MATH 4332/6313  Spring 2018
Introduction to Real Analysis, II
Course Info. View syllabus.
Week 1. Metric spaces. Open and closed
sets. Closure and interior.
Homework Set 1, due
February 1.
Week 2. Compactness. Completeness.
Homework Set 2, due
February 8.
Week 3. Compactness vs. sequential compactness.
Homework Set 3, due
February 15.
Week 4. Continuity and its
characterization. Continuity and compactness.
Homework Set 4, due
February 22.
Week 5. Metric completion. Baire's theorem.
Exam 1, March 1, in class. When you feel prepared,
try a practice
run.
Special office hours: Wed, Feb 28, 9:3010:50am.
Week 6. The Contraction Mapping Theorem.
Homework Set 5, due
March 8.
Week 7. Newtons's method and
finding solutions to ordinary differential equations.
Homework Set 6, due
March 22.
Week 8. Polynomial
approximation.
Taylor polynomials and Taylor series.
Homework Set 7, due
March 29.
Week 9.
Weierstrass theorem.
Exam 2, April 5, in class. When you feel prepared,
try a practice
run.
Week 10. Best
approximation with
polynomials. Equioscillation
condition. Chebyshev polynomials.
Homework Set 8, due
April 19.
Week
11.
Approximation with
trigonometric
polynomials. Fourier
series.
Homework Set 9, due
April 26.
Final
Exam, May 8, 8am11am. When you feel prepared,
try a practice
run.
MATH 4331/6312  Fall 2017
Introduction to Real Analysis
Course Info. View syllabus.
Office hours: PGH 604, Tu 11:30am12:30pm, We 12pm.
Week 1. The topology of
R^{n}. Cauchy sequences and completeness. Open and closed
sets.
Homework Set 1 (Solutions), due
date deferred to September 7.
Week 2. Closure of a set. Compactness.
Homework Set 2 (Solutions), due
September 14.
Week 3. Heine Borel property and other
properties of compact sets. Limits and continuity of functions.
Homework Set 3 (Solutions), due
September 21.
Week 4. Discontinuous
functions. Uniform continuity.
Connected sets.
Homework Set 4 (Solutions), due
September 28.
Week 5. Connectedness and the
Intermediate Value Theorem in higher dimensions.
Summary of the material in a handout.
Exam 1, October 5, in class. When you feel prepared,
try a practice
run (Solutions).
Week 6. Differentiablity and the
Mean Value Theorem.
Homework Set 5 (Solutions), due October
19.
Week 7. The Riemann integral and its
properties.
Homework Set 6 (Solutions), due October
26.
Week 8. The Fundamental Theorem of Calculus,
see handout. Normed vector spaces.
Homework Set 7 (Solutions), due November
2.
Week 9. Inner product spaces. CauchySchwarz
inequality. Relation between inner product and norm.
Exam 2, November 9, in class. Covers material
from the Intermediate Value Theorem (and its
higherdimensional generalization) to normed vector spaces.
When you feel prepared,
try a practice
run.
Week 10. Hölder and Minkowski
inequalities for functions and sequences.
Homework Set 8 (Solutions), due Tuesday, November
21.
Week 11. Limits of sequences of
functions. Uniform convergence. Completeness of C(K).
Homework Set 9 (Solutions), due November
30.
Week 12. Equicontinuity. Total
boundedness. Characterization of compact subsets in C(K).
Final exam. Dec. 12, 11am2pm, in
classroom. Material from Homework Sets 19. When you feel prepared,
try a practice
run (Solutions).
Review session on Tuesday, Dec 5, 10amnoon, PGH 646.

MATH 7321  Spring 2017
Functional Analysis II
Course Info. View syllabus.
Week 1. Recap of topologicval vector
spaces. Duality, quotient spaces and subspaces.
(Notes by Robert Mendez).
A characterization of separable Banach spaces (Notes by Worawit
Tepsan).
Week 2. Adjoints and annihilators
(Notes by Dylan DomelWhite).
Ranknullity, quotient spaces and reflexivity of subspaces (Notes by Zainab Alshair).
Week 3. Properties of reflexivity
(Notes by Nikolaos Mitsakos).
Characterization of reflexivity, weak sequential compactness (Notes by Wilfredo Molina).
Week 4. Properties of reflexivity
(Notes by Chandi Bhandari).
Consequences of weak sequential compactness for optimization problems
and
for dynamical systems (Notes by Wilfredo Molina).
Week 5.
Characterization of surjectivity in terms of the adjoint
(Notes by Qianfan Bai).
Operators on Banach spaces and Banach algebras (Notes by Worawit Tepsan).
Week 6.
Resolvent and spectrum
(Notes by Chandi Bhandari).
Projections and complemented subspaces (Notes by Dylan DomelWhite).
Week 7.
More examples of complemented subspaces
(Notes by Zainab Alshair).
Compact operators (Notes by Nikolaos Mitsakos).
Week 8.
Properties of compact operators
(Notes by Nikos Karantzas).
Approximating compact operators by finite rank ones (Notes by Qianfan Bai).
Week 9.
RieszFredholm theory
(Notes by Robert Mendez).
Approximating compact operators by finite rank ones (Notes by Wifredo Molina).
Week 10.
Spectral properties of compact operators and operators on
Hilbert spaces
(Notes by Chandi Bhandari).
Approximating compact operators by finite rank ones (Notes by Nikos Karantzas).
Week 11.
Test functions
(Notes by Worawit Tepsan).
The topology of the test function space (Notes by Dylan DomelWhite).
MATH 7320  Fall 2016
Functional Analysis
Course Info. View syllabus.
Office hours: PGH 604, Mo 1011am, We 11am12pm.
Week 1. Essentials of topology. From
semimetric to normed spaces, with examples
(Notes by Bernhard
Bodmann).
Continuity of linear maps and boundedness (Notes by Kazem
Safari).
Week 2. Completeness. Examples of Banach spaces
(Notes by Yaofeng Su).
Completions of metric spaces (Notes by Wilfredo Molina).
Week 3. Completions and extensions of bounded maps on normed spaces.
Generating topologies with maps (Notes by Worawit Tepsan).
Convergence of nets (Notes by Adrian
Radillo).
Week 4. From nets to filterbases (Notes by Sabrine Assi).
Countability and compactness (Notes by Nikolaos
Mitsakos).
Week 5. Topological vector spaces (Notes by Nikolaos Karantzas).
Separation properties (Notes by Dylan DomelWhite).
Week 6. Balanced neighborhoods of 0 (Notes by Duong Nguyen).
Finite dimensional subspaces, closedness and linear maps (Notes by Robert
Mendez).
Week 7. Finite dimensional subspaces (Notes by Qianfan Bai).
Seminorms and local bases (Notes by Jason
Duvall).
Week 8. From convex balanced local bases to seminorms (Notes by Grant Getzelman).
Characterization of locally convex TVS in terms of families of
seminorms (Notes by Robert
Mendez).
Week 9. Metrization (Notes by Robert
Mendez and by Kazem Safari).
Completeness and Baire categories (Notes by Dylan
DomelWhite
and by Nickos
Karantzas).
Week 10. Open mapping theorem (Notes by Yaofeng
Su, by Wilfredo Molina
and Qianfan Bai).
Closed graph theorem (Notes by Worawit
Tepsan
and by
Duong
Nguyen).
Week 11. Hahn Banach theorem (Notes by Worawit Tepsan
Su and by Adrian Radillo.
Masur's separation theorem (Notes by Adrian Radillo
and by
Sabrine
Assi).
Week 12. Strict separation (Notes by Qianfan Bai and by Yoafeng Su).
Weak topology (Notes by Duong Nguyen
and by
Sabrine Assi).
Week 13. Weak* topology (Notes by Dylan DomelWhite, by Jason Duvall, and by
Wilfredo
Molina).
Week 14. The KreinMilman theorem (Notes by Jason Duvall and by Kazem
Safari). Compactness, total boundedness and extreme points
(Notes by Jason Duvall, by Nikolaos Karantzas, and by
Nikolaos Mitsakos).
MATH 4332/6313  Spring 2016
Introduction to Real Analysis
Course Info. View syllabus.
Office hours: PH 604, Tu noon1pm, We 12pm.
Week 1. Metric spaces. Open and closed
sets.
Homework Set 1, due January
28 (Solutions).
Week 2. Boundedness and Cauchy sequences. Completeness.
Homework Set 2, due February
4 (Solutions).
Week 3. Compactness, sequential compactness
and total boundedness, completeness. Characterization of continuity.
Homework Set 3, due
February 11 (Solutions).
Week 4. The completion of a metric
space. Nowhere dense sets. Homework Set 4, due
February 18 (Solutions).
Week 5. Review on February 23 and Midterm
Exam 1 on February 25, 2:30pm, in class. Please
arrive early
to avoid traffic due to the debate.
After reviewing the material, do a practice
run (Solutions).
Week 6. The
contraction mapping theorem, Newton's method and the
solution to ordinary differential equations. Homework Set 5,
due March 10 (Solutions).
Week 7. Approximation by polynomials. Taylor series.
Homework Set 6,
due March 24 (Solutions).
Week 8. Review on March 29 and Midterm
Exam 2 on March 31, 2:30pm, in class.
After reviewing the material, do a practice
run (Solutions).
Week 9. Best approximation by
polynomials. Chebyshev polynomials.
Homework Set 7,
due April 14 (Solutions).
Week 10. Fourier series. Best approximation by
trigonometric polynomials. Convergence of Fourier series.
Homework Set
8, due April 21 (Solutions).
Week 11 Multivariable differential
calculus. Directional derivatives and differentiability. Local
inverse functions.
Homework Set
9, due May 3 (Solutions).
Review for the Final Exam on May 5, 2:304pm. Final
Exam on Tuesday, May 10, 25pm.
After reviewing the material, do a practice
run (Solutions).


MATH 4331  Fall 2015
Introduction to Real Analysis
Course Info. View syllabus.
Office hours: PGH 604, Tu 10:30amnoon, We 12pm.
Week 1. The topology of
R^{n}. Cauchy sequences and completeness. Open and closed
sets. Compactness.
Homework Set 1, due September 3.
Week 2. Limits of functions and continuity. Properties of continuous
functions.
Homework Set 2, due September 10.
Week 3. Uniform continuity. Compactness and
extremal values.
Connectedness and the Intermediate Value Theorem. Handout 1
contains a summary of the material on connectedness and
continuity.
Homework Set 3, due
September 17.
Week 4. Differentiation and
the Mean Value Theorem.
Homework Set 4, due
September 24.
Week 5. The Riemann integral and its properties.
Homework Set 5, due
October 1.
Week 6. Review on October 6 and Midterm
Exam on October 8, 2:30pm, in class. Please
arrive early
to avoid traffic due to the football game at 7pm.
After reviewing the material, do a practice
run.
Week 7. More properties of the
Riemann integral. The Fundamental Theorem of
Calculus.
Homework Set 6, due
October 22.
Week 8. Normed vector
spaces. Convergence and topology.
Homework Set 7, due
October 29.
Week 9. Inner
product spaces.
CauchySchwarz inequality and
consequences.
L^{p}norms.
Hölder and Minkowski inequalities.
Homework Set 8, due
November 5.
Week 10. Review on November 10 and Midterm
Exam 2 on November 12, 2:30pm, in class.
After reviewing the material, do a practice
run.
Week 11.
Convergence of sequences of
functions. Equicontinuous
families and compactness of subsets of C(K).
Homework Set 9, due
December 3.
Week 12. Review on December 8, 2:30pm and Final
Exam on December 10, 2pm, CBB 108.
After reviewing the material, do a practice
run.

MATH 1451H  Spring 2015
Accelerated Calculus, part II
Course Info. View syllabus.
Office: 604 PGH, (713) 743 3581; Hours: Mo, We 1:302:30pm.
Follow latest news and course status on Twitter @AccelCalcUH
Tentative Course Calendar (updated 50415)
Latest news:
Practice final posted. Special review session for the final on Thursday, May 7, 2:304pm,
in AH
106.
Week 
Sections to
be read, Exam dates 
Suggested homework problems 

12.112.2 
12.1
113 odd, 25, 29, 35
12.2 113 odd, 19, 23, 33

Jan 2630

12.312.5 
12.3 317 odd, 23, 25, 29, 43, 47, 49, 51
12.4
1, 3, 9, 23, 25, 29, 33, 35, 41, 45
12.5 119 odd, 23, 27, 31, 39, 47, 53, 67, 69

Feb 26 
13.113.3 
13.1
115 odd, 1923 odd, 27, 35, 39
13.2 5, 9, 11, 17, 19, 23, 29, 31, 35
13.3 1, 3, 11, 17, 19, 43, 45, 49, 59

Feb 913 
14.1, 14.2, 14.3 
14.1 1, 13, 25, 27, 31, 39, 47, 5559 odd
14.2
717 odd, 23, 27, 29, 31,
37
14.3
59 odd, 15, 17, 25, 33, 37, 43, 45, 49, 57, 61, 87, 95 bd 
Feb 1620 
14.4, 14.5 
14.4 1, 3, 13, 15, 25, 27, 33, 35, 39
14.5
1, 3, 7, 13, 15, 19, 23, 25, 27, 31, 35, 39, 49, 55 
Feb 2327


Review
Ch. 12 Ex 11, 15, 17, 19, 21
Review
Ch. 13 Ex 1, 3, 5, 9, 11, 15,
21
Review
Ch. 14 Ex 1, 5, 9, 13, 17, 23, 25, 39
14.6
1, 5, 7, 19, 27, 33, 37, 39, 43, 47 
Mar 26 
14.7, 14.8, 12.6 
14.7 1, 3, 9, 19, 21, 27, 29, 31, 37, 39, 41, 51
14.8 111 odd, 19, 21, 41, 45
12.6 1, 3, 9, 2127 odd 
Mar 913 
15.1, 15.2, 15.3 
15.1
11, 13
15.2 5, 13, 15, 19, 23, 25, 27
15.3 3, 9, 15, 25, 31, 33, 3943 odd, 47, 49, 51, 53, 55, 61

Mar 2327 
15.4, 15.5, 15.6 
15.4
1, 3, 5, 9, 11, 15, 17, 19, 25, 29, 31, 35
15.5 3, 7, 13, 27, 29
15.6 3, 11, 15, 21, 23 a, 29, 33, 39, 41, 51 
Mar 30Apr 3 
15.7, 15.8

15.7
1, 3, 5, 9, 15, 17, 21, 27
15.8 1, 3, 5, 9, 13, 15, 17, 19, 21, 23, 39

Apr 610 
Review, Apr 6
15.9

Review
Ch. 14 Ex 43, 45, 47, 51, 55, 59, 61
Review
Ch. 15 Ex 9, 13, 15, 17, 23, 25, 29, 31, 35, 39, 43, 45
15.9 1, 3, 5, 7, 11, 13, 23

Apr 1317 
16.1, 16.2, 16.3 
16.1 3, 11, 13, 15, 17, 23, 25
16.2
5, 11, 15, 17, 29(a), 41, 43
16.3 1, 3, 9, 11, 13, 21, 23, 27, 31


16.4, 16.5, 16.6 
16.4 1, 5, 7, 13, 17, 23, 29
16.5
1, 3, 5, 7, 9, 13, 15, 17, 19, 25, 29
16.6 1, 3, 11, 13, 15, 23, 39, 41, 43

Apr 27May 1 
16.7, 16.8, 16.9 
16.7 5, 7, 13, 17, 19, 21, 23, 27
16.8
1, 3, 5, 7, 9, 11(a,c), 13, 17
16.9 1, 5, 7, 11, 19, 23, 27

