MATH 6040 (Fall '17)
Time & Place: TR 9:40AM - 11:00AM, NS 202
Instructor: Firas Rassoul-Agha
Phone: (801) 585-1647, E-Mail: firas@math.utah.edu
Office Hour: Tuesdays 11AM-12PM or by appointment, at LCB 209

Textbook: email me for access to the textbook and supplementary lecture notes.

There will be a LOT OF HOMEWORK.
It is mostly out of the textbook and will be posted below.
Make sure you work on it in a timely fashion, otherwise you WILL fall behind.

There will be no midterm exams. Whether or not there will be a final exam will be decided next week.

Homework Problems:

Review (not due, but very recommended):

1.1, 1.3, 1.9, 1.11, 1.12, 1.13, 1.16, 1.22, 1.24, 1.28
2.1, 2.3, 2.4, 2.5, 2.7

Due Thursday Sep 7:

Prove Lemmas 3.4, 3.9, and 3.11 (one of the two claims needs finiteness, which one?!)
Prove that the Borel sigma-algebra on (0,1] is the same as the sigma-algebra generated by intervals of the form (a,b].

3.2, 3.9, 3.10, 3.11, 3.13, 3.14, 3.17(this one shows that existence of nonmeasurable sets relies on the axiom of choice!)

Due Thursday Sep 21:

4.1, 4.6, Construct a counter example to Fatou's lemma, 4.16, 4.18, 4.28, 4.29

Not due, but very recommended:

4.9-4.11, 4.17, 4.21, 4.22, 4.23, 4.24, 4.27

Due Thursday Oct 5:

5.2, 5.3, 5.4, 5.7, 5.13, 5.17

Not due, but very recommended:

5.5, 5.8, 5.14, 5.18

Due Tuesday Oct 24:

6.2, 6.4, 6.5, 6.7, 6.9, 6.10, 6.14, 6.19, 6.27, 6.28, 6.29, 6.30, 6.37

Not due, but very recommended:

6.3, 6.6, 6.16, 6.17, 6.20, 6.23, 6.25, 6.26, 6.32

Due Thursday Nov 16:

7.2, 7.7, 7.10, 7.16, 7.18 (can use 7.14(3)), 7.19, 7.20, 7.25, 7.26, 7.30, 7.32, 7.40

Not due, but very recommended:

7.3, 7.6, 7.8, 7.14(3), 7.15, 7.21, 7.42, 7.44

Due Monday Dec 11:

8.3, 8.4, 8.8, 8.14, 8.15, 8.25, 8.31, 8.34, 8.35, 8.39, 8.40, 8.51
8.14 continued: prove that $S_n$ is recurrent when $p=1/2$ and transient when $p\not=1/2$, i.e. $P(S_n\not=0\ \forall n\ge1)=0$ for $p=1/2$ and $>0$ for $p\not=1/2$.
Prove also that when $p=1/2$, $S_n$ is in fact null recurrent, i.e. that $E[\text{return time to }0]=\infty$. (When this is finite, the process is called positive recurrent)

Not due, but very recommended:

8.16, 8.20, 8.26, 8.27, 8.30, 8.36, 8.38, 8.43, 8.47, 8.48, 8.56