Safety Procedures
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Wear Safety glasses
Make all adjustments with the machine at a dead stop.
Examine cutting procedure; plan cuts to avoid backing out of a kerf.
If the machine is variable speed, adjust blade speed as recomme
MATH 614
Dynamical Systems and Chaos
Lecture 10:
Bifurcation theory.
Bifurcation theory
The object of bifurcation theory is to study changes that
maps undergo as parameters change.
In the context of one-dimensional dynamics, we consider a
one-parameter fa
MATH 614
Dynamical Systems and Chaos
Lecture 11:
Maps of the circle.
Circle S 1 .
S 1 = cfw_(x, y ) R2 : |x|2 + |y |2 = 1
S 1 = cfw_z C : |z| = 1
T1 = R/Z
T1 = R/2Z
: S 1 [0, 2),
angular coordinate
: S 1 R/2Z R
(multi-valued function)
1
: R S 1,
(x) =
MATH 614
Dynamical Systems and Chaos
Lecture 15:
Markov partitions.
Solenoid.
General symbolic dynamics
Suppose f : X X is a dynamical system. Given a partition
of the set X into disjoint subsets X , A indexed by
elements of a nite set A, we can dene the
MATH 614
Dynamical Systems and Chaos
Lecture 12:
Maps of the circle (continued).
Subshifts of nite type (revisited).
Maps of the circle
T : S 1 S 1,
T an orientation-preserving homeomorphism.
Rotation number
Suppose T : S 1 S 1 is an orientation-preservin
MATH 614
Dynamical Systems and Chaos
Lecture 8:
Structural stability.
Sharkovskiis theorem.
Structural stability
Informally, a dynamical system is structurally
stable if its structure is preserved under small
perturbations. To make this notion formal, one
MATH 614
Dynamical Systems and Chaos
Lecture 4:
Itineraries.
Cantor sets.
Logistic map
The logistic map is any of the family of quadratic maps
F (x) = x(1 x) depending on the parameter R.
2
If > 1, then for any x < 0 the orbit x, F (x), F (x), . . .
is de
MATH 614
Dynamical Systems and Chaos
Lecture 2:
Periodic points.
Hyperbolicity.
Orbit
Let f : X X be a map dening a discrete dynamical
system. We use notation f n for the n-th iteration of f dened
inductively by f 1 = f and f n = f n1 f for n = 2, 3, . .
MATH 614
Dynamical Systems and Chaos
Lecture 3:
Classication of xed points.
Logistic map.
Periodic points
Denition. A point x X is called a xed point
of a map f : X X if f (x) = x.
A point x X is called a periodic point of a map
f : X X if f m (x) = x for
MATH 614
Dynamical Systems and Chaos
Lecture 7:
Compact sets.
Topological conjugacy (continued).
Denition of chaos (revisited).
Compact sets
Denition. A subset E of a topological space X is compact if
any covering of E by open sets admits a nite subcover.
MATH 614
Dynamical Systems and Chaos
Lecture 5:
Cantor sets (continued).
Metric and topological spaces.
Symbolic dynamics.
Cantor Middle-Thirds Set
General Cantor sets
Denition. A subset of the real line R is called a (general)
Cantor set if it is
nonemp
MATH 614
Dynamical Systems and Chaos
Lecture 6:
Symbolic dynamics (continued).
Topological conjugacy.
Denition of chaos.
Interior and boundary
Let X be a topological space. Any open set of the topology
containing a point x X is called a neighborhood of x.
MATH 614
Dynamical Systems and Chaos
Lecture 13:
Dynamics of linear maps.
Hyperbolic toral automorphisms.
Linear transformations
Any linear mapping L : Rn Rn is represented as
multiplication of an n-dimensional column vector by
a nn matrix: L(x) = Ax, whe
MATH 614
Dynamical Systems and Chaos
Lecture 14:
The horseshoe map.
Invertible symbolic dynamics.
Stable and unstable sets.
The Smale horseshoe map
Stephen Smale, 1960
F
F
The Smale horseshoe map
F
F
The Smale horseshoe map
F
The map F is contracting on D
Which of the following does the full faith and credit clause ensure?
a.
b.
c.
d.
Ensures that official governmental actions of one state is denied by other states.
All 50 states are required to run credit checks prior to granting residency.
States must gi
Cleveland Bailey
ECON 2302
06/27/2016
Assignment 3
1.
1. For each of the following absolute values of price elasticity of demand, indicate whether
demand is elastic, inelastic, perfectly elastic, perfectly inelastic, or unit elastic. In addition, determin
Cleveland Bailey
ECON 2302
06/27/2016
Assignment 3
1.
1. For each of the following absolute values of price elasticity of demand, indicate whether
demand is elastic, inelastic, perfectly elastic, perfectly inelastic, or unit elastic. In addition, determin
Cleveland Bailey
Technical Writing 2314
06/16/2016
Outline
My subject is Mutual Fund Investing (Road to Financial Independence). In my article will have
informative Information that will keep readers engaged with the material.
Competent financial literacy
Solution of homework assignment 2
Problem 1.
Let Xt and Yt be stationary processes with means X , Y and covariance functions X and Y respectively. We have for Zt = aXt + bYt : EZt = aEXt + bEYt = aX + bY = const, and
Z (h)
=
cov(Zt , Zt+h ) = cov(aXt + bY
MATERIALS 101
INTRODUCTION TO STRUCTURE AND PROPERTIES
WINTER 2012
Problem Set 6
Due: Tuesday, March 6, 11:00 AM
1.
TTT Diagrams
Based on the transformation diagrams for eutectoid steel shown below, what microstructure
would result from the following cool
Chapter 3
Motion in Two and Three Dimensions
Conceptual Problems
1
[SSM] Can the magnitude of the displacement of a particle be less
than the distance traveled by the particle along its path? Can its magnitude be
more than the distance traveled? Explain.
Quiz 7
1.
The correlation coefficient between the variables EcoFree and GDPpc imples that:
A. As one variable increases, the other decreases
B. As one variable decreases, the other increases
C. There is no association between the variables
D. As one varia
Amath 301
Homework 4: Due Friday 8/12 at 4pm
Summer 2016
Warning: This homework is long, do not wait until the last minute to
start or you will not finish. Also, given its length, it is probably a good idea
to clear all between exercises to free up memory
CS 3113 / Math 3403, Fall 2011
Assignment 1
Comments and Solutions
1. Write a Octave (or MATLAB) function to compute using the series
=4
X
1 1 1
(1)n
= 4 1 + + . .
2n + 1
3 5 7
n=0
The input to your program consist of a single tolerance , and the return v
MATH 614
Dynamical Systems and Chaos
Lecture 1:
Examples of dynamical systems.
A discrete dynamical system is simply a transformation
f : X X . The set X is regarded the phase space of the
system and the map f is considered the law of evolution over a
per