Exam period 
Review, May 7

Review
Ch. 16 Ex 3,
5, 7, 11, 13, 17, 19, 25, 27, 29, 33, 35, 41 
MATH 1450H  Fall 2014
Accelerated Calculus, part I
Course Info. View syllabus.
Office: 604 PGH, (713) 743 3581; Hours: Mo, We 1:302:30pm.
Follow latest news and course status on Twitter @AccelCalcUH
To make sure you are well prepared for the course,
please review the formula sheet
which contains essential facts you should know.
Visit the page on frequently asked questions
and answers to find out what is on the mind of your peers.
Course Calendar (updated 111014)
Latest news: Special review session for the Final on Monday, Dec 8, 121:30pm,
in AH 301.
Week 
Sections to
be read, Exam dates 
Suggested homework problems 
Aug 2529

Introduction, Review
1.3, 1.6, 2.32.5 
1.3
7, 11, 21, 31, 35, 43 1.6
17, 23, 25, 53, 71
2.3 3, 5, 7, 9, 15, 19,
25, 35, 57, 41
2.4 3, 13 a,b, 15, 17, 19,
25, 37, 41, 43
2.5 1, 3, 5, 11, 13,
17, 21, 23, 37, 39, 41, 43 a,b, 45, 65


3.13.2 
3.1
9,
11, 15, 19, 23, 25, 31, 35, 45, 51, 55, 59, 61, 73, 77
3.2 3,
5, 7, 11, 17, 21, 25, 27, 35 a, 43, 45, 51

Sep
812 
3.33.5 
3.3 323
odd, 29, 33, 37, 39, 41, 45
3.4
5,
11, 13, 19, 23, 25, 31, 35, 37, 39, 49, 59, 61, 71, 75, 79,
89
3.5 7,
11, 15, 21, 25, 27, 35, 39, 41, 45, 47, 65

Sep 1519 
4.4, 3.73.9 
4.4
5,
15, 21, 15, 29, 43, 69
3.7 7,
9, 11, 15, 17, 19, 21, 31
3.8 3, 5, 7, 9, 11, 13,
15, 17, 19
3.9 5, 7, 13, 15, 23, 25,
27, 29, 33, 35, 39, 43

Sep 2226 
4.14.3, 4.7 
4.1
9, 11, 19, 21, 27,
33, 39, 41, 53, 57, 59, 61, 75
4.2 3, 5, 11, 15, 17, 19, 21 a, 25 (explain
briefly), 27, 31, 35
4.3 5, 7, 11, 13, 15, 25, 27, 35, 41, 43,
51, 67, 81
4.7 11, 13,
19, 25, 31, 35, 37, 49, 65, 67

Sep
29Oct 3 

Review
Ch. 3 Ex 123 odd, 37, 39, 49, 53, 59, 69, 83, 87, 93, 97, 99
Review
Ch. 4 Ex 1, 5, 17, 25, 29, 37, 40, 47,
51, 75, 79, 81, 83
5.1 3, 5, 11, 13, 15, 17, 19,
21
5.2
1, 3, 7, 17, 19, 21, 23, 37, 39, 41, 45 (use
geometric series formula), 49,
53, 69, 71 
Oct
610 
5.3, 5.4, 5.5 
5.3 3,
5, 717 odd, 21, 29, 43, 53, 55, 63
5.4
9,
23, 33, 41, 43, 53, 63
5.5 3,
5, 717 odd, 21, 29, 43, 53, 55, 6369 odd,
75, 81 
Oct
1317 
7.17.3 
7.1
1, 3, 5, 7, 9, 15, 17, 19, 23, 27, 31,
43, 47
7.2 5, 7, 9, 15, 19, 23, 31, 37,
43, 47
7.3 1, 3, 7, 13, 17, 27, 29, 35 
Oct 2024 
7.4, 7.5, 7.8 
7.4
1, 3, 5, 11, 15, 17, 19, 25, 29, 39,
41, 59
7.5 3, 7, 9, 13, 17, 21, 23, 27,
31, 33, 41, 45, 49, 61
7.8 5, 9, 11,
17, 21, 25, 37, 49, 53, 59, 75

Oct
2731 

Review
Ch. 5 Ex 3, 5, 921 odd, 35, 43, 45, 49, 53, 61, 67
Review
Ch. 7 Ex 1, 3, 7, 13, 17, 29, 33, 37, 41, 43, 45
6.1
113 odd, 31, 45 ab, 49, 53
6.2
111 odd, 17, 31, 33, 35, 51, 53, 63, 65

Nov 37 
6.3, 8.1, 8.2 
6.3
3, 5, 7,
9, 11, 13, 21, 23, 25, 37, 39, 41
8.1 1, 5, 7, 9, 11, 13, 19,
31, 35
8.2 1, 3, 9, 13,
15, 25, 31, 33 
Nov 1014 
11.111.3

11.1
1,
5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37,
49, 51, 53, 59, 63
11.2 9, 11, 17, 25,
31, 35, 41, 47, 49, 59
11.3 3, 7, 11, 17,
27, 31, 33, 39

Nov 1721 
Review, Nov 17


11.4, 11.5 

Review
Ch. 6 Ex 1, 3, 5, 7, 9, 11, 13, 25
Review
Ch. 8 Ex 1, 3, 7, 15
Review
Ch. 11 Ex 115 odd, 27
11.4
1, 3, 6, 8, 12, 13, 15, 19, 27, 29,
30, 39, 42, 43, 45, 46
11.5 1, 3, 4, 6, 7, 9, 13, 17, 19, 21, 22, 23 

11.6, 11.7 
11.6
1, 4, 11, 13, 15, 19, 25, 27, 29, 31,
33, 35
11.7 1, 3, 5, 8, 9, 13, 17, 25, 27, 35, 39 
Dec 15 
11.811.10 
11.8
1,
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 33, 35, 37
11.9 1,
3, 5, 7, 9, 11, 13, 14, 15, 17, 21, 23
11.10
119 odd, 25, 27, 33, 55 
Dec 8, 11 
Review, Dec 8

Review
Ch. 11 Ex 17, 21, 23, 25, 31, 35, 37, 41, 43, 45, 47, 51, 59 

MATH 6398  Fall 2014
Information Theory with Applications
Course Info, Introduction. View syllabus
at the beginning of notes.
Office: 604 PGH, (713) 743 3581; Hours: Mo, We 1:302:30pm.
Week 1. A brief history of information theory. Probability basics.
Entropy as a measure of uncertainty. Conditional entropy.
Week 2. Additivity of entropy; entropy inequalitites; concavity of entropy
(notes).
Relative entropy (divergence). Relative entropy and the Neyman Pearson hypothesis test
(notes). Relative entropy
and data processing. Pinsker's inequality (notes).
Week 3. Mutual information and its properties. Markov chains and
mutual information (notes). Data processing for Markov chains. Asymptotic equipartitioning principle
for discrete memoryless sources. Block codes (notes).
Homework Set 1 is due on Tuesday, September 16.
Week 4. Block coding theorem. Converse of block coding (notes). Asymptotic
equipartitioning for stationary ergodic processes (notes).
Week 5. Lossless coding. Separable and prefix codes. Kraft inequality (notes).
Entropy bound for average codeword length. Code trees and optimality
(notes).
Week 6. The Huffman code and its construction. Discrete memoryless channels.
Jointly typical sets (notes). Channel coding
(notes).
Homework Set 2 is deferred to Tuesday, October 14.
Week 7. Channel coding theorem, continued (notes). Examples of channel capacities. Weak converse
to channel coding (notes).
Week 8. Rate distortion theory. Distortion measures and distortion typical sets
(notes).
Week 9. More rate distortion theory (notes). Continous sources. Differential entropy and
its properties
(notes).
Week 10. More properties of differential entropy (notes). Asymtpotic equipartitioning for continuous sources, lossy compression (notes).
Week 11. Channel coding for continuous channels. The additive white Gaussian noise (AWGN) channel
(notes). AWGN as worst case scenario (notes).
Homework Set 3 is due on Tuesday, November 11.
Week 12. Parallel additive white Gaussian noise channels. Independent noise components
and noise with correlated components (notes).
The capacity of the AWGN channel with fixed linear encoding (notes).
Week 13. Linear codes for parallel additive white noise channels (notes). Frames as codes (notes,
notes).
Homework Set 4 is due on Tuesday, December 9.

