# All Questions

**1**

vote

**1**answer

108 views

### Subgroups generated by opposite root groups

Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its centralizer. For ...

**0**

votes

**3**answers

219 views

### orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates

$Z_p$:=cyclic group of order $p$.
I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates.
For ...

**1**

vote

**0**answers

146 views

### In set theory, is there a name for a function which maps the empty set to zero and all the others to one? [on hold]

I would like to avoid inventing something which might be standard.
Thus, I'am asking if there is a name for a function which is defined as $f$:
Let $S$ be any set, then $f(S)=0$ if $S$ is empty and ...

**6**

votes

**2**answers

348 views

### Texts about Dwork's work

I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...

**3**

votes

**2**answers

217 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

**11**

votes

**1**answer

568 views

### Applications of set theory in physics

In the introduction of the paper "Links between physics and set theory", the following quote of Eris Chric is stated:
...

**4**

votes

**1**answer

130 views

### Simply connected Lie groups homeomorphic to R^n are solvable

I have found the following claim in many proofs "Simply connected Lie groups homeomorphic to $\mathbb{R}^n$ are solvable". But the universal covering of $SL(2,\mathbb{R})$ satisfies the hypothesis of ...

**2**

votes

**0**answers

67 views

### Homotopical categories, the 2-out-of-6 property, and saturation

A homotopical category is a cateogry with a class $\mathcal W$ of arrows called weak equivalences which satisfies the 2-out-of-6 property.
The nlab article shows a deep connection between $\mathcal ...

**4**

votes

**1**answer

150 views

### Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

I asked this question on math.stackexchange a week ago, but did not get an answer.
First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general ...

**5**

votes

**2**answers

485 views

### Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...

**4**

votes

**0**answers

75 views

### Combinatorics of palindromic decompositions

This is sort of a companion to my question Number of trivializations of a trivial word in the free group (which in turn is motivated by my earlier question here). It turns out that that question may ...

**9**

votes

**2**answers

175 views

### A “quadratic” triangular inequality

In a Euclidian space (Hermitian as well), say $\ell^2_n$, the following inequality holds true
$$(QI)\qquad |b|\cdot|c-a|\le|c|\cdot|a-b|+|a|\cdot|b-c|,\qquad\forall a,b,c\in\ell^2_n.$$
In other words, ...

**1**

vote

**0**answers

36 views

### Concise disambiguation of Voronoi boundaries

Say that $x_1,\dots,x_n$ are points in the plane, with a Voronoi diagram $V_1,\dots,V_n$. The Voronoi diagram is typically defined by $$V_i = \{x:\|x-x_i\|\leq \|x-x_j\|~\forall j\}~.$$ Is there any ...

**0**

votes

**1**answer

86 views

### A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...

**0**

votes

**1**answer

65 views

### Double coset separability and the existence of vanishing sequences for surface group

Definition: Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite ...

**-4**

votes

**2**answers

240 views

### Brocard's problem [on hold]

According to Brocard's problem $$x^{2}=n!+1 \Rightarrow (x+1)(x-1)=n!$$, where $(x+1)/2$ , $(x-1)/2$ are consecutive integers, assume one of them is z, another is (z-1). z and (z-1) have all primes ...

**2**

votes

**1**answer

155 views

### Loop defects in Walker-Wang model

My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model.
Elementary excitations in the WW model can be point particles, loop defects and more ...

**2**

votes

**1**answer

155 views

### Smoothing transverse intersections

Let $S$ be a complex surface with ample canonical class. Let $C_1$ and $C_2$ be smooth complex curves in $S$ that intersect transversally at $n $ points. Furthermore, assume that the self-intersection ...

**3**

votes

**1**answer

126 views

### Is the derived category of perfect complexes idempotent complete?

Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any ...

**2**

votes

**1**answer

128 views

### Two equivalent descriptions of a physical system yielding a non-trivial mathematical formula

First I would like to admit that this question may not be entirely appropriate for this site, but I will give it a go none the less.
One often hears stories about how string dualities lead to highly ...

**6**

votes

**1**answer

117 views

### Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...

**0**

votes

**1**answer

70 views

### Mathematical statistical qm book-recommendation

I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum ...

**5**

votes

**0**answers

46 views

### Anything known about the Grundy Ordinal of Sylver's Coinage

Sylver's coinage is an example of an unbounded finite (if slightly modified) combinatorial impartial game. Quoth wikipedia:
The two players take turns naming positive integers that are not the
...

**0**

votes

**0**answers

22 views

### Pascal and Brianchon's theorems generalized for hyperbolic paraboloid

I know that giving a general version of these two theorems for quadrics can be quite tricky, but if we restrict ourselves to a verssion that holds for the hyperbolic paraboloid only it should be ...

**2**

votes

**0**answers

129 views

### One question about iteration on groups

Let $G$ be a finitely generated group, $H$ a subgroup of $G$ of index $n$, with $a_i$ a set of coset representatives and $$G=\displaystyle\bigcup_{i=1}^nH{a_i}.$$
Let $\phi:H\rightarrow G$ be a ...

**1**

vote

**1**answer

129 views

### Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...

**3**

votes

**0**answers

117 views

### A property of the Frechet filter and every ultrafilter

(Joint question with Piotr Szewczak.)
Definitions and notation. By filter we mean a filter on $\omega$ containing the cofinite sets at least.
For a filter $\mathcal{F}$, let ...

**11**

votes

**1**answer

209 views

### A classic cardinal characteristic of the continuum in disguise?

We believe the answer to the following question, that is relevant to a joint research project with Piotr Szewczak, should be known. We would appreciate any help or pointer.
Needed definitions may be ...

