Recent content by Oats

Hello, I am a second year PhD student in mathematics currently studying logic. I have recently begun questioning staying in the field, and I really think I may want to work in Machine Learning, at the very least in the theory of it. I am willing to self study from machine learning books, but...

Yeah I was considering perhaps something along statistics maybe, but I don't really know what it entails on a research level. I think it's a useful skill to have regardless. I know a tiny bit of programming. I would think I'd like to stay in Academia, rather than work in industry, but I can't...

Hello, I am a second year PhD student in pure mathematics, and I'm going through a bit of a dilemma. Up to this point I felt that I wanted to work in Mathematical Logic. To be honest, I think working in any field of math would be fun, but logic had extra cool topics like computability theory...

Hello, I am a first-year mathematics PhD student. I am completely initiated in rigorous mathematics, and have a strong intuitive and working understanding of basic mathematics concepts across analysis and algebra. I also have a strong interest in physics, and want to learn enough to be quite...

Hello, Every website that I go on that has MathJax support displays math in a font that almost looks like the default Computer Modern ##\LaTeX## font. The following is taken from Math Stack Exchange: I heard that MathJax usually looks for the STIX font located on your computer, and if it does...

I am familiar with the following formulation of the principle of recursive definition. Now, in certain proofs in analysis, there are times where a recursive definition for a function is used. Here are two examples. ##\textbf{Proof:}## Let ##p## be a limit point of ##E##, let ##\epsilon > 0##...

Aha, yes! I didn't even consider finding what ##f(c)## equals. Having ##|f(x)| \leq (x - c)^2## for all ##x \in I## forces ##f(c) = 0##. Now, to find ##f'(c)## we must now find $$\lim_{x \to c} \frac{f(x)}{x - c}$$, which we can show to be ##0##, if we can alternatively show that $$\lim_{x \to...

Homework Statement [/B] 2. The attempt at a solution I'm not really sure where to start. We just want to show that ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = 0##. I see that ##\lim_{x \to c} (x - c)^2 = 0##. I feel that this may be a simple trick of inequalities, but I am having a complete...

Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...

1. The problem statement: Let ##f:[a, b] \rightarrow \mathbb{R}##. Prove that if ##f## is continuous, then ##f## is bounded. 2. Relevant Information This is the previous exercise. I have already proved this result, and the book states to use it to prove the next exercise. It also hints to use...

Hello, I have read a fair chunk of Munkres' Topology book and took a short introductory course during undergraduate, but I would like to learn point-set topology a little better. I have quite a bit of mathematical maturity, so that isn't an issue for me. I had a larger list of potential books to...