# All Questions

**0**

votes

**0**answers

3 views

### Tensor calculus on the frame bundle

Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its ...

**1**

vote

**0**answers

14 views

### Application of Stickelberger's Theorem to Quadratic field

I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can ...

**0**

votes

**1**answer

27 views

### About direct limit of groups

Let $G_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi_i$ be a monomorphism from $G_i$ to $G_{i+1}$.
Let $\Sigma$ be the direcet limits of $G_i$ under the embeddings of $\phi_i$.
Let ...

**-1**

votes

**0**answers

14 views

### What is a $\mathbb P$-name in Forcing theory

I am reading about forcing.
If I understood correctly, we take a countable transitive ZFC model, with a partial relation $\mathbb P$ defined in it. We add a $\mathbb P$ generic filter $G$ where $G ...

**0**

votes

**0**answers

89 views

### Carlson's translatability

I asked the following on MSE a few weeks ago but I did not get any answer :
http://math.stackexchange.com/questions/1039593/carlsons-translatability-are-theses-characterisations-equivalent
Given a ...

**-1**

votes

**0**answers

24 views

### Minimum rank non-negative matrix summations

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$ of rank $r$.
What is minimum $k$ such that
$$\mathscr{A}[b,k]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(Q)\leq k\}$$ contains $R,S$ ...

**0**

votes

**1**answer

166 views

### How would I apply Wick's theorem to the time-ordered product of three fields?

I think I know how to apply Wick's theorem in order to expand the time-ordered product of quantum fields, but I just want to verify my understanding. Could someone perform it for the arbitrary ...

**1**

vote

**0**answers

43 views

### Bi-epimorphisms

A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
DEFINITION A morphism $\ f:X\rightarrow Y\ $ is a ...

**0**

votes

**0**answers

11 views

### Density of push-forward distribution

Let $\mathbb P$ be a probability distribution and let $X \sim \mathbb P$ be a random vector taking values in $\mathbb R^n$. Define $Y := \phi(X)$, where $\phi: \mathbb R^n \to \mathbb R$ is a ...

**-4**

votes

**0**answers

23 views

### Binomial Theorem [on hold]

Hello I am trying to figure out how Binomial theorem formula works when the one of its elements is zero.
For example in the formula above if I choose |a| = 2 and |b| = 0 then b^k does not look ...

**3**

votes

**1**answer

153 views

### How to implement linear constraints that include several absolute values

Dear all,
I am trying to implement a linear constraint that includes several absolute values in the form: Abs(A) + Abs(B) + Abs(C) + Abs(D) + ... = 1
Since the minimization problem includes quite a ...

**1**

vote

**1**answer

69 views

### Properties of Integral Closure

Definition(Integral closure): Let $R$ be a ring and $I$ an ideal of $R$. An element $x$ is said to be integral over $I$ if $x$ satisﬁes a monic equation
$x^n + i_1x^{n−1} + ··· + i_n = 0$ such ...

**-3**

votes

**0**answers

34 views

### I don't get how -6cos3xsin3x becomes -3sin6x in the later part [on hold]

y = cos²3x
dy/dx = 2cosx(-sin3x)(3)
= -6cos3xsin3x
= -3sin6x
I found this answer key in my guidebook but I can't find any trigonometric function's or differentiation formula ...

**0**

votes

**0**answers

12 views

### Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared:
$$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{\frac{1}{2}(\sqrt{4s(s-1)+1}-1)}(2\rho-1) da,$$
...

**5**

votes

**0**answers

128 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...

**9**

votes

**4**answers

1k views

### How to determine the homotopy groups of the suspension of a space?

Let $SX$ be the suspension of CW complex. What are some results available to determine the homotopy groups of $\pi_n(SX)$?

**45**

votes

**22**answers

9k views

### How to respond to “I was never much good at maths at school.” [closed]

We've all heard it. I even got it in Norwegian recently. It's number 1 on the list of responses to the statement "I'm a mathematician.". Does anyone have any good comebacks? What other responses ...

**0**

votes

**0**answers

12 views

### Unimodular triangulation and affine toric variety

Let $\mathcal{K}$ be a $pointed$ rational cone in $\mathbb{R}^d$ with extremal rays generated by $r_1,r_2,\dots, r_m\in \mathbb{Z}^d$.
Here, pointed means that all $r_i$ lie strictly on one side of ...

**-2**

votes

**1**answer

132 views

### Degree of a rational function [on hold]

I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...

**0**

votes

**1**answer

168 views

### Some quantities which definitions are (somehow) similar to the classical Divergence

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...

**1**

vote

**0**answers

34 views

### Convex hull of the union of two parameterized curves in $\mathbb{R}^3$

My goal is to find a way to calculate the convex hull of the union of some parameterized curves.
For instance, I had to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup ...

**8**

votes

**2**answers

252 views

### Non-Forcing and Independence

I asked this question about two weeks ago on MSE and haven't gotten an answer, so I thought I would post the question here.
Do there exists sentences which are independent of ZFC, cannot be shown to ...

