Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with big-list
Search options answers only
not deleted
user 766
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
381
votes
Examples of common false beliefs in mathematics
The closure of the open ball of radius $r$ in a metric space, is the closed ball of radius $r$ in that metric space.
In a somewhat related spirit: the boundary of a subset of (say) Euclidean space ha …
325
votes
Awfully sophisticated proof for simple facts
An example that came up in my measure theory class today:
The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges, because otherwise the functions $f_n := \frac{1}{n} 1_{[0,n]}$ would be dominat …
252
votes
Examples of unexpected mathematical images
The third image below was certainly unexpected for my soon-to-be-collaborators, Emmanuel Candes and Justin Romberg. They started with a standard image in signal processing, the Logan-Shepp phantom:
…
244
votes
Why is a topology made up of 'open' sets?
The textbook presentation of a topology as a collection of open sets is primarily an artefact of the preference for minimalism in the standard foundations of the basic structures of mathematics. Thi …
234
votes
Accepted
What is convolution intuitively?
I remember as a graduate student that Ingrid Daubechies frequently referred to convolution by a bump function as "blurring" - its effect on images is similar to what a short-sighted person experiences …
207
votes
Examples of common false beliefs in mathematics
Some false beliefs in linear algebra:
If two operators or matrices $A$, $B$ commute, then they are simultaneously diagonalisable.
(Of course, this overlooks the obvious necessary condition that each …
194
votes
Thinking and Explaining
I find there is a world of difference between explaining things to a colleague, and explaining things to a close collaborator. With the latter, one really can communicate at the intuitive level, beca …
141
votes
Intuitive crutches for higher dimensional thinking
I can't help you much with high-dimensional topology - it's not my field, and I've not picked up the various tricks topologists use to get a grip on the subject - but when dealing with the geometry of …
103
votes
Examples of common false beliefs in mathematics
In order to show that a polynomial $P \in F[x_1,\ldots,x_n]$ vanishes, it suffices to show that $P(x_1,\ldots,x_n) = 0$ for all $x_1,\ldots,x_n \in F$. True in infinite fields, but very false for sma …
94
votes
Accepted
Are there proofs that you feel you did not "understand" for a long time?
As an undergraduate, I learned the Sylow theorems in my algebra classes but could never retain either the statement or proof of these theorems in memory except for short periods of time (and in partic …
91
votes
Colloquial catchy statements encoding serious mathematics
"Can you hear the shape of a drum?"
This was Kac's famous way of asking whether the shape of a two-dimensional domain could be reconstructed from the spectrum of the Laplacian on that domain. (The …
88
votes
Examples of common false beliefs in mathematics
This is perhaps a misunderstood definition rather than a false belief, but:
"A subnet of a net $( x_\alpha )_{\alpha \in A}$ takes the form
$( x_\alpha )_{\alpha \in B}$ for some subset $B$ of $A$. …
73
votes
Still Difficult After All These Years
Difficulty is not additive, and measuring the difficulty of proving a single result is not a good measure of the difficulty of understanding the body of work in a given field as a whole.
Suppose for …
68
votes
Accepted
Why are proofs so valuable, although we do not know that our axiom system is consistent?
If you like, you can view proofs of a statement in some formal system (e.g. ZFC) as a certificate that a counterexample cannot be found without demonstrating the inconsistency of ZFC, which would be a …
68
votes
Results true in a dimension and false for higher dimensions
Keller's conjecture asserts that whenever one tiles ${\bf R}^n$ by unit cubes, there must be two cubes which share a common face. True when $n \leq 6$, false for $n\geq 8$, and still open for $n=7$.
…