Most Popular
1500 questions
160
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
- 1,793
160
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
- 2,597
160
votes
2 answers
Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$
Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$?
Thank you.
Myshkin
- 35,974
- 27
- 154
- 332
159
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…
Jared
- 31,451
159
votes
3 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…
Dan Rust
- 30,108
159
votes
7 answers
Is Apple ipad / tablet good for mathematics students?
I am a math student. I'd like to find out if tablets (iPads, Galaxy Note 10.1) are worth the cost.
How good are tablets for the purposes of reading textbooks as PDF and writing mathematics with a stylus?
For writing math in TeX, it looked like the…
T. Webster
- 3,052
159
votes
25 answers
Most ambiguous and inconsistent phrases and notations in maths
What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts?
For instance, a function $f$:
$f^{-1}(x)$ can be an inverse and a…
Frank Vel
- 5,339
158
votes
7 answers
Does a "cubic" matrix exist?
Well, I've heard that a "cubic" matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant - and other properties? I'm a young student, so... please don't get mad at me if I'm talking something…
Ian Mateus
- 7,431
158
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
- 8,810
158
votes
7 answers
Intuitive interpretation of the Laplacian Operator
Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian Operator (a.k.a. divergence of gradient)?
koletenbert
- 3,970
158
votes
1 answer
Classification of prime ideals of $\mathbb{Z}[X]$
Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$.
My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types?
If yes, how would you prove this?
$(0)$.
$(f(X))$, where $f(X)$ is an irreducible…
Makoto Kato
- 42,602
157
votes
5 answers
What is the Riemann-Zeta function?
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
157
votes
4 answers
Why is learning modern algebraic geometry so complicated?
Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the strong point of modern algebraic geometry. I'm reading…
Dubious
- 13,350
- 12
- 53
- 142
157
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
- 31,155
156
votes
8 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 $\det(A-\lambda I)$.…
onimoni
- 6,376