## 2018-12-10

I thought about some of the equivalences in this linear algebra summary table.

I proved some of the propositions in Linear Algebra Done Right.

I asked a question on Math Stack Exchange about the use of “conversely” in some proofs in Linear Algebra Done Right.

I thought about how to show that “divide the data points by the standard deviation” and “scale the data points by a constant so that the new standard deviation is 1” are the same.

I thought about some stuff related to composition of limits (in analysis), and did one exercise from Spivak’s Calculus.

I made some edits to the page Comparison of concepts in computability theory.

## 2018-12-09

I tried to reconstruct a proof (that I read a day or two ago) that there exists a pair of recursively enumerable but recursively inseparable sets (I think I succeeded).

I did problem 8 in chapter 8 of Spivak’s Calculus.

I did problem 1 in section 13 of Munkres’s Topology. I also started on problem 2 but after I got the hang of it I stopped.

I think I continued reading a bit of Rogers’s computability book.

## 2018-12-08

I didn’t do much math. I thought again about Stillwell’s proof that there is a computable infinite tree whose infinite paths are uncomputable (which I still don’t understand). (I think this is called the Kleene tree?)

While reading about recursively inseparable sets, I realized that this pattern of words was similar to saying a series is “absolutely divergent”, so I made a page to track similar terms.

I looked a bit at Hartley Rogers’s text on recursive functions. I think this book goes into computability in more depth than the text by Boolos, Burgess, and Jeffrey, so I might want to look at this text more. (Fun fact: Rogers was Stillwell’s advisor.)

## 2018-12-07

I continued reading Stillwell’s Reverse Mathematics. Stillwell’s book goes over some results in computability, so I went back a bit to thinking about some computability stuff.

I wrote a page called Tiers of learning in mathematics.

## 2018-12-06

I did more more topology from Munkres. I read through section 12 (topological spaces) but haven’t started on the exercises yet.

I spent some time thinking about the intermediate value theorem, the boundedness theorem, and the extreme value theorem in analysis.

I started doing some conditional probability problems out of Intermediate Counting & Probability (David Patrick). This isn’t really material that is new to me (actually I think some of the material in this book is new to me) but I want to get better at solving these types of problems (especially under time pressure).

I continued reading Stillwell’s Reverse Mathematics.

I started working on a dependency graph of results in analysis. This was inspired by (1) reading different textbooks that all give different proofs of the main results; and (2) reading Stillwell’s book.

## 2018-12-05

I continued reading Munkres’s Topology. I decided to skip the first chapter for now, since I am more excited about jumping into the actual topology. I plan to come back to the first chapter as needed. I started reading chapter 2, and thought about the initial definitions.

I tried proving the Heine–Borel theorem for the real line again, which I learned a while back.

I made some edits to the page Understanding definitions.

I discovered John Stillwell’s book Reverse Mathematics and started reading it a little. The book seems pretty interesting but I don’t know if I should continue reading it (it might be tangential to my goals, but it might also strengthen me mathematically or cement my understanding of real analysis).

## 2018-12-04

I spent some time reading about the Riemann–Stieltjes integral in Apostol’s Mathematical Analysis. (I wanted to see how some other books defined this.)

I also spent some time thinking about the nested intervals theorem.

I started reading James Munkres’s Topology. I finished reading section 1.1 (set theory and logic) and thought about/did some of the exercises (they are the kind of exercises I am familiar with from other texts, so I don’t plan to work through all of them). One particular exercise caught my eye, which was exercise 1.1.2(k)–(l); this is because I had seen the same exercise in Tao’s book a while back. Since I fell into the trap on my first attempt (when working through Tao’s book), I had wanted to “get back at” the problem by explaining it clearly (I didn’t find any excellent explanations online). This seemed like the right opportunity, so I started the page Conjunction of subset statements versus Cartesian product subset statement.

## 2018-12-03

I spent some time thinking about why a partition of an interval is finite. I didn’t come up with a definite conclusion. I can understand that finite sums are easier to deal with than countably infinite sums, but I couldn’t exhibit any problems with a countably infinite “partition” except by making the function unbounded.

I finished the final problem in section 11.2 and worked through section 11.3 (did not finish working through the exercises).

I started a page called Understanding definitions.

I submitted some corrections for Tao’s book Analysis I (they were added to the page on 2018-12-05).

## 2018-12-02: more analysis

I worked through the exercises in section 10.5 (l’Hopital’s rule), then worked through sections 11.1 (Riemann integral: partitions) and 11.2 (Riemann integral: piecewise constant functions).

## 2018-12-01: not much math

I figured out exercise 10.4.3(b) in Analysis I only after seeing the trick in this answer (linked from this answer) of substituting in a quotient to get the general case; basically, since part (a) proved the case of $x \to 1$, in part (b) the trick is to see that $x/x_0 \to 1$ as $x \to x_0$. And then some algebra to get the thing to work. I had seen this sort of trick before, but didn’t think of it when I needed it, so I’ll have to keep it in mind more.

I read section 10.5 (l’Hopital’s rule) but didn’t do the exercises (yet).

Today was spent mostly relaxing and doing “first day of the month” bureaucracies (updating the front page of my website, updating Timelines Wiki pageviews table, submitting contract work hours to Vipul).

## A note on scope of updates

I previously mentioned that I don’t talk about my daily Anki routine on this blog. I should also probably clarify the scope of this blog in other ways.

As I stated at the beginning, this blog came about because I wanted to make available information about “what I’m up to” in cases where “what I’m up to” doesn’t naturally lead to public updates. So I think it’s not so important to record things on here when the same information can be obtained through other sources (i.e. my public activity on other websites).

However, I also want to have some kind of conceptual coherence about this blog. And since the overlap between “my activity that doesn’t naturally lead to public updates but where I would like to make public updates available” and “my AI safety learning” is nearly perfect, I think I will track the former by writing about the latter. What does this mean in practice? It means:

1. There will be some AI safety-related stuff on this blog that can be tracked elsewhere. For instance, for now I am planning to blog about wiki pages I update in the course of my studying.
2. Some stuff not related to AI safety, even in cases where they can’t be tracked publicly, won’t be on this blog. This will mostly be one-off projects that I think up one day and decide to work on for a few hours or a day. Will these ever be made available? I think if they lead to something interesting I will try to publish them somehow.

## 2018-11-30

I worked on several wiki pages:

I also asked a question on Math Stack Exchange about point-free notation for limits.

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