Hilbert s Metric and Gromov Hyperbolicity

Size: px
Start display at page:

Download "Hilbert s Metric and Gromov Hyperbolicity"

Transcription

1 Hilbert s Metric and Gromov Hyperbolicity Andrew Altman May 13,

2 1 HILBERT METRIC 2 1 Hilbert Metric The Hilbert metric is a distance function defined on a convex bounded subset of the n-dimensional Euclidean space R n. It was introduced as an example of a metric for which Euclidean straight lines are geodesics, and it generalizes Cayley s distance formula in the Cayley-Klein model of hyperbolic geometry. Two notable applications of the Hilbert metric are to Perron-Frobenius theory and to constructing Gromov hyperbolic spaces. This paper will focus on the latter of these applications. Definition. Let D be a bounded convex domain in R n, and for distinct x, y D, let a, b be the points at which the line xy intersects the boundary D of D where the order of the points is a, x, y, b. We define the Hilbert distance h as follows ( ) bx ay h(x, y) = log by ax where xy is the usual Euclidean distance. Furthermore by defining h(x, x) = 0 we obtain a metric on D. It is important to note t hat if one of the points x, y is on the boundary of D then h(x, y) = as is shown by a limiting process in We will denote the cross-ratio of four collinear points by [a, x, y, b] = bx ay by ax. Note that since the cross-ratio is preserved by linear fractional transformations, the Hilbert metric coincides with the Cayley-Klein metric on the open unit ball. 2 Gromov Hyperbolic The aim of this paper is to show, using the Hilbert metric, that a bounded convex domain which satisfies a certain intersecting chord property is Gromov hyperbolic (or δ-hyperbolic). By definition a Gromov hyperbolic space is a geodesic metric space in which all geodesic triangles are δ-thin, hence the notion of δ-hyperbolic.

3 3 INTERSECTING CHORDS PROPERTY 3 Definition. The Gromov product relative to z is (x, y) z = 1 (d(x, z) + d(y, z) d(x, y)). 2 Definition. A geodesic metric space (X, d) is said to be Gromov hyperbolic if for some δ 0 and for all w, x, y, z X (x, y) z min{(x, w) z, (w, y) z } δ We further expand the inequality which will be useful for later computations. Note we first multiplied both sides by 2 to take care of the 1 2 factor from the Gromov product. h(x, z) + h(z, y) h(x, y) min{h(x, z) + h(z, w) h(x, w), h(w, z) + h(z, y) h(w, y)} 2δ and rearranging the terms we obtain = min{h(x, z) h(x, w), h(z, y) h(w, y)} + h(w, z) 2δ h(x, y) + h(z, w) 2δ + h(x, z) + h(z, y) min{h(x, z) h(x, w), h(z, y) h(w, y)} = 2δ + h(x, z) + h(z, y) + max{h(x, w) h(x, z), h(w, y) h(z, y)} = 2δ + max{h(x, w) + h(z, y), h(w, y) + h(x, z)}. 3 Intersecting Chords Property Let c 1, c 2 be two intersecting chords in D; it is clear that these two interesting chords define a plane. Let l 1, l 1 and l 2, l 2 denote the respective lengths of the segments for which the chords c 1, c 2 have been divided. Then the domain D satisfies the intersecting chord property if for all δ > 0 there exists a constant C = C(D, δ) such that 1 C l 1l 1 C. l 2 l 2 We now consider the property of Menger curvature of any triple of points. Note that three points x, y, z in a plane not all collinear lie on a unique circle with radius R(x, y, z) = c 2 sin θ

4 4 THEOREM 4 where c is the length of the side of the triangle and θ the opposing angle in the triangle. Definition. Menger curvature of three points is defined to be the reciprocal of the radius R above and is denoted K(x, y, z). Proofs of the following statements can be found in [6]. Proposition. Let x 1, y 1 and x 2, y 2 be the respective endpoints of the chords c 1, c 2 defined above. Then l 1 l 1 l 2 l 2 = K(x 1, x 2, y 2 )K(y 1, x 2, y 2 ) K(x 2, x 1, y 1 )K(y 2, x 1, y 1 ) Corollary. Let D be a bounded convex domain in R n. Assume there is a constant C > 0 such that K(x, y, z) K(x, y, z ) C for any two distinct sets of triples in D all lying in the same 2-dimensional plane. Then D satisfies the intersecting chords property. 4 Theorem Theorem 1. Let h be the Hilbert metric and D be a bounded convex domain in R n satisfying the intersecting chords property. Then the metric space (D, h) is Gromov hyperbolic. Proof. Let D be as described such that the intersecting chord property holds for a constant C. Let y be a fixed reference point and x, z, w D any other three points. Then we obtain six chords which we will identify with by their cross-ratio, c 1 = [y, y, z, z ] c 2 = [y, y, w, w ] c 3 = [y, y, x, x ] c 4 = [w, w, x, x ] c 5 = [x, x, z, z ] c 6 = [w, w, z, z ]

