M3A23/M4A23. Specimen Paper


 Geoffrey Weaver
 1 years ago
 Views:
Transcription
1 UNIVERSITY OF LONDON Course: M3A23/M4A23 Setter: J. Lamb Checker: S. Luzzatto Editor: Editor External: External Date: March 26, 2009 BSc and MSci EXAMINATIONS (MATHEMATICS) MayJune 2008 M3A23/M4A23 Specimen Paper Setter s signature Checker s signature Editor s signature
2 UNIVERSITY OF LONDON BSc and MSci EXAMINATIONS (MATHEMATICS) MayJune 2008 This paper is also taken for the relevant examination for the Associateship. M3A23/M4A23 Specimen Paper Date: examdate Time: examtime Credit will be given for all questions attempted but extra credit will be given for complete or nearly complete answers. Calculators may not be used.
3 1. Let X be a metric space with metric (distance function) d : X X R. List the properties that d must satisfy in order to be a metric. (b) Give the definition of a contraction F : X X, with respect to the metric d. (d) (e) For each of the combination of properties listed below of a contraction F on a metric space X, verify whether the situation can arise, and if so give one explicit example of F : X X that realizes the listed properties: (i) F has a unique fixed point and X is complete. (ii) F has no fixed point and X is complete. (iii) F has a unique fixed point and X is not complete. (iv) F has no fixed point and X is not complete. Prove the following variant of the Contraction Mapping Theorem: Theorem. Let F : X X be a Lipschitz map on a complete metric space X. Suppose that F is not a contraction, but F m is a contraction for some m > 1. Then F has a unique fixed point and all x X converge to this fixed point with exponential speed. Give an example of a map F that satisfies the conditions set out in the Theorem stated under (d). 2. Give a definition of topological mixing for a circle map F : S 1 S 1. (b) (d) (e) Show that invertible circle maps are never topologically mixing. [You may use results that were derived in the course without proof, but any such results should be stated carefully.] Show that all expanding circle maps are topologically mixing. Describe the condition on the parameters a, b and c that must be satisfied for F (x) = (ax + b cos 2 (2πx) + c) mod 1 to be an orientation preserving and expanding circle map. Show that if the circle map F of part (d) is orientation preserving and expanding then F k has a k 1 fixed points. Show that this implies that F has periodic orbits with period k for all k Z +.
4 3. Let F : S 1 S 1 defined by F (x) = (ax + b cos 2 (2πx) + c) mod 1 with x [0, 1) = S 1 diffeomorphism. and parameters a, b, c R be an orientation preserving circle (b) (d) (e) Determine what conditions on the parameters a, b, and c must be satisfied for F to be an orientation preserving circle diffeomorphism. Give a formula for the rotation number ρ(f ) of F. Show that although this formula refers to a specific point x S 1, ρ(f ) is independent of the choice of this point x. Prove the following proposition: Proposition. If ρ(f ) is irrational then for all x S 1 and any m, n N with m > n the following holds: for every y S 1 the forward orbit {F k (y) k N} intersects the interval I = [F m (x), F n (x)]. [You may use without proof results about properties of circle maps with rational rotation number, but any such result should be stated carefully.] Give the definition of the ωlimit set ω(x) for the map F and show that ω(x) is independent of x if ρ(f ) is irrational. [Hint: use the Proposition stated in part of this question.] Discuss the structure of the ωlimit set of F when ρ(f ) is irrational. [You may use results from the course without proof, but any such results should be stated carefully.]
5 4. Consider the cubic map with λ > 0. f λ : R R, x λx(1 x 2 ), For small λ > 0, x = 0 is an asymptotically stable fixed point. Determine the parameter value at which this fixed point loses stability. Determine the type of local bifurcation that occurs. Is it a generic (typical) local bifurcation of onedimensional maps with one parameter? [Motivate your answer.] (b) (i) Find the smallest value of λ > 0 at which a periodic doubling bifurcation occurs (from a fixed point). Note that at this parameter value two period doubling bifurcations occur at the same time. Verify that the branches of period two orbits have the explicit expression ±1 λ = x 1 x. 2 (ii) (iii) Show that in a typical period doubling bifurcation, the amplitude of the period doubled orbit grows as a the square root of the bifurcation parameter. Verify whether such growth is also observed in the period doubling bifurcation of f λ in part (i) of this question. Discuss how the map f λ in the neighbourhood of a perioddoubling bifurcation point can be considered as the Poincaré return map of a flow on a twodimensional surface. Describe (and sketch!) the local geometry of this suspension flow and illustrate the perioddoubling bifurcation on it (in pictures). Determine for which values of the parameter λ, the map f λ restricted to the set of initial conditions for which orbits do not escape from [0, 1] is topologically conjugate to a full shift on two symbols. [You do not need to prove the conjugacy, but need to show by what criteria you determine λ and how the conjugacy is defined.]
6 5. Let Σ 3 denote the set of infinite sequences {ω i } i N whose entries ω i are taken from a set of three symbols {1, 2, 3}. (Σ 3, d µ ) is a metric space with d µ (ω, ω ) := m N δ(ω m, ω m) µ m, where µ > 1 and δ(a, b) = 0 if a = b and δ(a, b) = 1 if a b. Determine for which values of µ R the cylinder C α0,...,α n 1 := {ω Σ 3 ω i = α i, 0 i < n}. is the open ball in Σ 3 around α = α 0... α n 1... with radius µ 1 n. (b) Consider the piecewise linear map F : [0, 1] [0, 1] given by 2x if x I 1 F (x) = 3 4x if x I 2 2x 3 if x I 2 3 where I 1 = [0, 1 2 ], I 2 = [ 1 2, 3 4 ], and I 3 = [ 3 4, 1]. (i) Show that the dynamics of this map is semiconjugate to a threestate Markov chain with a Markov graph as given in the figure that is depicted below, where points x [0, 1] are coded as ω Σ 3 with ω n = k if F n (x) I k. [Hint: Show that I ω0...ω n 1 ( 2) n, where I ω0...ω n 1 denotes the set of initial conditions in [0, 1] that are represented by a symbolic string in Σ 3 starting with ω 0... ω n 1.] (ii) Determine whether the following statements are correct the Markov chain in (i) is topologically transitive F is topologically transitive Motivate your answer. [You may use results from the course without proof, but any such results should be stated carefully.] (iii) Show that F 4 has 34 fixed points by a different method than writing out all admissible periodic words of length 4 for the Markov chain.
UNIVERSITY OF BRISTOL. Mock exam paper for examination for the Degrees of B.Sc. and M.Sci. (Level 3)
UNIVERSITY OF BRISTOL Mock exam paper for examination for the Degrees of B.Sc. and M.Sci. (Level 3) DYNAMICAL SYSTEMS and ERGODIC THEORY MATH 36206 (Paper Code MATH36206) 2 hours and 30 minutes This paper
More informationDYNAMICAL SYSTEMS PROBLEMS. asgor/ (1) Which of the following maps are topologically transitive (minimal,
DYNAMICAL SYSTEMS PROBLEMS http://www.math.uci.edu/ asgor/ (1) Which of the following maps are topologically transitive (minimal, topologically mixing)? identity map on a circle; irrational rotation of
More informationM3/4A16. GEOMETRICAL MECHANICS, Part 1
M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 1 of 5 UNIVERSITY OF LONDON Course: M3/4A16 Setter: Holm Checker: Gibbons Editor: Chen External: Date: January 27, 2008 BSc and MSci EXAMINATIONS (MATHEMATICS)
More informationProof: The coding of T (x) is the left shift of the coding of x. φ(t x) n = L if T n+1 (x) L
Lecture 24: Defn: Topological conjugacy: Given Z + d (resp, Zd ), actions T, S a topological conjugacy from T to S is a homeomorphism φ : M N s.t. φ T = S φ i.e., φ T n = S n φ for all n Z + d (resp, Zd
More informationMATH 614 Dynamical Systems and Chaos Lecture 16: Rotation number. The standard family.
