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1500 questions
83
votes
5 answers
Theorems that disappointed mathematicians
I was looking for a list of theorems that disappointed mathematicians, in the sense that some disgruntlement regarding the fact that the theorem in consideration being true/false was notable. An example could be a theorem whose negation was notably…
QED
- 929
83
votes
1 answer
In $n>5$, topology = algebra
During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when " topology = algebra ". I don't know what is the meaning of topology=algebra. Can someone clear the sentence to me?
Sepideh Bakhoda
- 6,044
83
votes
4 answers
How would you discover Stokes's theorem?
Let $S$ be a smooth oriented surface in $\mathbb R^3$ with boundary $C$ , and let $f: \mathbb R^3 \to \mathbb R^3$ be a continuously differentiable vector field on $\mathbb R^3$ . Stokes's theorem states that $$ \int_C f \cdot dr = \int_S (\nabla…
littleO
- 54,768
83
votes
3 answers
Escaping infinitely many pursuers
The fugitive is at the origin. They move at a speed of 1. There's a guard at every integer coordinate except the origin. A guard's speed is 1/100. The fugitive and the guards move simultaneously and continuously. At any moment, the guards only move…
Eric
- 1,929
83
votes
6 answers
If a two variable smooth function has two global minima, will it necessarily have a third critical point?
Assume that $f:\mathbb{R}^2\to\mathbb{R}$ a $C^{\infty}$ function that has exactly two minimum global points. Is it true that $f$ has always another critical point? A standard visualization trick is to imagine a terrain of height $f(x,y)$ at the…
Jyrki Lahtonen
- 142,653
83
votes
6 answers
How do you compute negative numbers to fractional powers?
My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a…
Kot
- 3,343
83
votes
7 answers
What exactly are eigen-things?
Wikipedia defines an eigenvector like this: An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a vector that differs from the original vector at most by a multiplicative scalar. So basically in layman…
Eigeneverything
- 833
83
votes
18 answers
Conjectures (or intuitions) that turned out wrong in an interesting or useful way
The question What seemingly innocuous results in mathematics require advanced proofs? prompts me to ask about conjectures or, less formally, beliefs or intuitions, that turned out wrong in interesting or useful ways. I have several in mind, but will…
Ethan Bolker
- 107,009
83
votes
8 answers
Multiple-choice: sum of primes below $1000$
I sat an exam 2 months ago and the question paper contains the problem: Given that there are $168$ primes below $1000$. Then the sum of all primes below 1000 is (a) $11555$ (b) $76127$ (c) $57298$ (d) $81722$ My attempt to solve it: We know that…
Sufaid Saleel
- 3,997
83
votes
3 answers
Why is $i! = 0.498015668 - 0.154949828i$?
While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i! . Curiously, Google's calculator dutifully informed me that $i!$ was, in fact, $0.498015668 - 0.154949828i$. Why…
growse
- 853
83
votes
1 answer
How was the Monster's existence originally suspected?
I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence. For example, in ncatlab's article , The Monster group was…
anon
- 156,297
83
votes
5 answers
What exactly is Laplace transform?
I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out that it gives "less familiar" frequency view. My…
hasExams
- 2,371
83
votes
7 answers
Is linear algebra more “fully understood” than other maths disciplines?
In a recent question , it was discussed how LA is a foundation to other branches of mathematics, be they pure or applied. One answer argued that linear problems are fully understood , and hence a natural target to reduce pretty much anything to.…
leftaroundabout
- 6,683
83
votes
0 answers
Is there a "ping-pong lemma proof" that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
Let $f,g\colon \mathbb R \to \mathbb R$ be the permutations defined by $f\colon x \mapsto x+1$ and $g\colon x \mapsto x^3$, or maybe even have $g\colon x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la Harpe, Topics in Geometric Group…
user29123
83
votes
5 answers
Completion of rational numbers via Cauchy sequences
Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
Derek Scavo
- 2,023