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Bernhard G. Bodmann
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    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 Domel-White).
          Rank-nullity, 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 Domel-White).
                    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. Riesz-Fredholm 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 Domel-White).

                            MATH 7320 - Fall 2016
                            Functional Analysis

                              Course Info. View syllabus.
                                Office hours: PGH 604, Mo 10-11am, We 11am-12pm.
                                  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 Domel-White).
                                            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 Domel-White 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 Domel-White, by Jason Duvall, and by Wilfredo Molina).
                                                            Week 14. The Krein-Milman 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 noon-1pm, We 1-2pm.
                                                                  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, 2:30-4pm. Final Exam on Tuesday, May 10, 2-5pm. 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 10:30am-noon, We 1-2pm.
                                                                                                              Week 1. The topology of Rn. 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. Cauchy-Schwarz inequality and consequences. Lp-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 1:30-2:30pm.
                                                                                                                                                              Follow latest news and course status on Twitter @AccelCalcUH

                                                                                                                                                              Tentative Course Calendar (updated 5-04-15)

                                                                                                                                                                Latest news: Practice final posted. Special review session for the final on Thursday, May 7, 2:30-4pm, in AH 106.
                                                                                                                                                                  Week Sections to be read, Exam dates Suggested homework problems

                                                                                                                                                                   Jan 20-23

                                                                                                                                                                  No class on Jan 19

                                                                                                                                                                   12.1-12.2 12.1 1-13 odd, 25, 29, 35
                                                                                                                                                                  12.2
                                                                                                                                                                  1-13 odd, 19, 23, 33
                                                                                                                                                                   Jan 26-30  12.3-12.5 12.3 3-17 odd, 23, 25, 29, 43, 47, 49, 51
                                                                                                                                                                  12.4
                                                                                                                                                                  1, 3, 9, 23, 25, 29, 33, 35, 41, 45
                                                                                                                                                                  12.5
                                                                                                                                                                  1-19 odd, 23, 27, 31, 39, 47, 53, 67, 69
                                                                                                                                                                   Feb 2-6  13.1-13.3 13.1 1-15 odd, 19-23 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 9-13  14.1, 14.2, 14.3 14.1 1, 13, 25, 27, 31, 39, 47, 55-59 odd
                                                                                                                                                                  14.2 7-17 odd, 23, 27, 29, 31, 37
                                                                                                                                                                  14.3 5-9 odd, 15, 17, 25, 33, 37, 43, 45, 49, 57, 61, 87, 95 b-d
                                                                                                                                                                   Feb 16-20  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 23-27

                                                                                                                                                                  Review, Feb 23

                                                                                                                                                                  Exam 1, Feb 24
                                                                                                                                                                  on Ch. 12, 13, 14.1-5
                                                                                                                                                                  See
                                                                                                                                                                  Sample
                                                                                                                                                                  test?

                                                                                                                                                                  14.6
                                                                                                                                                                  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 2-6  14.7, 14.8, 12.6 14.7 1, 3, 9, 19, 21, 27, 29, 31, 37, 39, 41, 51
                                                                                                                                                                  14.8
                                                                                                                                                                  1-11 odd, 19, 21, 41, 45
                                                                                                                                                                  12.6
                                                                                                                                                                  1, 3, 9, 21-27 odd
                                                                                                                                                                   Mar 9-13  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, 39-43 odd, 47, 49, 51, 53, 55, 61
                                                                                                                                                                   Mar 23-27  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 30-Apr 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 6-10  Review, Apr 6
                                                                                                                                                                  Exam 2, Apr 7
                                                                                                                                                                  on Chs. 14.6-8, 15.1-8
                                                                                                                                                                  See
                                                                                                                                                                  Sample
                                                                                                                                                                  test?
                                                                                                                                                                   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 13-17  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

                                                                                                                                                                   Apr 20-24

                                                                                                                                                                   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 27-May 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
                                                                                                                                                                  Final Exam, May 12
                                                                                                                                                                  on Chs. 12-16
                                                                                                                                                                  See
                                                                                                                                                                  Sample
                                                                                                                                                                  test?
                                                                                                                                                                  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 1:30-2:30pm.
                                                                                                                                                                        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 11-10-14)

                                                                                                                                                                              Latest news: Special review session for the Final on Monday, Dec 8, 12-1:30pm, in AH 301.
                                                                                                                                                                                Week Sections to be read, Exam dates Suggested homework problems

