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Results tagged with big-list
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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.
12
votes
Vector spaces without natural bases
The solution space of a homogeneous (ordinary or partial) linear differerential equation has no natural basis.
8
votes
Do you read the masters?
I enjoyed reading Gauss's Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Fläche auf einer andern gegebnen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in den kleinsten Theile …
2
votes
Why is a topology made up of 'open' sets?
First, note that a mapping between metric spaces is continuous if and only if the inverse image of an open set is always open. There are various concepts for metric spaces that you can likewise find e …
31
votes
What are the worst notations, in your opinion?
I rather dislike the notation $$\int_{\Omega}f(x)\,\mu(dx)$$ myself. I realize that just as the integral sign is a generalized summation sign, the $dx$ in $\mu(dx)$ would stand for some small measurab …
2
votes
Various concepts of "closure" or "completion" in mathematics
Just about every form of compactification. The compactification of a compact space is itself, and a compactification had better be compact or it shouldn't be called a compactification.
The same thing …
115
votes
Probabilistic proofs of analytic facts
One nice example is Bernstein's proof of the Weierstrass theorem. This proof analyses a simple game: Let $f$ be a continuous function on $[0,1]$, and run $n$ independent yes/no experiments in which th …
- 9,366
61
votes
Favorite popular math book
Title: Logicomix: An epic search for truth
Authors: Apostolos Doxiadis, Christos Papadimitriou
Artists: Alecos Papadatos, Annie Di Donna
Short description: A comic book biography of Bertrand Russe …
52
votes
Favorite popular math book
Title: Mathematics: A very short introduction
Author: Timothy Gowers
Short description: As the title says, very short. Gives the non-mathematical reader a good idea what mathematics is all about in …
36
votes
Famous mathematical quotes
“Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions.”
– Th. Bröcker & K. Jänich, Introduction to differential topology (p.25)
56
votes
What is convolution intuitively?
What is the operator $C_f\colon g\mapsto f*g$? Consider the translation operator $T_y$ defined by $T_y(g)(x)=g(x-y)$, and look at $f*g(x)=\int_{\mathbb{R}}f(y)g(x-y) \, dy$. Rewriting this as an opera …
21
votes
Dimension leaps
The wave equation behaves differently in even and odd space dimensions. In odd-dimensional space, radial waves satisfy a modified version of the one-dimensional wave equation. In particular, Huygens' …