**5**

votes

**2**answers

128 views

### Rademacher average based Hoeffding Inequality

I am following these lecture notes:
Given the i.i.d. $\mathcal{Z}$-valued random variables $Z_1,\dotsc,Z_m$ and $\mathcal{G}$ is a set of bounded functions $g\colon \mathcal{Z}\to[a,b]$.
Corollary ...

**1**

vote

**0**answers

31 views

### Probability of non-negative matrix relaxation

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, take $\mathscr{M}[M]=\{Q\in R_{\geq0}^{n\times n}:Q[ij]>0\iff M[ij]=1\}$.
Does ...

**-2**

votes

**0**answers

13 views

### Weighted Least square estimate problem with projection theorem [on hold]

I have been trying to solve the following problem.Granted I do not understand it much and I have to solve it. Can anyone help?
Question : Find the least squares estimate of the quantity k using the ...

**0**

votes

**0**answers

34 views

### Alternating quotients of (2,3,7;10)

It was shown that the only quotient of the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ of the form PSL(2, q) is PSL(2, 41). That lead me to consider other families of simple ...

**2**

votes

**0**answers

73 views

### Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...

**2**

votes

**2**answers

479 views

### Automorphisms of $\mathbb C_p$

I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.
If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that ...

**0**

votes

**0**answers

16 views

### Is it possible (or even valid) to obtain an eigensystem from a set of recursive equations? [on hold]

Let us start with the Golden Ratio, which is the number $\varphi \approx 1.6180339887\cdots$, and it can be defined in several ways, one of them is through a recurrent process involving the Fibonacci ...

**2**

votes

**0**answers

153 views

### When does a perverse sheaf occur in the decomposition theorem?

Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f_*\mathbb Q_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a ...

**1**

vote

**1**answer

235 views

### Is there a non-tempered representation of U(2)?

I am wondering why the first well known example of non-tempered irreducible admissible representation of $p$-adic group $U(n)$ should be $U(3)$. Because, Gelbart and Rogawski suggested the ...

**-4**

votes

**0**answers

26 views

### formula for turning star reviews into upvotes [on hold]

I want to turn reviews of up to 5 stars and the number of reviews into upvotes. Whats a good algorithm for doing this?
A venue with 10 reviews total with a 5 star average rating should obviously get ...

**2**

votes

**0**answers

49 views

### For which quiver varieties is Kirwan surjectivity known?

The cohomology of Nakajima quiver varieties is a quite interesting object. It's equipped with some natural classes given by the Chern classes of the tautological bundles associated to the spaces in ...

**118**

votes

**43**answers

60k views

### Magic trick based on deep mathematics

I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to ...

**2**

votes

**4**answers

346 views

### Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$

Let $(X,\tau)$ be a topological space. Let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$.
What can be said about spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$? For ...

**0**

votes

**1**answer

141 views

### On the upper bound of Hermitian matrices

Suppose we are given a Hermitian matrix $A$, how to describe the following set of Hermitian
$S=\{X:X\geq \pm A\}$, where $Y\geq B$ is $Y-B$ is semidefinite matrix.
This is of course a convex set, and ...

**51**

votes

**2**answers

951 views

### History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...

**21**

votes

**2**answers

720 views

### fake $S^{2k}\times S^{2k}$

Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...

**3**

votes

**0**answers

106 views

### Is the ring of invariants Noetherian?

Let $R$ be a complete regular local ring whose residue field is perfect. Suppose that a finite group $G$ acts on $R$ by ring automorphisms in such a way that the induced action on the residue field is ...

**4**

votes

**0**answers

125 views

### Can an abelian variety/Q have no points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...

**0**

votes

**1**answer

62 views

### Totally non fixed point property

Edit: According to the comment of Pietro Majer, I revise the question
Is there a non singleton compact connected Hausdorff topological space $X$ for which the following property hold?:
"Constant ...

**2**

votes

**1**answer

88 views

### Unitary representation with fixed Casimir

Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of ...

**45**

votes

**16**answers

10k views

### f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential

The question is about the function f(x) so that f(f(x))=exp (x)-1.
The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here ...

**0**

votes

**1**answer

107 views

### A question about open subsets of Hilbert space whose complements are compact sets

Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. In other ...

**4**

votes

**0**answers

37 views

### flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...

**9**

votes

**1**answer

176 views

### Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^* \to B(X)^*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, ...

**1**

vote

**0**answers

30 views

### Relative nonarchimedean disks and annuli

Let $A$ be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring $A_0$ which is adic and has finitely generated ideal of definition.
Is there a good notion of closed disk of ...

**-2**

votes

**0**answers

48 views

### What does this graph notation mean? E\S [on hold]

I am studying graph theory right now but I am confused what E\S means where both E and S are sets of edges. What does the "\" indicate?

**1**

vote

**1**answer

63 views

### Is there a straightforward way to define a differentiable structure on a localic manifold?

I'd ideally like a categorical definition of differentiability that can then be trivially translated into locales. Barring this, I'm still interested in whether the notion make sense for locales.

**0**

votes

**0**answers

24 views

### Are there compact riemannian manifolds whit Q-curvature negative?

Are there known examples of compact compact riemannian manifolds whit Q-curvature negative?