Lecture Notes, Exercises etc
https://statmathbc.wordpress.com
Notes and exercises for graduate and undergraduate studentsSat, 17 Mar 2018 16:26:15 +0000enhourly1http://wordpress.com/https://s2.wp.com/i/buttonw-com.pngLecture Notes, Exercises etc
https://statmathbc.wordpress.com
Exercises in Complex Analysis 200-270
https://statmathbc.wordpress.com/2018/02/22/exercises-in-complex-analysis-200-270/
https://statmathbc.wordpress.com/2018/02/22/exercises-in-complex-analysis-200-270/#respondThu, 22 Feb 2018 09:58:43 +0000http://statmathbc.wordpress.com/?p=791CAP
]]>https://statmathbc.wordpress.com/2018/02/22/exercises-in-complex-analysis-200-270/feed/0statmathbcCatalog
https://statmathbc.wordpress.com/2018/02/22/november-2017-catalog/
https://statmathbc.wordpress.com/2018/02/22/november-2017-catalog/#respondThu, 22 Feb 2018 09:46:58 +0000http://statmathbc.wordpress.com/?p=655Catalog of notes available here (updated monthly) I have not included the problems I post occasionally. Thanks to Nishant and Tiju for suggestions. A few links do not work. Please use the search feature on the site for these.
For hints/solutions to exercises as well as for comments/suggestions send mail to kavi.ramamurthy@gmail.com
1. Banach – Stone Theorem
2. Basic Wavelet Theory
3. Brunn – Minkowsky Inequality
4. Caratheodory Extension Theorem
5. Compactness in Metric Spaces
6. Continuous Hilbert Transform
7. Fourier Analysis on Groups
8. Fourier Series and Fourier Transforms
9. Gelfand – Naimark Theorem
10. Infinitely Divisible and Stable Laws Vol. 1
11. Infinitely Divisible and Stable Measures V. II
12. Laurent Schwartz’ Theory of Distributions
13. Marcinkiewicz Interpolation Theorem
14. Mergelyan’s Approximation Theorem
15. Notes on Fixed Point Theorems
16. Notes on Functional Analysis
17. Notes on Rado’s Theorem
18. The Stone – Weierstrass Theorem
19. Wiener’s Tauberian Theorem
20. Infinitely Divisible and Stable Measures V. III (Type and Cotype)
21. Exercises in Probability Theory
22. Exercises in Analysis – 1 to 100
23. Exercises in Analysis – 101 to 200
24. Exercises in Analysis – 201 to 300
25. Exercises in Analysis – 301 to 400
26. Notes on Ergodic Theory
27. The Banach Stone Theorem
28. “Anti-Calculus Proposition” of Erdos
29. Exercises in Analysis – 401 to 500
30. Exercises in Analysis 501-600
31. Exercises in Analysis 601-700
32. Exercises in Complex Analysis 1-100
33. Exercises in Complex Analysis 101-200
34. Exercises in Complex Analysis 200-
]]>https://statmathbc.wordpress.com/2018/02/22/november-2017-catalog/feed/0statmathbcAn exercise on i.i.d. sequences
https://statmathbc.wordpress.com/2018/02/16/an-exercise-on-i-i-d-sequences/
https://statmathbc.wordpress.com/2018/02/16/an-exercise-on-i-i-d-sequences/#respondFri, 16 Feb 2018 09:32:32 +0000http://statmathbc.wordpress.com/?p=784Let {X_n} be an i.i.d sequence and assume that X_1 is not a bounded random variable. Show that there is a sequence of continuous functions {f_n} on the real line converging uniformly to 0 on compact subsets such that $\latex P{f_n(X_n) \to 0}=0$.