5 4 THEOREM 5 Let H(x, z) = h(u, v)+h(w, y) h(u, w) h(v, y) then we must show that there is a constant independent of x, z, w such that min{h(x, z), H(z, w)} 2δ. First we can expand using the definition of h, ( ) xz zx wy yw xx ww zz yy H(x, z) = log xx zz ww yy xw wx zy yz ( ) xx xz yy yw zz zx ww wy = log xx xw yy yz zz zy zy yz By the intersecting chords property xx xx fractions), so that we have H(x, z) C + 2 log yw wy yz zy C xz xw ( ) xz yw zx wy. xw yz yz wx (and similarly for the other Furthermore y is a fixed point so is bounded as well giving ( ) xz H(x, z) C zx + 2 log. xw wx Hence we it suffices to check that { } xz zx min xw wx, zx xz zw wz is bounded. Without loss of generality we may assume zz xx by symmetry. Case 1: (xw xx or zw xx ) If xw zw then xz zx xz + zz )(zx + xx ) xw wx (xw) 2 (xw + wz + zz )(zw + wx + xx ) (xw) 2 (3xw)2 (xw) 2 = 9 Similarly the same argument applies for zw xw.

6 4 THEOREM 6 Case 2: (xw xx and zw xx ) Then since xx xz C(xx xw ), xz zx xw wx C xx zx xx wx C xx (xw + wz + xx ) wx xx 3C where zx = zx + xx xw + wz + xx since the sum of lengths of two sides of the triangle xwz are greater than or equal to the length of the third. Lastly we note that if D is the n-dimensional open unit ball B n = {(x 1,..., x n ) R n : n x 2 i < 1}. then by the Corollary (which is aided by intuition from the Proposition), the intersecting chords property holds for C = 1, and consequently the Theorem implies that the Cayley-Klein model of n-dimensional hyperbolic space is Gromov hyperbolic. i=1

7 REFERENCES 7 References [1] A. F. Beardon. The dynamics of contractions. Ergodic Theory Dynam. Systems, 17(6): , [2] Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, [3] Herbert Busemann. The geometry of geodesics. Academic Press Inc., New York, N. Y., [4] Herbert Busemann and Paul J. Kelly. Projective geometry and projective metrics. Academic Press Inc., New York, N. Y., [5] Ren Guo. A characterization of hyperbolic geometry among Hilbert geometry. J. Geom., 89(1-2):48 52, [6] Anders Karlsson and Guennadi A. Noskov. The Hilbert metric and Gromov hyperbolicity. Enseign. Math. (2), 48(1-2):73 89, 2002.

THE HILBERT METRIC AND GROMOV HYPERBOLICITY

THE HILBERT METRIC AND GROMOV HYPERBOLICITY THE HILBERT METRIC AND GROMOV HYPERBOLICITY by Anders Karlsson 1 ) and Guennadi A. Noskov ) Abstract. We give some sufficient conditions for Hilbert s metric on convex domains D to be Gromov hyperbolic.

More information

Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations

Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.

More information

COMP 175 COMPUTER GRAPHICS. Lecture 04: Transform 1. COMP 175: Computer Graphics February 9, Erik Anderson 04 Transform 1

COMP 175 COMPUTER GRAPHICS. Lecture 04: Transform 1. COMP 175: Computer Graphics February 9, Erik Anderson 04 Transform 1 Lecture 04: Transform COMP 75: Computer Graphics February 9, 206 /59 Admin Sign up via email/piazza for your in-person grading Anderson@cs.tufts.edu 2/59 Geometric Transform Apply transforms to a hierarchy

