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. Banach-Steinhaus 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 L1 into c0, but not onto!
Fourth week. Hahn-Banach 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. Radon-Nikodym(-Lebesgue) theorem.
Sixth week. The continuity in absolute continuity. Polar decomposition. Hahn decomposition.
Midterm exam. Tu, Mar 8, in class, with take-home 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, 3:30-5:30pm, in SEC 203. Office hours are extended to Tuesday 9:30-11am.
Eighth week. Duality between Lp and Lq, including p=1. C0(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 C0, 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, 2-5pm, in class. Review session on May 9,
4-7pm, 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 1-5, 7, 9; Chapter 5: 2, 6, 8, 9, 10, 11, 16, 17, 18;
Chapter 6: 2, 3, 4, 5, 10 a-b, 13; Chapter 7: 1, 10, 11, 12a-b, 14, 23;
Chapter 8: 2, 3, 4, 5a-d, 12, 14, 15. Chapter 9: 2, 6, 8.
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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 a-c, 6 a-c, 7 a-c, 9, 14 (use notation z=(x,y)), 15, 17, 19 (use z=(x,y)); Ch. 1.2: 3, 4 only plot points for z = 3-2i, 7 c-d (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. n-th roots, n-th roots of unity. Geometric series and n-th 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 a-d, 7 f-h, 13; Ch. 1.4: 1 a-c, 12 a-b , 13 a-b; Ch. 1.5: 4, 5 d-f, 10.
Third week. Limits and continuity. Rules for limits and continuity. Zeros and continuity. Differentiability.
Differentiation rules. Cauchy-Riemann Differential Equations and differentiability. Harmonic functions.
Homework Set 3, due Thursday, Sep 16, 11:30am. Ch. 2.1: 1 a,c,f, 3 a-b, 8 a-c; Ch 2.2:
7 a-c,f, 11 b-c, 17, 21 a-d; Ch 2.3 7 b-d, 9 a-b, 11 a-b,f (discuss differentiability and conclude about analyticity); Ch 2.4: 1 a-c, 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 a-c, 3 a,b,d,e, 8 a-c, 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 a-d, 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,
5-7pm, 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 a-d; 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 a-c, 2 a-d, 7 a-c, 11 a-c; 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. Term-by-term 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 a-d,
6 a-c, 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, 5:30-7:30pm, 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 a-d, 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, 11:30am-12:50pm. Material
from Homework Sets 1-8 (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, 5-7:30pm, PGH 646.
Final exam, December 14, 11am-1:30pm, 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. Set-theoretic notation. Topologies, bases, metric spaces. Sigma-algebras. Generating sigma-algebras. Measurable functions. Borel sets. Borel-measurability. Continuity. Compositions of functions. Other measurability-preserving 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 non-negative measurable functions. Properties of integrals. Monotone convergence.
Third week. More properties of integrals. L1 space of integrable functions, vector space property. Functions vs. measures.
Fourth and fifth week. Halmos's approach to measures. Rings, sigma-rings, 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 Rd via Riesz representation theorem.
Midterm exam. Tu, Oct 26, 5:30-7:30, AH 15. Closed book. To see how well you are prepared, take a
practice exam. Office hours are extended to Tuesday 9:30-11am.
Seventh week. Jensen's, Hölder and Minkowski's inequalities.
Eighth week. Essential supremum. Space of essentially bounded functions. Completeness of Lp-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, 2:30-5pm, 350 PGH. Closed book. To see how well you are prepared,
take a practice exam.
MATH 6397 - Spring 2010
High-dimensional measures and geometry
Course Info. View syllabus.
First week. The surface measure on high-dimensional spheres and the standard Gaussian
measures (notes). Projections onto subspaces and length
(notes).
Second week. The Johnson-Lindenstrauss 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). Higher-dimensional 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. Prekopa-Leindler inequality, isoperimetric inequality (notes).
Brunn-Minkowski inequality. Concentration on the sphere and on strictly convex surfaces (notes).
Tenth week. Concentration for strictly log-concave measures (notes).
Eleventh week. Reverse Holder (notes)
and reverse Jensen-type 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. 3.1-3.5, 4.1-4.3 and 5.1-5.2.
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 Calculus-Lab 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. 3.6-3.9. Likelihood, Bernoulli experiments and binomial distribution, Ch 4.8, 4.9, 5.1-5.6.
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 5.1-5.6, 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., chi-square distribution,
Gosset's t-distribution. Confidence intervals for binomial
distribution: Wald interval, Agresti-Coull interval.
Homework Set 5,
due Thursday, Oct 1, 2009.
Week 7. Confidence interval for binomials distributions, continued. Independent group comparisons
with t-distribution confidence interval, equal and unequal variances [Ch. 8.5, 8.7].
