# All Questions

**3**

votes

**1**answer

34 views

### Units of $\mathbf Z[X,Y]/(P(X,Y))$

Let $P(X,Y)\in \mathbf Z[X,Y]$ be an irreducible polynomial and let $A$ denote the quotient ring $\mathbf Z[X,Y]/(P)$.
What is known about the group of units of $A$?
It's not even clear to me that ...

**8**

votes

**0**answers

45 views

### Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?

In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...

**2**

votes

**1**answer

24 views

### Symplectic block-diagonalization of a complex symmetric matrix

This is a follow-up question to the one asked here:
Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...

**2**

votes

**1**answer

15 views

### Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional
$$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$
I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...

**3**

votes

**0**answers

48 views

### A nilpotent quotient of free groups

Let $F$ denote the free group on $n$ generators $g_1,\ldots, g_n$. Consider its quotient $Q$ by the universal relation $[x,[x,y]]$ (a "Serre relation" familiar from Lie theory). This group is ...

**1**

vote

**1**answer

32 views

### Results about the existence of solutions in groups

Let $G$ be a group. Consider an arbitrary equation given by $w(\vec{g})=e$, where $w: G^n \to G$ takes an $n$-tuple $(g_1,...,g_n)$ to some expression involving products of the $g_i$, their inverses ...

**0**

votes

**0**answers

41 views

### Triangle groups [on hold]

I am having a hard time finding references(apart from wikipedia) for the geometric interpretation of triangle groups T_a,b,c = $<x,y | |x|=a, |y|=b, |xy|=c>$. How can these groups be visualized ...

**14**

votes

**0**answers

141 views

### improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...

**4**

votes

**2**answers

83 views

### A Scalar Curvature Computation in Brendle Marques Neves' Min-Oo Conjecture paper

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition:
Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...

**2**

votes

**0**answers

29 views

### Triangulations of special polyhedra

Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...

**1**

vote

**0**answers

17 views

### A question for the inverse orbit in the construction of conformal measure

Recently, I read a theorem of existence of conformal measure for the rational map.
I did not understand two places in the proof. The author claims that
there exists an open set $V\subset ...

**-3**

votes

**0**answers

21 views

### Calculate minimal number of nodes? [on hold]

Calculate minimal number of nodes? in a loopless simple undirected pi-partite graph. that has exacatly 144 nodes

**-3**

votes

**0**answers

24 views

### Directed and undriected trees [on hold]

How many different directed trees can be obtained if we assign all possible orientation to the edges of an undirected tree having exactly 7 nodes? how many of them will be rooted(directed) trees?

**-3**

votes

**0**answers

23 views

### Can you give me an example of signed distance function [on hold]

Can you give me an example of signed distance function？
Thank you！

**0**

votes

**0**answers

26 views

### Determine number of directed trees and rooted trees obtainable [on hold]

I've been doing some exercices about graph theory and I find myself stuck on this one with no idea of to proceed.
Here's the question :
how many different directed trees can be obtained if we assign ...

**0**

votes

**0**answers

102 views

### About the proof of the Morse lemma

In the Chang's book "Infinite dimensional Morse theory and multiple solution problems"
the Morse lemma is a special case of the spliting lemma but i dont understand in the proof why ...

**1**

vote

**2**answers

35 views

### Linear Programm with matrix [on hold]

Is there a name for problems like this
min norm(Cx)
Ax = b
where C is a matrix and norm is the maximum norm.
This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...

**0**

votes

**0**answers

32 views

### Collecting terms of a linear expression with nested sums and combinatorics in coefficients

I need to collect the $\Pr(\cdot)$ terms of the following expression:
$\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left(
1-\theta \right) }\right) ^{m}}\left[ ...

**-5**

votes

**0**answers

33 views

### How can I compute the singular homotopy? [on hold]

Let D2 be a 2-dimensional disc and M be the Mobius strip. Note that the boundary of
both D2 and of M is homeomorphic to the circle S1.
(a) Consider the space X := (D2 ∪D2) /~ where ~ is the ...

**7**

votes

**1**answer

143 views

### Questions about Prikry forcing and Cohen forcing

I have two unrelated questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...

**0**

votes

**0**answers

50 views

### How to define the distributional Hessian for a convex function on a $C^0$ Riemannian manifold?

M is a $C^1$ manifold with a $C^0$ Riemannian metric, f is a convex function on M. How to define a functional on M which can represent $Hessf$?
For example: for $\Delta f$ we can define the ...

**0**

votes

**0**answers

35 views

### Hom-Lie algebras induced by derivations

Let $(\mathfrak{g},[,])$ be a Lie algebra, and $D\colon \mathfrak{g} \rightarrow \mathfrak{g}$ be a linear vector space map satisfying the Hom-Jacobi identity
$$
...

**4**

votes

**1**answer

69 views

### Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?

Allcock(2006) proved that
there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$).
His main technique of ...

**0**

votes

**0**answers

36 views

### Is there such a thing as cyclic Hasse diagram for posets?

If so can you name one ? If not how to prove that there is none? Thanks !

**0**

votes

**1**answer

55 views

### State of the Art in Approximating Fresnel Integrals

Background of my question is, that I need to calculate Clothoids and I found an AMS article "Chebyhev Approximations for Fresnel Integrals" by W.J. Cody from 1968 ...