MATH 614 Dynamical Systems and Chaos Lecture 16: Rotation number. The standard family. Maps of the circle T : S 1 S 1, T an orientationpreserving homeomorphism. Suppose T : S 1 S 1 is an orientationpreserving
More informationRobustly transitive diffeomorphisms
Robustly transitive diffeomorphisms Todd Fisher tfisher@math.byu.edu Department of Mathematics, Brigham Young University Summer School, Chengdu, China 2009 Dynamical systems The setting for a dynamical
More informationDynamical Systems and Ergodic Theory PhD Exam Spring Topics: Topological Dynamics Definitions and basic results about the following for maps and
Dynamical Systems and Ergodic Theory PhD Exam Spring 2012 Introduction: This is the first year for the Dynamical Systems and Ergodic Theory PhD exam. As such the material on the exam will conform fairly
More informationProblem: A class of dynamical systems characterized by a fast divergence of the orbits. A paradigmatic example: the Arnold cat.
À È Ê ÇÄÁ Ë ËÌ ÅË Problem: A class of dynamical systems characterized by a fast divergence of the orbits A paradigmatic example: the Arnold cat. The closure of a homoclinic orbit. The shadowing lemma.
More informationDiscrete Math I Exam II (2/9/12) Page 1
Discrete Math I Exam II (/9/1) Page 1 Name: Instructions: Provide all steps necessary to solve the problem. Simplify your answer as much as possible. Additionally, clearly indicate the value or expression
More information1 Introduction Definitons Markov... 2
Compact course notes Dynamic systems Fall 2011 Professor: Y. Kudryashov transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Introduction 2 1.1 Definitons...............................................
More informationOne dimensional Maps
Chapter 4 One dimensional Maps The ordinary differential equation studied in chapters 13 provide a close link to actual physical systems it is easy to believe these equations provide at least an approximate
More informationOn the smoothness of the conjugacy between circle maps with a break
On the smoothness of the conjugacy between circle maps with a break Konstantin Khanin and Saša Kocić 2 Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4 2 Department of Mathematics,
More information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integrodifferential equations
More informationTiling Dynamical Systems as an Introduction to Smale Spaces
Tiling Dynamical Systems as an Introduction to Smale Spaces Michael Whittaker (University of Wollongong) University of Otago Dunedin, New Zealand February 15, 2011 A Penrose Tiling Sir Roger Penrose Penrose
More informationApril 13, We now extend the structure of the horseshoe to more general kinds of invariant. x (v) λ n v.
April 3, 005  Hyperbolic Sets We now extend the structure of the horseshoe to more general kinds of invariant sets. Let r, and let f D r (M) where M is a Riemannian manifold. A compact f invariant set
More information( ) ( ) ( ) ( ) Given that and its derivative are continuous when, find th values of and. ( ) ( )
1. The piecewise function is defined by where and are constants. Given that and its derivative are continuous when, find th values of and. When When of of Substitute into ; 2. Using the substitution, evaluate
More information2. Metric Spaces. 2.1 Definitions etc.
