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1500 questions
72
votes
3 answers
How and why does Grothendieck's work provide tools to attack problems in number theory?
This is probably a horrible question to experts, but I think it is reasonable from someone who knows nothing. I have always been fascinated with Grothendieck and the way he did mathematics. I've heard Mochizuki's work on the abc conjecture heavily…
Arrow
- 14,560
72
votes
5 answers
What is so interesting about the zeroes of the Riemann $\zeta$ function?
The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$ The Riemann hypothesis asserts that all the…
Karna
- 739
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
- 17,047
72
votes
3 answers
If I know the order of every element in a group, do I know the group?
Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ have the same number of elements for each order,…
Stanley
- 3,254
72
votes
14 answers
Pseudo Proofs that are intuitively reasonable
What are nice "proofs" of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples of what I have in mind: $s=\sum_{i=0}^\infty…
Michael Greinecker
- 35,828
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
- 5,749
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
- 1,143
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
- 1,519
72
votes
6 answers
Foundation for analysis without axiom of choice?
Let's say I consider the Banach–Tarski paradox unacceptable, meaning that I would rather do all my mathematics without using the axiom of choice. As my foundation, I would presumably have to use ZF, ZF plus other axioms, or an approach in which sets…
user13618
71
votes
5 answers
Are functions of independent variables also independent?
It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed. If I have two independent random variables, $X_1$ and $X_2$, then I define two other random variables $Y_1$ and $Y_2$, where $Y_1$…
LLS
- 877
71
votes
4 answers
What lies beyond the Sedenions
In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: $$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions}…
Willem Noorduin
- 2,759
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
- 2,852
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 follows:…
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 one…
user9464
71
votes
5 answers
Why are two permutations conjugate iff they have the same cycle structure?
I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
Han Solo
- 711