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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.
8
votes
Looking for book with good general overview of math and its various branches
Penrose's The Road to Reality covers large portions of mathematical physics. This isn't a textbook, and omits many details, but it is as meaty as GEB.
12
votes
Looking for book with good general overview of math and its various branches
The classic answer to this is Courant and Robbins, What is Mathematics? A bit dated, but certainly worth looking at if you haven't yet.
5
votes
Asymptotic Methods in Combinatorics
At a lower level than Flajolet and Sedgewick, Chapter 9 of Concrete Mathematics (Graham, Knuth and Patashnik) is a good introduction to elementary methods.
9
votes
Asymptotic Methods in Combinatorics
At a lower level than Flajolet and Sedgewick, Chapter 5 of generatingfunctionology by Wilf is a good introduction to complex analytic methods. (Yes, my two answers look very similar. As usual in a big …
5
votes
Asymptotic Methods in Combinatorics
If you want to know about quantities which (1) have nice generating functions and (2) depend on more than one parameter, the most thorough guide will be found in the papers of Robin Pemantle. to the b …
24
votes
Undergraduate Level Math Books
Concrete Mathematics, Graham, Knuth and Patashnik. Extremely useful, very good exercises, and a sense of humor that appeals to me.
17
votes
Examples of prime numbers in nature
Cicadas spend most of their lives underground, emerging to mate every $k$ years where the integer $k$ varies from species to species. Biologists have observed that $k$ tends to be a prime number -- fo …
34
votes
Facts from algebraic geometry that are useful to non-algebraic geometers
If $p_1$, $p_2$, ..., $p_m$ are polynomials in $n$ variables, with $m>n$, then there is a polynomial $q$ such that $q(p_1, p_2, \ldots, p_m)$ is identically zero.
41
votes
Suggestions for good notation
Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG
$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^ …
4
votes
Computer algebra errors
We found some interesting bugs in Mathematica's integration software on this thread.
To wit, set
integral[m_,n_] = Integrate[Log[2+Cos[2Pi x]+Cos[2Pi y]] Cos[2Pi m x] Cos[2Pi n y],
…
3
votes
Suggestions for good notation
Since this one is on the front page again: In my personal notes, I have started writing sums/integrals over complicated index sets as $\sum \left( \text{summand} \mid \text{condition} \right)$, rather …
30
votes
Examples of common false beliefs in mathematics
I'm not sure how common this is, but it confused me for years. Let $f : \mathbb{C} \to \mathbb{C}$ be an analytic function and $\gamma$ a path in $\mathbb{C}$. In your first class in complex analysis, …
2
votes
Examples of common false beliefs in mathematics
If a matrix $A$ is self-adjoint/skew-self-adjoint with respect to a symmetric bilinear form, then it is diagonalizable.
True for matrices over $\mathbb{R}$, with respect to a positive definite inner …
10
votes
Examples of common false beliefs in mathematics
Multiplication of differential forms is inherently anti-commutative. Thus, if $x$ and $y$ are coordinates on a surface, then $dx \wedge dy$ makes sense but $(dx)^2+(dy)^2$ is either nonsense or, if it …
29
votes
Examples of common false beliefs in mathematics
I'm not sure that anyone holds this as a conscious belief but I have seen a number of students, asked to check that a linear map $\mathbb{R}^k \to \mathbb{R}^{\ell}$ is injective, just check that each …