2. Metric Spaces 2.1 Definitions etc. The procedure in Section for regarding R as a topological space may be generalized to many other sets in which there is some kind of distance (formally, sets with
More informationPreCalculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and
PreCalculus: 1.1 1.2 Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and finding the domain, range, VA, HA, etc.). Name: Date:
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More information(Non)Existence of periodic orbits in dynamical systems
(Non)Existence of periodic orbits in dynamical systems Konstantin Athanassopoulos Department of Mathematics and Applied Mathematics University of Crete June 3, 2014 onstantin Athanassopoulos (Univ. of
More informationMASLOV INDEX. Contents. Drawing from [1]. 1. Outline
MASLOV INDEX JHOLOMORPHIC CURVE SEMINAR MARCH 2, 2015 MORGAN WEILER 1. Outline 1 Motivation 1 Definition of the Maslov Index 2 Questions Demanding Answers 2 2. The Path of an Orbit 3 Obtaining A 3 Independence
More informationTopological conjugacy and symbolic dynamics
Chapter 4 Topological conjugacy and symbolic dynamics Conjugacy is about when two maps have precisely the same dynamical behaviour. Equivalently it is about changing the coordinate system we use to describe
More informationMapping Class Groups MSRI, Fall 2007 Day 2, September 6
Mapping Class Groups MSRI, Fall 7 Day, September 6 Lectures by Lee Mosher Notes by Yael Algom Kfir December 4, 7 Last time: Theorem (Conjugacy classification in MCG(T. Each conjugacy class of elements
More informationOn low speed travelling waves of the KuramotoSivashinsky equation.
On low speed travelling waves of the KuramotoSivashinsky equation. Jeroen S.W. Lamb Joint with Jürgen Knobloch (Ilmenau, Germany) MarcoAntonio Teixeira (Campinas, Brazil) Kevin Webster (Imperial College
More informationMore On Exponential Functions, Inverse Functions and Derivative Consequences
More On Exponential Functions, Inverse Functions and Derivative Consequences James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 10, 2019
More informationUNIVERSITY OF MANITOBA
Question Points Score INSTRUCTIONS TO STUDENTS: This is a 6 hour examination. No extra time will be given. No texts, notes, or other aids are permitted. There are no calculators, cellphones or electronic
More informationThree hours THE UNIVERSITY OF MANCHESTER. 31st May :00 17:00
Three hours MATH41112 THE UNIVERSITY OF MANCHESTER ERGODIC THEORY 31st May 2016 14:00 17:00 Answer FOUR of the FIVE questions. If more than four questions are attempted, then credit will be given for the
More information1.3 Group Actions. Exercise Prove that a CAT(1) piecewise spherical simplicial complex is metrically flag.
Exercise 1.2.6. Prove that a CAT(1) piecewise spherical simplicial complex is metrically flag. 1.3 Group Actions Definition 1.3.1. Let X be a metric space, and let λ : X X be an isometry. The displacement
More informationCopyright 2018 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. 2/10
Prep for Calculus This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (281 topics + 125 additional topics) Real
More informationThe dynamics of circle maps
The dynamics of circle maps W. de Melo demelo@impa.br W. de Melo demelo@impa.br () Dynamical Systems, august 2010 1 / 18 Dynamics in the circle Circle Diffeomorphisms critical circle mappings covering
More informationM3P11/M4P11/M5P11. Galois Theory
BSc and MSci EXAMINATIONS (MATHEMATICS) MayJune 2014 This paper is also taken for the relevant examination for the Associateship of the Royal College of Science. M3P11/M4P11/M5P11 Galois Theory Date:
More informationOn Periodic points of area preserving torus homeomorphisms
On Periodic points of area preserving torus homeomorphisms Fábio Armando Tal and Salvador AddasZanata Instituto de Matemática e Estatística Universidade de São Paulo Rua do Matão 11, Cidade Universitária,
More informationMath Review for AP Calculus
Math Review for AP Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet
More informationMath Prep for Statics
Math Prep for Statics This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular
More informationORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY
ORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY AND ωlimit SETS CHRIS GOOD AND JONATHAN MEDDAUGH Abstract. Let f : X X be a continuous map on a compact metric space, let ω f be the collection of ωlimit
More informationHW 3 Solution Key. Classical Mechanics I HW # 3 Solution Key
Classical Mechanics I HW # 3 Solution Key HW 3 Solution Key 1. (10 points) Suppose a system has a potential energy: U = A(x 2 R 2 ) 2 where A is a constant and R is me fixed distance from the origin (on
More informationMTH4100 Calculus I. Lecture notes for Week 2. Thomas Calculus, Sections 1.3 to 1.5. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 2 Thomas Calculus, Sections 1.3 to 1.5 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Reading Assignment: read Thomas
More informationFunctions and relations
Functions and relations Relations Set rules & symbols Sets of numbers Sets & intervals Functions Relations Function notation Hybrid functions Hyperbola Truncus Square root Circle Inverse functions 2 Relations
More informationLecture 4. Entropy and Markov Chains
preliminary version : Not for diffusion Lecture 4. Entropy and Markov Chains The most important numerical invariant related to the orbit growth in topological dynamical systems is topological entropy.
