Math 582: GMT I (Fractals and Dynamics)
Lecture 7, Jan 22, 2014, Summary
6. Orthogonal projections (cont.)
Theorem (Marstrandsupercritical case) Let E R2 be a Borel set. Consider the
orthogonal projection P : R2 R to the line making angle with the horizon
Lecture 21: Principal Component Analysis
c Christopher S. Bretherton
Winter 2014
Ref: Hartmann Ch. 4
21.1
The covariance matrix and principal component analysis
Suppose S is an m n data matrix, in which the rst dimension is the spacelike dimension and the
Lecture 22: PCA: Implementation
c Christopher S. Bretherton
Winter 2014
Ref: Hartmann Ch. 4
22.1
Normalization
SVD of the data matrix gives left singular vectors that describe the patterns of
variability across locations, and right singular vectors that d
Lecture 24: Pattern recognition
c Christopher S. Bretherton
Winter 2014
24.1
Motivation
When we see and hear, we instantly recognize words, people, or other things.
This is a data processing task that we can also train computers to do. For
instance, imagi
Lecture 25: Model-data fusion: Sequential
estimation and a simple Kalman lter
c Christopher S. Bretherton
Winter 2014
Ref: Hartmann, Ch. 8
25.1
Motivation
Commonly, we need to use data to help control, forecast, or estimate governing
equations for a compl
Lecture 27: Kalman ball-tracking
c Christopher S. Bretherton
Winter 2014
27.1
Model equations and Kalman notation
Following Lecture 25, imagine we are tracking a batted ball moving under gravity. Let (rn , zn ) be the horizontal and vertical components of
Lecture 26: Theory of Kalman ltering
c Christopher S. Bretherton
Winter 2014
Ref: Hartmann, Ch. 8
26.1
Tracking a ball
Were playing center eld in a baseball game. The batter hits the ball toward
us. We need to quickly judge where it is going to land, so w
Math 582, Winter 2005
Assignment 2. Due Wednesday, Jan. 26.
Reading: Horn and Johnson, secs. 1.01.3, 1.5. Trefethen and Embree, secs. 12.
1. (a) Determine precisely the field of values of the n by n Jordan block with eigenvalue : 1 . . J = . . . 1 [Hint:
Math 582, Winter 2005
Assignment 3. Due Friday, Feb. 11.
Reading: Generalizations of the Field of Values .
1. Let A be a normal matrix whose eigenvalues lie on a line in the complex plane. Show that the polynomial numerical hull of degree 2 of A is equal
Math 582, Winter 2005
Assignment 4. Due Friday, Feb. 25.
Reading: Secs. 56 and 57 of T & E.
1. Use your 1-norm pseudospectral code to compute pseudospectra of the decay matrix associated with the Ehrenfest urn / random walk on a hypercube problem, taking,
Math 582: Special Topics in Numerical Analysis, Winter 2005
Some Possible Project Topics:
1. Consider an application area that you are interested in and where you think that nonnormality might have a signicant eect on the matrices or operators involved. I
Lecture 20: Multivariate data and the Singular
Value Decomposition
c Christopher S. Bretherton
Winter 2014
Ref: Hartmann Ch. 4
20.1
Multivariate data
Data is often multivariate, i. e. several unknowns varying as an sampling
coordinate is changed:
Space-ti
Lecture 17: Wavelet Analysis
c Christopher S. Bretherton
Winter 2014
Refs: Matlab Wavelet Toolbox help.
17.1
Introduction
Wavelets are an ecient tool for analyzing data that varies on a wide range
of scales, especially when the data is statistically non-s
Lecture 19: Wavelet compression of time series
and images
c Christopher S. Bretherton
Winter 2014
Ref: Matlab Wavelet Toolbox help.
19.1
Wavelet compression of a time series
The last section of wavelet leleccum notoolbox.m demonstrates the use of
wavelet
Math 582: GMT I (Fractals and Dynamics)
Lecture 8, Jan 24, 2014, Summary
II Intro to Ergodic Theory
For details, see the online text by Karl Petersen, referred below as [Pet].
Ergodic Theory is the theory of measure-preserving transformations and more gen
Math 582: GMT I (Fractals and Dynamics)
Lecture 3, Jan 10, 2014, Summary
2. Iterated Function Systems (cont.)
Let (X, ) be a complete metric space. An iterated function system (IFS) is a family of
contractions cfw_f1 , . . . , fm on X .
Symbolic represen
Math 582: GMT I (Fractals and Dynamics)
Lecture 2, Jan 8, 2014, Summary
1. Hausdor measure and dimension
For this material, see any of the reserve books, or the Bishop-Peres [BP] text, pp.5-7.
Let E Rd and > 0. A collection of sets (dont have to be open o
Math 582: GMT I (Fractals and Dynamics)
Lecture 1, Jan 6, 2014, Summary
Fractal geometry studies sets and measures which are very irregular, such as Cantortype sets and measures, nowhere dierentiable curves, etc. They naturally appear in many
branches of
Math 582: GMT I (Fractals and Dynamics)
Lecture 4, Jan 13, 2014, Summary
3. Open Set Condition
Denition. An IFS cfw_fj m is said to satisfy the Open Set Condition (OSC) if there
j =1
exists a non-empty open set O such that fj (O) O and fi (O) fj (O) = for
Math 582: GMT I (Fractals and Dynamics)
Lecture 6, Jan 17, 2014, Summary
5. Dimension of product sets (cont.)
Theorem. For A Rd and B Rn we have
dimH (A) + dimH (B ) dimH (A B ) dimH (A) + dimM (B ).
We proved the right inequality, which is elementary. I
Math 582: GMT I (Fractals and Dynamics)
Lecture 9, Jan 27, 2014, Summary
II Intro to Ergodic Theory (cont.)
Poincar recurrence theorem. Let (X, M, , T ) be a m.-p. s. and let E M with
e
(E ) > 0. Then for -a.e. x E there exists an innite sequence nj such
Math 582A
Assignment 1, due Friday, January 31
Winter 2014
You can do as much or as little as you like.
I. Connectedness of attractors
1. The attractor of an IFS cfw_f1 , f2 , consisting of two maps, is either connected (when f1 (K )
f2 (K ) = ), or tota
Math 582: GMT I (Fractals and Dynamics)
Lecture 5, Jan 15, 2014, Summary
4. Potential-theoretic methods (cont.)
For a nite Borel measure on Rd dene
d (y )
(x) =
and I () =
|x y |
(x) d(x) =
d (x) d(y )
.
|x y |
The latter is called the -energy of .
Prop
Lecture 18: Mutiresolution Wavelet Analysis
c Christopher S. Bretherton
Winter 2014
Refs: Matlab Wavelet Toolbox help.
18.1
Interpretation of Haar wavelets in terms of
lters
We can interpret the level-1 Haar detail and average vectors as ltering with
the
Math 582, Winter 2005
Assignment 1. Due Wednesday, Jan. 19.
Reading: Horn and Johnson, secs. 1.01.3, 1.5.
1. Let A be an n by n matrix. Show that the following two statements are equivalent: (a) A has a complete set of orthonormal eigenvectors; that is, A