                                                                                                                                                                                 Aug 25-29

                                                                                                                                                                                 Introduction, Review
                                                                                                                                                                                1.3, 1.6, 2.3-2.5
                                                                                                                                                                                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

                                                                                                                                                                                 Sep 2-5

                                                                                                                                                                                No class on Sep 1

                                                                                                                                                                                 3.1-3.2 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 8-12  3.3-3.5 3.3 3-23 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 15-19  4.4, 3.7-3.9 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 22-26  4.1-4.3, 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 29-Oct 3

                                                                                                                                                                                Review, Sep 29

                                                                                                                                                                                Exam 1, Sep 30
                                                                                                                                                                                on Ch. 3-4
                                                                                                                                                                                See
                                                                                                                                                                                Sample
                                                                                                                                                                                test?

                                                                                                                                                                                5.1-5.2
                                                                                                                                                                                Review Ch. 3 Ex 1-23 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 6-10  5.3, 5.4, 5.5 5.3 3, 5, 7-17 odd, 21, 29, 43, 53, 55, 63
                                                                                                                                                                                5.4 9, 23, 33, 41, 43, 53, 63
                                                                                                                                                                                5.5 3, 5, 7-17 odd, 21, 29, 43, 53, 55, 63-69 odd, 75, 81
                                                                                                                                                                                 Oct 13-17  7.1-7.3 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 20-24  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 27-31

                                                                                                                                                                                Review, Oct 27

                                                                                                                                                                                Exam 2, Oct 28
                                                                                                                                                                                on Chs. 5, 7
                                                                                                                                                                                See
                                                                                                                                                                                Sample
                                                                                                                                                                                test?

                                                                                                                                                                                6.1, 6.2
                                                                                                                                                                                Review Ch. 5 Ex 3, 5, 9-21 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 1-13 odd, 31, 45 a-b, 49, 53
                                                                                                                                                                                6.2 1-11 odd, 17, 31, 33, 35, 51, 53, 63, 65
                                                                                                                                                                                 Nov 3-7  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 10-14  11.1-11.3
                                                                                                                                                                                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 17-21
                                                                                                                                                                                 Review, Nov 17
                                                                                                                                                                                Exam 3, Nov 18
                                                                                                                                                                                on Ch. 6, 8, 11
                                                                                                                                                                                See
                                                                                                                                                                                Sample
                                                                                                                                                                                test?
                                                                                                                                                                                 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 1-15 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

                                                                                                                                                                                 Nov 24, 25

                                                                                                                                                                                Thanks-
                                                                                                                                                                                giving

