# All Questions

**2**

votes

**1**answer

85 views

### Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?

(I've asked this in MSE but nobody had an idea since dec 14...)
(Roughly related, but generalizing, of this earlier MSE question)
Background: ...

**0**

votes

**1**answer

120 views

### Algebraic Hecke characters with a given infinite part

I'm needing to find out if there exists an algebraic Hecke character for a number field F, $\phi: \mathbb{A}_F \rightarrow \mathbb{C}$, for a fixed infinite part $\phi_\infty$ and a fixed component ...

**2**

votes

**4**answers

569 views

### Intrinsic definition of arc length [closed]

Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?

**1**

vote

**0**answers

65 views

### Complex Hessian Signature

It' all, simply, about the signature of a matrix.
Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...

**4**

votes

**0**answers

78 views

### How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of G. D. Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...

**3**

votes

**2**answers

308 views

### Is this a $C^0$ foliation of $\mathbb{R}^2$?

Let $f(x)=\frac{1}{\sin(\pi x)}$ for $x\in (0, 1)$ and let
$\Gamma=\left\{(x,f(x)): x\in (0, 1)\subset \mathbb{R}^2\right\}$ be its graph.
For any set $X\subset \mathbb{R}^2$ and $\lambda>0$ and ...

**1**

vote

**1**answer

92 views

### Homotopy invariance of Kan nerve of simplicial categories

The following question concerns the well-known paper of Dwyer and Kan "Localization of Simplicial Categories". They define a nerve for simplicial categories (with fixed set of objects $O$), by the ...

**4**

votes

**0**answers

64 views

### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...

**4**

votes

**0**answers

48 views

### Expected size of determinant of $AA^T$ for non-square random Toeplitz $A$

If $A$ is chosen uniformly at random over all possible $m$ by $n$ Toeplitz (0,1)-matrices, what is the expected size of the value of the determinant of $AA^T$? We can assume $m \leq n$ and all ...

**18**

votes

**2**answers

444 views

### Why is every variety of bands determined by a single identity?

A band is a semigroup where every element is idempotent: $a a = a$. A collection of bands is a variety if it is closed under taking subobjects (subsemigroups), quotient objects (images of ...

**2**

votes

**1**answer

87 views

### On the vertices of a Coxeter complex

Let $A$ be a Coxeter complex which is euclidean, so I assume that $A$ is an affine space over the reals on which a Coxeter group $(W,S)$ acts, the elements of $S$ are reflections and I assume the ...

**8**

votes

**1**answer

257 views

### Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections

I previously asked this on Math.SE but didn't receive a satisfactory answer.
Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: ...

**95**

votes

**107**answers

32k views

### Most memorable titles [closed]

Apparently, for a large number of readers, the choice whether they select to read a paper or not is often strongly influenced by the title.
I was wondering if the MO-users would be willing to share ...

**1**

vote

**1**answer

131 views

### Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research:
$$\Delta f - e^f \partial_s f = E(s,t)\,,$$
where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...

**83**

votes

**90**answers

11k views

### What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...

**5**

votes

**0**answers

83 views

### Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...

**0**

votes

**1**answer

180 views

### Worst case difference in rank by column-row swapping

Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...

**22**

votes

**3**answers

2k views

### Modern Source for Spectra (including Ring Spectra)

I am looking for a modern introduction to Spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what has been ...

**2**

votes

**0**answers

110 views

### Subfields of $\mathbb{Q}\bigl(\sqrt[n]{a}\bigr)$ for $a>0$

This is related to a question on Math Stack Exchange.
Given a rational number $a>0$ and an $n\in\mathbb{N}$ such that $x^n - a$ is irreducible over $\mathbb{Q}$, it is known that every subfield of ...

**0**

votes

**0**answers

52 views

### An additive question on polynomials

Consider $S\cup T=\{0,1\}^n$ where $S\cap T=\emptyset$.
Consider real multilinear (only monomials of form $x_ix_jx_k$) polynomials $P,Q$ such that:
$$Q(S)=0\quad Q(T)\neq0\quad P(S)\neq0\quad ...

**20**

votes

**2**answers

637 views

### Is every closed curve in 3D a geodesic on a genus-0 surface?

Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.
Q. Does there always exist a smooth, embedded, genus-zero surface
$S \subset \mathbb{R}^3$
such that $\gamma$ is a ...

**8**

votes

**2**answers

277 views

### Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...

**2**

votes

**0**answers

35 views

### Noncommutative fusion categories

Although noncommutativeness is almost a defining trait for fusion categories, offhand I recall only the extended Haagerup N-N (rank 8). It's a two minute computation to find that even a based ring ...

**0**

votes

**1**answer

123 views

### Chow groups of rational varieties

Let $X$ be a smooth projective rational variety over a field $k$. Let $CH^i(X)$ denote the Chow group of codimension $i$ algebraic cycles on $X$ modulo rational equivalence. What can one say about ...

**0**

votes

**0**answers

55 views

### Morphism of modules of sections and pullback bundles

I'v asked this question on StackExchange but unfortunately nobody answered. I thought that maybe it would be more apropriate to post it here:
so suppose that we have a morphism $\theta: \Gamma(B,E_1) ...

**4**

votes

**2**answers

482 views

### Which vector bundle are the Christoffel symbols sections of?

The collection of Christoffel symbols $\Gamma_{ij}^k$ of a connection (or of a metric) on a smooth manifold $M$ is not the collection of components of a tensor field in some local chart, i.e. they ...

**5**

votes

**1**answer

103 views

### The (Hecke) double coset von Neumann algebra

It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on ...

**5**

votes

**1**answer

249 views

### A survey for various $K$-homology theories and their relationship

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology ...

**2**

votes

**0**answers

23 views

### MLE and CRLB with mismatched likelihoods

Suppose that I can do a Karhunen-Loeve expansion of a log-likelihood function $p(\bf{x};\theta)$ into N terms and that these accounts for a fraction $1-\delta$ of the total energy. Now consider ...

**-3**

votes

**1**answer

294 views

### Asymptotic formula for $\prod_{p\leq x} (1-p^{-1})$ [closed]

Does there exists a good asymptotic formula for
$$A(x) := \prod_{p\leq x}(1-\frac 1p).$$
By using a heuristic argument one can guess:
$$A(x) \sim \frac{1}{2\,\mathrm{ln}(x)}.$$
Here is the ...

**2**

votes

**1**answer

119 views

### global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS):
$$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$
$$\phi (x,0) = ...

**6**

votes

**3**answers

255 views

### Borel cross section

It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes.
A short elementary proof is given in ...

**4**

votes

**1**answer

143 views

### The groups with symmetric subgroups lattice

Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice.
If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the ...

**3**

votes

**3**answers

126 views

### Integrals involving the Tricomi hypergeometric function

I am looking for a reference for the two following equalities involving the Tricomi function $U$ and the Meijer function $G$. I have found these formulas on the website http://functions.wolfram.com/, ...

**0**

votes

**1**answer

279 views

### Is there a Gröbner basis analogue that exists for vector spaces?

Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$.
...

**9**

votes

**2**answers

174 views

### What's the most simple proof of Kostant's version of Borel-Weil-Bott for Lie Algebra cohomology?

Besides Kostant's original proof (in http://www.math.tamu.edu/~jml/kostant61.pdf) of the above mentioned theorem (using the Lie Algebra Laplacian), there are a few other approaches:
...

**3**

votes

**0**answers

73 views

### Implications between different covering properties of spaces

Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a set, we define the ...

**88**

votes

**41**answers

30k views

### Where does a math person go to learn quantum mechanics?

My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn ...

**20**

votes

**22**answers

15k views

### What are the worst notations, in your opinion ? [closed]

With which notation do you feel uncomfortable ?

**1**

vote

**1**answer

178 views

### Integrating Poisson groups

Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ ...

**10**

votes

**1**answer

714 views

### Examples of Einstein four-manifolds of negative sectional curvature

Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, ...

**60**

votes

**21**answers

8k views

### Probabilistic Proofs of Analytic Facts

What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...

**33**

votes

**30**answers

11k views

### Most intricate and most beautiful structures in mathematics

In the December 2010 issue of Scientific American, an article "A Geometric Theory of
Everything" by A. G. Lisi and J. O. Weatherall states "... what is arguably the most
intricate structure known to ...

**2**

votes

**1**answer

126 views

### Locally conformal Kahler manifolds with SU(4) structure

I would like to know if there exist eight-dimensional manifolds such that:
It has SU(4)-structure.
It is locally conformal Kahler.
It is not a Calabi-Yau four-fold.
A weaker question that also ...

**7**

votes

**1**answer

285 views

### Products of subgroups of a free group

Let $F$ be a free group, and let $A,B \leq F$ be two subgroups such that $AB$ contains a nontrivial normal subgroup of $F$. Must either $A$ or $B$ contain a nontrivial normal subgroup of $F$?
What if ...

**1**

vote

**1**answer

135 views

### Singularities in minimal surfaces [closed]

There is a theorem by Jean Taylor that says that an almost minimal set in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times ...

**0**

votes

**1**answer

54 views

### Link between Harris recurrence and Ergodicity

Is it possible to obtain Harris recurrent Markov chain from Ergodic chain (in Birkhoff sense) under certain assumption? That is, suppose we know a Markov chain is ergodic (in Birkhoff sense); is it ...

**3**

votes

**2**answers

238 views

### N-th root of unity in N-th division field of abelian variety?

Let K be a number field and A/K an abelian variety over it. Can it be that K(A[n]) does not contain a primitive n-th rooth of unity? If the answer is yes is it always possible to bound the bad n ...

**-2**

votes

**2**answers

77 views

### does a lattice have a minimal item [closed]

A lattice (L, ≤) is a partially ordered set, where each two elements of the set have a least upper bound and a greatest lower bound. Let's consider the situation where L is finite.
I think that ...

**0**

votes

**1**answer

186 views

### The weighting function for the infinite product of necklaces

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$ where $N(n,a)$ is the number of fixed necklaces of length $n$ composed of $a$ types of beads.
Let's rewrite the product in a way ...