All Questions
18,177 questions
51
votes
3
answers
4k
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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 ...
- 19.2k
51
votes
0
answers
2k
views
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 ...
- 1,010
50
votes
7
answers
16k
views
Way to memorize relations between the Sobolev spaces?
Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel ...
- 2,935
49
votes
13
answers
24k
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Why is it so cool to square numbers (in terms of finding the standard deviation)?
When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why ...
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48
votes
8
answers
5k
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A sudden smiley? :-)
This is a vague question, and I will no doubt be (properly!) chastised for posing it.
I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which
has the ...
- 151k
48
votes
7
answers
12k
views
What's the use of a complete measure?
A complete measure space is one in which any subset of a measure-zero set is measurable.
For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...
- 481
48
votes
6
answers
7k
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Is there an "elegant" non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?
Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway...
Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "...
47
votes
6
answers
6k
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Can we actually find any fixed points with Brouwer's theorem?
Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
- 15.5k
47
votes
7
answers
5k
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Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...
- 11.4k
46
votes
7
answers
10k
views
Conway's game of life for random initial position
What is the behavior of Conway's game of life when the initial position is random? -- We can ask this question on an infinite grid or on an $n$ by $n$ table (planar or on a torus). Specifically ...
- 24.7k
46
votes
0
answers
2k
views
Set-theoretic reformulation of the invariant subspace problem
The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
- 42.8k
45
votes
7
answers
9k
views
What's an example of a space that needs the Hahn-Banach Theorem?
The Hahn-Banach theorem is rightly seen as one of the Big Theorems in functional analysis. Indeed, it can be said to be where functional analysis really starts. But as it's one of those "there ...
- 26.8k
45
votes
7
answers
16k
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What is an intuitive view of adjoints? (version 2: functional analysis)
After realising that I don't have an intuitive understanding of adjoint functors, I then realised that I don't have an intuitive understanding of adjoint linear transformations!
Again, I can use 'em, ...
- 26.8k
45
votes
5
answers
6k
views
Nonstandard analysis in probability theory
I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
...
- 1,302
45
votes
1
answer
4k
views
Rolling a random walk on a sphere
A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...
- 151k
45
votes
1
answer
6k
views
Anti-concentration bound for permanents of Gaussian matrices?
In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of i....
- 9,823
45
votes
1
answer
2k
views
Existence and uniqueness of Haar measure on compacta; a cohomological approach
I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group.
I think the best way of introducing the idea I am pursuing is ...
user30211
44
votes
5
answers
7k
views
Heuristically false conjectures
I was very surprised when I first encountered the Mertens conjecture. Define
$$ M(n) = \sum_{k=1}^n \mu(k) $$
The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...
44
votes
10
answers
47k
views
Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
44
votes
1
answer
4k
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Example of a compact set that isn't the spectrum of an operator
This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity:
Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty ...
- 5,743
43
votes
8
answers
3k
views
How to quantify noncommutativity?
If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
- 1,890
43
votes
3
answers
9k
views
Why the name 'separable' space?
It is well known that a separable space is a topological space that has a countable dense subset. I am wondering how is this related to the name 'separable'? Any intuition where the name come from?
- 1,157
43
votes
1
answer
5k
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Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?
Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
- 559
42
votes
3
answers
5k
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The probability for a symmetric matrix to be positive definite
Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
- 52.3k
41
votes
6
answers
4k
views
Measures of non-abelian-ness
Let $G$ be a finite non-abelian group of $n$ elements.
I would like a measure that intuitively captures the
extent to which $G$ is non-commutative.
One easy measure is a count of the non-commutative ...
- 151k
41
votes
4
answers
16k
views
Product of Borel sigma algebras
If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I ...
- 31.5k
41
votes
4
answers
2k
views
What is the probability two random maps on n symbols commute?
It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
- 38.6k
41
votes
2
answers
2k
views
Topple height of randomly stacked bricks
What is the expected height of a stack of unit-length bricks, each one
stacked on the previous with a uniformly random shift within $\pm \delta$?