]]>https://statmathbc.wordpress.com/2018/02/16/an-exercise-on-i-i-d-sequences/feed/0statmathbcExercises in Complex Analysis 101-200
https://statmathbc.wordpress.com/2018/02/05/exercises-in-complex-analysis-101-200/
https://statmathbc.wordpress.com/2018/02/05/exercises-in-complex-analysis-101-200/#respondMon, 05 Feb 2018 09:40:43 +0000http://statmathbc.wordpress.com/?p=780CA101200
]]>https://statmathbc.wordpress.com/2018/02/05/exercises-in-complex-analysis-101-200/feed/0statmathbcAn exercise involving a functional equation
https://statmathbc.wordpress.com/2018/01/29/an-exercise-involving-a-functional-equation/
https://statmathbc.wordpress.com/2018/01/29/an-exercise-involving-a-functional-equation/#respondMon, 29 Jan 2018 09:07:03 +0000http://statmathbc.wordpress.com/?p=776Let f be a function from R to R such that f(x+y^{n})=f(x)+{f(y)}^{n} for all x,y where n is a given positive integer greater than 1. Show that f(x)=cx where either c=0 it c is an n-th root of unity.
No hypothesis on continuity or measurability of f!
]]>https://statmathbc.wordpress.com/2018/01/29/an-exercise-involving-a-functional-equation/feed/0statmathbcSolution to a problem posted on S-E
https://statmathbc.wordpress.com/2018/01/24/solution-to-a-problem-posted-on-s-e/
https://statmathbc.wordpress.com/2018/01/24/solution-to-a-problem-posted-on-s-e/#respondTue, 23 Jan 2018 22:43:29 +0000http://statmathbc.wordpress.com/?p=773Solution
]]>https://statmathbc.wordpress.com/2018/01/24/solution-to-a-problem-posted-on-s-e/feed/0statmathbcExercises in Complex Analysis 1-100
https://statmathbc.wordpress.com/2018/01/03/exercises-in-complex-analysis-1-100/
https://statmathbc.wordpress.com/2018/01/03/exercises-in-complex-analysis-1-100/#respondWed, 03 Jan 2018 09:49:17 +0000http://statmathbc.wordpress.com/?p=767CA12200
]]>https://statmathbc.wordpress.com/2018/01/03/exercises-in-complex-analysis-1-100/feed/0statmathbcAlmost everywhere continuity
https://statmathbc.wordpress.com/2017/12/27/almost-everywhere-continuity/
https://statmathbc.wordpress.com/2017/12/27/almost-everywhere-continuity/#respondWed, 27 Dec 2017 09:11:10 +0000http://statmathbc.wordpress.com/?p=763Let f be a map from to . Prove that the following are equivalent:
a) f is almost everywhere equal to a function g which is almost everywhere continuous
b) there is a set A of (Lebesgue) measure 0 such that the restriction of f to the complement of A is continuous in the relative topology.
]]>https://statmathbc.wordpress.com/2017/12/27/almost-everywhere-continuity/feed/0statmathbcExercises in Analysis 601-700
https://statmathbc.wordpress.com/2017/12/11/exercises-in-analysis-601-700/
https://statmathbc.wordpress.com/2017/12/11/exercises-in-analysis-601-700/#respondMon, 11 Dec 2017 09:17:08 +0000http://statmathbc.wordpress.com/?p=756601to700
]]>https://statmathbc.wordpress.com/2017/12/11/exercises-in-analysis-601-700/feed/0statmathbcAn exercise in calculus
https://statmathbc.wordpress.com/2017/12/04/an-exercise-in-calculus/
https://statmathbc.wordpress.com/2017/12/04/an-exercise-in-calculus/#respondMon, 04 Dec 2017 07:50:51 +0000http://statmathbc.wordpress.com/?p=754Let f,f_1,f_2,… be functions from R into itself. Suppose f_n(x_n) converges to f(x) whenever x_n converges to x. Show that f is continuous. Caution: it is not given that f_n’s are continuous!
]]>https://statmathbc.wordpress.com/2017/12/04/an-exercise-in-calculus/feed/0statmathbc