More information

A NOTION OF NONPOSITIVE CURVATURE FOR GENERAL METRIC SPACES

A NOTION OF NONPOSITIVE CURVATURE FOR GENERAL METRIC SPACES A NOTION OF NONPOSITIVE CURVATURE FOR GENERAL METRIC SPACES MIROSLAV BAČÁK, BOBO HUA, JÜRGEN JOST, MARTIN KELL, AND ARMIN SCHIKORRA Abstract. We introduce a new definition of nonpositive curvature in metric

More information

Pythagorean Property and Best Proximity Pair Theorems

Pythagorean Property and Best Proximity Pair Theorems isibang/ms/2013/32 November 25th, 2013 http://www.isibang.ac.in/ statmath/eprints Pythagorean Property and Best Proximity Pair Theorems Rafa Espínola, G. Sankara Raju Kosuru and P. Veeramani Indian Statistical

More information

Results from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2000

Results from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2000 2000k:53038 53C23 20F65 53C70 57M07 Bridson, Martin R. (4-OX); Haefliger, André (CH-GENV-SM) Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles

More information

Ceva s and Menelaus Theorems Characterize the Hyperbolic Geometry Among Hilbert Geometries

Ceva s and Menelaus Theorems Characterize the Hyperbolic Geometry Among Hilbert Geometries Ceva s and Menelaus Theorems Characterize the Hyperbolic Geometry Among Hilbert Geometries József Kozma and Árpád Kurusa Abstract. If a Hilbert geometry satisfies a rather weak version of either Ceva s

More information

Available online at J. Nonlinear Sci. Appl., 10 (2017), Research Article

Available online at   J. Nonlinear Sci. Appl., 10 (2017), Research Article Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2719 2726 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa An affirmative answer to

More information

Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions

Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions David R. Wilkins Copyright c David R. Wilkins 2000 2010 Contents 4 Vectors and Quaternions 47 4.1 Vectors...............................

More information

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math (001) - Term 181 Recitation (1.1)

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math (001) - Term 181 Recitation (1.1) Recitation (1.1) Question 1: Find a point on the y-axis that is equidistant from the points (5, 5) and (1, 1) Question 2: Find the distance between the points P(2 x, 7 x) and Q( 2 x, 4 x) where x 0. Question

More information

arxiv: v2 [math.dg] 3 Sep 2014

arxiv: v2 [math.dg] 3 Sep 2014 HOLOMORPHIC HARMONIC MORPHISMS FROM COSYMPLECTIC ALMOST HERMITIAN MANIFOLDS arxiv:1409.0091v2 [math.dg] 3 Sep 2014 SIGMUNDUR GUDMUNDSSON version 2.017-3 September 2014 Abstract. We study 4-dimensional

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

New York Journal of Mathematics. Natural maps between CAT(0) boundaries

New York Journal of Mathematics. Natural maps between CAT(0) boundaries New York Journal of Mathematics New York J. Math. 19 (2013) 13 22. Natural maps between CAT(0) boundaries Stephen M. Buckley and Kurt Falk Abstract. It is shown that certain natural maps between the ideal,

More information

Intermediate Math Circles Wednesday October Problem Set 3

Intermediate Math Circles Wednesday October Problem Set 3 The CETRE for EDUCTI in MTHEMTICS and CMPUTIG Intermediate Math Circles Wednesday ctober 24 2012 Problem Set 3.. Unless otherwise stated, any point labelled is assumed to represent the centre of the circle.

More information

GENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS

GENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS GENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS PALLAVI DANI 1. Introduction Let Γ be a finitely generated grou and let S be a finite set of generators for Γ. This determines a word metric on

More information

Fuchsian groups. 2.1 Definitions and discreteness

Fuchsian groups. 2.1 Definitions and discreteness 2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

More information

The Sphere OPTIONAL - I Vectors and three dimensional Geometry THE SPHERE

The Sphere OPTIONAL - I Vectors and three dimensional Geometry THE SPHERE 36 THE SPHERE You must have played or seen students playing football, basketball or table tennis. Football, basketball, table tennis ball are all examples of geometrical figures which we call "spheres"

More information

Lagrange Multipliers

Lagrange Multipliers Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each

More information

A strong Schottky Lemma for nonpositively curved singular spaces

A strong Schottky Lemma for nonpositively curved singular spaces A strong Schottky Lemma for nonpositively curved singular spaces To John Stallings on his sixty-fifth birthday Roger C. Alperin, Benson Farb and Guennadi A. Noskov January 22, 2001 1 Introduction The classical

