3
votes
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
3answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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 $s\in S$, $$ ...

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