# All Questions

**0**

votes

**0**answers

33 views

### What is the growth of the rank of a power of a finite simple group?

Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?

**3**

votes

**1**answer

63 views

### Adding sets not containing arithmetic progressions of length three by forcing

Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where:
1) $s\in 2^{<\omega}$,
2) $N\in \mathbb{N}$,
3) (by identifying $s$ with a subset of $lh(s)$) $s$ ...

**0**

votes

**0**answers

19 views

### separating parameters in generalized quadratic Gauss sum

The normalized generalized quadratic Gauss sum is defined by
$$
G(a,b,c)=\frac{1}{c}\sum_{n=1}^ce\left(\frac{an^2+bn}{c}\right)
$$
where $e(x)=\exp(2\pi ix)$.
Under what conditions on $c$ can we ...

**0**

votes

**0**answers

24 views

### Tree property and singular strong limit cardinals

I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question:
Theorem. It is consistent, relative to the existence of large cardinals, that ...

**0**

votes

**0**answers

16 views

### When is a word metric on a CAT(-1) group a bounded distance from some CAT(-k) metric?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...

**3**

votes

**0**answers

55 views

### Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group
${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use
the low index subgroups ...

**-4**

votes

**0**answers

23 views

### Which monoids contains word a^2 b^2? [on hold]

I have a problem with one simple exercise. I don't know how I should start. The question is:
Which monoids contains word $a^2b^2$:
(a) {a,b}*
(b) {abb,a}*
(c) {aa,bb}*
(d) {ba,ab}*
(e) ...

**0**

votes

**1**answer

98 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 ...

**2**

votes

**0**answers

67 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 ...

**1**

vote

**1**answer

107 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 ...

**0**

votes

**0**answers

19 views

### Density of push-forward distribution [on hold]

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 ...

**-1**

votes

**0**answers

35 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

**0**answers

26 views

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

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

**-4**

votes

**0**answers

48 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

21 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 ...

**1**

vote

**0**answers

91 views

### Sum-epimorphisms and prod-monomorphisms

Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
...

**2**

votes

**1**answer

66 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 ...

**0**

votes

**0**answers

53 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 ...

**0**

votes

**0**answers

19 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 ...

**-6**

votes

**0**answers

31 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

61 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 ...

**4**

votes

**1**answer

188 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 ...

**7**

votes

**0**answers

159 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 ...

**1**

vote

**0**answers

35 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 ...

**5**

votes

**0**answers

46 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, ...

**8**

votes

**2**answers

329 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 ...

**1**

vote

**0**answers

35 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

51 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?

**2**

votes

**1**answer

94 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 ...

**1**

vote

**1**answer

88 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 ...

**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?

**5**

votes

**0**answers

110 views

### 3D models of the unfoldings of the hypercube?

There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the
tesseract, into 3D.1
These unfoldings (or "nets") are analogous to the 11 unfoldings of
the 3D cube into the plane.2
...

**-2**

votes

**0**answers

82 views

### Lie algebra and Lie groups [on hold]

A complex Lie algebra $L$ has a representation on $Der(L)$ by just putting $x.D=-ad_{Dx}$. For semisimple Lie algebras, by Weyl's theorem, $Der(L)$ decomposes into irreducible subspaces $D_i$. Then my ...

**-2**

votes

**0**answers

27 views

### Population and date values [on hold]

Use the population data values below.
North= 18,200 South=12,900 East=17,600 West=13,300
If there are 26 representatives for all districts how many ...

**-2**

votes

**1**answer

86 views

### Combinatorical meaning of such expression [on hold]

Any combinatorical meaning or interpretation of
$$1^{\alpha_1}2^{\alpha_2}3^{\alpha_3}...s^{\alpha_s}\alpha_1!\alpha_2!...\alpha_s!$$
for partition ...

**0**

votes

**1**answer

72 views

### Completeness of a set of propositional formulas [on hold]

A set $\sum$ of formulas in propositional logic is complete if for each propositional formula $\phi$ either $\sum \vdash \phi$ or $\sum \vdash \neg \phi$. Clearly every inconsistent set of formulas ...

**2**

votes

**1**answer

59 views

### Classification of ergodic measures for circle expanding maps

Let us consider the classical self-covering of the circle $S^1=\mathbb{R}/\mathbb{Z}$ given by
$$\times_d(x) = dx \mod 1$$
where the degree $d$ is any integer greater than $1$.
There are a wealth of ...

**0**

votes

**1**answer

64 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

**0**answers

127 views

### Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial

Let $f:X\longrightarrow B$ be a family of curves, with $f$ relatively minimal, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all ...

**5**

votes

**0**answers

60 views

### A jump operator for Borel equivalence relations

It is well-known that with respect to Borel reducibility the class of Borel equivalence relations on a standard Borel space does not admit a maximal element. We can use the well-known Friedman-Staley ...

**8**

votes

**1**answer

178 views

### Translates of meager sets

Does there exist a meager set of reals M such that every meager set can be covered by countably many translates of M? This is the category analogue of the following.

**-1**

votes

**0**answers

52 views

### Name and theorems of transitive/Galois groups of quadratic (even) and cubic power polynomials: cyclic group extensions [on hold]

What is the transitive group details of a polynomial where only the third power terms occur? That is ${x}^{3n}+{a}_{n−1}{x}^{3(n−1)}+\cdots+{a}_{1}{x}^{3}+{a}_{0}$. I need the basic theorems that ...

**2**

votes

**2**answers

141 views

### boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$

Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...

**-1**

votes

**0**answers

37 views

### Orthogonal vector to arbitrary vector in R3 [on hold]

I got a vector $(0, 0, 0)^T \neq v \in R^3$. Now I want a closed formula for some orthogonal vector to $w$ (I don't care which).
My problem is that if I, for instance, fix $w_1$ and $w_2$ then $w_3 = ...

**-5**

votes

**0**answers

66 views

### Prove an equation is always false [on hold]

How can I prove an equation is always false?
For example:
b = b + 1
is false for all values of b. Very simple to see.
Now given a more complicated equation, such as:
b = sin(sin(b) - .56))
...

**0**

votes

**1**answer

66 views

### How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper.
Equation:
$min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$,
where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...

**5**

votes

**1**answer

104 views

### Symmetry type of non-cohomological automorphic forms

By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on ...

**21**

votes

**5**answers

785 views

### On an example of an eventually oscillating function

For $x\in(0,1)$, put
$$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$
This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...

**0**

votes

**0**answers

62 views

### Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...

**2**

votes

**0**answers

163 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 ...