More information

IYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas

IYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas IYGB Special Paper U Time: 3 hours 30 minutes Candidates may NOT use any calculator Information for Candidates This practice paper follows the Advanced Level Mathematics Core Syllabus Booklets of Mathematical

More information

The fundamental group of a visual boundary versus the fundamental group at infinity

The fundamental group of a visual boundary versus the fundamental group at infinity The fundamental group of a visual boundary versus the fundamental group at infinity Greg Conner and Hanspeter Fischer June 2001 There is a natural homomorphism from the fundamental group of the boundary

More information

1. Projective geometry

1. Projective geometry 1. Projective geometry Homogeneous representation of points and lines in D space D projective space Points at infinity and the line at infinity Conics and dual conics Projective transformation Hierarchy

More information

MATH 304 Linear Algebra Lecture 19: Least squares problems (continued). Norms and inner products.

MATH 304 Linear Algebra Lecture 19: Least squares problems (continued). Norms and inner products. MATH 304 Linear Algebra Lecture 19: Least squares problems (continued). Norms and inner products. Orthogonal projection Theorem 1 Let V be a subspace of R n. Then any vector x R n is uniquely represented

More information

GLOBAL, GEOMETRICAL COORDINATES ON FALBEL S CROSS-RATIO VARIETY

GLOBAL, GEOMETRICAL COORDINATES ON FALBEL S CROSS-RATIO VARIETY GLOBAL GEOMETRICAL COORDINATES ON FALBEL S CROSS-RATIO VARIETY JOHN R. PARKER & IOANNIS D. PLATIS Abstract. Falbel has shown that four pairwise distinct points on the boundary of complex hyperbolic -space

More information

0, otherwise. Find each of the following limits, or explain that the limit does not exist.

0, otherwise. Find each of the following limits, or explain that the limit does not exist. Midterm Solutions 1, y x 4 1. Let f(x, y) = 1, y 0 0, otherwise. Find each of the following limits, or explain that the limit does not exist. (a) (b) (c) lim f(x, y) (x,y) (0,1) lim f(x, y) (x,y) (2,3)

More information

MATH 52 FINAL EXAM SOLUTIONS

MATH 52 FINAL EXAM SOLUTIONS MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }

More information

Isometries of the Hilbert Metric

Isometries of the Hilbert Metric arxiv:1411.1826v1 [math.mg] 7 Nov 2014 Isometries of the Hilbert Metric Timothy Speer Abstract On any convex domain in R n we can define the Hilbert metric. A projective transformation is an example of

More information

2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time is

2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time is . If P(A) = x, P = 2x, P(A B) = 2, P ( A B) = 2 3, then the value of x is (A) 5 8 5 36 6 36 36 2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time

More information

General Relativity ASTR 2110 Sarazin. Einstein s Equation

General Relativity ASTR 2110 Sarazin. Einstein s Equation General Relativity ASTR 2110 Sarazin Einstein s Equation Curvature of Spacetime 1. Principle of Equvalence: gravity acceleration locally 2. Acceleration curved path in spacetime In gravitational field,

More information

7a3 2. (c) πa 3 (d) πa 3 (e) πa3

7a3 2. (c) πa 3 (d) πa 3 (e) πa3 1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

More information

Finite Presentations of Hyperbolic Groups

Finite Presentations of Hyperbolic Groups Finite Presentations of Hyperbolic Groups Joseph Wells Arizona State University May, 204 Groups into Metric Spaces Metric spaces and the geodesics therein are absolutely foundational to geometry. The central

More information

SYSTEM OF CIRCLES If d is the distance between the centers of two intersecting circles with radii r 1, r 2 and θ is the

SYSTEM OF CIRCLES If d is the distance between the centers of two intersecting circles with radii r 1, r 2 and θ is the SYSTEM OF CIRCLES Theorem: If d is the distance between the centers of two intersecting circles with radii r 1, r 2 and θ is the 2 2 2 d r1 r2 angle between the circles then cos θ =. 2r r 1 2 Proof: Let

More information

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016 Announcements Monday October 3: Discussion Assignment

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 31 OCTOBER 2018

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 31 OCTOBER 2018 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY OCTOBER 8 Mark Scheme: Each part of Question is worth marks which are awarded solely for the correct answer. Each

More information

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ

More information

Random Walks on Hyperbolic Groups III

Random Walks on Hyperbolic Groups III Random Walks on Hyperbolic Groups III Steve Lalley University of Chicago January 2014 Hyperbolic Groups Definition, Examples Geometric Boundary Ledrappier-Kaimanovich Formula Martin Boundary of FRRW on

More information

11.1 Three-Dimensional Coordinate System

11.1 Three-Dimensional Coordinate System 11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

MA651 Topology. Lecture 10. Metric Spaces.

