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18 votes
0 answers
584 views

Human Knot game [duplicate]

In the popular game "Human Knot", a group of people stands in a circle and each person grabs another person's hands at random (one with the left hand and one with the right hand). The goal is to ...
Michal Kotowski's user avatar
18 votes
1 answer
452 views

Is defining measures as functionals ever insufficiently general in practice?

Crossposting from Math Stack Exchange, as it has yet to receive any answers there; the original question is here. The way I learned measures was as set functions on a $\sigma$-algebra with certain ...
Justin Toyota's user avatar
18 votes
3 answers
2k views

What are the right categories of finite-dimensional Banach spaces?

This is inspired partly by this question, especially Tom Leinster's answer. Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
Mark Meckes's user avatar
  • 11.4k
17 votes
4 answers
4k views

How much does the absolute value of an operator behave like an absolute value?

Recall that the absolute value of a bounded operator $T$ on a Hilbert space $H$ is the unique positive operator $|T|$ such that $$\||T|x\|=\|Tx\|$$ for all $x\in H$. It can be defined using the ...
Iian Smythe's user avatar
  • 3,115
17 votes
2 answers
1k views

Is measure preserving function almost surjective?

Let $F:[0,1]\to[0,1]$ be a Lebesgue measure preserving function. Is $F$ almost surjective, i.e., the image of $F$ has interior measure one? This question is motivated by the following observation. If ...
Zuofeng Shang's user avatar
17 votes
1 answer
622 views

Longest of random worm-like paths in $\mathbb{Z}^2$

Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$, with coordinates $\equiv 2 \bmod 3$, we place, with equal probability, one of these six patterns:       The result ...
Joseph O'Rourke's user avatar
17 votes
1 answer
1k views

Can this probability be obtained by a combinatorial/symmetry argument?

Suppose that $a_1,\dots,a_n,b_1,\dots,b_n$ are iid random variables each with a symmetric non-atomic distribution. Let $p$ denote the probability that there is some real $t$ such that $t a_i \ge b_i$ ...
Iosif Pinelis's user avatar
17 votes
1 answer
10k views

Conjugate prior of the Dirichlet distribution?

What is the conjugate prior distribution of the Dirichlet distribution? Edit: Since I asked this question many years ago, I've written a Python library for working with exponential families. Maximum ...
Neil's user avatar
  • 598
17 votes
5 answers
7k views

A counter example to Hahn-Banach separation theorem of convex sets.

I'm trying to understand the necessity for the assumption in the Hahn-Banach theorem for one of the convex sets to have an interior point. The other way I've seen the theorem stated, one set is closed ...
Dorian's user avatar
  • 2,641
17 votes
1 answer
912 views

$(1+\epsilon)$-injective Banach spaces, complex scalars

It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964, download here). Using common ...
Fred Dashiell's user avatar
17 votes
3 answers
736 views

Probability that a word in the free group becomes (much) shorter?

Let $w$ be a word of length $2\ell$ chosen at random on the alphabet $\{x_1,x_1^{-1},x_2,x_2^{-1},\dotsc,x_k,x_k^{-1}\}$. By the reduction $\rho(w)$ I mean what you obtain by deleting substrings of ...
H A Helfgott's user avatar
  • 20.2k
17 votes
1 answer
910 views

Randomly switching street lights, in a square city

This is a combinatorics-probability question, best stated however in "recreational" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each ...
Richard's user avatar
  • 1,363
17 votes
1 answer
2k views

Which Fréchet manifolds have a smooth partition of unity?

A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is: Which Fréchet manifolds have a smooth partition of unity? How is the ...
Konrad Waldorf's user avatar
17 votes
5 answers
4k views

Brownian motion and spheres

Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$: $$ W\left(\frac{k}...
Alekk's user avatar
  • 2,133
16 votes
3 answers
1k views

A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$. Motivation: A lot! For example, in game theory $S$ can be a set of ...
Bogdan's user avatar
  • 161
16 votes
6 answers
2k views

Finding closed subspaces whose sum isn't closed

Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum ...
Nik Weaver's user avatar
  • 42.8k
16 votes
1 answer
526 views

Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm. If $X$ is compact, Atiyah-Jänich proved that $$[X,\mathcal{F}]\...
Bo Liu's user avatar
  • 673
16 votes
1 answer
929 views

A simple stochastic game

An individual, henceforth called the runner starts at the center of an open two dimensional square $\Omega$ of side length $r \geq 2$. At each turn, a vector $x \in S^1$ is chosen uniformly at random, ...
Nate River's user avatar
  • 6,215
16 votes
3 answers
2k views

A random walk on random lines

I am wondering if this random walk remains finite with positive probability. Start with three lines $A,B,C$ that are extensions of an equilateral triangle. Let $p_0$ be one corner. Generate a line $...
Joseph O'Rourke's user avatar
16 votes
1 answer
1k views

Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...
Mohammad Golshani's user avatar
16 votes
4 answers
1k views

Continuity on a measure one set versus measure one set of points of continuity

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$? Now more carefully, with some notation: Suppose $(X, d_X)$ ...
Nate Ackerman's user avatar
16 votes
6 answers
2k views

Optimal pebble-packing shape

Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes. Q. ...
Joseph O'Rourke's user avatar
16 votes
4 answers
1k views

Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random polynomial:...
Joseph O'Rourke's user avatar
16 votes
2 answers
996 views

Perturbation of unbounded self-adjoint operators

In the paper "A CRITERION FOR THE NORMALITY OF UNBOUNDED OPERATORS AND APPLICATIONS TO SELF-ADJOINTNESS" by M. H MORTAD (http://arxiv.org/pdf/1301.0241.pdf), the author states the following theorem ...
m.gn's user avatar
  • 163
16 votes
3 answers
2k views

Solving a modified birthday problem at a glance

Modified Birthday Problem: a bunch of people line up, and the winner is the first person who shares their birthday with someone lined up ahead of them. What position in the line is optimal? Three (...
Benjamin Dickman's user avatar
16 votes
3 answers
5k views

Integration of differential forms using measure theory?

Setup: Let $(M,g)$ be a (possibly non-compact) Riemannian manifold with volume density $d_gV$. Then one may think of $(M,g)$ as a measure space $(\Omega,\mathcal{A},\mu)$, where $\Omega:=M$, $\mathcal{...
Meneldur's user avatar
  • 408
15 votes
3 answers
6k views

Is there a central limit theorem for bounded non identically distributed random variables?

I have a sequence of centered independent random variables $X_i$ that are all bounded by one in absolute value. They are not identically distributed, though. I would like to know if the central limit ...
Caroline Fontaine's user avatar
15 votes
4 answers
1k views

Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game

The game Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. Lucy picks $n$ different balls in any way Lucy likes, and paint them red. Lucy then giftwrap all the balls so that it is ...
Irvan's user avatar
  • 215
15 votes
2 answers
1k views

self-avoidance time of random walk

How many steps on average does a simple random walk in the plane take before it visits a vertex it's visited before? If an exact formula does not exist (as seems likely), then I'm interested in good ...
James Propp's user avatar
  • 19.7k
15 votes
1 answer
889 views

Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix. For complex numbers $a_1,\ldots,a_n$, I will use the notation $$ \mbox{...
Eusebio Gardella's user avatar
15 votes
2 answers
810 views

Are extensions of nuclear Fréchet spaces nuclear?

Consider the category of Fréchet spaces, the morphisms being continuous linear maps with closed image. Suppose that we have a short exact sequence in that category: $0 \rightarrow V_1 \rightarrow ...
Ralf's user avatar
  • 261
15 votes
1 answer
1k views

In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?

This question is a variation of the return to the origin problem. Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = (...
Sebastien Palcoux's user avatar
15 votes
4 answers
2k views

Naive questions about "matrices" representing endomorphisms of Hilbert spaces.

This is a very basic question and might be way too easy for MO. I am learning analysis in a very backwards way. This is a question about complex Hilbert spaces but here's how I came to it: I have in ...
Kevin Buzzard's user avatar
15 votes
3 answers
13k views

Maximum of two normal random variables

The main purpose of the following question is to get some intuition and deeper understanding why the presented method works which would hopefully help me in trying to adapt it to the setting I am ...
user0820's user avatar
  • 311
15 votes
0 answers
477 views

Expanding disks lead to what packing of the plane?