**13**

votes

**7**answers

1k views

### Are there any Algebraic Geometry Theorems that were proved using Combinatorics?

I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of Combinatorics and Algebraic Geometry, and gave some examples like the ...

**-2**

votes

**1**answer

97 views

### About structure of the set of perfect matchings of $K_{n,n}$

Are there any special properties known about the set of perfect matchings of $K_{n,n}$? Like any global structure of this set? Some natural way to partition it? Like is there some algebraic structure ...

**0**

votes

**0**answers

68 views

### absolutely continuous of two probability measures

Suppose $X_t$ satisfies
$$X_t=\int_0^t b(X_s)ds+ L_t,\quad t\in[0,1]$$ where $L_t, t\in[0,1]$ is a $\alpha-$stable process. Let $P_L$ be the law of $L$, $P_X$ be the law of $X$. ($P_L, P_X$ are ...

**-4**

votes

**0**answers

67 views

### Probability- My homework is confusing me [closed]

The question is that there is a game, it has 38 congruent pieces, 18 are orange, 18 are blue, and 2 are white. To win you have to get either orange or blue and you get 2$, to play you pay 1 dollar, ...

**1**

vote

**0**answers

71 views

### Poincare-Lefschetz duality, universal coefficients, and middle cohomology [migrated]

Sorry if the question is too simple, algebraic topology is not my strong suit. Let $(M,\partial M)$ be a $2n$-dimensional manifold with boundary, with one-dimensional middle cohomology. By ...

**7**

votes

**0**answers

83 views

### Is there a universal straightedge and compass construction of a segment incommensurable to a given one in the hyperbolic plane?

"Universal" means that the construction steps are independent of the length of the given segment. In the Euclidean plane one can take the diagonal of a square built on it. Without the "universal" the ...

**1**

vote

**1**answer

56 views

### Injecting premises into two implicational premises connected by a tensor (multiplicative conjunction) in linear logic

I have another question regarding linear logic: I want to get to the proof E, using the premises in (1-4). Is this at all possible?
1: $A$
2: $C$
3: $(A\multimap B)\otimes(C\multimap D)$
4: ...

**4**

votes

**0**answers

124 views

### Schoenflies and symplectic topology

The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the ...

**-2**

votes

**0**answers

98 views

### Proving that (e^x+1)^(1/3) has no elementary antiderivative [migrated]

How should one prove that $\int (e^x + 1)^\frac{1}{3}dx$ is non-elementary? (In case that is really is)

**5**

votes

**0**answers

108 views

### Non-embeddable varieties

Suppose that $k$ is a perfect field of characteristic $p>0$, $\mathcal{V}$ is a complete discrete valuation ring with residue field $k$ and quotient field $K$, of characteristic $0$.
Then when ...

**-4**

votes

**0**answers

46 views

### probability distribution of balls in an urn [closed]

So I have the following question in "probability":
An urn contains three balls: white, blue and red. At each stage a ball is picked up randomly and, if it is not red, it is returned to the urn. The ...

**0**

votes

**0**answers

74 views

### A relative version of Urysohn's Lemma?

Let $f:Y\to X$ be a continuous surjective map between locally compact Hausdorff spaces. Assume there is a continuous section $s:X\to Y$ which has closed image and is a homeomorphism to the image.
I ...

**2**

votes

**0**answers

154 views

### How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?

How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
In particular, does Corollary 4.4 from SGA III Exp. VIB hold for G/S being merely a group space? Here the ...

**-4**

votes

**0**answers

93 views

### sequences and series [closed]

I think it is interesting, if we have the formula
$$\frac{n (n + 1) (2 n + 1)}{6} = 1^2 + 2^2 + \cdots + n^2 .$$
If the difference between the closest numbers is smaller (let's call is a) we ...

**4**

votes

**2**answers

69 views

### About trigonometric series of the Lip $\alpha$ class

Assume we have a trigonometric series
$$
f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1.
$$
Is there anything we can say about the series
$$
g(x)=\sum_{n=1}^{\infty} ...

**2**

votes

**0**answers

41 views

### Localized eigenfunctions of drift Laplacians

I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...

**1**

vote

**1**answer

112 views

### Arbitrary chains of prime ideals in $R[X]$

For a commutative ring $S$ of finite Krull dimension $d$, we have $1+d\leq \dim(S[X])\leq 2d+1$. One proof of this uses the fact that if $Q_1\subset Q_2\subset Q_3$ is a chain of prime ideals of ...

**-3**

votes

**0**answers

34 views

### Divide 4D cube into tetrahedrons/simplices [closed]

I am trying to figure out how can I possibly divide a 4D cube into 16 tetrahedrons/simplices.
Thanks.

**8**

votes

**2**answers

612 views

### How to pack 3D boxes into a bigger box?

Given a box of given size $L\times M\times N$ and a list of smaller boxes of given sizes $(l_i,m_i,n_i)$, decide whether the smaller boxes altogether fit into the big box (and produce such a packing ...

**6**

votes

**0**answers

217 views

### Mass Transportation Through Wonderful Roller

There is a wonderful roller that transports mass from A to B and there is a pile of sand with weight $W$ in point A and we want to transport it to B.
Wonderfulness of roller comes from this property ...

**0**

votes

**0**answers

32 views

### Differentiability and the Maximum Theorem [closed]

Suppose that in Berge's Maximum Theorem the argmax correspondence happens to be single-valued. Are there assumptions that will guarantee that this argmax function will be smooth (e.g., continuously ...

**1**

vote

**1**answer

77 views

### A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...