More informationSection 4.1: Polynomial Functions and Models
Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial
More informationSolutions to Problem Set 1
Solutions to Problem Set 1 18.904 Spring 2011 Problem 1 Statement. Let n 1 be an integer. Let CP n denote the set of all lines in C n+1 passing through the origin. There is a natural map π : C n+1 \ {0}
More informationThe Mandelbrot Set. Andrew Brown. April 14, 2008
The Mandelbrot Set Andrew Brown April 14, 2008 The Mandelbrot Set and other Fractals are Cool But What are They? To understand Fractals, we must first understand some things about iterated polynomials
More informationMath 354 Transition graphs and subshifts November 26, 2014
Math 54 Transition graphs and subshifts November 6, 04. Transition graphs Let I be a closed interval in the real line. Suppose F : I I is function. Let I 0, I,..., I N be N closed subintervals in I with
More informationNATIONAL BOARD FOR HIGHER MATHEMATICS. Research Scholarships Screening Test. Saturday, January 20, Time Allowed: 150 Minutes Maximum Marks: 40
NATIONAL BOARD FOR HIGHER MATHEMATICS Research Scholarships Screening Test Saturday, January 2, 218 Time Allowed: 15 Minutes Maximum Marks: 4 Please read, carefully, the instructions that follow. INSTRUCTIONS
More informationMath 273 (51)  Final
Name: Id #: Math 273 (5)  Final Autumn Quarter 26 Thursday, December 8, 266: to 8: Instructions: Prob. Points Score possible 25 2 25 3 25 TOTAL 75 Read each problem carefully. Write legibly. Show all
More informationIntroduction to Ergodic Theory
Lecture I Crash course in measure theory Oliver Butterley, Irene Pasquinelli, Stefano Luzzatto, Lucia Simonelli, Davide Ravotti Summer School in Dynamics ICTP 2018 Why do we care about measure theory?
More informationPart II. Dynamical Systems. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2
More information(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);
STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend
More informationDynamical Systems and Ergodic Theory
MATH3606  MATHM606 Dynamical Systems and Ergodic Theory Teaching Block, 07/8 Lecturer: Prof. Alexander Gorodnik PART I: LECTURES 7 course web site: people.maths.bris.ac.uk/ mazag/ds7/ Copyright c University
More informationObjectives List. Important Students should expect test questions that require a synthesis of these objectives.
MATH 1040  of One Variable, Part I Textbook 1: : Algebra and Trigonometry for ET. 4 th edition by Brent, Muller Textbook 2:. Early Transcendentals, 3 rd edition by Briggs, Cochran, Gillett, Schulz s List
More informationClass IX Chapter 1 Number Sustems Maths
Class IX Chapter 1 Number Sustems Maths Exercise 1.1 Question Is zero a rational number? Can you write it in the form 0? and q, where p and q are integers Yes. Zero is a rational number as it can be represented
More informationDifferential equations: Dynamics and Chaos. Pierre Berger
Differential equations: Dynamics and Chaos Pierre Berger 2 Contents 1 Introduction 5 1.1 What is a dynamical system?.............................. 5 1.2 Plan............................................