Suppose one sprinkles points uniformly at random on the infinite Euclidean plane, with some density $\rho$ per unit area. View the points as disks of radius zero. Now the radii $r$ of all disks grows ...
Joseph O'Rourke's user avatar
15 votes
1 answer
2k views

Inductive tensor product and smooth functions

Given complete, locally convex Hausdorff vector spaces $E$ and $F$, let $$ E \otimes_i F, \qquad E \otimes_\pi F$$ denote the (completed) inductive and projective tensor products respectively. The ...
Allan Yashinski's user avatar
15 votes
0 answers
749 views

Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$

I would like to prove that $$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$ for any $\omega > 0$ and $...
Tanya Vladi's user avatar
15 votes
3 answers
2k views

Disintegrations are measurable measures - when are they continuous?

This is a sequel to another question I have asked. The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
Tom LaGatta's user avatar
  • 8,512
15 votes
1 answer
4k views

Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?

The Open Mapping Theorem, the Bounded Inverse Theorem, and the Closed Graph Theorem are equivalent theorems in that any can be easily obtained from any other. The Closed Graph Theorem also easily ...
Bruce Blackadar's user avatar
15 votes
2 answers
734 views

On sums of independent random variables in Banach spaces

Let $(\xi_n)_{n\ge 1}$, $(\eta_n)_{n\ge 1}$ be independent mean-zero random variables with values in a Banach space $X$ such that $$\sum_n\mathbb P(\xi_n\in A)\le\sum_n\mathbb P(\eta_n\in A)$$for any ...
Lviv Scottish Book's user avatar
15 votes
2 answers
2k views

Range of completely positive projection

Let $A$ be a C*-algebra. Suppose that $P:A \rightarrow A$ is a contractive completely positive projection. Does the range $P(A)$ is completely order isomorphic to a $C^*$-algebra? In the case where ...
BigBill's user avatar
  • 1,222
15 votes
6 answers
8k views

Any sum of 2 dice with equal probability

The question is the following: Can one create two nonidentical loaded 6-sided dice such that when one throws with both dice and sums their values the probability of any sum (from 2 to 12) is the same. ...
jakab922's user avatar
  • 261
15 votes
1 answer
660 views

Which limit to take as a key applied math decision

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...
15 votes
3 answers
2k views

Probability that product is a perfect square

The probability a given integer in $[0,n]$ is a square is $\frac1{\sqrt n}$. What is the probability that if you take two integers uniformly then their product is square? I know the main term is $\...
Turbo's user avatar
  • 13.9k
15 votes
2 answers
5k views

What areas of algebra could be interesting to probability theorists?

I would like to find some topic of algebra (beyond linear algebra; algebraic number theory is fine) that would be interesting both to a student that wants to specialize in probability theory and to me ...
Mikhail Bondarko's user avatar
15 votes
1 answer
1k views

Convolution algebras for double groupoids?

There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for ...
Ronnie Brown's user avatar
  • 12.3k
15 votes
4 answers
2k views

Can one do without Riesz Representation?

In more detail, can one establish that the continuous linear dual of a Hilbert space is again a Hilbert space without appealing to the Riesz Representation Theorem? For me, the Riesz Representation ...
Andrew Stacey's user avatar
15 votes
3 answers
8k views

What is an isomorphism of Banach spaces?

The nLab page on Banach spaces (http://ncatlab.org/nlab/show/Banach%20space) was recently criticised as being, in effect, too heavily biased to category theory (not of the Baire kind) and not enough ...
Andrew Stacey's user avatar
15 votes
3 answers
2k views

Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$

I've been trying to find an asymptotic expansion of the following series $$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$ and $$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{...
Trax's user avatar
  • 153
15 votes
3 answers
2k views

Riesz's representation theorem for non-locally compact spaces

Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...
Alex M.'s user avatar
  • 5,407

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