**8**

votes

**1**answer

263 views

### Computing $\Pi_p(\frac{p^2-1}{p^2+1})$ without the zeta function?

We see that $\frac{2}{5}=\frac{36}{90}=\frac{6^2}{90}=\frac{\zeta(4)}{\zeta(2)^2}=\Pi_p\frac{(1-\frac{1}{p^2})^2}{(1-\frac{1}{p^4})}=\Pi_p(\frac{(p^2-1)^2}{(p^2+1)(p^2-1)})=\Pi_p(\frac{p^2-1}{p^2+1})$
...

**-1**

votes

**0**answers

34 views

### Context Free Grammar [on hold]

does anyone know how to find the Context Free Grammar for this language?
L = {anbm | n > m}

**1**

vote

**1**answer

149 views

### Blow-ups in Motivic Homotopy Theory

I have what I hope is an easy question in motivic homotopy theory:
Let $X$ be a smooth scheme over a field $k$, and let $Z\subset X$ be a closed sub-scheme of codimension $>1$. Let $Bl_Z(X)$ ...

**5**

votes

**0**answers

58 views

### Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?

Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...

**13**

votes

**1**answer

356 views

### Is anything known about which numbers appear in the continued fraction expansion of $\pi$?

This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in ...

**2**

votes

**0**answers

31 views

### Almost-Monotone Kernels - Examples and/or Covering Theorems

I am looking for examples (or, if it exists, a theory) of almost-monotone kernels. First, a bit of notation.
Recall that if $(\leq, \Omega)$ is a partially ordered set, then the set of measures ...

**0**

votes

**0**answers

24 views

### adjacent matrix directed or undirected [on hold]

I'm having trouble seeing how you can determine if a graph is directed or directed based off of the adjacent matrix. Can someone explain to me how to determine ths? Thanks!

**0**

votes

**0**answers

7 views

### Is a constant such as 8 considered an expression? [migrated]

The question asked was "Which of the following expressions are considered polynomials?"
8 was one of the answers, and though it is clearly a monomial, it was part of the answer and I'm confused as to ...

**0**

votes

**0**answers

21 views

### Finding a particular solution to the non-homogenous system [on hold]

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$
Step 1) Find the Eigenvalues ...

**3**

votes

**2**answers

135 views

### Real character modular forms: Fourier coefficient real?

Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$.
Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real?
What I knew is that if $N=1$ and ...

**4**

votes

**0**answers

102 views

### NP Problems with unique solution [migrated]

Is there any class of NP problems that have one unique solution?
I'm asking that, because when I was studying cryptography I read about the knapsack and I found very interesting the idea.

**0**

votes

**0**answers

34 views

### Extensions on Higher-dimensional local fields

I have the following question:
Let $M/L$ b a finite extension of n-dimensional local fields and $t_1,\dots, t_n$ a system of local parameters of $L$ with valuation $v$. Let us fix an $1\leq i \leq ...

**1**

vote

**0**answers

31 views

### Valuations in Higher-dimensional local fields

I have the following question which I believ should be true but I would like to have a different opinion about it:
Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and ...

**0**

votes

**0**answers

47 views

### Reductive Group Actions and Completion

Suppose that $A$ is a Noetherian (not necessarily commutative) $\mathbb{C}[h]$-algebra equipped with a rational action of an affine reductive group $G$, i.e., $A$ is a $\mathbb{C}[G]$-comodule and ...

**2**

votes

**0**answers

169 views

### Can we prove an open affine subscheme of a noetherian scheme is noetherian without Axiom of Choice?

I'm interested in proving basic results of algebraic geometry without Axiom of Choice.
As for why I think this is interesting, please see Pete L. Clark's answer to this question.
To state my problem, ...

**4**

votes

**3**answers

126 views

### On formal solutions to differential equations

Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and ...

**1**

vote

**0**answers

40 views

### Collecting terms with nested sums and combinatorics

I need to collect the $\Pr(\cdot)$ terms of the following expression:
$\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left(
1-\theta \right) }\right) ^{m}}\left[ ...

**3**

votes

**0**answers

49 views

### Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map
$D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...

**3**

votes

**1**answer

205 views

### Can we sometimes define the parity of a set?

I have accepted the best (and only) answer but the problem is still open.
Suppose that ${n\choose k}, {n-1\choose k-1}, \ldots, {n-k+1\choose 1}$ are all even. (This happens for example if ...

**3**

votes

**1**answer

35 views

### variation of the Lieb concavity theorem

A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B:
$$
(A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \}
$$
for $s \in ...

**-2**

votes

**1**answer

94 views

### Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field? [on hold]

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...

**5**

votes

**1**answer

106 views

### Areas of Triangles in (Non-Riemannian) Metric spaces?

I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this ...

**-1**

votes

**0**answers

77 views

### Time derivative of an integral on a moving surface? [on hold]

I need to take the time derivative inside the surface integral,
$$\displaystyle\dfrac{\mathrm{d}}{\mathrm{d}t}\left(\oint_{\partial B} \left(\mathbf{x} \times ( \mathbf{n} \times \mathbf{u} ) ...

**1**

vote

**0**answers

108 views

### Quadratic - Ternary Forms

Hi I have the following problems concerning quadratic and ternary forms. Any help would be greatly appreciated.
$3\displaystyle\sum_{x, y\in\mathbb{Z}}q^{x^2+xy+7y^2}=3\displaystyle\sum_{x, ...

**2**

votes

**0**answers

65 views

### Are semigroups with finite-to-one right multiplication “moving”?

A semigroup $S$ is moving if $S$ is infinite, and for all finite
$F\subseteq S$ and infinite $A\subseteq S$, there are $a_{1},\dots,a_{k}\in A$ such that,
for all but finitely many $s\in S$,
$$
...