                                                                                                                                                                                 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 1-5  11.8-11.10 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 1-19 odd, 25, 27, 33, 55
                                                                                                                                                                                 Dec 8, 11 Review, Dec 8
                                                                                                                                                                                Final Exam, Dec 11
                                                                                                                                                                                on Chs. 3-8, 11
                                                                                                                                                                                See
                                                                                                                                                                                Sample
                                                                                                                                                                                test?
                                                                                                                                                                                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 1:30-2:30pm.
                                                                                                                                                                                        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 code-word 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 a-d, 4, 8, 12, 16 a-b, 22 a-e.
                                                                                                                                                                                                                            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 m-files 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 m-files 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 m-file 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 m-file.
                                                                                                                                                                                                                                  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 in-class exam on Thursday, Oct 3, 2013, 4:00-5:20pm.
                                                                                                                                                                                                                                    Bring a blue book and your student ID!
                                                                                                                                                                                                                                    Review: Sections 1.1-1.5, 1.7-1.9, 2.1-2.3.
                                                                                                                                                                                                                                    Review session on Friday, Sep 27, 2013, 2:30-4:30pm; 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 a-e, 34.
                                                                                                                                                                                                                                            Section 4.4, Exercises: 4, 8, 14, 16 a-c, 28 (use basis {1,t,t2, t3}), 36.
                                                                                                                                                                                                                                            Section 4.5, Exercises: 6, 8, 12, 18, 22.
                                                                                                                                                                                                                                            Optional: Use the m-files 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 m-files.
                                                                                                                                                                                                                                              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 a-d, 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 10-by-10 matrix A with entries 0 or 1. Test using rref whether the matrix A has full rank. If not, repeat generating random 10-by-10 matrices with entries 0 and 1 until it has full rank. Record how often you had to repeat.
                                                                                                                                                                                                                                              2*) Next, form m-by-n 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 m-by-n 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 in-class exam on Thursday, Nov 14, 2013, 4:00-5:20pm.
                                                                                                                                                                                                                                                Bring a blue book and your student ID!
                                                                                                                                                                                                                                                Review: Sections 3.1-3.3, 4.1-4.7.
                                                                                                                                                                                                                                                Review session on Friday, Nov 8, 2013, 2:30-4:30pm, 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 a-e, 24.
                                                                                                                                                                                                                                                    Section 6.2, Exercises: 2, 10, 12, 14, 20.
                                                                                                                                                                                                                                                    Section 6.3, Exercises: 2, 10, 12, 18, 22 a-d.
                                                                                                                                                                                                                                                    Section 6.5, Exercises: 2, 10.
                                                                                                                                                                                                                                                    Extra credit problem in Matlab/Octave (8 points): 20 equations, 50 unknowns.
                                                                                                                                                                                                                                                    1*) Prove that a matrix-vector 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 non-zero 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 non-zero entries in x by checking whether for two columns aj and aj' 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 non-zero, 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, 5:00-7:50pm.
                                                                                                                                                                                                                                                      Bring two blue books and your student ID!
                                                                                                                                                                                                                                                      Test your understanding of concepts with the True-False Marathon from class. Review on Thursday, Dec 12, 2-4pm, 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, 11:30-1pm..
                                                                                                                                                                                                                                                                      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 Riemann-Stieltjes 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, 11am-2pm, in our usual classroom. Closed book. Review session on Thursday, Dec 12, 10-11:30 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 square-norm convergence. Homework: Assignment 6 is due Tuesday, Mar 5.
                                                                                                                                                                                                                                                                                                          Midterm exam: March 7, 2:30-4:30pm, 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 Vj or Vj-1 and Wj-1, 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, 2:30-4pm. Final Exam: Tu, May 7, 2:00-5pm (in our usual classroom, as scheduled by registrar), bring Student ID, pen, pencil. Formula sheet will be provided.
                                                                                                                                                                                                                                                                                                                      Exam topics: Inner product spaces, L2(R) and l2(Z), orthonormal bases, orthogonal projections onto subspaces (with given orthonormal basis or vector-space basis), least squares approximations, convergence for sequences or series of functions (in L2, pointwise, or uniform), Fourier series on [-π,π] or on [-a,a], real and complex form, symmetries, convergence of Fourier series (unif, pointwise, in L2), Parseval's identity, Fourier transform, properties of the FT, sampling theorem for bandlimited functions, convolutions, causality, digital and analog filters, low-pass 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, Vj and Wj, Haar reconstruction algorithm, expression in terms of filtering (discrete convolution) and up/downsampling, multiresolution analysis, properties of scaling functions, two-scale relation, from scaling coefficients {pk} 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 skew-Hermitian 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. Courant-Fischer theorem. Weyl's theorem for eigenvalue estimates of low-rank perturbations
                                                                                                                                                                                                                                                                                                                                          Eighth week. Weyl's eigenvalue estimates for sums of Hermitian matrices.
                                                                                                                                                                                                                                                                                                                                            Ninth week. An example for low-rank perturbations. Eigenvalue interlacing for principal submatrices More eigenvalue interlacing. Generalized Rayleigh-Ritz 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. Least-squares problem. The normal equation for linear systems. Abbreviated singular value decomposition.
                                                                                                                                                                                                                                                                                                                                                  Twelfth week. Matrix norms. From the Hilbert-Schmidt 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, 10-11am, 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. Chi-squared distribution and confidence intervals for the variance.
                                                                                                                                                                                                                                                                                                                                                                                      Homework Set 6, due Thursday, October 6, 1pm.
                                                                                                                                                                                                                                                                                                                                                                                        Seventh week. T-distribution 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, 1:30-3pm, PGH 646.
                                                                                                                                                                                                                                                                                                                                                                                              Eighth week. Hypothesis testing. Z-score. P-value. T-test. Equivalence between hypothesis testing (with two-sided 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 two-sided alternatives. Computing p-values 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. Chi-squared testing for equality of proportions and for independence, for 2x2 tables from case-control 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. Non-parametric tests and Monte-Carlo methods.
                                                                                                                                                                                                                                                                                                                                                                                                                Review session for the final on Friday, Dec 9, 3-5pm, 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. 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.
                                                                                                                                                                                                                                                                                                                                                                                                                                                    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.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  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, 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 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.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  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, 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.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  Bernhard G. Bodmann,   University of Houston    ---    Last modified:  April 20 2007 - 13:48:50
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