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Results tagged with big-list
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user 1916
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.
4
votes
Proving theorems by using functions with fixed points.
There is a very slick proof (discussed here on MO) that every prime $p=4k+1$ is a sum of two squares, which looks at the set $S= \{(x,y,z) \in N^3: x^2+4yz=p \}$ and shows that a particular involution …
9
votes
What are the most overloaded words in mathematics?
Complete/Completion
complete metric spaces,
complete measure spaces,
completing a ring at an ideal,
complete graph
complete category
complete lattice
and many more uses (a lot in computation th …
49
votes
28
answers
8k
views
Problems where we can't make a canonical choice, solved by looking at all choices at once
It's a common theme in mathematics that, if there's no canonical choice (of basis, for example), then we shouldn't make a choice at all. This helps us focus on the heart of the matter without giving o …
1
vote
Various concepts of "closure" or "completion" in mathematics
The radical of an ideal - we have $\sqrt{\sqrt{I}}=\sqrt{I}$.
1
vote
Various concepts of "closure" or "completion" in mathematics
The closure of a set in a topology - we have $\overline{\bar{X}}=\bar{X}$.
1
vote
Various concepts of "closure" or "completion" in mathematics
The total ring of fractions for a ring - we have $Q(Q(R))\simeq Q(R)$.
6
votes
11
answers
3k
views
Various concepts of "closure" or "completion" in mathematics
Out of idle curiosity, I'm wondering about all the various idempotent constructions we have in mathematics (they seem to be generally referred to as a "closure" or "completion"), and how some of them …
19
votes
Famous mathematical quotes
"So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality."
- Albert Einstein
(Personally, I'd take certainty over being a …
192
votes
Jokes in the sense of Littlewood: examples?
Let "$\int$" denote $\int_0^x$. We want to find the solution to
$$\int f = f-1.$$
We simply "factor out" $f$, getting $1=\left(1-\int\right)f$. Thus, $f=(1-\int)^{-1}1$.
Using the geometric series …
10
votes
Which book would you like to see "texified"?
Atiyah + Macdonald, Introduction to Commutative Algebra.
12
votes
Errata for Atiyah–Macdonald
On page 29, the example at the top has two typos: it says "$(x)=2x$", when it should be "$f(x)=2x$", and the exact sequence at the end of that same line says "$0\rightarrow\mathbb{Z}\otimes \stackrel{ …
6
votes
Errata for Atiyah–Macdonald
On page 91, the second line in the second Example should refer to Proposition 8.8, not Theorem 8.7.
16
votes
Errata for Atiyah–Macdonald
On page 8, the proof of part ii of Proposition 1.11 begins "Suppose $\mathfrak{p}\not\subseteq\mathfrak{a}_i$ for all $i$." It should be $\not\supseteq$.
8
votes
Errata for Atiyah–Macdonald
On page 31, the first line refers to Proposition 2.11, when it should be 2.12.
53
votes
7
answers
14k
views
Good lattice theory books?
A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - …