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Results tagged with big-list
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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.
15
votes
Slick ways to make annoying verifications
One can often work out the exponents in some identity or inequality by using dimensional analysis or by plugging in key examples, and this is often faster than deriving the identity or inequality pain …
9
votes
Problems where we can't make a canonical choice, solved by looking at all choices at once
Zagier's one-sentence proof of Fermat's two square theorem (discussed previously on MathOverflow here) seems to qualify.
28
votes
What are some examples of proving that a thing exists by proving that the set of such things...
Szemerédi's theorem asserts that every set $A$ of integers of positive upper density (thus $\limsup_{N \to \infty} \frac{|A \cap [-N,N]|}{|[-N,N]|} > 0$) contains arbitrarily long arithmetic progressi …
29
votes
Cardinalities larger than the continuum in areas besides set theory
One of the quickest ways to demonstrate that there exist Lebesgue measurable subsets of the real line that are not Borel measurable is to compute the cardinality of the Lebesgue $\sigma$-algebra and t …
- 114k
8
votes
Examples of statements that provably can't be proved using a promising looking method
When solving an initial value problem for a PDE, with initial data in some function space class, one popular way to proceed is to apply the contraction mapping (or Picard iteration) method in a suitab …
30
votes
Important (but not too well known) inequalities
The class of concentration of measure inequalities is a fundamental tool in modern probability (and any field that uses probability, e.g., random matrix theory, theoretical computer science, statistic …
23
votes
When forgetting structure doesn't matter
By the positive solution to Hilbert's fifth problem (as well as the theorem of Cartan that continuous homomorphisms between Lie groups are automatically smooth), the forgetful functor from the categor …
60
votes
The half-life of a theorem, or Arnold's principle at work
If $p$ is a prime, then every minor of the Fourier matrix $(e^{2\pi i jk/p})_{1 \leq j,k \leq p}$ is non-singular.
This fact was proven by Chebotarev in 1927 (answering a question of Ostrowski), Dani …
26
votes
Examples of statements that provably can't be proved using a promising looking method
There are examples (possibly due to Hecke?) of zeta-like functions which obey almost all of the known properties of the Riemann zeta function, such as the functional equation, Euler product, and asymp …
15
votes
What are some examples of proving that a thing exists by proving that the set of such things...
The Chevalley-Warning theorem asserts that if a system of polynomial equations in $r$ variables over a finite field of characteristic $p$ has total degree less than $r$, then the number of solutions t …
49
votes
Suggestions for good notation
I am fond of subscripting asymptotic notation with the parameters that the implied constant is allowed to depend on (and on the asymptotic parameter, if needed). e.g.
$X = O_k(Y)$ (or $X \ll_k Y$, o …
21
votes
Demonstrating that rigour is important
One can rigorously prove that pyramid schemes cannot run forever, and that no betting system with finite monetary reserves can guarantee a profit from a martingale or submartingale.
But there are cou …
8
votes
Why are so few operations with arity bigger than 2?
In higher order Fourier analysis, there are d-dimensional parallelopiped structures which can be viewed as a $2^d-1$-ary relation of the form "given all but one vertex of a parallelopiped as input, re …
23
votes
Suggestions for good notation
One can decorate a subscript or superscript by additional symbols to indicate what the subscript or superscript is doing. For instance, consider a truncation $f 1_{|f| \leq N}$ of a function to its v …
34
votes
Proofs that require fundamentally new ways of thinking
It seems that certain problems seem to induce this sort of new thinking (cf. my article "What is good mathematics?"). You mentioned the Fourier-analytic proof of Roth's theorem; but in fact many of t …