MATH 2331  Fall 2013
Linear Algebra
Course Info. View syllabus.
First week.
Reading assignment: Sections 1.1 and 1.2.
Overview. Linear systems and solutions. Reduced row echelon form.


Homework, due September 5, 4pm (Solution):
Section 1.1, Exercises 6, 8, 12, 20, 24, 30, 32. Section 1.2, Exercises 2 ad, 4, 8, 12, 16 ab, 22 ae.
Optional: You may use Matlab or Octave to perform elementary row operations.
In this case, install Matlab or Octave following the instructions in the
matlab tutorial
or in the Octave manual.
Download the mfiles in this directory (glitches have been fixed) to the directory where you installed
the software, start it and
use the diary on and diary off command to store
your work in a file, see the example diary.
The use of the elementary row operations is explained in the readme.
Print the diary file and attach the printout to your homework. 
Second week.
Reading assignment: Sections 1.3, 1.4 and 1.5.
Reduced row echelon form, pivot positions and columns, basic and free variables, parametric description of the solution set (another example of a diary showing the forward elimination phase). Vectors, arithmetic with vectors. Linear combinations, span of vectors. Relationship between linear systems and the span.
 
Homework, due September 12, 4pm (Solution):
Section 1.3, Exercises 2, 4, 6, 10, 12, 14, 16, 26. Section 1.4, Exercises 4, 8, 12, 14, 16, 18, 22, 26, 38.
The use of a software package for solving Problem 38 is recommended. You are allowed to eliminate unknowns in a pivot column in one step without showing all the elementary row operations separately. For this purpose, download
gauss.m and bgauss.m and place them with the other mfiles on your installation of Matlab or Octave. Read the updated
readme for the use of gauss.m (forward elimination) and bgauss.m (backward elimination). 
Third week.
Reading assignment: Sections 1.7, 1.8, 1.9.
Solution sets of linear systems. Linear independence. Linear transformations.
 
Homework, due September 19, 4pm (Solution):
Section 1.5, Exercises 2, 6, 10, 14, 16, 24 a, b and d, 30. Section 1.7, Exercises: 2, 4, 10, 18, 20, 22 a, c, d, 32.
Section 1.8, Exercises: 2, 4, 12.
Optional: Use the mfile transUH.m to see how linear transformations coming from
2x2 matrices behave. The command plots a polygonal path in the shape of UH and shows to which points the vectors on
this path are transformed. As usual, the readme has been updated with a description
of the new mfile.

Fourth week.
Reading assignment: Sections 2.1, 2.2.
Matrix operations, the matrix inverse.
 
Homework, due September 26, 4pm (Solution):
Section 1.8, Exercises 20, 22, 32, 38 (use of gauss.m and bgauss.m recommended).
Section 1.9, Exercises: 2, 16, 18, 20, 26, 38.
Section 2.1, Exercises: 4, 6, 8, 10.
Section 2.2, Exercises: 2, 10, 32.
You can use the Matlab command inv(A) to check your answer when inverting a matrix A.

Fifth week.
Reading assignment: Section 2.3, 3.1 and 3.2
Equivalent formulations of invertibility. Determinants
and how to compute them.

First inclass exam on Thursday, Oct 3, 2013, 4:005:20pm. Bring a blue book and your student ID!
Review: Sections 1.11.5, 1.71.9, 2.12.3.
Review session on Friday, Sep 27, 2013, 2:304:30pm; SEC 102.
After feeling confident about the material,
take a practice exam
(solution)
to see whether more review is needed.

Sixth week.
Reading assignment: Section 3.2 and 3.3
Properties of determinants. Cramer's rule, inverse formula and area/volume.

Homework, due October 10, 4pm (Solution):
Section 3.1, Exercises 10, 14, 16, 18, 38.
Section 3.2, Exercises: 6, 8, 12, 22, 24, 38.
Section 3.3, Exercises: 4, 6, 10, 12, 20, 24, 32.
You can use the Matlab command det(A) to check your answer when computing the determinant of a matrix A.

Seventh week.
Reading assignment: Section 4.1 to 4.3.
Vector spaces and subspaces, null space and column space.
See notes from the remainder of class.

Homework, due October 17, 4pm (Solution):
Section 4.1, Exercises 2, 6, 8, 12, 16, 18, 36.
Section 4.2, Exercises: 6, 10, 12, 14, 16, 24, 26, 40.
Section 4.3, Exercises: 8, 10.

Eighth week.
Reading assignment: Sections 4.3 and 4.4.
Bases and coordinates.

No homework this week!

Ninth week.
Reading assignment: Sections 4.4 and 4.5.
Coordinate mapping. Dimension of a vector space.

Homework, due October 31, 4pm (Solution):
Section 4.3, Exercises 14, 16, 20, 22 ae, 34.
Section 4.4, Exercises: 4, 8, 14, 16 ac, 28 (use basis {1,t,t^{2}, t^{3}}), 36.
Section 4.5, Exercises: 6, 8, 12, 18, 22.
Optional: Use the mfiles plotaxes.m,
plotvector.m, tip1.m and
plotbasisgrid.m to plot a coordinate system
and populate it with vectors, and to plot basis vectors and visualize the integer linear combinations
they produce. As usual, the readme has been updated with a description
of the new mfiles.

Tenth week.
Reading assignment: Sections 4.6 and 4.7.
Rank and row space. Change of basis.

Homework, due November 7, 4pm (Solution):
Section 4.6, Exercises 2, 4, 6, 8, 10, 14, 16, 18 ad, 20, 22, 24, 30.
Section 4.7, Exercises: 2, 6, 12, 14.
Extra credit problem in Matlab/Octave (5 points): 1*) Use the commands rand and round to make a random 10by10
matrix A with entries 0 or 1. Test using rref whether the matrix A has full rank. If not, repeat generating
random 10by10 matrices with entries 0 and 1 until it has full rank. Record how often you had to repeat.
2*) Next, form mbyn submatrices out of the first m
rows and the first n columns of A, for at least 6 different choices of m and n (neither m nor n repeats), by B=A(1:m,1:n). Comment on the rank of
the resulting matrices and explain why this happens. You do not need to submit a printout of your experiments,
just explain why the ranks of the submatrices come out the way they do.
Hints: What is the maximal possible rank of an mbyn matrix? Is this rank always assumed for the submatrices?

Eleventh week.
Reading assignment: Sections 5.1 and 5.2
Eigenvalues and eigenvectors. Characteristic polynomial.

Second inclass exam on Thursday, Nov 14, 2013, 4:005:20pm. Bring a blue book and your student ID!
Review: Sections 3.13.3, 4.14.7.
Review session on Friday, Nov 8, 2013, 2:304:30pm, in AH 104.
After feeling confident about the material,
take a practice exam
(solution)
to see whether more review is needed.

Twelfth week.
Reading assignment: Sections 5.2 and 5.3.
Bases of eigenvectors. Diagonalization.

Homework, due November 21, 4pm (Solution):
Section 5.1, Exercises 6, 8, 14, 18, 20, 22, 38 (use eig(A) to get eigenvalues of A).
Section 5.2, Exercises: 4, 6, 12, 14, 20, 28.
Section 5.3, Exercises: 2, 6, 18, 22.

Thirteenth week.
Reading assignment: Sections 6.1 to 6.3.
Orthogonality. Orthogonal projections.

