Search Results

Stallings's theorem on ends of groups, together with some applications (my favorite would be the fact that groups of cohomological dimension one are free, but that might be too algebro-topological for …

My favorite way to think about Mayer–Vietoris is via sheaf cohomology. So let $F$ be a sheaf of abelian groups on a locally compact Hausdorff space $X$. …

Countable union of countable sets is one of my all-time favorites (and surely one of the all-time best): here It's a picture of Cantor's pairing function.

Here is my favorite one (though not so elementary). Theorem (Grothendieck). Let $X$ be a smooth projective variety over an algebraically closed field $k$. …

The only branch where I think this is explicitly recognized in the literature is topology, where for example Munkres is careful to point out and discuss his favorite counterexamples in his book, and Counterexamples … So: what are your favorite examples of counterexamples that really illuminate something about some aspect of a subject? …

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. …

My favorite example is the paper of Dubuc, which proves the conjecture is true if one of the transformations is an $n$-cycle. …

[1] One of my favorite quotes in "The C programming language": "C is not a big language, and it is not well served by a big book". -- Brian Kernighan [2] I never got the hang of object orientation, and …

In fact, this is a general construction: take your favorite paracompact Hausdorff locally compact space, then double a non-isolated point. …

So, my solution was to follow my favorite general principle: "If you don't know how to do something, do it at random". …

To see this, consider any real number $ x $ (in your favorite system of real numbers $ \mathbb R $). Let $ b _ n = \frac { | x | } { 2 ^ { n - 1 } } $ for all natural numbers $ n $. …

(I've edited this question: I originally asked "what is your favorite way to go about it?", but was told that was too subjective. …

Let $X_t=Z$ for all $t$ where $Z=W^2+1$ for your favorite random variable $W$. Then $X_t$ is a continuous martingale and $G(t)=t\mathbb P(X_1\geq t)=t$ for $t\in [0,1]$. …

My favorite example is the space of scalar-valued simple functions on a $\sigma$-algebra $\mathcal{A}$ (or a measurable space $(X,\mathcal{A})$ if you want). …

My favorite reference about such things is Cai, Chen, Lipton and Lu's paper "On Tractable Exponential Sums". …