MA651 Topology. Lecture 10. Metric Spaces. MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky

More information

LESSON 7.1 FACTORING POLYNOMIALS I

LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of

More information

1 Solving equations 1.1 Kick off with CAS 1. Polynomials 1. Trigonometric symmetry properties 1.4 Trigonometric equations and general solutions 1.5 Literal and simultaneous equations 1.6 Review 1.1 Kick

More information

THE RING OF POLYNOMIALS. Special Products and Factoring

THE RING OF POLYNOMIALS. Special Products and Factoring THE RING OF POLYNOMIALS Special Products and Factoring Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely Special Products - rules

More information

LECTURE 10, MONDAY MARCH 15, 2004

LECTURE 10, MONDAY MARCH 15, 2004 LECTURE 10, MONDAY MARCH 15, 2004 FRANZ LEMMERMEYER 1. Minimal Polynomials Let α and β be algebraic numbers, and let f and g denote their minimal polynomials. Consider the resultant R(X) of the polynomials

More information

KIRSZBRAUN-TYPE THEOREMS FOR GRAPHS

KIRSZBRAUN-TYPE THEOREMS FOR GRAPHS KIRSZBRAUN-TYPE THEOREMS FOR GRAPHS NISHANT CHANDGOTIA, IGOR PAK, AND MARTIN TASSY Abstract. A well-known theorem by Kirszbraun implies that all 1-Lipschitz functions f : A R n R n with the Euclidean metric

More information

Part IB GEOMETRY (Lent 2016): Example Sheet 1

Part IB GEOMETRY (Lent 2016): Example Sheet 1 Part IB GEOMETRY (Lent 2016): Example Sheet 1 (a.g.kovalev@dpmms.cam.ac.uk) 1. Suppose that H is a hyperplane in Euclidean n-space R n defined by u x = c for some unit vector u and constant c. The reflection

More information

ON A NEW SPECIES OF IMAGINARY QUANTITIES CONNECTED WITH A THEORY OF QUATERNIONS. William Rowan Hamilton

ON A NEW SPECIES OF IMAGINARY QUANTITIES CONNECTED WITH A THEORY OF QUATERNIONS. William Rowan Hamilton ON A NEW SPECIES OF IMAGINARY QUANTITIES CONNECTED WITH A THEORY OF QUATERNIONS By William Rowan Hamilton (Proceedings of the Royal Irish Academy, 2 (1844), 424 434.) Edited by David R. Wilkins 1999 On

More information

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Spring Quarter 2016 Announcements Project 1 due next Friday at

More information

A NOTE ON RANDOM HOLOMORPHIC ITERATION IN CONVEX DOMAINS

A NOTE ON RANDOM HOLOMORPHIC ITERATION IN CONVEX DOMAINS Proceedings of the Edinburgh Mathematical Society (2008) 51, 297 304 c DOI:10.1017/S0013091506001131 Printed in the United Kingdom A NOTE ON RANDOM HOLOMORPHIC ITERATION IN CONVEX DOMAINS FILIPPO BRACCI

More information

Group, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S,

Group, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, Group, Rings, and Fields Rahul Pandharipande I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ

More information

Existence Postulate: The collection of all points forms a nonempty set. There is more than one point in that set.

Existence Postulate: The collection of all points forms a nonempty set. There is more than one point in that set. Undefined Terms: Point Line Distance Half-plane Angle measure Area (later) Axioms: Existence Postulate: The collection of all points forms a nonempty set. There is more than one point in that set. Incidence

More information

MAT 419 Lecture Notes Transcribed by Eowyn Cenek 6/1/2012

MAT 419 Lecture Notes Transcribed by Eowyn Cenek 6/1/2012 (Homework 1: Chapter 1: Exercises 1-7, 9, 11, 19, due Monday June 11th See also the course website for lectures, assignments, etc) Note: today s lecture is primarily about definitions Lots of definitions

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

More information

Mathematics for Economists

Mathematics for Economists Mathematics for Economists Victor Filipe Sao Paulo School of Economics FGV Metric Spaces: Basic Definitions Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb. 2017 1 / 34 Definitions and Examples