Homework, due December 5, 4pm (Solution):
Section 6.1, Exercises 6, 12, 14, 20 ae, 24.
Section 6.2, Exercises: 2, 10, 12, 14, 20.
Section 6.3, Exercises: 2, 10, 12, 18, 22 ad.
Section 6.5, Exercises: 2, 10.
Extra credit problem in Matlab/Octave (8 points): 20 equations, 50 unknowns.
1*) Prove that a matrixvector equation Ax=b has a solution if the rank of the coefficient matrix A
equals the rank of the augmented matrix [A b].
2*) Use the commands rand and round (and other ways to manipulate entries) to generate a 20x50 random matrix
with entries 1 and 1, each occurring with probability 1/2. Test whether each 4 columns of this matrix are
linearly independent with the matlab file
full_submatrix_rank.m. If not, repeat generating random matrices until this is satisfied.
Document this matrix and the result of full_submatrix_rank.m in a printout.
3*) Pick a vector x in R^50 with two nonzero entries and compute b=Ax with your matrix A.
Your task is to recover x based only on the knowledge of A and b!
Step 1: Write a function find_columns.m which takes arguments A and b and outputs the two indices j and j'
of the nonzero entries in
x by checking whether for two columns a_{j} and a_{j'} of A,
b is in their span (use a strategy similar to the loops in full_submatrix_rank.m).
Step 2: Based on knowing *which* entries of x are nonzero, recover their values from A and b.
Document this by printing out your function find_columns.m and its result and show how you
recover x.

Fourteenth week.
Reading assignment: Sections 6.5.
Least squares. Interpolation vs. curve fitting.

Final exam on Thursday, Dec 19, 2013, 5:007:50pm. Bring two blue books and your student ID! Test your understanding of concepts with the TrueFalse Marathon from class. Review on Thursday, Dec 12, 24pm, in SEC 104. Practice with a mock final
(solution).


MATH 4331  Fall 2013
Introduction to Real Analysis
Course Info. View syllabus.
First week. Overwiew and review. Metric spaces. Homework: Assignment 1
is due on Thursday, September 5, 10am.
Second week. Open sets in metric spaces. Open balls, uniformly equivalent metrics.
Homework: Assignment 2
is due on Thursday, September 12, 10am.
Third week. Closed sets. Closed balls. Convergence.
Homework: Assignment 3
is due on Tuesday, September 24, 10am.
Fourth week. Closure and interior, boundary. No office
hours this Wednesday, Sept 18. Instead, we can discuss questions on Thursday, Sept 19, 11:301pm..
Fifth week. Convergence in Euclidean spaces, boundedness, complete metric spaces.
Homework: Assignment 4
is due on Thursday, October 3, 10am.
Sixth week. Compactness. Sequential compactness. Total boundedness. A summary of the material up to
now is available here.
Seventh week. Heine Borel and Bolzano Weierstrass. First Midterm Exam on Thursday, October 10, in class. Material covered
as in the 4 homework assignments, up to and including the definition of compactness and examples.
After reviewing, test your skills in a practice run.
Eighth week. Separability and continuity. Homework: Assignment 5
is due on Thursday, October 24, 10am.
Ninth week. Continuity and Euclidean spaces. Continuity of elementary functions.
Notes on continuity. Homework: Assignment 6
is due on Thursday, October 31, 10am.
Tenth week. Intermediate value theorem and contraction mapping pinciple.
Notes on connectedness and intermediate values.
Homework: Assignment 7
is deferred to Tuesday, November 12, 10am.
Eleventh week. Applications of the contraction mapping principle: Newton's method and
solutions of ordinary differential equations, see Notes.
Homework: Assignment 8
is due on Thursday, November 14, 10am.
Twelfth week. Riemann and Riemann Stieltjes integral. Notes on integration.
Second midterm exam on Thursday, November 21, in class. Material covered ranges from Assignment 5 to 8. View practice exam here.
Thirteenth week. Properties of the Riemann Stieltjes integral
(Notes).
Application of RiemannStieltjes integration in probability theory. Homework: Assignment 9
is due on Thursday, December 5, 10am.
Fourteenth week. The Riemann Stieltjes integral in probability theory and the Fundamental Theorem of Calculus.
Final exam, as scheduled by the registrar: Thursday, Dec 19, 11am2pm, in our usual classroom. Closed book. Review session on Thursday, Dec 12, 1011:30 in
our classroom. To see how well you are prepared,
take a practice exam.