More informationRudiments of Ergodic Theory
Rudiments of Ergodic Theory Zefeng Chen September 24, 203 Abstract In this note we intend to present basic ergodic theory. We begin with the notion of a measure preserving transformation. We then define
More informationMA4H4  GEOMETRIC GROUP THEORY. Contents of the Lectures
MA4H4  GEOMETRIC GROUP THEORY Contents of the Lectures 1. Week 1 Introduction, free groups, pingpong, fundamental group and covering spaces. Lecture 1  Jan. 6 (1) Introduction (2) List of topics: basics,
More informationObjectives List. Important Students should expect test questions that require a synthesis of these objectives.
MATH 1040  of One Variable, Part I Textbook 1: : Algebra and Trigonometry for ET. 4 th edition by Brent, Muller Textbook 2:. Early Transcendentals, 3 rd edition by Briggs, Cochran, Gillett, Schulz s List
More informationMATHEMATICS AS/M/P1 AS PAPER 1
Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks MATHEMATICS AS PAPER 1 Bronze Set B (Edexcel Version) CM Time allowed: 2 hours Instructions to candidates:
More informationOne Dimensional Dynamical Systems
16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with onedimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar
More informationSeminar In Topological Dynamics
Bar Ilan University Department of Mathematics Seminar In Topological Dynamics EXAMPLES AND BASIC PROPERTIES Harari Ronen May, 2005 1 2 1. Basic functions 1.1. Examples. To set the stage, we begin with
More informationMAS331: Metric Spaces Problems on Chapter 1
MAS331: Metric Spaces Problems on Chapter 1 1. In R 3, find d 1 ((3, 1, 4), (2, 7, 1)), d 2 ((3, 1, 4), (2, 7, 1)) and d ((3, 1, 4), (2, 7, 1)). 2. In R 4, show that d 1 ((4, 4, 4, 6), (0, 0, 0, 0)) =
More information1 The Existence and Uniqueness Theorem for FirstOrder Differential Equations
1 The Existence and Uniqueness Theorem for FirstOrder Differential Equations Let I R be an open interval and G R n, n 1, be a domain. Definition 1.1 Let us consider a function f : I G R n. The general
More information11 /2 12 /2 13 /6 14 /14 15 /8 16 /8 17 /25 18 /2 19 /4 20 /8
MAC 1147 Exam #1a Answer Key Name: Answer Key ID# Summer 2012 HONOR CODE: On my honor, I have neither given nor received any aid on this examination. Signature: Instructions: Do all scratch work on the
More informationFinal Exam Solutions
Final Exam Solutions Laurence Field Math, Section March, Name: Solutions Instructions: This exam has 8 questions for a total of points. The value of each part of each question is stated. The time allowed
More informationLecture 11 Hyperbolicity.
Lecture 11 Hyperbolicity. 1 C 0 linearization near a hyperbolic point 2 invariant manifolds Hyperbolic linear maps. Let E be a Banach space. A linear map A : E E is called hyperbolic if we can find closed
More informationPre AP Algebra. Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra
Pre AP Algebra Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra 1 The content of the mathematics standards is intended to support the following five goals for students: becoming
More informationAlgebra II CP Final Exam Review Packet. Calculator Questions
Name: Algebra II CP Final Exam Review Packet Calculator Questions 1. Solve the equation. Check for extraneous solutions. (Sec. 1.6) 2 8 37 2. Graph the inequality 31. (Sec. 2.8) 3. If y varies directly
More informationNational Quali cations
H 08 X747/76/ National Quali cations Mathematics Paper (NonCalculator) THURSDAY, MAY 9:00 AM 0:0 AM Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only to
More informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationPuiseux series dynamics and leading monomials of escape regions
Puiseux series dynamics and leading monomials of escape regions Jan Kiwi PUC, Chile Workshop on Moduli Spaces Associated to Dynamical Systems ICERM Providence, April 17, 2012 Parameter space Monic centered
More informationPart 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 nspace R n defined by u x = c for some unit vector u and constant c. The reflection
More informationPreCalculus. Curriculum (447 topics additional topics)
PreCalculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.
