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1500 questions
72
votes
12 answers
Why do we not have to prove definitions?
I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? It seems like some definitions should have a…
Zduff
- 4,380
72
votes
6 answers
Nice expression for minimum of three variables?
As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function.
$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive explanation to go along with this: If we go to the…
Oscar Cunningham
- 16,939
72
votes
17 answers
Why is a circle 1-dimensional?
In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position of a circle, but apparently a circle can be…
mr eyeglasses
- 5,709
72
votes
4 answers
Conjugate subgroup strictly contained in the initial subgroup?
Let $G$ be a group, $H\subseteq G$ a subgroup and $a\in G$ an element of the group. Is it possible that $aHa^{-1} \subset H$, but $aHa^{-1} \neq H$?
If $H$ has finite index or finite order, this is not possible.
Sasha
- 1,133
72
votes
5 answers
How can a probability density be greater than one and integrate to one
Wikipedia says:
The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.
and it also says.
Unlike a probability, a probability density function can take on values greater than one; for…
zenna
- 1,509
71
votes
4 answers
Learning Roadmap for Algebraic Topology
I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them.
I have a solid background in Abstract Algebra, and also have knowledge on Homological Algebra(in…
Arsenaler
- 4,020
71
votes
5 answers
Understanding the Laplace operator conceptually
The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it?
Any good essays (combining both history and conceptual understanding) on the Laplace operator, and…
bzm3r
- 2,812
71
votes
6 answers
Origin of the dot and cross product?
Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product…
Curiosity
- 1,628
71
votes
1 answer
Differential forms on fuzzy manifolds
This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me.
Question
In the usual setting of open subsets of $\mathbb{R}^n$, differential forms are defined as…
Isaac Sledge
- 719
71
votes
4 answers
How do people apply the Lebesgue integration theory?
This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas.
In a lot of calculus textbooks, there is usually a chapter about "applications" after the…
user9464
71
votes
24 answers
Theorems' names that don't credit the right people
The point of this question is to compile a list of theorems that don't give credit to right people in the sense that the name(s) of the mathematician(s) who first proved the theorem doesn't (do not) appear in the theorem name.
For instance the…
Git Gud
- 31,706
71
votes
2 answers
Number of monic irreducible polynomials of prime degree $p$ over finite fields
Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$?
Thanks!
IBS
- 4,335
71
votes
7 answers
Proof that the largest eigenvalue of a stochastic matrix is $1$
The largest eigenvalue of a stochastic matrix (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$.
Wikipedia marks this as a special case of the Perron-Frobenius theorem, but I wonder if there is a simpler (more direct)…
koletenbert
- 4,150
71
votes
6 answers
"Gaps" or "holes" in rational number system
In Rudin's Principles of Mathematical Analysis 1.1, he first shows that there is no rational number $p$ with $p^2=2$. Then he creates two sets: $A$ is the set of all positive rationals $p$ such that $p^2<2$, and $B$ consists of all positive…
Larry
- 751
71
votes
2 answers
What is the difference between minimum and infimum?
What is the difference between minimum and infimum?
I have a great confusion about this.
Manoj
- 1,827