More information

On Shalom Tao s Non-Quantitative Proof of Gromov s Polynomial Growth Theorem

On Shalom Tao s Non-Quantitative Proof of Gromov s Polynomial Growth Theorem On Shalom Tao s Non-Quantitative Proof of Gromov s Polynomial Growth Theorem Carlos A. De la Cruz Mengual Geometric Group Theory Seminar, HS 2013, ETH Zürich 13.11.2013 1 Towards the statement of Gromov

More information

Ball Versus Distance Convexity of Metric Spaces

Ball Versus Distance Convexity of Metric Spaces Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 45 004, No., 48-500. Ball Versus Distance Convexity of Metric Spaces Thomas Foertsch Institute for Mathematics, University

More information

5. Introduction to Euclid s Geometry

5. Introduction to Euclid s Geometry 5. Introduction to Euclid s Geometry Multiple Choice Questions CBSE TREND SETTER PAPER _ 0 EXERCISE 5.. If the point P lies in between M and N and C is mid-point of MP, then : (A) MC + PN = MN (B) MP +

More information

Math 31CH - Spring Final Exam

Math 31CH - Spring Final Exam Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate

More information

MATH 31BH Homework 1 Solutions

MATH 31BH Homework 1 Solutions MATH 3BH Homework Solutions January 0, 04 Problem.5. (a) (x, y)-plane in R 3 is closed and not open. To see that this plane is not open, notice that any ball around the origin (0, 0, 0) will contain points

More information

REFLECTIONS IN A EUCLIDEAN SPACE

REFLECTIONS IN A EUCLIDEAN SPACE REFLECTIONS IN A EUCLIDEAN SPACE PHILIP BROCOUM Let V be a finite dimensional real linear space. Definition 1. A function, : V V R is a bilinear form in V if for all x 1, x, x, y 1, y, y V and all k R,

More information

14.1. Multiple Integration. Iterated Integrals and Area in the Plane. Iterated Integrals. Iterated Integrals. MAC2313 Calculus III - Chapter 14

14.1. Multiple Integration. Iterated Integrals and Area in the Plane. Iterated Integrals. Iterated Integrals. MAC2313 Calculus III - Chapter 14 14 Multiple Integration 14.1 Iterated Integrals and Area in the Plane Objectives Evaluate an iterated integral. Use an iterated integral to find the area of a plane region. Copyright Cengage Learning.

More information

AREA OF IDEAL TRIANGLES AND GROMOV HYPERBOLICITY IN HILBERT GEOMETRY

AREA OF IDEAL TRIANGLES AND GROMOV HYPERBOLICITY IN HILBERT GEOMETRY AREA OF IDEAL TRIANGLES AND GROMOV HYPERBOLICITY IN HILBERT GEOMETRY B. COLBOIS, C. VERNICOS AND P. VEROVIC Introduction and statements The aim of this paper is to show, in the context of Hilbert geometry,

More information

1.1 Line Reflections and Point Reflections

1.1 Line Reflections and Point Reflections 1.1 Line Reflections and Point Reflections Since this is a book on Transformation Geometry, we shall start by defining transformations of the Euclidean plane and giving basic examples. Definition 1. A

More information

DOMAINS WHICH ARE LOCALLY UNIFORMLY LINEARLY CONVEX IN THE KOBAYASHI DISTANCE

DOMAINS WHICH ARE LOCALLY UNIFORMLY LINEARLY CONVEX IN THE KOBAYASHI DISTANCE DOMAINS WHICH ARE LOCALLY UNIFORMLY LINEARLY CONVEX IN THE KOBAYASHI DISTANCE MONIKA BUDZYŃSKA Received October 001 We show a construction of domains in complex reflexive Banach spaces which are locally

More information

Note: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2

Note: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2 Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and

More information

A crash course the geometry of hyperbolic surfaces

A crash course the geometry of hyperbolic surfaces Lecture 7 A crash course the geometry of hyperbolic surfaces 7.1 The hyperbolic plane Hyperbolic geometry originally developed in the early 19 th century to prove that the parallel postulate in Euclidean

More information

arxiv:math/ v1 [math.gr] 24 Oct 2005

arxiv:math/ v1 [math.gr] 24 Oct 2005 arxiv:math/0510511v1 [math.gr] 24 Oct 2005 DENSE SUBSETS OF BOUNDARIES OF CAT(0) GROUPS TETSUYA HOSAKA Abstract. In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group