MATH 4355  Spring 2013
Mathematics of Signal Representations
Course Info. View syllabus.
First week. Overview. Vector spaces. Functions as vectors. Linear independence. Bases. A
Matlab Cheatsheet may help with elementary Matlab operations needed for the homework, see also
the more verbose guide
Getting Started with Matlab. Homework: Assignment 1
is due on Thursday, Jan 24.
Second week. Fundamental inequalities in inner product spaces. Orthogonality, orthogonal projections.
Least squares property of orthogonal projections. Orthogonal projections and orthogonal subspaces. Homework: Assignment 2
is due on Thursday, Jan 31.
Third week. Fourier series and orthogonality. Trigonometric identities. Fourier coefficients of even
or odd functions. Homework: Assignment 3
is due on Thursday, Feb 7.
Fourth week. Conditions for pointwise convergence of Fourier series. Dirichlet kernel. Consequence of pointwise convergence: series expression for pi. Uniform convergence of Fourier series. Homework: Assignment 4
is deferred until Tuesday, Feb 19.
Fifth week. Fourier series on other intervals. Complex form of Fourier series. Parseval identity.
Convergence in square norm. The updated (3/5/13) course notes give a brief summary of the material up to the Fourier transform. Homework: Assignment 5
is deferred until Tuesday, Feb 26.
Sixth week. Fourier transform. Relating FT of related functions.
Plancherel theorem. Orthogonality of shifted copies of the sinc function.
Sampling theorem. Uniform and squarenorm convergence.
Homework: Assignment 6
is due Tuesday, Mar 5.
Midterm exam: March 7, 2:304:30pm, AH 12 (basement of
Agnes Arnold Hall), bring Student ID, pen, pencil. Formula sheet will be provided.
The material covers everything up to and including the Fourier transform
properties, and the Plancherel theorem. The sampling theorem is NOT included.
A practice exam may be
useful to check whether your review was successful.
Seventh week. Convolutions and filters. Convolutions and the Fourier transform. Causality.
Homework: Assignment 7
is deferred until Tuesday, April 2.
Eighth week. Analog versus digital filters. Decay of analog impulse response vs. decay
of coefficients for digital convolution. Oversampling. Homework:
Assignment 8
is deferred until Tuesday, April 9.
Ninth week. Spaces of piecewise constant functions, resolution levels,
and the Haar wavelet. Homework: Assignment 9
is deferred until Tuesday, April 16 .
Tenth week. Haar decomposition and reconstruction algorithms. Relationship between
coefficients in expansions using orthonormal bases for V_{j} or V_{j1} and W_{j1},
in terms of filtering and up/downsampling. Block diagrams.
Homework: Assignment 10
is due Thursday, April 18.
Eleventh week. Properties of Haar wavelets, decomposition and reconstruction. Filtering with wavelets. Vanishing coefficients in subband decomposition for piecewise constant functions. Multiresolution Analysis. Daubechies wavelets. Homework: Assignment 11
is due Thursday, April 25.
Review session: Tu, April 30, 2:304pm. Final Exam: Tu, May 7,
2:005pm (in our usual classroom, as scheduled by registrar), bring Student ID, pen, pencil. Formula sheet will be provided.
Exam topics: Inner product spaces,
L^{2}(R) and l^{2}(Z), orthonormal bases, orthogonal projections onto subspaces (with given orthonormal basis or vectorspace basis),
least squares approximations, convergence for sequences or series of functions (in L^{2}, pointwise, or uniform), Fourier series on [π,π] or on [a,a], real and complex form, symmetries, convergence
of Fourier series (unif, pointwise, in L^{2}), Parseval's identity,
Fourier transform, properties of the FT, sampling theorem for bandlimited functions, convolutions, causality, digital and analog filters, lowpass filters, Butterworth filters, (excluded material: periodic sequences, Discrete Fourier Transform and its properties), spaces of piecewise constant functions and corresponding orthogonal projections, Haar decomposition, Haar scaling function and wavelet, V_{j} and W_{j}, Haar reconstruction algorithm, expression in terms of filtering (discrete convolution) and up/downsampling,
multiresolution analysis, properties of scaling functions, twoscale relation, from scaling coefficients {p_{k}} to P(z), quadrature mirror filter condition. (Excluded: Daubechies wavelet).
MATH 6304  Fall 2012
Theory of Matrices
Course Info. View syllabus.
First week. Overwiew and review. Matrices as linear maps, range and kernel, rank and nullity.
Dot product and orthogonality. Orthogonal projection. Gram Schmidt procedure.
Trace and determinant. Eigenvalues and eigenvectors. Similarity.
Second week. Determinant and invertibility. Diagonalization. Sufficient and necessary
conditions for diagonalizability. Algebraic and geometric
multiplicity of eigenvalues.
Simultaneous diagonalization. Commuting matrices.
Third week. Invariant subspaces. Commuting families having a common eigenvector.
Families of diagonalizable matrices, simultaneous diagonalization and commutativity.
Hermitian and skewHermitian matrices. Polarization identity.
Unitary matrices. Householder transformations. Unitary equivalence.
Fourth week. Schur triangularization. Normal matrices. Unitary diagonalizability and normality.
Cayley Hamilton. Block diagonalization
with triangular blocks. Diagonalizable perturbations.
Fifth week. QR factorization and QR algorithm. Cholesky factorization.
Real matrices. Orthogonoal diagonalization for symmetric matrices. Block triangularization
for real matrices.
Sixth week. Block diagonalization for real orthogonal
matrices. Jordan normal form for nilpotent matrices.
Jordan form. Applications to matrix exponentials.
Seventh week. Variational characterization of eigenvalues for
Hermitian matrices. CourantFischer theorem. Weyl's theorem for eigenvalue
estimates of lowrank perturbations
Eighth week. Weyl's eigenvalue estimates for sums of Hermitian matrices.
Ninth week. An example for lowrank perturbations. Eigenvalue interlacing for principal submatrices
More eigenvalue interlacing. Generalized RayleighRitz principle.
Majorization.
Tenth week. Majorization of eigenvalues by diagonal entries.
Singular value decomposition. Singular values vs. eigenvalues.
Interlacing of singular values for submatrices.
Eleventh week. Interlacing of singular values for sums of matrices. Polar decomposition.
Leastsquares problem.
The normal equation for linear systems. Abbreviated singular value decomposition.
Twelfth week. Matrix norms. From the HilbertSchmidt inner product to the Euclidean/Frobenius norm.
Matrix norm and spectral radius.
Thanksgiving week. Gelfand formula and Gersgorin's circle theorem.
Fourteenth week. Refined estimates for Gersgorin disks. Consequences of Gersgorin for invertibility
of matrices. Introduction to frame theory.
MATH 4310/BIOL6317  Fall 2011
Biostatistics
Course Info. View syllabus.
First week. Overwiew. Probability measures. Computing probabilities.
Slides from first class.
Homework Set 1, due Thursday, September 1, 1pm.
Second week. Random variables. Cumulative distribution function. Quantiles.
Mean and variance. Chebyshev inequality. Independence. Optional recap on calculus essentials,
Wed, 1011am, 646 PGH.
Classroom change. From Tuesday, September 6, we are in Farish Hall, FH 135.
Homework Set 2, due Thursday, September 8, 1pm.
Third week. Variance of the sample mean. Standard error of the sample mean. Sample variance.
Sample standard error. Conditional probabilities. Bayes's rule. Sensitivity and specificity of a diagnostic test.
Homework Set 3, deferred to Tuesday, September 20, 1pm.
Fourth week. Diagnostic likelihood ratios and intepreting the outcome of a test result. Likelihoods. Likelihood ratios.
Example: coin flips and biasedness hypotheses. Binomial distribution and maximum likelihood estimator.
Introduction to R. For the next homework and in the future, you may find the notes on R by Dr. Peters, Basics,
Graphics,
Statistics Functions and Regression, etc
helpful.
Homework Set 4, due Thursday, September 22, 1pm.
Fifth week. Normal and standard normal distributions. Conversion between quantiles.
Computing probabilities in the standardized form. Maximum likelhood estimation of the mean for i.i.d normal random
variables with known variance. MLE for mean and variance. Law of large numbers and consistent estimators.
Central limit theorem.
Homework Set 5, deferred to Tuesday, October 4, 1pm.
Sixth week. Central limit theorem and estimating probabilities for finding sample averages in a given
interval. Confidence intervals for unknown means. Chisquared distribution and confidence intervals
for the variance.
Homework Set 6, due Thursday, October 6, 1pm.
Seventh week. Tdistribution and confindence intervals for small sample sizes.
Confidence intervals for paired observations.
Confidence intervals for the success probability of small sequences of Bernoulli trials.
Summary of lectures.
Homework Set 7, due Thursday, October 13, 1pm.
Midterm exam. October 20, 1pm. Note the date was changed by unanimous vote of all students in class! Bring your ID, a scientific calculator, and a pen. The Summary of the lectures relevant for the midterm has been updated.
Review session on Wednesday, Oct 19, 1:303pm, PGH 646.
Eighth week. Hypothesis testing. Zscore. Pvalue. Ttest. Equivalence between hypothesis testing (with twosided
alternative) and computing the confidence interval. Computing power. Paired observations.
Homework Set 8, due Thursday, November 3, 1pm. For students enrolled
in Biol6317, start working on Project 1 as part of the assignment. The project
is due November 10.
Ninth week. Testing for independent groups. Testing for equal variance.
Testing of binomial proportions. Testing for equality of binomial proportions between
independent groups.
Homework Set 9, due Thursday, November 10, 1pm. For students enrolled
in Biol6317, complete Project 1 as part of the assignment.
Tenth week. Relative risk. Delta method for estimating standard errors. Odds ratio.
Fisher's exact test. Hypergeometric distribution. One and twosided alternatives.
Computing pvalues with the Monte Carlo method.
Homework Set 10, due Thursday, November 17, 1pm. For students enrolled
in Biol6317, complete Project 2 as part of the assignment.
Eleventh week. Chisquared testing for equality of proportions and for independence, for
2x2 tables from casecontrol studies and larger tables with more categories/relative proportions.
Homework Set 11, due Thursday, December 1, 1pm. For students enrolled
in Biol6317, complete Project 3 as part of the assignment.
Twelfth week. Familywise error and Bonferroni procedure. False discovery rate. Nonparametric tests
and MonteCarlo methods.
Review session for the final on Friday, Dec 9, 35pm, PGH 646. The duration of the final is
3 hours. The Summary of the lectures has been updated to include material up to the final.
MATH 6321  Spring 2011
Theory of functions of a real variable, part II
Course Info. View syllabus.
First week. Banach spaces. Bounded linear maps. Baire's theorem.
Second week. BanachSteinhaus theorem. Open mapping theorem. Theorem of bounded inverese.
Closed graph theorem.
Third week. Application to convergence of Fourier series. Fourier series as a map
from L^{1} into c_{0}, but not onto!
Fourth week. HahnBanach theorem (real and complex version). Uniqueness of extensions. The disk algebra and
the Poisson kernel.
Fifth week. Complex measures. Total variantion measure. Lebesgue decomposition.
Absolute continuity. RadonNikodym(Lebesgue) theorem.
Sixth week. The continuity in absolute continuity. Polar decomposition. Hahn decomposition.
Midterm exam. Tu, Mar 8, in class, with takehome part due on Thursday, Mar 10, 2:30pm. Closed book.
To see how well you are prepared, take a
practice exam. Review session Friday, March 4, 3:305:30pm, in SEC 203. Office hours are extended to Tuesday 9:3011am.
Eighth week. Duality between L^{p} and L^{q}, including p=1. C_{0}(X), the space of continuous
functions vanishing at infinity, on a locally compact Hausdorff space X.
Ninth week. Regularity of complex measures. Duality between regular complex measures and C_{0}, another version
of the Riesz representation theorem. Consequence of the Riesz representation theorem (Ch. 6, Ex. 4).
Tenth week. Differentiation. Lebesgue points. Maximal function. Fundamental theorem of calculus.
Eleventh week. Product algebras and product measures. Fubini's theorem. Convolution.
Product measures and completion.
Twelfth weeek. Fourier transform. Elementary properties. Inversion theorem.
Final exam, as scheduled by the registrar: May 10, 25pm, in class. Review session on May 9,
47pm, PGH 348. Closed book. To see how well you are prepared,
take a practice exam.
To study for the prelim, please review the course material and work through recommended problems: Chapter 1: 1, 4,
5, 7, 8, 12; Chapter 2: 1, 2, 3, 5, 7, 11, 21, 22;
Chapter 3: 1, 4, 5, 7, 10, 14 a and d; Chapter 4: Problems 15, 7, 9; Chapter 5: 2, 6, 8, 9, 10, 11, 16, 17, 18;
Chapter 6: 2, 3, 4, 5, 10 ab, 13; Chapter 7: 1, 10, 11, 12ab, 14, 23;
Chapter 8: 2, 3, 4, 5ad, 12, 14, 15. Chapter 9: 2, 6, 8.