More informationVersion B QP114,1824, Calc ,App BD
MATH 00 Test Fall 06 QP,8, Calc..,App BD Student s Printed Name: _Key_& Grading Guidelines CUID: Instructor: Section # : You are not permitted to use a calculator on any portion of this test. You are
More informationStudy Guide for Benchmark #1 Window of Opportunity: March 411
Study Guide for Benchmark #1 Window of Opportunity: March 11 Benchmark testing is the department s way of assuring that students have achieved minimum levels of computational skill. While partial credit
More informationDifferential Topology Solution Set #2
Differential Topology Solution Set #2 Select Solutions 1. Show that X compact implies that any smooth map f : X Y is proper. Recall that a space is called compact if, for every cover {U } by open sets
More informationTest 3, Linear Algebra
Test 3, Linear Algebra Dr. Adam GrahamSquire, Fall 2017 Name: I pledge that I have neither given nor received any unauthorized assistance on this exam. (signature) DIRECTIONS 1. Don t panic. 2. Show all
More informationAlgebra 2. Curriculum (384 topics additional topics)
Algebra 2 This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.
More informationUNIVERSITY OF LONDON IMPERIAL COLLEGE LONDON
UNIVERSITY OF LONDON IMPERIAL COLLEGE LONDON BSc and MSci EXAMINATIONS (MATHEMATICS) MAY JUNE 23 This paper is also taken for the relevant examination for the Associateship. M3S4/M4S4 (SOLUTIONS) APPLIED
More informationHonors Advanced Math Final Exam 2009
Name Answer Key. Teacher/Block (circle): Kelly/H Olsen/C Olsen/F Verner/G Honors Advanced Math Final Exam 009 Lexington High School Mathematics Department This is a 90minute exam, but you will be allowed
More informationAlgebra II Vocabulary Alphabetical Listing. Absolute Maximum: The highest point over the entire domain of a function or relation.
Algebra II Vocabulary Alphabetical Listing Absolute Maximum: The highest point over the entire domain of a function or relation. Absolute Minimum: The lowest point over the entire domain of a function
More informationTernary Expansions of Powers of 2
Ternary Expansions of Powers of 2 Jeff Lagarias, University of Michigan Workshop on Discovery and Experimentation in Number Theory Fields Institute, Toronto (September 25, 2009) Topics Covered Part I.
More information27. Topological classification of complex linear foliations
27. Topological classification of complex linear foliations 545 H. Find the expression of the corresponding element [Γ ε ] H 1 (L ε, Z) through [Γ 1 ε], [Γ 2 ε], [δ ε ]. Problem 26.24. Prove that for any
More informationVersion A QP114,1824, Calc ,App BD
MATH 100 Test 1 Fall 016 QP11,18, Calc1.11.3,App BD Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed
More informationMathematics AKS
Integrated Algebra I A  Process Skills use appropriate technology to solve mathematical problems (GPS) (MAM1_A20091) build new mathematical knowledge through problemsolving (GPS) (MAM1_A20092) solve
More information62. Absolute Value, Square Roots, and Quadratic Equations. Vocabulary. Lesson. Example 1 Solve for x: x  4 = 8.1. Mental Math
Chapter 6 Lesson 62 Absolute Value, Square Roots, and Quadratic Equations BIG IDEA Geometrically, the absolute value of a number is its distance on a number line from 0. Algebraically, the absolute value
More informationMATH642. COMPLEMENTS TO INTRODUCTION TO DYNAMICAL SYSTEMS BY M. BRIN AND G. STUCK
MATH642 COMPLEMENTS TO INTRODUCTION TO DYNAMICAL SYSTEMS BY M BRIN AND G STUCK DMITRY DOLGOPYAT 13 Expanding Endomorphisms of the circle Let E 10 : S 1 S 1 be given by E 10 (x) = 10x mod 1 Exercise 1 Show
More informationEssential hyperbolicity versus homoclinic bifurcations. Global dynamics beyond uniform hyperbolicity, Beijing 2009 Sylvain Crovisier  Enrique Pujals
Essential hyperbolicity versus homoclinic bifurcations Global dynamics beyond uniform hyperbolicity, Beijing 2009 Sylvain Crovisier  Enrique Pujals Generic dynamics Consider: M: compact boundaryless manifold,
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationAlgebra 2 Honors Curriculum Pacing Guide
SOUTH CAROLINA ACADEMIC STANDARDS FOR MATHEMATICS The mathematical processes provide the framework for teaching, learning, and assessing in all high school mathematics core courses. Instructional programs
More informationCOARSE HYPERBOLICITY AND CLOSED ORBITS FOR QUASIGEODESIC FLOWS
COARSE HYPERBOLICITY AND CLOSED ORBITS FOR QUASIGEODESIC FLOWS STEVEN FRANKEL Abstract. We prove Calegari s conjecture that every quasigeodesic flow on a closed hyperbolic 3manifold has closed orbits.
More informationPrep for College Algebra
Prep for College Algebra This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (219 topics + 85 additional topics)
More informationPrep for College Algebra
Prep for College Algebra This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (219 topics + 85 additional topics)
More informationHow well do I know the content? (scale 1 5)
Page 1 I. Number and Quantity, Algebra, Functions, and Calculus (68%) A. Number and Quantity 1. Understand the properties of exponents of s I will a. perform operations involving exponents, including negative
More informationMath 41 Second Exam November 4, 2010
Math 41 Second Exam November 4, 2010 Name: SUID#: Circle your section: Olena Bormashenko Ulrik Buchholtz John Jiang Michael Lipnowski Jonathan Lee 03 (1111:50am) 07 (1010:50am) 02 (1:152:05pm) 04 (1:152:05pm)
More informationComparison of Virginia s College and Career Ready Mathematics Performance Expectations with the Common Core State Standards for Mathematics
Comparison of Virginia s College and Career Ready Mathematics Performance Expectations with the Common Core State Standards for Mathematics February 17, 2010 1 Number and Quantity The Real Number System
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationAlgebra 2 with Trigonometry
Algebra 2 with Trigonometry This course covers the topics shown below; new topics have been highlighted. Students navigate learning paths based on their level of readiness. Institutional users may customize
More informationA strongly invariant pinched core strip that does not contain any arc of curve.
A strongly invariant pinched core strip that does not contain any arc of curve. Lluís Alsedà in collaboration with F. Mañosas and L. Morales Departament de Matemàtiques Universitat Autònoma de Barcelona
More informationDefinition We say that a topological manifold X is C p if there is an atlas such that the transition functions are C p.
13. Riemann surfaces Definition 13.1. Let X be a topological space. We say that X is a topological manifold, if (1) X is Hausdorff, (2) X is 2nd countable (that is, there is a base for the topology which
More informationDYNAMICAL SYSTEMS. I Clark: Robinson. Stability, Symbolic Dynamics, and Chaos. CRC Press Boca Raton Ann Arbor London Tokyo
DYNAMICAL SYSTEMS Stability, Symbolic Dynamics, and Chaos I Clark: Robinson CRC Press Boca Raton Ann Arbor London Tokyo Contents Chapter I. Introduction 1 1.1 Population Growth Models, One Population 2
More informationPrep for College Algebra with Trigonometry
Prep for College Algebra with Trigonometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (246 topics +
More informationPart I: Multiple Choice Questions
Name: Part I: Multiple Choice Questions. What is the slope of the line y=3 A) 0 B) 3 ) C) 3 D) Undefined. What is the slope of the line perpendicular to the line x + 4y = 3 A) / B) / ) C)  D) 3. Find
More information