Special session on set-theoretic problems and on computing with
random variables or their densities (early homework), Thursday 11am-noon, PGH 646.
Homework Set 6,
due Thursday, Oct 8, 2009.
Week 8. Displaying data: Histogram, stem and leaf plot, box plot, dot charts, qq-plots. 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. 7.1-7.7]. Z and T scores and associated tests. One and two-sided alternatives.
P-value 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]. F-test.
Hypothesis testing for binomial proportions [Ch. 7.10]. Wilson's score
and interval. Comparing two binomial proportions [Ch. 10.1, 10.2, 13.1-13.3].
Fisher's exact test [Ch. 10.3, 10.6-10.9].
Homework Set 9,
due Thursday, Nov 12, 2009.
Week 12. Chi-squared testing for equality of proportions and for independence [Ch. 10.2, 10.3, 10.6-9].
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. 9.2-9.4].
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, 3-4:30pm, in 646 PGH.
The final exam will be on Tuesday, Dec 15, 2-5pm. 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.
Assignment 1 Solution.
Homework Assignment 2 is due Wednesday, February 11 (extended to Monday, February 16). Assignment 2 Solution.
Homework Assignment 3 is due Wednesday, February 18. Assignment 3 Solution.
Homework Assignment 4 is due Wednesday, March 4.
Assignment 4 Solution.
A review session for the midterm will be held on Friday, March 6, 6-7:30pm, 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 (and Solution) could be helpful for finding out how well prepared you are.
The midterm exam will be on March 11, 5:30-7:30pm, in 345 PGH. Bring a pen or pencil and eraser. No calculators or other materials allowed. Midterm Exam (and Solution).
Homework Assignment 5 is due Wednesday, April 1. Assignment 5 Solution.
Homework Assignment 6 is deferred until Monday, April 13. Assignment 6 Solution.
Homework Assignment 7 is deferred until Monday, April 20.
Assignment 7 Solution.
Homework Assignment 8 is due Wednesday, April 29.
Read the supplementary notes beforehand. (Assignment 8 Solution.)
A review session for the final exam will be held on Friday, May 1, 6-8:00pm, 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, 5:30-8:30pm, in 345 PGH. Bring a pen or pencil and eraser. No calculators or other materials allowed.
MATH 4377 - Fall 2008
Advanced Linear Algebra
Course Info. View syllabus.
First four homework assignments, see
here and Solutions to Set 1, Set 2,
Set 3 and
Set 4.
Fifth homework set, due Sep 25, 2008. View
Homework Set 5
and Solution.
Sixth homework set, due Oct 2, 2008. View
Homework Set 6 and
Solution.
Seventh homework set, due Oct 7, 2008. View
Homework Set 7 and
Solution.
First in-class exam on Tuesday, Oct 14, 2008. Bring a blue book!
Review session on Friday, Oct 10, 2008, 3-4pm; PGH 343.
A selection of review topics is posted here.
Exam 1. View
Problems and Solutions.
Eighth homework set, due Oct 28, 2008. View
Homework Set 8 and
Solution.
Ninth homework set, due Nov 6 (postponed), 2008. View
Homework Set 9 and
Solution.
Tenth homework set, due Nov 11, 2008. View
Homework Set 10 and
Solution.
Second in-class exam on Thursday, Nov 20, 2008. Bring a blue book and ID!
Review session Monday, Nov 17, 5:30-7pm; 343 PGH. A selection of review topics is posted
here.
Exam 2. View
Problems and Solutions.
Eleventh homework set, due Dec 9, 2008. View
Homework Set 11 and
Solution.
Final exam, Thursday, Dec 18, 2008; 2-5pm. Bring a blue book and ID!
A selection of review topics is posted here. Review session Tuesday, Dec 16, 5:30-7pm; 348 PGH.
MATH 6397 - Fall 2008
Information Theory with Applications
Course Info. View syllabus.
First homework set, due Sep 23, 2008. View
Exercise Sheet 1 and Solutions.
Second homework set, due Oct 16, 2008. View
Exercise Sheet 2 and Solutions.
Third homework set, due Nov 18, 2008. View
Exercise Sheet 3 and Solutions.
Fourth homework set, due Dec 11, 2008. View
Exercise Sheet 4.
MATH Special - Fall 2008
Wavelets in 60 Slides - Short Course
Lecture slides. View here.
MATH 6397 - Spring 2008
Stochastic Processes
Course Info. View syllabus.
First homework set, due Feb 28, 2008. View
Exercise Sheet 1 and Solution.
Practice problems for the Midterm Exam and Solutions
Second homework set, due April 22, 2008. View
Exercise Sheet 2 and Solution.
Practice problems and Solutions
as preparation for the Final Exam, April 29, 9am; 646 PGH.
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