More information

arxiv: v1 [math.dg] 28 Jun 2008

arxiv: v1 [math.dg] 28 Jun 2008 Limit Surfaces of Riemann Examples David Hoffman, Wayne Rossman arxiv:0806.467v [math.dg] 28 Jun 2008 Introduction The only connected minimal surfaces foliated by circles and lines are domains on one of

More information

Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg

Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg Undefined Terms: Point, Line, Incident, Between, Congruent. Incidence Axioms:

More information

Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)

Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a) Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5

More information

Plane hyperbolic geometry

Plane hyperbolic geometry 2 Plane hyperbolic geometry In this chapter we will see that the unit disc D has a natural geometry, known as plane hyperbolic geometry or plane Lobachevski geometry. It is the local model for the hyperbolic

More information

AN ELEMENTARY PROOF OF THE GROUP LAW FOR ELLIPTIC CURVES

AN ELEMENTARY PROOF OF THE GROUP LAW FOR ELLIPTIC CURVES AN ELEMENTARY PROOF OF THE GROUP LAW FOR ELLIPTIC CURVES Abstract. We give a proof of the group law for elliptic curves using explicit formulas. 1. Introduction In the following K will denote an algebraically

More information

2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3).

2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3). Circle 1. (i) Find the equation of the circle with centre ( 7, 3) and of radius 10. (ii) Find the centre of the circle 2x 2 + 2y 2 + 6x + 8y 1 = 0 (iii) What is the radius of the circle 3x 2 + 3y 2 + 5x

More information

Topological properties

Topological properties CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological

More information

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles.

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles. » ~ $ ) 7 x X ) / ( 8 2 X 39 ««x» ««! «! / x? \» «({? «» q «(? (?? x! «? 8? ( z x x q? ) «q q q ) x z x 69 7( X X ( 3»«! ( ~«x ««x ) (» «8 4 X «4 «4 «8 X «x «(» X) ()»» «X «97 X X X 4 ( 86) x) ( ) z z

More information

(x, y) = d(x, y) = x y.

(x, y) = d(x, y) = x y. 1 Euclidean geometry 1.1 Euclidean space Our story begins with a geometry which will be familiar to all readers, namely the geometry of Euclidean space. In this first chapter we study the Euclidean distance

More information

Math 341: Convex Geometry. Xi Chen

Math 341: Convex Geometry. Xi Chen Math 341: Convex Geometry Xi Chen 479 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca CHAPTER 1 Basics 1. Euclidean Geometry

More information

Solving equations UNCORRECTED PAGE PROOFS

Solving equations UNCORRECTED PAGE PROOFS 1 Solving equations 1.1 Kick off with CAS 1. Polynomials 1.3 Trigonometric symmetry properties 1.4 Trigonometric equations and general solutions 1.5 Literal equations and simultaneous equations 1.6 Review

More information

The Nearest Doubly Stochastic Matrix to a Real Matrix with the same First Moment

The Nearest Doubly Stochastic Matrix to a Real Matrix with the same First Moment he Nearest Doubly Stochastic Matrix to a Real Matrix with the same First Moment William Glunt 1, homas L. Hayden 2 and Robert Reams 2 1 Department of Mathematics and Computer Science, Austin Peay State

More information

Practice Problems for the Final Exam

Practice Problems for the Final Exam Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of

More information

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt

Jim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain

More information

Core Mathematics C12

Core Mathematics C12 Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C12 Advanced Subsidiary Tuesday 12 January 2016 Morning Time: 2 hours

More information

Core Mathematics C12

Core Mathematics C12 Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C12 Advanced Subsidiary Tuesday 12 January 2016 Morning Time: 2 hours

More information

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere. MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.