MATH 3364  Fall 2010
Introduction to complex analysis
Course Info. View syllabus.
First week. Algebra of complex numbers. Point representation. Modulus, triangle inequalities.
Complex conjugation. Polar form of complex numbers.
Homework Set 1, due Thursday, Sep 2, 11:30am. Ch. 1.1: 5 ac, 6 ac, 7 ac, 9, 14 (use notation z=(x,y)), 15, 17, 19 (use z=(x,y)); Ch. 1.2: 3, 4 only plot points for z = 32i, 7 cd (start explanation by writing equations for x and y).
Second week. Trigonometric identities and the complex exponential, de Moivre's identity. Integrating powers of trigonometric functions. The Mandelbrot set. nth roots, nth roots of unity. Geometric series and nth roots. Sets in the complex plane. Domain and range of complex functions. The exponential function. Limits.
Homework Set 2, due Thursday, Sep 9, 11:30am. Ch. 1.3: 3, 5 ad, 7 fh, 13; Ch. 1.4: 1 ac, 12 ab , 13 ab; Ch. 1.5: 4, 5 df, 10.
Third week. Limits and continuity. Rules for limits and continuity. Zeros and continuity. Differentiability.
Differentiation rules. CauchyRiemann Differential Equations and differentiability. Harmonic functions.
Homework Set 3, due Thursday, Sep 16, 11:30am. Ch. 2.1: 1 a,c,f, 3 ab, 8 ac; Ch 2.2:
7 ac,f, 11 bc, 17, 21 ad; Ch 2.3 7 bd, 9 ab, 11 ab,f (discuss differentiability and conclude about analyticity); Ch 2.4: 1 ac, 3, 5, 10.
Fourth and fifth week. Level curves of real and imaginary parts of analytic functions.
Polynomials and rational functions. Complex trigonometric functions. The logarithm. Inverse
trigonometric functions.
Homework Set 4, due Thursday, Sep 30, 11:30am. Ch. 2.5: 1 ac, 3 a,b,d,e, 8 ac, 12;
Ch. 3.2: 7, 9 a,c,e, 12 a, 17 a,b,c; Ch. 3.3: 1 a,b,c,d, 16; Ch. 3.5: 1 ad, 10.
First midterm exam, October 5, in class. Material up to and including Homework Set 4.
Bring pen, pencil, student ID but no calculator! Review session on Thursday, Sep 30,
57pm, in AH 16.
Sixth week. Smooth arcs, curves, contours. Parametrization. Contour integrals. Reparametrization invariance.
Fundamental theorem of calculus.
Homework Set 5, due Thursday, Oct 21, 11:30am. Ch. 4.1: 1 a,b,d, 8; Ch. 4.2: 3 a,b,c, 5, 6, 8, 9;
Ch. 4.3: 1 a,b,d,e, 4 (explain briefly); Ch. 4.4: 3 a,b,d, 10 a,b,c,e, 13, 15, 17.
Seventh week. Cauchy formulas and their consequences: Liouville's theorem, maximum modulus and
fundamental theorem of algebra. Maxima/minima of harmonic functions.
Homework Set 6, due Thursday, Oct 28, 11:30am. Ch. 4.4: 18 ad; Ch. 4.5: 1, 3 a,b,c,f, 4 a,b, 5, 6, 7;
Ch. 4.6: 1, 2, 3, 5, 16, 17, 19.
Eighth week. Sequences and series of complex numbers. Convergence tests. Absolute convergence.
Sequences and series of functions. Taylor series and its convergence.
Homework Set 7, due Thursday, Nov 4, 11:30am. Ch. 5.1: 1 ac, 2 ad, 7 ac, 11 ac; Ch. 5.2: 1 a,b,e,
2 (for a,b,e only), 5 a,b,e, 7, 11 a, b, 18 a.
Ninth week. Power series. Radius of convergence. Uniform convergence. Relation to Taylor series. Termbyterm differentiation
and integration.
Homework Set 8, due Thursday, Nov 11, 11:30am. Ch. 5.2: 3 a,b,c, 13; Ch. 5.3: 2, 3 a,b,c,d,f, 5 ad,
6 ac, 7, 10, 13 a,b.
Tenth week. Solutions to differential equations by power series.
Laurent series. Evaluating contour integrals by Laurent series. Residues.
Second midterm exam, November 16, in class.
Material up to and including Homework Set 8.
Bring pen, pencil, student ID but no calculator.
Review session on Monday, Nov 15, 5:307:30pm, 348 PGH.
Eleventh week. Integrals of trigonometric functions and rational functions.
Zeros and poles. Integrals involving exponentials.
Homework Set 9, due Monday, Dec 6, 11:30am, in PGH 604. Ch. 5.6: 1 a,b,d, 2; Ch. 6.1: 1 ad, 3 a,b,e;
Ch. 6.2: 1, 4, 8; Ch. 6.3: 1, 2, 3, 10 a, 11.
Makeup Midterm for eligible students. Tuesday, Dec 7, 11:30am12:50pm. Material
from Homework Sets 18 (covering both midterms). To be eligible, submit
documentation for the missed midterm no later than Thursday, Dec 2.
Review for final exam. Thursday, Dec 9, 57:30pm, PGH 646.
Final exam, December 14, 11am1:30pm, AH 108. Bring pen, pencil, student ID but
no calculator! Cell phones will need to be switched off during the exam.
MATH 6320  Fall 2010
Theory of functions of a real variable
Course Info. View syllabus.
First week. Settheoretic notation. Topologies, bases, metric spaces. Sigmaalgebras. Generating sigmaalgebras. Measurable functions. Borel sets. Borelmeasurability. Continuity. Compositions of functions. Other measurabilitypreserving manipulations of functions. Lim inf and lim sup. Pointwise limits of measurable functions.
Second week. Measures. Properties of measures. Integrals of simple functions. Monotonicity. Integrals of nonnegative measurable functions. Properties of integrals. Monotone convergence.
Third week. More properties of integrals. L^{1} space of integrable functions, vector space property. Functions vs. measures.
Fourth and fifth week. Halmos's approach to measures. Rings, sigmarings, monotone class. Sigma rings and monotone classes
generated by rings. The Lebesgue measure.
Sixth week. Topological preliminaries. Riesz representation theorem. Regularity of Borel measures. Lebesgue measure
on R^{d} via Riesz representation theorem.
Midterm exam. Tu, Oct 26, 5:307:30, AH 15. Closed book. To see how well you are prepared, take a
practice exam. Office hours are extended to Tuesday 9:3011am.
Seventh week. Jensen's, Hölder and Minkowski's inequalities.
Eighth week. Essential supremum. Space of essentially bounded functions. Completeness of L^{p}spaces.
Approximation properties.
Ninth and tenth week. Hilbert spaces. Riesz representation theorem
for bounded linear functionals on Hilbert spaces. Closed subspaces and orthogonal projections.
Orthonormal bases. Fourier series.
Eleventh week. Banach spaces.
Final exam. Th, Dec 9, 2:305pm, 350 PGH. Closed book. To see how well you are prepared,
take a practice exam.
MATH 6397  Spring 2010
Highdimensional measures and geometry
Course Info. View syllabus.
First week. The surface measure on highdimensional spheres and the standard Gaussian
measures (notes). Projections onto subspaces and length
(notes).
Second week. The JohnsonLindenstrauss Lemma (notes).
Bounds for the Laplace transform on the boolean cube (notes).
Third week. The martingale method for estimating Laplace transforms (notes). Concentration around the mean.