The stack topples if the center of gravity of the top $...
- 151k
40
votes
5
answers
5k
views
"Entropy" proof of Brunn-Minkowski Inequality?
I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...
- 22.8k
40
votes
5
answers
10k
views
Is there a natural measures on the space of measurable functions?
Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
- 1,475
40
votes
5
answers
6k
views
Probabilities in a riddle involving axiom of choice
The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question):
The Riddle:
We assume ...
- 1,341
40
votes
4
answers
4k
views
Polynomials on the Unit Circle
I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...
- 3,626
40
votes
1
answer
6k
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The human body's random number generator
I remember learning in microbiology that the human body generates antibodies using a random process so that an enormous variety of antibodies can be produced with a simple genetic code.
Now that I'm ...
- 3,359
40
votes
1
answer
5k
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When should we expect Tracy-Widom?
The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
- 2,135
39
votes
3
answers
14k
views
Is the Invariant Subspace Problem interesting?
There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...
- 1,241
39
votes
2
answers
2k
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Random sequence of integers in $\{1, 2, \dots, n \}$ which is "everywhere probably increasing" - how long can it be?
Let $D=(d_1,d_2,\dots,d_k)$ be a sequence of correlated random variables. $D$ is "everywhere $r$-probably increasing" if the event $d_j > d_i$ has probability $\geq r$ for all $j > i$.
Fix $r \...
- 1,927
39
votes
2
answers
4k
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Can random variables that almost surely solve equations be repaired to surely solve these equations?
Let $(X_\alpha)_{\alpha \in A}$ be a family of boolean random variables $X_\alpha: \Omega \to \{0,1\}$ on a probability space $\Omega = (\Omega, {\mathcal F}, {\mathbf P})$. Let ${\mathcal S}$ be a ...
- 114k
39
votes
3
answers
4k
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Manifold of probability measures: connections between two types of metrics
The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with ...
- 1,127
39
votes
9
answers
3k
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The shortest path in first passage percolation
Update (January 17): The problem has now been solved by Daniel Ahlberg and Christopher Hoffman. (Thanks to Matt Kahle for informing us.)
Consider a square planar grid. (The vertices are pair of ...
- 24.7k
39
votes
1
answer
1k
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Modeling question: how often does "the world's oldest person" die?
This story yesterday (no need to follow the link to understand the question!)
http://www.cnn.com/2011/US/02/01/texas.oldest.person.dies/index.html?hpt=T2
reminds me that I've often wondered about ...
- 17.6k
38
votes
2
answers
13k
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What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
37
votes
3
answers
3k
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On Mathematical Analysis of MathSciNet & MathOverflow
This question has two original motivations: mathematical and social.
The mathematical motivation is mainly based on what I have seen about Zipf's law here and there. The Zipf's law simply states ...
37
votes
4
answers
4k
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Which differential equations allow for a variational formulation?
Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\...
- 7,583
37
votes
3
answers
3k
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An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
- 11.4k
37
votes
2
answers
2k
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Moving one family of commuting self-adjoint operators to another without losing commutativity on the way
This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving ...
- 61.9k
36
votes
3
answers
4k
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the following inequality is true，but I can't prove it
The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...
- 363
36
votes
2
answers
13k
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Mean minimum distance for N random points on a one-dimensional line
Let's say that I have a one-dimensional line of finite length 'L' that I populate with a set of 'N' random points. I was wondering if there was a simple/straightforward method (not involving long ...
- 811
36
votes
4
answers
2k
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Determinant of the random matrix $X^2+Y^2$
$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$.
i) When $n=2,3,4$, one ...
- 3,741
36
votes
0
answers
2k
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Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
- 188k
35
votes
7
answers
6k
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Why is conformal invariance only possible for massless theories?
I'm conscious that this isn't necessarily a research level question, but I've asked this question on mathstackexchange, and received no answer. So I'm trying it here.
A usual mantra in field theories ...
- 914