Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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7
votes
2answers
1k views

What is the name for a shape that is like a capsule, but with two different radii?

I'm looking for the name of a shape that is like a capsule, but where each circle can have different radii. The shape could be described using two circles (two centers and two radii). Something like ...
12
votes
4answers
7k 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?
10
votes
3answers
2k views

If and only if, which direction is which?

I can never figure out (because the English language is imprecise) which part of "if and only if" means which implication. ($A$ if and only if $B$) = $(A \iff B)$, but is the following correct: ($A$ ...
3
votes
6answers
8k views

What's the difference between open and closed sets?

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!
8
votes
2answers
533 views

Why are Hopf algebras called quantum groups?

Why are noncommutative nonassociative Hopf algebras called quantum groups? This seems to be a purely mathematical notion and there is no quantum anywhere in it prima facie.
11
votes
3answers
2k views

Correct usage of the phrase “In the sequel”? History? Alternatives?

While I feel quite confident that I've inferred the correct meaning of "In the sequel" from context, I've never heard anyone explicitly tell me, so first off, to remove my niggling doubts: What does ...
51
votes
7answers
2k views

Why is compactness in logic called compactness?

In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or ...
4
votes
2answers
216 views

$n$ out of $ m$ theorems (some imply the rest)

Is there symbolism (or even a name) for groups of statements in which any fixed-number of them imply all the rest? For example, in linear algebra, a basis is sometimes defined as a set of ...
5
votes
2answers
353 views

Is this the right categorical generalisation of dual space

Motivatation I am presently looking at a structure that I am trying to pin down- my strategy being to pull the thing up into the greatest possible generality (based on the bits I'm sure about) and ...
24
votes
5answers
4k 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.
103
votes
5answers
11k views

What is the Riemann-Zeta function?

In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
5
votes
2answers
865 views

Unital homomorphism

What is a unital homomorphism? Why are they important?
43
votes
10answers
6k views

Is $0$ a natural number?

Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in ...
18
votes
4answers
11k views

What exactly does it mean for a function to be “well-behaved”?

Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the ...
63
votes
11answers
6k views

Why 1 is not considered to be a prime number?

Why $1$ is not considered to be a prime number? Or why definition of prime numbers is given for integers greater than $1$?
7
votes
1answer
420 views

What are gradients and how would I use them?

I keep seeing this symbol $\nabla$ around and I know enough to understand that it represents the term "gradient." But what is a gradient? When would I want to use one mathematically?
24
votes
5answers
17k views

What are the differences between rings, groups, and fields?

Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
9
votes
10answers
3k views

What is a real number (also rational, decimal, integer, natural, cardinal, ordinal…)?

In mathematics there seem to be a lot of different types of numbers. What exactly are: Real numbers Integers Rationals numbers Decimals Complex numbers Natural numbers Cardinals Ordinals And as ...