Most Popular
1500 questions
52
votes
5 answers
Tetris-like falling sticky disks
Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, if the disk center reaches $y=0$, the disk…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
52
votes
3 answers
Properties of functors and their adjoints
I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are different types of functors and different types…
David Jordan
- 6,053
51
votes
2 answers
vector balancing problem
I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct.
This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest possible proof strategies?
I have vectors $v_1,…
Nik Weaver
- 42,041
51
votes
3 answers
What is the difference between holonomy and monodromy?
What is the difference between holonomy and monodromy?
And what is the simplest example in which one is trivial and the other is not?
James Propp
- 19,363
51
votes
2 answers
Which colimits commute with which limits in the category of sets?
Given two categories $I$ and $J$ we say that colimits of shape $I$ commute with limits of shape $J$ in the category of sets, if for any functor $F : I \times J \to \text{Set}$ the canonical map $$\textrm{colim}_{i\in I} \text{lim}_{j\in J} F(i,j)…
Omar Antolín-Camarena
- 5,189
51
votes
3 answers
What is the sandpile torsor?
Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all vertices gives a homomorphism from Div(G) to Z which…
JSE
- 19,081
51
votes
1 answer
Does $2^X=2^Y\Rightarrow |X|=|Y|$ imply the axiom of choice?
The Generalized Continuum Hypothesis can be stated as $2^{\aleph_\alpha}=\aleph_{\alpha+1}$. We know that GCH implies AC (Jech, The Axiom of Choice, Theorem 9.1 p.133).
In fact, a relatively weak formulation: $|X|\le|Y|< 2^X\implies |X|=|Y|$ would…
Asaf Karagila
- 38,140
51
votes
6 answers
What does Mellin inversion "really mean"?
Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is defined by the function
$$M(f)(s) = \int_0^{\infty}…
Frank Thorne
- 7,199
51
votes
6 answers
What does actually being a CW-complex provide in algebraic topology?
From time to time, I pretend to be an algebraic topologist. But I'm not really hard-core and some of the deeper mysteries of the subject are still ... mysterious. One that came up recently is the exact role of CW-complexes. I'm very happy with…
Andrew Stacey
- 26,373
51
votes
5 answers
How to resolve a disagreement about a mathematical proof?
I am having a problem which should not exist. I am reading what I believe to be an important paper by a person - let me call him/her $A$ - whom I believe to be a serious and talented mathematician. A lemma in this paper is proven by means of an…
Umbra
- 61
- 2
- 6
51
votes
8 answers
Motivating the category of chain complexes
Let $R$ be a commutative ring. For awhile I have been trying to motivate to myself more fully the definition of and various structures on the category $\text{Ch}(R)$ of chain complexes of $R$-modules (and various subcategories thereof). One…
Qiaochu Yuan
- 114,941
51
votes
6 answers
Which nonlinear PDEs are of interest to algebraic geometers and why?
Motivation
I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a number of important achievements in the study of…
mathphysicist
- 5,233
51
votes
7 answers
Why forgetful functors usually have LEFT adjoint?
for forgetful functors, we can usually find their left adjoint as some "free objects", e.g. the forgetful functor: AbGp -> Set, its left adjoint sends a set to the "free ab. gp gen. by it". This happens even in some non-trivial cases. So my question…
Yuhao Huang
- 4,982
51
votes
0 answers
Alternating colors on a line: infinitely often or converge?
Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all blue intervals distributed as $\mathbb{P}_B$.
Once…
Ngoc Mai Tran
- 1,010
51
votes
25 answers
Theorems for nothing (and the proofs for free)
Some theorems give far more than you feel they ought to: a weak hypothesis is enough to prove a strong result. Of course, there's almost always a lot of machinery hidden below the waterline. Such theorems can be excellent starting-points for…
Andrew Stacey
- 26,373