More information

Y. D. Chai and Young Soo Lee

Y. D. Chai and Young Soo Lee Honam Mathematical J. 34 (01), No. 1, pp. 103 111 http://dx.doi.org/10.5831/hmj.01.34.1.103 LOWER BOUND OF LENGTH OF TRIANGLE INSCRIBED IN A CIRCLE ON NON-EUCLIDEAN SPACES Y. D. Chai and Young Soo Lee

More information

c i x (i) =0 i=1 Otherwise, the vectors are linearly independent. 1

c i x (i) =0 i=1 Otherwise, the vectors are linearly independent. 1 Hilbert Spaces Physics 195 Supplementary Notes 009 F. Porter These notes collect and remind you of several definitions, connected with the notion of a Hilbert space. Def: A (nonempty) set V is called a

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include PUTNAM TRAINING POLYNOMIALS (Last updated: December 11, 2017) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

Math 234 Final Exam (with answers) Spring 2017

Math 234 Final Exam (with answers) Spring 2017 Math 234 Final Exam (with answers) pring 217 1. onsider the points A = (1, 2, 3), B = (1, 2, 2), and = (2, 1, 4). (a) [6 points] Find the area of the triangle formed by A, B, and. olution: One way to solve

More information

Identity. "At least one dog has fleas" is translated by an existential quantifier"

Identity. At least one dog has fleas is translated by an existential quantifier Identity Quantifiers are so-called because they say how many. So far, we've only used the quantifiers to give the crudest possible answers to the question "How many dogs have fleas?": "All," "None," "Some,"

More information

Detailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors

Detailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9 MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)

More information

SOLUTIONS TO ADDITIONAL EXERCISES FOR II.1 AND II.2

SOLUTIONS TO ADDITIONAL EXERCISES FOR II.1 AND II.2 SOLUTIONS TO ADDITIONAL EXERCISES FOR II.1 AND II.2 Here are the solutions to the additional exercises in betsepexercises.pdf. B1. Let y and z be distinct points of L; we claim that x, y and z are not

More information

Inexact alternating projections on nonconvex sets

Inexact alternating projections on nonconvex sets Inexact alternating projections on nonconvex sets D. Drusvyatskiy A.S. Lewis November 3, 2018 Dedicated to our friend, colleague, and inspiration, Alex Ioffe, on the occasion of his 80th birthday. Abstract

More information

Zangwill s Global Convergence Theorem

Zangwill s Global Convergence Theorem Zangwill s Global Convergence Theorem A theory of global convergence has been given by Zangwill 1. This theory involves the notion of a set-valued mapping, or point-to-set mapping. Definition 1.1 Given

More information

arxiv: v2 [math.mg] 6 Jul 2014

arxiv: v2 [math.mg] 6 Jul 2014 arxiv:1312.3214v2 [math.mg] 6 Jul 2014 The Chen-Chvátal conjecture for metric spaces induced by distance-hereditary graphs Pierre Aboulker and Rohan Kapadia Concordia University, Montréal, Québec, Canada

More information

A NOTE ON BILLIARD SYSTEMS IN FINSLER PLANE WITH ELLIPTIC INDICATRICES. Milena Radnović

A NOTE ON BILLIARD SYSTEMS IN FINSLER PLANE WITH ELLIPTIC INDICATRICES. Milena Radnović PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 74(88) (2003), 97 101 A NOTE ON BILLIARD SYSTEMS IN FINSLER PLANE WITH ELLIPTIC INDICATRICES Milena Radnović Communicated by Rade Živaljević

More information

Weak sharp minima on Riemannian manifolds 1

Weak sharp minima on Riemannian manifolds 1 1 Chong Li Department of Mathematics Zhejiang University Hangzhou, 310027, P R China cli@zju.edu.cn April. 2010 Outline 1 2 Extensions of some results for optimization problems on Banach spaces 3 4 Some

More information

Dynamics of Hilbert nonexpansive maps

Dynamics of Hilbert nonexpansive maps Dynamics of Hilbert nonexpansive maps Anders Karlsson February 19, 2013 Abstract In his work on the foundations of geometry, Hilbert observed that a formula which appeared in works by Beltrami, Cayley,

More information

MATHS 267 Answers to Stokes Practice Dr. Jones

MATHS 267 Answers to Stokes Practice Dr. Jones MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the

More information

Harmonic quadrangle in isotropic plane

Harmonic quadrangle in isotropic plane Turkish Journal of Mathematics http:// journals. tubitak. gov. tr/ math/ Research Article Turk J Math (018) 4: 666 678 c TÜBİTAK doi:10.3906/mat-1607-35 Harmonic quadrangle in isotropic plane Ema JURKIN

More information

On self-circumferences in Minkowski planes

On self-circumferences in Minkowski planes extracta mathematicae Article in press On self-circumferences in Minkowski planes Mostafa Ghandehari, Horst Martini Department of Mathematics, University of Texas at Arlington, TX 76019, U.S.A. Faculty

More information