Tagged Questions
6
votes
3answers
137 views
How to write $\pi$ as a set in ZF?
I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, ...
6
votes
2answers
117 views
It is possible to define our intuitive notion for probability in subsets of $[0,1]$
I've always heard and read the sentence:
If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$.
What is the meaning for that? Is this the "real" ...
0
votes
1answer
99 views
How information works?
I am really confused after reading wikipedia...
What I don't get is how can something "bring" information, and in mathematics, how a mathematical object (like a set) can "have" information.
For ...
1
vote
1answer
49 views
The relationship between inner automorphisms, commutativity, normality, and conjugacy.
An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$
I have three somewhat broad questions about this:
Why is it related to ...
2
votes
1answer
48 views
What are central automorphisms used for?
A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$.
It's not difficult to prove that the set of central automorphisms forms a subgroup of ...
5
votes
1answer
114 views
Axioms vs. Universal Constructions/Properties
What (exactly) is the difference between defining a mathematical object by it's axioms and by a universal construction ? Please take my 3 opinions into consideration, as they also contain more ...
3
votes
4answers
97 views
What is a minimal polynomial of a group element, and why would we care if it was quadratic?
EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question.
I am trying to understand the definition of a p-stable group. The ...
6
votes
2answers
174 views
Understanding induced representations
Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these.
My question is: ...
2
votes
1answer
118 views
What is the intuitive meaning of the adjugate matrix?
The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything.
What is the intuitive meaning of this matrix?
Are there examples of applications which may ...
7
votes
2answers
183 views
Is there an idea of “primeless isomorphism” studied somewhere in finite group theory?
What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't ...
1
vote
2answers
72 views
visualisation of pointwise boundedness
A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$
...
6
votes
1answer
126 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
229 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 ...
2
votes
1answer
170 views
Free boolean algebra
Consider the following definition:
Let $X$ be a set and $e : X \mapsto A$ a mapping to a boolean algebra
$A.$ We say that $A$ is free over $X$ (with respect to $e$) if for
every mapping $f:X ...
33
votes
8answers
1k views
What makes elementary functions elementary?
Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
0
votes
0answers
193 views
What is “essentially equivalent”?
I thought that "essentially equivalent" for two sets means that there are a bijection between these sets.
But recently I thought that every two sets of the same cardinality are essentially equivalent ...
15
votes
5answers
948 views
Chain Rule Intuition
We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
1
vote
3answers
88 views
Should one think of a network as a connected graph ? (Or: What is the right way to think of a network?)
In the definition of a network, are we only considering connected graphs ?
Because I keep encountering definitions that don't assume explicitly that we deal with connected graphs, but which would be ...
14
votes
5answers
1k views
Why do we require a topological space to be closed under finite intersection?
In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this?
I'm assuming ...
7
votes
6answers
2k views
What is the Direction of a Zero (Null) Vector?
To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...
