The prismatic power measured at the prism reference point shall not exceed 0.33Δ or the prism reference point shall not be more than 1.0 mm away from its specified position in any direction. -
Area, Volume and Surface Area
The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 YEARS 810 Area, Volume and Surface Area (Measurement and Geometry: Module 11) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 810 {4} A guide for teachers AREA, VOLUME AND SURFACE AREA ASSUMED KNOWLEDGE • Knowledge of the areas of rectangles, triangles, circles and composite figures. • The definitions of a parallelogram and a rhombus. • Familiarity with the basic properties of parallel lines. • Familiarity with the volume of a rectangular prism. • Basic knowledge of congruence and similarity. • Since some formulas will be involved, the students will need some experience with substitution and also with the distributive law. -
Riemann Surfaces
RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them. -
Visual Cortex: Looking Into a Klein Bottle Nicholas V
776 Dispatch Visual cortex: Looking into a Klein bottle Nicholas V. Swindale Arguments based on mathematical topology may help work which suggested that many properties of visual cortex in understanding the organization of topographic maps organization might be a consequence of mapping a five- in the cerebral cortex. dimensional stimulus space onto a two-dimensional surface as continuously as possible. Recent experimental results, Address: Department of Ophthalmology, University of British Columbia, 2550 Willow Street, Vancouver, V5Z 3N9, British which I shall discuss here, add to this complexity because Columbia, Canada. they show that a stimulus attribute not considered in the theoretical studies — direction of motion — is also syst- Current Biology 1996, Vol 6 No 7:776–779 ematically mapped on the surface of the cortex. This adds © Current Biology Ltd ISSN 0960-9822 to the evidence that continuity is an important, though not overriding, organizational principle in the cortex. I shall The English neurologist Hughlings Jackson inferred the also discuss a demonstration that certain receptive-field presence of a topographic map of the body musculature in properties, which may be indirectly related to direction the cerebral cortex more than a century ago, from his selectivity, can be represented as positions in a non- observations of the orderly progressions of seizure activity Euclidian space with a topology known to mathematicians across the body during epilepsy. Topographic maps of one as a Klein bottle. First, however, it is appropriate to cons- kind or another are now known to be a ubiquitous feature ider the experimental data. of cortical organization, at least in the primary sensory and motor areas. -
Graphs on Surfaces, the Generalized Euler's Formula and The
Graphs on surfaces, the generalized Euler's formula and the classification theorem ZdenˇekDvoˇr´ak October 28, 2020 In this lecture, we allow the graphs to have loops and parallel edges. In addition to the plane (or the sphere), we can draw the graphs on the surface of the torus or on more complicated surfaces. Definition 1. A surface is a compact connected 2-dimensional manifold with- out boundary. Intuitive explanation: • 2-dimensional manifold without boundary: Each point has a neighbor- hood homeomorphic to an open disk, i.e., \locally, the surface looks at every point the same as the plane." • compact: \The surface can be covered by a finite number of such neigh- borhoods." • connected: \The surface has just one piece." Examples: • The sphere and the torus are surfaces. • The plane is not a surface, since it is not compact. • The closed disk is not a surface, since it has a boundary. From the combinatorial perspective, it does not make sense to distinguish between some of the surfaces; the same graphs can be drawn on the torus and on a deformed torus (e.g., a coffee mug with a handle). For us, two surfaces will be equivalent if they only differ by a homeomorphism; a function f :Σ1 ! Σ2 between two surfaces is a homeomorphism if f is a bijection, continuous, and the inverse f −1 is continuous as well. In particular, this 1 implies that f maps simple continuous curves to simple continuous curves, and thus it maps a drawing of a graph in Σ1 to a drawing of the same graph in Σ2. -
The Hairy Klein Bottle
Bridges 2019 Conference Proceedings The Hairy Klein Bottle Daniel Cohen1 and Shai Gul 2 1 Dept. of Industrial Design, Holon Institute of Technology, Israel; [email protected] 2 Dept. of Applied Mathematics, Holon Institute of Technology, Israel; [email protected] Abstract In this collaborative work between a mathematician and designer we imagine a hairy Klein bottle, a fusion that explores a continuous non-vanishing vector field on a non-orientable surface. We describe the modeling process of creating a tangible, tactile sculpture designed to intrigue and invite simple understanding. Figure 1: The hairy Klein bottle which is described in this manuscript Introduction Topology is a field of mathematics that is not so interested in the exact shape of objects involved but rather in the way they are put together (see [4]). A well-known example is that, from a topological perspective a doughnut and a coffee cup are the same as both contain a single hole. Topology deals with many concepts that can be tricky to grasp without being able to see and touch a three dimensional model, and in some cases the concepts consists of more than three dimensions. Some of the most interesting objects in topology are non-orientable surfaces. Séquin [6] and Frazier & Schattschneider [1] describe an application of the Möbius strip, a non-orientable surface. Séquin [5] describes the Klein bottle, another non-orientable object that "lives" in four dimensions from a mathematical point of view. This particular manuscript was inspired by the hairy ball theorem which has the remarkable implication in (algebraic) topology that at any moment, there is a point on the earth’s surface where no wind is blowing. -
Natural Cadmium Is Made up of a Number of Isotopes with Different Abundances: Cd106 (1.25%), Cd110 (12.49%), Cd111 (12.8%), Cd
CLASSROOM Natural cadmium is made up of a number of isotopes with different abundances: Cd106 (1.25%), Cd110 (12.49%), Cd111 (12.8%), Cd 112 (24.13%), Cd113 (12.22%), Cd114(28.73%), Cd116 (7.49%). Of these Cd113 is the main neutron absorber; it has an absorption cross section of 2065 barns for thermal neutrons (a barn is equal to 10–24 sq.cm), and the cross section is a measure of the extent of reaction. When Cd113 absorbs a neutron, it forms Cd114 with a prompt release of γ radiation. There is not much energy release in this reaction. Cd114 can again absorb a neutron to form Cd115, but the cross section for this reaction is very small. Cd115 is a β-emitter C V Sundaram, National (with a half-life of 53hrs) and gets transformed to Indium-115 Institute of Advanced Studies, Indian Institute of Science which is a stable isotope. In none of these cases is there any large Campus, Bangalore 560 012, release of energy, nor is there any release of fresh neutrons to India. propagate any chain reaction. Vishwambhar Pati The Möbius Strip Indian Statistical Institute Bangalore 560059, India The Möbius strip is easy enough to construct. Just take a strip of paper and glue its ends after giving it a twist, as shown in Figure 1a. As you might have gathered from popular accounts, this surface, which we shall call M, has no inside or outside. If you started painting one “side” red and the other “side” blue, you would come to a point where blue and red bump into each other.