1
vote
1answer
47 views

What does it mean for a function to “quickly” approach $0$?

We can talk about how "quickly" an infinite series approaches $0$ by talking about an asymptotic bound on its terms - a series that is $O(1/x)$ converges more slowly than one that is $O(1/x^2)$, etc. ...
1
vote
5answers
212 views

Reflexivity: How can something be related to itself?

Background: I'm a philosophy student. I'm comfortable with math, but don't have much of a background in it. One of the topics I'm writing about (I-relation in theories of identity) closely mirrors ...
1
vote
1answer
94 views

Is there a way to mathematically describe “surprise”?

Let's say that there are ten people entered into a random drawing, the winner gets some large prize. If I were one of those ten people, and I were to win, then I would be pleasantly surprised. If ...
3
votes
3answers
159 views

Are statements like “Every time I've done X, Y has happened” (vacuously) true if I've never done X?

I've recently been wondering about vacuous truths. I know a statement like "I've never been beaten in a race" is true if I've never been in a race, but what I'm wondering is if the following ...
3
votes
2answers
133 views

How to describe discretization to a novice?

While going through some C++ code about stochastic processes, I came across this concept of discretization repeatedly. I have checked the Wikipedia link but description goes into deeper details too ...
7
votes
1answer
136 views

When do modifiers denote sub or super? Pseudo-, quasi-, ultra-, strong-, well-, pre-, c0- …

One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- ...
3
votes
2answers
292 views

What exactly is a manifold?

Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth. Does this mean that in mathematics a manifold is essentially a representation of something that ...
43
votes
5answers
2k views

Why “characteristic zero” and not “infinite characteristic”?

The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
3
votes
3answers
2k views

Bilinearity: what does it mean?

What does bilinear really mean? Everytime I heard the word, I think it should be "linear in 2 ways?" For example, from the definition of inner product (taken from Appendix A of "Wavelets For ...
10
votes
2answers
232 views

What is the difference between probability and statistics?

Is it that probability is top-down (going from pure distributions to predictions about events) and statistics is bottom-up (going from specific events to predicting pure distributions?) I'm pretty ...
5
votes
1answer
152 views

Sorting through “algebra of random variables,” vs. “probability space,” etc

I have been reading through Wikipedia pages, and I'm still really confused. What is the difference between "algebra of random variables" and "probability space."? Are they just different words for ...
8
votes
2answers
171 views

In/out equivalent to left/right “chirality”

Apologies if this is off-topic, but we're having a problem over on English Language with this question, and I thought you guys might be able to help. Basically it's a matter of topology. We know the ...
42
votes
12answers
5k views

I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]

I didn't know how to phrase the question properly so I am going to explain how this came about. I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
3
votes
2answers
172 views

Relationship between torsion modules and topology

I was reviewing my class notes and found the following: "The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band" In our notes we used the following definition ...
9
votes
4answers
3k views

Intuitive Way To Understand Principal Component Analysis

I know that this is meant to explain variance butthe description on Wikpiedia stinks and it is not clear how you can explain variance using this technique Can anyone explain it in a simple way?
1
vote
5answers
3k views

What's the difference between open and closed sets?

Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!
14
votes
5answers
1k views

What is a Markov Chain?

What is a intuitive explanation of a Markov Chain, and how they work? Please provide at least one practical example.