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1500 questions
166
votes
20 answers
Online tool for making graphs (vertices and edges)?
Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to draw. (why do I have so many? It's for pathing in a game) Only tool…
f20k
- 1,763
165
votes
16 answers
What's new in higher dimensions?
This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3". What I am wondering about is what new geometrical phenomena are there in higher dimensions. When I say new I mean phenomena…
Martin Hurtado
- 1,843
164
votes
1 answer
What functions can be made continuous by "mixing up their domain"?
Definition . A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ such that $f\circ \phi$ is continuous. So one could say a potentially continuous (p.c.) function is "a continuous…
M. Winter
- 30,997
163
votes
4 answers
The direct sum $\oplus$ versus the cartesian product $\times$
In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times A$; however in, for example, the fundamental…
Sputnik
- 3,917
162
votes
18 answers
How do you describe your mathematical research in layman's terms?
"You do research in mathematics! Can you explain your research to me?" If you're a research mathematician, and you have any contact with people outside of the mathematics community, I'm sure you've been asked this question many times. For years now,…
Jared
- 32,247
162
votes
20 answers
Are there any open mathematical puzzles?
Are there any (mathematical) puzzles that are still unresolved? I only mean questions that are accessible to and understandable by the complete layman and which have not been solved, despite serious efforts, by mathematicians (or laymen for that…
Řídící
- 3,288
162
votes
7 answers
Show that the determinant of $A$ is equal to the product of its eigenvalues
Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic polynomial of the matrix $\det(A-\lambda I)$. But,…
onimoni
- 6,816
162
votes
2 answers
Slice of pizza with no crust
The following question came up at a conference and a solution took a while to find. Puzzle. Find a way of cutting a pizza into finitely many congruent pieces such that at least one piece of pizza has no crust on it. We can make this more concrete,…
Dan Rust
- 31,243
162
votes
15 answers
Are there real-life relations which are symmetric and reflexive but not transitive?
Inspired by Halmos ( Naive Set Theory ) . . . For each of these three possible properties [reflexivity, symmetry, and transitivity], find a relation that does not have that property but does have the other two. One can construct each of these…
000
- 5,808
162
votes
4 answers
Sum of random decreasing numbers between 0 and 1: does it converge??
Let's define a sequence of numbers between 0 and 1. The first term, $r_1$ will be chosen uniformly randomly from $(0, 1)$, but now we iterate this process choosing $r_2$ from $(0, r_1)$, and so on, so $r_3\in(0, r_2)$, $r_4\in(0, r_3)$... The set of…
Carlos Toscano-Ochoa
- 2,653
161
votes
21 answers
Best book of topology for beginner?
I am a graduate student of math right now but I was not able to get a topology subject in my undergrad... I just would like to know if you guys know the best one..
gg1
- 111
161
votes
6 answers
Connection between Fourier transform and Taylor series
Both Fourier transform and Taylor series are means to represent functions in a different form. What is the connection between these two? Is there a way to get from one to the other (and back again)? Is there an overall, connecting (geometric?)…
vonjd
- 9,110
161
votes
31 answers
Stopping the "Will I need this for the test" question
I am a college professor in the American education system and find that the major concern of my students is trying to determine the specific techniques or problems which I will ask on the exam. This is the typical "will this be on the test?"…
Wintermute
- 3,940
161
votes
7 answers
Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant. Along similar lines, you can show that $\mathbb{R^2}$ isn't…
user7530
- 50,865
160
votes
28 answers
Simple theorems that are instances of deep mathematics
So, this question asks about how useful computational tricks are to mathematics research, and several people's response was "well, computational tricks are often super cool theorems in disguise." So what "computational tricks" or "easy theorems" or…
Stella Biderman
- 31,615