Application of the martingale method to the boolean cube. Concentration in product spaces
and law of large numbers (notes).
Fourth week. Optimal asymptotics for the coin toss (notes).
General results in product spaces (notes).
Fifth week. Back to the fair and unfair coin, and Gaussians as limits of projected spherical
measures (notes). Higherdimensional Gaussians as
projected spherical measures (notes).
Sixth week. Concentration about the median for spheres. Concentration about the mean
for Gaussian measures (notes).
Seventh week. Finishing concentration about the mean for Gaussians (notes) and deduce
concentration about the mean for spheres (notes).
Eighth week. Concentration on subspaces (notes). Compressive sensing
(notes).
Ninth week. PrekopaLeindler inequality, isoperimetric inequality (notes).
BrunnMinkowski inequality. Concentration on the sphere and on strictly convex surfaces (notes).
Tenth week. Concentration for strictly logconcave measures (notes).
Eleventh week. Reverse Holder (notes)
and reverse Jensentype inequalities for norms (notes).
Twelfth week. Approximating the ball with polytopes (notes). Edge counts and the graph Laplacian (notes).
Thirteenth week. Growth rates of subsets of graphs (notes). Concentration on graphs (notes).
MATH 4397/6397  Fall 2009
Biostatistics
Course Info. View syllabus.
Week 1. We are covering parts of Rosner, Ch. 3.13.5, 4.14.3 and 5.15.2.
Students who were absent during this week may want to consult
notes for week 1 to see
a summary of the material.
Week 2. Still covering above sections in Rosner and, in addition, 4.4, 4.5.
Homework Set 1,
due Thursday, Sep 3, 2009.
Week 3. Completing 4.4, 4.5, and 4.9. A CalculusLab for anyone who wants to
brush up a little will be held on Tuesday, Sep 8, 1:30pm. Either be at my office
(PGH 604) before 1:30 or come to PGH 646 directly.
Homework Set 2,
due Thursday, Sep 10, 2009.
Week 4. Conditional probability, Bayes's rule, diagnostic testing, Ch. 3.63.9. Likelihood, Bernoulli experiments and binomial distribution, Ch 4.8, 4.9, 5.15.6.
Homework Set 3.
For this homework and in the future, you may find the notes on R by Dr. Peters, Basics,
Graphics,
Statistics Functions and Regression, etc
helpful.
Week 5. Maximum likelihood estimates for binomial and normal random variables. Law of large numbers and
central limit theorem, Ch 5.15.6, 6.1, 6.2, 6.5.
Homework Set 4,
due Thursday, Sep 24, 2009,.
Week 6. Confidence intervals for the mean and variance of a normal r.v., chisquare distribution,
Gosset's tdistribution. Confidence intervals for binomial
distribution: Wald interval, AgrestiCoull interval.
Homework Set 5,
due Thursday, Oct 1, 2009.
Week 7. Confidence interval for binomials distributions, continued. Independent group comparisons
with tdistribution confidence interval, equal and unequal variances [Ch. 8.5, 8.7].
Special session on settheoretic problems and on computing with
random variables or their densities (early homework), Thursday 11amnoon, PGH 646.
Homework Set 6,
due Thursday, Oct 8, 2009.
Week 8. Displaying data: Histogram, stem and leaf plot, box plot, dot charts, qqplots. Review.
Homework Set 7,
due Wednesday, Oct 14, 2009, at 2:30pm in 604 PGH.
The Midterm will be held on Tuesday, Oct 20, in class. Bring a pen or pencil, a scientific calculator,
and your student ID. The collection of
review topics might be hepful.
Week 9 and 10. Hypothesis testing [Ch. 7.17.7]. Z and T scores and associated tests. One and twosided alternatives.
Pvalue and its interpretation. Power.
Homework Set 8,
due Thursday, Nov 5, 2009. For students
enrolled in Math 6397, Project 1
is part of the assignment. .
Week 11. Independent group tests with unequal variance [Ch. 8.6, 8.7]. Ftest.
Hypothesis testing for binomial proportions [Ch. 7.10]. Wilson's score
and interval. Comparing two binomial proportions [Ch. 10.1, 10.2, 13.113.3].
Fisher's exact test [Ch. 10.3, 10.610.9].
Homework Set 9,
due Thursday, Nov 12, 2009.
Week 12. Chisquared testing for equality of proportions and for independence [Ch. 10.2, 10.3, 10.69].
Controlling the Familywise Error (Bonferroni) and the expected False Discovery Rate (Benjamini and Hochberg)
[Ch. 12.4].
Homework Set 10,
due Thursday, Nov 19, 2009. For students
enrolled in Math 6397, Project 2
is part of the assignment.
Week 13. Nonparametric tests: sign test and Wilcoxon's signed rank, rank sum tests [Chs. 9.29.4].
Homework Set 11,
due Thursday, Dec 3, 2009. For students
enrolled in Math 6397, Project 3
is part of the assignment.
A review session for the final exam will be held on Thursday, Dec 10, 34:30pm, in 646 PGH.
The final exam will be on Tuesday, Dec 15, 25pm. Bring pen/pencil, calculator, ID, and
a sheet with your favorite formulas or insights. To prepare, you may find the
summary (updated 12/8!) of the course topics
useful.
MATH 4355  Spring 2009
Mathematics of Signal Representations
Course Info. View syllabus.
Homework Assignment 1 is due Wednesday, February 4.
Homework Assignment 2 is due Wednesday, February 11 (extended to Monday, February 16).
Homework Assignment 3 is due Wednesday, February 18.
Homework Assignment 4 is due Wednesday, March 4.
A review session for the midterm will be held on Friday, March 6, 67:30pm, in 345 PGH.
To prepare you may consult Course Notes giving a brief outline of the material. Additional material is available in the slides of a short course, up to page 17. A
Practice Midterm could be helpful for finding out how well prepared you are.
The midterm exam will be on March 11, 5:307:30pm, in 345 PGH. Bring a pen or pencil and eraser. No calculators or other materials allowed.
Homework Assignment 5 is due Wednesday, April 1.
Homework Assignment 6 is deferred until Monday, April 13.
Homework Assignment 7 is deferred until Monday, April 20.
Homework Assignment 8 is due Wednesday, April 29.
Read the supplementary notes beforehand.
A review session for the final exam will be held on Friday, May 1, 68:00pm, in 345 PGH.
To prepare you may consult the updated Course Notes giving a brief outline of the material. Additional material is available in the slides of a short course, up to page 40.
The final exam will be on May 4, 5:308:30pm, in 345 PGH. Bring a pen or pencil and eraser. No calculators or other materials allowed.


