*Update: I ended up not taking the Quals and leaving the Ph.D. program because of the pandemic; and, because I realized that I no longer agree with nor enjoy most of academia.*

In preparation for my Quals(RA and NLA) I have grabbed 4 books to study with:

- Measure and Integral an Introduction to Real Analysis by Antoni Zygmund and Richard L Wheeden,
- Real Analysis (4th Edition) by Halsey Royden, Patrick Fitzpatrick,
- Real and Complex Analysis by Walter Rudin, and
- Numerical Linear Algebra by Lloyd N. Trefethen, David Bau III

I’ve collected problems from other grad students who have taken the exams in the past, and from my professors:

I’ve begun “LaTex-ing” all of my solutions and rearranging the provided problems and solving their many variations.

Next, I plan on sending my solutions to my professors(past and present), to get their input on my solutions. And to reach out to peers at other institutions to get their advice and thoughts.

I’m also on the hunt for other texts to study with. I want to leave no stone unturned here.

Some study material my officemate and I have come across:

- https://www.math.tamu.edu/~keifler/resources/Real_solutions.pdf
- https://web.mst.edu/~jcmcfd/5215-notes.pdf
- https://people.cas.uab.edu/~mosya/teaching/Problems.pdf
- https://math.stackexchange.com/questions/267554/on-ph-d-qualifying-exams/270467
- https://www.math.mcgill.ca/~labute/courses/255w03/L10.pdf
- https://math.stackexchange.com/questions/163367/lebesgues-criterion-for-riemann-integrability-of-banach-space-valued-functions/163409#163409

Here’s a copy of the (draft)work that my officemate and I have done thus far(AND, that I’ve managed to get “LaTex’ed”).

Please keep in mind that **these are prone to ERRORS and TYPOS!!!!**(I’ve actually found several typos and corrected them. Let me know if you find any errors in the proofs and I’ll happily correct them. Again, some of these proofs are wrong… they are not all 100% correct, this was/is only a starting point.) We’re trying to write up our solutions first. Then, after we think we’ve “finished” all the problems, we plan on checking them. It’s only after then, that we plan on sending them to our profs. for review and critiques. If we get the thumbs up, then we plan on making our own example questions and “testing” each other with “mock quals.”:

Our professor overseeing the Real Analysis Qualifying Exam declined to review our solutions for the “review problems.” The professor overseeing the NLA Qualifying Exam, however, did not. And, after going through his feed back and making the appropriate adjustments to our proofs, we’ve now moved on to giving each other “mock quals.” To do this, I’ve written four sample quals using the sample questions, and questions from past quals. Each of our mock exams have one more question than what will be on the actual exams. The plan is for us to take them in the allotted time and get comfortable with the exams in order to help minimize the role anxiety may play on our performance the day of the exams.

Personally, I want to walk away from these exams knowing and understanding NLA and RA so well that I will still be stating proofs in my sleep when I’m in a retirement home.

Here are some questions that I’ve put to StackExchange for various reasons… though, usually it’s because my officemate and I disagree over some detail:

- https://math.stackexchange.com/questions/3963002/real-analysis-qualifying-exam-sample-question/3963115#3963115
- https://math.stackexchange.com/questions/3960198/nla-qualifying-exam-sample-question

On a separate yet related note… Death to qualifying exams!

- https://www.karenpanetta.com/blog/2018/7/13/death-to-phd-qualifying-exams
- https://gradadmissions.mit.edu/blog/are-qualifying-exams-waste-time

A brief rebuke of qualifying exams: